The
exogenous growth model, also known as the
neo-classical growth model or
Solow–Swan growth model is a term used to sum up the contributions of various authors to a model of long-run
economic growthEconomic growth is a term used to indicate the increase of total GDP. It is often measured as the rate of change of gross domestic product . Economic growth refers only to the quantity of goods and services produced; it says nothing about the way in which they are produced...
within the framework of
neoclassical economicsNeoclassical economics is a term variously used for approaches to economics focusing on the determination of prices, outputs, and income distributions in markets through supply and demand, often as mediated through a hypothesized maximization of income-constrained utility by individuals and of...
.
Development of the model
The neo-classical model was an extension to the 1946 Harrod–Domar model that included a new term,
productivityProductivity is a measure of output from a production process, per unit of input. For example, labor productivity is typically measured as a ratio of output per labor-hour, an input. Productivity may be conceived of as a metric of the technical or engineering efficiency of production. As such, the...
growth. The most important contribution was probably the work done by
Robert SolowRobert Merton Solow is an American economist particularly known for his work on the theory of economic growth. He was awarded the John Bates Clark Medal and the 1987 Nobel Memorial Prize in Economic Sciences.-Biography:...
; in 1956, Solow and
T.W. SwanTrevor W. Swan was an Australian economist. He is best known for his work on the neoclassical model of economic growth, published simultaneously with that of Robert Solow, for his work on integrating internal and external balance, represented by the Swan diagram and for pioneering work in...
developed a relatively simple growth model which fit available data on
USThe United States of America is a federal constitutional republic comprising fifty states and a federal district...
economic growth with some success. Solow received the 1987 Nobel Prize in Economics for his work on the model.
Solow also was the first to develop a growth model with different vintages of capital. The idea behind Solow's vintage capital growth model is that new capital is more valuable than old (vintage) capital because capital is produced based on known technology and because technology is improving. Both
Paul RomerPaul Michael Romer is an American economist, entrepreneur, and activist. He is also a Senior Fellow at Stanford University's Center for International Development and the Stanford Institute for Economic Policy Research. He is considered an expert on economic growth.Romer earned a B.S. in physics...
and
Robert Lucas, Jr.Robert Emerson Lucas, Jr. is an American economist at the University of Chicago. He received the Nobel Memorial Prize in Economic Sciences in 1995 and is consistently indexed among the top 10 economists in the Research Papers in Economics rankings. He is married to economist Nancy Stokey.He...
subsequently developed alternatives to Solow's neo-classical growth model. Today, economists use Solow's sources-of-growth accounting to estimate the separate effects on economic growth of technological change, capital, and labor.
Extension to the Harrod–Domar model
Solow extended the Harrod–Domar model by:
- Adding labor as a factor of production;
- Requiring diminishing returns
In economics, diminishing returns refers to how the marginal production of a factor of production, in contrast to the increase that would otherwise be normally expected, actually starts to progressively decrease the more of the factor are added...
to labor and capital separately, and constant returns to scale for both factors combined;
- Introducing a time-varying technology variable distinct from capital and labor.
The capital-output and capital-labor ratios are not fixed as they are in the Harrod–Domar model. These refinements allow increasing
capital intensityCapital intensity is the term in economics for the amount of fixed or real capital present in relation to other factors of production, especially labor...
to be distinguished from technological progress.
Short run implications
- Policy measures like tax
To tax is to impose a financial charge or other levy upon a taxpayer by a state or the functional equivalent of a state such that failure to pay is punishable by law.Taxes are also imposed by many subnational entities...
cuts or investment subsidiesA subsidy is a form of financial assistance paid to a business or economic sector. Most subsidies are made by the government to producers or distributors in an industry to prevent the decline of that industry or an increase in the prices of its products or simply to encourage it to hire more...
can affect the steady state level of output but not the long-run growth rate.
- Growth is affected only in the short-run as the economy converges to the new steady state output level.
- The rate of growth as the economy converges to the steady state is determined by the rate of capital accumulation
Most generally, the accumulation of capital refers simply to the gathering or amassment of objects of value; the increase in wealth; or the creation of wealth...
.
- Capital accumulation is in turn determined by the savings rate (the proportion of output used to create more capital rather than being consumed
Consumption is a common concept in economics, and gives rise to derived concepts such as consumer debt. Generally consumption is defined by opposition to production. But the precise definition can vary because different schools of economists define production quite differently...
) and the rate of capital depreciationDepreciation is a term used in accounting, economics and finance to spread the cost of an asset over the span of several years.In simple words we can say that depreciation is the reduction in the value of an asset due to usage, passage of time, wear and tear, technological outdating or...
.
Long run implications
In neoclassical growth models, the long-run rate of growth is
exogenousExogenous refers to an action or object coming from outside a system. It is the opposite of endogenous, something generated from within the system....
ly determined – in other words, it is determined outside of the model. A common prediction of these models is that an economy will always
convergeThe catch-up effect, also called the theory of convergence, states that poorer economies tend to grow at faster rates than richer economies. Therefore, all economies should in the long run converge in terms of per capita income and productivity...
towards a steady state rate of growth, which depends only on the rate of technological progress and the rate of labor force growth.
A country with a higher saving rate will experience faster growth, e.g.
SingaporeSingapore , officially the Republic of Singapore, is an island city-state located at the southern tip of the Malay Peninsula, lying north of the equator, south of the Malaysian state of Johor and north of Indonesia's Riau Islands. At , Singapore is a microstate and the smallest nation in Southeast...
had a 40% saving rate in the period 1960 to 1996 and annual
GDPThe gross domestic product or gross domestic income is a basic measure of a country's economic performance and is the market value of all final goods and services made within the borders of a country in a year...
growth of 5-6%, compared with
KenyaThe Republic of Kenya is a country in East Africa. Lying along the Indian Ocean, at the equator, Kenya is bordered by Ethiopia , Somalia , Tanzania , Uganda plus Lake Victoria , and Sudan . The capital city is Nairobi. Kenya spans an area about 85% the size of France or Texas...
in the same time period which had a 15% saving rate and annual GDP growth of just 1%. This relationship was anticipated in the earlier models, and is retained in the Solow model; however, in the very long-run capital accumulation appears to be less significant than technological innovation in the Solow model.
Assumptions
The key assumption of the neoclassical growth model is that capital is subject to
diminishing returnsIn economics, diminishing returns refers to how the marginal production of a factor of production, in contrast to the increase that would otherwise be normally expected, actually starts to progressively decrease the more of the factor are added...
. Given a fixed stock of labor, the impact on output of the last unit of capital accumulated will always be less than the one before. Assuming for simplicity no technological progress or labor force growth, diminishing returns implies that at some point the amount of new capital produced is only just enough to make up for the amount of existing capital lost due to depreciation. At this point, because of the assumptions of no technological progress or labor force growth, the economy ceases to grow.
Assuming non-zero rates of labor growth complicates matters somewhat, but the basic logic still applies – in the short-run the rate of growth slows as diminishing returns take effect and the economy converges to a constant "steady-state" rate of growth (that is,
no economic growth per-capita).
Including non-zero technological progress is very similar to the assumption of non-zero workforce growth, in terms of "effective labor": a new steady state is reached with constant output per
worker-hour required for a unit of output. However, in this case, per-capita output is growing at the rate of technological progress in the "steady-state" (that is, the rate of
productivityProductivity is a measure of output from a production process, per unit of input. For example, labor productivity is typically measured as a ratio of output per labor-hour, an input. Productivity may be conceived of as a metric of the technical or engineering efficiency of production. As such, the...
growth).
Variations in productivity's effects
Within the Solow growth model, the
Solow residualThe Solow residual is a number describing empirical productivity growth in an economy from year to year and decade to decade. Robert Solow defined rising productivity as rising output with constant capital and labor input. It is a "residual" because it is the part of growth that cannot be explained...
or
total factor productivityIn economics, total-factor productivity is a variable which accounts for effects in total output not caused by inputs. For example, a year with unusually good weather will tend to have higher output, because bad weather hinders agricultural output...
is an often used measure of technological progress. The model can be reformulated in slightly different ways using different productivity assumptions, or different measurement metrics:
- Average Labor Productivity (ALP) is economic output per labor hour.
- Multifactor productivity
Multifactor productivity measures the changes in output per unit of combined inputs. Indexes of MFP are produced for the private business, private nonfarm business, and manufacturing sectors of the economy...
(MFP) is output divided by a weighted average of capital and labor inputs. The weights used are usually based on the aggregate input shares either factor earns. This ratio is often quoted as: 33% return to capital and 66% return to labor (in Western nations), but Robert J. GordonRobert James "Bob" Gordon is an American economist. He is the Stanley G. Harris Professor of the Social Sciences at Northwestern University. He is known for his work on productivity, growth, the causes of unemployment, and airline economics....
says the weight to labor is more commonly assumed to be 75%.
In a growing economy, capital is accumulated faster than people are born, so the denominator in the growth function under the MFP calculation is growing faster than in the ALP calculation. Hence, MFP growth is almost always lower than ALP growth and growth. (Therefore, measuring in ALP terms increases the apparent
capital deepeningCapital deepening is a term used in economics to describe an economy where capital per worker is increasing. A process of increasing the amount of capital per worker. It is an increase in the capital intensity. Capital deepening is often measured by the capital stock per labour hour. Overall, the...
effect.)
Technically, MFP is measured by the "
Solow residualThe Solow residual is a number describing empirical productivity growth in an economy from year to year and decade to decade. Robert Solow defined rising productivity as rising output with constant capital and labor input. It is a "residual" because it is the part of growth that cannot be explained...
", not ALP..
Empirical evidence
A key prediction of neoclassical growth models is that the income levels of poor countries will tend to
catch upThe catch-up effect, also called the theory of convergence, states that poorer economies tend to grow at faster rates than richer economies. Therefore, all economies should in the long run converge in terms of per capita income and productivity...
with or
converge towards the income levels of rich countries as long as they have similar characteristics – like for instance saving rates. Since the 1950s, the opposite empirical result has been observed
on average. If the average growth rate of countries since, say, 1960 is plotted against initial GDP per capita (i.e. GDP per capita in 1960), one observes a positive relationship. In other words, the developed world appears to have grown at a faster rate than the developing world, the opposite of what is expected according to a prediction of convergence. However, a few formerly poor countries, notably
Japanis an island country in East Asia. Located in the Pacific Ocean, it lies to the east of the Sea of Japan, People's Republic of China, North Korea, South Korea and Russia, stretching from the Sea of Okhotsk in the north to the East China Sea and Taiwan in the south...
, do appear to have converged with rich countries, and in the case of Japan actually exceeded other countries' productivity, some theorize that this is what has caused Japan's poor growth recently –
convergent growth rates are still expected, even after convergence has occurred; leading to over-optimistic investment, and actual
recessionIn economics, a recession is a general slowdown in economic activity over a long period of time, or a business cycle contraction. During recessions, many macroeconomic indicators vary in a similar way...
.
The evidence is stronger for convergence within countries. For instance the per-capita income levels of the southern states of the United States have tended to converge to the levels in the Northern states. These observations have led to the adoption of the
conditional convergence concept. Whether convergence occurs or not depends on the characteristics of the country or region in question, such as:
- Institutional arrangements
- Free market
A free market describes a market without economic intervention and regulation by government except to regulate against force or fraud. The terminology is used by economists and in popular culture. A free market requires protection of property rights, but no regulation, no subsidization, no single...
s internally, and trade policyFree trade is a type of trade policy that allows traders to act and transact without interference from government. According to the law of comparative advantage the policy permits trading partners mutual gains from trade of goods and services....
with other countries.
- Education
Education in its broadest sense is any act or experience that has a formative effect on the mind, character or physical ability of an individual...
policy
Evidence for conditional convergence comes from multivariate, cross-country regressions.
If productivity were associated with high technology then the introduction of information technology should have led to a noticeable productivity acceleration over the past twenty years; but it has not:
see: Solow computer paradox.
Econometric analysis on Singapore and the other "
East Asian TigersThe term Four Asian Tigers or Asian Tigers refers to the highly developed economies of: Hong Kong Singapore South Korea TaiwanThey are also known as Asia's Four Little Dragons in Chinese, as these countries and territories have at various points in history been under the Chinese cultural sphere of...
" has produced the surprising result that although output per worker has been rising, almost none of their rapid growth had been due to rising per-capita productivity (they have a low "
Solow residualThe Solow residual is a number describing empirical productivity growth in an economy from year to year and decade to decade. Robert Solow defined rising productivity as rising output with constant capital and labor input. It is a "residual" because it is the part of growth that cannot be explained...
").
Criticisms of the model
Empirical evidence offers mixed support for the model. Limitations of the model include its failure to take account of entrepreneurship (which may be catalyst behind economic growth) and strength of institutions (which facilitate economic growth). In addition, it does not explain how or why technological progress occurs. This failing has led to the development of
endogenous growth theoryIn economics, endogenous growth theory or new growth theory was developed in the 1980s as a response to criticism of the neo-classical growth model....
, which endogenizes technological progress and/or knowledge accumulation.
Some critics suggest that Schumpeter’s 1939 Theory of Business Cycles, modern Institutionalism and Austrian economics offer an even better prospect of explaining how long run economic growth occur than the later Lucas/Romer models.
Graphical representation of the model
The model starts with a neoclassical production function Y/L = F(K/L), rearranged to y = f(k), which is the orange curve on the graph. From the production function; output per worker is a function of capital per worker. The production function assumes diminishing returns to capital in this model, as denoted by the slope of the production function.
n = population growth rate
d =
depreciationDepreciation is a term used in accounting, economics and finance to spread the cost of an asset over the span of several years.In simple words we can say that depreciation is the reduction in the value of an asset due to usage, passage of time, wear and tear, technological outdating or...
k = capital per worker
y = output/income per worker
L = labor force
s = saving rate
Capital per worker change is determined by three variables:
- Investment (saving) per worker
- Population growth, increasing population decreases the level of capital per worker.
- Depreciation – capital stock declines as it depreciates.
When
sy > (
n +
d)
k, in other words, when the savings rate is greater than the population growth rate plus the depreciation rate, when the green line is above the black line on the graph, then capital (k) per worker is increasing, this is known as
capital deepeningCapital deepening is a term used in economics to describe an economy where capital per worker is increasing. A process of increasing the amount of capital per worker. It is an increase in the capital intensity. Capital deepening is often measured by the capital stock per labour hour. Overall, the...
. Where capital is increasing at a rate only enough to keep pace with population increase and depreciation it is known as capital widening.
The curves intersect at point
A, the "steady state". At the steady state, output per worker is constant. However total output is growing at the rate of
n, the rate of population growth.
The optimal savings rate is called the golden rule savings rate and is derived below. In a typical Cobb–Douglas production function the golden rule savings rate is alpha.
Left of point A, point k
1 for example, the saving per worker is greater than the amount needed to maintain a steady level of capital, so capital per worker increases. There is capital deepening from
y1 to
y0, and thus output per worker increases.
Right of point A where
sy < (
n +
d)
k, point
y2 for example, capital per worker is falling, as investment is not enough to combat population growth and depreciation. Therefore output per worker falls from
y2 to
y0.
The model and changes in the saving rate
The graph is very similar to the above, however, it now has a second savings function
s1y, the blue curve. It demonstrates that an increase in the saving rate shifts the function up. Saving per worker is now greater than population growth plus depreciation, so capital accumulation increases, shifting the steady state from point
A to
B. As can be seen on the graph, output per worker correspondingly moves from
y0 to
y1. Initially the economy expands faster, but eventually goes back to the steady state rate of growth which equals
n.
There is now permanently higher capital and productivity per worker, but economic growth is the same as before the savings increase.
The model and changes in population
This graph is again very similar to the first one, however, the population growth rate has now increased from
n to
n1, this introduces a new capital widening line (
n1 +
d)
Mathematical framework
The Solow growth model can be described by the interaction of five basic macroeconomic equations:
- Macro-production function
- GDP equation
- Savings function
- Change in capital
- Change in workforce
Macro-production function
This is a Cobb–Douglas function where
Y represents the total production in an economy.
A represents
multifactor productivityMultifactor productivity measures the changes in output per unit of combined inputs. Indexes of MFP are produced for the private business, private nonfarm business, and manufacturing sectors of the economy...
(often generalized as technology),
K is capital and
L is labor.
An important relation in the macro-production function:
which is the macro-production function divided by
L to give total production per capita
y and the capital intensity k
.
Savings function
This function depicts savings, I
as a portion s
of the total production Y.
The model's solution
First we'll need to define some growth functions.
1. Growth in capital
2. Growth in the GDP
3. Growth function for
capital intensityCapital intensity is the term in economics for the amount of fixed or real capital present in relation to other factors of production, especially labor...
Solution assuming no multifactor productivity growth
This simplification makes the solution's derivation more comprehensible, as it allows the following calculations:
When there is no growth in A then we can assume the following based on the first calculation:
Moving on:
Divide the fraction by L
and you will see that
By subtracting gL from gK we end up with:
If k
is known in the year t
then this formula can be used to calculate k
in any given year.
In the first segment on the right side of the equation we see that
Deriving the Steady-state equation:
where k
= K
/L
and k
denotes capital per worker.
Differentiating we obtain:
which is
we know that
is the population growth rate over time denoted by n.
Furthermore we know that
where is the depreciation rate of capital.
Hence we obtain:
which is the fundamental Solow equation. The same can be done if technological progress is included.
In the steady state the change in must be 0.
The steady state consumption will then be:
A simple explanation
Consider a simple case Cobb–Douglas production function:
where is level output, level of capital, level of employment (given, fixed), and is relative capital intensity (given, fixed). Net capital accumulation per capita in period is given by:
where is the savings rate, and the depreciation rate. The economy reaches a steady state level of output and capital when net capital accumulation per capita is zero. That is,
the amount of total investment (left side) is equal to the amount of capital depreciation (right side) in any given period. From the production function we know that output per capita is given by:
which implies that the steady state levels of capital and output, denoted by asterisks, are:
and
for given values of ,, and .
Now consider output as a Cobb–Douglas function of capital and effective labor
:
where increases in technology positively affect output by improving the efficiency of labor . If technology grows at a constant positive rate of , and labor at , then their product grows at a rate approximately equal to . Consequently, the steady state level of output per unit of effective labor (derived from the original steady state condition)
is actually declining since output is by definition growing at zero in the steady state (left side numerator), whereas (in the denominator) effective labor is growing at . Therefore, in order to offset this additional source of per unit erosion in steady state output, the steady state condition must be modified to read:
total investment (left side) must equal the amount of growth in effective labor in addition to the amount of capital depreciation. This modification implies that the steady state level of output per unit of effective labor is
.
Similarly, the steady state level of capital per unit of effecitve labor is
.
Note: Although per unit growth is zero, the absolute levels of output and capital in the steady state are still growing at a constant positive rate . This result is sometimes referred to as balanced growth
. Also note that neither
the savings rate , the depreciation rate , nor
the relative intensity of capital affects the rate of growth of output in the steady state, although it does still contribute to the initial level
of output and capital at the start of a period of balanced growth.
The golden rule savings rate maximizes the steady state level of aggregate consumption per unit of effective labor, as defined by the national income (GDP) identity:
.
Assuming that the steady state level of investment equals , the golden rule savings rate solves the unconstrained maximization problem
.
Since
,
this implies
,
setting equal to zero and simplifying,
,
finally,
.
Note: This implies that aggregate consumption per unit of effective labor in the steady state is maximized when the savings rate is exactly equal to the relative intensity of capital in the production function.
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