The Rise of the Machines: Automation, Horizontal

Innovation and Income Inequality

David Hemous and Morten Olsen ∗

March 2018 (first draft: September 2013)

Abstract

We construct an endogenous growth model with automation and horizontal in-

novation in an economy with low- and high-skill workers. Automation enables the

replacement of low-skill workers with machines, increasing the skill premium and

possibly decreasing low-skill wages. Horizontal innovation increases both wages.

Higher low-skill wages increase incentives to automate so that automation plays a

bigger role as an economy develops. Our model is consistent with a permanently

increasing skill premium, a temporary drop in low-skill wages and a drop in the

labor share. We calibrate it and show that taxing automation innovation reduces

low-skill wages in the long run.

JEL: O41, O31, O33, E23, E25.

KEYWORDS: Endogenous growth, automation, horizontal innovation, directed tech-

nical change, income inequality.

∗David Hemous, University of Zurich and CEPR, david.hemous@econ.uzh.ch, Morten Olsen, Uni-versity of Copenhagen, mgo@econ.ku.dk. Morten Olsen gratefully acknowledges the financial supportof the European Commission under the Marie Curie Research Fellowship program (Grant AgreementPCIG11-GA-2012-321693) and the Spanish Ministry of Economy and Competitiveness (Project ref:ECO2012-38134). We thank Daron Acemoglu, Philippe Aghion, Ufuk Akcigit, Pol Antras, TobiasBroer, Steve Cicala, Per Krusell, Brent Neiman, Jennifer Page, Andrei Shleifer, Che-Lin Su, FabrizioZilibotti and Joachim Voth among others for helpful comments and suggestions. We also thank seminarand conference participants at IIES, University of Copenhagen, Warwick, UCSD, UCLA Anderson,USC Marshall, Barcelona GSE Summer Forum, the 6th Joint Macro Workshop at Banque de France,Chicago Harris, the 2014 SED meeting, the NBER Summer Institute, the 2014 EEA meeting, EcolePolytechnique, the University of Zurich, NUS, London School of Economics, the 2015 World Congressof the Econometric Society, ECARES, Columbia University, EIEF, Brown University, Boston Univer-sity, Yale University, College de France and Washington University. We thank Ria Ivandic and MartonVarga for excellent research assistance.

1 Introduction

How does the automation of production drive economic growth and affect the distri-

bution of income? Conversely, how do wages shape technological progress? Developed

economies have seen dramatic changes in the income distribution which are often at-

tributed to skill-biased technical change, notably automation. By allowing for the use of

machines in some tasks, automation increases economic output, but also reduces the de-

mand for labor in those tasks. As the range of tasks performed by machines is expanding,

the general public increasingly worries about the negative consequences of technological

progress. Yet, economists often argue that technological development also creates new

products and tasks, which boost the demand for labor: certainly many of today’s jobs

did not exist just a few decades ago.1 Surprisingly, the economics literature lacks a

dynamic framework to analyze the interaction between automation and the creation of

new products or tasks. This paper provides the first model to do so.

We build our model to be consistent with three stylized facts of the evolution of the

income distribution in the United States over the past 50 years (shown in Figure 1).

Since 1963, the college premium (a proxy for the skill premium) has increased by 31%

even though the relative skill supply increased substantially. Wages at the bottom of the

income distribution have stagnated with accumulated growth between 1963 and 2007 of

13% for non-college educated workers. Finally, over the same period the labor share

has declined. As a result, our model does not feature balanced growth: as the economy

develops, labor income inequality increases and the labor share declines.

Of course, a large literature exists that relates exogenous technical change to the

income distribution (e.g. Goldin and Katz, 2008, and Krusell, Ohanian, Rıos-Rull and

Violante, 2002). Previous attempts at endogenizing the direction of technical change

rely on factor-augmentation and exogenous shocks to the skill supply (Acemoglu, 1998)

and feature balanced growth. The novelty of our approach is that we present an en-

dogenous growth version of a task framework in the vein of Autor, Levy and Murnane

(2003) and Acemoglu and Autor (2011), in which the direction of innovation evolves en-

dogenously. Endogenizing technological change matters, not only because it is consistent

with empirical evidence, but also, as we show, because the response of an economy with

endogenous technological change to policy interventions (such as taxes on automation

or on machines) differs from that of an economy with exogenous technological progress.

1For instance, the introduction of the telephone led to the creation of new jobs. In 1970 there were421 000 switchboard operators in the United States. This occupation has largely been automated today.

1

1960 1970 1980 1990 2000 2010

Years

1

1.2

1.4

1.6

1.8

2

2.2

Ski

ll-P

rem

ium

(ra

tio)

Panel A - Composition-adjustedcollege/non-college weekly wage ratio

1960 1970 1980 1990 2000 2010

Years

50

55

60

65

70

75

80

Per

cen

t

Panel B - Labor share of GDP

1960 1970 1980 1990 2000 2010

Years

80

90

100

110

120

130

140

(inde

x 19

63=

100)

Panel C - Composition-adjustedreal weekly wage for non-college wages

Figure 1: The US skill-premium, labor share and real wage growth for low-skill workers.Panel A and C are taken from Acemoglu and Autor (2011), college educated work-ers correspond to those with a college degree and half of those with some college,non-college educated workers are the rest. Panel B is from Koh, Santaeulalia-Llopisand Zheng (2016). See further details in Section 4.

Formally, we consider an expanding variety growth model with low-skill and high-

skill workers. Horizontal innovation, modeled as in Romer (1990), increases the demand

for both low- and high-skill workers. Automation allows for the replacement of low-skill

workers with machines in production and takes the form of a secondary innovation in

existing product lines.2 Within a firm, automation increases the demand for high-skill

workers but reduces the demand for low-skill workers. “Non-automated” products only

use low-skill and high-skill labor. Once invented, a specific machine is produced with

the same technology as a consumption good.

We initially take technical progress as given and study the effect of technological

change on wages. An increase in the number of products increases all wages, while

an increase in automation both increases the overall productivity of the economy and

allows for the substitution away from low-skill workers, resulting in an ambiguous net

effect on low-skill wages. Yet, we show that for very general processes of horizontal and

automation innovation, the asymptotic growth rate of low-skill wages must be positive,

albeit strictly lower than that of high-skill wages. Therefore, the introduction of new

(non-automated) products is not sufficient to guarantee balanced growth.

We then endogenize innovation. Low-skill wages play a key role in determining

technological change: as they increase the cost advantage of an automated over a non-

automated firm increases and the incentive to automate is not constant. Instead, an

2Secondary innovations in a growth model were introduced by Aghion and Howitt (1996) who studythe interplay between applied and fundamental research.

2

economy with an initially low level of technology first goes through a phase where growth

is mostly generated by horizontal innovation and the skill premium and the labor share

are constant. Only when low-skill wages are sufficiently high, do firms invest in au-

tomation. During this second phase, the share of automated products increases, the

skill premium rises, the labor share drops and low-skill wages might temporarily de-

crease—replicating the stylized facts of Figure 1. Finally, the economy moves towards

its asymptotic steady state. The share of automated products stabilizes as the entry

of new, non-automated products compensates for the automation of existing ones. The

skill-premium keeps rising but more slowly. The economy will then have endogenously

shifted from a Cobb-Douglas aggregate production function to a nested CES.

Finally, we extend our baseline model to include a capital stock and calibrate it

to match the evolution of the skill premium, the labor share, productivity and the

equipment to GDP ratio the 1960s. As is common in the literature and only for this

exercise, we identify skill groups with education groups, such that high-skill workers

correspond to college-educated workers. Our model is able to reproduce the trends in the

data quantitatively. In particular, the impact of automation on low-skill wages and the

skill-premium decreases in the 90s and 2000s even though expenditures on automation

remain nearly constant; a response to a point of critique of the skill-biased technological

change hypothesis put forward by Card and DiNardo (2002). We use these parameters

to discipline our model for policy experiments. We find that a tax on machine has a

positive effect on low-skill wages even more so due to the endogeneity of technology. In

addition, a tax on automation innovation initially increases low-skill wages but ends up

having a negative impact after a few years.

Our modeling of automation as high-skill-biased is motivated by a large empiri-

cal literature. Autor, Katz and Krueger (1998) and Autor, Levy and Murnane (2003)

demonstrate that computerization is associated with relative shifts in demand for college-

educated workers. Bartel, Ichniowski and Shaw (2007) present similar evidence at the

firm level. Autor, Katz, and Kearney (2006, 2008) and Autor and Dorn (2013) show

that the more recent phenomenon of wage and job polarization, the relative decline of

wages and employment in the middle of the income distribution, can be explained by

the computer-driven automation of routine tasks often performed by middle-skill work-

ers (storing, processing and retrieving information).3 Graetz and Michaels (2018) and

Acemoglu and Restrepo (2017b) find that the introduction of industrial robots leads to

3See also Spitz-Oener (2006) and Goos, Manning and Salomons (2009) for job polarization in Europe,and Feng and Graetz (2016) for a model of why automation targets middle-skill workers.

3

a reduction in the demand for (mostly) low-skill workers. For our purpose, we will not

distinguish between low- and middle-skill workers since both have often performed tasks

which have later on been automated.4

The idea that high wages might incentivize technological progress in the form of au-

tomation dates back to Habakkuk (1962). The associated empirical literature is smaller,

but Lewis (2001) finds that low-skill immigration slows down the adoption of automation

technology and Hornbeck and Naidu (2014) find that the emigration of black workers

from the American South favored the adoption of modern agricultural production tech-

niques there. A large literature shows that the direction of innovation is endogenous in

other contexts (see Newell, Jaffe and Stavins, 1999, or Aghion, Dechezlepretre, Hemous,

Martin and Van Reenen, 2016).

There is a small theoretical literature on labor-replacing technology. In Zeira (1998),

exogenous increases in TFP raise wages and encourage the adoption of a capital-intensive

technology analogous to automation in this paper. Acemoglu (2010) shows that labor

scarcity induces labor-saving innovation. Neither paper analyzes labor-saving innovation

in a fully dynamic model nor focuses on income inequality. Peretto and Seater (2013)

build a dynamic model where innovation allows firms to replace labor with capital. Since

wages are constant over time, so is the incentive to automate. In addition, they do not

focus on income inequality. In subsequent work, Acemoglu and Restrepo (2017a) also

develop a growth model where technical change involves automation and the creation of

new tasks. Automation plays a similar role in both papers (although in their baseline

version, there is only one type of labor). Yet, while in our model all tasks are symmetric

(except for whether they are automated), in theirs, new tasks are exogenously born with

a higher labor productivity. As a result, their model features a balanced growth path

and their focus is on the self-correcting elements of the economy after a technological

shock. In contrast, our model does not feature a balanced growth path and we focus on

accounting for secular trends such as the rise in the skill premium.5

A large literature argues that skill-biased technical change can explain the increase in

the relative demand for skilled workers since the 1970’s. This literature can be divided

into three strands. The first emphasizes Nelson and Phelps (1966)’s hypothesis that

more skilled workers are better able to adapt to technological change (see Lloyd-Ellis,

4A previous version of our paper, Hemous and Olsen (2016) presented an extension of our modelseparating low- and middle- skill workers.

5Benzell, Kotlikoff, LaGarda and Sachs (2017), following Sachs and Kotlikoff (2012) build a modelwhere a code-capital stock can substitute for labor, and show that a technological shock which favorsthe accumulation of code-capital can lead to lower long-run GDP.

4

1999, Caselli, 1999, Galor and Moav, 2000,, Aghion, Howitt and Violante, 2002, and

Beaudry and Green, 2005).6 While such theories mostly explain transitory increases in

inequality, our model features permanent and widening inequality. Yet, we borrow the

idea of a shift in production technology spreading through the economy.

A second strand sees the complementarity between capital and skill as the source for

the increase in the skill premium. Krusell, Ohanian, Rıos-Rull and Violante (2000) find

that the observed increase in the stock of capital equipment can account for most of the

variation in the skill premium. Our model also features capital-skill complementarity but

our focus is different since we seek to explain why innovation has been directed towards

automation and analyze the interactions between automation and horizontal innovation.

Finally, a third branch of the literature, building on Katz and Murphy (1992), consid-

ers technology to be either high-skill or low-skill labor augmenting. Using this framework,

Goldin and Katz (2008) find that technical change has been skill-biased throughout the

20th century in the United States (Katz and Margo, 2014, argue that the relative demand

for white-collar workers has been increasing since 1820). Further, the directed technical

change literature (Acemoglu, 1998, 2002 and 2007) endogenizes the bias of technology.

Such models of factor-augmenting technical change deliver important insights about

inequality and technical change, but have no role for labor-replacing technology and

therefore cannot generate declines in low-skill wages (a point emphasized in Acemoglu

and Autor, 2011). Our model is also a directed technical change framework but de-

viates from the assumption of factor-augmenting technologies. It explicitly allows for

labor-replacing automation, generating the possibility for (temporary) absolute losses

for low-skill workers, and permanently increasing income inequality.

Section 2 describes the baseline model with exogenous technology and shows the

effects of technological change on wages. Section 3 endogenizes the path of technology

and describes the evolution of the economy. Section 4 calibrates an extended version

of the model and conducts policy exercises. Section 5 concludes. Appendix 6 presents

the main extensions of our model and discusses the identification of our parameters.

Appendix 7 presents proofs, other extensions and details the calibration exercise.

6Relatedly, Beaudry, Green and Sand (2016) model the IT revolution as an exogenous increase inthe demand for organizational capital, which is built by cognitive labor. Once the capital stock reachesits steady-state, the demand for cognitive tasks decreases. We reproduce a somewhat similar patternbut in the growth rate of the skill premium instead of the level of demand for cognitive tasks and withan endogenous increase in the benefits from automation.

5

2 A Baseline Model with Exogenous Innovation

This section presents a baseline model with exogenous technology to study the conse-

quences of automation and horizontal innovation on factor prices. Section 2.3 derives

comparative statics results and section 2.4 the asymptotic behavior of wage for general

paths of technology. We discuss some of our modeling assumptions in section 2.5.

2.1 Preferences and production

We consider a continuous time infinite-horizon economy populated by H high-skill and

L low-skill workers. Both types of workers supply labor inelastically and have identical

preferences over a single final good of:

Uk,t =

∫ ∞t

e−ρ(τ−t)C1−θk,τ

1− θdτ,

where ρ is the discount rate, θ ≥ 1 is the inverse elasticity of intertemporal substitution

and Ck,t is consumption of the final good at time t by group k ∈ H,L. The final good

is produced by a competitive industry combining a set of intermediate inputs, i ∈ [0, Nt]

using a CES aggregator:

Yt =

(∫ Nt

0

yt(i)σ−1σ di

) σσ−1

,

where σ > 1 is the elasticity of substitution between these inputs and yt(i) is the use

of intermediate input i at time t. As in Romer (1990), an increase in Nt represents

a source of technological progress. Throughout the paper, we use interchangeably the

terms “intermediate input” and “product”.

We normalize the price of Yt to 1 at all points in time and drop time subscripts when

there is no ambiguity. The demand for each intermediate input i is:

y(i) = p(i)−σY, (1)

where p(i) is the price of intermediate input i and the normalization implies that the

ideal price index, [∫ N

0p(i)1−σdi]1/(1−σ) equals 1.

Each intermediate input is produced by a monopolist who owns the perpetual rights

of production. She can produce the intermediate input by combining low-skill labor, l(i),

6

high-skill labor, h(i), and, possibly, type-i machines, x(i), using the production function:

y(i) =[l(i)

ε−1ε + α(i) (ϕx (i))

ε−1ε

] εβε−1

h(i)1−β, (2)

where α(i) ∈ 0, 1 is an indicator function for whether or not the monopolist has access

to an automation technology which allows for the use of machines. If the product is not

automated (α(i) = 0), production takes place using a Cobb-Douglas production function

with only low-skill and high-skill labor and a low-skill factor share of β. If the product

is automated (α(i) = 1) machines can be used in the production process. We allow

for perfect substitutability, in which case ε = ∞ and the production function is y(i) =

[l(i) + α(i)ϕx (i)]β h(i)1−β. The parameter ϕ is the relative productivity advantage of

machines over low-skill workers and G denotes the share of automated products.

Since each input is produced by a single firm, we identify each input with its firm

and refer to a firm which uses an automated production process as an automated firm.

We refer to the specific labor inputs provided by high-skill and low-skill workers in the

production of different inputs as “different tasks” performed by these workers, so that

each product comes with its own tasks. It is because α(i) is not fixed, but can change

over time, that our model captures the notion that machines can replace low-skill labor

in new tasks. A model with a fixed α(i) for each product would only allow for machines

to be used more intensively in production, but always for the same tasks. Although,

we will refer to x as “machines”, our interpretation also includes any form of computer

inputs, algorithms, the services of cloud-providers, etc.

For now, machines are an intermediate input. Once invented, machines of type i

are produced competitively one for one with the final good, such that the price of an

existing machine for an automated firm is always equal to 1 and technological progress

in machine production follows that in the rest of the economy. Yet, our model can

capture the notion of a decline in the real cost of equipment, as automation for firm i

can equivalently be interpreted as a decline of the price of machines i from infinity to 1.

2.2 Equilibrium wages

In this section we derive how wages are determined in equilibrium, taking as given the

technological levels N (the number of products), G (the share of automated products)

and the employment of high-skill workers in production, HP ≡∫ N

0h(i)di (we letHP ≤ H

to accommodate later sections where high-skill labor is used to innovate).

7

First, note that all automated firms are symmetric and therefore behave in the sameway. Similarly all non-automated firms are symmetric. This gives aggregate output of:

Y = N1

σ−1×

(1−G)1σ

((LNA)β(HP,NA)1−β︸ ︷︷ ︸

)T1

σ−1σ

+G1σ

([(LA)

ε−1ε + (ϕX)

ε−1ε ]

εβε−1 (HP,A)1−β︸ ︷︷ ︸

)σ−1σ

T2

σσ−1

,

(3)

where LA (respectively LNA) is the total mass of low-skill workers in automated (re-

spectively non-automated) firms, HP,A (respectively HP,NA) is the total mass of high-

skill workers hired in production in automated (respectively non-automated) firms and

X =∫ N

0x(i)di is total use of machines. The aggregate production function takes the

form of a nested CES between two sub-production functions. The first term T1 cap-

tures the classic case where production takes place with constant shares between factors

(low-skill and high-skill labor), while the second term T2 represents the factors used

within automated products and features the substitutability between low-skill labor and

machines. G is the share parameter of the “automated” products nest and therefore an

increase in G is T2-biased (as σ > 1). N1

σ−1 is a TFP parameter. Besides the functional

form the aggregate production function (3) differs from the often assumed aggregate

CES production function in two ways: First, instead of the often-used factor-augmenting

technical in an aggregate production function, we explicitly model automation of tasks

and derive the aggregate production function from the cost function of individual firms.

Second, with an endogenous G we will be able to capture effects that the usual focus on

an exogenous aggregate production function cannot.7

The unit cost of intermediate input i is given by:

c(wL, wH , α(i)) = β−β(1− β)−(1−β)(w1−εL + ϕα(i))

β1−εw1−β

H , (4)

where ϕ ≡ ϕε, wL denotes low-skill wages and wH high-skill wages. When ε < ∞,

c(·) is strictly increasing in both wL and wH and c(wL, wH , 1) < c(wL, wH , 0) for all

wL, wH > 0 (automation reduces costs). Price is set as a markup over costs: p(i) =

σ/(σ − 1) · c(wL, wH , α(i)). Using Shepard’s lemma and equations (1) and (4) delivers

7At the firm level, this model features an elasticity of substitution between high-skill labor and ma-chines equal to that between high-skill and low-skill labor. This, however, does not hold at the aggregatelevel, consistent with Krusell et al. (2000), who argue that the aggregate elasticity of substitution be-tween high-skill and low-skill labor is greater than that between high-skill labor and machines.

8

the demand for low-skill labor of a single firm.

l(wL, wH , α(i)) = βw−εL

w1−εL + ϕα(i)

(σ − 1

σ

)σc(wL, wH , α(i))1−σY, (5)

which decreases in wL and wH . The effect of automation on demand for low-skill labor in

a firm is generally ambiguous. This is due to the combination of a negative substitution

effect (automation allows for substitution between machines and low-skill workers) and a

positive scale effect (automation decreases costs, lowers prices and increases production).

As we focus on labor-substituting innovation, we impose the condition ε > 1 +β (σ − 1)

throughout the paper which is necessary and sufficient for the substitution effect to

dominate and ensures l(wL, wH , 1) < l(wL, wH , 0) for all wL, wH > 0.

Let x (wL, wH) denote the use of machines by an automated firm. The relative use

of machines and low-skill labor for such a firm is then:

x(wL, wH)/l(wL, wH , 1) = ϕwεL, (6)

which increases in wL as the wage is also the price of low-skill labor relative to machines.

The iso-elastic demand (1), coupled with constant mark-up σ/(σ − 1), implies that

revenues are given by R(wL, wH , α(i)) = ((σ − 1) /σ)σ−1 c(wL, wH , α(i))1−σY and profits

are a fixed share of revenue: π(wL, wH , α(i)) = R(wL, wH , α(i))/σ. We define µ ≡β(σ − 1)/(ε − 1) < 1 (by our assumption on ε). Using (4), the relative revenues (and

profits) of non-automated and automated firms are given by:

R(wL, wH , 0)/R(wL, wH , 1) = π(wL, wH , 0)/π(wL, wH , 1) = (1 + ϕwε−1L )−µ, (7)

which is a decreasing function of w. As non-automated firms rely more heavily on

low-skill labor, their relative market share drops with higher low-skill wages.

Since firms’ profits are a constant share of firms’ revenues, aggregate profits are a

constant share 1/σ of output Y . Similarly, the share of firms’ revenues accruing to high-

skill labor in production is the same for all firms and given by νh = (1 − β)(σ − 1)/σ.

Therefore payment to high-skill labor in production is a constant share of output:

wHH = (1− β)σ − 1

σN [GR(wL, wH , 1) + (1−G)R(wL, wH , 0)] = (1− β)

σ − 1

σY. (8)

Using factor demand functions, the share of revenues accruing to low-skill labor is given

by νl(wL, wH , α(i)) = σ−1σβ(1 + ϕwε−1

L α(i))−1

, and decreases with automation. Using

9

labor market clearing (∫ N

0l(i)di = L), we obtain total wages of low-skill workers as:

wLL = N [GR(wL, wH , 1)νl(wL, wH , 1) + (1−G)R(wL, wH , 0)νl(wL, wH , 0)] . (9)

Equations (7), (8) and (9) give the high-skill to low-skill labor share in production as:8

wHHP

wLL=

1− ββ

G+ (1−G)(1 + ϕwε−1L )−µ

G(1 + ϕwε−1

L

)−1+ (1−G)(1 + ϕwε−1

L )−µ. (10)

This expression gives the relative demand curve for high-skill and low-skill labor. We

represent this relationship for given technology levels and factor supply ratio L/HP by

a curve in the (wL, wH) space in Figure 2. For G = 0, the curve is a straight line, with

slope (1−β)L/(βHP ), reflecting the constant factor shares in a Cobb-Douglas economy.

For G > 0, the right-hand side of (10) increases in w, so that the relative demand

curve is non-homothetic and rotates counter-clockwise as wL grows. Intuitively, higher

low-skill wages both induce more substitution towards machines in automated firms (as

reflected by the term (1 + ϕwε−1L )−1 in equation (10)) and improve the cost-advantage

and therefore the market share of automated firms (term (1 + ϕwε−1L )−µ ) and increases

the ratio of high-skill to low-skill labor share in production. This immediately implies

that in the long-run low-skill wages and high-skill wages cannot grow at the same rate,

i.e. we cannot have positive balanced growth.

Figure 2: Relative demand curve and isocost curve for different values of N and G.

8When ε = ∞, the skill premium is given by wHwL

= 1−ββ

LHP

if wL < ϕ−1 such that no firm uses

machines, and wHwL

= 1−ββ

LHP

G+(1−G)(ϕwL)−1

(1−G)(ϕwL)−1 if wL > ϕ−1.

10

With constant mark-ups, the cost equation (4) and the price normalization give:

σ

σ − 1

N1

1−σ

ββ (1− β)1−β

(G(ϕ+ w1−ε

L

)µ+ (1−G)w

β(1−σ)L

) 11−σ

w1−βH = 1. (11)

This relationship defines the unit isocost curve in figure 2. It shows the positive re-

lationship between real wages and the level of technology given by N , the number of

intermediate inputs, and G the share of automated firms. Together (10) and (11) deter-

mine real wages uniquely as a function of N,G and HP .

Given the amount of resources devoted to production (L,HP ), the static equilibrium

is closed by the final good market clearing condition:

Y = C +X (12)

where C = CL + CH is total consumption. GDP includes the payment to labor and

aggregate profits so GDP and the total labor share become

GDP ≡ 1

σY + wLL+ wHH, LS = 1− 1

1 + (σ − 1) (1− β)(

wLLwHHP + H

HP

) , (13)

where the second equality uses (8). Therefore, the labor share decreases in the skill

premium for a given mass of high-skill workers in production HP .

2.3 Technological change and wages

The consequences of technological changes on the level of wages are most easily seen

with the help of Figure 2. An increase in the number of products, N , pushes out the

isocost curve and increases both low-skill and high-skill wages. When G = 0, both types

of wages grow at the same rate as the relative demand curve is a straight line, but for

G > 0, the demand curve is non-homothetic and the skill premium grows. Therefore,

an increase in N at constant G is high-skill biased.

An increase in the share of automated products G has a positive effect on high-skill

wages and the skill premium but an ambiguous effect on low-skill wages: Higher au-

tomation increases the productive capability of the economy and pushes out the isocost

curve (an aggregate productivity effect), which increases low-skill wages, but it also al-

lows for easier substitution away from low-skill labor which pivots the relative demand

curve counter-clockwise (an aggregate substitution effect), decreasing low-skill wages.

11

Therefore automation is always high-skill labor biased (wH/wL increases) but low-skill

labor saving (wL decreases) if and only if the aggregate substitution effect dominates

the aggregate productivity effect. Formally, one can show (proof in Appendix 7.1):9

Proposition 1. Consider the equilibrium (wL, wH) determined by equations (10) and

(11). Assume that ε <∞, it holds that

A) An increase in the number of products N (keeping G and HP constant) leads to

an increase in both high-skill (wH) and low-skill wages (wL). Provided that G > 0, an

increase in N also increases the skill premium wH/wL and decreases the labor share.

B) An increase in the share of automated products G (keeping N and HP constant)

increases the high-skill wages wH , the skill premium wH/wL and decreases the labor

share. Its impact on low-skill wages is generally ambiguous, but low-skill wages are

decreasing in G if i) 1 ≤ (σ − 1)(1− β) or if ii) N and G are high enough.

Automation is low-skill labor saving if (1− β) (σ − 1) ≥ 1. The aggregate substitu-

tion effect is larger than the scale effect when i) σ is large (as then the increased demand

for the newly automated product does not lead to a large increase in demand for other

products). ii) when the cost share of the low-skill labor-machines aggregate β is small (as

the cost-saving effect of automation is smaller in that case). iii) when G is high as most

of the aggregate productivity gains are realized for low G and for high G the automation

of one more firm hurts low-skill workers more, as there are fewer non-automated firms

left. iv) when N is large and the higher wages make the substitution effect stronger.

One could also consider the effect of an increase in the number of non-automated

products (that is an increase in N keeping GN constant), which corresponds to the

“horizontal innovation” introduced in section 3. Such technological change pushes out

the iso-cost curve but also makes the relative demand curve rotate clockwise. This

increases both low-skill and high-skill wages (see proof in Appendix 7.1). In addition,

for N large enough or ε < σ, the rotation of the relative demand curve is sufficiently

strong for this form of technical change to be low-skill labor biased.10

In section 3, when we specify the innovation process, we will show that as the number

of products N increases, the share of automated products endogenously increases from

9In the perfect substitute case, ε = ∞, wH increases in N and weakly increases in G, wH/wLweakly increases in N and G and wL weakly increases in N and weakly decreases in G provided that1/ (1− β) ≤ σ − 1 or G is large enough. When ε = ∞ and G = 1, the isocost curve has a horizontalarm and the relative demand curve a vertical one.

10In the perfect substitute case, an increase in the number of non-automated products increases wH ,weakly increases wL and, if G < 1 and N is large enough, decreases the skill premium.

12

an initial level close to 0. As a result, growth will progressively become unbalanced

with a rising skill premium (and accordingly a decline in the labor share), and for some

parameter values, low-skill wages will temporarily decline.

2.4 Asymptotics for general technological processes

We study the asymptotic behavior of the model for given paths of technologies and mass

of high-skill workers in production. For any variable at (such as Nt), we let gat ≡ at/at

denote its growth rate and ga∞ = limt→∞gat if it exists. In Appendix 7.2.1 we derive:

Proposition 2. Consider three processes [Nt]∞t=0, [Gt]

∞t=0 and [HP

t ]∞t=0 where (Nt, Gt, HPt ) ∈

(0,∞) × [0, 1] × (0, H] for all t. Assume that Gt, gNt and HP

t all admit limits G∞, gN∞

and HP∞ with gN∞ > 0 and HP

∞ > 0.

A) If G∞ ∈ (0, 1), the asymptotic growth of high-skill wages wHt and output Yt are:

gwH∞ = gY∞ = gN∞/ ((1− β)(σ − 1)) , (14)

and the asymptotic growth rate of wLt is given by

gwL∞ = gY∞/ (1 + β(σ − 1)) . (15)

B). If G∞ = 1, the asymptotic growth rates of wHt and Yt also obey (14). If Gt

converges sufficiently fast (so that limt→∞

(1−Gt)Nψ(1−µ) ε−1

εt exists and is finite) then :

-i) If ε <∞ the asymptotic growth of wLt is positive at :

gwL∞ = gY∞/ε. (16)

-ii) If low-skill workers and machines are perfect substitute then limt→∞

wLt is finite and

weakly greater than ϕ−1 (equal to ϕ−1 when limt→∞

(1−Gt)Nψt = 0).

C) If G∞ = 0 and Gt converges sufficiently fast (so that limt→∞

GtNβt exists and is

finite), then the asymptotic growth rates of wLt, wHt and Yt obey:

gwL∞ = gwH∞ = gY∞ = gN∞/(σ − 1). (17)

This proposition first relates the growth rate of output and high-skill wages to the

growth rate of the number of products. Without automation, that is if Gt converges to 0

sufficiently fast, Yt is proportional to N1/(σ−1)t as in a standard expanding-variety model.

13

Automation introduces machines as an additional reproducible input such that a higher

level of productivity leads to a higher supply of machines further increasing output when

G∞ > 0. This multiplier effect is increasing in the asymptotic share of machines, β.

Second, with positive growth in Nt, mild assumptions are sufficient for asymptotic

positive growth in low-skill wages. wLt only remains bounded when there is economy-

wide perfect substitution, i.e. low-skill workers and machines are perfect substitutes,

ε = ∞, and all products are automated asymptotically (Gt converges to 1 sufficiently

fast). Even then low-skill wages are bounded below by ϕ−1, as a lower wage would

imply that no firm would use machines. In general, the processes of Nt and Gt depend

on the rate at which new products are introduced, the extent to which they are initially

automated, and the rate of automation. As long as new non-automated products are

continuously introduced, and the intensity at which non-automated firms are automated

is bounded, the share of non-automated products is always positive, i.e. G∞ < 1 (see

proof in Appendix 7.2.2). This ensures that there is no economy-wide perfect substitu-

tion between low-skill workers and machines.

With aggregate imperfect substitution (because G∞ < 1 or ε <∞), a growing stock

of machines and a fixed supply of low-skill labor imply that the relative price of a worker

(wLt) to a machine (pxt ) must grow at a positive rate. Since machines are produced

with the same technology as the consumption good, pxt = pCt , where pCt is the price

of the consumption good (1 with our normalization), the real wage wLt = wLt/pCt =

(wLt/pxt )(p

xt /p

Ct ) must also grow at a positive rate.

Third, the proposition shows that as long as G∞ > 0, low-skill wages cannot grow

at the same rate as output. This is easily seen from the aggregate production function

(3), which asymptotically is a nested CES with constant share parameters and where

technological change in the form of an increase in Nt is not labor-augmenting (unless

G∞ = 0). Therefore, following Uzawa’s theorem, balanced growth is impossible. Within

our framework, this holds when automation intensity is bounded away from 0 which

ensures that the asymptotic share of automated products is positive and rules out case

C (see Appendix 7.2.2 for a proof).

If ε <∞ and G∞ = 1 (sufficiently fast), low-skill workers derive their income asymp-

totically from automated firms and the asymptotic growth rate depends on the elasticity

of substitution between machines and low-skill workers in automated firms, ε.

In contrast, when G∞ ∈ (0, 1), the demand for low-skill labor increasingly comes

from the non-automated firms (as automation is labor-saving at the firm level). With

14

growing wages, the relative market share of non-automated firms decreases in proportion

with (1 +ϕwε−1Lt )−µ ∼ ϕ−µw

−β(σ−1)Lt , while most of the demand for high-skill labor comes

from automated firms. Then, the growth rate of low-skill wages is a fraction of the

growth rate of high-skill wages given by (15). The ratio between high-skill and low-skill

wage growth rates increases with a higher importance of low-skill workers (a higher β) or

a higher substitutability between automated and non-automated products (a higher σ)

since both imply a faster loss of competitiveness of the non-automated firms. Yet, it is

independent of the elasticity of substitution between machines and low-skill workers, ε or

of the exact asymptotic share of automated products G∞. In this case, non-automated

products provide employment opportunities for low-skill workers which limits the relative

losses of low-skill workers compared to high-skill workers (their wages grow according to

(15) instead of (16) and ε > 1 + β(σ − 1)).

2.5 Discussion

Proposition 2 establishes general conditions under which low-skill wages asymptotically

grow but slower than high-skill wages. We now discuss the robustness of this result.

First, our assumption that machines are an intermediate input is innocuous: Section 4

relaxes this assumption and lets machines take the form of capital with no qualitative

change of result. Second, Appendix 6.4 relaxes the assumption of an exogenous stock

of labor and considers a Roy model where workers are heterogeneous in the quantity

of high-skill labor they can supply. Proposition 2 generalizes to this case, although the

relative growth rate of low-skill wages is higher and asymptotically all workers supply

high-skill labor. Third, Appendix 7.4 presents a model where the production technologies

for machines and the consumption good differ allowing for negative growth in pxt /pCt . In

this case, low-skill wages may decline asymptotically. Fourth, even if some of the tasks

(but not all) performed by high-skill workers are automatable, our results would remain

similar as long as high-skill workers remain essential in production.11

Appendix 6.1 breaks the assumption of symmetry and assumes that new products

have a higher productivity. Higher productivity comes in the form of TFP improvements

(which augments the productivity of all factors, including machines) or higher (low-skill

and high-skill) labor productivity. Productivity increases exponentially, so we assume

that Nt grows linearly to maintain non-explosive growth, and non-automated products

11If all labor tasks are automatable, infinite production is possible in finite time once N is largeenough. Factors such as natural resources or land are then likely to be the scarce factor.

15

still have a positive probability of becoming automated. We show that, if all productivity

improvements are labor-augmenting, horizontal innovation becomes more low-skill biased

and balanced growth is possible; a result similar to that of Acemoglu and Restrepo

(2017a). However, if machines also become partly more productive in new tasks (once

those are automated), balanced growth is impossible. Therefore, the invention of new

non-automated tasks is not enough to ensure balanced growth, what is required is that

labor but not potential machines are more productive in these new tasks.12

3 Endogenous innovation

We now model automation and horizontal innovation as a the result of investment. This

allows us first to look at the impact of wages on technological change (the reverse of

Proposition 1), second to study the transitional dynamics of the system, and third to

explore the interactions between the two innovation processes. Section 3.1-3.5 character-

izes the model and its solution, section 3.6 relates it to historical experience and section

3.7 provides additional results and comparative statics.

3.1 Modeling innovation

A non-automated firm can hire hAt (i) high-skill workers to perform automation re-

search which will result in the firm becoming automated as a Poisson process with

rate ηGκt

(Nth

At (i)

)κ. Once a firm is automated it remains so forever. η > 0 denotes

the productivity of the automation technology, κ ∈ (0, 1) measures the concavity of

the automation technology, Gκt , κ ∈ [0, κ], represents possible knowledge spillovers from

the share of automated products, and Nt represents knowledge spillovers from the total

number of intermediate inputs. The spillovers in Nt ensure that both automation and

horizontal innovation may take place in the long-run; they exactly compensate for the

mechanical reduction in the amount of resources for automation available for each prod-

uct (namely high-skill workers) when the number of product increases.13 Automation is

12We have also abstracted from the accumulation of low-skill human capital. “Traditional” humancapital would be equivalent to augmenting low-skill labor in (3), and, from Uzawa’s theorem, wouldbe insufficient to guarantee balanced growth when G∞ > 0 and Nt grows exponentially. Grossman etal. (2017) show how non-traditional human capital can lead to balanced growth when technologicalchange is not purely labor-augmenting. He and Liu (2008) show that the endogenous accumulation ofskill through a “schooling” technology fits the rise in the skill-ratio and the skill-premium well.

13These spillovers can be micro-funded as follows: let there be a fixed mass one of firms indexedby j each producing a continuum Nt of products indexed by i so that production is given by Yt =

16

undertaken by the incumbent firm, but we could accommodate automation by entrants

as long as the incumbent also automates with positive probability or captures a share of

the surplus created by the automation innovation.

New intermediate inputs are developed by high-skill workers in a standard manner

according to a linear technology with productivity γNt. With HDt high-skill workers

pursuing horizontal innovation, the mass of intermediate inputs evolves according to:

Nt = γNtHDt .

We assume that firms do not exist before their product is created and therefore cannot

invest in automation. As a result, new products are born non-automated, which means

that “horizontal innovation” corresponds to an increase in Nt keeping GtNt constant

and (following our discussion in section 3.7) is low-skill biased under certain conditions.

This is motivated by the idea that when a task is new and unfamiliar, the flexibility

and outside experiences of workers allow them to solve unforeseen problems. Only as

the task becomes routine and potentially codefiable a machine (or an algorithm) can

perform it (Autor, 2013). Our results carry through if only a share of the new products

are born non-automated as discussed in section 3.7.14

Therefore the rate and direction of innovation will depend on the equilibrium allo-

cation of high-skill workers between production, automation and horizontal innovation.

Defining the total mass of high-skill workers working in automation as HAt ≡

∫ Nt0hAt (i)di,

we get that high-skill labor market clearing leads to

HAt +HD

t +HPt = H. (18)

3.2 Innovation allocation

We denote by V At the value of an automated firm, by rt the economy-wide interest rate

and by πAt ≡ π(wLt, wHt, 1) the profits at time t of an automated firm. The asset pricing

(∫ 1

0

∫ Nt(j)0

yt(i, j)σ−1σ didj)

σσ−1 . When a firm hires HA

t (j) high-skill workers in automation each of its non-

automated products gets independently automated with a Poisson rate of ηGκt [HAt (j)/(1 − Gt(j)))]κ.

The aggregate economy would be identical to ours and have the same social planner allocation (thedecentralized equilibrium would qualitatively behave qualitatively similarly but the externality in theautomation technology from the number of products would be internalized).

14The model predicts that the ratio of high-skill to low-skill labor in production is higher for automatedthan non-automated firms, though not overall since non-automated firms also hire high-skill workersfor the purpose of automating. In particular, new firms do not always have a higher ratio of low tohigh-skill workers (and at the time of its birth a new firm only relies on high-skill workers).

17

equation for an automated firm is given by

rtVAt = πAt + V A

t . (19)

This equation states that the required return on holding an automated firm, V At , must

equal the instantaneous profits plus appreciation. An automated firm only maximizes

instantaneous profits and has no intertemporal investment decisions to make.

A non-automated firm invests in automation. Denoting by V Nt the value of a non-

automated firm and letting πNt ≡ π(wLt, wHt, 0), we get the asset pricing equation:

rtVNt = πNt + ηGκ

t

(Nth

At

)κ (V At − V N

t

)− wHthAt + V N

t , (20)

where hAt is the mass of high-skill workers in automation research hired by a single non-

automated firm. This equation is similar to equation (19), but profits are augmented

by the instantaneous expected gain from innovation ηGκt

(Nth

At

)κ (V At − V N

t

)net of

expenditure on automation research, wHthAt . This gives the first order condition:

κηGκtN

κt

(hAt)κ−1 (

V At − V N

t

)= wHt. (21)

hAt increases with the difference in value between automated and non-automated firms,

and thereby current and future low-skill wages—all else equal.

Since non-automated firms get automated at Poisson rate ηGκt

(Nth

At

)κ, and since

new firms are born non-automated, the share of automated firms obeys:15

Gt = ηGκt

(Nth

At

)κ(1−Gt)−Gtg

Nt . (22)

Free-entry in horizontal innovation guarantees that the value of creating a new firm

cannot be greater than its opportunity cost:

γNtVNt ≤ wHt, (23)

with equality whenever there is strictly positive horizontal innovation (Nt > 0).

The low-skill and high-skill representative households’ problems are standard and

15Using that by symmetry the total amount of high-skill workers hired in automation research isHAt = (1−Gt)NthAt , we can also write this expression as Gt = ηGκt

(HAt

)κ(1−Gt)1−κ −GtgNt .

18

lead to Euler equations which in combination give

Ct/Ct = (rt − ρ) /θ, (24)

with a transversality condition requiring that the present value of all time-t assets in the

economy (the aggregate value of all firms) is asymptotically zero:

limt→∞

(exp

(−∫ t

0

rsds

)Nt

((1−Gt)V

Nt +GtV

At

))= 0.

3.3 Equilibrium characterization

Following Proposition 2, asymptotically high-skill wages output and consumption grow

proportionately to Nψt where ψ ≡ ((1− β) (σ − 1))−1 when G∞ > 0 (hence ψ is the

asymptotic elasticity Yt with respect to Nt). Therefore to study the behavior of the

system we introduce the normalized variables vt ≡ wHtN−ψt and ct ≡ ctN

−ψt . As hAt

mechanically tends to 0 as the mass of non-automated firms grows we also introduce

hAt ≡ NthAt . We further define the auxiliary variable χt ≡ ct

θ/vt which allows us to

simplify the system (χt is related to the mass of high-skill workers in production and

therefore, given hAt , to HDt and the growth rate of Nt). Since the economy does not

feature balanced growth, we also need to keep track of the level of Nt, we do this by

introducing nt ≡ N−β/[(1−β)(1+β(σ−1))]t , which tends toward 0 as Nt tends toward infinity.

Finally, we define ωt ≡ (wLtN−ψ/(1+β(σ−1))t )β(1−σ) which asymptotes a finite positive

number. The equilibrium can then be characterized by a system of differential equations

with two state variables nt, Gt, two control variables, hAt , χt and an auxiliary equation

defining ωt (see equations (31), (32), (39), (40), (45) and (47)-(49) in Appendix 6.2).

This system admits a steady-state as stipulated below (proof in Appendix 7.5.1):

Proposition 3. Assume that

κ−κ (γ(1− κ)/ρ)κ−1 ρ/η + ρ/γ < ψH, (25)

then the system of differential equations admits a steady state (n∗, G∗, hA∗, χ∗) with n∗ =

0, 0 < G∗ < 1 and positive growth(gN)∗> 0.

We will refer to the steady state (n∗, G∗, hA∗, χ∗) as as an asymptotic steady state

for our original system of differential equations. In addition, the assumption that θ ≥

19

1 ensures that the transversality condition always holds.16 For the rest of the paper

we restrict attention to parameters such that there exists a unique saddle-path stable

steady state (n∗, G∗, hA∗, χ∗) with n∗ = 0, G∗ > 0. Then, for an initial pair (N0, G0) ∈(0,∞) × [0, 1] sufficiently close to the asymptotic steady state, the model features a

unique equilibrium converging towards it.17

The following section describes the equilibrium and explains how the economy reaches

this asymptotic steady-state even starting far from it. We focus on the evolution of the

automation incentives, which, we show, are crucially linked to the level of low-skill wages.

3.4 Innovation incentives along the transitional path

A distinctive feature of this economy is that the path of technological change itself will

be unbalanced through the transitional dynamics. In the following, we elaborate on this

by showing that the economy goes through three phases: a Phase 1 where the incentive

to automate is low and the economy behaves close to a Romer model, a Phase 2 where

automation increases the share of automated products Gt and a Phase 3 where the

economy approaches the steady state. This section helps guide the intuition, and formal

proofs of all the results mentioned are contained in Appendices 7.5.3, 7.5.4 and 7.5.5.

These are “quantitative phases” in that the incentive to automate in Phase 1 is small

but not exactly 0 and in Phase 3, Gt is only approximately constant.

Following (21), the mass of high-skill workers in automation (HAt = (1−Gt)Nth

At )

and therefore the automation intensity rate, given by ηGκt

(HAt / (1−Gt)

)κ, depends on

the ratio between the gain in firm value from automation V At −V N

t , and its effective cost

namely the high-skill wage divided by the number of products wHt/Nt, that is:

HAt = (1−Gt)

(κηGκ

t

V At − V N

t

wHt/Nt

)1/(1−κ)

. (26)

Crucially, as the number of products in the economy increases, the ratio(V At − V N

t

)/ (wHt/Nt)

changes value. To see this, we combine (19), (20) and (21), and integrate over the dif-

16To understand equation (25), let the efficiency of the automation technology η be arbitrarily largesuch that the model approaches a Romer model with only automated firms. Then equation (25) becomesρ/γ < ψH, which mirrors the condition for growth in a Romer model with linear innovation technology.With a smaller η the present value of a new product is reduced and the condition is more stringent.

17Multiple asymptotic steady states with G∗ > 0 are technically possible but are not likely for rea-sonable parameter values (see Appendix 7.5.2). In addition, with two state variables (nt and Gt) saddlepath stability requires exactly two eigenvalues with positive real parts. In our numerical investigation,for all parameter combinations which satisfy the previous restrictions, this condition was always met.

20

ference in value between an automated and a non-automated firm to get:

V At − V N

t =

∫ ∞t

exp

(−∫ τ

t

rudu

)(πAτ − πNτ −

1− κκ

wHτhAτ

)dτ, (27)

such that the difference in value between the two types of firms is given by the discounted

difference of the profit flows adjusted for the cost and probability of automation. Further,

by the assumptions of Cobb-Douglas and iso-elastic demand, high-skill wages (for given

HPt ) and aggregate profits are both proportional to aggregate output. Therefore, wHt/Nt

is proportional to average profits: wHt/Nt =[Gtπ

At + (1−Gt) π

Nt

]/[ψHP

t

]. As a result,

the mass of high-skill workers in automation essentially depends on the discounted flow

of profits of automated versus non-automated firms divided by the average profits made

by firms. Intuitively — from equation (27) — with a positive discount rate, as a first

approximation V At − V N

t will move like πAt − πNt , so that one gets

V At − V N

t

wHt/Nt

∝ πAt − πNtGtπAt + (1−Gt) πNt

=1−

(1 + ϕwε−1

Lt

)−µGt + (1−Gt)

(1 + ϕwε−1

Lt

)−µ , (28)

where we used πAt −πNt =[(

1 + ϕwε−1Lt

)µ − 1]πNt from (7) and where ∝ denotes“approxi-

mately proportional”. This highlights low-skill wages (relative to the inverse productivity

of machines ϕ−1) as the key determinant of automation innovations. Note, that when

wLt ≈ 0 the incentive for automation innovation is very low, whereas when wLt → ∞it approaches 1/Gt > 0. This price effect bears similarity with Zeira (1998), where the

adoption of a labor-saving technology also depends on the price of labor.18

First Phase. When the number of products, Nt, is sufficiently low that wLt is small

relative to ϕ−1, the difference in profits between automated and non-automated firms

is small relative to average profits. Following (26) and (28), the allocation of high-skill

labor to automation, HAt , is low and automation intensity is low. Consequently, growth

is driven by horizontal innovation and the behavior of the economy is close to that of a

Romer model with a Cobb-Douglas production function between low-skill and high-skill

labor, and both wages approximately grow at a rate gNt / (σ − 1). This corresponds to

what we label as Phase 1 of the economy (naturally, if Gt is not initially low, it must

depreciate during this period following equation (22)).

18Beyond a focus on different empirical phenomena (an increase in inequality vs. cross-countryproductivity differences), there are two important differences between our model and Zeira (1998).Zeira (1998) assumes exogenous technological progress (while we model endogenous innovation) andhere the innovation cost changes over time while Zeira (1998) has constant adoption cost.

21

Second Phase. As wLt grows relative to ϕ−1, the term (V At − V N

t )/(wHt/Nt) in-

creases, in fact, following (28) it grows like(1 + ϕwε−1

Lt

)µ−1 when Gt is low. This raises

the incentive to innovate in automation. Without the externality in the automation

technology (κ = 0), (26) directly implies that HAt must raise significantly above zero,

and with it the Poisson rate of automation, η(HAt / (1−Gt)

)κand thereby the share of

automated products, Gt. For κ > 0, the depreciation in the share of automated products

during Phase 1 might gradually make the automation technology less effective which can

delay or even potentially prevent the take-off of automation.19

In line with Proposition 1, both the increase in Gt and Nt lead to an increase in the

skill premium. As we show below, the low-skill wage may temporarily decline. We will

label this time period where the share of automated products in the economy increases

sharply the second phase (the transition between phases is smooth and therefore the

exact limits are arbitrary). Arguably, this time period is the one where our model differs

the most from the rest of the literature.

Third Phase. As the share of automated products Gt is no longer near zero, the

gain from automation V At − V N

t and its effective cost wHt/Nt grow at the same rate

(the right-hand side in (28) is now close to 1/Gt). As a result, the normalized mass of

high-skill workers in automation research (NthAt ) stays bounded (see (26)), and so does

the Poisson rate of automation, which implies that Gt converges to a constant below 1.

The economy then converges toward the asymptotic steady-state which features a

constant share of automated products strictly below zero and a constant growth rate in

the number of products. Following Proposition 1, part A, the skill premium grows, but

the continued existence of non-automated products ensures that low-skill wages grow at

a positive rate asymptotically given by gwL∞ = gwH∞ / (1 + β(σ − 1)) . In this phase, the

profits made by a non-automated firm are negligible relative to the horizontal innovation

cost, therefore it is the prospect of future automation which guarantees the entry of new

products. This plays an important role in the interaction between the two innovation

technologies. Besides, the growth rate in the number of products is always lower in Phase

3 than in Phase 1 in particular because automation reduces the share of output which

goes to profits of new firms (see Appendix 7.5.5 for more details). Though in Phase 3

the share parameters of the nested CES function are constant, the model continues to

differ from a generic capital deepening model in that long-run growth is endogenized

19Automation takes off if either G0 and N0 are not too low or, for any values of N0, G0 > 0 wheneverκ(1− β) + κ < 1—see Appendix 7.5.4. If we were to assume instead that the automation technology isgiven by max

ηGκt , η

(Nth

At

)κ, then automation would always take off.

22

and depends on its interaction with automation.

3.5 An illustration of the transitional dynamics

We now illustrate our previous result and further analyze the behavior of our economy

through the use of numerical simulations.20 Thereafter, we relate our theoretical results

to the historical experience of the US economy. Unless, otherwise specified, and in

line with the results of Sections 2.3 and 3.4, the broad patterns described below do

not depend on specific parameter choices and we simply choose “reasonable” parameters

(Table 1). Appendix 7.7.6 gives a systematic exploration of the parameter space and

section 4 calibrates a richer model to the U.S. data.

Table 1: Baseline Parameter Specification

σ ε β H L θ η κ ϕ ρ κ γ N0 G0

3 4 2/3 1/3 2/3 2 0.2 0.5 0.25 0.02 0 0.3 1 0.001

Baseline Parameters. Total stock of labor is 1 with L = 2/3 and β = 2/3 such

that absent automation and if all high-skill workers were in production the skill premium

would be 1. The initial mass of products is set low at N0 = 1 to ensure we begin in Phase

1. The initial share of automated products is low, G0 = 0.001, but would initially decline

had we chosen a higher level. We set σ = 3 to capture an initial labor share close to 2/3.

We set ϕ = 0.25 and ε = 4, so that at t = 0, the profits of automated firms relative to

non-automated firms are only 0.004%. The innovation parameters (γ, η, κ) are chosen

such that GDP growth is close to 2% both initially and asymptotically. There is no

externality from the share of automated products in the automation technology, κ = 0.

ρ and θ are chosen such that the interest rate is around 6% initially and asymptotically.

Figure 3 plots the evolution of the economy. Based on the behavior of the automation

expenditures (Panel C) we delimit Phase 1 as corresponding to the first 100 years and

Phase 2 as the period between year 100 and year 250.

Innovation and growth. Initially, low-skill wages and hence the incentive to auto-

mate—proportional to (V At − V N

t )/(wHt/Nt)—are low (Panel B) and so is the share of

automated firms Gt (Panel C). With growing low-skill wages, the incentive to automate

picks up a bit before year 100. Then the economy enters Phase 2 as automation expenses

sharply increase (up to 4% of GDP). This leads to an increase in the share of automated

20We employ the so-called “relaxation” algorithm for solving systems of discretized differential equa-tions (Trimborn, Koch and Steger, 2008). See Appendix 7.6 for details.

23

products Gt which eventually stabilizes at a level strictly below 1. There is no simple

one-to-one link between automation spending and rising inequality in our model: here,

automation spending is higher in Phase 3 than in Phase 2 (Panel C in figure 3), yet the

growth in the skill premium is slower.21

Panel A: Growth Rates of Wages and GDPPhase 1 Phase 2 Phase 3

0 100 200 300 400Year

0

1

2

3

4

Per

cen

t

gGDP

gwL

gwH

0 100 200 300 4000

1

2

3

4

5

automation incentive (left)

Panel B: Automation Incentive and Skill PremiumPhase 1 Phase 2 Phase 3

0 100 200 300 400Year

0

2

4

6

log

unit

skill premium (right)

0 100 200 300 4000

5

10

15

Per

cen

t

Autom./GDP (left)Hori./GDP (left)

Panel C: Research Expenditures and G

Phase 1 Phase 2 Phase 3

0 100 200 300 400Year

0

20

40

60

80

100

G (right)

Panel D: Factor SharesPhase 1 Phase 2 Phase 3

0 100 200 300 400Year

0

20

40

60

80

100

Per

cen

t

Total labor shareLow-skill labor share

Figure 3: Transitional Dynamics for baseline parameters. Panel A shows growth rates forGDP, low-skill wages (wL) and high-skill wages (wH), Panel B the incentive to au-tomate,

(V At − V Nt

)/ (wHt/Nt), and the skill premium, Panel C the total spending

on horizontal innovation and automation as well as the share of automated prod-ucts (G), and Panel D the wage share of GDP for total wages and low-skill wages.

As shown in Panel C, spending on horizontal innovation as a share of GDP declines

during Phase 2 and for any parameter values ends up being lower in Phase 3 than Phase

1. Despite this, the growth rate of GDP is roughly the same in Phases 1 and 3 because

the lower rate of horizontal innovation in Phase 3 is compensated by a higher elasticity of

GDP wrt. Nt (1/ [(σ − 1) (1− β)] instead of 1/ (σ − 1)). As a result, the phase of intense

automation—which also contributes to growth—is associated with a temporary boost

of growth. This is, however, specific to parameters (Appendix 7.7.2 gives a counter-

example). That intense automation need not be associated with a sharp increase in

growth is important because the lack of an acceleration in GDP growth in recent decades

has often been advanced in opposition to the hypothesis that a technological revolution

21Intuitively the elasticity of the skill-premium with respect to the skill-bias of technology is notconstant in our model, contrary to a CES framework with factor-augmenting technologies.

24

explains the recent increase in the skill-premium (Acemoglu, 2002a).

Wages. In the first phase, and referring to figure 2, the relative demand curve stays

close to the straight dotted line with slope 1−ββ

LHP associated with a Cobb-Douglas pro-

duction. Continuous horizontal innovation pushes the isocost curve towards the North

East so that both wages grow at around 2% (Panel A).

As rising low-skill wages trigger the second phase, the relative demand curve pivots

counter-clockwise and bends upwards increasing the growth rate of high-skill wages to

almost 4% and suppressing the growth rate of low-skill wages to around 1%. Though

our parameter values satisfy the conditions of Proposition 1 B.ii and any increase in

Gt has a negative impact on wLt, the growth in Nt is sufficient to ensure that low-skill

wages grow at a positive rate throughout. Section 3.7 demonstrates that this need not

be the case. It is precisely this movement of the relative demand curve that allows our

model to capture labor-saving innovation; a feature which a model with constant G or

a capital deepening model would not include. In the third phase, the relative demand

curve no longer moves, but the continuous push of the isocost curve keeps increasing the

skill-premium albeit more slowly than previously.22 Yet, at the same time, the decline

in horizontal innovation reduces it. Appendix 7.7.1 presents a growth decomposition

exercise where we compute the instantaneous contribution of horizontal innovation and

automation to low-skill and high-skill wages.23

Factor shares. Panel D of figure 3 plots the labor share and the low-skill labor

share. With machines as intermediate input, capital income corresponds to aggregate

profits, which are a constant share of output. High-skill labor in production also earns a

constant share of output. Both correspond to a rising share of GDP in Phase 2 as during

this time period, the ratio Y/GDP increases since machines expenditures are excluded

from GDP . Low-skill labor earns (close to) a constant share of output in Phase 1 but

its share declines with automation in Phase 2 and approaches 0 in Phase 3. As a result,

the labor share of GDP , constant in Phase 1, declines in Phase 2 and stabilizes at a

22Changes in the mass of high-skill workers in production, HPt , also affect the skill premium: increas-

ing Gt requires hiring more high-skill workers in automation innovation (in the same vein as the GeneralPurpose Technology literature; notably Beaudry et al. (2016)). Here, this effect is counteracted by adecrease in horizontal innovation and the net effect is small.

23At the aggregate level, our model boils down to a nested CES production function (see equation(3)), and Phase 2 corresponds to a period where the share parameter of the composite which featuressubstitutability between machines and low-skill labor, Gt, rises. This change in the share parametershould not be confused with an increase in the elasticity of substitution between machines and low-skilllabor (in fact, the Morishima’s elasticity of substitution between these two factors is symmetric anddeclines in Phase 2 from a value close to ε to a value close to 1 + β(σ − 1)).

25

lower level than before in Phase 3. Working contrary to this is the increase in innovation

as only high-skill workers work in innovation, but the net effect is an increase in the

capital share. For different parameter values, since HPt is not constant, the drop in the

labor share can be delayed relative to the rise in the skill premium (see Appendix 7.7.3).

The ratio of wealth to GDP also increases during Phase 2 and asymptotes a constant

in Phase 3 (see Appendix 7.7.1).

3.6 Comparison with the historical experience

Our model suggests that technological progress is overall biased against low-skill labor,

especially at later stages when low-skilled wages are sufficiently high. This bias manifests

as a process of automation which causes a widening skill premium and a decline in the

labor share. This is in line with the experience of the United States since the 1980s,

where the college premium (considered to be a good proxy for the skill premium over

that time period) has been steadily increasing (our stylized fact 1 in the introduction)

and the labor share has declined by around 5 percentage points (stylized fact 3, see

Karabarbounis and Neiman, 2013, or Elsby, Hobijn and Sahin, 2013). This contrasts

our paper with most of the growth literature which features a balanced growth path and

therefore does not have permanently increasing labor inequality.24

Extrapolating further in the past is more complicated, but it is worth discussing

it briefly since automation did not start in the 1980s. Goldin and Katz (2008) argue

that technological change has been skill-biased throughout the 20th century as periods

of decline in the skill premium (such as the 1970s) can be accounted for by exogenous

changes in the relative supply of skills. Even before, Katz and Margo (2014) argue that

the relative demand for highly skilled workers (in professional, technical and manage-

rial occupations) has increased steadily from perhaps as early as 1820 to the present.

This shift in relative demand may have been (partly) compensated for by changes in

relative skill supply (see Appendix 6.4) and human capital accumulation (see Section

2.5) making growth more balanced. However, automation did not always replace the

least skilled individuals: the mechanization of the 19th century replaced skilled artisans,

and the computerization of the last 30 years displaced more severely middle-skill work-

ers. Therefore our model offers a framework to shed light on the historical experience

24For instance, in Acemoglu (1998), low-skill and high-skill workers are imperfect substitutes inproduction. Yet, since the low-skill augmenting technology and the high-skill augmenting technologygrow at the same rate asymptotically, the relative stocks of effective units of low-skill and high-skilllabor is constant, leading to a constant relative wage.

26

provided that the definition of ‘high-skill’ is narrow.

The capital share has followed a U-curve in the 20th century (Piketty and Zucman,

2014 and Piketty, 2014), which our baseline model cannot account for. As the decline of

the labor share in manufacturing is more persistent (9 percentage points from 1960 to

2005; Alvarez-Cuadrado, Van Long and Poschke, 2018), one way to reconcile our model

with the historical pattern on factor shares would be to extend it to include multiple

sectors (agriculture, manufacturing and services) with an elasticity of substitution less

than 1, experiencing automation waves at different points in time. In such a model, as

one sector automates, spending on the other sectors would increase (as in Acemoglu and

Guerrieri, 2008) securing a higher labor share.

3.7 Additional results

We now exploit the endogenous nature of endogenous change in our model and examine

the interaction between horizontal innovation and automation

Declining low-skill wages. Empirical evidence suggests that low-skill wages have

been stagnating and perhaps even declining in recent periods (Stylized fact 2 ). Because

our model features a labor saving innovation, it can accommodate declining low-skill

wages in Phase 2, unlike a model with fixed G, a capital deepening model with perfectly

elastic capital or in the previous DTC literature. Here, it happens when the relative

demand curve pivots sufficiently fast counter-clockwise compared with the movement

of the isocost curve in Figure 2. We can ensure this, for instance, by introducing ex-

ternalities in the automation function and setting κ = 0.49. This delays the onset of

automation, but intensifies it once it takes off, so that the relative demand curve moves

quickly enough compared to the isocost curve for a decline in low-skill wages.25 Yet,

this drop must be temporary because lower low-skill wages discourage automation. See

Appendix 6.3 for details and another example with κ = 0.

Effect of innovation parameters. In Appendix 7.5.6, we establish:

Proposition 4. The asymptotic growth rates of GDP gGDP∞ and low-skill wages gwL∞

increase in the productivity of automation η and horizontal innovation γ.

Therefore, in the long-run, a better automation technology (a higher η) actually

benefits low-skill wages: the reason is that firms automate faster which encourages hor-

izontal innovation. During the transition, however, a higher η also means that Phase 2

25Horizontal innovation drops during this intense automation phase because its cost increases as thereis a high demand for high-skill workers in automation innovation.

27

starts sooner, leading to lower low-skill wages at that point and a higher skill premium

(see a numerical example in Appendix 7.7.4). These result preview those on the effect

of taxes on automation in section 4.2.

Further results and extensions in the Appendix. i) Our model features ele-

ments of self-correction in the presence of exogenous shocks (as Acemoglu and Restrepo,

2017). For instance, with no automation externality (κ = 0), a positive exogenous shock

to Gt will be followed by a period where automation is relatively less intense (as the skill

premium would decline), so that eventually the asymptotic share of automated prod-

ucts stays the same (see Appendix 7.7.5). ii) Appendix 7.8 studies the social planner’s

problem. The optimal allocation is qualitatively similar to the equilibrium we described,

which shows that our results are not driven by the market structure we imposed. iii) Ap-

pendix 7.9 presents a setting in which automation can only be undertaken before a firm

starts production. The transition of the economy is qualitatively identical, which shows

that the main results of the paper depend only on the feature that higher wLt creates in-

centives to automate and not on the assumption that firms are born non-automated, iv)

Appendix 6.4 presents the transitional dynamics with an endogenous supply response.

4 Quantitative Exercise and Policy Experiments

In this section, we conduct a quantitative exercise to compare empirical trends for the

United States with the predictions of our model. This allows us to dicipline the param-

eters of our model and subsequently conduct policy experiments.

4.1 Quantitative exercise

To match the data quantitatively, we need to modify the baseline model. First, since the

share of high-skill workers has dramatically increased, we let H and L vary over time and

here take as given their path from the data (as opposed to endogenizing labor supply as

in Appendix 6.4). Second, we assume that producers rent machines from a capital stock.

Capital can also be used used as structures in both automated and non-automated firms.

Third, we allow for the possibility that low-skill workers are replaced by a composite of

machines and high-skill workers. The production function (2) is then replaced by

y(i) = [l(i)ε−1ε + α(i)(ϕhe(i)

β4ke(i)1−β4)

ε−1ε ]

εβ1ε−1hs(i)

β2ks(i)β3 , (29)

28

where β1 +β2 +β3 = 1 and β4 ∈ [0, 1). The central difference between equations (2) and

(29) is the introduction of he(i) as high-skill labor which—along with machines—perform

the newly automated tasks (“e” for equipment). This feature is necessary to capture a

relatively low drop in the labor share. ks(i) is structures and ke(i) and ks(i) are both

rented from the same capital stock Kt. Kt increases with investment in final goods and

depreciates at a fixed rate ∆, so that (12) is replaced by:

Kt = Yt − Ct −∆Kt. (30)

The cost advantage of automated firms now depends on the ratio between low-skill

wage and the price of the high-skill labor capital aggregate namely wLt/(wβ4Htr

1−β4t ) where

rt = rt + ∆ is the gross rental rate of capital. The logic of the baseline model directly

extends to this case. Proposition 1 still holds and Proposition 2.A) holds with gwH∞ =

gY∞ = gN∞/ ((β2 + β1β4)(σ − 1)) and gwL∞ = gY∞ (1 + (σ − 1) β1β4) / (1 + β1(σ − 1)) . An

equivalent to Proposition 3 holds but the system of differential equations includes three

control variables and three state variables and Proposition 4 holds as well. The tran-

sitional dynamics are similar to that of the baseline model but automation innovation

now depends on wLt/(wβ4Htr

1−β4t ). It is low in a first phase, increases in a second phase

and stabilizes in the third phase as the economy approaches its asymptotic steady-state.

The capital share and the capital output ratio increase in Phase 2 as equipment replaces

low-skill labor in production. Details and proofs are provided in Appendix 7.11.

We match our extended model to the data (details on the data and the method in Ap-

pendix 7.12). Because of data availability and to make our exercise easily comparable to

the rest of the literature, we identify low-skill workers with non-college educated workers

and high-skill workers with college educated workers and focus on the years 1963-2007.

We match the skill-premium and take the empirical skill-ratio as given (and normal-

ize total population to 1). We also match the growth rate of real GDP/employment

and the labor share. We further associate the use of machines with private equipment

(excluding transport) and software. As pointed out by Gordon (1990) the NIPA price

indices for real equipment are likely to understate quality improvements in equipment

and therefore growth in the real stock of equipment. Hence, we use the adjusted price

index from Cummins and Violante (2002) for equipment, and build (private) equipment

29

and software to GDP ratios from 1963 to 2000.26

Our exercise has similarities with Goldin and Katz (2008) who attempt to explain

the skill-premium using a constant trend in the skill bias of technical change or Krusell

et al. (2000) who feed in the empirical path of equipment to explain the increase in the

skill-premium. But, it is more demanding: both the technology path and the equipment

stock must be endogenous.

Our model is not stochastic and cannot be directly estimated. Instead, we take a

parsimonious approach and choose parameters to minimize the weighted squared log-

difference between observed and predicted paths. We start the simulation 40 years before

1963 to force N1963 and G1963 to be consistent with the long-run behavior of our model.27

Since the model requires the skill-ratio before and beyond the time period we estimate,

we fit a generalized logistical function to the path of the log of the skill-ratio and use

the predicted values outside 1963-2007 (over that time period, the fit is excellent).

The model features a total of 14 parameters (ϕ can be normalized to 1 without loss

of generality). Instead of exogenously restricting parameters we will fit the model with

all parameters freely (other than the economically motivated boundaries imposed by the

model itself) and then assess whether these parameter estimates fit with other similar

estimates. Table 2 gives the resulting parameters. The elasticity of substitution across

products σ is estimated at 6.7, consistent with Christiano, Eichenbaum and Evans (2005)

who find that observed markups are consistent with a sigma of around 6. The elasticity

of substitution between machines and workers is estimated at almost 5. κ is estimated at

0.58 implying a substantial automation externality; a force which causes an accelerated

Phase 2. Finally, we find β1—the factor share of machines/low-skill workers—of 0.62

which implies sizable room for automation, though a β4 of 0.73 means that the share

of high-skill workers in the composite that replaces low-skill workers is of substantial

importance. The preference parameters are within standard estimates with a ρ of 3.7%

and the implied θ resulting in log-preferences. The only parameter that is estimated

outside a common range is the depreciation rate ∆ (but it is not precisely identified).

Appendix 6.5 discusses in details how the parameters are identified.

26The series on the skill premium, the skill ratio, GDP/employment and the labor share directlypredict a series for low-skill wages. Since this series differs substantially from the low-skill wages seriesof Acemoglu and Autor (2011) depicted in Figure 1 (particularly because of compositional effects), wedo not attempt to match the latter.

27Initial values for Gt have little impact on the state of the economy several years later (see e.g.Figure 22 in Appendix 7.7.6). And the same is true for initial values of Kt. Therefore by simulatingthe economy 40 years prior, we ensure that the simulated moments are nearly independent of the initialvalues for Gt and Kt. We fix K1923 at its steady-state value in a model with no automation.

30

Table 2: Parameters from quantitative exercise

Parameter σ ε β1 γ κ θ η κ ρ β2 ∆ β4 N1963 G1963

Value 6.7 4.97 0.62 0.64 0.58 1 0.41 0.58 0.037 0.18 0.014 0.73 21.6 0.02

1960 1970 1980 1990 2000 2010 2020Years

1

1.2

1.4

1.6

1.8

2

2.2

Ski

ll-P

rem

ium

(ra

tio)

Panel A - Composition-adjustedcollege/non-college weekly wage ratio

Predicted ValuesEmpirical Values

1960 1970 1980 1990 2000 2010 2020Years

50

55

60

65

70

75

80

Per

cen

t

Panel B - Labor share of GDP

1960 1970 1980 1990 2000 2010 2020Years

50

100

150

200

250300350400

Inde

x (1

963

= 1

00)

Panel C - GDP / Employment

1960 1970 1980 1990 2000 2010 2020Years

0.25

0.5

1

2

3

Equ

ipm

ent /

GD

P

Panel D - Equipment / GDP

Figure 4: Predicted and empirical time paths

Figure 4 further shows the predicted path of the matched data series along with

their empirical counterparts. Panel A demonstrates that the model matches the rise in

the skill premium from the early 1980s and the flat skill premium in the period before

reasonably well. Though less pronounced than in the data, our model also includes the

more recent decline in the growth rate of the skill premium, which, computed over a

5 years moving window, peaks in 1984 at 1.3% and drops to 0.64% in the 2000s. The

decline in the labor share is matched from 1970 onward. The average growth rate of

the economy is matched completely as shown in Panel C. Although, the model largely

captures the average growth rate of capital equipment over GDP during the period, the

predicted path differs somewhat from its empirical counterpart as shown in Panel D on

log-scale. Whereas the empirical path is close to exponential, the predicted path tapers

off somewhat towards the end of the period.28 In Appendix 6.5.5, we reproduce the same

28 Interestingly, more recent data show a slow down in software investment (see Beaudry et al., 2016).Moreover, the ratio of equipment to GDP in the data is only a proxy for the ratio of machines to GDPin the model, since all equipment may not be used to replace low-skill workers.

31

1980 2000 2020 2040 2060Year

0

0.5

1

1.5

2

2.5

Per

cen

tPanel A: Growth Rates of Wages and GDP

gGDP

gwL

gwH

1980 2000 2020 2040 206050

55

60

65

70

75

Per

cen

t

labor share(left)

1980 2000 2020 2040 2060Year

0.3

0.4

0.5

0.6

0.7

0.8

0.9

log

unit

Panel B: Labour share and Skill Premium

skill premium (right)

1980 2000 2020 2040 2060Year

0

1

2

3

4

5

6

7

8

9

Per

cen

t

Autom./GDP (left)Hori./GDP (left)

1980 2000 2020 2040 2060Year

0

20

40

60

80

100

Per

cen

t

Panel C: Research Expenditures and G

G (right)

Figure 5: Transitional Dynamics with calibrated parameters (the growth rates are computedover a 5 year moving average).

exercise but only matching the first 30 years, the parameters are nearly identical and

the calibrated model matches well the rest of the sample period.

Figure 5 plots the transitional dynamics from 1963 to 2063. Panel A shows that

GDP growth slows down past 2007, Panel B that the skill premium keeps growing albeit

at a slower rate than in the 1980s, while the labor share smoothly declines toward its

steady-state value of 56%. Panel C shows that the share of automated products increases

sharply through the 1963-2007 time period: In 1963 the value is 0.02 (yet this value is

not precisely identified) but rises to 0.20 by the 1980s and 0.49 by 2007 to finally settle

at 0.88 asymptotically—in this case, the marginal effect of automation unambiguously

reduces low-skill wages on the entire path. Besides, the growth rate of the skill premium

decreases in the 1990s and 2000s even though automation expenditures are roughly

constant between 1980 and 2000. This constitutes a response to the critique of the

literature on skill-biased technical change put forward by Card and DiNardo (2002),

who argue that inequality rising the most in the early to mid 1980s and technological

change continuing in the 1990s, squares poorly with the predictions of a framework based

on skill-biased technological change.

4.2 Automation taxes

Among the many policy proposals to address rising income inequality, is a tax on the

use of automation technology or a “robot tax”. In the following we examine the conse-

quences of two taxes: either on the use of machines—in the form of a tax on the rental

rate of equipment—or the innovation of new machines—in the form of taxing high-skill

workers in automation innovation. In either cases we consider the permanent unexpected

32

introduction of a tax of 50% (See Appendix 7.11 for details).

First, consider a tax on the use of machines. To clarify the role of endogenous

technology we also simulate the economy holding technology, Nt andGt and thereforeHPt

at the baseline level. Figure 6 reports the results. The immediate effect is to discourage

the use of machines and consequently low-skill wages rise by 4% on impact (Panel B)

with a corresponding lower skill premium (Panel C). The endogeneity of technology

amplifies the effect of the tax over time (in panel B, the gap between the endogenous

and the exogenous cases widens). This results from two effects. First, the tax discourages

automation innovation leading to a lower G (Panel E). Second, since high-skill workers

and machines are complements, the tax reallocates high-skill workers toward horizontal

innovation, increasing N (Panel D). Consequently, the positive effect on low-skill wages

is eventually larger than the initial 4%. Even output will asymptotically be higher than

the baseline (extending Panel A).29

2007 2027 2047 2067 2087Year

-8

-6

-4

-2

0

2

Per

cen

t

Panel A: Output deviation

Machine tax(endogenous technology)

Machine tax(exogenous technology)

Automation Innovation tax

2007 2027 2047 2067 2087Year

-4

-2

0

2

4

6

Per

cen

t

Panel B: Low-skill wages deviation

2007 2027 2047 2067 2087Year

-12

-10

-8

-6

-4

-2

0

Per

cen

t

Panel C: Skill premium deviation

2007 2027 2047 2067 2087Year

-20

-10

0

10

20

30

Per

cen

t

Panel D: N deviation

2007 2027 2047 2067 2087Year

-12

-10

-8

-6

-4

-2

0

Per

cen

t

Panel E: G deviation

Figure 6: Effects of a machine tax and an automation innovation tax relative to baseline.

A tax on automation innovation has very different implications: First, high-skill

workers reallocate from innovation in automation toward production which, on impact,

boosts output and marginally low-skill wages. As the share of automated product G

29Asymptotically, a machine tax has no effect on G or on the growth rate of N : as using low-skillworkers instead of machines becomes prohibitively expensive, the allocation of high-skill workers remainsundistorted by the presence of a finite tax. See Proposition 12 in Appendix 7.11.

33

decreases, low-skill wages further increase though very modestly. However, discouraging

automation innovation also discourages horizontal innovation which eventually reduces

low-skill wages. The intuition is similar to that behind Proposition 4: a tax on automa-

tion innovation has similar effects to that of reducing the effectiveness of the automation

technology. The skill-premium is reduced as the economy grows at a slower rate.

This exercise highlights the importance of endogenous technology: Though both

forms of “robot” taxes increase low-skill wages on impact, the long-run effects depend

crucially on whether the tax is designed to encourage or discourage overall innovation.

Of course, this exercise is only a first pass and analyzing the welfare consequences of

these policies or others, say minimum wage legislation, is of interest for future research.

5 Conclusion

This paper introduces automation in a horizontal innovation growth model. In such

a framework, the economy undertakes a structural break. After an initial phase with

stable income inequality and stable factor shares, automation picks up. During this

second phase, the skill premium increases, low-skill wages stagnate and possibly decline,

the labor share drops—all consistent with the US experience in the past 50 years—and

growth starts relying increasingly on automation. In a third phase, the share of auto-

mated products stabilizes, but the economy still features a constant shift of low-skill

employment from recently automated tasks to as of yet non-automated tasks. Low-skill

wages grow in the long-run but slower than high-skill wages. A calibrated version of our

model shows that a tax on machine use increases low-skill wages more than in a model

with exogenous technology, while a tax on automation innovation eventually reduces

low-skill wages.

A lesson from our framework is that if tasks performed by a scarce factor (say labor)

can be automated but it is not presently profitable to do so, then, in a growing economy,

the return to this factor will eventually increase sufficiently to make it profitable. In

other words, there is a long-run tendency for technical progress to displace substitutable

labor (a point made by Ray, 2014,), but this only occurs if the relevant wages are large

relative to the price of machines. This in turn can only happen under three scenarios:

either automation itself increases the wages of these workers (the scale effect dominates

the substitution effect), or there is another source of technological progress (here, hori-

zontal innovation), or technological progress allows a reduction in the price of machines

34

relative to the consumption good (as in Appendix 7.4). Importantly, when machines

are produced with a technology similar to the consumption good, automation can only

reduce wages temporarily: a prolonged drop in wages would end the incentives to auto-

mate in the first place. Although, we focus on a general equilibrium model with low-skill

labor, these insights extend to subsectors of the economy and other scarce factors.

Fundamentally, the economy in our model undertakes an endogenous structural

change when low-skill wages become sufficiently high. This distinguishes our paper

from most of the literature, which seeks to explain changes in the distribution of income

inequality through exogenous changes: an exogenous increase in the stock of equipment

as per Krusell et al. (2000), a change in the relative supply of skills, as per Acemoglu

(1998), or the arrival of a general purpose technology as in the associated literature.

This makes our paper closer in spirit to the work of Buera and Kaboski (2012), who

argue that the increase in income inequality is linked to the increase in the demand for

high-skill intensive services, which results from non-homotheticity in consumption.

The present paper is only a step towards a better understanding of the links be-

tween automation, growth and income inequality. Given that automation has targeted

either low- or middle-skill workers and that artificial intelligence may now lead to the

automation of some high-skill tasks, a natural extension of our framework would include

more skill heterogeneity. Another natural next step would be to add firm heterogeneity

and embed our framework into a quantitative firm dynamics model. Our framework

could be used to study the recent phenomenon of “reshoring”, where US companies that

had offshored their low-skill intensive activities to China, now start repatriating their

production to the US after having further automated their production process.

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39

6 Main Appendix (For Online Publication)

6.1 A combined HO-AR model

We now assume that intermediate i is produced according to

y (i) = ((b (i) l (i)) + α (i) b (i)ς x (i))β

(b (i)h (i))1−β

instead of (2). That is we restrict attention for simplicity to the case where low-skill

labor and machines are perfect substitute in automated firms and we normalize ϕ to 1.

We assume that b (i) = exp (Bi) for some B > 0 and ς ∈ [0, 1]. ς represents the share of

technological progress in new varieties which is TFP-augmenting while 1−α is the share

which is purely labor augmenting (for both forms of labor). We assume that Nt grows

exogenously and linearly, so that Nt = nt for some n > 0. Further, once invented a good

has an exogenous probability η of becoming automated, and all high-skill workers are

hired in production (HP = H).

We find that in this model low-skill and high-skill wages grow at the same rate if

and only if the form of technological progress which is associated from moving from

an intermediate i to an intermediate i′ with a higher index is purely labor augmenting

(ς = 0). As soon as newer intermediates also feature more productive machines if

automated (ς > 0) then growth will be asymptotically unbalanced. This result is in the

spirit of Uzawa’s theorem but differs in so far as it refers to technological progress from

one intermediate to another instead of aggregate technological progress (which here

features increasing the range of intermediate inputs and automating some of them).

Formally, we establish in Appendix 7.3:

Proposition 5. Low-skill and high-skill wages grow asymptotically at the same rate if

and only if ς = 0. Otherwise, high-skill wages grow asymptotically faster than low-skill

wages: gwH > gwL .

6.2 Formal description of the normalized system of differential

equations

We first define an equilibrium as follows:

Definition 1. A feasible allocation is defined by time paths of stock of prod-

ucts and share of those that are automated, [Nt, Gt]∞t=0, time paths of use of low-

40

skill labor, high-skill labor, and machines in the production of intermediate inputs

[lt(i), ht(i), xt(i)]∞i∈[0,Nt],t=0, a time path of intermediate inputs production [yt(i)]

∞i∈[0,Nt],t=0,

time paths of high-skill workers engaged in automation [hAt (i)]∞i∈[0,Nt],t=0, and in hori-

zontal innovation [HDt ]∞t=0, time paths of final good production and consumption levels

[Yt, Ct]∞t=0 such that factor markets clear ((18) holds) and good market clears ((12) holds).

Definition 2. An equilibrium is a feasible allocation, a time path of intermediate

input prices [pt(i)]∞i∈[0,Nt],t=0, a time path for low-skill wages, high-skill wages, the interest

rate and the value of non-automated and automated firms [wLt, wHt, rt, VNt , V

At ]∞t=0 such

that [yt(i)]∞i∈[0,Nt],t=0 maximizes final good producer profits, [pt(i), lt(i), ht(i), xt(i)]

∞i∈[0,Nt],t=0

maximize intermediate inputs producers’ profits, [hAt (i)]∞i∈[0,Nt],t=0 maximizes the value

of non-automated firms, [HDt ]∞t=0 is determined by free entry, [Ct]

∞t=0 is consistent with

consumer optimization and the transversality condition is satisfied.

We now derive the system of differential equations satisfied by the normalized vari-

ables (nt, Gt, ht, χt). The definition of nt immediately gives:

nt = − β

(1− β)(1 + β(σ − 1))gNt nt. (31)

Rewriting (22) with hAt gives:

Gt = ηGκt

(hAt

)κ(1−Gt)−Gtg

Nt . (32)

Defining normalized profits πAt ≡ N1−ψt πAt and πNt ≡ N1−ψ

t πNt and the normalized values

of firms V At ≡ N1−ψ

t V At and V N

t ≡ N1−ψt V N

t , then we can rewrite (19) and (20) as

(rt − (ψ − 1) gNt

)V At = πAt +

·

V At , (33)

(rt − (ψ − 1) gNt

)V Nt = πNt + ηGκ

t

(hAt

)κ (V At − V N

t

)− vthAt +

·

V Nt . (34)

Equation (21) can similarly be rewritten as:

κηGκt

(hAt

)κ−1 (V At − V N

t

)= vt. (35)

Equation (23) with equality implies that and V Nt = vt/γ, therefore using (35) into (34),

41

we get: (rt − (ψ − 1) gNt

)vt = γπNt + γ

1− κκ

vthAt +

·vt. (36)

Taking the difference between (33) and (34) and using (35)we obtain:

(rt − (ψ − 1) gNt

) (V At − V N

t

)= πAt − πNt −

1− κκ

vthAt +

( ·

V At −

·

V Nt

).

Using again (35) we get,

(rt − (ψ − 1) gNt

)= κηGκ

t

(hAt

)κ−1[πAt − πNt

vt− 1− κ

κhAt

]

+κηGκt

(hAt

)κ−1 d

dt

(hAt

)1−κ

κηGκt

+

·vtvt.

Using (36), we can rewrite this expression a

γ

(πNtvt

+1− κκ

hAt

)= κηGκ

t

(hAt

)κ−1[πAt − πNt

vt− 1− κ

κhAt

]

+κηGκt

(hAt

)κ−1 d

dt

(hAt

)1−κ

κηGκt

.

Using (32), this leads to:

γ

(πNtvt

+1− κκ

hAt

)= κηGκ

t

(hAt

)κ−1[πAt − πNt

vt− 1− κ

κhAt

](37)

+ (1− κ)

·

h

A

t

hAt− κ

Gt

(ηGκ

t

(hAt

)κ(1−Gt)−Gtg

Nt

).

From the definition of ωt and nt, we get that wβ(1−σ)Lt = ωtnt, so that

πNt = ωtnt

(ϕ+ (ωtnt)

1µ

)−µπAt (38)

42

We can then reorder terms in (37) and use (38) to obtain:

·

h

A

t =γhAt

1− κ

(ωtnt

(ϕ+ (ωtnt)

1µ

)−µ πAtvt

+1− κκ

hAt

)(39)

−κηGκ

t

(hAt

)κ1− κ

(1− ωtnt

(ϕ+ (ωtnt)

1µ

)−µ) πAtvt

+ηGκt

(hAt

)κ+1

+κhAt

1− κ

(ηGκ−1

t

(hAt

)κ(1−Gt)− gNt

)Rewriting (24) using the definition of ct, leads to

rt = ρ+ θ

·ctct

+ θψgNt .

Combining this equation with (36) and (38), and using the definitions of χt, vt and πAt

leads to

χt = χt

(γωtnt

(ϕ+ (ωtnt)

1µ

)−µ πAtvt

+ γ1− κκ

hAt − ρ− (θψ − ψ + 1) gNt

). (40)

Together equations (31), (32), (39) and (40) form a system of differential equations which

depends on ωt, πAt /vt and gNt . To determine πAt /vt, recall that (as proved in the text),

profits are given by

π (wL, wH , α (i)) =(σ − 1)σ−1

σσc (wL, wH , α (i))1−σ Y. (41)

Using (4) and the definition of ωt, one gets:

πAt =(σ − 1)σ−1

σσ

(ββ (1− β)1−β

)σ−1 (ϕ+ (ωtnt)

1µ

)µw−ψ

−1

Ht Yt. (42)

Rearranging terms in (11) gives

vt =

(σ − 1

σ

) 11−β

ββ

1−β (1− β)(G(ϕ+ (ωtnt)

1µ

)µ+ (1−G)ωtnt

)ψ. (43)

Using (8), one further gets:

Yt = σψvtHPt N

ψt . (44)

43

Therefore, rewriting (42) with (43) and (44), one gets:

πAtvt

=ψ(ϕ+ (ωtnt)

1µ

)µHPt

G(ϕ+ (ωtnt)

1µ

)µ+ (1−G)ωtnt

, (45)

which still requires finding HPt . Using (4), (5), (6) and aggregating over all automated

firms, one gets the following expression for the total demand of machines:

Xt = βGtNtϕ

(σ − 1

σ

)σ (ββ (1− β)(1−β)

)σ−1 (ϕ+ (ωtnt)

1µ

)µ−1

w−ψ−1

Ht Yt.

Using (43), this expression can be rewritten as:

Xt =σ − 1

σ

βGtϕ(ϕ+ (ωtnt)

1µ

)µ−1

G(ϕ+ (ωtnt)

1µ

)µ+ (1−G)ωtnt

Yt. (46)

This together with (44) implies that ct obeys

ct =

1− σ − 1

σ

βGtϕ(ϕ+ (ωtnt)

1µ

)µ−1

G(ϕ+ (ωtnt)

1µ

)µ+ (1−G)ωtnt

σψvtHPt .

Combining this equation with the definition of χt and (43), leads to

HPt =

(σ−1σ

) 11−β ( 1

θ−β)

×(1−β)

1θ β

β1−β ( 1

θ−1)χ

1θt

(Gt

(ϕ+(ωtnt)

1µ

)µ+(1−Gt)ωtnt

)ψ( 1θ−1)+1

Gt

((1−β σ−1

σ )ϕ+(ωtnt)1µ

)(ϕ+(ωtnt)

1µ

)µ−1

+(1−Gt)ωt

. (47)

Using the definition of HDt and hAt , one can rewrite (18) for high-skill workers as:

gNt = γ(H −HP

t − (1−Gt) hAt

). (48)

Together (45), (47) and (48) determine πAt /vt and gNt as a function of the original

variables nt, Gt, hAt , χt and of ωt, which still needs to be determined. To do so, combine

44

(10) and (11), and use the definitions of nt and ωt to obtain an implicit definition of ωt:

ωt =

(σ−1σβ) 1

1−β HPt

L

(Gt

(ϕ+ (ωtnt)

1µ

)µ−1

(ωtnt)1−µµ + (1−Gt)

)×(Gt

(ϕ+ (ωtnt)

1µ

)µ+ (1−Gt)ωtnt

)ψ−1

β(1−σ)

1+β(σ−1)

. (49)

Therefore eventually the system of differential equations satisfied by nt, Gt, hAt , χt is

defined by (31), (32), (39) and (40), with πAt /vt, HPt , gNt and ωt given by (45), (47), (48)

and (49).

6.3 Negative growth for low-skill wages

This section presents two examples with negative growth low-skill wages. Further results

on the transitional dynamics derived using simulations are presented in Appendix 7.7.

We ensure temporary negative growth in low-skill wages in figure 7 by setting κ =

0.49, thereby introducing the externality in automation.30 Initially Gt is small and the

automation technology is quite unproductive. Hence, Phase 2 starts later, even though

the ratio(V At − V N

t

)/ (wHt/Nt) has already significantly risen (Panel B). Yet, Phase 2 is

more intense once it gets started, partly because of the sharp increase in the productivity

of the automation technology (following the increase in Gt) and partly because low-skill

wages are higher. Intense automation puts downward pressure on low-skill wages. At

the same time, horizontal innovation drops considerably, both because new firms are less

competitive than their automated counterparts, and because the high demand for high-

skill workers in automation innovations increases the cost of inventing a new product.

This results in a short-lived decline in low-skill wages. Indeed, the decline in wLt (and

increase in high-skill wage wHt) lowers the incentive to automate (Panel B), which in

return reduces automation.

30We choose this value for κ instead of 0.5, because in that case there is no horizontal innovation forsome time periods (that is (23) holds with a strict inequality). This is not an issue in principle butsimulating this case would require a different numerical approach.

45

0 100 200 300 400Year

0

2

4

6

8

10

Per

cen

t

Panel A: Growth Rates of Wages and GDP

gGDP

gwL

gwH

0 100 200 300 4000

5

10automation incentive (left)

0 100 200 300 400Year0

2

4

6

log

unit

Panel B: Automation Incentive and Skill Premium

skill premium (right)

0 100 200 300 4000

5

10

15

Per

cen

t

Autom./GDP (left)Hori./GDP (left)

0 100 200 300 400Year

0

20

40

60

80

100Panel C: Research Expenditures and G

G (right)

0 100 200 300 400Year

0

20

40

60

80

100

Per

cen

t

Panel D: Factor Shares

Total labor shareLow-skill labor share

Figure 7: Transitional Dynamics with temporary decline in low-skill wages with an automa-tion externality. Note: same as for Figure 3 but with an automation externality ofκ = 0.49.

Importantly, low-skill wages can drop for κ = 0—albeit for a small parameter set—as

shown in Figure 8 where low-skill wages slightly decline for a short time period (our

numerical investigation suggests that larger declines in the absence of an automation

externality need to be associated with periods where horizontal innovation completely

ceases). The associated parameters are given in Table 3.

Table 3: Baseline Parameter Specification

σ ε β H L θ η κ ϕ ρ κ γ

3 73 0.72 0.35 0.65 2 0.2 0.97 0.25 0.022 0 0.28

The crucial parameter change is an increase in κ, such that the automation technology

is less concave. This delays Phase 2, which is then more intense and leads to a sharp

increase in high-skill wages, reducing considerably horizontal innovation (note that in the

period where low-skill wages decline the share of high-skill workers hired in production

increases slightly, which has a positive contribution on low-skill wages’ growth rates).

46

0 50 100 150 200 250 300 350 400Year

-1

0

1

2

3

4

5

6

7

Per

cen

t

Panel A: Growth Rates of Wages and GDP

gGDP

gwL

gwH

0 50 100 150 200 250 300 350 400Year

0

2

4

6

8

10

12

14

16

Per

cen

t

Autom./GDP (left)Hori./GDP (left)

0 50 100 150 200 250 300 350 400Year

0

20

40

60

80

100

Per

cen

t

Panel B: Research Expenditures and G

G (right)

Figure 8: Transitional dynamics with temporary decline in low-skill wages without an au-tomation externality.

6.4 An endogenous supply response in the skill distribution

We present here an extension of the baseline model with an endogenous supply re-

sponse in the skill distribution. Specifically, let there be a unit mass of heterogeneous

individuals, indexed by j ∈ [0, 1] each endowed with lH units of low-skill labor and

Γ (j) = H (1+q)qj1/q units of high-skill labor (the important assumption here is the exis-

tence of a fat tail of individuals with low ability). The parameter q > 0 governs the shape

of the ability distribution with q → ∞ implying equal distribution of skills and q < ∞implying a ranking of increasing endowments of high-skill on [0, H(1+q)/q]. Proposition

3 can be extended to this case and in fact the steady state values (G∗, hA∗, gN∗, χ∗) are

the same as in the model with a fixed high-skill labor supply H. Proposition 2 also ap-

plies except that the asymptotic growth rate of low-skill wages is higher (see Appendix

7.10):

gwH∞ = gY∞ = ψgN∞ and gwL∞ =1 + q

1 + q + β(σ − 1)gY∞. (50)

At all points in time there exists an indifferent worker (jt) where wLt = (1+q)/q(jt)1/qwHt,

with all j ≤ jt working as low-skill workers and all j > jt working as high-skill work-

ers. This introduces an endogenous supply response as the diverging wages for low- and

high-skill workers encourage shifts from low-skill to high-skill jobs, which then dampens

the relative decline in low-skill wages. Hence, besides securing themselves a higher fu-

ture wage growth, low-skill workers who switch to a high-skill occupation also benefit

the remaining low-skill workers. Since all changes in the stock of labor are driven by

demand-side effects, wages and employment move in the same direction.

Figure 9 shows the transitional dynamics for this model when the common parameters

47

are the same as in Table 1, H = 1/3 (so that G∗, hA∗, gN∗, χ∗ are the same as in the

baseline model), l = 1 and q = 0.3. The figure looks similar to Figure 3, but the gap

in steady-state between the low-skill growth rate and the high-skill growth rate is a bit

smaller. In addition Panel B shows that the skill ratio increases from Phase 2 and Panel

A shows that the growth rate is lower in Phase 1 as the mass of high-skill workers is

lower then.

0 100 200 300 400Year

0

0.5

1

1.5

2

2.5

3

3.5

Per

cen

t

Panel A: Growth Rates of Wages and GDP

gGDP

gwL

gwH

0 100 200 300 400Year

-1

0

1

2

3

4

log

units

Panel B: Skill ratio and Skillpremium

skill ratioskillpremium

0 100 200 300 400Year

0

5

10

15

Per

cen

t

Autom./GDP (left)Hori./GDP (left)

0 100 200 300 400Year

0

20

40

60

80

100

Per

cen

t

Panel C: Research expenditures and G

G (right)

0 100 200 300 400Year

0

20

40

60

80

100

Per

cen

t

Panel D: Factor Shares

Wage shareLow-skill labor share

Figure 9: Transitional Dynamics for model with endogenous skill supply. Panel A showsgrowth rates for GDP, low-skill wages (wL) and high-skill wages (wH), Panel Bthe skill ratio and the skill premium, Panel C the total spending on horizontalinnovation and automation as well as the share of automated products (G), andPanel D the wage share of GDP for total wages and low-skill wages.

6.5 Parameters identification

In this section we discuss how our parameters are identified, first by carrying a back-

of-the-envelope calibration, second by computing the elasticities of the initial and final

values of the series we match with respect to the parameters, and third by computing how

precisely each parameter is identified. We then discuss specifically how κ is determined

and finally carry an out-of-sample prediction exercise, where we only use the first 30

years of the data to calibrate our parameters.

48

6.5.1 Back-of-the-envelope calibration

We first study how the production parameters σ, β1, β2, β4 and ∆ would be identified

under a naive back-of-the-envelope calibration, where we assume that in 1963 the U.S.

economy was in the first phase while in 2007, it was in the third phase. Since both

assumptions are actually not met in our estimation, this naive calibration gives param-

eters that are still far from those which we actually estimate. Nevertheless, the exercise

is informative to understand how these production parameters are related to moments

in the data.

Assuming that the economy in 1963 is close to the first phase, and using (159), we

get that the skill premium must obey:

wH1963

wL1963

≈ β2

β1

L1963

H1963 − 1γgN1963

.

Further, using that most high-skill workers work in production, such that 1γgN1963 is small

relative to H, we obtainβ2

β1

≈ wH1963H1963

wL1963L1963

, (51)

so that the ratio β2/β1 is determined by the ratio between the high-skill wage bill and

the low-skill wage bill. Because the economy is in fact not in the first phase in 1963

(with an equipment stock to GDP ratio which is not 0), this approximation is likely to

overstate the ratio β2/β1. Similarly, using (159), (160), and (165), we get that the labor

share in 1963 should obey

ls1963 ≈β2

HH− 1

γgN1963

+ β1

σσ−1

+ β2

1γgN1963

H− 1γgN1963

,

which simplifies into

ls1963 ≈σ − 1

σ(β2 + β1) , (52)

if most high-skill workers are in production. Therefore, given σ, the initial labor share

determines β3, the ‘external’ capital share. We can then combine (51) and (52) to obtain

β1 ≈1

wH1963H1963

wL1963L1963+ 1

ls1963

1− 1σ

, (53)

49

so that β1 which the Cobb-Douglas share for low-skill workers in phase 1 is given by the

labor share in 1963 and the ratio between the high-skill wage bill and the low-skill wage

bill, and σ which determines mark-ups.

Combining (168) and (168), we get that if the economy is close to its asymptotic

steady-state in 2007, the growth rate of the skill premium is given by

gsp2007 ≈β1 (σ − 1) (1− β4)

1 + β1 (σ − 1)gGDP2007 . (54)

Using (159), (160) (165), the labor share now obeys:

ls2007 ≈ H

[σH

(σ − 1) (β2 + β1β4)−(

σ

(σ − 1) (β2 + β1β4)− 1

)(1

γgN1963 +HA

1963

)]−1

,

which under the assumption that most high-skill workers are in production would sim-

plify again into

ls2007 ≈σ − 1

σ(β2 + β1β4) . (55)

Combining (51), (52) and (55) we obtain:

β4 ≈ 1−(

1− ls2007

ls1963

)(wH1963H1963

wL1963L1963

+ 1

). (56)

Therefore, in this approximation, β4 is identified through the decline in the labor share

and the initial wage bill ratio between high-skill and low-skill workers. In the data the

labor share does not monotonically decline. To understand how the parameters are

identified, we replace ls2007 by the lowest value over 1983-2007 (which is 62.2%) and

ls1963 by the highest value (68.8%). With wH1963H1963

wL1963L1963= 0.562, we then obtain β4 ≈ 0.85.

This is higher than the value we actually end up finding (β4 = 0.73), mostly because

the economy is still far from its steady-state in 2007 (so that ls2007 is higher than the

asymptotic value of the labor share).

Using (53), (54) and (56) we obtain:

σ ≈ 1

ls1963

[gGDP2007

gsp2007

(1− ls2007

ls1963

)− 1

wH1963H1963wL1963L1963

+1

] .that is given the initial wage bill ratio and the labor shares in 1963 and 1964, which

informs us about β1, β2 and β4, σ is determined by the ratio between the growth rate

50

of GDP and that of the skill-premium in the third phase. The larger is σ, the more

automated firms gain over non-automated ones and therefore the more the skill premium

rises relative to GDP: hence a lowergGDP2007

gsp2007is associated with a larger σ. When using the

last 5 years to determinegGDP2007

gsp2007, we find that σ ≈ 5.53, while our estimation procedure

leads to σ = 6.7.

Given σ, one can then find β1 using (53), we find β1 ≈ 0.54, below but not too far

from the estimated value of 0.62 (this approximation is not too sensitive on σ provided

that σ is large enough). Using (51), we then obtain β2 ≈ 0.3 which is higher than the

estimated value of 0.18, in line with the fact that (51) gives an overestimate of β2/β1.

To get a proxy for ∆, we look at the steady-state value for the equipment to GDP

ratio. Using (184), (185) and the definition of GDP, we obtain that

K∗

GDP∗ =

1

r∗β3 + β1 (1− β4)

σσ−1

+ (β2 + β1β4)(

HHP∗ − 1

) .Denote by Keq the stock of capital used as equipment, we get

K∗eq =β1 (1− β4)

β3 + β1 (1− β4)K∗,

since in steady-state the economy is Cobb-Douglas with a total physical capital share of

β3 + β1 (1− β4) and an equipment of share β1 (1− β4). Using (183), we obtain

K∗eq

GDP∗ =

1

ρ+ ∆ + θgGDP∗β1 (1− β4)

σσ−1

+ (β2 + β1β4)(

HHP∗ − 1

) .Therefore, assuming that in 2000 (the last year for which we have data on the equipment

to GDP ratio), we are close to the steady-state, and that most high-skill workers are in

production, we get

ρ+ ∆ + θgGDP2000 =

(σ − 1

σ

)β1 (1− β4)

Keq,2000GDP2000

≈ 0.0443 (57)

using the values computed above. It is therefore not surprising that we find a low ∆

in the estimation. This is due in particular to the high-level of Keq,2000GDP2000

= 1.5 (with the

actual estimated values for σ, β1 and β4 we would still find that ρ+∆+θgGDP2000 ≈ 0.094).

Yet, as explained in footnote 28, this level is somewhat arbitrary.

51

6.5.2 Parameters’ effect on empirical moments

In this section we illustrate the role that parameters have on the empirical moments

which allows us to identify what features of the data pin down the parameters. Taking

as our starting point the parameter estimates of Section 4, we iteratively change each

on the parameters by 2 per cent and illustrate the resulting effect on the initial (1963)

and the final (2007) value of each of the four empirical paths. Table 4 gives the resulting

elasticities (note that β3 is completely determined by β1 and β2).

The initial skill premium is most strongly affected by the production function pa-

rameters β1, β2 and β4: A higher share of high-skill workers in production, β2, directly

increases the skill-premium. A higher value of β4 makes automation more expensive,

which increases the demand for low-skill workers and reduces the skill premium. A

higher β1 implies a lower β3 which reduces the role of structural capital, reduces the

rental rate of capital, which increases the use of capital and thereby increases the skill-

premium. β2 has the opposite effect on the skill premium in 2007. A higher β2 reduces

the multiplier of Nt on output, Yt which reduces the growth rate of the economy. The

automation technology parameters, κ, κ, η also have a large effect on the skill premium

in 2007.

The initial labor share depends on β1, β2 and β4, the latter having a much larger

effect in 2007 since the share of automated products is much larger.

GDP/labor is mechanically affected negatively by higher σ since we keep the stock of

products in 1963 constant. Both β1 and β2 reduce the importance of structural capital

and thereby have a negative effect on GDP/labor in 1963 as the stock of capital is

sufficiently large. In 2007 σ, β1,β2, β4 all reduce the multiplier of Nt on Yt and therefore

GDP/labor. The innovation parameters γ, η lead to higher growth and therefore higher

GDP/labor in 2007, though naturally not in 1963.

Capital equipment / GDP in 1963 depends positively on β1 and negatively on β4

because the initial capital stock is fixed. For 2007, a higher β4 increases the cost of

automation and thereby reduces the stock of Keq/GDP . The horizontal innovation pro-

ductivity, γ, encourages more innovation. This drives up the wage of high-skill workers

in 1963, makes automation more expensive and reduces Keq/GDP . It further increases

the growth rate of the economy and reduces G2007 such that Keq/GDP in 2007 is lower

as well.

52

Parameters Skill premium Labor share GDP/labor Keq/GDP1963 2007 1963 2007 1963 2007 1963 2007

σεβ1

γκθηκρβ2

∆β4

N1963

G1963

−0.1 0.1−0.1 −0.2

0.4 0.00.1 −0.2−0.2 −0.4

0.0 −0.10.1 0.4−0.1 −0.4−0.1 −0.1

0.5 −0.20.0 −0.1−0.7 −1.9

0.0 0.10.1 0.0

0.1 0.00.0 0.00.7 0.60.0 0.10.0 0.10.0 0.00.0 0.00.0 0.00.0 0.00.2 0.30.0 0.00.1 0.60.0 0.00.0 0.0

−0.6 −1.90.0 −0.1−0.8 −2.3

0.0 0.60.0 −0.20.0 −0.20.0 0.20.0 −0.30.0 −0.4−0.5 −1.4

0.0 −0.1−0.1 −1.3

0.2 0.20.0 0.0

2.1 1.1−0.2 −0.1

8.3 1.8−0.2 −0.5

0.3 −0.40.1 −0.2−0.1 0.2

0.3 −0.20.2 −0.31.1 −0.10.0 −0.2−5.1 −3.7−0.2 0.0

0.5 0.0

Table 4: The effect of parameters on the four empirical paths (numbers refer to elasticitiesof empirical value wrt. parameter)

6.5.3 The precision of parameters

In the following we calculate the effect the parameters have on the aggregate final mo-

ment. We do this allowing for all the other parameters to adjust, illustrating how

precisely each of the parameters are determined. Since deviations from the minimum

parameter values are naturally second order we do not compute elasticities. Instead for

a given parameter θi consider

V (θi, θ−i(θi)),

where θ−i(θi) are the parameters that minimize V for any given θi and θi = argminθiV (θi, θ−i(θi))

is the minimizing value of θi. Consequently, a Taylor expansion around θi yields:

V (θi, θ−i(θi))− V (θi, θ−i(θi))

V (θi, θ−i(θi))/

(θi − θiθi

)2

≈ 1

2

1

V (θi, θ−i(θi))

d2V (θi, θ−i(θi))

dθ2i

θ2i .

We compute the expression on the left. The results are in Table 5 for a 5% shock on

the parameter of interest. It shows that the parameters that govern the production

function: (σ, β1, β2, β4) are the hardest to vary and consequently the ones most precisely

identified. The exception is ε, the elasticity between low-skill labor and machines, which

as Proposition 2 makes clear, does not govern the asymptotic growth of income inequality.

53

Parameters σ ε β1 γ κ θ η κ ρ β2 ∆ β4 N1963 G1963

Curvature 82.0 2.7 219.3 3.7 1.7 4.2 3.0 47.1 4.8 12.8 2.4 2142.1 8.3 2.0

Table 5: The “curvature” of deviating from the optimal parameter

Parameters σ ε β1 γ κ θ η κ ρ β2 ∆ β4 N1963 G1963

Curvature 6.7 4.95 0.62 0.64 0.57 1.03 0.40 0.57 0.038 0.17 0.015 0.75 20.4 0.02

Table 6: Parameters (only matching the first 30 years)

ρ, θ, η, γ all govern the growth rate of the economy and are weakly identified individually.

∆ is also not well identified because it mostly affects the growth rate of the capital stock

which also depends on ρ and θ (equation 57). Given that this parameter is the one

estimated outside a common range this is a reassuring finding.

6.5.4 The role of the automation externality

To analyze more specifically the role played by the automation externality, we recali-

brate our model but without the automation externality (i.e., we impose that κ = 0).

Figure 10 reports the results. The model still reproduces the paths for the labor share,

GDP/employment and equipment/GDP. Yet, it does not capture the evolution of the

skill premium. Indeed, the fast rise in the skill premium in the 1980s and 1990s require

an “accelerated Phase 2” which, given the moderate decline in the labor share and the

stable economic growth, can only be brought about by a positive automation externality.

The data clearly favor a positive automation externality (even though the exact value

of κ is not precisely estimated, see Table 5).

6.5.5 Out-of-sample prediction

Finally, we reproduce our calibration exercise but only trying to match the first 30 years

of data. Figure 11 reports the results and Table 6 gives the new parameters. The model

behaves quite well out-of-sample. The parameter estimates are very similar and the

predicted path are close. The largest difference is in the skill premium, where the model

calibrated over the first 30 years captures the upward trend but underestimates the pace.

In addition, the parameters are very similar to those in the full sample calibration of

Table 2.

54

1960 1970 1980 1990 2000 2010 2020Years

1

1.2

1.4

1.6

1.8

2

2.2

Ski

ll-P

rem

ium

(ra

tio)

Panel A - Composition-adjustedcollege/non-college weekly wage ratio

Predicted ValuesEmpirical Values

1960 1970 1980 1990 2000 2010 2020Years

50

55

60

65

70

75

80

Per

cen

t

Panel B - Labor share of GDP

1960 1970 1980 1990 2000 2010 2020Years

50

100

150

200

250300350400

Inde

x (1

963

= 1

00)

Panel C - GDP / Employment

1960 1970 1980 1990 2000 2010 2020Years

0.25

0.5

1

2

3

Equ

ipm

ent /

GD

P

Panel D - Equipment / GDP

Figure 10: Predicted and empirical time paths for κ = 0

1960 1970 1980 1990 2000 2010 2020Years

1

1.5

2

Ski

ll-P

rem

ium

(ra

tio)

Panel A - Composition-adjustedcollege/non-college weekly wage ratio

Predicted ValuesEmpirical Values

1960 1970 1980 1990 2000 2010 2020Years

50

60

70

80

Per

cen

t

Panel B - Labor share of GDP

1960 1970 1980 1990 2000 2010 2020Years

100

200

300

400

Inde

x (1

963

= 1

00)

Panel C - GDP / Employment

1960 1970 1980 1990 2000 2010 2020Years

0.25

0.5

1

2

3

Equ

ipm

ent /

GD

P

Panel D - Equipment / GDP

Figure 11: Predicted and empirical time paths only matching the first 30 years

55

7 Secondary Appendix (For Online Publication)

7.1 Relationship between wages and N and G

7.1.1 Imperfect substitute case: ε <∞.

We first focus on the imperfect substitute case. Rewrite (10) as

wHwL

=1− ββ

L

HP

G(1 + ϕwε−1

L

)µ+ 1−G

G(1 + ϕwε−1

L

)µ−1+ 1−G

. (58)

Since 0 < µ < 1, (58) establishes wH as a function of G,HP and wL (but notN) such that

wH is increasing in wL and G and decreasing in HP , with wH/wL > (1− β) /β ×L/HP

for G > 0. (11) similarly establishes wH as a function of N,G and wL (but not HP ), wH

is decreasing in wL and increasing in N and G. It is then immediate that wH , wL are

jointly uniquely determined by (58) and (11) for given N,G and HP , both increase in

N , and wH increases in G (in addition, (11) traces an iso-cost curve in the input prices

plan which is convex).

In addition, (58) shows that wH/wL increases with wL. Since wL increases in N ,

then wH/wL increases in N as well (and following (13) the labor share decreases in N).

Assume that wL decreases in G, then since wH increases in G, we immediately get that

wH/wL increases in G. Assumes now that wL increases in G, then the right-hand side of

(58) increases with G both directly and because wL increases, this ensures that wH/wL

increases in G (and following (13) the labor share decreases in G). Therefore both an

increase in N and an increase in G are skill-biased.

Comparative statics of wL with respect to G. We now analyze how wL changes

with G (for given N and HP ). To do so, we combine both equations to get:

wL =σ−1σβ(HP

L

)(1−β)

N1

σ−1

(G(1 + ϕwε−1

L

)µ−1+ (1−G)

)1−β

×(G(1 + ϕwε−1

L

)µ+ (1−G)

) 1σ−1−(1−β)

, (59)

56

Log differentiating with respect to G one obtains:

wL =

1

σ − 1

(1 + ϕwε−1

L

)µ − 1

G(1 + ϕwε−1

L

)µ+ (1−G)︸ ︷︷ ︸

scale effect

− (1− β)

(1−

(1 + ϕwε−1

L

)µ−1

G(1 + ϕwε−1

L

)µ−1+ (1−G)

+

(1 + ϕwε−1

L

)µ − 1

G(1 + ϕwε−1

L

)µ+ (1−G)

)︸ ︷︷ ︸

substitution effect

GG

Den

(60)

where

Den ≡ 1− βϕwε−1L(

1 + ϕwε−1L

) G(1 + ϕwε−1

L

)µG(1 + ϕwε−1

L

)µ+ (1−G)

+ϕwε−1

L (ε− 1) (1− β)(1 + ϕwε−1

L

) (µG(1 + ϕwε−1

L

)µG(1 + ϕwε−1

L

)µ+ 1−G

+(1− µ)G

(1 + ϕwε−1

L

)µ−1

G(1 + ϕwε−1

L

)µ−1+ 1−G

).

Den > 0 as ε > 1, µ ∈ (0, 1) andβϕwε−1

L

1+ϕwε−1L

G(1+ϕwε−1L )

µ

G(1+ϕwε−1L )

µ+(1−G)

< 1. In (60) the scale effect

term is positive as(1 + ϕwε−1

L

)µ − 1 > 0. This term comes from the differentiation of

(11) with respect to G at constant wH (hence it represents the shift right of the isocost

curve). The substitution effect term is negative because 1 −(1 + ϕwε−1

L

)µ−1> 0 since

µ < 1, it comes from the differentiation of (10) with respect to G.

First note that if 1σ−1≤ 1−β, the scale effect is always dominated by the substitution

effect. Hence wL is decreasing in G.

If on, the other hand 1σ−1

> 1−β, then the scale effect is dominated by the substitu-

tion effect provided that1−(1+ϕwε−1

L )µ−1

G(1+ϕwε−1L )

µ−1+1−G

/(1+ϕwε−1

L )µ−1

G(1+ϕwε−1L )

µ+1−G

is large enough. From (59)

we get:

wL =σ − 1

σβ

(HP

L

)(1−β)

N1

σ−1

1−G(

1−(1 + ϕwε−1

L

)µ−1)

G((

1 + ϕwε−1L

)µ − 1)

+ 1

1−β (G((

1 + ϕwε−1L

)µ − 1)

+ 1) 1σ−1

>σ − 1

σβ

(HP

L

)(1−β)

N1

σ−1

((1 + ϕwε−1

L

)−1)1−β

,

57

where the last line uses that G ∈ [0, 1]. We then obtain that

wL(1 + ϕwε−1

L

)1−β>σ − 1

σβ

(HP

L

)(1−β)

N1

σ−1 ,

which ensures that limN→∞

wL =∞ uniformly with respect to G (i.e. for any wL > 0, there

exist N such that for any N > N and any G, w > wL). Since

limwL→∞,G→1

1−(1+ϕwε−1L )

µ−1

G(1+ϕwε−1L )

µ−1+1−G

(1+ϕwε−1L )

µ−1

G(1+ϕwε−1L )

µ+1−G

=∞,

we get that

limN→∞,G→1

1−(1+ϕwε−1L )

µ−1

G(1+ϕwε−1L )

µ−1+(1−G)

(1+ϕwε−1L )

µ−1

G(1+ϕwε−1L )

µ+(1−G)

=∞.

Therefore for N and G large enough the substitution effect dominates. This achieves

the proof of Proposition 1.

Increase in the number of non-automated products. To study the effect of an

increase in the number of non-automated products only, we log differentiate (59) with

respect to both N and G and obtain:

wL =

( 1σ−1−(1−β))((1+ϕwε−1

L )µ−1)

G(1+ϕwε−1L )

µ+(1−G)

−(1−β)

(1−(1+ϕwε−1

L )µ−1

)G(1+ϕwε−1

L )µ−1

+(1−G)

GG+1

σ − 1N

1

Den

In that case NG is a constant, so that G = −N , therefore denoting wNTL (NT for “new

tasks”), the change in wL, we get:

wNTL =

(1−β)G((1+ϕwε−1L )

µ−1)

G(1+ϕwε−1L )

µ+(1−G)

+(1−β)G

(1−(1+ϕwε−1

L )µ−1

)G(1+ϕwε−1

L )µ−1

+(1−G)

+ (σ−1)−1

G(1+ϕwε−1L )

µ+(1−G)

N

Den. (61)

Hence low-skill wages always increase with the arrival of non-automated products. Log-

58

differentiating (10), one gets:

wH − wL =

(µG(1 + ϕwε−1

L

)µG(1 + ϕwε−1

L

)µ+ 1−G

+(1− µ)G

(1 + ϕwε−1

L

)µ−1

G(1 + ϕwε−1

L

)µ−1+ 1−G

)(ε− 1)ϕwε−1

L

1 + ϕwε−1L

wL

+

G((

1 + ϕwε−1L

)µ − 1)

G(1 + ϕwε−1

L

)µ+ 1−G

+G(

1−(1 + ϕwε−1

L

)µ−1)

G(1 + ϕwε−1

L

)µ−1+ 1−G

G.

Using (61) and that G = −N , then we get that following a change in the mass of

non-automated products (keeping the mass of automated products constant):

wNTH =

((+

(1− µ)G(1 + ϕwε−1

L

)µ−1

G(1 + ϕwε−1

L

)µ−1+ (1−G)

)(ε− 1)ϕwε−1

L

1 + ϕwε−1L

+ 1

)

×

(1−β)G((1+ϕwε−1L )

µ−1)

G(1+ϕwε−1L )

µ+(1−G)

+(1−β)G

(1−(1+ϕwε−1

L )µ−1

)G(1+ϕwε−1

L )µ−1

+(1−G)

+ (σ−1)−1

G(1+ϕwε−1L )

µ+(1−G)

N

Den

−

G((

1 + ϕwε−1L

)µ − 1)

G(1 + ϕwε−1

L

)µ+ (1−G)

+G(

1−(1 + ϕwε−1

L

)µ−1)

G(1 + ϕwε−1

L

)µ−1+ (1−G)

N .

We then obtain:

wNTH =

((

µG(1+ϕwε−1L )

µ

G(1+ϕwε−1L )

µ+1−G

+(1−µ)G(1+ϕwε−1

L )µ−1

G(1+ϕwε−1L )

µ−1+1−G

)(ε−1)ϕwε−1

L

1+ϕwε−1L

+ 1

)(σ−1)−1

G(1+ϕwε−1L )

µ+1−G

+β

(G((1+ϕwε−1

L )µ−1)

G(1+ϕwε−1L )

µ+1−G

+G(

1−(1+ϕwε−1L )

µ−1)

G(1+ϕwε−1L )

µ−1+1−G

)[G(1+ϕwε−1

L )µ

G(1+ϕwε−1L )

µ+1−G

ϕwε−1L

(1+ϕwε−1L )− 1

] N

Den

=(σ − 1)−1

G(1 + ϕwε−1

L

)µ+ 1−G

(1 +

G (1− µ)(1 + ϕwε−1

L

)µ−1

G(1 + ϕwε−1

L

)µ−1+ 1−G

(ε− 1)ϕwε−1L

1 + ϕwε−1L

)N

Den

Therefore an increase in the mass of non-automated products leads to higher high-skill

wages. Finally we obtain

wNTH − wNTL =

(σ−1)−1

G(1+ϕwε−1L )

µ+1−G

((1−µ)(1+ϕwε−1

L )µ−1

G(1+ϕwε−1L )

µ−1+1−G

(ε−1)ϕwε−1L

1+ϕwε−1L

)− (1− β)

[(1+ϕwε−1

L )µ−1

G(1+ϕwε−1L )

µ+1−G

+1−(1+ϕwε−1

L )µ−1

G(1+ϕwε−1L )

µ−1+1−G

] GN

Den

59

wNTH − wNTL =

[(ε−1σ−1− β

)1

1+ϕwε−1L

− (1− β)] (

1 + ϕwε−1L

)µ−1ϕwε−1

L(G(1 + ϕwε−1

L

)µ+ 1−G

) (G(1 + ϕwε−1

L

)µ−1+ 1−G

) GNDen

.

Therefore an increase in the mass of non-automated products reduces the skill premium

(and increases the labor share) if and only if 1− β >(ε−1σ−1− β

)1

1+ϕwε−1L

. This in turn is

true for wL sufficiently large (that is N large enough) or for ε < σ.

7.1.2 Perfect substitute case:ε =∞

In the perfect substitute case, there are three possibilities. Case i) wL < ϕ−1: automated

firms only use low-skill workers and low-skill wages are given by

wL =σ − 1

σβ

(HP

L

)1−β

N1

σ−1 , (62)

with a skill premium obeying wHwL

= 1−ββ

LHP .

Case ii) wL = ϕ−1: automated firms use machines but also possibly workers, in which

case high-skill wages can be obtained from (11) which is now written as:

σ

σ − 1

N1

1−σ

ββ (1− β)1−β ϕ−βw1−β

H = 1.

Case iii) wL > ϕ−1 and all automated firms use machines only, in that case, we get

that (59) is replaced by

wL =σ − 1

σβ

(HP

L

)(1−β)

N1

σ−1 (1−G)1−β(G (ϕwL)β(σ−1) + 1−G

) 1σ−1−(1−β)

, (63)

and the skill premium obeys:

wHwL

=1− ββ

L

HP

G (wLϕ)β(σ−1) + 1−G1−G

. (64)

60

One can rewrite (63) as

w1−βL =

σ − 1

σβ

(HP

L

)(1−β) N1

σ−1 (1−G)1−β(G+ (1−G) (ϕwL)−β(σ−1)

) 1σ−1

ϕβ(G (ϕwL)β(σ−1) + 1−G

)1−β .

The left-hand side increases in wL and the right-hand side decreases in wL, hence this

expression defines wL uniquely, in addition, the solution is greater than ϕ−1 if and only

if

N1

σ−1 (1−G)1−β >σ

(σ − 1) βϕ

(L

HP

)(1−β)

.

Hence wL and wH are defined uniquely as functions of N,G and HP . If N1

σ−1 <σ

(σ−1)βϕ

(LHP

)(1−β), we are in case i), if N

1σ−1 (1−G)1−β ≤ σ

(σ−1)βϕ

(LHP

)(1−β) ≤ N1

σ−1

then we are in case ii) and if N1

σ−1 (1−G)1−β > σ(σ−1)βϕ

(LHP

)(1−β), we are in case iii).

It is then direct to show that wH increases in N and weakly increases in G, that

wH/wL is weakly increasing in N and G (weakly because of case i)), and that wL is

weakly increasing in N (weakly because of case ii)).

Comparative statics of wL with respect to G. Furthermore, (63) shows that

wL is decreasing in G in case iii) if 1σ−1≤ 1 − β. Therefore wL is weakly decreasing in

G if 1σ−1≤ 1− β.

Assume now that 1σ−1

> 1− β. Log-differentiating (63), one gets:

wL =

( 1

σ − 1− (1− β)

) ((ϕwL)β(σ−1) − 1

)G

G (ϕwL)β(σ−1) + (1−G)− (1− β)

G

1−G

G

Den(65)

where

Den ≡ 1− β G (ϕwL)β(σ−1)

G (ϕwL)β(σ−1) + 1−G+

(1− β) β (σ − 1)G (ϕwL)β(σ−1)

G (ϕwL)β(σ−1) + 1−G.

We have

(1

σ − 1− (1− β)

) ((ϕwL)β(σ−1) − 1

)G

G (ϕwL)β(σ−1) + 1−G− (1− β)

G

1−G

<1

σ − 1− 1− β

1−G,

61

which is negative for G large enough. Hence we obtain that for G high enough, wL is

weakly decreasing in G (strictly in case iii)).

Increase in the number of non-automated products. In case i) an increase in

the mass of non-automated products leads to an increase in wH and wL while wH/wL is

constant. In case ii), wH increases, wH/wL increases and wL is constant.

Log-differentiating (63) with respect to both N and G, one gets:

wL =

( 1

σ − 1− (1− β)

) ((ϕwL)β(σ−1) − 1

)G

G (ϕwL)β(σ−1) + 1−G− (1− β)

G

1−G

G+1

σ − 1N

1

Den.

For an increase in the mass of non-automated products, G = −N , so that the change in

wL in that case is given by:

wNTL =

(1− β)

(

(ϕwL)β(σ−1) − 1)G

G (ϕwL)β(σ−1) + 1−G+

G

1−G

+1

σ − 1

1−

((ϕwL)β(σ−1) − 1

)G

G (ϕwL)β(σ−1) + 1−G

N

Den.

Therefore wL increases.

Log-differentiating (64), we get:

wH =G(

(ϕwL)β(σ−1) − 1)

G (ϕwL)β(σ−1) + 1−GG+

G

1−GG+

(1 +

β (σ − 1)G (ϕwL)β(σ−1)

G (ϕwL)β(σ−1) + 1−G

)wL.

Therefore, for an increase in the mass of non-automated products, one gets:

wNTH =1

σ − 1

1−

((ϕwL)β(σ−1) − 1

)G

G (ϕwL)β(σ−1) + 1−G

N

Den,

which ensures that wH increases with the mass of non-automated products. Finally,

wNTH − wNTL = − (1− β)G (ϕwL)β(σ−1)(G (ϕwL)β(σ−1) + 1−G

)(1−G)

N

Den.

Therefore an increase in the mass of non-automated products decreases the skill premium

in case iii).

Overall we get that an increase in the mass of non-automated products weakly in-

62

creases wL, increases wH and decreases wH/wL if N is large enough but G 6= 1 (so that

we are in case iii)).

7.2 Proofs of the asymptotic results

7.2.1 Proof of Proposition 2

Case with G∞ > 0 (Parts A and B). To see that wLt is bounded from below, assume

that liminfwLt=0. Then using that HPt and Gt admit positive limits, (10) implies that

liminfwHt = 0. Plugging this further in (11) gives liminfNt = 0, which is impossible

since gNt admits a positive limit. Therefore, wLt must be bounded below, so that (11)

gives gwH∞ = ψgN∞. Further, using that HPt admits a limit and (8) gives the growth rate

of Yt. We now derive the asymptotic growth rate of wLt. To do so we consider in turn

the case where ε <∞, and the case where ε =∞.

Subcase with ε < ∞. We use equation (59) which gives wLt as a function of

Nt, Gt and HPt . Note that assuming that wLt is bounded above leads to a contradiction,

therefore limwLt =∞.

Assume first that G∞ < 1, then, since limwLt =∞, (59) implies

wLt ∼

((σ − 1

σβ

) 11−β

(1−G∞)HP∞L

(G∞ϕµ)ψ−1

) 11+β(σ−1)

Nψ

1+β(σ−1)

t ,

where for xt and yt (possibly with no limits), xt ∼ yt signifies xt/yt → 1. This delivers

Part A).

Consider now the case where G∞ = 1. Note that (59) gives:

wLt ∼

((σ − 1

σβ

) 11−β HP

∞Lϕµ(ψ−1)

) 1ε

Nψεt

(ϕµ−1 + (1−Gt)w

(ε−1)(1−µ)Lt

) 1ε. (66)

Following the assumption of Part B in Proposition 2, we assume that lim (1−Gt)Nψε

(ε−1)(1−µ)t

exists and is finite. Suppose first that lim sup (1−Gt)w(ε−1)(1−µ)Lt =∞, then there must

exist a sequence of t’s, denoted tn for which:

wLtn ∼

((σ − 1

σβ

) 11−β HP

∞L

(ϕµ)ψ−1

) 11+β(σ−1) (

(1−Gtn)Nψtn

) 11+β(σ−1)

.

63

Yet, this implies

(1−Gtn)w(ε−1)(1−µ)Ltn

∼

((σ−1σβ) 1

1−β HP∞L

(ϕµ)ψ−1) (ε−1)(1−µ)

1+β(σ−1)

×(

(1−Gtn)Nψ(ε−1)(1−µ)

εtn

) ε1+β(σ−1)

,

the left-hand side is assumed to be unbounded, while the right-hand side is bounded:

there is a contradiction. Therefore, lim sup (1−Gt)w(ε−1)(1−µ)Lt <∞.

Consider now the possibility that lim (1−Gt)w(ε−1)(1−µ)Lt = 0, then (66) implies

wLt ∼

((σ − 1

σβ

) 11−β HP

∞Lϕµψ−1

) 1ε

Nψεt .

Therefore we get that gwL∞ = ψεgN∞ = 1

εgY∞.

31

Alternatively, lim sup (1−Gt)w(ε−1)(1−µ)Lt is finite but strictly positive (given by λ1).

In this case, there exists a sequence of t′s, denoted tm such that

wLtm ∼

((σ − 1

σβ

) 11−β HP

∞Lϕµ(ψ−1)

(ϕµ−1 + λ1

)) 1ε

Nψεtm . (67)

This leads to

λ1 ∼

((σ − 1

σβ

) 11−β HP

∞Lϕµ(ψ−1)

(ϕµ−1 + λ1

)) (ε−1)(1−µ)ε

(1−Gtm)Nψε

(ε−1)(1−µ)tm ,

which is only possible if lim (1−Gt)Nψε

(ε−1)(1−µ)t > 0. We denote such a limit by λ.

Then (66) leads to

(wεLtN

−ψt

)∼(σ − 1

σβ

) 11−β HP

∞Lϕµ(ψ−1)

(ϕµ−1 + λ

(N−ψt wεLt

) (ε−1)(1−µ)ε

),

which defines uniquely the limit of wεLtN−ψt . We then obtain that gwL∞ = ψ

εgN∞. This

completes the poof of part B).

Subcase with ε = ∞. Low skill wages are now defined as described in Appendix

31Expressions regarding the asymptotic growth rates (here and below) assume existence of the limitsbut expressions on equivalence (∼) or orders of magnitude (O) do not.

64

7.1.2. First consider the case where G∞ < 1, then Part A) immediately follows. Assume

now that G∞ = 1 and that lim (1−Gt)Nψt exists and is finite. Note first that (62)

implies that wLt must be bounded weakly above ϕ in the long-run. As a result, (63)

leads to

wLt ∼

((σ − 1

σβϕβ(1−ψ−1)

) 11−β HP

∞L

) 11+β(σ−1) (

(1−Gt)Nψt

) 11+β(σ−1)

if wLt > ϕ.

Since lim (1−Gt)Nψt exists and is finite, wLt also admits a finite limit. In particular, if

lim (1−Gt)Nψt = 0, then wL∞ = ϕ.

Case where G∞ = 0 (Part C). If limGt = 0 then (59) implies that for ε <∞:

wLt ∼σ − 1

σβ

(HP∞L

)(1−β)

N1

σ−1

t

(Gt

(1 + ϕwε−1

Lt

)µ+ 1) 1σ−1−(1−β)

.

This expression directly implies that limwLt = ∞ (otherwise there is a subsequence

where the left-hand side is bounded while the right-hand side is unbounded). Therefore

we actually get:

wLt ∼σ − 1

σβ

(HP∞L

)(1−β)

N1

σ−1

t

(Gtw

β(σ−1)Lt ϕµ + 1

) 1σ−1−(1−β)

. (68)

Note that if ε = ∞, then we must be in case iii) when G∞ = 0 and (63) also directly

implies (68) (as ϕµ = ϕβ(σ−1) in that case).

Assume that limt→∞

GtNβt = λ exists and is finite. Then (68) implies:

wLtN− 1σ−1

t ∼ σ − 1

σβ

(HP∞L

)(1−β)(λ

(wLtN

− 1σ−1

t

)β(σ−1)

ϕµ + 1

) 1σ−1−(1−β)

,

which implies that limt→∞

wLtN− 1σ−1

t exists and is finite as well. Therefore one gets that

gwL∞ = gN∞/(σ − 1). Using (58) then immediately implies (17).

7.2.2 Sufficient conditions for Part A of Proposition 2.

We prove the following Lemma:

Lemma 1. Consider processes [Nt]∞t=0, [Gt]

∞t=0 and [HP

t ]∞t=0 , such that gNt and HPt admit

65

strictly positive limits. If i) the probability that a new product starts out non-automated

is bounded below away from zero and ii) the intensity at which non-automated firms are

automated is bounded above and below away from zero, then any limit of Gt must have

0 < G∞ < 1.

Note that GtNt is the mass of automated firms and let ν1,t > 0 be the intensity at

which non-automated firms are automated at time t and 0 ≤ ν2,t < 1 be the fraction

of new products introduced at time t that are initially automated. Then ˙(GtNt) =

ν1,t(1 − Gt)Nt + ν2,tNt such that Gt = ν1,t(1 − Gt) − (Gt − ν2,t)gNt . First assume that

G∞ = 1, then if ν1,t < ν1 < ∞ and ν2,t < ν2 < 1, we get that Gt must be negative for

sufficiently large t, which contradicts the assumption that G∞ = 1. Similarly if G∞ = 0,

then having ν1,t > ν for all t, gives that Gt must be positive for sufficiently large t, which

also implies a contradiction. Hence a limit must have 0 < G∞ < 1.

7.3 Proof of Proposition 5

We denote by g (i, t) the share of products indexed by i which are automated by time t.

The Poisson process for automation implies that

g (i, t) = 1− exp

(−η(t− i

n

))for t ≥ i

n, (69)

since a product i must be born at time t = i/n (and it is born non-automated).

The unit cost function of intermediate i can be written as:

ci (wL, wH , α (i)) =

(min

(wL

b (i)1−ς ,1

α (i)

))βw1−βH

b (i)1−β(1−ς) ββ (1− β)1−β .

Therefore an automated firm (α (i) = 1) will use machines instead of low-skill labor if

wL > b (i)1−α. This implies that two cases must be considered, if wLt > exp (B (1− ς)nt),then all automated firms will use machines. Otherwise, there exists a It ∈ [0, Nt) such

that wLt = exp (B (1− ς) It) and automated firms with an index i < It use machines

while those with an index i > It use low-skill workers instead. We fix It = Nt if

wLt ≥ exp(B (1− ς)nt.The resolution of the system then follows that in the baseline case, firms charge a

mark-up σ/ (σ − 1) and revenues of firm i are given by

Ri (wL, wH , α (i)) = ((σ − 1) /σ)σ−1 ci (wL, wH , α (i))1−σ Y.

66

A share (1− β) (σ − 1) /σ of firms’ revenues accrue to high-skill workers, while low-skill

workers obtain a share β (σ − 1) /σ if the firm is non-automated or has an index i > It

and 0 otherwise. Therefore

wHtH = (1− β)σ − 1

σ

∫ Nt

0

(1− g (i, t))Ri (wLt, wHt, 0) + g (i, t)Ri (wLt, wHt, 1) di,

wLtL = βσ − 1

σ

∫ Nt

0

(1− g (i, t))Ri (wL, wH , 0) + 1i>Itg (i, t)Ri (wLt, wHt, 1) di,

where 1i>It denotes the index function for i > It. Taking the ratio between these two

expressions, we obtain:

wHtH

wLtL=

1− ββ

[1 + w

β(σ−1)Lt

∫ Nt0

1i<Itg (i, t) b (i)(1−β(1−ς))(σ−1) di∫ Nt0

(1− 1i<Itg (i, t)) b (i)(σ−1) di

], (70)

which traces the relative demand curve (and replaces (10)).

Similarly, using the price normalization, we obtain:

σ

σ − 1

w1−βHt

ββ (1− β)1−β

(∫ Nt

0

((1− 1i<Ig (i, t))w

β(1−σ)Lt b (i)(σ−1)

+1i<Ig (i, t) b (i)(1−β(1−ς))(σ−1)

)di

) 11−σ

= 1, (71)

which replaces (11). We must then consider two cases in turn: It = Nt and It < Nt:

Case 1: It = Nt. Using (69) and the expression for b (i) we can compute the integral

in (70) and obtain:

wHtH

wLtL=

1− ββ

1 + wβ(σ−1)Lt

exp(B(1−β(1−ς))(σ−1)nt)−1B(1−β(1−ς))(σ−1)

− exp(B(1−β(1−ς))(σ−1)nt)−exp(−ηt)(B(1−β(1−ς))(σ−1)+ η

n)exp(B(σ−1)nt)−exp(−ηt)

(B(σ−1)+ ηn)

.For t large enough (and since (1− β (1− ς)) (σ − 1) > 0), we get:

wHtH

wLtL∼t→∞

1− ββ

[1 +

(B (σ − 1) + η

n

)ηn

(wL exp (−B (1− ς)nt))β(σ−1)

(B (1− β (1− ς)) (σ − 1))((B (1− β (1− ς)) (σ − 1) + η

n

))] .(72)

67

Similarly (71) gives

σ

σ − 1

w1−βHt

ββ (1− β)1−β

=

wβ(1−σ)Lt

exp(B(σ−1)nt)−exp(−ηt)B(σ−1)+ η

n+ exp(B(1−β(1−ς))(σ−1)nt)−1

B(1−β(1−ς))(σ−1)

− exp(B(1−β(1−ς))(σ−1)nt)−exp(−ηt)(B(1−β(1−ς))(σ−1)+ η

n)

1σ−1

.

From this, we obtain:

wHt ∼t→∞

exp

(B (1− β (1− ς))nt

1− β

)(1− β)

(σ − 1

σββ) 1

1−β

(73)(

exp(B(1−ς)nt)wLt

)β(σ−1)

B (σ − 1) + ηn

+ηn

B (1− β (1− ς)) (σ − 1)(B (1− β (1− ς)) (σ − 1) + η

n

)ψ

.

with ψ ≡ 1/ [(σ − 1) (1− β)] as before.

Since It = Nt, then we must have wLt ≥ exp (B (1− ς)nt). Therefore (73) implies

that

wHt = O

(exp

(B (1− β (1− ς))n

1− βt

)).

Further (72) implies that

wHtwLt

= O(

(wL exp (−B (1− ς)nt))β(σ−1)),

from which we get that

wLt = O

(exp

((1− β (1− ς) + (1− β) β (σ − 1) (1− ς)

(1 + β (σ − 1)) (1− β)

)Bnt

)).

We need to verify that wLt ≥ exp (B (1− ς)nt). Note that(1− β (1− ς) + (1− β) β (σ − 1) (1− ς)

(1 + β (σ − 1)) (1− β)

)Bn ≥ B (1− ς)n⇐⇒

ς ≥ 0.

Therefore, if ς > 0, then wLt > exp (B (1− ς)nt) is verified for sure for large t, and we

get that gwH∞ and gwL∞ exist with gwH∞ = (1−β(1−ς))Bn1−β and gwL∞ = 1−β(1−ς)+(1−β)β(σ−1)(1−ς)

(1+β(σ−1))(1−β)Bn

68

and we can verify that gwH∞ > gwL∞ .

In contrast if ς = 0, then we have gwH∞ = gwL∞ = Bn but this case only applies if there

exist constant wH and wL such that

wH = (1− β)

(σ − 1

σββ) 1

1−β(

wβ(1−σ)L

B (σ − 1) + ηn

+ηn

B (1− β) (σ − 1)(B (1− β) (σ − 1) + η

n

))ψ

,

(74)

wHH

wLL=

1− ββ

[1 + w

β(σ−1)L

(B (σ − 1) + η

n

)ηn

B (1− β) (σ − 1)((B (1− β) (σ − 1) + η

n

))] , (75)

with wL ≥ 1, in which case wHt ∼t→∞

wH exp (Bnt) and wLt ∼t→∞

wL exp (Bnt) and

It = Nt is a possibility.

Case 2: It < Nt. Then (70) implies

wHtH

wLtL=

1− ββ

1 + wβ(σ−1)Lt

exp(B(1−β(1−ς))(σ−1)It)−1B(1−β(1−ς))(σ−1)

− exp (−ηt) exp((B(1−β(1−ς))(σ−1)+ ηn)It)−1

(B(1−β(1−ς))(σ−1)+ ηn)

exp(B(σ−1)nt)−exp(B(σ−1)It)B(σ−1)

+ exp (−ηt) exp((B(σ−1)+ ηn)It)−1

(B(σ−1)+ ηn)

.Note that we must have lim It =∞, otherwise, there would be periods where wLt remain

bounded even though an arbitrarily large number of products use low-skill workers.

Therefore, using wLt = exp ((1− ς)BIt), the previous equation leads to:

wHtH

wLtL∼t→∞

1− ββ

1 +

1B(1−β(1−ς))(σ−1)

− exp(− ηn

(nt− It))

1B(1−β(1−ς))(σ−1)+ η

n

exp(B(σ−1)(nt−It))−1B(σ−1)

+ exp(− ηn

(nt− It))

1B(σ−1)+ η

n

,Since nt ≥ It, then we must have that wHt = O (wLt) = O (exp ((1− ς)BIt)). Similarly

(71) now implies:

σw1−βHt

(σ − 1) ββ (1− β)1−β =

wβ(1−σ)Lt

exp(B(σ−1)nt)−exp(B(σ−1)It)B(σ−1)

+ wβ(1−σ)Lt exp (−ηt) exp((B(σ−1)+ η

n)It)−1

B(σ−1)+ ηn

+ exp(B(1−β(1−ς))(σ−1)It)−1B(1−β(1−ς))(σ−1)

− exp (−ηt) exp((B(1−β(1−ς))(σ−1)+ ηn)It)−1

(B(1−β(1−ς))(σ−1)+ ηn)

1

σ−1

.

Using that It →∞ and that wLt = exp ((1− ς)BIt), we then get

σw1−βHt

(σ − 1) ββ (1− β)1−β ∼t→∞ exp ((1− (1− ς) β)BIt)

exp(B(σ−1)(nt−It))−1B(σ−1)

+exp(− ηn (nt−It))B(σ−1)+ η

n

+ 1B(1−β(1−ς))(σ−1)

− exp(− ηn (nt−It))B(1−β(1−ς))(σ−1)+ η

n

1

σ−1

.

69

The left-hand side is of order exp ((1− ς − β (1− ς))BIt) while the the right-hand side

is of order at least exp ((1− (1− ς) β)BIt), therefore this situation is only possible if

ς = 0. Moreover, if nt − It is unbounded then the ratio of the right-hand side to the

left-hand side is also unbounded, which is a contradiction. Therefore it must be that

nt− It remains bounded. In that case, we must then have that wLt and wHt are of the

same order as exp (BIt).

We then obtain an equilibrium with It < Nt if there exist wH , wL < 1 (which implies

limnt− It = − logwLB

) such that:

wHH

wLL=

1− ββ

1 +

1B(1−β)(σ−1)

− wηBnL

1B(1−β)(σ−1)+ η

n

w(1−σ)L −1

B(σ−1)+ w

ηBnL

1B(σ−1)+ η

n

, (76)

wH = (1− β)

(σ − 1

σββ) 1

1−β

wL

w(1−σ)L −1

B(σ−1)− w

ηBnL Bβ(σ−1)

(B(σ−1)+ ηn)(B(1−β)(σ−1)+ η

n)

+ 1B(1−β)(σ−1)

ψ

. (77)

(76) can be rewritten as wH = 1−ββ

LHs1 (wL) with

s1 (wL) ≡ wL

1 +

1B(1−β)(σ−1)

− wηBnL

1B(1−β)(σ−1)+ η

n

w(1−σ)L −1

B(σ−1)+ w

ηBnL

1B(σ−1)+ η

n

.We obtain (after a lot of algebra):

s′1 (wL) =

(ηn( η

Bn+1)

(B(σ−1)+ ηn)(B(1−β)(σ−1)+ η

n)+ 1

B(1−β)(σ−1)+

β(σ−1)1−β +1+ β

B(1−β)ηn

B(σ−1)+ ηn

)w

ηBnL

w1−σL −1

B(σ−1)

+

(w1−σL −1

B(σ−1)+

1−wηBnL

B(1−β)(σ−1)

)w1−σL −1

B(σ−1)+

1−wηBnL

B(1−β)(σ−1)

w1−σL

B+ 1+β(σ−1)

B(1−β)(σ−1)

wηBnL

B(σ−1)+ ηn

+ηnw

ηBnL

(B(σ−1)+ ηn)(B(1−β)(σ−1)+ η

n)

(w1−σL

B+

wηBnL

B(σ−1)+ ηn

)

(w1−σL −1

B(σ−1)+

wηBnL

B(σ−1)+ ηn

)2 ,

which is positive when wL ≤ 1 (as then w1−σL − 1 ≥ 0 and 1−w

ηBnL ≥ 0), so that s1 is in-

creasing in wL for wL ≤ 1. Similarly (77) can be rewritten as wH = (1− β)(σ−1σββ) 1

1−β s2 (wL)ψ,

70

with

s2 (wL) ≡ w(σ−1)(1−β)L

(w

(1−σ)L − 1

B (σ − 1)− w

ηBnL Bβ (σ − 1)(

B (σ − 1) + ηn

) (B (1− β) (σ − 1) + η

n

) +1

B (1− β) (σ − 1)

).

We get

s′2 (wL) = −w(σ−1)(1−β)−1L

[(B (σ − 1) β

B (σ − 1) + ηn

)w

ηBnL

B+ β

w1−σL − 1

B

],

which is negative for wL ≤ 1. Therefore (75) and (76) together trace a positive relation-

ship between wH and wL (it is easy to show continuity between the two expressions) and

similarly (74) and (77) trace a positive relationship (continuity is also easily verified),

checking the limits when wL → 0,∞, we obtain that there exist a single solution. This

ensures that asymptotically when ς = 0, the asymptotic equilibrium described above

exists, and depending on other parameters it is in case 1 or case 2.

7.4 Alternative production technology for machines

The assumption of identical production technologies for consumption and machines im-

poses a constant real price of machines once they are introduced. As shown in Nordhaus

(2007) the price of computing power has dropped dramatically over the past 50 years

and the declining real price of computers/capital is central to the theories of Autor and

Dorn (2013) and Karabarbounis and Neiman (2013). As explained in section 2, it is

possible to interpret automation as a decline of the price of a specific equipment from

infinity (the machine does not exist) to 1. Yet, our assumption that once a machine is

invented, its price is constant, is crucial for deriving the general conditions under which

the real wages of low-skill workers must increase asymptotically in Proposition 2. We

generalize this in what follows.

Let there be two final good sectors, both perfectly competitive employing CES pro-

duction technology with identical elasticity of substitution, σ. The output of sector 1,

Y , is used for consumption. The output of sector 2, X, is used for machines. The two

final good sectors use distinct versions of the same set of intermediate inputs, where we

denote the use of intermediate inputs as y1(i) and y2(i), respectively, with i ∈ [0, N ].

The two versions of intermediate input i are produced by the same intermediate input

71

supplier using production technologies that differ only in the weight on high-skill labor:

yk(i) =[lk(i)

ε−1ε + α(i)(ϕxk(i))

ε−1ε

] εβkε−1

hk(i)1−βk ,

where a subscript, k = 1, 2, refers to the sector where the input is used. Importantly, we

assume β2 ≥ β1, such that the production of machines relies more heavily on machines

as inputs than the production of the consumption good. Continuing to normalize the

price of final good Y to 1, such that the real price of machines is pxt , and allowing

for the natural extensions of market clearing conditions, we derive below the following

generalization of Proposition 2 (where ψk = (σ − 1)−1(1− βk)−1).

Proposition 6. Consider three processes [Nt]∞t=0, [Gt]

∞t=0 and [HP

t ]∞t=0 where (Nt, Gt, HPt ) ∈

(0,∞)× [0, 1]× (0, H] for all t. Assume that Gt, gNt and HP

t all admit strictly positive

limits, then:

gpx

∞ = −ψ2 (β2 − β1) gN∞

gGDP∞ =

[ψ1 + ψ1

β1 (β2 − β1)

1− β2

]gN∞, (78)

and if G∞ < 1 then the asymptotic growth rate of wLt is32

gwL∞ =1

1 + β1(σ − 1)

1− β2 + β1 (β2 − β1)(1− ψ−1

1

)1− β2 + β1 (β2 − β1)

gGDP∞ . (79)

Proposition 6 naturally reduces to Proposition 2 for the special case of β2 = β1.

When β2 > β1, the productivity of machine production increases faster than that of the

production of Y , implying a gradual decline in the real price of machines. For given

gN∞, a faster growth in the supply of machines increases the (positive) growth in the

relative price of low-skill workers compared with machines, wL/px, but simultaneously,

it reduces the real price of machines, px. The combination of these two effects always

implies that low-skill workers capture a lower fraction of the growth in Y . Low-skill wages

are more likely to fall asymptotically for higher values of the elasticity of substitution

between products, σ, as this implies a more rapid substitution away from non-automated

products.

32If Gt tends towards 1 sufficiently fast such that limt→∞(1−Gt)Nψ2(1−µ1)

ε−1ε

t is finite, then gwL∞ =1ε

(1− (β2−β1)(ε−1)

(1−β2+β1)

)gGDP∞ ≥ gp

x

∞ whether ε is finite or not. It is clear that there always exists an ε

sufficiently high for the real wage of low-skill workers to decline asymptotically.

72

Proof. The analysis follows similar steps as in the baseline model. The cost function (4)

now becomes

ck (α (i)) = β−βκk (1− βk)−(1−βk) (w1−εL + ϕ (px)1−ε α (i)

) βk1−ε w1−βk

H , (80)

for k ∈ 1, 2 indexing, respectively, the production of final good and machines. As

before aggregating (80) and the price normalization gives a “productivity” condition,

which replaces (11).

(G(w1−εL + ϕ (px)1−ε)µ1 + (1−G)w

β1(1−σ)L

) 11−σ

w1−β1H =

σ − 1

σββ11 (1− β1)1−β1N

1σ−1 ,

(81)

where we generalize the definition of µ: µk ≡ βk(σ−1)ε−1

. Following the same methodology

for the production of machines, we get

(G(w1−εL + ϕ (px)1−ε)µ2 + (1−G)w

β2(1−σ)L

) 11−σ

w1−β2H =

σ − 1

σββ22 (1− β2)1−β2N

1σ−1px.

(82)

Taking the ratio between these two expressions, we get

(G

((wLpx

)1−ε+ ϕ

)µ2+ (1−G)

(wLpx

)β2(1−σ)) 1

1−σ

wβ1−β2H(G

((wLpx

)1−ε+ ϕ

)µ1+ (1−G)

(wLpx

)β1(1−σ)) 1

1−σ=ββ22 (1− β2)1−β2 (px)1−β2+β1

ββ11 (1− β1)1−β1.

(83)

The share of revenues accruing to machines in the production of intermediate input i for

the usage-k (i.e for use in the final sector or the machines sector) is given by

νk,x (α (i)) =σ − 1

σα (i) βk

ϕ (px)1−ε

w1−εL + ϕ (px)1−ε , (84)

aggregating over all intermediates inputs and denoting Rk (α (i)) the revenues generated

through usage k by a firm of type α (i), we get that the total expenses in machines are

given by

pxX = NG (R1(1)ν1,x(1) +R2(1)ν2,x(1)) . (85)

The zero profit condition in the machines sector gives

pxX = N (GR2 (1) + (1−G)R2 (0)) . (86)

73

Revenues themselves are given by

R1 (α (i)) =

(σ − 1

σ

)σ−1

c1 (α (i))1−σ Y and R2 (α (i)) =

(σ − 1

σ

)σ−1

c2 (α (i))1−σ pxX,

(87)

so that (7) still holds but separately for revenues occurring from each activity and with

µk replacing µ. Combining (7), (84), (85) and (86), we get(G(

1− σ−1σβ2

ϕ(px)1−ε

w1−εL +ϕ(px)1−ε

)+ (1−G)

(1 + ϕ

(wLpx

)ε−1)−µ2)

R2(1)R1(1)

= Gσ−1σβ1

ϕ(px)1−ε

w1−εL +ϕ(px)1−ε

, (88)

which determines the revenues ratio as a function of input prices solely.

To derive low-skill wages, we compute the share of revenues accruing to low-skill

labor in the production of intermediate input i for the usage-k as:

νk,l (α (i)) =σ − 1

σβk

(1 + α (i)ϕ

(wLpx

)ε−1)−1

,

so that total low-skill income can be written as:

wLL = N (GR1(1)ν1,l(1) + (1−G)R1(0)ν1,l(0) +GR2(1)ν2,l(1) + (1−G)R2(0)ν2,l(0)) .

(89)

The share of revenues going to high-skill workers is given by νk,h = σ−1σ

(1− βk) both in

automated and non-automated firms. As a result

wHHP = N (ν1,h (GR1(1) + (1−G)R1(0)) + ν2,h (GR2(1) + (1−G)R2(0))) , (90)

74

Take the ratio between (89) and (90), and use (7) to obtain:

wLL

wHHP=

β1

(G

(1 + ϕ

(wLpx

)ε−1)−1

+ (1−G)

(1 + ϕ

(wLpx

)ε−1)−µ1)

+β2R2(1)R1(1)

(G

(1 + ϕ

(wLpx

)ε−1)−1

+ (1−G)

(1 + ϕ

(wLpx

)ε−1)−µ2)

(1− β1)

(G+ (1−G)

(1 + ϕ

(wLpx

)ε−1)−µ1)

+ (1− β2) R2(1)R1(1)

(G+ (1−G)

(1 + ϕ

(wLpx

)ε−1)−µ2)

(91)

Together (81), (83), (88) and (91) determine wL, wH , px and R (2) /R (1) given N,G and

HP .

Asymptotic behavior for ε < 1. As the supply of machines is going up and there

is imperfect substitutability in production between machines and low-skill labor, any

equilibrium must feature wL∞/px∞ =∞ even if wL∞ <∞. Applying this to (83), we get

(pxt )1−β2+β1 ∼ ββ11 (1− β1)1−β1

ββ22 (1− β2)1−β2ϕµ2−µ11−σ wβ1−β2Ht . (92)

Further plugging this last relationship in (81), we get:

wHt ∼(σ−1σ

)1+β1

1−β2 ϕψ2µ1ββ11 (1− β1)1−β1

(β

β21−β22 (1− β2)

)β1Gψ1

(1+

β1(β2−β1)(1−β2)

)t N

ψ1

(1+

β1(β2−β1)(1−β2)

)t

, (93)

Hence

gwH∞ = ψ1

(1 +

β1 (β2 − β1)

(1− β2)

)gN∞. (94)

Through (88), the revenues of the machines sector and the final good sector are of the

same order, which implies that Y , pxX and wH grow at the same rate. Therefore

gGDP∞ = gY∞ = gwH∞ = ψ1

(1 +

β1 (β2 − β1)

(1− β2)

)gN∞.

In fact (88) givesR2,t (1)

R1,t(1)∼

σ−1σβ1

1− σ−1σβ2

. (95)

75

Using (92) and (93), one further gets:

pxt ∼ββ11 (1− β1)1−β1(

ββ22 (1− β2)1−β2) 1−β1

1−β2

ϕψ2µ1

(β1−β2)β1

(σ

σ − 1

)β2−β11−β2

G−ψ2(β2−β1)t N

−ψ2(β2−β1)t ,

therefore

gpx∞ = −ψ2 (β2 − β1) gN∞ < 0, (96)

since β2 > β1. Using that wL∞/px∞ =∞ and (95) in (91) leads to:

wLt

(wLtpxt

)ε−1

∼

wHtHPt

β1

(Gt + (1−Gt)

(ϕ(wLtpxt

)ε−1)1−µ1

)

+β2

σ−1σβ1

1−σ−1σβ2

(Gt + (1−Gt)

(ϕ(wLtpxt

)ε−1)1−µ2

)

ϕGtL(

1− β1 + (1− β2)σ−1σβ1

1−σ−1σβ2

) . (97)

Since β2 > β1, then (1−Gt)(wLtpxt

)(ε−1)(1−µ1)

dominates (1−Gt)(wLtpxt

)(ε−1)(1−µ2)

asymp-

totically regardless of the value ofG∞ (in other words, we can always ignore (1−Gt)(wLtpxt

)(ε−1)(1−µ2)

in our analysis).

The reasoning then follows that of Appendix 7.2.1. If G∞ < 1, then (97) implies

w1+β1(σ−1)Lt ∼ (pxt )

(σ−1)β1 wHtHPβ1 (1−Gt)

ϕµ1GtL(

1− β1 + (1− β2)σ−1σβ1

1−σ−1σβ2

) , (98)

which, together with (94) and (96) gives (79).

Alternatively assume that G∞ = 1 and that lim (1−Gt)Nψ2(1−µ1) ε−1

εt exists and is

finite. Suppose first that lim sup (1−Gt)(wLtpxt

)(ε−1)(1−µ1)

= ∞, then there must be a

sub-sequence where (98) is satisfied, which with (94) and (96) leads to a contradiction

with the assumption that lim (1−Gt)Nψ2(1−µ) ε−1

εt exists and is finite.

If lim (1−G)(wLtpx

)(ε−1)(1−µ1)

= 0, then (97) gives

wεLt ∼(pxt )

ε−1wHtHPt

(β1 + β2

σ−1σβ1

1−σ−1σβ2

)ϕL(

1− β1 + (1− β2)σ−1σβ1

1−σ−1σβ2

) ,

76

which implies with (94) and (96) that:

gwL∞ =1

ε

(1− (β2 − β1) (ε− 1)

(1− β2 + β1)

)gGDP∞ . (99)

Finally, if lim sup (1−Gt)w(ε−1)(1−µ)Lt is finite but strictly positive, then as in Ap-

pendix 7.2.1, one can show that this requires that lim (1−Gt)Nψ2ε

(ε−1)(1−µ1)t > 0, from

which we can derive that (99) also holds in that case. This proves Proposition 6 and the

associated footnote in the imperfect substitutes case.

Perfect substitutes case. In the perfect substitutes case, (81) becomes:

(Gϕβ1(σ−1) (px)β1(1−σ) + (1−G)w

β1(1−σ)L

) 11−σ

w1−β1H

= σ−1σββ11 (1− β1)1−β1N

1σ−1 for wL > px/ϕ

, (100)

wβ1L w1−β1H =

σ − 1

σββ11 (1− β1)1−β1N

1σ−1 for wL < px; (101)

(83) becomes

(Gϕβ2(σ−1) (px)β2(1−σ) + (1−G)w

β2(1−σ)L

) 11−σ

wβ1−β2H(Gϕβ1(σ−1) (px)β1(1−σ) + (1−G)w

β1(1−σ)L

) 11−σ

=ββ22 (1− β2)1−β2px

ββ11 (1− β1)1−β1for wL > px/ϕ,

(102)

px =ββ11 (1− β1)1−β1

ββ22 (1− β2)1−β2wβ2−β1L wβ1−β2H for wL < px/ϕ; (103)

(88) becomes(G

(1− σ − 1

σβ2

)+ (1−G) ϕβ2(1−σ)

(wLpx

)β2(1−σ))R2 (1)

R1(1)= G

σ − 1

σβ1 for wL > px/ϕ,

(104)

with R2 (1) = 0 for wL < px/ϕ; and (91) becomes

wLL

wHHP= (1−G)

β1

(ϕwLpx

)β1(1−σ)

+ β2R2(1)R1(1)

(ϕwLpx

)β2(1−σ)(1− β1)

(G+ (1−G)

(ϕwLpx

)β1(1−σ))

+ (1− β2) R2(1)R1(1)

(G+ (1−G)

(ϕwLpx

)β2(1−σ))

for wL > px/ϕ,

(105)

77

wLL

wHHP=

β1

1− β1

for wL < px/ϕ. (106)

Together (101), (103) and (106) show that we must have wLt ≥ pxtϕ

for t large enough,

which delivers (94) and (96).

Assume that G∞ < 1, then (105) gives (98) from which we get that (79) is satisfied.

Now consider the case where G∞ = 1 and lim (1−Gt)Nψ2t exists and is finite. Then

(105) and (104) imply

wLt ∼ (1−Gt)wHt

(ϕwLtpxt

)β1(1−σ)

(β1 + β2

σ−1σβ1

1−σ−1σβ2

(ϕwLt

pxt

)−(β2−β1)(σ−1))

1− β1 + (1− β2)σ−1σβ1

1−σ−1σβ2

HPt

Lfor wL > px/ϕ.

We can then derive that ϕwLtpxt

must have a finite (and positive) limit, so that

gwL∞ = gpx

∞ = − β2 − β1

1− β2 + β1

gGDP∞ .

This proves Proposition 6 and its associated footnote in the perfect substitutes case.

7.5 Proofs and analytical results for the baseline dynamic model

7.5.1 Proof of Proposition 3

We look for a steady state with positive long-run growth for the system defined by

(31), (32), (39) and (40) and we denote such a (potential) steady state n∗, G∗, hA∗, χ∗

(more generally we denote all variables at steady state with a ∗).33 Following (31), we

immediately get that n∗ = 0. Using (32), we get that G∗ obeys:

G∗ =η (G∗)κ

(hA∗)κ

η (G∗)κ(hA∗)κ

+ gN∗. (107)

We focus on a solution with G∗ > 0 (when κ > 0, G∗ = 0 is also a solution), this

equation implies that with(gN)∗> 0, G∗ < 1 . Then, recalling that µ ∈ (0, 1), (49),

implies that:

33Note that if gNt = 0, the economy does not obey this system of equations but that it is also impossibleto achieve positive long-run growth, as production is bounded by the production of an economy whichhas Gt = 1.

78

ω∗ =[ (

σ−1σβ) 1

1−β HP∗

L(1−G∗) (G∗ϕµ)ψ−1

] β(1−σ)1+β(σ−1)

.

Using (45), (40) implies that in steady state,

hA∗ =κ

γ (1− κ)

(ρ+ ((θ − 1)ψ + 1) gN∗

)(108)

which uniquely defines defines hA∗ as a linear and increasing function of gN∗ (recall that

θ ≥ 1). Note that if gN∗ > 0, then hA∗ > 0.

Then, for G∗ > 0, (107) combined with (108), defines G∗ uniquely as an increasing

function of gN∗. (48) also uniquely defines HP∗ as a function of gN∗:

HP∗ = H − gN∗

γ− (1−G∗) hA∗. (109)

(45) and (107) allows to rewrite (39) in steady state as:

ηκ (G∗)κ−1(hA∗)κ

1− κψHP∗ =

γ

κ

(hA∗)2

+ ηGκt

(hA∗)κ+1

. (110)

Since G∗, hA∗ and HP∗ are functions of gN∗, one can rewrite (110) as an equation

determining gN∗. A steady state with positive growth-rate is a solution to

f(gN∗)≡ 1− κ

κ

γG∗hA∗

ψHP∗

(1

κη (G∗)κ

(hA∗)1−κ

+1

γ

)= 1, (111)

with gN∗ > 0. Indeed, (47) simply determines χ∗ as:

χ∗ =

(σ

σ − 1

) 11−β (1−θβ)

(1− β σ−1

σ

)θ (HP∗)θ

(1− β) ββ

1−β (1−θ) (G∗ϕµ)ψ(1−θ), (112)

which achieves the characterization of a steady state for the system of differential equa-

tions defined by (31), (32), (39) and (40).

In order to establish the sufficiency of equation (25) for positive growth. Note that

as gN∗ → 0, then equations (108), (107) and (109) imply that

f (0) =ρ

ψH

(1

ηκκ (1− κ)1−κ

(ρ

γ

)1−κ

+1

γ

).

79

In addition, gN∗

γ+(1−G∗) hA∗ is always greater than gN∗

γ, therefore for a sufficiently large

gN∗ (smaller than γH), HP∗ is arbitrarily small, while for the same value G∗ and hA∗

are bounded below and above. This establishes that for gN∗ large enough, f(gN∗)> 1.

Therefore a sufficient condition for the existence of at least one steady state with positive

growth and positive G∗ is that f (0) < 1 (such that f(gN∗)

= 1 has a solution), which

is equivalent to condition (25).

7.5.2 Uniqueness of the steady state

Generally the steady state is not unique. Nonetheless, consider the special case in which

κ = 0. Then f can be rewritten as

f(gN∗)

=1− κκ

γG∗hA∗

ψHP∗

(1

κη

(hA∗)1−κ

+1

γ

), (113)

note that HP∗ is decreasing in gN∗ and hA∗ is increasing in gN∗, so a sufficient condition

for f to be increasing in gN∗ is that G∗hA∗ is also increasing in gN∗. With κ = 0, using

(108), (107), we get:

G∗hA∗ =η(

κγ(1−κ)

)κ+1 (ρ+ ((θ − 1)ψ + 1) gN∗

)κ+1

η(

κγ(1−κ)

)κ(ρ+ ((θ − 1)ψ + 1) gN∗)κ + gN∗

.

Therefore

d(G∗hA∗

)dgN∗

=

η( κγ(1−κ))

κ+1(ρ+((θ−1)ψ+1)gN∗)

κ

(η( κγ(1−κ))

κ(ρ+((θ−1)ψ+1)gN∗)κ+gN∗)

2

×

(η(

κγ(1−κ)

)κ((θ − 1)ψ + 1)

(ρ+ ((θ − 1)ψ + 1) gN∗

)κ−ρ+ gN∗κ ((θ − 1)ψ + 1)

) .

Since gN∗ > 0, we get thatd(G∗hA∗)dgN∗

> 0 (so that the steady state is unique) if(1−κ)κγκ

ηκκρ1−κ < (θ − 1)ψ + 1. This condition is likely to be met for reasonable pa-

rameter values as long as the automation technology is not too concave: ρ is a small

number, θ ≥ 1 and γ and η being innovation productivity parameters should be of the

same order (it is indeed met for our baseline parameters).

80

7.5.3 Transitional dynamics and the first phase

Combining (27) and (26), we can write:

NthAt =

(κηGκ

t

(∫ ∞t

exp

(−∫ τ

t

rudu

)(Nt

wHt

(πAτ − πNτ

)dτ − 1− κ

κ

Nt

Nτ

wHτwHt

(Nτh

Aτ

))dτ

)) 11−κ

.

Using (8) and that aggregate profits Πt = Nt

(Gtπ

At + (1−Gt) π

Nt

)are a share 1/σ of

output, we can rewrite this equation as:

hAt =

(κηGκ

t

(∫ ∞t

exp

(−∫ τ

t

rudu

)(ψHP

t

πAτ − πNτGtπAt + (1−Gt) πNt

dτ − 1− κκ

Nt

Nτ

vτvthAτ

)dτ

)) 11−κ

.

(114)

Recalling (7), we can write:

hAt =

κηGκt

∫ ∞t

ψHPt

(1+ϕwε−1Lτ )

µ−1

Gt(1+ϕwε−1Lt )

µ+1

exp(∫ τ

t

(gπ

N

u − ru)du)

−1−κκ

exp(∫ τ

t

(gvu − gNu − ru

)du)hAτ

dτ

1

1−κ

.

Consider a fixed t > 0. Then for an arbitrarily large T , if wL0 is sufficiently small

relative to ϕ−1, we will have that wLt is small relative to ϕ−1 over(0, t+ T

). For any

τ ∈(0, t+ T

), we have that

(1+ϕwε−1Lτ )

µ−1

Gt(1+ϕwε−1Lt )

µ+1

= µϕwε−1Lτ + o

(ϕwε−1

Lτ

). The notation o (z)

denotes negligible relative to z (that is f (z) = o (z), if limz→0

f (z) /z = 0) and O (z) will

denote of the same order or negligible in front of z (f (z) = O (z) if lim supz→0

|f (z) /z| <

∞). Then for any t ∈(0, t)

(hAt

)1−κ≤ κηGκ

t

∫ t+Tt

ψHPt

(µϕwε−1

Lτ + o(ϕwε−1

Lτ

))exp

(∫ τt

(gπ

N

u − ru)du)dτ

+∫∞t+T

ψHPt

(1+ϕwε−1Lτ )

µ−1

Gt(1+ϕwε−1Lt )

µ+1

exp(∫ τ

t

(gπ

N

u − ru)du)dτ

.

Further, we know that ru = ρ + θgCu with θ ≥ 1. In addition Cu = Yu − Xu, with Xu

the aggregate spending on machines (initially negligible and later on a share of output

bounded away from 1) and πNu initially grows like Yu/Nu (and from then on will grow

81

slower), therefore we have that ru − gπN

u > ρ. Hence one can write:(hAt

)1−κ

≤ κηGκt

(∫ t+T

t

µψHPt ϕw

ε−1Lτ exp

(∫ τ

t

(gπ

N

u − ru)du

)dτ + o

(ϕwε−1

Lt+T

)+ o

(e−ρ(T+t−t)

))

Since ru − gπN

u > ρ, there exists a φ > 0, such that

∫ t+T

t

exp

(∫ τ

t

(gπ

N

u − ru)du

)dτ ≤

∫ t+T

t

e−(ρ+φ)(τ−t)dτ

≤ 1

ρ+ φ

(1− e−(ρ+φ)(t+T−t)

).

This allows us to rewrite:

(hAt

)1−κ≤ κηGκ

t

(µψHP

t ϕwε−1

Lt+T

ρ+ o

(ϕwε−1

Lt+T

)+ o

(e−ρT

))

Therefore, since T is large and ϕwε−1

Lt+Tis small, then hAt must be small too. In fact, we

get that hAt = O

((ϕwε−1

Lt+T

) 11−κ)

+ o(e−ρT

).

For any t ∈(0, t), we can then rewrite (40) as

χtχt

= γψHP − ρ− (θψ − ψ + 1) gNt +O

((ϕwε−1

Lt+T

) 11−κ)

+ o(e−ρT

). (115)

Using (46) we obtain

Ct = Yt −Xt =(1 +O

(Gtϕw

ε−1Lt

))Yt.

Next (5) and the corresponding equation for high-skill labor demand in production imply:

LNA

LA=

(1−Gt)(1 + ϕwε−1

Lt

)−µ−1

Gt

andHP,NA

HP,A=

(1−Gt)(1 + ϕwε−1

Lt

)−µGt

.

Using (3), we can then write

Yt = N1

σ−1Lβ(HPt

)1−β ×

82

Gt

[1 +O

(ϕwε−1

Lt

)+

(O(ϕwε−1

Lt

)ϕ

1εYtL

) ε−1ε

] εβε−1

σ−1σ (

1 +O(ϕwε−1

Lt

))+ 1−G+O

(ϕwε−1

Lt

)σσ−1

Note that we have wLt = O (Yt/L) therefore ϕ1εYt/L = O

(ϕ

1εwLt

). Therefore

Yt =(1 +O

(ϕwε−1

Lt

))N

1σ−1Lβ

(HPt

)1−β.

From this, using (8), one obtains that high-skill wages obey:

wHt =(1 +O

(ϕwε−1

Lt

)) σ − 1

σ(1− β)N

1σ−1

t Lβ(HPt

)−β,

while for low-skill wages, we get

wLt =(1 +O

(ϕwε−1

Lt

)) σ − 1

σβN

1σ−1

t Lβ−1(HPt

)1−β. (116)

Therefore using the definition of χt, we obtain that

χt =(1 +O

(ϕwε−1

Lt

))σψLβ(θ−1)

(HPt

)(1−β)θ+βN

(1−θ)β(σ−1)(1−β)t .

Differentiating and plugging into (115) and using (48), we get (recalling (116) so that

d ln(1 +O

(ϕwε−1

Lt

))/dt will be of order O

(ϕwε−1

Lt

)as well).

((1− β) θ + β)

·HPt

HPt

= γψHPt −ρ−

(θ − 1

σ − 1+ 1

)γ(H −HP

t

)+O

((ϕwε−1

Lt+T

) 11−κ)

+o(e−ρT

),

(117)

we dropped terms in ϕwε−1Lt since there will negligible in front of

(ϕwε−1

Lt+T

) 11−κ

. The

exact counterpart of this system admits a BGP with HPt constant, and as in the Romer

(1990), there is no transitional dynamics. Therefore, here, we must have over the interval(0, t)

HPt =

(θ−1σ−1

+ 1)H + ρ

γ

ψ + θ−1σ−1

+ 1+O

((ϕwε−1

Lt+T

) 11−κ)

+ o(e−ρT

)and gNt =

γHψ − ρψ + θ−1

σ−1+ 1

+O

((ϕwε−1

Lt+T

) 11−κ)

+ o(e−ρT

)(118)

which is positive under assumption (25). We then have that for Nt low, (22) can be solved

83

as Gt = G0 exp

(− γHψ−ρψ+ θ−1

σ−1+1t

)+ O

((ϕwε−1

Lt+T

) 11−κ)

+ o(e−ρT

). This characterizes the

solution during Phase 1. We can summarize our discussion in the following Proposition.

Proposition 7. For N0 low enough that wL0 is small relative to ϕ1/(ε−1), then there

is an interval(0, t)

during which the automation rate ηGκt

(hAt

)κis small, the share of

automated products Gt depreciates and the economy behaves similarly to that of a Romer

model where automation is impossible.

7.5.4 Transition from the first to the second phase

We prove the following Proposition:

Proposition 8. If κ(1− β) + κ < 1, then Gt cannot go toward 0.

If κ = 0, Phase 1 cannot last forever as at some point, Nt and therefore wLt will

become large. Since, the Poisson rate is η(hAt

)κ= O

(ϕwε−1

Lt

). This implies that Gt

must start growing at a positive rate and that we enter the second phase.

When κ > 0 (and G0 6= 0, otherwise automation is impossible), however, whether the

Poisson rate of automation becomes negligible or not depends on a horse race between the

drop in the share of automated products (and therefore the efficiency of the automation

technology) and the rise in the low-skill wages (which, through horizontal innovation can

become arbitrarily large). We look for a sufficient condition under which the Poisson

rate will take off.

First assume that Gtwβ(σ−1)Lt does not tend towards 0. Then from (114) we obtain

that:

hAt = hAt(Gκ−1t

) 11−κ =⇒ ηGκ

t

(hAt

)κ= O

(G

κ−κ1−κt

)(119)

Since κ ≤ κ, we obtain that the Poisson rate of automation diverges: a contradiction.

Assume now thatGtwβ(σ−1)Lt does tend towards 0. This ensures that wLt = O

(N

1σ−1

t

).

Moreover,πAt −πNt

GtπAt +(1−Gt)πNt= O

(wβ(σ−1)Lt

). Then using this in (114), we obtain

hAt = O(Gκtw

β(σ−1)Lt

) 11−κ

.

Note that hAt must remain bounded otherwise high-skill labor market clearing is violated.

Therefore, we must haveGκtw

β(σ−1)Lt bounded (which implies thatGtw

β(σ−1)Lt tends towards

84

0). Therefore the Poisson rate obeys:

ηGκt

(hAt

)κ= O

(G

κ1−κt N

βκ1−κt

)Plugging this in (32) we get:

·Gt = O

(G

κ1−κt N

βκ1−κt

)− gNt Gt

To obtain that the share Gt is going towards 0, it must first be that Gκ

1−κt N

βκ1−κt declines

at the same rate or faster than Gt.

Consider first the case where, Gκ

1−κt N

βκ1−κt and Gt are of the same order. In that case,

we must have:

Gt = O

(N

βκ1−κ−κt

)This cannot go towards 0 if 1−κ− κ > 0. In addition, recall that this reasoning assumed

that Gκtw

β(σ−1)Lt remains bounded. We have

Gκtw

β(σ−1)Lt = O

(N

β(1−κ)(1−κ)1−κ−κ

t

),

which is indeed declining if 1 − κ − κ < 0. Furthermore, in that case we must have·Gt ≥ −gNt Gt, that is Gtshould not decline at a rate faster than N−1

t . This implies that

we must have βκκ+κ−1

≤ 1⇐⇒ κ(1− β) + κ ≥ 1.

Alternatively, if Gκ

1−κt N

βκ1−κt goes towards 0 faster than Gt then Gt will be declining

at the rate gNt , so that we have Gt = O(N−1t

). This then implies

Gκ

1−κt N

βκ1−κt /Gt = O

(N

βκ+1−κ−κ1−κ

t

).

As soon as κ(1− β) + κ < 1 then this cannot go towards 0.

Therefore κ(1 − β) + κ < 1 is a sufficient condition which ensures that the Poisson

rate of automation must take off.

7.5.5 Transitional dynamics in the third phase

In this appendix, we prove two results:

85

Lemma 2. If Gt is bounded above 0 then hAt is bounded.

Proposition 9. If Gt and gNt admit positive limits then the economy converges toward

a steady-state as described in Proposition 3,

We then provide details on the behavior of the economy close to the steady-state.

Proof of Lemma 2. Assume that Gt is bounded above 0 (in fact the analysis of

Phase 2 in Appendix 7.5.4 shows that as long as κ (1− β) + κ < 1, it is impossible

to have G∞ = 0). Note that HPt must be bounded below otherwise there would be

arbitrarily large welfare gains from increasing consumption at time t and reducing it at

later time periods. As HPt is also bounded above (by H), then we must have (following

a reasoning similar to that in Appendix 2.4), that wHt = Θ(Nψt

), Ct = Θ

(Nψt

)and

wLt is bounded below, so that vt and ct are bounded above and below and ωt must be

bounded above.

Integrating (19), using the transversality condition and dividing by wHt/Nt, we get:

V At

wHt/Nt

=

∫ ∞t

exp

(−∫ s

t

r (u) du

)πAs

wHt/Nt

ds,

using the Euler equation (24), this leads to:

V At

wHt/Nt

=

∫ ∞t

exp (−ρ (t− s))(C (s)

C (t)

)−θπAs N

ψ−1s

vsNψ−1t

ds.

Rewriting this expression with the normalized variables and using (45), we get:

V At

wHt/Nt

=

∫ ∞t

exp (−ρ (s− t))(Ns

Nt

)−(1+(θ−1)ψ)c (s)θ

c (t)θ

ψ(ϕ+ (ωsns)

1µ

)µHPs

Gs

(ϕs + (ωsns)

1µ

)µ+ (1−Gs)ωsns

vsvtds.

Note that c(s)c(t)

,ψ

(ϕ+(ωsns)

1µ

)µHPs

G

(ϕs+(ωsns)

1µ

)µ+(1−Gs)ωsns

and vsvt

are all bounded and that Ns is weakly

increasing, therefore we get that

V At

wHt/Nt

≤∫ ∞t

exp (−ρ (s− t))Mds,

for some constant M . This ensures thatV At −V NtwHt/Nt

must remain bounded, and following

(21), hAt must be bounded as well.

86

Proof of Proposition 9. Assume now that Gt has a limit G∞ and that gNt also

has a positive limit gN∞. Then (32) implies that G∞ < 1. Following Proposition (2), we

then get that wLt = O

(N

ψ1+β(σ−1)

t

)or ωt = O (1). Therefore, we can rewrite the system

as (32),

˙hAt = γ

1

κ

(hAt

)2

−ηκGκ

t

(hAt

)κ1− κ

πAtvt

+ ηGκt

(hAt

)κ+1

+κ

1− κ

(ηGκ−1

t

(hAt

)κ+1

(1−Gt)− gNt hAt)

+O (nt) ,

χt = χt

(γ

1− κκ

hAt − ρ− (θψ − ψ + 1) gNt

)+O (nt) .

Knowing that

HPt =

(σ − 1

σ

) 11−β ( 1

θ−β) (1− β)

1θ β

β1−β ( 1

θ−1)χ

1θt (Gtϕ

µ)ψ( 1

θ−1)

Gtϕµ(1− β σ−1

σ

) +O

(n

1µ

t

), (120)

πAtvt

=ψHP

t

Gt

+O

(n

1µ

t

), (121)

and (48). Using that gNt and Gt have limits in (32) implies that hAt must also have

a limit. Using (48), this implies that HPt must also have a limit and therefore using

(120) that χt must have a limit. In other words, the equilibrium path tends toward the

steady-state(hA∗, G∗, χ∗

)with nt → 0 defined in Proposition 3.

Behavior close to the steady-state. In this steady-state, using (108) we get

gN∗ =1

(θ − 1)ψ + 1

(γ (1− κ)

κhA∗ − ρ

).

Therefore in the third phase, we obtain that Nt grows at rate gNt = gN∗+ o (1), that the

share of automated product obeys Gt = G∗ + o (1), with the mass of high-skill workers

in automation given by HAt = (1−G∗)hA∗ + o (1) and the mass of high-skill workers in

production given by HPt = HP∗+ o (1), with HP∗ given by (109). Using (43), we obtain

87

that wages obey:

wHt = (1− β)

(σ − 1

σββ) 1

1−β

(G∗ϕµ)ψNψt + o

(Nψt

),

and

wLt = (ω∗)1

β(1−σ) Nψ

1+β(σ−1)

t + o

(N

ψ1+β(σ−1)

t

).

Using (121), the profit made by an automated firm are given by

πAt =1

σ

((σ − 1) βϕ

εε−1

σ

) β1−β

HP∗ (G∗)ψ−1Nψ−1t + o

(Nψ−1t

)while the profits made by a non-automated firm πNt are negligible in front of πAt . Using

(33), the value of an automated firm is then simply given by:

V At =

πAtr∗ − (ψ − 1) gN∗

+ o(Nψ−1t

), (122)

where

r∗ = ρ+ θψgN∗ (123)

is the steady-state interest rate. Following (34) and (35), the normalized value of a

non-automated firm obeys:

(rt − (ψ − 1) gNt

)V Nt = πNt + (1− κ) ηGκ

t hAt

(V At − V N

t

)+

·

V Nt .

Therefore, one gets that for large Nt,

V Nt =

(1− κ) ηG∗κhA∗

r∗ − (ψ − 1) gN∗ + (1− κ) ηG∗κhA∗V At + o

(Nψ−1t

)(124)

so that asymptotically all the value of a new firm comes from the profits it makes once

automated (obviously we still get that in equilibrium V Nt = wHt/Nt).

Comparing the growth rate in the number of products in Phase 1 and

Phase 3. Use (122), (124) and (123) to get that in steady-state:

V N∗ = fV A∗

88

with

f =(1− κ) ηG∗κhA∗

ρ+ (ψ (θ − 1) + 1) g∗N + (1− κ) ηG∗κhA∗, (125)

V A∗ =πAt

ρ+ (ψ (θ − 1) + 1) g∗N.

Using V N∗ = v∗/γ and (121), we then obtain:

f1

ρ+ (ψ (θ − 1) + 1) g∗N

ψHP∗

G∗=

1

γ.

Rearranging terms, this leads to

gN∗ =

fG∗γψ(H − (1−G∗) hA∗

)− ρ

fG∗ψ + θ−1

(σ−1)(1−β)+ 1

, (126)

while from (118) the growth rate in the first period is approximately given by

gN1 =γHψ − ρψ + θ−1

σ−1+ 1

. (127)

The two expressions differ by three terms: In the numerator, H − (1−G∗) hA∗ in (126)

replaces H in (127), since some high-skill workers are hired to automate in Phase 3, the

pool of high-skill workers available for horizontal innovation or production is smaller,

and this force pushes toward gN∗ < gN1. In the denominator, θ−1(σ−1)(1−β)

in (126) replacesθ−1σ−1

in (127), this reflects the fact that the growth rate in the number of products has a

larger impact on the economy growth rate in phase 3 than in phase 1, which in return

reduces the present value of an automated firm (it increases the effective interest rate).

This also pushes toward gN∗ < gN1. Finally the term f/G∗ in (126) does not exist

in (127). Note that ∂gN∗/ (∂f/G∗) > 0, so that this term reflects two different forces:

on one hand in phase 3, the value of a new firm is a fraction f < 1 of the value of an

automated firm and this reduces the growth rate in the number of products, on the other

hand a lower G∗ reduces the demand for high-skill workers and therefore high-skill wages

compared to the profits an automated firm, which increases the asymptotic growth rate

89

in the number of products. Combining (32), (107) and (125), we get

f

G∗< 1⇔ 1

G(1− κ) ηG∗κhA∗ < ρ+ (ψ (θ − 1) + 1) g∗N + (1− κ) ηG∗κhA∗

⇔ 1−GG

(1− κ) ηG∗κhA∗ < ρ+ (ψ (θ − 1) + 1) g∗N ,

⇔ (1− κ) gN∗ < ρ+ (ψ (θ − 1) + 1) g∗N .

Since θ ≥ 1 and κ < 1, this equation necessarily holds. Therefore f < G∗ and it is

always the case that gN∗ > gN1.

7.5.6 Comparative statics

In this section, we prove Proposition 4. The proposition is established when the steady

state is unique but it extends to the case of the steady states with the highest and lowest

growth rates when there is multiplicity. Recall that the steady state is characterized as

the solution to an equation f(gN∗)

= 1 through (111), where G∗, hA∗ and HP∗ can all

be written as functions of gN∗ and parameters. Moreover, when there is a single steady

state (as well as for the steady states with the highest and the lowest growth rates in

case of multiplicity), f must be increasing in the neighborhood of gN∗.

Comparative static with respect to γ. (108) implies that hA∗ is inversely pro-

portional to γ (for given gN∗). Formally, we have:

∂hA∗

∂γ= − h

A∗

γ. (128)

Differentiating (107) and using (128) leads to:

∂G∗

∂γ=

−κgN∗G∗

γ(η (G∗)κ

(hA∗)κ

+ (1− κ) gN∗) , (129)

so that for a given gN∗, G∗ is also decreasing in γ. Using (109), (128) and (129), we get:

∂HP∗

∂γ=

1

γ

gN∗

γ+ (1−G∗) (1− κ) hA∗

+(1−κ)κhA∗G∗(gN∗)

2

(η(G∗)κ(hA∗)κ)(η(G∗)κ(hA∗)

κ+(1−κ)gN∗)

> 0

90

so that HP∗ is increasing in γ. Note that f , defined in (111), can be rewritten as

f(gN∗)

=1− κκ

1

ψHP∗

(G∗)1−κ(hA∗)1−κ (

γhA∗)

κη+G∗hA∗

,

which shows that f is decreasing in γ for a given gN∗ (HP∗ is increasing, G∗ and hA∗ are

decreasing, and γhA∗ is constant). Since f is increasing in gN∗ at the equilibrium value,

(111) implies that gN∗ increases in γ.

Comparative static with respect to η. For given gN∗, (108) implies that hA∗

does not depend on η. Differentiating (107), we get:

∂ lnG∗

∂ ln η=

gN∗

η (G∗)κ(hA∗)κ

+ (1− κ) gN∗, (130)

so for given gN∗, G∗ increases in η. (109) implies then that

∂ lnHP∗

∂ ln η=G∗hA∗

HP∗gN∗

η (G∗)κ(hA∗)κ

+ (1− κ) gN∗.

Using this equation together with (130) and (111), we obtain:

∂ ln f

∂ ln η=

gN∗

η(G∗)κ(hA∗)κ

+(1−κ)gN∗

(1− G∗hA∗

HP∗ − κ1

κη(G∗)κ (hA∗)1−κ

1

κη(G∗)κ (hA∗)1−κ

+ 1γ

)−

1

κη(G∗)κ (hA∗)1−κ

1

κη(G∗)κ (hA∗)1−κ

+ 1γ

.

Using (108), we can rewrite this as:

∂ ln f

∂ ln η=

− 1

η(G∗)κ(hA∗)κ

+(1−κ)gN∗

×

gN∗G∗hA∗

HP∗ + ρ+((θ−1)ψ+κ)gN∗

γ(1−κ)

(1

ηLκGκL

(hA∗)1−κ

+ 1γ

) ,

so that f is decreasing in η. This implies that gN∗ must be increasing in η. Since hA∗

only depends on η through gN∗, we also get that hA∗ increases in η.

91

7.6 Simulation technique

In the following we describe the simulation techniques employed for the baseline model

presented in 2. The approach for the extensions follow straightforwardly. Let xt ≡(nt, Gt, h

At , χt, ωt) and note that equation (49) defines ω implicitly. We can therefore

write equations (31), (32), (39) and (40) as a system of autonomous differential equations

(nt, Gt,˙hAt , χt) = F (xt) with initial conditions on state variables as (n0, G0) and an

auxiliary equation of ωt = ϑ(xt).

For the numerical solution, we specify a (small) time period of dt > 0 and a (large)

number of time periods T . Using this we approximate the four differential equations by

(T − 1)× 4 errors as:

st = (nt+1 − nt

dt,Gt+1 −Gt

dt,hAt+1 − hAt

dt,χt+1 − χt

dt)− F ((xt + xt+1)/2), t = 1, ...T − 1

with T corresponding errors for ωt :

sωt = ωt − ϑ(xt), t = 1, ..., T.

As shown in Appendix 7.5.1 for a set of parameter values, the system admits an

asymptotic steady state. We assume that the system has reached this asymptotic steady

state by time T and restrict hAT and χT accordingly. Together with the initial conditions

(n1 = nstart and G1 = Gstart) this leads to a vector of errors:

sT ≡ (n1 − nstart, G1 −Gstart, hAT − hA∗, χT − χ∗)′.

Letting x = xtTt=1, we then stack errors to get a vector, S(x), of length 5T and solve

the following problem:

x = argminxS(x)′WS(x),

for a 5T × 5T diagonal weighting matrix, W , and the 5T vector x. For dt → 0 and

T → ∞ S(x)′WS(x) → 0. For the simulations we set dt = 2 and T = 2000. We

accept the solution when S(x)′WS(x) < 10(−7), but the value is typically less than

10(−20). The choice of weighting matrix matters somewhat for the speed of convergence,

but is inconsequential for the final result. With the solution xt, ωtTt=1 in hand, it is

straightforward to find remaining predicted values.

92

7.7 Complements on simulation

7.7.1 Additional results on the baseline simulation

0 100 200 300 4000

1

2

3

4

Year

Per

cen

t

Panel A: Growth Rate of Consumption

gC

0 100 200 300 400

0

5

10

Year

units

Panel B: Log Consumption per Capita

Low−skill consumptionHigh−skill consumption

0 100 200 300 40040

60

80

100

120

140Panel C: Share of Assets Held by Low−Skill Workers

Year

Per

cen

t

% Share of Assets held by L

0 100 200 300 4002

4

6

8

10

Year

units

Panel D: Wealth Ratio

Wealth ratio

Figure 12: Consumption and wealth for baseline parameters. Panel A shows yearly growthrates for consumption, Panel B log consumption of high-skill workers and low-skillworkers (per capita), Panel C the share of assets held by low-skill workers andPanel D the wealth to GDP ratio.

Wealth and consumption. Figure 12 shows the evolution of wealth and consumption

for the baseline parameters both in the aggregate and for each skill group. Panel A

shows that consumption growth follows a pattern very similar to that of GDP growth

(displayed in Figure 3.A), which is in line with a stable ratio of total R&D expenses over

GDP across the three phases (Figure 3.D). In the absence of any financial constraints,

low-skill and high-skill consumption must grow at the same rate, with high-skill workers

consuming more since they have a higher income (Panel B). Since low-skill labor income

becomes a negligible share of GDP , while the high-skill labor share increases, a con-

stant consumption ratio can only be achieved if high-skill workers borrow from low-skill

workers in the long-run. This is illustrated in Panel C, which shows the share of assets

held by low-skill workers, under the assumption that initially assets holdings per capita

are identical for low-skill and high-skill workers (so that low-skill workers hold 2/3 of

the assets in year 0, since with these parameters H/L = 1/2). Initially, low-skill and

93

high-skill income grow at a constant rate so that the share of assets held by low-skill

workers is stable; but, in anticipation of a lower growth rate for low-skill wages than for

high-skill wages, low-skill workers start saving more and more, and the share of assets

they hold increases. This share eventually reaches more than 100%, meaning that the

high-skill workers net worth becomes negative. As claimed in the text, Panel D shows

that since profits become a higher share of GDP (an effect which dominates a temporary

increase in the interest rate in Phase 2), the wealth to GDP ratio increases in phase 2,

such that its steady state value is nearly 3 times higher than its original, which still need

to be automated (foll value.

The accumulation of asset holdings by low-skill workers predicted by the model seems

counter-factual, it results from our assumptions of infinitely lived agents with identical

discount rates and no financial constraints. Reversing these unrealistic assumptions

would change the evolution of the consumption side of the economy but should not alter

the main results which are about the production side.

0 100 200 300 400Year

-3

-2

-1

0

1

2

3

4

Per

cen

t

Panel A: Low-skill wages growth decomposition

Total gwL

Horiz. innov. contributionAutomation contribution

0 100 200 300 400Year

-3

-2

-1

0

1

2

3

4

Per

cen

t

Panel B: High-skill wages growth decomposition

Total gwH

Horiz. innov. contributionAutomation contribution

Figure 13: Growth decomposition. Panel A: The growth rate of low-skill wages and theinstantaneous contribution from horizontal innovation and automation, respec-tively. Panel B is analogous for high-skill wages. See text for details.

Growth decomposition. Figure 13 performs a growth decomposition exercise for

low-skill and high-skill wages by separately computing the instantaneous contribution

of each type of innovation. We do so by performing the following thought experiment:

at a given instant t, for given allocation of factors, suppose that all innovations of a

given type fail. By how much would the growth rates of wL and wH change?34 In Phase

34More specifically we can write wLt = f(Nt, Gt, HPt ), using equations (10) and (11). Differentiating

with respect to time and using equation (32) gives:

gwLt =

(NtwLt

∂f

∂N− GtwLt

∂f

∂G

)γHD

t +1

wLt

∂f

∂GηGκt (1−Gt)(hAt )κ +

1

wLt

∂f

∂HPHPt .

94

1, there is little automation, so wage growth for both skill-groups is driven almost en-

tirely by horizontal innovation. In Phase 2, automation sets in. Low-skill labor is then

continuously reallocated from existing products which get automated, to new, not yet

automated, products. The immediate impact of automation on low-skill wages is neg-

ative, while horizontal innovation has a positive impact, as it both increases the range

of available products and decreases the share of automated products. The figure also

shows that automation plays an increasing role in explaining the instantaneous growth

rate of high-skill wages, while the contribution of horizontal innovation declines. This is

because new products capture a smaller and smaller share of the market and therefore

do not contribute much to the demand for high-skill labor. Consequently, automation is

skill-biased while horizontal innovation is unskilled-biased. We stress that this growth

decomposition captures the immediate effect of automation or horizontal innovation.

This should not be interpreted as “automation being harmful” to low-skill workers in

general. In fact, as we demonstrate in Section 3.7, an increase in the effectiveness of the

automation technology, η, will have positive long-term consequences. A decomposition

of gGDPt would look similar to the decomposition of gwHt : while instantaneous growth is

initially almost entirely driven by horizontal innovation, automation becomes increas-

ingly important in explaining it (long-run growth, however, is ultimately determined by

the endogenous rate of horizontal innovation).

7.7.2 Decreasing growth rate

Figure 14 shows a case where growth is substantially lower in Phase 3 than in Phase

1 and in fact continuously decreases from the middle of Phase 2. The parameters are

identical to the baseline case except for: σ = 2.5, β = 0.55, η = 0.1, γ = 0.23 and

N0 = 344.25, these parameters lower the growth rate of the economy particularly in

the asymptotic steady state because automation consumes more resources and is less

effective as high-skill workers have a larger factor share in production (N0 is higher so

as to shorten Phase 1 in the graph). As argued in the text, this case illustrates that our

model does not necessarily predict that intense automation needs to be associated with a

boost in economic growth. Conversely, making the automation technology more effective

(say by reducing the cost share of high-skill workers, β) could create the opposite pattern

of a low initial growth rate followed by a higher eventual growth rate.

Figure 13 plots the first two terms as the growth impact of expenses in horizontal innovation andautomation, respectively. The third term ends up being negligible. We perform a similar decompositionfor wHt.

95

0 100 200 300 400Year

0

1

2

3

4P

er c

ent

Panel A: Growth Rates of Wages and GDP

gGDP

gwL

gwH

0 100 200 300 4000

5

10

15

Per

cen

t Autom./GDP (left)Hori./GDP (left)

0 100 200 300 400Year

0

20

40

60

80

100Panel B: Research Expenditures and G

G (right)

Figure 14: Transitional dynamics with a low growth rate in Phase 2.

7.7.3 A delayed decline in the labor share

Empirically, the drop in the labor share is a more recent phenomenon than the increase

in the skill premium. In Figure 15, we choose parameters such that this happens.

The automation technology is more productive η = 0.4; the automation technology

is less concave κ = 0.9 (so that a higher incentive to automate is required to get a

significant share of high-skill workers in automation innovation); and all other parameters

are identical to the baseline case. In this case, more high-skill workers get allocated to

automation during Phase 2: as shown in Panel C automation expenditures represent a

much larger share of GDP during Phase 2 than they do in the baseline case. The mass of

high-skill workers engaged in production declines during Phase 2. This results first in a

sharper increase in the skill premium (the skill premium condition moves further to the

left). In addition, the drop in the labor share is delayed since innovation spending are

part of GDP while capital income is a constant share of output Y . The growth rates of

low-skill and high-skill wages start diverging significantly from around year 135 and by

year 150, the high-skill wage growth rate is 2pp higher than the low-skill wage growth

rate, while the total labor share only start declining from around year 150 and in fact

increases slightly before.

96

0 100 200 300 400Year

0

1

2

3

4

5

Per

cen

t

Panel A: Growth Rates of Wages and GDP

gGDP

gwL

gwH

0 100 200 300 400Year

0

1

2

3 automation incentive (left)

0 100 200 300 400Year

0

2

4

6

8

log

unit

Panel B: Automation Incentive and Skill Premium

skill premium (right)

0 100 200 300 400Year

0

5

10

15

Per

cen

t

Autom./GDP (left)Hori./GDP (left)

0 100 200 300 400Year

0

20

40

60

80

100

Per

cen

t

Panel C: Research Expenditures and G

G (right)

0 100 200 300 400Year

0

20

40

60

80

100

Per

cen

t

Panel D: Factor Shares

Total labor shareLow-skill labor share

Figure 15: Transitional dynamics with a delayed drop in capital share

7.7.4 The effect of the innovation parameters

0 200 400

0

50

100

Year

Per

cen

t

Panel A: GDP deviation

High γHigh η

0 200 400

0

50

100

Year

Per

cen

t

Panel B: Low−skill wages deviation

High γHigh η

0 200 400

0

50

100

Year

Per

cen

t

Panel C: Skill premium deviation

High γHigh η

Figure 16: Deviations from baseline model for more productive horizontal innovation tech-nology (γ) and more productive automation technology (η).

Figure 16 shows the impact (relative to the baseline case) of increasing productivity in

the automation technology to η = 0.4 (from 0.2) and the productivity in the horizontal

innovation technology to γ = 0.32 (instead of 0.3). A higher η initially has no impact

during Phase 1, but it moves Phase 2 forward as investing in automation technology is

profitable for lower level of low-skill wages. Since automation occurs sooner, the absolute

level of low-skill wages drops relative to the baseline case (Panel B), which leads to a fast

increase in the skill premium. However, as a higher η means that new firms automate

97

100 150 200 250Year

-20

-10

0

10

20

Per

cen

tPanel A: High-skill workers

in automation relative to baseline

-50% shock on G150

+50% shock on G150

100 150 200 250Year

-50

-40

-30

-20

-10

0

10

20

30

40

50

Per

cen

t

Panel B: Share of automated products relative to baseline

-50% shock on G150

+50% shock on G150

100 150 200 250Year

-10

-5

0

5

10

Per

cen

t

Panel C: skill premium relative to baseline

-50% shock on G150

+50% shock on G150

Figure 17: Technological shocks on Gt in year 150.

faster, it encourages further horizontal innovation. A faster rate of horizontal innovation

implies that the skill premium keeps increasing relative to the baseline, but also that

low-skill wages are eventually larger than in the baseline case. A higher productivity for

horizontal innovation, γ, implies that GDP and low-skill wages initially grow faster than

in the baseline. Therefore Phase 2 starts sooner, which explains why the skill premium

jumps relative to the baseline case before increasing smoothly.

7.7.5 Technological shocks

Figure 17 shows the effect of a technological shock in the form of an exogenous change

in Gt in year 150 (for the same parameters as in the baseline case). Panel A shows that

after an increase in G, the mass of high-skill workers in automation innovation declines,

this guarantees that the share of automated products goes back to its initial path (panel

B). The skill premium is reduced by the shock but thereafter increases relative to the

baseline level (panel C). It does not converge back to exactly the same level as before,

however, because the shock to Gt and the following decrease in HAt implies that for some

time more high-skill workers undertake horizontal innovation which increases a bit the

number of automated products relative to the baseline.

7.7.6 Systematic comparative statics

In this section we carry a systematic comparative exercise with respect to the parameters

of the model, namely σ, ε, β, ρ, θ, ϕ, η, κ, κ,γ,H/L (we keep H + L = 1), N0, G0. We

show the evolution of the growth rate of high-skill and low-skill wages and the share of

98

automated products for the baseline parameters and two other values for one parameter,

keeping all the other ones fixed. In all cases, the broad structure of the transitional

dynamics in three phases is maintained.

0 100 200 300 400

2

4

6

Year

Per

cen

t

Panel A: Growth rate of high−skill wages

σ=3.5baseline: σ=3σ=2.5

0 100 200 300 4000

2

4

Year

Per

cen

t

Panel B: Growth rate of low−skill wages

σ=3.5baseline: σ=3σ=2.5

0 100 200 300 4000

50

100

Year

Per

cen

t

Panel C: Share of automated products

σ=3.5baseline: σ=3σ=2.5

0 100 200 300 400

2

4

6

Year

Per

cen

t

Panel D: Growth rate of high−skill wages

ε=6baseline: ε=4ε=3

0 100 200 300 4000

2

4

Year

Per

cen

tPanel E: Growth rate of low−skill wages

ε=6baseline: ε=4ε=3

0 100 200 300 4000

50

100

Year

Per

cen

t

Panel F: Share of automated products

ε=6baseline: ε=4ε=3

0 100 200 300 400

2

4

6

Year

Per

cen

t

Panel G: Growth rate of high−skill wages

β=3/4baseline: β=2/3β=1/2

0 100 200 300 4000

2

4

Year

Per

cen

t

Panel H: Growth rate of low−skill wages

β=3/4baseline: β=2/3β=1/2

0 100 200 300 4000

50

100

YearP

er c

ent

Panel I: Share of automated products

β=3/4baseline: β=2/3β=1/2

Figure 18: Comparative statics with respect to the elasticity of substitution acrossproducts (σ), the elasticity of substitution between machines and low-skillworkers in automated firms (ε) and the factor share of low-skill workersand machines in production (β).

Figures 18.A,B,C show that a higher elasticity of substitution across products σ

reduces the growth rate of the economy (the elasticity of output with respect to the

number of products is lower), which leads to a delayed transition. The asymptotic

growth rate of low-skill wages is a smaller fraction of that of high-skill wages (following

Proposition 2), since automated products are a better substitute for non-automated

ones. Figures 18.D,E,F show that the elasticity of substitution between machines and

low-skill workers in automated firms, ε, plays a limited role (as long as the assumption

µ < 1 is kept), a higher elasticity reduces the growth of low-skill wages and increases

that of high-skill wages during Phase 2. Figures 18.G,H,I show that a lower factor share

in production for high-skill workers (a higherβ) increases the growth rate of the economy

(high-skill wages are lower which favors innovation). As a result Phase 2 occurs sooner.

In addition, following Proposition 2, the asymptotic growth rate of low-skill wages is a

99

lower fraction of that of high-skill wages (the cost advantage of automated firms being

larger).

0 100 200 300 400

2

4

6

Year

Per

cen

t

Panel A: Growth rate of high−skill wages

ρ=0.03baseline: ρ=0.02ρ=0.01

0 100 200 300 4000

2

4

Year

Per

cen

t

Panel B: Growth rate of low−skill wages

ρ=0.03baseline: ρ=0.02ρ=0.01

0 100 200 300 4000

50

100

Year

Per

cen

t

Panel C: Share of automated products

ρ=0.03baseline: ρ=0.02ρ=0.01

0 100 200 300 400

2

4

6

Year

Per

cen

t

Panel D: Growth rate of high−skill wages

θ=2.5baseline: θ=2θ=1.5

0 100 200 300 4000

2

4

Year

Per

cen

t

Panel E: Growth rate of low−skill wages

θ=2.5baseline: θ=2θ=1.5

0 100 200 300 4000

50

100

Year

Per

cen

t

Panel F: Share of automated products

θ=2.5baseline: θ=2θ=1.5

0 100 200 300 400

2

4

6

Year

Per

cen

t

Panel G: Growth rate of high−skill wages

ϕ=0.35

baseline: ϕ=0.25

ϕ=0.15

0 100 200 300 4000

2

4

Year

Per

cen

t

Panel H: Growth rate of low−skill wages

ϕ=0.35

baseline: ϕ=0.25

ϕ=0.15

0 100 200 300 4000

50

100

Year

Per

cen

t

Panel I: Share of automated products

ϕ=0.35

baseline: ϕ=0.25

ϕ=0.15

Figure 19: Comparative statics with respect to the discount rate (ρ), the inverse elasticityof intertemporal substitution (θ) and the productivity of machines (ϕ)

Figures 19.A,B,C show that a higher discount rate ρ reduces the growth rate of the

economy, which slightly postpones Phase 2. At the time of Phase 2, the growth rate of

low-skill wages is not affected much by the discount rate: on one hand, since low-skill

wages are lower Phase 2 is postponed, which favor low-skill wages’ growth, but on the

other hand, horizontal innovation is lower which negatively affects low-skill wages. A

lower elasticity of intertemporal substitution (a higher θ) has a similar effect on the

economy’s growth rate (Figures 19.D,E,F). Figures 19.G,H,I show that the productivity

of machines (ϕ) only affects the timing of Phase 2 (Phase 2 occurs sooner when machines

are more productive).

100

0 100 200 300 400

2

4

6

Year

Per

cen

tPanel A: Growth rate of high−skill wages

η=0.4baseline: η=0.2η=0.1

0 100 200 300 4000

2

4

Year

Per

cen

t

Panel B: Growth rate of low−skill wages

η=0.4baseline: η=0.2η=0.1

0 100 200 300 4000

50

100

Year

Per

cen

t

Panel C: Share of automated products

η=0.4baseline: η=0.2η=0.1

0 100 200 300 400

2

4

6

Year

Per

cen

t

Panel D: Growth rate of high−skill wages

κ=0.75baseline: κ=0.5κ=0.25

0 100 200 300 4000

2

4

YearP

er c

ent

Panel E: Growth rate of low−skill wages

κ=0.75baseline: κ=0.5κ=0.25

0 100 200 300 4000

50

100

Year

Per

cen

t

Panel F: Share of automated products

κ=0.75baseline: κ=0.5κ=0.25

0 100 200 300 4002

4

6

8

Year

Per

cen

t

Panel G: Growth rate of high−skill wages

κ=0.49κ=0.25baseline κ=0

0 100 200 300 400−1

0

1

2

3

Year

Per

cen

tPanel H: Growth rate of low−skill wages

κ=0.49κ=0.25baseline κ=0

0 100 200 300 4000

50

100

Year

Per

cen

t

Panel I: Share of automated products

κ=0.49κ=0.25baseline κ=0

Figure 20: Comparative statics with respect to the automation productivity (η), the concav-ity of the automation technology (κ) and the automation externality (κ)

The comparative statics with respect to the automation technology shown in Figures

20.A,B,C follow the pattern described in the text. A less concave automation technology

(higherκ) delays Phase 2 and reduces the economy’s growth rate. It particularly affects

the growth rate of low-skill wages in Phase 2 (as the increase in automation expenses

comes more at the expense of horizontal innovation)—see Figures 20.D,E,F. The role

of the automation externality has already been discussed in the text, Figures 20.G,H,I

reveal that for a mid-level of the automation externality (κ = 0.25), the economy looks

closer to the economy without the automation externality than to the economy with a

large automation externality.

101

0 100 200 300 400

2

4

6

Year

Per

cen

t

Panel A: Growth rate of high−skill wages

γ=0.35baseline: γ=0.3γ=0.25

0 100 200 300 4000

2

4

Year

Per

cen

t

Panel B: Growth rate of low−skill wages

γ=0.35baseline: γ=0.3γ=0.25

0 100 200 300 4000

50

100

Year

Per

cen

t

Panel C: Share of automated products

γ=0.35baseline: γ=0.3γ=0.25

0 100 200 300 400

2

4

6

8

Year

Per

cen

t

Panel D: Growth rate of high−skill wages

H=1/2 L=1/2baseline: H=1/3 L=2/3H=1/4 L=3/4

0 100 200 300 4000

2

4

YearP

er c

ent

Panel E: Growth rate of low−skill wages

H=1/2 L=1/2baseline: H=1/3 L=2/3H=1/4 L=3/4

0 100 200 300 4000

50

100

Year

Per

cen

t

Panel F: Share of automated products G

H=1/2 L=1/2baseline: H=1/3 L=2/3H=1/4 L=3/4

Figure 21: Comparative statics with respect to the horizontal innovation productivity (η)and the skill ratio (H/L)

Figures 21.A,B,C show the impact of the horizontal innovation parameter γ, which

was already discussed in the text. Figures 21.D,E,F show that a higher ratio H/L nat-

urally leads to a higher growth rate, which implies that Phase 2 occurs sooner. Figures

22.A,B,C show that a higher initial number of products simply advance the entire evo-

lution of the economy. Figures 22.D,E,F show that a higher initial value for the share

of automated products (even as high as the steady-state value G∗) barely affects the

evolution of the economy, the share of automated products initially drops quickly as

there is little automation to start with.

7.8 Social planner problem

This section presents the solution to the social planner problem. After having set-up the

problem, we derive the optimal allocation, emphasizing in particular the different inef-

ficiencies in our competitive equilibrium. Then, we show the optimal allocation for our

baseline parameters. Finally, we derive how the optimal allocation can be decentralized.

7.8.1 Characterizing the optimal allocation

We introduce the following notations: NAt (respectively NN

t ) denotes the mass of auto-

mated (respectively non-automated) firms, LAt (respectively LNt ) is the mass of low-skill

workers hired in automated (respectively non-automated) firms, and HP,At (respectively

HP,Nt ) is the mass of high-skill workers hired in production in automated (respectively

non-automated) firms. The social planner problem can then be written as (we write the

102

0 100 200 300 400Year

1

2

3

4

5P

er c

ent

Panel A: Growth rate of high-skill wages

N(1)=2baseline: N(1)=1N(1)=0.5

0 100 200 300 400Year

0

1

2

3

Per

cen

t

Panel B: Growth rate of low-skill wages

N(1)=2baseline: N(1)=1N(1)=0.5

0 100 200 300 400Year

0

20

40

60

80

100

Per

cen

t

Panel C: Share of automated products

N(1)=2baseline: N(1)=1N(1)=0.5

0 100 200 300 400Year

1

2

3

4

5

Per

cen

t

Panel D: Growth rate of high-skill wages

G(1)=G ss

G(1)=0.3baseline: G(1)=0.001

0 100 200 300 400Year

0

1

2

3

Per

cen

t

Panel E: Growth rate of low-skill wages

G(1)=G ss

G(1)=0.3baseline: G(1)=0.001

0 100 200 300 400Year

0

20

40

60

80

100

Per

cen

t

Panel F: Share of automated products G

G(1)=G ss

G(1)=0.3baseline: G(1)=0.001

Figure 22: Comparative statics with respect to the initial number of products N0 and theinitial share of automated products G0

Lagrange multipliers next to each constraint):

max

∫ ∞0

e−ρtC1−θt

1− θ

such that

λt : Ct +Xt = F(LAt , H

P,At , Xt, L

Nt , H

P,Nt , NA

t , NNt

),

with

F ≡

(NAt

) 1σ

((ϕX

ε−1ε

t +(LAt) ε−1

ε

) εε−1

β (HP,At

)1−β)σ−1

σ

+(NNt

) 1σ

((LNt)β (

HP,Nt

)1−β)σ−1

σ

σσ−1

,

wt : LAt + LNt = L,

vt : HP,At +HP,N

t +HAt +HD

t = H,

ζt :·NN

t = γ(NAt +NN

t

)HDt − η

(NAt

)κ (NNt +NA

t

)κ−κ (HAt

)κ (NNt

)1−κ,

ξt :·NA

t = η(NAt

)κ (NNt +NA

t

)κ−κ (HAt

)κ (NNt

)1−κ,

103

HDt ≥ 0.

The first order condition with respect to Ct gives

C−θt = λt

To denote the ratio of the Lagrange parameter of each constraint with respect to λt

(that is the shadow value expressed in units of final good at time t), we remove the tilde

(hence wLt ≡ wLt/λt is the shadow wage of low-skill workers).

The first order conditions with respect to Xt implies that

∂F

∂Xt

= 1, (131)

so that the shadow price of a machine must be equal to 1. First order conditions with

respect to LAt , LNt , HP,At , HP,N

t lead to

wLt =∂F

∂LAt=

∂F

∂LNtand wHt =

∂F

∂HP,At

=∂F

∂HP,Nt

, (132)

so that labor inputs are paid their marginal product in aggregate production. This is

not the case in the competitive equilibrium, where labor inputs are paid their marginal

product in the production of intermediates, while intermediates themselves are priced

with a mark-up as they are provided by a monopolist. It is easy to show that for a given

HPt , the optimal provision of machines and allocation of high-skill and low-skill workers

across firms can be obtained if the purchase of all intermediate inputs is subsidized by

at rate 1/σ (a lump-sum tax finances the subsidy).

The first-order conditions with respect to NNt and NA

t are given by:

ρζt −·

ζt =λt

∂F∂NN

t+ ζtγH

Dt +

(ξt − ζt

)η(HAt

)κ (NNt

)−κ×(NAt

)κ ((1− κ)NN

t + (1− κ)NAt

) (NNt +NA

t

)κ−κ−1, (133)

ρξt −·

ξt =λt

∂F∂NA

t+ ζtγH

Dt +

(ξt − ζt

)η(HAt

)κ×(NNt

)1−κ (NAt

)κ−1 (κNN

t + κNAt

) (NNt +NA

t

)κ−κ−1. (134)

Interestingly, ∂Ft∂NN

tand ∂F

∂NAt

correspond to the profits realized by a non-automated and

an automated firm respectively in the equilibrium once the subsidy to the use of inter-

104

mediates is implemented. Therefore we denote

πNt =∂Ft∂NN

t

and πAt =∂Ft∂NA

t

Further the (shadow) interest rate is given by rt = ρ + θ·CtCt

= ρ −·λtλt

. Using that

HAt = (1−Gt)Nth

At , we can rewrite (133) and (134) as:

rtζt = πNt + ζtgNt + (ξt − ζt) η (Gt)

κNκt

(hAt)κ

((1− κ) (1−Gt) + (1− κ)Gt) +·ζt, (135)

rtξt = πAt + ζtgNt + (ξt − ζt) η (Gt)

κNκt

(hAt)κ

(1−Gt)

(κ

1−Gt

Gt

+ κ

)+·ξt. (136)

These expressions parallel equations (19) and (20) in the paper. The rental social value

of a non-automated firm (rtζt) consists of the current value of one intermediate (which

equals the profits when the optimal subsidy to the use of intermediates inputs is in

place), its positive impact on the horizontal innovation technology (the productivity of

which is γNt), its positive impact on the automation technology (which results from

the direct externality embedded in the automation technology from the number of firms

diminished by the additional externality coming from the share of automated products),

the expected increase in its value if it becomes automated minus the cost of the resources

required (the difference between these two terms is positive since the automation technol-

ogy is concave) and the change in its value. The rental social value of an automated firms

(rtξt) is the sum of the profits, its impact on horizontal innovation (through the same

externality as non-automated firm), its impact on the automation technology (which

results from two externalities as both the number of firms and the share of automated

products improve the automation technology), and the change in its value.

The first order condition with respect to HDt gives (together with the condition that

HDt ≥ 0):

wHt ≥ ζtγNt, (137)

with equality when HDt > 0. This equation is the counterpart of (23) in the equilibrium

case, it stipulates that when horizontal innovation takes place the social value of a non-

automated intermediate equals the cost of creating one. The first-order condition with

respect to HAt gives:

wHt = (ξt − ζt)κη (Gt)κNκ

t

(hAt)κ−1

. (138)

105

This equation is the counterpart of (21) in the equilibrium case. Everything else given,

ξt − ζt increases with πAt − πNt , which increases with wLt, therefore this equation shows

that automation increases with low-skill wages (everything else given), just as in the

equilibrium case.

7.8.2 System of differential equations and steady state

After having introduced the same variables as in the equilibrium case, one can follow

the same steps and derive a system of differential equation in(nt, Gt, h

At , χt

)which

characterizes the solution (when there is positive growth). Equations (31) and (32) still

hold, while equations (39) and (40) are replaced with

·

h

A

t =

γhAt1−κ

(ωtnt

(ϕ+ (ωtnt)

1µ

)−µπAtvt

+ 1−κ+(κ−κ)(1−Gt)κ

hAt

)−ηκGκt

1−κ

(htA)κ(

1− ωtnt(ϕ+ (ωtnt)

1µ

)−µ)πAtvt

+ ηGκt

(htA)κ+1

+ 1−κ1−κg

Nt h

At

,

(139)·χt = χt

(γωtnt

(ϕ+ (ωtnt)

1µ

)−µ πAtvt

+ γ1− κ+ (κ− κ) (1−Gt)

κhAt − ρ− (θ − 1)ψgNt

).

gNt is still given by (48),πAtvt

, HPt and ωt are now given by

πAtvt

=ψ(ϕ+ (ωtnt)

1µ

)µHPt

Gt

(ϕ+ (ωtnt)

1µ

)µ+ (1−Gt)ωtnt

,

HPt =

(1− β)1θ β

β1−β ( 1

θ−1)χ

1θt

(Gt

(ϕ+ (ωtnt)

1µ

)µ+ (1−Gt)ωtnt

)ψ( 1θ−1)+1

Gt

((1− β)ϕ+ (ωtnt)

1µ

)(ϕ+ (ωtnt)

1µ

)µ−1

+ (1−Gt)ωtnt

,

ωt =

β1

1−β HPt

L

(Gt

(ϕ+ (ωtnt)

1µ

)µ−1

(ωtnt)1−µµ + (1−Gt)

)×(Gt

(ϕ+ (ωtnt)

1µ

)µ+ (1−Gt)ωtnt

)ψ−1

β(1−σ)

1+β(σ−1)

,

which replace (45), (47) and (49).

One can then solve for a steady state of this system with G∗ > 0 (and(gN)∗> 0 so

106

that n∗ = 0). (107) and (109) still apply, but (108) is replaced with

hA∗ =κ

γ

ρ+ (θ − 1)ψgN∗

1− κ+ (1−G∗) (κ− κ), (140)

and (111) with

f sp(gN∗)≡ ρ+ (θ − 1)ψgN∗

ψHP∗

(hA∗)1−κ

ηκ (G∗)κ−1+

1

γ

,

which is obtained by fixing·

h

A

t = 0 in (139) using (107) and (140). For gN∗ large enough

(but finite—and, in particular smaller than γH), HP∗ is arbitrarily small, while for the

same value G∗ and hA∗ are bounded below and above. As before, this establishes that

for gN∗ large enough f sp(gN∗)> 1. Furthermore f sp (0) = f (0), therefore condition 25

is also a sufficient condition for the existence of a steady state with positive growth and

G∗ > 0 for the system of differential equations.

7.8.3 Decentralizing the optimal allocation

We have already seen that the “static” optimal allocation given HPt is identical to the

equilibrium allocation once a subsidy to the use of intermediates 1/σ is in place. The

“dynamic” part of the problem consists of the allocation of high-skill workers across the

two types of innovation and production. Therefore, we postulate that a social planner

can decentralize the optimal allocation using the subsidy to the use of intermediate

inputs and subsidies (or taxes) for high-skill workers hired in automation(sAt)

and in

horizontal innovation(sHt). Let us consider such an equilibrium and introduce the

notations ΩAt ≡ 1− sAt and ΩH

t similarly defined. In this situation, the law of motion for

the private value of an automated firm, V At , is still given by (19), for a non-automated

firm it obeys:

rtVNt = πNt − ΩA

t wHtht + η (Gt)κNκ

t

(hAt)κ (

V At − V N

t

)+·VN

t , (141)

instead of (20), the first-order condition for automation is given by:

κη (Gt)κNκ

t

(hAt)κ−1 (

V At − V L

t

)= ΩA

t wHt, (142)

107

instead of (21), while the free entry condition, when gNt > 0, is given by

γNtVNt = ΩH

t wHt, (143)

instead of (23). For ΩAt and ΩH

t to decentralize the optimal allocation it must be that

these 4 equations hold together with (135), (136), (137) and (138).

Using (137) and (143), we then get that ΩHt must satisfy

ΩHt ζt = V N

t , (144)

similarly, using (138) and (142), we get

ΩAt (ξt − ζt) = V A

t − V Lt . (145)

Plugging (144) and (145) in (141), we get that

rtζt =πNtΩHt

− ΩAt

ΩHt

wHtht + η (Gt)κNκ

t

(hAt)κ ΩA

t

ΩHt

(ξt − ζt) +

·ΩH

t

ΩHt

ζt +·ζt. (146)

Similarly, using (145) and the difference between (19) and (141) gives:

rt (ξt − ζt) =πAt − πNt

ΩAt

+wHtht−η (Gt)κNκ

t

(hAt)κ

(ξt − ζt)+

·ΩA

t

ΩAt

(ξt − ζt)+·ξt−

·ζt. (147)

Combining (146) with (135), using (138) and (137) and the definition of ΩAt and ΩH

t , we

get:

·sH

t =γπNtvtsHt −

(1− sHt

)gNt

+γhAtκ

((1− sAt

)(1− κ) +

(1− sHt

)(κ (1−Gt) + κGt − 1)

) . (148)

Similarly combining (147) with the difference between (136) and (135) and using

(137) gives:

·sA

t

(hAt

)1−κ

ηGκt

= κπAt − πNt

vtsAt − κ

(1− sAt

)hAt

1−Gt

Gt

. (149)

108

Therefore, in steady state, we have

sA∞ =κhA∞ (1−G∞)

κψHP∞ + κhA∞ (1−G∞)

≥ 0.

Note from (149) that the share of automated products, sAt , must always be non-negative,

otherwise it cannot converge to a positive value, therefore sAt ≥ 0 everywhere (and in

fact > 0 if κ 6= 0). Furthermore, if κ = 0, sAt = 0 everywhere, the only externality

in automation comes from the total number of products, therefore the equilibrium fea-

tures the optimal amount of automation investment (when the monopoly distortion is

corrected and the optimal subsidy to horizontal innovation is implemented).

(148) gives the steady state value of the subsidy to horizontal innovation as:

sH∞ = 1−γhA∞ (1− κ)

(1− sA∞

)κgN∞ + γhA∞ (1− κ (1−G∞)− κG∞)

.

In addition, knowing that sAt > 0, imposes that sHt > 0—as sHt < 0 would lead to·sH

t < 0.

7.8.4 Transitional dynamics for the social planner case

Figure 23 plots the transitional dynamics for the optimal allocation in our baseline case

(which features κ = 0) and in the case where κ = 0 analyzed in Figure 7. As shown

in Panel A and C, the economy also goes through three phases as a higher (shadow)

low-skill wage leads to more automation over time and a transition from a small share

to a high share of automated products. Relative to Figure 3.A and Figure 7.A, the

overall dynamics look quite similar but the growth rates are higher in the social planner

case, and the transition to phase 2 now happens roughly at the same time with and

without the automation externality, while in the equilibrium it is considerably delayed

in the presence of the externality (as, effectively, the productivity of the automation

technology is initially very low). In both cases, the social planner maintains a positive

subsidy to horizontal innovation. When κ = 0 (without the automation externality), the

subsidy to automation is 0, while when κ > 0 there is a positive subsidy to automation,

which is the largest in Phase 1. This subsidy explains why Phase 2 now starts at around

the same time.

109

0 100 200 300 400Year

0

2

4

6

8

Per

cen

t

Without the Automation ExternalityPanel A: Growth Rates of Wages and GDP

gGDP

gwL

gwH

0 100 200 300 400Year

0

2

4

6

8

Per

cen

t

With the Automation ExternalityPanel B: Growth Rates of Wages and GDP

gGDP

gwL

gwH

0 100 200 300 400Year

0

20

40

60

80

100

Per

cen

t

Panel C: Research Subsidies

automationhorizontal innovation

0 100 200 300 400Year

0

20

40

60

80

100

Per

cen

t

Panel D: Research Subsidies

automationhorizontal innovation

Figure 23: Transitional Dynamics in the Social Planner Case. Panel A and B, baseline case.Panel C and D, with κ = 0.5

7.9 Alternative model with automation at the entry-stage

To highlight that the evolution of the economy through three phases does not depend on

our assumption that new products are born non-automated, we present in this section a

model where, instead, we assume that automation can only take place at the entry stage.

That is, when a new firm is born, it can hire hAt workers to automate it, in which case

it is successful with probability min(η(Nth

At

)κ, 1)

(we abstract from the automation

externality for simplicity). Ex-ante a firm does not know whether it will succeed or not,

therefore, the free-entry condition can now be written as

wHt ≥ γNtVt,

where

Vt = min(η(Nth

At

)κ, 1)V At +

(1−min

(η(Nth

At

)κ, 1))V Nt − wHthAt .

is the expected value of a new firm. Since we used similar functional forms we have that

hAt obeys (21) unless κη1κNt

(V At − V L

t

)> wHt, in which case Nth

At = η−

1κ . Afterward a

110

firm never becomes automated so that the law of motion for the value of an automated

and a non-automated firms both follow (19). In addition, the law of motion for Gt is

now given by·Gt = gNt

(η(Nth

At

)κ −Gt

).

The resolution of the model follows the same steps as in the baseline case, and under

the appropriate condition on the discount rate, there exists an asymptotic steady state

with gNt > 0.

An important difference is that G∗ may be equal to 1 since all new products may

choose to be automated in steady state. In fact, one can derive that hA∗ = min(η−

1κ , κ

1−κ1γ

).

Therefore G∗ < 1, if and only if η(

κ1−κ

1γ

)κ< 1. When G∗ < 1, we will have that

G∞ = G∗ < 1, so that, following Proposition 2,

gwL∞ =1

1 + β (σ − 1)gwH∞ .

On the contrary, if G∗ = 1, then G∞ = 1, and following Proposition 2, we get that

gwL∞ = gwH∞ /ε.

Figure 24 draws the transitional dynamics for the same parameters as in the baseline

case (even though the automation technology parameters have a different meaning here).

These parameters satisfy η(

κ1−κ

1γ

)κ< 1, and the figure shows that the economy goes

through three phases as in our baseline model.

7.10 Supply response in the skill distribution: details

The supply of low-skill and high-skill labor are now endogenous. This does not affect

(11) which still holds. (10) also holds with Lt replacing L and knowing that HPt obeys

(18) but with Ht instead of H in the right-hand side. Because workers are ordered such

that a worker with a higher index j supplies relatively more high-skill labor, then at all

point in times there exists a threshold jt such that workers j ∈(0, jt

)supply low-skill

labor and workers j ∈(jt, 1

)supply high-skill labor. As a result, we get that the total

mass of low-skill labor is:

Lt = lHjt, (150)

111

0 100 200 300 400Year

0

0.5

1

1.5

2

2.5

3

3.5

4

Per

cen

tGrowth Rates of Wages and GDP

gGDP

gwL

gwH

0 100 200 300 400Year

0

2

4

6

8

10

12

14

16

18

Per

cen

t

Autom./GDP (left)Hori./GDP (left)

0 100 200 300 400Year

0

20

40

60

80

100

Per

cen

t

Research Expenditures and G

G (right)

Figure 24: Transitional Dynamics for an alternative model where automation only happensat entry. Baseline parameters.

and the mass of high-skill labor is

Ht = H

(1− j

1+qq

t

)≤ H. (151)

The cut-off jt obeys lHwLt = Γ(jt)wHt, that is

jt =

(q

1 + q

lwLtwHt

)q. (152)

jt decreases as the skill premium increases and q measures the elasticity of jt with respect

to the skill premium.

7.10.1 Asymptotic growth rates

We consider processes(Nt, Gt, H

Pt

)such that gNt , Gt and HP

t admit strictly positive

limits. Plugging (152) and (150) in (10), we get:

wHtwLt

= l

(1− ββ

H

HPt

(q

1 + q

)q Gt + (1−Gt)(1 + ϕwε−1

Lt

)−µGt

(1 + ϕwε−1

Lt

)−1+ (1−Gt)

(1 + ϕwε−1

Lt

)−µ) 1

1+q

,

(153)

112

which together with (11) determines wHt and wLt for given(Nt, Gt, H

Pt

). From then on

the reasoning follows that of Appendix 7.2.1. First, we derive that wL∞ > 0, such that

gwH∞ = gGDP∞ = ψgN∞, and that we must have gwL∞ < gwH∞ , such that j∞ = 0. Second, we

study the asymptotic behavior of wLt both when ε <∞ and when ε =∞.Case with ε <∞. Plugging (153) in (11) gives wLt in function of Nt, Gt and HP

t :

wLt =

σ−1σβ

1+βq1+q

((1− β)1+q

q

) (1−β)q1+q 1

l1−β

(HPt

H

) 1−β1+q

N1

σ−1

×(Gt(1+ϕwε−1

Lt )µ−1

+(1−Gt)) 1−β

1+q

(Gt(ϕwε−1Lt +1)

µ+1−Gt)

11−σ+

1−β1+q

, (154)

which replaces (59). It is direct that when G∞ < 1, we obtain (50). In this case, we

further have

gj∞ = q (gwL∞ − gwH∞ ) = − qβ (σ − 1)

1 + q + β (σ − 1)gGDP∞ . (155)

Case with ε =∞. In this case, (154) becomes

wLt =

σ−1σβ(

1+qq

)(1−β)q (HPt

lH

)1−βN

1σ−1 (1−Gt)

1−β1+q

×(Gt (ϕwLt)

β(σ−1) + (1−Gt)) 1σ−1− 1−β

1+q, if wLt > ϕ−1,

wLt =σ − 1

σβ

(1 + q

q

)(1−β)q (HPt

lH

)1−β

N1

σ−1 , if wLt < ϕ−1.

Once again, following the steps of Appendix 7.2.1, we get that if G∞ < 1, (50) applies

(and accordingly we also get (155)).

7.10.2 Dynamic system

It is convenient to redefine nt ≡ N−β

(1−β)(1+q)

1+q+β(σ−1)

t , we can then write the entire dynamic

system as a system of differential equations in(nt, Gt, h

At , χt

)with two auxiliary vari-

ables ωt and jt ≡ jtn− q

1+q

t . Equations (31) is now given by

·nt = − β

1− β1 + q

1 + q + β (σ − 1)gNt nt,

113

(32), (39), (40), (45), (47) still apply and equation (48) as well provided that H is

replaced by Ht given by (151). ωt is implicitly defined by:

ωt =

(σ−1σ

) 1+q1−β β

1+βq1−β

((1− β)1+q

q

)qHPt

l1+qH

(Gt

(1 + ϕ (ωtnt)

− 1µ

)µ−1

+ (1−Gt)

)×(Gt

(ϕ+ (ωtnt)

1µ

)µ+ (1−Gt)ωtnt

)ψ(1+q)−1

β(1−σ)

1+q+β(σ−1)

,

which replaces (49) and is a rewriting of (154) and jt is given by

jt =

ωt q

1 + q

β

1− β

Gt

(1 + ϕ (ωtnt)

− 1µ

)µ−1

+ 1−Gt

Gt

(ϕ+ (ωtnt)

1µ

)µ+ (1−Gt)ωtnt

HPt

H

q

1+q

which is derived using (152) and (153).

The steady state for this system involves n∗ = 0 and therefore ω∗and j∗ are positive

constant (so that j∗

= 0: in steady-state all workers are high-skill). As a result H∗ = H,

so that the steady state values of(gN∗, G∗, hA∗, χ∗

)are identical to the baseline case

with H replacing H.

7.11 Machines as a capital stock

7.11.1 Set-up

To avoid repetitions, we already include the taxes of section 4.2, namely, we assume

that there is a tax τm on the rental rate of equipment and a tax τa on high-skill workers

in automation innovations. The solution follows similar steps to the baseline case. We

denote by rt the gross rental rate of machines and by ∆ their depreciation rate, such

that:

rt = rt + ∆. (156)

The Euler equation (24) still applies and the capital accumulation equation is given by

(30). The unit cost of intermediate input i is now given by

c (wL, wH , r, α (i)) =

(w1−εL + α(i)ϕ

(r1−β4wβ4H

)1−ε) β1

1−ε

wβ2H rβ3

ββ11 ββ22 β

β33

(157)

114

instead of (4) where ϕ ≡ ϕε(ββ44 (1− β4)1−β4

)ε−1

. Define µ ≡ β1 (σ − 1) / (ε− 1), we

can then derive the isocost curve as:

N1

1−σσ

σ − 1

wβ2H rβ3

ββ11 ββ22 β

β33

(G

(ϕ(

((1 + τm) r)1−β4 wβ4H

)1−ε+ w1−ε

L

)µ+ (1−G)w

β1(1−σ)L

) 11−σ

= 1.

(158)

The same steps as before allows us to obtain the relative demand for high-skill versus

low-skill workers as:

wHHP

wLL(159)

=

G

(β2 +

β1β4ϕ(

((1+τm)r)1−β4wβ4H

)1−εw1−εL +ϕ

(((1+τm)r)1−β4w

β4H

)1−ε)(

ϕ(

((1 + τm) r)1−β4 wβ4H

)1−ε+ w1−ε

L

)µ+ β2 (1−G)w

β1(1−σ)L

β1

(Gw1−ε

L

(ϕ(

((1 + τm) r)1−β4 wβ4H

)1−ε+ w1−ε

L

)µ−1

+ (1−G)wβ1(1−σ)L

) .

Similarly, taking the ratio of income going to high-skill workers in production over income

going to machines owners, we obtain a relationship linking the gross rental rate of capital

and high-skill wages:

rK

wHHP(160)

=

G

(β3 +

β1(1−β4)ϕ(

((1+τm)r)1−β4wβ4H

)1−ε(1+τm)

(w1−εL +ϕ

(((1+τm)r)1−β4w

β4H

)1−ε))(

ϕ(

((1 + τm) r)1−β4 wβ4H

)1−ε+ w1−ε

L

)µ+β3 (1−G)w

β1(1−σ)L

G

(β2 +

β1β4ϕ(

((1+τm)r)1−β4wβ4H

)1−εw1−εL +ϕ

(((1+τm)r)1−β4w

β4H

)1−ε)(

ϕ(

((1 + τm) r)1−β4 wβ4H

)1−ε+ w1−ε

L

)µ+ β2 (1−G)w

β1(1−σ)L

.

7.11.2 Effect of technology on wages

First note that one can rewrite (159)

wHHP

wLL=G(β2 + β1β4

Φ1+Φ

)(Φ + 1)µ + β2 (1−G)

β1

(G (Φ + 1)µ−1 + (1−G)

) , (161)

115

where we defined

Φ ≡ ϕ

(wL

((1 + τm) r)1−β4 wβ4H

)ε−1

= ϕ

((wL

(1 + τm) r

)1−β4 (wLwH

)β4)ε−1

.

In (161), the RHS is increasing in wL and decreasing in wH/wL for given G, r. Therefore,

this equation defines the relative demand curve in the wL, wH space as rotating counter-

clockwise (when G > 0) when wL increases. Plugging (161) in (158) then defines wL

uniquely as a function of N , G, r and HP . We can then derive the effect of changes in

G and N for given HP and r (i.e. when K is perfectly elastically supplied) on wages,

the skill premium and the labor share as follows:

Proposition 10. Consider the equilibrium (wL, wH) determined by equations (161) and

(158). Assume that ε <∞, it holds that

A) An increase in the number of products N (keeping G and HP constant) leads to

an increase in both high-skill (wH) and low-skill wages (wL). Provided that G > 0, an

increase in N also increases the skill premium wH/wL and decreases the labor share for

H ≈ HP .

B) An increase in the share of automated products G (keeping N and HP constant)

increases the high-skill wages wH , the skill premium wH/wL and decreases the labor

share for H ≈ HP . Its impact on low-skill wages is ambiguous.

Proof. One can rewrite (158) as:

N1

1−σσ

σ − 1

rβ3

ββ11 ββ22 β

β33

(wHwL

)β2wβ2L

G

(ϕ(

((1 + τm) r)1−β4 wβ4H

)1−ε+ w1−ε

L

)µ+ (1−G)w

β1(1−σ)L

1

1−σ

= 1.

(162)

Using that (161) establishes wHwL

as an increasing function of wL otherwise independent

of N , we get that (162) implies that wL and therefore wH/wL (when G > 0) and wH

itself must increase in N .

(161) also establishes that wHwL

increases in G for a given wL. Therefore if wL is

increasing in G, then it is direct that wHwL

and wH both also increase in G. Assume on

the contrary that wL decreases in G, then in (158) the direct effect of an increase in G

is to decrease the LHS (because

(ϕ(

((1 + τm) r)1−β4 wβ4H

)1−ε+ w1−ε

L

)µ> w

β1(1−σ)L ), in

addition an increase in G would reduce wL which further reduces the LHS. To maintain

116

the inequality, it must be that wH increases. Therefore in this case too, wH increases in

G and so does wH/wL.

This model is isomorphic to the previous one when β4 = β3 = 0 (with ϕ ((1 + τm) r)1−ε

replacing ϕ), therefore the impact of a change of G on wL is also ambiguous.

The labor share is now given by

LS =wLL+ wHH

Y + (1 + τa)wH (H −HP ).

As before profits are a share 1σ

of output so that

Y =σ

σ − 1

(wLL+ wHH

P + rK + Tm), (163)

where Tm denotes the tax proceeds from the tax on equipment. We have

TmwHHP

(164)

=

Gτmβ1(1−β4)ϕ

(((1+τm)r)1−β4w

β4H

)1−ε(1+τm)

(w1−εL +ϕ

(((1+τm)r)1−β4w

β4H

)1−ε)(ϕ(

((1 + τm) r)1−β4 wβ4H

)1−ε+ w1−ε

L

)µG

(β2 +

β1β4ϕ(

((1+τm)r)1−β4wβ4H

)1−εw1−εL +ϕ

(((1+τm)r)1−β4w

β4H

)1−ε)(

ϕ(

((1 + τm) r)1−β4 wβ4H

)1−ε+ w1−ε

L

)µ+ β2 (1−G)w

β1(1−σ)L

.

Then, we obtain:

LS =wLL+ wHH

σσ−1

(wLL+ wHHP + rK + Tm) + (1 + τa)wH (H −HP ). (165)

Assume that H = HP , then we get that

LS =σ − 1

σ

(1 +

rK + TmwLL+ wHH

)−1

.

117

Using (159), (160), (164) and the definition of Φ, we obtain:

rK + TmwLL+ wHH

(166)

=G(β3 + β1 (1− β4) Φ

1+Φ

)(Φ + 1)µ + β3 (1−G)

G(β2 + β1β4 + β1

(1−β4)Φ+1

)(Φ + 1)µ + (β1 + β2) (1−G)

=β3

β2 + β1

+1

β2 + β1

G (1− β4) β1

G(β2 + β1β4 + β1

(1−β4)Φ+1

)+ (β1 + β2) (1−G) (Φ + 1)−µ

Φ

1 + Φ.

This expression is increasing in Φ. From (161), Φ moves like wH/wL, therefore the labor

share decreases in N (the opposite of wH/wL) when H ≈ HP (this result may not extend

if HP is far from H when β4 is close to 1).

Further, we can rewrite (161) as:

wHHP

wLL=β2

β1

+(β2 + β1β4)

β1

GΦ (Φ + 1)µ−1

G (Φ + 1)µ−1 + (1−G).

We have already derived that an increase inG increases wH/wL, therefore, this expression

shows that it will increase GΦ(Φ+1)µ−1

G(Φ+1)µ−1+(1−G). We can then rearrange terms in (166) and

write:

rK + TmwLL+ wHH

=β3

β2 + β1

+(1− β4) β1

β2 + β1

(β1β4 + β2 + (β2 + β1)

G (Φ + 1)µ−1 + (1−G)

GΦ (1 + Φ)µ−1

)−1

.

The right hand side is an increasing function of GΦ(Φ+1)µ−1

G(Φ+1)µ−1+(1−G), which ensures that the

labor share decreases in G when H ≈ HP .

7.11.3 Asymptotic behavior

The asymptotic behavior is in line with Proposition 2 but the fact that automation

now replaces low-skill workers with a Cobb-Douglas aggregate of capital and high-skill

workers limit the ratio between the growth rate of high-skill and low-skill wages. In

addition, we here need to consider the long-run behavior of the gross rental rate r. Since

r is determined by the Euler equation, then on a path where consumption growth is

asymptotically constant, then r is also asymptotically constant (see (156)). We focus on

118

the case where G∞ ∈ (0, 1) (although results analogous to those in Proposition 2 could

be derived when G∞ ∈ 0, 1) and prove:

Proposition 11. Consider four processes [Nt]∞t=0, [Gt]

∞t=0,

[HPt

]∞t=0

and [rt]∞t=0 where(

Nt, Gt, HPt , rt

)∈ (0,∞) × [0, 1] × (0, H] × (0,∞) for all t. Assume that Gt, g

Nt , HP

t

and rt all admit positive and finite limits with G∞ ∈ (0, 1). Then the asymptotic growth

rate of high-skill wages wHt and output Yt are

gwH∞ = gY∞ = gN∞/ [(σ − 1) (β2 + β1β4)] , (167)

and the asymptotic growth rate of low-skill wages is

gwL∞ =1 + (σ − 1) β1β4

1 + (σ − 1) β1

gwH∞ . (168)

Proof. For simplicity we assume that the limits gwH∞ , gwL∞ and gY∞ exist (although we could

show that formally as we did in Appendix 7.2.1). Suppose that gwL∞ ≤ β4gwH∞ . Then

Φt must either tend toward a positive constant or toward 0, in either case (161) implies

that gwL∞ = gwH∞ , which is a contradiction as β4 < 1. Hence it must be that gwL∞ > β4gwH∞ ,

which ensures that Φt →∞. Using this in (158), we obtain:

wβ2+β1β4Ht ∼

t→∞

σ − 1

σ

ββ11 ββ22 β

β33 (G∞ϕ)

1σ−1

(1 + τm)(1−β4)β1 rβ3+(1−β4)β1∞

N1

σ−1

t .

This establishes gwH∞ = gN∞/ [(σ − 1) (β2 + β1β4)], from which we can obtain that gY∞ =

gwH∞ = gN∞/ [(σ − 1) (β2 + β1β4)] (using that HPt admits a positive limit).

Moreover (161) now implies

wHtHP∞

wLtL∼t→∞

G∞ (β2 + β1β4) Φµt

β1 (1−G∞),

=⇒ w1+β1(σ−1)Lt ∼

t→∞

β1 (1−G∞) (1 + τm)(1−β4)β1(σ−1) r(1−β4)β1(σ−1)∞ HP

∞G∞ (β2 + β1β4)ϕµL

w1+β4β1(σ−1)Ht ,

which implies (168). Since 1+(σ−1)β1β41+(σ−1)β1

> β4, we verify that gwL∞ > β4gwH∞ .

7.11.4 Dynamic equilibrium

We can solve for the dynamic equilibrium as in the baseline model. The long-run

elasticity of output with respect to the number of products is now given by ψ ≡

119

1/ [(σ − 1) (β2 + β1β4)]. We then introduce the same normalized variables as in the

baseline model: V At , V N

t , πAt , πNt , hAt , ct and vt. We also introduce Yt ≡ YtN−ψ and

Kt ≡ KtN−ψ. Finally we now define

nt ≡ N− 1−β4

1+β1(σ−1)β1

β2+β1β4t

and

ωt ≡

(N

1−β4(σ−1)(β2+β1β4)(1+β1(σ−1))

t

r1−β4t wβ4HtwLt

)β1(σ−1)

,

so that (wL

r1−β4t wβ4H

)β1(1−σ)

= ωtnt.

The transitional dynamics can then be expressed as a system of differential equations

in xt ≡(nt, Gt, Kt, h

At , vt, ct

)where the first three variables are state variables and the

last three control variables.

Equation (41) still applies, therefore, we get using (157) that

πAt =(σ − 1)σ−1

σσ

(ϕ(

((1 + τm) rt)1−β4 wβ4H

)1−ε+ w1−ε

L

)µ(wβ2H r

β3

ββ11 ββ22 β

β33

)1−σ

Yt.

We can rewrite this as

πAt =(σ − 1)σ−1

σσ

(ββ11 β

β22 β

β33

)σ−1 (r

(1−β4)β1+β3t vβ2+β4β1

t

)1−σ (ϕ (1 + τm)(1−β4)(1−ε) + (ωtnt)

1µ

)µYt.

(169)

We can derive πNt similarly and we find

πNt = ωtnt

(ϕ (1 + τm)(1−β4)(1−ε) + (ωtnt)

1µ

)−µπAt (170)

(31) is now replaced by

·nt = − 1− β4

1 + β1 (σ − 1)

β1

β2 + β1β4

gNt nt. (171)

(32) still applies and so does (33). Because of the automation tax (34) is replaced by

(rt − (ψ − 1) gNt

)V Nt = πNt + ηGκ

t

(hAt

)κ (V At − V N

t

)− (1 + τa) vtht +

·

V Nt (172)

120

and (35) by

κηGκt

(hAt

)κ−1 (V At − V N

t

)= (1 + τa) vt. (173)

Combining (170), (172), (173) and (23) with equality, we now obtain:

·vt = vt

(rt −∆− (ψ − 1) gNt − γωtnt

(ϕ (1 + τm)(1−β4)(1−ε) + (ωtnt)

1µ

)−µ πAtvt− γ (1 + τa)

1− κκ

hAt

).

(174)

Following the same steps as those used to derive (39), we now obtain:

·

hAt (175)

= γhAt

1− κ

(ωtnt

(ϕ (1 + τm)(1−β4)(1−ε) + (ωtnt)

1µ

)−µ πAtvt

+ (1 + τa)1− κκ

hAt

)

−κηGκ

t

(hAt

)κ(1− κ) (1 + τa)

(1− ωtnt

(ϕ (1 + τm)(1−β4)(1−ε) + (ωtnt)

1µ

)−µ) πAtvt

+ ηGκt

(hAt

)κ+1

+κ

1− κ

(η(hAt

)κ+1

Gκ−1t (1−Gt)− gNt hAt

)Further, (24) still applies and we can rewrite it as:

·ct =

ctθ

(rt −

(ρ+ ∆ + θψgNt

)). (176)

Finally, we can rewrite (30) as

·

Kt = Yt − ct −(∆ + ψgNt

)Kt (177)

Equations (171), (32), (174), (175), (176) and (177) form a system of differential

equations which depend on Yt, πAt , rt and gNt .

(158) implies

σ

σ − 1

vβ2+β4β1 rβ3+β1(1−β4)

ββ11 ββ22 β

β33

(G(ϕ (1 + τm)(1−β4)(1−ε) + (ωtnt)

1µ

)µ+ (1−G)ωtnt

) 11−σ

= 1,

121

so that

r =

[σ − 1

σ

ββ11 ββ22 β

β33

vβ2+β4β1

(G(ϕ (1 + τm)(1−β4)(1−ε) + (ωtnt)

1µ

)µ+ (1−G)ωtnt

) 1σ−1

] 1β3+β1(1−β4)

,

(178)

which defines r as a function of xt and ωt. (160) can be written as:

HPt =

rtKt

vt

Gt

(β2 + β1β4ϕ(1+τm)(1−β4)(1−ε)

(ωtnt)1µ+(1+τm)(1−β4)(1−ε)

)(ϕ (1 + τm)(1−β4)(1−ε) + (ωtnt)

1µ

)µ+ β2 (1−Gt)ωtnt

Gt

(β3 + β1(1−β4)ϕ(1+τm)(1−β4)(1−ε)−1

(ωtnt)1µ+ϕ(1+τm)(1−β4)(1−ε)

)(ϕ (1 + τm)(1−β4)(1−ε) + (ωtnt)

1µ

)µ+ β3 (1−Gt)ωtnt

,

(179)

which gives, together with (178), HPt as a function of xt and ωt. gNt still obeys (48),

which then defines it as a function of xt and ωt.

Combine (163), (159), (160) and (164) to obtain:

Y

wHHP

=

σσ−1

(G

(ϕ(

((1 + τm) r)1−β4 wβ4H

)1−ε+ w1−ε

L

)µ+ (1−G)w

β1(1−σ)L

)G

(β2 +

β1β4ϕ(

((1+τm)r)1−β4wβ4H

)1−εw1−εL +ϕ

(((1+τm)r)1−β4w

β4H

)1−ε)(

ϕ(

((1 + τm) r)1−β4 wβ4H

)1−ε+ w1−ε

L

)µ+ β2 (1−G)w

β1(1−σ)L

,

which we can rewrite as

Yt =σ

σ − 1

[Gt

(ϕ (1 + τm)(1−β4)(1−ε) + (ωtnt)

1µ

)µ+ (1−G)ωtnt

]vtH

Pt

G

(β2 + β1β4ϕ(1+τm)(1−β4)(1−ε)

ϕ(1+τm)(1−β4)(1−ε)+(ωtnt)1µ

)(ϕ (1 + τm)(1−β4)(1−ε) + (ωtnt)

1µ

)µ+ β2 (1−G)ωtnt

.

(180)

This expression, with the previous equations, gives Yt as a function of xt and ωt. (169)

then ensures that πAt is defined as a function xt and ωt.

Finally, from (159) we obtain:

ωt =

(vtrt

)1−β4 HPt

Lβ1

(G(ϕ (1 + τm)(1−β4)(1−ε) (ωtnt)

− 1µ + 1

)µ−1

+ (1−G)

)G

(β2 + β1β4ϕ(1+τm)(1−β4)(1−ε)

(ωtnt)1µ+ϕ(1+τm)(1−β4)(1−ε)

)(ϕ (1 + τm)(1−β4)(1−ε) + (ωtnt)

1µ

)µ+ β2 (1−G)ωtnt

β1(1−σ)

1+β1(σ−1)

,

(181)

122

which implicitly defines ωt as a function of xt. Hence, together with (178), (179), (48),

(180), (169) and (181), the system formed by (171), (32), (174), (175), (176) and (177)

describes the dynamic equilibrium. We then obtain

Proposition 12. Assume that

ρ

((1 + τa)

κ

κκ (1− κ)1−κ η

(ρ

γ

)1−κ

+1

γ

)< ψH (182)

is satisfied, then the economy admits a steady-state(n∗, G∗, K∗, hA∗, v∗, c∗

)with n∗ = 0,

G∗ ∈ (0, 1) and gN∗ > 0. gN∗, G∗ and hA∗ are independent of τm.

Proof. As before, we directly get that in a steady-state with gN∗ > 0, we must have

n∗ = 0. (181) then implies that ω∗ is a constant defined by

ω∗ =

[(v∗

r∗

)1−β4 HP∗

L

β1 (1−G∗) (1 + τm)(1−β4)(σ−1)

G∗ (β2 + β1β4)ϕµ

] β1(1−σ)1+β1(σ−1)

.

This guarantees that in such a steady-state, wLt ∼ ω∗ 1β1(1−σ) r∗1−β4 v∗β4N

1+β4β1(σ−1)1+β1(σ−1)

1(σ−1)(β2+β1β4)

t

such that gwL∞ = 1+β4β1(σ−1)1+β1(σ−1)

gwH∞ as stipulated in Proposition 11.

In addition, (176) implies that in steady-state,

r∗ = ρ+ ∆ + θψgN∗. (183)

(179) implies that

HP∗ =r∗K∗

v∗(β2 + β1β4)

(β3 + β1 (1− β4)). (184)

Then (180) implies that

Y ∗ =σ

(σ − 1) (β2 + β1β4)v∗HP∗. (185)

We then get that (169) implies that

πA∗

v∗=

(σ − 1)σ−2

σσ−1

(ββ11 β

β22 β

β33

)σ−1

(β2 + β1β4)

(r∗(1−β4)β1+β3 v∗β2+β4β1 (1 + τm)(1−β4)β1

)1−σϕµHP∗.

(186)

123

(178) gives

r∗ =

[σ − 1

σ

ββ11 ββ22 β

β33

v∗β2+β4β1

(G∗ϕµ)1

σ−1

(1 + τm)(1−β4)β1

] 1β3+β1(1−β4)

. (187)

Therefore (186) simplifies intoπA∗

v∗= ψ

HP∗

G∗, (188)

just as in the baseline model. Then (174) and (183) together imply that

hA∗ =κ

γ (1 + τa) (1− κ)

(ρ+ ((θ − 1)ψ + 1) gN∗

). (189)

This defines hA∗ as an increasing function of gN∗. Further, in steady-state G∗ still obeys

(107) and HP∗ obeys (109), which imply that G∗ and HP∗ also be defined as function

of gN∗.

(188), (175), (107), (189) then lead to

1− κκ

γG∗ (1 + τa) hA∗

ψHP∗

(1 + τaκηGκ

t

(hA∗)1−κ

+1

γ

)= 1, (190)

which up to the term 1+τa is the same as (111) in the baseline case. Therefore following

the same reasoning, there exists a steady-state with gN∗ > 0 and G∗ ∈ (0, 1) as long as

(182) is satisfied. As (190), (107), (109) and (189) are independent of τm, so are gN∗,

hA∗ (now given by (189)), G∗ (given by (107)) and HP∗ (given by (109)).

We further obtain r∗ through (183), which must be independent of τm as well. We

then get v∗ through (187) as

v∗ =

[σ − 1

σ

ββ11 ββ22 β

β33

r∗β3+β1(1−β4)

(G∗ϕµ)1

σ−1

(1 + τm)(1−β4)β1

] 1β2+β4β1

.

We then get K∗ through (184) and c∗ from (177) which, using (185), implies:

c∗ =σ

(σ − 1) (β2 + β1β4)v∗HP∗ −

(∆ + ψgN

∗)K∗.

Further if τa = τm = 0, gN∗, G∗, hA∗ are determined by the same equations are in

the baseline model except that the definition of ψ has changed. It is then direct that

124

Proposition 4 extends to this case.

7.12 Quantitative Exercise

We choose parameters to minimize the log-deviation of predicted and observed variables

for the four time paths of the skill-premium, the labor-share of GDP, stock of equipment

over GDP and an index of GDP per hours worked. That is, for a given set of parameters

b the model produces predicted output of Yi =Yi,t

Tit=1

for each of these four paths from

1963 and until 2007 for the labor share, skill-premium, and GDP per hour, and 2000

for equipment over GDP (due to data limitations from Cummins and Violante, 2002).

We let Y (b) = Yi(b)4i=1 as the combined vector of these paths and make explicit the

dependency on the parameters b. Y is the corresponding vector of actual values. We

then solve :

minb(log(Y (b))− log(Y ))′W (log(Y (b))− log(Y )),

where W is a diagonal matrix of weights. In a previous version of the paper (Hemous

and Olsen, 2016) we articulated a stochastic version of our model by introducing auto-

correlated measurement errors. Here we choose a much simpler approach and simply

choose “reasonable” weights based on how easily the model matches the path. In partic-

ular, the diagonal elements are 4 for the skill-premium, though 10 for the first 5 years,

10 for the labor share, 1 for GDP/hours and 2 for equipment over GDP. For a given

starting value of b we then run 12 estimations based on“nearby”randomly chosen param-

eters. We choose the best fit of these 13 (12 plus the original starting point), take that

value as the next starting value and repeat the step. We continue this process until 100

steps (1200 nearby simulations) have not improved the fit. We do this for 10 (substan-

tially) different starting points. They all give the same result. There is little substantial

difference between the Bayesian approach taken previously and the one pursued here.

7.12.1 Data

We do not seek to match the skill-ratio H/L but take it as exogenously given. We

normalize H + L = 1, throughout. The skill-ratio is taken from Acemoglu and Autor

(2012). However, since our estimation requires a skill-ratio both before and after the pe-

riod 1963-2007 we match the observed path of the log of the skill-ratio to a “generalized”

125

logistical function of the form:

α

1 + exp(µ−ts

) + β,

where (α, β, µ, s) are parameters to be estimated. We use the observed skill-ratio in the

period 1963− 2007 and the predicted values outside of this time interval. Yet, the fit is

so good that there is no visual difference in the match of the four time periods between

this approach and using the predicted value in the interval 1963− 2007.

The skill-premium is taken directly from Acemoglu and Autor (2012) specifically the

data underlying their Figures 1 and 2 from David Autor’s website. The labor share is

taken from Koh, Santaeulalia-Llopis and Zheng (2016). We take GDP per hours worked

from the series on non-farm business from the BLS (series PRS85006092).

Capital equipment is calculated as follows. We follow Krusell et al. (2000) and use

quality-adjusted price indices of equipment from Cummins and Violante (2002) who

update the series from Gordon (1990). We combine two different series. First, we

use NIPA data on private investment in equipment excluding transportation equipment

(Tables 1.5 and 5.3.5. from NIPA). We iteratively construct an index for the stock

of private real capital equipment by assuming a depreciation rate of 12.5 per cent (as

Krusell et al., 2000) and using the price index for private equipment from Cummins

and Violante (2002). We start this approach in 1947 but only use the stock from 1963

onwards. We combine this with the growth rate of real private GDP to get an index

for equipment over GDP. We match this index to the NIPA private equipment capital

stock (excluding transportation) over (private) GDP number for 1963 to get a series in

absolute value. To this, we add software, but following the suggestion of Cummins and

Violante (2002), we use the NIPA data on the stock of software over GDP (table 2.1 from

NIPA). We add these two values to get our combined stock of equipment (+software)

over GDP.

126