Empirical Applications of “Neoclassical” Growth Models

(The Solow model and the Ramsey-Cass-Koopmans model are

neoclassical)

1. Level differences accounted for by differences in factor accumulation.

i. Savings rates and their determinants

ii. Population growth

2. Growth rate differences mainly a transitory phenomenon.

Countries are in different positions relative to their eventual balanced

growth path (steady-state).

92

Empirical focus on two types of questions:

1. Do differences in accumulation account well for differences in per capita

income?

2. Do countries “converge” in terms of either income levels or growth rates?

In order to address these questions, we have to be more explicit about what

is meant by “accumulation”.

So far, endogenous factor accumulation has been limited to saving.

This contributes to the accumulation of physical capital: machines, vehicles,

buildings, etc..

93

But people also acquire skills and knowledge

This contributes to the accumulation of human capital.

Human and physical capital differ in a number of ways:

1. Human capital cannot typically be bought and sold.

An implication of this is that it typically cannot be used as collateral for

debt.

2. The two types of capital may “depreciate” differently.

3. Human capital may be more difficult to measure because it is not

observable in the same sense as, say, a factory building.

94

Human and physical capital are also alike in some key aspects:

1. Both can be accumulated through investment activities.

2. Both take time to accumulate.

We will think of the primary input to human capital accumulation as time.

Time spent in school or in learning on the job.

• Time in school is time is not spend working and earning income that can

either be consumed or saved.

• Workers who are learning on the job are typically less productive than

regular workers.

95

Mankiw, Romer, and Weil (1992):

Add human capital to the Solow model:

Y = Kα(AH)1−α (79)

•: Y , K , A: just as before.

•: H = eψuL: human capital/skilled labour

u: time devoted to acquiring skills

ψ: effectiveness of the schooling function:

ψ =d lnH

du(80)

L: As before; hours devoted to work

96

Physical capital is accumulated in the usual way:

K = sKY − dK,

sK : savings rate (rate of capital accumulation)

Proceeding as before:

Y = Kα(AH)1−α

y = kα(Ah)1−α where h = eψu (81)

y =y

Ah= kα (82)

All that has changed is the definition of effective labour units.

97

The capital accumulation equation is completely unchanged by the addition

of human capital:

˙k = sK y − (n+ g + d)k (83)

We can again represent the model:

˙k = sK kα − (n+ d+ g)k (84)

Along the balanced growth path k = 0, so we can solve:

sK kα = (n+ d+ g)k (85)

for the steady-state level of capital per unit of effective labour:

98

k =

[sK

n+ d+ g

] 1

1−α

(86)

y =

[sK

n+ d+ g

] α

1−α

(87)

y(t) = yhA(t) =

[sK

n+ d+ g

] α

1−α

eψuA0egt (88)

The last equation gives the time path for income per capita along the

balanced growth path.

From it we can see that the growth rate is equal to g (the rate of

technological progress), and that the level of y(t) depends on several

factors:

99

1. The parameters α, sK , n, d, and g all have the same effects as in the

basic Solow model.

2. Income per capita is increasing in the initial technology level, A0.

3. Income per capita is increasing in both:

u: the effort devoted to acquiring human capital (schooling time?)

ψ: the effectiveness of that effort (education quality?)

We now want to ask to what extent this model can account for observed

differences in both levels and growth rates across countries.

100

It is useful to express variables relative to those of a base country (Jones

uses the U.S.):

y =y

yUS=

[sK

n+g+d

] α

1−α

eψuA(t)[

sUS

K

(n+d+g)US

] α

1−α

(eψu)USA(t)US

(89)

=

[sKx

] α

1−α

hA(t) (90)

where x = n+ d+ g and all variables are measured relative to their U.S.

counterparts.

We now use data on all the variables on the right-hand side of this equation

to construct measures of per capita income relative to that of the U.S. that

are predicted by the model.

101

We then compare them to actual per capita income for each of these

countries.

Assumptions:

1. Symmetry across countries except with regard to sK , n, and u.

Differences explained by differences in saving, population growth, and

schooling.

2. α = .33

Physical capital’s share of national income is one-third.

3. g + d = .075, A = 1

Both growth and the state of technology equal across countries.

4. ψ = .1

An additional year of schooling raises a worker’s wage by 10%102

In Figure 3.1:

The main failure is that the model predicts poor countries to be richer than

they actually are.

Perhaps the assumption of equal technology levels across countries has

something to do with this:

Y = Kα(AH)1−α

= Kα(AhL)1−α (91)

y = kα(Ah)1−α (92)

Solve this for A(t):

A(t) =[y

k

] α

α−1 y(t)

h. (93)

104

Overall this version of the neoclassical growth model correctly predicts:

1. Countries that accumulate both physical and human capital at a high rate

are relatively rich.

2. Countries that use these inputs productively (that is, have a high level of

technology or total factor productivity) are rich.

But, the model does not explain why these countries accumulate so much

or are so productive.

The main overall problem with the model is that it predicts that poor

countries will be richer than they are.

Another way to look at this is to say that the model over-predicts these

countries levels of total factor productivity.106

Differences in Growth Rates and “Convergence”

According to the model, countries at a low level of per capita income (having

low levels of both physical and human capita) should tend to accumulate

factors quickly and grow at a rapid pace in the short-run.

Over time, they will converge to the rich countries:

1. in growth rates (to the common rate of technological progress, g).

2. in levels conditional on all the parameter differences (as we have seen).

107

• Convergence appears to be taking place among the rich countries of the

world (OECD countries).

• But, the poor countries are not catching up to the rich ones.

Poor and rich countries may, however, be converging conditionally. That is,

they may be converging in growth rates and to very different steady-states.

According to the model, a poor country with a low level of k will grow at a

lower rate than a rich country with a high k if it is closer to its steady-state:

112

.06

.08

.12

.15

.18

.21

Mar

gina

l pro

duct

and

n+

d+g

1 1.5 2.22 3 4

Capital per unit of effecitve labour (k/A)

Pop. growth, depr., and tech prog. Marginal product 1Marginal Product 2

Growth equals red minus blueThe Solow Model

Another way to look at the issue of convergence is to consider the evolution

of the world-wide income distribution.

By some measures, this suggests convergence; by others it doesn’t

Overall, the neoclassical growth model appears to do a fairly good job of

accounting for differences in both levels of per capita income across

countries and different countries growth experiences.

115

Fig. 3.10