Lecture notes on modern growth theory

Part 3

Mario Tirelli

Very preliminary material

Not to be circulated without the permission of the author

November 13, 2017

Contents

1. Introduction 1

2. Preliminary concepts: discrete time versus continuous 2

3. Endogenous growth driven by capital accumulation 3

3.1. CRTS on capital (Rebelo, 1987) 3

3.2. Introducing human capital (Lucas, 1988) 6

4. Endogenous growth driven by technological progress 10

4.1. Romer’s analysis 13

4.2. Literature directions 15

4.3. A Shumpeterian Growth Model (Aghion & Howitt, 1992) 16

4.4. A more general growth model with quality improving innovation 22

4.5. Scale effect 26

4.6. Shumpeterian growth and development 28

4.7. Property right protection versus competition 31

1. Introduction

Ramsey-Cass-Koopmans (RCK) growth theory provides empirical predictions that are in

line with Solow-Swan’s. The main difference between the two models is that RCK allows

to pin down, endogenously, the saving rate; hence, to determine a unique efficient path of

long-run economic activity (output, consumption, capital stock) as a function of the economic

fundamentals (essentially, preferences, technology, initial stocks).

These contributions form the heart of the exogenous-growth literature, explaining growth

per-capita as a function of (exogenous) ‘technological progress’ or, more mysteriously, of ‘ef-

fectiveness of labor’ (captured by A and its growth rate µ). Instead, endogenous capital ac-

cumulation cannot account for a large part of either long-run growth or cross-country income

differences: if µ = 0 the only balanced growth of the RCK economy is one with no capital

accumulation (in per-capita terms).

We shall argue that this result is essentially due to the assumption of decreasing return to

scale to capital. This assumption, not only has consequences on long-run growth predictions,

but also affects transitional dynamics in a way that is inconsistent with most of the empirical

evidence. Indeed, if we consider two countries (or a country in different time periods) with

similar fundamentals but different levels of output per-capita, RCK would require unrealistically

large differences in capital per-worker and real interest rates. Following Romer (par. 1.6, 1996)

consider the US economy within the last 100 years period (an alternative example is to compare

India and US today).1 In the US, during the period considered, per-worker output has increased

by a of factor 10, y = 10y0; also, k/y has been roughly constant over time, hence k, y have

roughly grown at the same rate, implying that k = 10k0. Now, putting aside technological

product (assuming A constant) and assuming a standard technology y = Akα,

k

k0=

(y

y0

) 1α

= 101α

with α = 1/3, k = 1000k0. In absence of technological progress, the model predicts a difference

in k higher than the true figure by a factor 100. Turning to interest rates,

r

r0≈(y

y0

)α−1α

= (10)1−αα = 100

this is a remarkably large change that does not match the fact that real interest rates have

remained roughly constant in the US during the last 100 years.

It seams that the marginal role of capital accumulation has to do with a low α, that is a low

level of marginal productivity of capital, the main driver of accumulation in RCK. To further

shed light on this issue, in the first section of these notes, we shall present some benchmark

models of endogenous growth, from the the second half of ‘980’s and early ‘990s. In doing so

we shall also introduce the important idea that physical capital might not be a good measure

of firms’ capital: tangible capital, defined in a broader sense, should not exclude human capital

(knowledge and training).

1David Romer, Advanced Macroeconomics, McGraw-Hill Co, 1996.

1

In the last section, we introduce a different strand of literature literature, from the same

time period, which focusses on an (endogenous) explanation of A. The idea is that A and its

dynamics may capture important phenomena, beyond capital accumulation, which could go

from the evolution of abstract human knowledge to technological progress, through research

and development (R&D), either accomplished by the private or by the public sector.

2. Preliminary concepts: discrete time versus continuous

Most of the economic growth models are in continuous time, something that is equivalent

to the discrete-time modeling when the time interval between two consecutive dates becomes

infinitesimal. Thus, we can interpret the growth prescriptions of continuous-time models as an

approximation of discrete-time.

Let ∆t be the time interval between any two consecutive dates t and t+1. The instantaneous,

absolute growth rate of a sequence x = (.., xt, ..) is,

lim∆t→0

xt+∆t − xt∆t

=∂xt∂t

=: xt

or simply x, when omitting the time index does not produce confusion. Accordingly, the

instantaneous rate of growth is xt/xt.2

For example, consider the accumulation equation, Kt+1 = It + (1 − δ)Kt. Its continuous

time version is,

∆Kt+1 = It − δKt

Define the time interval ∆t such that ∆Kt+1 = Kt+∆t −Kt. Then, as above, take the limit on

the interval and attain,

Kt = It − δKt orK

K=

I

K− δ

As an exercise, show that in the RCK model with exogenous growth η := n + µ, one attains

the following law of motion of capital in efficiency units,

k

k=y

k− (δ + η)− c

k

Analogously, you can show that in the Solow-Swan’s model, with CRTS, Cobb-Douglas tech-

nology, the latter equation reduces to,

k

k= skα−1 − (δ + η)

A last useful remark is about continuous time interest rate compounding and discounting.

We shall briefly explain why the continuous time version of the household intertemporal utility

of consumption is,

(U) U(c) =

∫Te−θtu(ct)dt

when the individual intertemporal discount factor is β = (1 + θ)−1, 0 < θ ≤ 1 measuring the

discount rate.

2It is useful to note that the instantaneous rate of growth of x at t is just the time-derivative of log(xt).

2

Let vt = (1 + r)t be the value of one unit of numeraire after t periods (say years) at the

compounded, constant, per-period (yearly) interest rate r. Next, suppose that interests are

payed out m times during the given (annual) period (m = 12 with monthly instalments).

Then, the value becomes,

vt,m =(

1 +r

m

)mtTo determine the instantaneous compounded return, take the limit,

limm→∞

vt,m = er

This result is not immediate; the proof is tedious (one has to apply the binomial theorem),

hence omitted. Finally, let θ = r and consider a compound discounting of a continuous-time

consumption plan c with instantaneous payoff at t, u(ct). It should be now clear why the

intertemporal utility takes the form in (U).

3. Endogenous growth driven by capital accumulation

The only source of balanced growth in the neoclassical theory due to RCK and Solow-Swan is

exogenous: in the long run (at a steady state) the rate of growth of the main economic variables

η equals the rate of growth of population and ‘technological progress’, n and µ, respectively.

Both variables are exogenously specified; hence, the term ‘exogenous growth models’.

We shall try to clarify the role played by physical capital and further explain why long-run

growth is independent of capital accumulation (remember that the equilibrium growth rate is

independent on the saving rate!). We shall argue that:

(1) physical capital accumulation does not effect long-run growth when the technology has

CRTS overall and DRTS on physical capital; assuming CRTS on capital and overall one

can attain endogenous growth, driven by physical capital accumulation (Rebelo, 1987);

(2) there might be other production factors, beyond physical capital, that accumulate;

accumulation affects growth when both physical and human capital accumulate and

the technology has CRTS overall, (Lucas, 1988).

3.1. CRTS on capital (Rebelo, 1987). Since we aim at focusing on the role of capital

accumulation on growth, let us assume that there is neither population growth nor technological

progress, n = µ = η = 0. Technology is CRTS to capital and overall (also called AK model),

yt = Akt

This implies that the marginal productivity of capital is constant.

In this context, (P) can be rewritten as follows,

max∑t∈T

βtu (ct)

such that, for all t in T ,

(F) ct = Akt + (1− δ)kt − kt+1, k0 > 0 given

3

Letting (λt)t∈T being a sequence of Lagrange multipliers, we form the Lagrangian L0 at t = 0

and derive necessary conditions for an interior solution of (P): at all t in T ,

∂L0

∂ct= βtu′(ct)− λt = 0

∂L0

∂kt+1= λt+1 (A+ 1− δ)− λt = 0

∂L0

∂λt= ct + kt+1 −Akt − kt(1− δ) = 0

Therefore, we attain,

βu′ (ct+1)

u′(ct)=λt+1

λt= (A+ 1− δ)−1

and the Euler equation,

βu′ (ct+1)

u′(ct)(A+ 1− δ) = 1(E)

where, we can define an implicit interest rate, rt := f ′(kt)− δ = A− δ, constant over time. As

for RCK, (E) (F ) and transversality fully describe an interior equilibrium of this economy.

To ease the exposition, hereafter assume the per-period utility is iso-elastic, u(c) = c1−σ/(1−σ), σ > 0, for which the intertemporal marginal rate of substitution is,

βu′(ct+1)

u′(ct)= β

(ct+1

ct

)−σRewriting the Euler equation, (

ct+1

ct

)σ= β (A+ 1− δ)

Finally, consider the continuous time version of this economy. Approximations yield,

cσt+1 = cσt + σcσ−1t ct,

(ct+1

ct

)σ= 1 + σ

ctct

We can use this to derive a new version of the Euler equation,

ctct

=β

σ(A+ 1− δ)− 1

σ

yielding,

ctct

=1

σ(A− δ − θ)(Eσ)

where β = (1 + θ)−1 and σ := σ(1 + θ). Hereafter, we shall drop the time index, unless it is

required for greater clarity.

A balanced growth requires, c/c = g; hence, by (Eσ),

(g) g =1

σ(A− δ − θ)

Clearly, g > 0 if and only if A > δ + θ. We now have to check that capital grows at the same

rate. For all t in T , the resource constraint is,

(F’)k

k= A− δ − c

k

4

Imposing, k/k = g, yields, at all dates,

c

k= A− δ − g

Since k0 > 0 is given, we conclude that there is only one candidate balanced growth with,

c0 = (A− δ − g)k0

Indeed, if it were that,

c0 > (A− δ − g)k0

(taking logs from both sides and differentiating with respect to time) you can check that

c0

c0>k0

k0

that is, the first initial growth rate of consumption is larger than that of capital. The latter

implies that c1/k1 > c0/k0 > (A − δ − g). Reiterating this argument over time, we find that

ct/kt is ever increasing; this, by feasibility in (F ′), requires that kt/kt has to decline over time.

Hence, there could not be convergence to a balance growth path. Next, consider the opposite

case,

c0 < (A− δ − g)k0

Now we find the opposite. The first initial growth rate would be k0/k0 > c0/c0. Then, ct/kt

decreases over time, to zero, and kt/kt (by (F ′)) tends to A + 1 − δ > θ + 1 > 0, violating

transversality.

We are now going to show that, unlike with DRTS to capital, the saving rate affects growth.

By definition, and (F ′),

s :=y − cy

= 1− 1

A

( ck

)= 1− 1

A(A− δ − g)

=δ + g

A

This shows that an increase in the saving rate results in an increase in the balanced growth.

To sum up,

• the economy can display balanced growth: it can have a positive, steady growth rate

of economic activity due to saving and accumulation of physical capital (even if there

is no population growth and technological progress, n = µ = 0).

• The only candidate balanced growth is one of an economy that starts right on the

balanced growth path (i.e. there are no efficient transitional dynamics).

• Countries with same fundamentals have the same balanced growth and saving rate.

However, since there are no efficient transitional dynamics, it is not possible to capture

any form of convergence (absolute or relative).

• I believe α = 1 should not be taken seriously, as capturing the true marginal pro-

ductivity of physical capital. However, it may be signaling that endogenous growth is

produced in a model in which capital could be formed by more that one component–

5

one being traditional physical capital– May be we need a model in which the technol-

ogy has CRTS on the overall capital components and decreasing return on each one of

them. This idea is followed up in the next subsection.

The last observation motivates our interest in theories which consider more than one factor ac-

cumulating; such as those considering human capital and technological progress (some economic

phenomenon producing growth of factor productivity).

3.2. Introducing human capital (Lucas, 1988). Mankiev-Romer-Weil (1992) (MRW) ar-

gued that, quantitatively, the Solow model (and by analogy the RCK one) does not satisfactorily

explain cross-country per-capita income differences. Estimates on different country groups re-

veal that the impacts of the saving rate and of the population growth rate are much larger

than the model predicts; and these imply that α ≈ 0.5, a much larger number than the 1/3

commonly estimated. For this reason MRW suggest to interpret ‘capital’ in a broader sense,

including human capital. Empirically, this seems to pay-off, although the two saving rates,

say s and sH are highly correlated with the productivity variable A across-countries. Such an

high correlation might be explained saying that s and sH capture cross-countries technological

differences, which are otherwise detected by differentials on A. This interpretation seams to

be confirmed by the fact that the coefficient associated to sH is too high compared with the

evidence given by micro-data. The question that is left, then, is how important are these techno-

logical differences captured by TFP cross-countries variations, in oder to explain the per-capita

income differentials observed across countries? The answer is quite a lot! Hall-Jones (1999)

and Caselli (2005) find cross-country TFP variation explains around 1/2 of the cross-country

GDP variation.3 This in hinted by Figure 1, in which a regression is run between the term

Aj , for each country j, of an Harrod-neutral-technology and GDP pre-worker yj . Clearly, this

motivates our interest on the strand of growth literature focusing on (endogenously) explaining

A, which we shall analyze in the next section.

Human capital is a generic term used to address a stock of knowledge, specific skills/abilities

(including creativity) and other attributes which can affect labor productivity.4 In analogy

with physical capital, it can be accumulated over time and can eventually depreciate, if it is

not mantained with further ‘investment’. Moreover, it yields a return in terms of the wage

increments, following those productivity growth due to human capital accumulation. A better

insight on what economists mean by human capital can be found in the empirical literature, in

which researchers are called to actually measure human capital. Just to mention two popular set

of measures, we find health conditions (e.g. daily food calories, life expectancy, expenditure for

medical services) and school achievements (e.g. years in education, school teachers’ credentials,

number of students per-teachers, test achievements).5 In both cases, some measures tend more

to detect the quantity human capital (stocks or flows), while others look more into its quality.

3Hall, R. E., & Jones, C. I. (1999). Why Do Some Countries Produce So Much More Output Per Worker Than

Others?. The Quarterly Journal of Economics, 114(1), 83-116. Caselli, F. (2005). Accounting for cross-country

income differences. Handbook of economic growth, 1, 679-741.4See Weil’s manual, chapter 6, for an introductory treatment on human capital.5For health conditions and growth see, for example, Weil, D. N. (2005). Accounting for the effect of health

on economic growth (No. w11455). National Bureau of Economic Research. For education achievements, see

6

Figure 1. Hall-Jones, 1999

For example, in the first class we find ‘years in primary education’, while in the second we find

‘achievements in standardized math tests’.

Lucas’ is the first to introduce human capital in the RCK model. Human capital can be used

in current production or spent to build up more human capital; in other words and in analogy

to physical capital, over ht units of human capital available at t, a fraction ξt can be used in

current production and the remaining (1− ξt) allocated (or invested) in a training activity that

produces an increase in the stock of human capital. More precisely, we assume that the law of

motion of human capital is,

ht+1 − ht = B(1− ξt)ht =: iht

where 0 < B ≤ 1 captures depreciation and iht is the ‘investment’ in human capital. The output

production function is,

F (kt, ξtht) = Akαt (ξtht)1−α

Clearly, this function is strictly increasing and concave in both arguments, it satisfies F (0, ·) =

F (·, 0) = 0; the technology it represents is of CRTS overall and has DRTS in each factor (h, k).

Thus, the only novelty, with respect to RCK’s, is that both factors accumulate. Being CRTS

to ‘capital’ (human and physical) one would expect that Lucas’ model has the same properties

of Rebelo’s. Yet, as you should learn next, this is not entirely true.

Lucas’ optimal growth model (P) can be written as,

max(ct,kt+1,ht+1,ξt)t∈T

∑t∈T

βtc1−σt

1− σ

Barro, R. J., & Lee, J. W. (2001). International data on educational attainment: updates and implications.

Oxford Economic papers, 53(3), 541-563.

7

such that, for all t in T ,

ct = F (kt, ξtht) + (1− δ)kt − kt+1,

ht+1 − ht = B(1− ξt)htk0 > 0, h0 > 0, given

We remark that, in every date t, the planner chooses the capital stocks to carry over to t + 1

(kt+1, ht+1), the fraction of current human capital ξt to devote to current output production

and consumption ct.

The time-0 Lagrangian L0, with multipliers (λt, γt) is,6

L0 =∑t∈T

βt{c1−σt

1− σ+ λt [F (kt, ξtht) + (1− δ)kt − kt+1 − ct] + γt [B(1− ξt)ht − ht+1 + ht]

}Therefore, first order conditions for an interior optimum, at all t in T , are,

∂L0

∂ct= c−σt − λt = 0

(FOC)

∂L0

∂kt+1= βλt+1 (Fk,t+1 + 1− δ)− λt = 0, Fk,t+1 := αA

(kt+1

ξt+1ht+1

)α−1

∂L0

∂ht+1= βλt+1Fh,t+1ξt+1 + βγt+1 [B(1− ξt+1) + 1]− γt = 0, Fh,t+1 := (1− α)A

(kt+1

ξt+1ht+1

)α∂L0

∂ξt= λtFh,t −Bγt = 0, Fh,t := (1− α)A

(ktξtht

)α∂L0

∂λt= ct + kt+1 − F (kt, ξtht)− kt(1− δ) = 0,

∂L0

∂γt= ht+1 − ht −B(1− ξt)ht = 0

Following the same transformation used in the previous subsection and passing to continuous

time, we attain,

ctct

=1

σ(Fk,t+1 − δ − θ),(E(σ))

ktkt

=ytkt− δ − ct

kt,(k)

htht

= B(1− ξt)(h)

Next, we check that a balanced growth exists. From (FOC),

gc =1

σ(Fk,t+1 − δ − θ), gk =

1

αFk,t+1 − δ −

ctkt, gh = B(1− ξt)

where (gc, gk, gh) are constant over time. From (h), ξt has to be constant, say equal to ξ. From

the first, Fk,t has to be constant, say equal to Fk. Then, by definition of marginal productivity

of capital Fk = αA[(kt/ht)(1/ξ)α−1, (kt/ht) is constant; hence. (kt, ht) grow at the same rate.

6Notice that by writing the problem in this ways, we should interpret a date−t Lagrange multiplier λt in

present value as βtλt.

8

Also, by the same reasoning, from the gk equation, (ct, kt) grow at the same rate. Therefore,

all activity variables grow at the same constant rate g.

Next, we want to determine g. Observe that, at an optimum,

λt+1

λt=

(ct+1

ct

)−σ≈ 1− σ ct

ct

Hence, at a balanced growth, we can take,

λt+1

λt= 1− σg or

λtλt

= −σg

Also, at balanced growth, our previous analysis tells us that Fh,t is constant; hence, by the

first-order optimality condition in ξ, λt/γt has to be constant, implying that γ also grows at

the rate −σg. Now, use these results in the third equation (FOC), after dividing through by

γt:

βλt+1

λt

λtγtFh,t+1ξt+1 + β

γt+1

γt[B(1− ξt+1) + 1]− 1 = 0

Passing to the balanced growth, and using γt/λt = Fh,t/B = Fh/B,

β(1− σg)Bξ + β(1− σg) [B(1− ξ) + 1]− 1 = 0

β(1− σg)[Bξ +B(1− ξ) + 1]− 1 = 0

β(1− σg)[B + 1]− 1 = 0

Hence, using β−1 = 1 + θ,

(g) g =1

σ

(B − θ1 +B

)positive as long as investing in human capital is sufficiently productive, B > θ. Indeed, recall

that investing in human capital implies subtracting labor to current production, something

that –everything else equal– reduces current production and consumption and increases future

one.

Finally, we observe that if the economy is initially not on the balanced growth path, it will

adjust, eventually, approaching it.7 In fact, at balanced growth, from the law of motion of

capital (k),

g =1

σ(Fk − δ − θ),

where Fk := αA(kξ·h

)α−1. Rearranging and using (h) to substitute for ξ, yields,

A

(k

ξh

)α−1

= σg + δ + θ

k

h= ξ

(A

σg + δ + θ

)1−α

k

h=(

1− g

B

)( A

σg + δ + θ

)1−α

7Whether or not the latter occurs is a matter of solution stability we shall not examine here.

9

Also considering (g), the last expression says that kt/ht has to be constant over time, at a

balanced growth. Therefore, unless its right hand side coincides with the initial condition

k0/h0, the economy would have to adjust both types of capital to get to the balanced growth

path. This adjustment describes transitional dynamics.

We are now ready to draw the main conclusions.

• The economy considered is one in which there is neither population growth nor tech-

nological progress (i.e. we assumed n = µ = 0) and the technology has CRTS. Yet the

economy has a balanced growth efficient path, with all the activity variables growing

at the a rate g > 0. The reason is that, unlike in Solow, RCK economies, there is a

second input, beyond physical capital, that accumulates. One could say that there is a

broader notion of capital, physical plus human, with respect to which the technology

has CRTS (in this respect, Lucas’ model is like Rebelo’s).

• The long-run growth rate of the economy is a function of the saving rate; something

that can be shown as before, in Rebelo’s. It is also a function of the saving/investment

rate in the human capital, 1 − ξ, which equals g/B, at balanced growth; this can be

used to argue that economies with higher investment in human capital tend to grow at

an higher rate g.

• Two economies with identical preferences and technologies, will display the same bal-

anced growth, regardless for initial conditions (k0, h0). Hence, Lucas’ model can display

conditional convergence. Indeed, one can prove that the balance growth has saddle path

properties (a proof is omitted, but can be found in Barro and Sala-i-Martin’s manual).

Now, initial conditions in human capital matter too, possibly reducing the role of phys-

ical capital differentials.

4. Endogenous growth driven by technological progress

Economists agree that cross-country differences in income per-capita (and growth rate) is

explained both by differences in capital stocks (accumulation) and productivity levels (rate of

change). We shall, respectively, call these two determinants the accumulation and productivity

components. Measuring productivity is hard, since this is not directly observable and it has to

do with how effectively inputs are combined and used to produce output. We have illustrated

earlier that a possible estimate is achieved using ‘Solow’s residuals’. Following this same logic,

if we compare two countries j = 1, 2, with production function,

(y) yj = Ajkαj h

1−αj

we can define their productivity differentials as,

A1

A2=

y1/y2(kα1 h

1−α1

)/(kα2 h

1−α2

)The right hand side of the latest is formed by observable variables, while the left hand side is

unobservable. Just as an illustration, we report data on per-capita variables (y, k, h) expressed

as a fraction of the corresponding variables measured for the United States.

10

yj kj hj

UK = 1 .7 .8 .82

India = 2 0.086 0.047 0.55

Letting α = 1/3, we find that productivity in the US is more than double the one in India:

A1

A2= 2.42

This approach is exposed to the same criticisms of the one based on Solow’s residuals;

residuals are a measure of ‘ignorance’, hence it is possible that A, so estimated, captures

various mistakes. Among such mistakes, a possibly relevant one concerns the measurement of

capital. Indeed, especially for undeveloped countries, statistics about investment expenditure

may not effectively account for the actual input used in production activities and may not be a

good base to construct measures of capital stock. An old joke, made popular by Pritchett, gives

the sense of how biased can be a measure of the stock from public infrastructure in countries

in which burocrats/politicians are corrupted.8

While on a foreign trip a government official of country A is visiting the pent-

house apartment of his friend B, a bureaucrat of a poor country. After A admires

the fine residence and furnishings, he says: ‘We’re friends. Be honest with me

B. I know that with your official salary you cannot possibly afford this. What

gives?’ Taking his friend to the window, B replies, ‘See that superhighway run-

ning through town? 10 percent.’ Some time later B has the occasion to visit the

even poorer country of his friend A and finds himself in an even larger and more

luxuriously appointed penthouse apartment. Says B, ‘I know your official salary

must be even lower than mine, yet your house is much nicer. What gives?’ Tak-

ing his friend to the window, A points and says, ‘See that superhighway running

out into the jungle?’ After straining his eyes for a minute B replies, ‘But there is

no highway out there.’ ‘Exactly,’ says A with a wink. ‘100 percent.’ [Pritchett,

2000, p. 361.]

Therefore, when capital stock is measured using data on investment expenditure, it may result

in an overstatement. If capital stock in country 2 is overstated, then productivity differentials

A1/A2 are overstated too. This is what one can fear especially in comparing underdeveloped

countries with developed ones. Despite criticisms, there is a large consensus among economists

that productivity differentials are important to explain both cross-country differentials in in-

come per-capita and in growth rates.

Let us go back to cross-country differential in income per-capita. By equation (y), these can

be written as,y1

y2=A1

A2× kα1 h

1−α1

kα2 h1−α2

or, in logs,

log y1 − log y2 = (logA1 − logA2) + α(k1 − k2) + (1− α)(h1 − h2)

8See Pritchett, L. (2000). The tyranny of concepts: CUDIE (cumulated, depreciated, investment effort) is

not capital. Journal of Economic Growth, 5(4), 361-384.

11

If we measure all differences with respect to a benchmark country, say the US, we can write

such equation for a country j as,

(y) yj = Aj + αkj + (1− α)hj

with the xj := log xj − log xUS . The corresponding (unrestricted) regression equation is,

(y) yj = γ0Aj + γ1kj + γ2hj + εj

The top richest countries (to which the US belongs) have a capital stock that is the 95%

of US, while productivity is only the 86%. For poorest countries the distance of the two

components with respect to US is reversed, respectively, 16% and 24%. Running regressions

based on (y), for country groups, confirms that the importance of productivity differences in

explaining differences in output per-capita; and how such correlation increases with the level

of income groups. On average, in the whole sample of 127 countries, the two components

have a very similar contribution: per-capita income differentials are explained for the 57% by

the accumulation component (physical and human capital) and by the 43% by productivity

differentials.9

A similar analysis can be done on the growth differentials. From equation (y), differentiating

through time, one can get a regression equation in terms of per-capita growth differentials and

growth rates related to the productivity component and the factor component. In this case,

empirical studies reproduce similar results to those obtained for income differentials. One finds

that, on average, the 58% of growth rate differentials are explained by differences in rate of

growth of productivity, while the rest is explained by differentials in the growth rates of the

accumulation component.

Next, we provide a very basic introduction of the potential role of technological progress

as a determinant of long-run economic growth. In the first part, the exposition closely fol-

lows Romer’s (chp. 3, 1996) manual. It is based on a simple model capturing the idea that

more resources devoted to accumulate knowledge or to increase R&D, yield an improvement

of knowledge and innovation that increase factors’ productivity. The model however does

not endogenously represent the determinants of this accumulation process, in as much as the

Solow-Swan’s model does not represent the determinants of saving decisions. More precisely,

the major simplifications introduced are that the saving rate on physical capital and the frac-

tion of labor and capital devoted to knowledge accumulation or technological innovation are

exogenous. As we argue in the end, relaxing these assumptions allows to capture and represent

the nature of the accumulation process, explaining the driving forces of important phenomena

such as R&D expenditure and expenditure on basic scientific research, technical education and

training. In the last part, after summarizing the main literature directions, we present a seminal

contribution to Shumpeterian growth theory, due to Aghion and Howitt.10 This consists in an

9See, for example, chapter 7 in D. Weil, Economic Growth, Pearson, 2013. A more detailed analysis can be

found in Hall, R. E., & Jones, C. I. (1999). Why Do Some Countries Produce So Much More Output Per Worker

Than Others?. The Quarterly Journal of Economics, 114(1), 83-116. See also Caselli, F. (2005). Accounting for

cross-country income differences. Handbook of Economic Growth, 1, 679-741.10Aghion, P., and Howitt, P. (1992). A Model Of Growth Through Creative Destruction. Econometrica,

60(2), 323-351.

12

endogenous growth model in which income dynamics are driven by technological progress, and

the occurrence of technological progress depends on agents’ decisions about R&D investment.

4.1. Romer’s analysis. The model is based on the following three equations,

Yt = [(1− aK)Kt]α[At(1− aL)Lt]

1−α, 0 < aK < 1, 0 < aL < 1, 0 < α < 1

At = [aKKt]β[aLLt]

γAθ, 0 < β, γ, 0 < θ

Kt = sYt, 0 < s < 1

The assumption of a CRTS, Cobb-Douglas, technology is for simplicity; motivated by our

interest to focus on the production of knowledge or R&D, captured buy the second equation.

In this equation, A/A captures the rate of technological progress or (if positive) the increase of

the set of discoveries/knowledge. The restrictions β, γ > 0 allow to consider different scenarios:

i) increasing returns on K,L could be motivated by saying that, certain forms of R&D have

fixed costs; ii) decreasing (or constant) returns, could be explained saying that a certain input

allocation successfully used to achieve some innovation, might not (or just suffices) to keep up

with the set of discoveries made. θ represents the effect of A on A/A, for example, the return

of a certain level of R&D on technological progress and innovation. More precisely, if θ = 1

there is no scale effect; if θ < 1 the rate (say of technological progress) tends to decrease as

the economy gets to an higher (technological) level (e.g. think at the case in which a country

approaches the world technological frontier, something called the ‘fishing-out’ effect); otherwise,

if θ > 1 the (technological) progress is self-enforcing.

Collecting constants, the first two conditions, can be written in a more compact form as,

Yt = BKαt (AtLt)

1−α, B := (1− aK)α(1− aL)1−α

At = GKβt L

γtA

θ, G := aβKaγL

By a simple substitution, the model reduces to,

Kt

Kt= s

(AtLtKt

)1−α, s := sB(∗)

AtAt

= GKβt L

γtA

θ−1

Let (gK,t, gA,t) :=(KtKt, AtAt

)be the instantaneous growth rates. A balanced growth is an

allocation satisfying gK,t = gY,t = g, in which gA,t is also constant, at all dates t in T . We

now derive growth rate dynamics from (∗), by taking logs and differentiating with respect to

t; these, for simplicity are reported without time subscripts,

gKgK

= (1− α) [gA + n− gK ](∗∗)

gAgA

= βgK + γn+ (θ − 1)gA

We study the existence of a balanced growth and its stability by mean of a phase diagram

in the plane of coordinates (gA, gK). At balanced growth the left hand sides in (∗∗) are zero;

13

hence, rearranging, we respectively obtain,

gK = gA + n

gA =γn

1− θ+

β

1− θgK

Both loci are linear, respectively, with a negative and a positive intercept.11 Their intersec-

tion (the solution of the system latest system) occurs at,

(ss) g∗A =(β + γ)n

1− (θ + β), g∗K = g∗A + n

(g∗A, g∗K) identifies the balanced growth, if it is a point in the positive orthant. Indeed, three

cases are possible,

(1) θ + β < 1 - balanced growth

(2) θ + β = 1 and n = 0 - balanced growth

(3) in the remaining cases [θ + β > 1 and θ + β = 1 with n > 0] - no balanced growth

In case 1 and 2 the solution is globally, asymptotically stable (for case 1, see figure 2).12 Case

3 is such that, for all initial condition (gA, gK) > 0, there will explosive dynamics in both K

and A. Notice that these conditions relate to the return on stocks that accumulate A,K. For

example, case 2 is equivalent to Rebelo’s model.

gA

gK

Figure 2. Case 1 θ + β < 1

Interpreting (ss), g∗A is increasing in n, as positive population growth sustains both gdp

growth and technological progress. However, if we look at countries with high n, the evidence

11To draw the second locus, expicitate it with respect to gK as,

gK =1− θβ

gA −γn

β

Notice that its slope is greater than one if and only if θ + β < 1.12Case 2 implies that the two loci in figure 2 that are superimposed.

14

often goes in the opposite direction. A different story is if we consider this model as one that

captures the importance of the world population growth on the worldwide economic growth.

Next, consider a simplified version of the above model, with no physical capital (β = 0).

The differential equation for A in (∗∗) reduces to,

gAgA

= γn+ (θ − 1)gA(?)

In this context, (ss), in case 1, reduces to g∗A = γn1−θ . Hence, with θ < 1 a balanced growth

exists [plot the graph of (?)]. We notice that an increase in the fraction of labor used in R&D,

aL (or G), has only a temporary effect on innovation and income growth, but no effect on

the balanced growth. This occurs because of the decreasing return to scale of knowledge in

the output technology. In perfect analogy to the Swan-Solow model, DRTS to the factor that

accumulates prevents the ‘saving rate’ aL to affect long-run growth. Instead, if θ = 1, we have

something similar to Rebelo’s CRTS model [See how the previous graph changes]. Now, going

back to the more general model, with β > 0, the same reasoning applies to the two factors

which accumulate, knowledge and physical capital. This should be enough to read the three

cases above and explain when a change of s, aL, aK has only short-run or also long-run effects

on economic growth.

4.2. Literature directions. As we anticipated, relaxing the assumption that of exogenous

allocation rules s, aL, aK , allows to explain the determinant of productivity and its dynamics.

This literature has moved in different directions. The following are the best known ones.

• Knowledge as a public good, for example, publicly disseminated government and aca-

demic research [Shell (1966, 1967), Phelps (1966), Nordhause (1967)]

• R&D and innovation driven by private incentives, such as patents [Romer (1990), Gross-

man and Helpman (1991), Aghion and Howit (1992)]

• Knowledge driven by experience, learning-by-doing (e.g. At depends on output pro-

duced (Ys)ts=0) [Arrow (1962)].

See Romer’s (1996) manual for more references and further discussion.

Problem 1. Consider the following Mathusian version of the Solow model, in which the pop-

ulation is proportionally increasing in aggregate income:

Lt = BYt, B ≥ 0

Technology is represented by, Yt = Kαt (AtLt)

1−α, 0 ≤ α ≤ 1, K0, L0 > 0 are given. For

simplicity At = 1 at all t and there is complete capital depreciation, δ = 1. Capital accumulates

according to, Kt = sYt, for 0 < s < 1 given.

a Write down the expression representing the population growth rate gL(t) and capital

growth rate gK(t) as functions of Lt and Kt.

b Show that there is a unique balanced growth equilibrium, gL(t) = gK(t) = g∗, at all t.

Draw the graph of the two loci gL = 0 & gK = 0 in the plane (gL, gK).

c Within the graph you have drawn in point b), study the stability of the balanced growth

equilibrium using the functional relationship between gK(t) & gL(t), which you have to

15

show being defined by,

gK(t) = s

(B

gL(t)

) 1−αα

What do you conclude on stability?

d Compute the balanced growth rate g∗ and show that it is increasing in s and decreasing

in K0/L0. Is this model one of endogenous growth?

4.3. A Shumpeterian Growth Model (Aghion & Howitt, 1992). The essential feature of

Schumpeterian-growth models is the incorporation of technological progress which is generated

by the endogenous introduction of product and/or process innovations. The term ‘endogenous’

refers to innovations that result from agents’ optimal decisions (typically firms). Although

Schumpeterian-growth theory contains a broader set of ideas, these growth models capture

an important one, related to the concept of ‘creative destruction’. Creative destruction was

described in the writings or Joseph Schumpeter (1928, 1942) and refers to the endogenous

introduction of new products and/or processes. For instance, in Capitalism, Socialism and

Democracy, chapter 8, Schumpeter states:

The essential point to grasp is that in dealing with capitalism we are dealing

with an evolutionary process [...] The fundamental impulse that sets and keeps

the capitalist engine in motion comes from the new consumer goods, the new

methods of production, or transportation, the new forms of industrial orga-

nization that capitalist enterprize creates [...] In the case of retail trade the

competition that matters arises not from additional shops of the same type,

but from the department store, the chain store, the mail-order house and the

super market, which are bound to destroy those pyramids sooner or later. Now

a theoretical construction which neglects this essential elements of the case ne-

glects all that is most typically capitalist about it; even if correct in logic as

well as in fact, it is like Hamlet without the Danish prince.

In this section we present a seminal model of Shumpeterian growth due to Aghion and Howit

(1992).13

The economy has three characteristics:

• growth is generated by technological innovation;

• innovation is uncertain; when it occurs, replaces the existent technology (Shumpeter’s

creative destruction);

• innovation results from R&D investments and yields a monopoly rent to the innovator.

We shall assume that time is discrete, although time length might be thought to be arbitrarily

short. Moreover, since in absence of any technological innovation, our economy would not

have any equilibrium dynamics, we shall reinterpret the time index as the actual number of

innovations achieved since time zero.

13Aghion, P., and Howitt, P. (1992). A Model Of Growth Through Creative Destruction. Econometrica,

60(2), 323-351. See also, Aghion, P., Akcigit, U., and Howitt, P. (2013). What do we learn from Schumpeterian

growth theory? (No. w18824). National Bureau of Economic Research. Chapter 2 of Aghion, P., and Howitt,

P. (2009). The economics of growth. MIT Press.

16

Agents. There is a continuum (mass 1) of identical, infinitely lived individuals. Think of such

individuals as risk neutral entrepreneurs who discount profits at a common, constant interest

rate r. Each individual, in every period, has one unit endowment of labor.

Commodities, technology and markets. A part for labor, there is one consumption/final

commodity and an intermediate one. The final commodity Y is produced out of the interme-

diate y with the technology,

(Y) Yt = Atyαt , 0 < α < 1.

The intermediate commodity is produced out of labor, with a linear production function (CRTS

technology)

yt = `t

There is a market for each commodity (labor, final and intermediate goods). Labor can either

be used to produce the intermediate good (in the amount `) or employed in R&D (in the

amount z); market clearing requires that, in every period t, 1 = `t + zt. There is perfect

competition in the labor and final good markets. Instead, the intermediate good market is a

monopoly, with the incumbent being the last successful innovator firm. The idea is that the

monopoly rent on the intermediate product is the prize awarded to the most recent innovator

to reward the fact that she is now providing an input of higher ‘quality’ (i.e. productivity).14

In this respect, the expenditure for the intermediate product by each final-good producer can

be interpreted as a fee associated to a patent licence.

At all dates t, we normalize the price of the final good to one and let the price of the

intermediate good and the wage rate, respectively, be (pt, wt).

Technological innovation. Technological innovation consists in an increment of A, the pro-

ductivity (or ‘quality’) of the intermediate input. Increments are deterministic: at all t, if an

innovation occurs, At increases by (a gross) rate η > 1, At+1 = ηAt. Every period t ends

with an innovation with probability, ξt := λzt, where 0 < λ < 1 is a stationary arrival rate;15

0 ≤ zt ≤ 1 captures the society effort in R&D. So, in this economy, the event that an innovation

occurs is uncertain, but its size (η) is certain.

Every innovation is patented and the innovator becomes the monopolist of in the intermediate

good sector and remains so as long as someone innovates (it is assumed that patents do not

expire). R&D rises the probability that somebody innovates and innovation raises A for the

whole economy; this configures a positive externality. Innovation does also determine a negative

externality, since it destroys the monopoly rent of the previous innovator (this effect is the

Shumpeterian ‘creative destruction’).

Decisions. Hereafter, we assume that the only agents who invest time/labor in R&D and have

a chance to become the next innovator, are the outsiders (not the incumbent monopolist, who

14Observe that we are implicitly assuming that the old input becomes unavailable; i.e. no producer can keep

using the old technology. This assumption can and will be relaxed later.15λ can be equivalently interpreted as the fraction of people that innovates in the interval [t, t+ 1), or as the

innovation flow to the single innovator entrepreneur. λ is also an index of the entrepreneurs’ (intrinsic) ability

to innovate.

17

instead works only in the intermediate sector). This assumption is without loss of generality,

as we shall argue later it always hold at a equilibrium (see remark 4.1 below).

At all t, let Vt+1 be the net present value of becoming the next innovator for an outsider

entrepreneur and πt+1 the profit associated of being a patent holder/monopolist in the inter-

mediate good sector. Preferences are recursive, so that,

(V) Vt+1 = πt+1 +1

1 + rEtVt+2

Since the probability that someone (different from the incumbent at t) innovates at t + 1 is

ξt+1 := λzt+1,

EtVt+2 = ξt+1 × 0 + (1− ξt+1)Vt+1 = (1− ξt+1)Vt+1

Keep in mind that if an innovator at t+ 1 remains an innovator in t+ 2, it means that no one

has innovated and he keeps getting Vt+1. This is so, as we are focusing in cases in which the

incumbent does not invest in R&D, hence we focus on situations in which the t+ 2−economy

will be identical to the one in t+ 1. Next, substituting the latest in (V ),

(V’) Vt+1 = πt+1 +1− ξt+1

1 + rVt+1

Collecting terms and using the definition of ξt,

Vt+1 =1 + r

r + λzt+1πt+1

Observe that Vt+1 is lower than the utility corresponding to the case in which the innovation

yields a perpetual monopoly position (say with λ = 0). Indeed, if innovating means that πt+1

is given with certainty, the payoff of innovating would be,

Vt+1 = πt+1 +1

1 + rVt+1 ⇔ Vt+1 =

1 + r

rπt+1

Clearly, Vt+1 > Vt+1 whenever zt+1λ > 0.

To determine πt and zt we proceed in two steps. First, let us look at the final good market.

This is perfectly competitive; hence, at all t, profit maximization (marginal revenue product of

yt equals its marginal cost–the price of the intermediate good pt),16

(∗) αAtyα−1t = pt

From which, the indirect demand of the intermediate good is a function,

(p) p(yt) = αAtyα−1t

In the intermediate good market, the monopolist at t, given the wage rate wt and indirect

demand p(yt), solves,

πt = maxy{p(yt)yt − wt`t}, s.t. yt = `t

16Given (Y ), this is just the sufficient first-order-cond. derived from the profit maximization of a final-good

producer:

maxyt{Atyαt − ptyt}

The solution yt is the demand of the intermediate good by this firm, at pt.

18

Substituting, the problem becomes,

maxy{αAtyα − wtyt}

and its first order conditions yields the supply function of the intermediate good,

α2Atyα−1 − wt = 0 ⇔ y∗t =

(α2Atwt

) 11−α

Substituting into the objective, we find the profit function,

πt =(αAty

α−1t − wt

)y∗t(π)

=

(αAt

(α2Atwt

)−1

− wt

)y∗t

=(wtα− wt

)y∗t

=

(1− αα

)wty

∗t

Notice that (1−α)/α is the mark-up over the production cost wt`∗t , where `∗t = y∗t is the labor

demanded by the monopolist. Substituting,

πt = wt

(1− αα

)(α2Atwt

) 11−α

Also, from (p), the price schedule of the intermediate good is,

p∗t = αAt(y∗)α−1 =

wtα

Finally, the supply of final good is,

Y ∗t = Aty∗αt = At

(α2Atwt

) α1−α

We are left to determine z∗t , the optimal supply of labor to R&D. This is immediate if we

impose labor market clearing,

z∗t = 1− `∗t = 1− y∗tWe now define the equilibrium and compute prices.

Equilibrium. An equilibrium is an allocation (Y ∗t , y∗t , `∗t , z∗t )t∈T and prices (w∗t , p

∗t )t∈T such

that, every individual optimize and, at all t in T ,

`∗t + z∗t = 1

p∗t =w∗tα

Since, by assumption, we are considering that the outsiders are the innovators, they will

choose to work in R&D a number of hours zt that maximizes the expected return from inno-

vation,

maxzt{λztVt+1 − wtzt}

where recall that: an unsuccessful R&D activity yields a zero payoff; the probability to innovate

is ξt = λzt; the outside option of working in R&D is to be employed in the intermediate sector

19

for an hourly wage of wt. Hence, each outsider works in R&D up to the point at which its

expected marginal return λVt+1 equals wt.17 Using (V) or (V’), we manipulate the solution as

follows:

wt = λVt+1

= λ1 + r

r + λztπt+1

= λ1 + r

r + λzt+1

(1− αα

)wt+1y

∗t+1

= λ1 + r

r + λ(1− y∗t+1)

(1− αα

)wt+1y

∗t+1

= λ1 + r

r + λ

[1−

(α2At+1

wt+1

) 11−α] (1− α

α

)wt+1

(α2At+1

wt+1

) 11−α

This is a dynamic equation (a first order, non-linear difference equation) in the wage rate,

describing the equilibrium evolution of this variable over time.

Easier equilibrium computation. The equilibrium can be more easily computed introducing

a change of variable: (ω, p) := (w/A, p/A), which are the productivity adjusted wage and the

intermediate-good price. Then, the equilibrium is described by the following conditions (1-

4,5,6).

y∗t = y(ωt) :=

(α2

ωt

) 11−α

(1-4)

π∗t =

(1− αα

)wty(ωt) = Atπ(ωt), π(ωt) :=

(1− αα

)ωty(ωt)

`∗t = y(ωt)

z∗t = 1− y(ωt)

Y ∗t = At[y(ωt)]α

wt = λVt+1 = λ1 + r

r + λz∗t+1

At+1π(ωt+1)

can be rewritten as,

(5) ωt = ηλ1 + r

r + λz∗t+1

π(ωt+1)

and from (∗),

(6) pt = α[y(ωt)]α−1

The equations (1-4,6) are static, the last (5) is a difference equation in ω.

17Notice that,

λVt+1 =∂

∂ztProb(success|zt)Vt+1

where, in our context, Prob(success|zt) = λzt, so that the marginal effect of R&D investment on the success

probability is exactly λ.

20

Balanced growth equilibrium. A balanced growth equilibrium can be described by constant

(z, ω) satisfying,

z = 1− y(ω)(ζ)

ω = ηλ1 + r

r + λzπ(ω)(ω)

A constant ω is equivalent to w increasing at the rate of technological progress η. You can verify

that, in the plane (z, ω), (ζ) is monotonically increasing and (ω) is monotonically decreasing.

Their unique intersection pins down the balanced growth equilibrium. All other equilibrium

variables can then be retrieved. In particular, notice that a constant ω implies a constant

production y = y(ω) and labor ` = y(ω); hence, z = 1 − y(ω) is also constant. Instead, final

output grows at the rate of technological process η:

Y ∗t+1

Y ∗t=At+1

At

[y(ω)

y(ω)

]α= η

Also pt grows at the rate η.

Finally, let us compute the expected, equilibrium, growth rate at t, gt+1. Letting zt be the

equilibrium aggregate labor in R&D,

EtYt+1 = λztYt+1 + (1− λzt)Yt

gt+1 :=EtYt+1 − Yt

Yt= λzt

Yt+1 − YtYt

= λzt(η − 1)

The expected growth rate increases in R&D effort zt and ability λ, as well as in the gross rate

of technological progress η. Since z is an endogenous variable (defined based on a decision

problem), the model is one of endogenous growth.

Remark 4.1 (At equilibrium, the incumbent is never the innovator). The agent who currently

is the monopolist (because has successfully innovated in the previous period) does not work in

the R&D sector; therefore she cannot be an innovator in the current period too. The reason

for this is that outsiders who innovates at At gets a payoff Vt+1, while for the incumbent payoff

is just the incremental value Vt+1 − Vt. This difference can be interpreted by saying that the

first receives a patent that she can sell at market price Vt+1, while the second gets a new patent

but looses the previous one. Moreover, since the R&D technology is linear, at equilibrium, the

outsiders are indifferent between working in the manufacturing and in R&D sector:

wt = λVt+1 > λ(Vt+1 − Vt),

implying that, the incumbent will value more working in the intermediate sector than in R&D.

This is an example of the ‘Arrow effect’ (Arrow, 1962) or ‘replacement effect’, an extreme

illustration of the lower incentive to innovate one typically attributes to patent holders.

Problem 2. Verify that, in the plane (z, ω), (ζ) is monotonically increasing and (ω) is mono-

tonically decreasing. Draw the graph and perform some comparative statics. In particular, with

reference to the he balanced growth values (z∗, ω∗) show that,

• z∗ decreases in r, increases in λ and η;

• z∗ is decreasing in the productivity parameter α.

21

Explain the economic intuition behind these comparative statics. Try to analyze them using

your graphical representation.

4.4. A more general growth model with quality improving innovation. 18

As in the economy of the previous section there are three commodities, a final-good, an

intermediate-good and labor. The intermediate good is produced in m different sectors (or

entrepreneurs’ types) indexed by i = 1, ..,m, which are ex-ante identical. An intermediate

good i has a quality Ai,t, which varies over time due to innovation. Unlike before, however,

innovation can be imitated at a fixed cost of χ > 1 units of the final good, which drops to 1

after one period. There is free-entry of entrepreneurs in every sector.

The final-good is produced according to the following,

Yt =

(m∑i=1

A1−αi,t yαi,t

)(L

m

)1−α

where L is the number of individuals, each of whom is endowed with one unit of homogeneous

labor. Labor is offered inelastically, hence fully employed in the production activity of the final

good only, equally across sectors (i.e. each sector has L/m workers).19 The final good, which

is again assumed to be the numeraire, is both used for the individuals’ consumption and, as

an input, in the production of the intermediate good and in R&D. As before, the probability

to innovate is increasing in R&D expenditure.

R&D, innovation and imitation. Let as first consider a typical intermediate sector i. Each

entrepreneur in t is involved in R&D and, if successful, is able to produce yi,t units of the

intermediate goods of quality Ai,t at a unit, constant marginal cost. In this case Ai,t determines

the frontier technology of sector i. Instead, the other entrepreneurs in i can either imitate the

innovation at a constant marginal cost χ > 1 or produce At−1 at a unit marginal cost (i.e.

imitation is costly only in the period in which the innovation occurs).20 Competition and

imitation imply that the innovator can at most charge a price χ. Therefore, the innovator’s

profits are,

πi,t = (χ− 1)yi,t

where yit is the quantity of intermediate good i demanded by the final-good sector.

To determine yi,t just recall that the demand of the intermediate-good input is such that

each firm in the final good sector will choose yi,t that equates marginal revenue with marginal

cost (the price of the intermediate good). If in t a firm i innovated, the marginal cost is χ and

the marginal revenue is,

MRi ≡∂Yt∂yi,t

= αA1−αi,t yα−1

i,t

(L

m

)1−α

18This section heavily borrow from Aghion, P., & Howitt, P. (2005). Growth with quality-improving innova-

tions: an integrated framework. Handbook of economic growth, 1, 67-110.19Notice that in a more general model, with elastic labor supply, we would have a ‘standard’ Harrod neutral

technology, Yt =∑mi=1(Ai,tLi,t)

1−αyαi,t.20χ can also be interpreted as a unit patent fee, which has been fixed by some market authority. Also,

At−1 = maxi{Ai,t−1} because imitation is costless after one period.

22

Equating (MRi = χ) and solving, yields the innovator supply of intermediate good i,

yi,t =

(α

χ

) 11−α Ai,tL

m

Substituting, for a successful entrepreneur i, we obtain,

(π) πi,t = (χ− 1)yi,t = σ(χ)Ai,tL

m, σ(χ) ≡ (χ− 1)

(α

χ

) 11−α

where you can verify that σ is a positive real-valued function with σ′ > 0.

All other entrepreneurs in i who did not innovate (and face a price of 1), supply,

yct ≡ (α)1

1−αAt−1L

m

and make zero profits.21

Innovation requires R&D expenditure that is measured in units of the final good. For sector

i and expenditure Zi,t, the productivity-adjusted R&D is defined as,

zi,t =Zi,t

ηAi,t−1

where ηAi,t−1 can be interpreted as a target productivity parameter for i. The probability that

i is the innovator and Ai,t = ηAt−1, is,

ξi,t = λf(zi,t), f(0) = 0, f ′ > 0, f ′′ < 0

an increasing function of R&D expenditure, with 0 < λ < 1 being an ‘ability’ parameter. The

use of adjusted expenditure captures the fact that as the quality increases, it becomes more

difficult to innovate, requiring an higher research effort/expenditure.22

We can now determine the optimal R&D expenditure, by a typical firm i, as the solution to,

maxZi,t{λf(zi,t)πi,t − Zi,t} s.t. zi,t =

Zi,tηAi,t−1

that is zi,t solving,

λf ′(zi,t)πi,tηAi,t−1

= 1

where πi,t assumes the value in (π) yielding,

(z) λf ′(zi,t)σ(χ)L/m = 1

21selling price = marg. revenue = 1 = marg. cost = avg. cost.22You can interpret f(z) = F (z, 1) with F being a CRTS function in the two inputs Z, ηA−1, where A−1 is

offered competitively. Most importantly, the role of z is to capture the so called ‘fishing-out’ effect: on average,

each quality improvement is harder than the previous one.

23

Macro dynamics. We now determine the economy consumption growth as follows. In doing

so, we have to consider that two or more sectors can innovate and improve on the intermediate

good quality. Hence, assume that Mt ≤ m is the number of entrepreneurs who innovated in t,

and Ct = m−Mt all the other. Starting with

Yt =

Mt∑i=1

A1−αi,t yαi,t +

Ct∑j=1

A1−αt−1 (yct )

α

( Lm

)1−α

making substitutions, rearranging and using,

At ≡1

Mt

Mt∑i=1

Ai,t

one finds,23

YtL

=

At(αχ

) α

1−α

Mt

m+(At−1 (α)

α

1−α) m−Mt

m

that is, per-capita gdp is approximately equal to the weighted-average productivity. The latest

can be interpreted as the expected productivity, by the LLN, if there is a large number of

sectors whose growth rates are statistically independent and identically distributed, according

23Here are the transformations,

Yt =

(Mt∑i=1

A1−αi,t yαi,t +

Ct∑j=1

A1−αt−1 (yct )

α

)(L

m

)1−α

=

(Mt∑i=1

A1−αi,t

((α

χ

) 11−α Ai,tL

m

)α+ CtA

1−αt−1

((α)

11−α

At−1L

m

)α)(L

m

)1−α

=

(Mt∑i=1

(Ai,t

L

m

)1−α((

α

χ

) 11−α Ai,tL

m

)α+ Ct

(At−1

L

m

)1−α((α)

11−α

At−1L

m

)α)

=

Mt∑i=1

Ai,tL

m

(α

χ

) α

1−α+ CtAt−1

L

m(α)

α

1−α

=

(1

Mt

Mt∑i=1

Ai,tL

(α

χ

) α

1−α)Mt

m+(At−1L (α)

α

1−α) Ctm

= L

[(At

(α

χ

) α

1−α)Mt

m+(At−1 (α)

α

1−α) Ctm

].

24

to f (i.e. Mt/m ≈ ξt ≡ λf(zt)). Since, At ≈ λf(zt)× ηAt−1,

YtL

=

At(αχ

) α

1−α

Mt

m+(At−1 (α)

α

1−α) m−Mt

m

=

(α

χ

) α

1−αλf(zt)ηAt−1 + (α)

α

1−α (1− λf(zt)ηAt−1)

=

(αχ

) α

1−αξt + (α)

α

1−α (1− ξt)

︸ ︷︷ ︸

≡ζ(zt)

At−1

= ζ(zt)At−1

Accordingly, the expected growth rate of productivity is,

at ≡ Et(

AtAt−1

)− 1 = λf(zt)× (η − 1) + (1− λ) f(zt)× 0 = λf(zt)(η − 1)

and the expected growth rate of gdp per-capita gt is approximately given by the sum of the

growth rate of ζ(zt) and at−1.

Balance growth. Balance growth entails, at all dates t, zt = z∗, ζ(zt) = ζ(z∗) and constant

growth rates,

(g) at = a∗ = λf(z∗)(η − 1), gt = g∗ = a∗

Qualitative predictions and policy considerations. Assuming f(z) =√

2z, from (z) we

find z∗ = (λσL/m)2/2, which used in the balance growth equation (g) yields,

(g’) g∗ = a∗ = λ2σ(χ)(η − 1)L/m

Equation (g′) delivers several comparative-statics results, each with important policy impli-

cations:

(1) Growth increases with the productivity/ability of innovations λ and with the supply of

(skilled) labor L;24 both results point to the importance of (higher quality) education as

a growth-enhancing factor. Countries that invest more in higher education will achieve

a higher productivity of research activities and also reduce the opportunity cost of R&D

by increasing the aggregate supply of skilled labor. An increase in the size of population

should also bring about an increase in growth by raising L. This ‘scale effect’ has been

challenged in the literature and will be discussed in the next section.

(2) Growth increases with the size of innovations, measured by η. This points to the

existence of a wedge between private and social innovation incentives: a decrease in η

would reduce the cost of innovation in proportion to the expected rents, leaving the

size of R&D investment z unaffected (see that in equation (z) η is absent). However,

24In the model presented, higher L increases the number of workers in the final-good industry, rising output

and the demand for intermediate products. This, fixed χ, increases innovator profits also in relative terms;

thereby, providing a further incentive to invest in R&D. The latest increases expected growth and innovation.

25

equation (g′) shows that the social benefit from R&D, in the form of enhanced growth,

is proportional not to η but to the ‘incremental size’ 1−η. This implies that, when η ≈ 1

it is not socially optimal to spend on R&D as when η > 1 and large, exactly because

it would have a little social benefit. Instead, in a market economy the same z would

emerge independently on the level of η − 1. This is not surprising if we think at R&D

has having an externality effect on expected economic growth that is not internalized

in the laissez-faire economy.

(3) Growth is decreasing with the degree of product market competition and/or with the

degree of imitation as measured inversely by χ. Thus patent protection (or, more

generally, better protection of intellectual property rights), will enhance growth by in-

creasing χ and therefore increasing the potential rewards from innovation. Instead,

pro-competition policies will tend to discourage innovation and growth by reducing

χ and thereby forcing incumbent innovators to charge a lower limit price. Existing

historical evidence supports the view that property rights protection is important for

sustained long-run growth; however the prediction that competition should be unam-

biguously bad for innovations and growth is questioned by all recent empirical studies,

starting with the work of Nickell (1996) and Blundell et al (1999). The Schumpeterian

framework outlined above can be extended so as to reconcile theory and evidence on

the effects of entry and competition on innovations, and that it also generates novel

predictions regarding these effects which are borne out by empirical tests.25

Problem 3. Assume that every agent is endowed with one unit of labor that she offers inelas-

tically, and that intermediate goods are produced using labor instead of the final good. Also,

assume that

• labor is hired at the market wage rate w,

• the technology used in the intermediate sector is one that produces xθ/θ units of inter-

mediate good i out of x units of labor, 0 < θ < 1,

• there is a continuum of intermediate goods with different quality/productivity and the

final-good is produced according to the following,

Yt =

∫ 1

0

(A1−αi,t yαi,t

)di

Then, derive again the demand and supply functions and discuss the effect of a change in the

wage rate (do some comparative statics).

4.5. Scale effect. In two influential papers, Jones (1995a, 1995b)26 argued that earlier Schum-

peterian growth models incorporate a scale-effect property: the rate of technological progress

and the growth rate of output is proportional to the level of R&D investment services. For

25For a recent reference, see for example Futagami, K., & Iwaisako, T. (2007). Dynamic analysis of patent

policy in an endogenous growth model. Journal of Economic Theory, 132(1), 306-334.26Jones, C. (1995a), Time Series Tests of Endogenous Growth Models, Quarterly Journal of Economics

110: 495-525. Jones, C. (1995b), R&D-Based Models of Economic Growth, Journal of Political Economy 103:

759-784.

26

instance, in the first model of section 4.3,

g∗ :=EtYt+1 − Yt

Yt= λzt(η − 1)

where zt is the fraction of the labor force employed in R&D, if one doubles zt, then the level

of R&D investment and the growth rate double as well. In the second model,

g∗ = a∗ = A/A = λf(z)(η − 1)

and, assuming f(z) =√

2z, we found z∗ = (λσL/m)2/2 and,

g∗ = a∗ = λ2σ(χ)(η − 1)L/m

Moreover, if we had positive population growth, the scale-effects property would imply that per

capita growth rate would increase over time and not converge to a balance growth equilibrium.

Jones argued that the scale-effects property is inconsistent with time-series, empirical evi-

dence from several advanced countries. This evidence shows that resources devoted to R&D

have increased exponentially, but the growth rates of total factor productivity and per capita

output has remained roughly constant over time. In the United States, for example, the number

of scientists and engineers engaged in R&D has grown by a factor of five since the 1950s with no

significant trend increase in productivity growth. Similarly, the fact that productivity-adjusted

R&D has grown substantially over the same period rejects the version of the model presented in

section 4.4 above in which productivity growth is a function of productivity-adjusted research.

Schumpeterian theory has dealt with this problem incorporating Youngs (1998) insight that

there is a kind of decreasing marginal ‘productivity’ of R&D in terms of its innovation capacity:

as an economy grows, proliferation of product varieties reduces the effectiveness of R&D aimed

at quality improvement, by causing it to be spread more thinly over a larger number of different

sectors.27 When technology is modified to account for this effect, the theory is consistent with

the above empirical facts; namely, the coexistence of a stationary TFP growth and rising R&D

input.

As an illustration, take the economy in section 4.4 and suppose that the number of sectors m

(also reflecting quality innovation) is proportional to the size of population L. For simplicity,

let m = L and assume f(z) =√

2z. Then, (g′) becomes,

g∗ = a∗ = λ2σ(χ)(η − 1)

It follows that all the qualitative predictions of the above model are confirmed except that

now the growth rate is independent of population size. This is because in this new version, at

balance growth, the growth-enhancing effect of rising R&D input is just offset by the negative

effect of product proliferation.

27Variants of this idea have been explored by van de Klundert and Smulders (1997), Peretto (1998), Dinopou-

los and Thompson (1998) and Howitt (1999).

27

4.6. Shumpeterian growth and development. How do Shumpeterian growth theory deal

with the empirical evidence on growth convergence-divergence? We are going to argue that

these models can also shed light on why some countries that were initially poor have managed to

grow faster than industrialized countries, whereas others have continued to fall further behind.

The history of cross-country income differences exhibits mixed patterns of convergence and

divergence. The most striking pattern over the long run is the ‘great divergence’ - the dramatic

widening of the distribution that has taken place since the early 19th Century. Pritchett

(1997) estimates that the proportional gap in living standards between the richest and poorest

countries grew more than five-fold from 1870 to 1990, and according to the tables in Maddison

(2001) the proportional gap of income per capita between the richest group of countries and

the poorest grew from 3 in 1820 to 19 in 1998.28 But over the second half of the twentieth

century this widening seems to have stopped, at least among a large group of nations, with

most countries are converging to parallel growth paths.29

However, the recent pattern of convergence is not universal. In particular, the gap between

the leading countries as a whole and the very poorest countries as a whole has continued to

widen. The proportional gap in per-capita income between Mayer-Foulkes’s (2002) richest and

poorest convergence groups grew by a factor of 2.6 between 1960 and 1995, and the proportional

gap between Maddison’s richest and poorest groups grew by a factor of 1.75 between 1950 and

1998. Thus as various authors have observed, the history of income differences since the mid

20th Century has been one of ‘club-convergence’; that is, all rich and most middle-income

countries seem to belong to one group, or ‘convergence club’, with the same long-run growth

rate, whereas all other countries seem to have diverse long-run growth rates, all strictly less

than that of the convergence club.

The explanation we develop in this section for club convergence follows Howitt (2000), who

took the cross-sectoral-spillovers variant of the closed-economy model described in section 4.4

above and allowed the spillovers to cross international as well as intersectoral borders. This

international spillover, or ‘technology transfer’, allows a backward sector in one country to

catch up with the current technological frontier whenever it innovates. Because of technology

transfer, the further behind the frontier a country is initially, the bigger the average size of

its innovations, and therefore the higher its growth rate for a given frequency of innovations.

As long as the country continues to innovate at some positive rate, no matter how small, it

will eventually grow at the same rate as the leading countries. (Otherwise the gap would

continue to rise and therefore the countrys growth rate would continue to rise.) However,

countries with poor macroeconomic conditions, legal environment, education system or credit

markets will not innovate in equilibrium and therefore they will not benefit from technology

transfer, but will instead stagnate. What makes difficult for poor countries to adopt or copy

foreign technologies is the fact that technology transfer requires the receiving country to invest

resources in order to master foreign technologies and adapt them to the local environment.

28The richest group was Western Europe in 1820 and the ‘European Offshoots’ (Australia, Canada, New

Zealand and the US) in 1998. The poorest group was Africa in both years.29In particular, the results of Barro and Sala-i-Martin (1992), Mankiw, Romer and Weil (1992) and Evans

(1996).

28

Although these investments may not fit the conventional definition of R&D, they play the

same role as R&D in an innovation-based growth model; that is, they use resources, including

skilled labor with valuable alternative uses, they generate new technological possibilities where

they are conducted, and they build on previous knowledge. While it may be the case that

implementing a foreign technology is somewhat easier than inventing an entirely new one,

this is a difference in degree, not in kind. For simplicity, Shumpeterian models tend to ignore

that difference in degree and treats the implementation and adaptation activities undertaken by

countries far behind the frontier as being analytically the same as the research and development

activities undertaken by countries on or near the technological frontier. For all countries they

assign to R&D the role that Nelson and Phelps (1966) assumed was played by human capital,

namely that of determining the countrys ‘absorptive capacity’.

In summary, what is interesting about this approach is that it does not only deliver a theory

of club convergence but also a theory of the worlds growth rate and of the cross-country

distribution of productivity, as well as of technological innovation and adoption.

A model of technological transfer. Consider one country in a world of H different countries.

This country looks just like the ones described in the basic model of section 4.4 above, except

that whenever an innovation takes place in any given sector the productivity parameter attached

to the new product will be an improvement over the pre-existing global frontier. That is, let

At−1 be the maximum productivity parameter over all countries in the sector at the end of

period t − 1; in other words the ‘frontier’ productivity at t − 1. Then an innovation at date

t will result in a new version of that intermediate sector whose productivity parameter is

At = ηAt−1, which can be implemented by the innovator in this country, and which becomes

the new global frontier in that sector. The frontier parameter will also be raised by the factor

η if an innovation occurs in that sector in any other country (i.e. there are cross-countries

technological spillovers). Therefore each country domestic productivity evolves according to:

lnAt =

{ln At−1 + ln η = ln At, with probability ξt = λf(zt);

lnAt−1, with probability 1− ξt.

and zt is the productivity-adjusted R&D, zt = Zt/(ηAt−1), with the targeted productivity

parameter that is defined with respect to the world technological frontier.

The global frontier advances by the factor η with every innovation regardless where in the

world it has taken place:

ln At =

{ln At−1 + ln η, with probability ξt =

∑Hh=1 ξ

ht =

∑Hh=1 λ

hf(zht );

ln At−1, with probability 1− ξt.

The long-run average growth rate of the productivity frontier is,

at = ξt(ln At − ln At−1) = ξt ln η

If there were no international trade in both the intermediate and final goods; then R&D

cost-benefits would be same as in the previous model, except that the domestic productivity

parameter At would differ from the global parameter At. Each innovation of the referent

29

country aims to improve on the actual distance from the productivity frontier,

dt−1 ≡ ln At−1 − lnAt−1

Hence, in the event a country innovates, the change of its productivity is,

lnAt − lnAt−1 = ln At−1 + ln η − lnAt−1

that is,

lnAt − lnAt−1 = dt−1 + ln η

As Gerschenkron (1952) argued when discussing the ‘advantage of backwardness’, the greater

the distance the larger the innovation. The average productivity growth rate will again be the

expected frequency of innovations times size:

at = Et−1 (ln η + dt−1)

which is larger than the distance from the frontier dt−1. The distance variable dt evolves

according to:

dt =

dt−1, with probability 1− ξt;ln η + dt−1, with probability ξt − ξt;0, with probability ξt.

That is, with probability 1− ξt there is no innovation at all, either globally or in this country,

so both domestic productivity and frontier productivity remain unchanged; with probability

ξt − ξt an innovation will occur in the considered sector of some other country, implying that

domestic productivity remains the same but the productivity gap dt−1 grows by the factor η;

and with probability ξt an innovation will occur in this sector in considered sector and country,

implying that the country itself moves up to the frontier (i.e., its productivity gap dt−1 goes

to zero). Therefore, the expected distance from the frontier evolves according to,

Et−1dt = (1− ξt)dt−1 + (ξt − ξt)(ln η + dt−1)

= (1− ξt)dt−1 + (ξt − ξt) ln η

Assume a constant R&D intensity z∗, implying constant probability to innovate ξ∗. If ξ∗ > 0

this is a stable difference equation with a unique rest point. That is, as long as the country

continues to perform R&D at a positive intensity z its distance to the frontier will stabilize,

meaning that its productivity growth rate will converge to that of the global frontier. Otherwise

the difference equation has no stable rest point and dt is expected to diverge to infinity. That is,

if the country stops innovating it will have a long-run productivity growth rate of zero because

innovation is a necessary condition for the country to benefit from technology transfer.

We now ask under which conditions we can expect a country R&D to be zero or positive.

Recall that z is decided so as to satisfy (z). More precisely if we account for the fact that a

solution may not be interior (i.e. z ≥ 0 may bind), then we have,

λf ′(z)σ(χ)L/m ≤ 1, with equality if z > 0

Moreover, generically, z = 0 only if,

λf ′(0)σ(χ)L/m < 1 ⇔ f ′(0) <1

λσ(χ)L/m

30

That is if the marginal probability to innovate is sufficiently large the country will always

find optimal to choose z > 0. In the case in which limz→0 f′(z) = +∞ (i.e. an Inada type of

condition holds), there will always be an interior solution z > 0. Under more general conditions

z > 0 for countries which have a small R&D ability λ (such as developing economies), a

relatively small labor force L/m which can be used in the R&D sector, a scarce level of property

right protection χ.

To sum up, countries may fall into two groups, corresponding to two convergence clubs:

(1) Countries with highly productive R&D, as measured by λ, or good educational systems

as measured by high λ or high L, or good property right protection as measured by a

high χ, will grow asymptotically at the frontier growth rate a.

(2) Countries with low R&D productivity, poor educational systems and low property right

protection will not grow at all, thereby increasing their distance from the frontier coun-

tries.

This simple theory of world growth distribution sheds also light on the importance of in-

stitutions in explaining some divergent patterns observed in the data. The relevance of the

educational (or more broadly of a training) system to prepare the labor force to develop and

‘absorb’ innovation is crucial in explaining countries potentials to catch-up. Also the law and

property right enforcement system is important, since it provides those economic incentives

that are necessary to enhance R&D.

A branch of literature have address the important issues on whether markets openness might

help relatively backward economies to grow more rapidly and catch up with more advanced

ones.

4.7. Property right protection versus competition. The prototype Shumpeterian model

we have described above unambiguously predicts that more property right protection increases

economic growth. This is clear from equation (g′) and is explained by the fact that protection

yields monopoly rent and monopoly rent incentivate R&D expenditure. This prediction, not

only the contradicts a common wisdom that goes back to Adam Smith, but it has also been

shown to be (partly) counterfactual. An increasing number of empirical studies have cast

doubt on this prediction. The empirical industrial organization (IO) literature on competition

and innovation starts with the pioneering work of Scherer (1965), followed by Cohen-Levin

(1967), and more recently by Geroski (1994).30 All these papers point to a positive correlation

between competition and growth. One of the strongest arguments in IO theory is that too

much protection might have detrimental effect on innovation, as it discourages incumbents

from investing in R&D. Indeed, if they had to innovate before their patent protection has

expired, they would loose the associated rents; in some cases, a perpetual protection would

drive patent holders’ R&D to zero.

In this section we adapt our basic Shumpeterian model to reproduce a situation in which the

relationship between expected growth and property right protection is non-monotonic: some

protection is desirable but too much is not. We do so showing that patent protection might

incentivate R&D expenditure and growth of backwards sector/countries but disincentivate R&D

30See also Nickell (1996), and Blundell et al (1999).

31

and innovation at the frontier. The theory developed in this section is based on Aghion-Harris-

Vickers (1997) and Aghion-Harris-Howitt-Vickers (2001), which reconcile theory and evidence

on the effects of competition and growth.

We start by considering an isolated country in a variant of the technology transfer model of

the previous section. At the beginning of date t there is a global technological frontier At−1

that is common to all sectors, and which is drawn on by all innovations. Technologies are

as above and, for simplicity, we let m/L = 1. The final-good is produced according to the

following,

Yt =

(m∑i=1

A1−αi,t yαi,t

)The final good, which is again assumed to be the numeraire, is both used for the individuals’

consumption and, as an input, in the production of the intermediate good. Any intermediate

good i of quality min{At−1, Ai,t} can be reproduced at a cost χ, where 0 < χ < 1/α < ηχ.

That is, firms can compete on the the production of i, which is either at or below the current

frontier At−1, but they have to pay a unit cost χ (e.g. a patent fee) to produce it. Instead,

an innovation over the global frontier technology, yielding At = ηAt−1 cannot be reproduced.

As above, competition nails down to χ the price of each quality i good that can be reproduced

(see section 4.4).

In each period t, there are three types of sectors and firms, indexed by i = 0, 1, 2. A type-i

sector starts up at the beginning of period t with productivity Ai,t−1 = At−1−i (i.e. i steps

behind the current frontier). So, type i = 0 is on the frontier, i = 1 is one step behind, at At−2,

and i = 2 is at At−3. Between the beginning and the end of the current period t, the incumbent

firm in any sector i has the possibility of innovating with positive probability ξit. Innovations

occur step-by-step, with fixed increments η − 1. Incumbent firms can affect the probability of

an innovation by investing more in R&D at the beginning of the period. For simplicity, we let

them directly choose their own innovation probabilities ξt, assuming that each of them faces a

quadratic investment cost,31

1

2ηAi,t−1ξ

2t Φ2, where Ψ2 = 0 if i = 2 and Ψ2 = 1 otherwise

In words, innovation is assumed to be automatic (occurs with probability one, at no cost) in

type-2 sector, the most underdeveloped one. This assumption is thought to reflect a knowledge

externality from more advanced sectors to type-2, which effectively limit the maximum distance

of any sector to the technological frontier.

A firm in type-1 sector. Consider a firm i that starts in a type-1 sector (i = 1), facing a

frontier At−1. Conditional, on innovating, her end-of-the-period productivity is,

Ai,t =

{ηAt−2 = At−1, with probability ξit;

At−2, with probability 1− ξit.

Innovation grants to i the possibility to produce the good of quality At−1 at a unit marginal

cost; she can produce any other intermediate good at a marginal cost of χ. As anyone else has

31This is a minor departure from the earlier model if one assumes that probabilities map one-to-one to R&D

expenditure according to the function f . That is, once ξ∗ is chosen z∗ = f−1(ξ∗) is uniquely pin down.

32

this competitive option, the innovation can at most be sold at a price χ. Therefore, profits

cannot exceed,

πi,t = Ai,tσ(χ), σ(χ) ≡ (χ− 1)(χα

) 1α−1

The net rent to innovate is the difference in profits across the two states,(At−1 − At−2

)σ(χ)

A firm i in sector 1 has the objective,

maxξ(1)

{ξ(1)

(At−1 − At−2

)σ(χ)− 1

2ηAt−2ξ(1)2

}A concave problem with solution,

ξ(1) =

(1− 1

η

)σ(χ)

Since σ′ > 0, as above, more protection (↑ χ) yields higher innovation incentives and higher

R&D effort ξ(1). This is the usual Shumpeterian effect: competition reduces innovation.

A firm in type-0 sector. We now consider a firm in a sector that is at the frontier technology.

Since innovating creates a novel good, no one can immediately copy it: the incumbent of a sector

j = 0 is a monopolist. Let us solve this firm problem. The firm faces a demand curve for y0,t

that is equal to the marginal product of this good in the final-good production:

∂Yt∂y0,t

= αA1−α0,t y

α−10,t

where, A0,t = At if the firm innovates and A0,t = At−1 otherwise. Upon observing that she has

innovated, the firm solves,

π0,t = maxy0,t

{(αA1−α

t yα−10,t − 1

)y0,t

}which yields the supply,

y0,t = At

(1

α

) 2α−1

a price p0,t = 1/α and profits,32

π0,t = Atσ(1/α)

Instead, if the firm does not innovate, her good can be copied, implying that,

π0,t = At−1σ(χ)

Therefore, a firm i in sector 0 has an R&D objective,

maxξ(0)

{ξ(0)

(Atσ(1/α)− At−1σ(χ)

)− 1

2ηAt−1ξ(1)2

}32You should verify all these computation, recalling that,

σ(1/α) = (1/α− 1)

(1

α2

) 1α−1

33

A concave problem with solution,

ξ(0) = σ(1/α)− 1

ησ(χ)

where ξ(0) > 0 if and only if,

0 < σ(1/α)− 1

ησ(χ)

= (1/α− 1)

(1

α2

) 1α−1

− 1

η(1/χ− 1)

(χα

) 1α−1

A sufficient condition for the latest to hold is that ηχ > 1/α, as we have assumed above.

Observe that ξ(0) is decreasing in the level of protection χ; that is, more competition pro-

duces an incentive of firms in a type-0 sector to innovate, essentially, to preserve their position

of ‘absolute’ monopoly (with no competitive fringe χ) rather than one of ‘relative’ market

power (with a competitive fringe χ). In fact, a firm in type-0 sector who succeeds, prices at

1/α > χ > 1 and attains the highest possible net rent. Therefore, more competition makes less

convenient to be a firm in type-1 sector, while it leaves unchanged the profit opportunities of

being on the productivity frontier; this is why more competition boosts the incumbent’s R&D

expenditure in type-0 sector. For this reason we shall call this the escape competition effect.

Long-run R&D incentives and competition. We have just seen that product market com-

petition tends to have opposite effects on frontier and lagging sectors, fostering innovation of

the firms already at the frontier and discouraging innovation of those in the less productive

sector. Next, we consider the impact of competition on the steady-state aggregate innovation

intensity I. We measure this ‘intensity’ as a weighted average of the innovation probabilities

of those firms who invest in R&D. Assuming that that qi is the steady-state fraction of type i

sector and recall that type i = 2 sector does not perform R&D,

(I) I = q0ξ0 + q1ξ1

To get a non-trivial steady-state fraction of type-0 firms, we need that the net flow out of state

0, q0(1− ξ0) (which corresponds to type-0 firms that fail to innovate in the current period), be

compensated by a net flow into state 0. We simply postulate such a inner flow occurs at the

end of any period t, with an exogenous probability ε, and leads type-2 firms to rich the new

frontier At: the flow into 0 is q2ε. This is the first condition in (g) below. Next, former type-0

firms who do not innovate fall into the type-1 sector. We impose that this inflow in sector 1,

q0(1 − ξ0), equal the outflow of firms that where in sector 1 who fail to innovate, q1(1 − ξ1).

This is the second condition in (g) below. Finally, the flow of type-2 firms who innovate and

move to the frontier, q2ε, is assumed to equate the inflow, given by the firms who were in sector

1 and did not innovate, q1(1− ξ1). Summarizing, at a steady state, given ε, (q0, q1, q2) satisfy

the following flow-restrictions,

q2ε = q0(1− ξ0)(q)

q0(1− ξ0) = q1(1− ξ1)

q1(1− ξ1) = q2ε

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along with the natural restriction,

q0 + q1 + q2 = 1

We can use all these condition to rewrite (I) as,33

I = 1− q2(1 + 2ε)

where,34

q2 =1

1 + ε1−ξ0 + ε

1−ξ1The overall effect of increased product market competition (a reduction of the patent pro-

tection χ) on R&D intensity I is ambiguous, as it produces opposite effects on innovation

probabilities of the firms operating in type-0 and type-1 sectors (i.e. negative on ξ0 and posi-

tive on ξ1). Moreover,

• the escape competition effect always dominates for η sufficiently close to one (q2 ↓, I ↑);• the Schumpeterian effect always dominates for η > 1 sufficiently large (q2 ↑, I ↓);• for intermediate values of η, the escape competition effect dominates when competition

is initially low (with χ ≤ 1/α), whereas the Schumpeterian effect dominates when

competition is initially high (with χ > 1/α and close to one). In this latter case, the

relationship between competition and innovation is inverted-U shaped with respect to

χ.

The inverted-U pattern can be explained as follows: at low initial levels of competition (i.e

high initial levels of σ(χ)), type-1 firms have strong incentives to innovate; it follows that

many firms in the intermediate (type-1) sectors end up being type-0 firms in steady-state (this

we refer to as the composition effect of competition on the relative equilibrium fractions of

type-0 and type-1). This translates into an increase of I if q2 falls as a result of an increase

in type-1 incentive to innovate. But, as there are more firms entering in 0, there will be

33To see this, first, we can rewrite the second and third conditions in (q), respectively, as,

q0ξ0 = q0 − q1 + q1ξ1

q1ξ1 = q1 − q2ε

Summing each member, using (I) and making obvious substitutions,

I = q0 − q1 + q1ξ1 + q1 − q2ε

= (1− q1 − q2) + q1ξ1 − q2ε

= 1− q1 − q2 + q1 − q2ε− q2ε

= 1− q2(1 + 2ε)

34To see this, note the following,

q2 = 1− q0 − q1

= −q11− ξ11− ξ0

− q1

= −(

1− 1− ξ11− ξ0

)q2ε

1− ξ1

Factoring q2 and simplifying, yields the result.

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more competition. If this increases competition it will make dominant the escape competition

effect, whereby more competition fosters innovation by type-0 firms. On the other hand, more

competition reduces innovation incentives of firms in type-1 sector; as χ decreases, they will

tend to remain for a longer time in type-1 and 2 sectors. Thus, for high levels of competition

the negative Schumpeterian effect of competition on innovation tends to dominate; as a result

q2 increases and I falls.

Empirical predictions. The above analysis generates several interesting qualitative implica-

tions:

(1) Innovation in sectors in which firms are close to the technology frontier, react positively

to an increase in product market competition;

(2) Innovation reacts less positively, or negatively, in sectors in which firms are further

below the technological frontier;

(3) The average fraction of frontier sectors decreases, that is the average technological gap

between incumbent firms and the frontier in their respective sectors increases, when

competition increases;

(4) The overall effect of competition on aggregate innovation, is inverted-U shaped.

These predictions have been tested by Aghion-Bloom-Blundell-Griffith-Howitt (2002) with UK

firm level data on competition and patenting, whose main finding are summarized in the sequel.

Their findings are very much in tune with our theoretical discussion in the previous pages: the

escape competition effect dominates in industries in which firms are closest to the technological

frontier. The Schumpeterian effect is also at work, and it dominates at high initial levels of

product market competition. This in turn reflects the ‘composition effect’ pointed out above:

namely, as competition increases, neck-and-neck firms engage in more intense innovation to

escape competition; therefore, the equilibrium fraction of frontier firms tends to decrease;

hence, the average impact of the escape competition effect decreases at the expense of the

counteracting Schumpeterian effect.

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