Solow Model Develompent Economics

ASSUMPTIONS

Consider a closed economy with no government with a large number of small firms producing a

single homogeneous good, Y, using two inputs: labour (N), and capital (K), which can be

produced and accumulated.

Population and the labour force are the same.

Inputs are hired from households, who are also the owners of the firms and the consumers in this

economy.

Households spend a fraction (1-s) of their income on consumptions, and save the remaining, to

buy new capital.

The capital stock depreciates at a constant rate,

In this model, the ability to accumulate capital (via savings) prevents output per capita from

declining when population increases.

Output is produced by neoclassical production function Y = F(K,L) that has usual properties

1. (diminishing marginal product, constant returns to scale).

The economy is closed, savings are equal to investments, S I.

2. ≡

Population growth is given exogenously (outside of the model).

3. The production function has diminishing return

to per capita capital.

As capital increases, output falls because of

shortage of labor.

It represents the technical knowledge of the

economy.

Net national product Y is a function of capital K

and labor L, Y =F(K,L).

This aggregate production function is fixed; how

the product depends on capital and labor does

not change as time passes.

The production function exhibits constant

returns to scale; doubling the capital and labor

doubles the output.

Depreciation increases at a constant rate as the

capital stock increases. —> the more capital you

have, the more capital depreciation you have.

Where does money for capital accumulation

came from? from saving and investments.

Depreciation is growing at the same rate as the

capital stock grows.

Each unit of capital creates creates an equal

amount of depreciation.

When I > depreciation:

Capital stock is growing, as the capital stock

grows, investment and depreciation intersect at

one point, the steady state level of capital

where I = depreciation.

When I < depreciation: some of the capital

stock needs repair but there isn’t enough

investments to do all the needed repairs and

capital stock shrinks, pushing back to the steady

In words, since the economy generates savings (and hence new investment) larger than the

amount needed to keep the amount of capital per worker constant, the capital-labour ratio will

increase.

The figure displays the production function in the intensive form, per capita savings and the break

even investment line.

The steady state occurs when the break even investment line crosses the schedule of per capita

savings.

If the economy starts out on the left (right) of the steady state, per capita savings will exceed (be

less than) the required to keep the capital labour ratio constant, so the capital will increase

(decrease).

If the economy starts out on the right of the steady state, per capita savings will be less than the

required to keep the capital labour ratio constant, so the capital will decrease.

TRANSITIONAL DYNAMICS

An important feature of the Solow model is that, if the economy is not in the steady state, it will

converge to the steady state.

But the economy will not jump instantaneously from one steady state to the other: since capital

accumulation is bounded by the availability of savings, there will be an adjustment period, during

which the economy approaches the new steady state.

HOW PARAMETERS AFFECT THE STEADY STATE

Savings rate

•

Although the saving rate s does raise the rate of economic growth in the short run, it has no effect

on the rate of growth in the long run.

A higher value s does raise the steady-state capital/labor ratio k.

Hence the steady-state output per capita rises.

K*/ Y* = S / n+d

The Figure shows how the steady state in the

Solow model changes with an exogenous

increase in the saving rate.

In the new steady state (point 1), the capital

labour ratio and per capita output are higher than

before, but the average product of capital (Y/K) is

lower, due to diminishing returns.

In the long term, per capita income is again

constant (the increase in the saving rate

produced a “level effect”).

Population

•

The rate of population growth sets the long-run growth rate of the economy.

If the population growth rate n rises, the capital-widening term nk rises.

Consequently the steady-state capital/labor ratio k falls and the steady-state output per capita falls.

In the steady state, the real interest rate is now higher, and the real wage is lower.

A fall in the population growth rate has a similar effect to that of a rise in the savings rate.

The difference is that the change in the steady state will be caused by a downward shift of the

break-even investment line.

Thus, both output per worker and capital per worker will increase, but this will happen only during

the transition from one steady state to the other.

In the new steady state, the average product of capital - and interest rate - will be lower than in the

initial steady state.

Suppose population growth changes from n1 to n2.

This shifts the line representing population growth and depreciation upward.

At the new steady state k*2 capital per worker and output per worker are lower

The model predicts that economies with higher rates of population growth will have lower levels of

capital per worker and lower levels of income.

Population growth has two effects:

effect: leaves the rate of growth unchanged, while

1.Level

shifting up or down the entire path traced out by the variable

over time. effect: effect that change the rate of growth of a

2.Growth

variable, typically income or per capita income.

SOLOW WITH TECHNICAL PROGRESS

In the absence of technical progress, a country cannot sustain per capita income growth

indefinitely.

For this to happen, capital must grow faster than population, but then diminishing return implies

that the marginal contribution of capital to output must decline, which forces a decline in the growth

rate of output and, therefore, of capital. At the point where both (k) and (y) are constant it must

be the case that,

Δk = s*f(k) – (δ+n+g)k = 0 this occurs at our

equilibrium point k*.

Like depreciation and population growth, the labour

augmenting technological progress rate causes the

capital stock per worker to shrink.

Suppose the worker efficiency growth rate changes

from g to g .

1 2

This shifts the line representing population growth,

depreciation, and worker efficiency growth upward.

At the new steady state k*2 capital per worker and output per worker are lower.

The model predicts that economies with higher rates of worker efficiency growth will have

lower levels of capital per worker and lower levels of income.

CONVERGENCE

At the heart of the Solow model is the prediction of convergence.

”Unconditional convergence" occurs when the income gap between two countries decreases

1. irrespective of countries characteristics (e.g., institutions, policies, technology or even

investments).

"Conditional convergence" occurs when the economic gap between two countries that are

2. similar in observable characteristics is becoming narrower over time.

Unconditional convergence

If we think that, in the long run, countries tend to have the same rate of technical progress,

savings, population growth, and capital depreciation.

In such a case, the Solow model predicts that in all countries, capital per efficiency unit of labor

converges to the common value k^, and this will happen irrespective of the initial state of each of

these of these economies. The graph illustrates unconditional

convergence. It plots the logarithm of income

against time, so that a constant rate of

growth appears as a straight line.

The line AB plots the time path of (log) per

capita income in steady state, where income

per efficiency unit of labor is precisely at the

level generated by kˆ⇤.

The path CD represents a country that

starts below the steady-state level per

eciency unit.

According to the Solow model, this country

will initially display a rate of growth that

exceeds the steady-state level, and its time

path of (log) per capita income will move

asymptotically toward the AB line as shown.

Over time, its growth rate will decelerate to the steady-state level.

Likewise, a country that starts o↵ above the steady state, say at E, will experience a lower rate of

growth, because its time path EF of (log) income flattens out to converge to the line AB from

above. Unconditional convergence, then, is indicated by a strong negative relationship between the

initial value of per capita income and subsequent growth rates of per capita income.

Unconditional Convergence: Evidence or Lack Thereof

Time Horizons: The first problem that arises when testing a hypothesis of this sort is the issue of

time horizons.

Ideally, we’d like to go back a century or more in history, but the systematic collection of data in

developing economies is a modern phenomenon.

There are just two choices: cover a relatively small number of countries over a large period of time

or cover a large number of countries over a short period of time.

The Baumol Study. That last sentence held true a fortiori in 1986, when William Baumol published

one of the first studies of long-run convergence.

At the time, there were just sixteen countries in Maddison’s database for which “reliable” estimates

of per-capita income existed (and I put “reliable” in quotes because this sort of historical detective

work must always be taken with a large pinch of salt).

These were, in order of poorest to richest in 1870: Japan, Finland, Sweden, Norway, Germany,

Italy, Austria, France, Canada, Denmark, the United States, the Netherlands, Switzerland, Belgium,

the United Kingdom, and Australia.

They are among the richest countries in the world today.

The graph plots 1870 per capita income for these sixteen countries on the horizontal axis, and the

growth rate of that income over 1870–1979 (measured by the difference in the logs of per capita

income over this period) on the vertical axis.

The convergence of these sixteen countries to one another, starting from widely different levels of

per capita income in 1870, is unmistakeable.

It appears, then, that Baumol’s finding supports the unconditional convergence hypothesis quite

strongly.

Selection Bias. Unfortunately, there’s a classic statistical pitfall lurking both in the picture and in

the study. The sixteen countries are the first to have historical records for good reason: they are

rich countries today!

Yet in 1870 they were all over the economic map. Japan is a perfect case in point. It is there

precisely because of hindsight: Japan is rich today, but in 1870, it was probably midway in the

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