Lesson 2

“First Generation” theories of economic
growth
 The Harrod-Domar Model
 The Solow “Neoclassical” Growth Model

Convergence

Developed by Sir Roy Harrod (1939) and
Evsey Domar (1946)

Economic growth is the result of sacrificing
current consumption

Households save from current income =>
firms invest these savings => creation of
capital goods => production of output
increases => economic growth

National income, Y(t), is either spent on
consumption goods, C(t), or saved, S(t):
Y(t)=C(t)+S(t)

In a closed economy, the supply of savings
must equal the demand for investment, I(t):
S(t)=I(t)

Investment increases the economy’s stock of
capital, while replacing the part of this stock that
is wearing out:
K (t )  I (t )   K ( t )
 Notation:
x(t )  dx(t ) dt
K (t ) : the economy's stock of capital goods
 : the rate of depreciation of capital

Assumptions of the model:
 Savings is a fixed fraction of income:
S (t )  sY (t ), 0  s  1
 The capital-output ratio is constant:
K (t )   Y (t )

The savings rate, s, and the capital-output ratio,
, are the two key variables in the model

The key equation in the Harrod-Domar model
is:
g
s


 Where, g  Y (t ) Y (t ) : the growth rate of output
 Note: Details of the above equation to be derived
in class

Adding population growth (n) to the model,
we get:
s
g   n 

 g* is now the rate of growth of per-capita output


This growth equation was instrumental in
shaping policies in socialist countries such as
India and the Soviet Union in the 1950s and
1960s

The key variables in the Harrod-Domar model
are:
 The savings rate
 The population growth rate
 The capital-output ratio

All these variables are assumed to be
exogenous, which is unrealistic

What determines these variables?

There is evidence to suggest that the savings rate (s)
rises with income:
 Poor people or countries close to subsistence do not or
cannot save much
 As income rises, there is more room for savings; but the
very rich have less need for savings
 Some inequality might also affect the savings rate: middle
class saves more than poor or rich

This implies that poor and rich countries should have
low growth rates, compared to middle-income
countries

Huge body of evidence suggests that population growth changes
with the level of development: Demographic Transition
 At low levels of per-capita income, death rates are very high. This
leads to high birth rates too. Combined effect is a low rate of
population growth
 As per-capita income rises, living standards improve and death rates
fall sharply. But birth rates don’t. So population growth is high
 At very high levels of per-capita income, birth rates fall as well, and
population growth declines

Relationship between per-capita income and population growth:
“inverse-U”

Developed by Nobel Prize winning
economist, Robert Solow (1955)

Extends the Harrod-Domar story in a very
important direction:
 Endogeneizes the capital-output ratio, .
 Based on the Law of Diminishing Returns
 The capital-output ratio depends on the relative
endowments of capital and labor in the economy

Rate of capital accumulation (per-capita):
k(t )  sy(t )  (  n)k (t )
 Where,

k (t )  K (t ) / L(t )  capital per unit of labor
y (t )  Y (t ) / L(t )  per - capita output
L(t )  Size of labor force
Key difference from Harrod-Domar story: Production
function: per-capita output is a function of per-capita
capital stock:
y  f k , f   0, f   0

Over the long-run, due to diminishing returns, the
per-capita stock of capital stops eventually stops
growing:
k(t )  0 as t  

This situation (stationary per-capita capital stock) is
called the steady-state equilibrium:
~
~
sf (k )    nk

Given the parameters , s, and n, we can determine
~
the unique steady-state per-capita stock of capital, k

There is no per-capita growth in output in the
steady-state (due to diminishing returns to
capital)

Total output and capital grow at the rate of
population growth:
Y K
 n
Y K

An increase in the population growth rate (n):
 Lowers the steady-state level of per-capita output
 Increases the growth rate of total output

Why the “dual” effect?
 Labor is both an input in production as well as a consumer
of final goods

Growth Effect: changes the growth rate of a variable

Level Effect: changes the level of a variable, but
leaves its growth rate unchanged

In the previous version, growth of per-capita
output cannot be sustained indefinitely
 Economy eventually converges to a stationary
equilibrium (constant per-capita output)

What happens if we introduce technical
progress?
 Production function shifts upwards over time as new
knowledge and efficiency is gained and applied
 Does this outweigh the effect of diminishing returns?
 The answer is …

Assumption: Labor-augmenting technical progress
 Technical progress enhances the efficiency or productivity
of the labor force

Let E(t) denote the level of labor productivity and let 
denote its growth rate

Then, the effective labor force is defined as:
E(t)L(t)

Note that the effective labor force is growing at the
rate (n + )

Now, define all quantities in terms of units of
effective labor (not per-capita):
kˆ  K EL : capital per effective worker
yˆ  Y EL : output per effective worker

The accumulation of capital per effective
worker is then given by:
kˆ  syˆ    n   kˆ

At the steady state, there is no growth in
capital per effective worker: kˆ  0 as t  

The level of the capital stock per effective
worker is stationary (constant) in the steadystate, and solves:
sf (kˆ)  n     kˆ

If the capital stock per effective worker is
constant in the steady state, then it must be
the case that:
 Per-capita output (and capital) is growing at the
rate of technical progress, .
 Total output (and capital) is growing at the rate
(n + )

Unconditional Convergence:
 If structural parameters are identical across countries,
then all countries must converge to a common level of
per-capita income, irrespective of initial conditions
 Poor countries have higher marginal product of
capital than rich countries, and therefore grow faster
(“catching-up”)
 History does not matter for the long run

This is one of the strongest predictions of the
Solow model

Baumol (1986)
 Examined growth rates of the world’s richest 16
countries, using data from 1870-1979
 Horizontal axis: plot per-capita income in 1870
(initial condition)
 Vertical axis: plot per-capita income growth from
1870-1979
 Prediction of the model: inverse relationship

Countries in Baumol’s sample:
 Richest countries in 1979
 Japan, Finland, Sweden, Norway, Germany, Italy, Austria,
France, Canada, Denmark, USA, Netherlands, Switzerland,
Belgium, UK, Australia
 But in 1870, Japan was not one of the richest countries;
Argentina, East Germany, and Portugal were rich, but are
excluded from the sample

Only those countries that were successful at the end
of the sample were chosen

Selection Bias: using wisdom after the event

A good test should look at a set of countries
that, ex-ante, seemed likely to converge to
high per-capita GDP levels over time

Use countries with similar initial conditions,
and not similar terminal conditions

De Long (1988) addressed this problem by
extending Baumol’s dataset to include 23
countries, but excluded Japan


Use a large set of countries, but over a short
time horizon
Parente and Prescott (1993)
 Studied 102 countries over the period 1960-85
 Each country’s per-capita GDP is expressed as a
fraction of US per-capita GDP in a given year
 Standard deviation of relative GDP for each year is
calculated
 If convergence hypothesis is true, then standard
deviations should decline over time

Structural parameters (technology, population
growth, savings) differ across countries

Countries converge to their own steady states, and
not to a common steady state

Although long-run levels of per-capita income can
vary across countries, there could be convergence in
growth rates, if technological progress is identical

In testing for convergence, one must control for all
the factors that are different across countries