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Theory of growthsamuelsons model

Few issues are as important to a country as the long-term growth and productivity trends facing their economy. The relative slow-down in the growth rates of the United States economy since 1973 has worried economists and politicians alike. Many possible causes have been put forth, though none is fully satisfactory.

Before discussing the theoretical models of growth it would be useful to study the data on growth that is currently available. As Nicholas Kaldor, in his influential article on growth ("Capital Accumulation and Economic Growth"; 1961) stated , a theorist ought to start with the summary of the facts that are immediately available, concentrating primarily on broad tendencies or "stylized facts." Theories can then be constructed to explain the facts. Listed below are the stylized facts as mentioned by Kaldor:

1. Output per worker shows continuous growth.

2. Capital per worker shows continuous growth.

3. The rate of return on capital is steady.

4. The capital-output ratio is steady.

5. Labor and capital receive constant shares of total income.

6. There are wide differences in the rate of growth across countries.

In addition to the above, other researchers have found additional features which are obscure for a wide array of data:

7. Average growth rates show no variations with the level of per-capita income.

8. Growth in trade is positively correlated with income levels.

9. Population growth rates are negatively correlated with income levels.

10. The rate of growth of factors inputs is never large enough to explain the rate of growth; that is, technical progress is essential to growth.

Angus Maddison, in his book, Phases of Capitalistic Development 1982, lists in great detail the empirical aspects of growth during the past two hundred years. This study extends out on the whole Kaldor's main observations. Both output and capital per worker has shown tremendous growth over time. Even though growth rates have slowed down since 1973, they are at levels still high by historic standards. Similarly, the constancy of the capital-output ratio is borne out by the statistical data of developed countries.

However, Kaldor's assertion about the constancy of labor and capital shares in total income has increasingly been disputed. As the figures below show:


Japan 1913-1938 40 Ohkawa and Rosovsky

1954-1964 31

United Kingdom 1856-1873 41 Matthews, Feinstein and

1873-1913 43 Odling-Smee

1913-1951 33

1951-1973 27

United States 1899-1919 35 Kendrick

1919-1953 25

1929-1953 29

The data suggests that the share of capital has declined from around 40% to 30% over the course of the century.

Lastly, the great differences in growth rates between countries over time indicates that there has been no tendency for the convergence in rates of growth, something that the neo-classical theory would predict.

Having discussed the empirical issues of economic growth, we now turn to the various theories that have been developed to explain the facts. Perhaps the three most important theories of growth are:

1. Harrod-Domar model

2. Neoclassical theory of growth

3. Endogenous growth models

The Harrod-Domar model is an offshoot of Keynes' macroeconomic model as stated in the "General Theory." Indeed, it can be viewed as an attempt to put Keynes' macroeconomic model of an economy in a dynamic context. From the fundamental relation that investment must equal saving in a closed economy, leads to the result that the rate of growth of an economy equals the product of the savings rate and the incremental output-capital ratio. The equations are:

I º S

But, DK=I

Also, S=sY (s: savings rate)

Define n=DY/DK (incremental output-capital ratio)

This implies, DK=sY

But, DK=DY/n

Therefore, gºDY/Y=s×n (gºDY/Y: rate of growth of output)

Unfortunately, the Harrod-Domar model has a "knife-edge" property which made it somewhat unrealistic. This meant that if an economy strayed from its optimal growth path it either exploded or imploded. This lead to the search for alternative models, the most famous being the neo-classical growth model, usually associated with the infamous Robert Solow.

The neo-classical growth model assumes that the economy converges towards a steady-state rate of growth. Given a neo-classical production function:

Y=A×F(K, N)

Assuming a constant rate of labor force growth (DN/N=n) and no technical progress (DA/A=0) then in a steady state rate of growth of output (DY/Y) equals rate of population growth which implies there is no growth in per capita income unless technical progress takes place.

A critical difference between the Harrod-Domar model and the neoclassical growth model lies in the effect the savings rate has on growth rates. In the Harrod-Domar model an increase in the savings rate increases the growth rate. However, in the neo classical model, an increase in the savings rate increases the per capita income but it does not result in a permanent (as compared to a temporary) increase in the growth rate.

To summarize, in the neo-classical model the rate of output growth equals the rate of growth of technical progress (DA/A) and the level per capita output is determined by the steady-state equation:


where s: savings rate

y: per capita output

d: depreciation rate of capital stock

n: population growth rate

k: per capita capital stock

While Solow's neo-classical model explains the first five out of the six stylized facts quite well, it cannot explain the fact that growth rates differ between countries for long periods of time. This model would suggest convergence in growth rates, something that does not seem to take place (see table).

To explain this problem, theorists have focused their attention on technical progress and have made attempts to make the growth rate endogenous (i.e. determined within the theory). Various endogenous growth theory models, proposed by economists like Robert Lucas and Paul Romer, have constructed a dynamic model where the rate of growth of output depends on aggregate stock of capital (both physical and human) and on the level of research and development in an economy. Many of the models are mathematically complex but do explain the persistent difference in growth rates between countries and the importance of research and human capital development in permanently increasing the growth rate of an economy.

Works Cited

Dornbusch, R. and Fischer, S. Macroeconomics. New York:McGraw Hill, 1994.

Kaldor, N. "Capital Accumulation and Economic Growth" in F.A. Lutz and D.C. Hague

(eds.), The Theory of Capital. New York: St. Martin's Press, 1961.

Maddison, A. Phases of Capatalist Development. Oxford: Oxford University Press, 1982.

Romer, P.M. "Capital Accumulation in the Theory of Long Run Growth." In R. Barro

(ed.) Modern Business Cycle Theory. Cambridge: Harvard University Press, 1989.

Testing Samuelson's Multiplier - Accelerator Interaction Model

The Fundamental Equations Are:

Yt = Ct + It + Go [Go is exogenous] --- Definitional Equation.

Ct = gYt-1 [0<g<1]

It = a(Ct - Ct-1) [a>0] --- Behavioral Equations.

·The model helps to demonstrate how the multiplier and accelerator interact to generate cyclical fluctuations in the economy.

·g is the marginal propensity to consume (MPC) and a is the accelerator, the marginal propensity to save (MPS).

·To test the behavioral equations we apply basic econometrics; using the t- statistic to verify its significance. The first behavioral equation:

Ct = -146.66 + 0.714Yt-1

(1.597) (38.57)

The figures in the brackets are the t- statistics. The higher the value of the t- statistics the more significant is the value of the parameter. This would suggest that the marginal propensity to consume 'g' where lagged income is regressed on current consumption is 0.714. If we chose another behavioral assumption that current is lagged on current investment then we get the equation:

Ct = -226.18 + 0.711Yt

(6.75) (106.75)

This would suggest that the marginal propensity to consume where current income is regressed on current consumption is 0.711 which is not very different from the value of 0.714 calculated earlier. So both behavioral assumptions lead to approximately the same value of 'g'.

·The second behavioral assumption:

It = a(Ct - Ct-1)

where a is the accelerator coefficient. The parameters for the above behavioral equation are:

It = 623.3 + 0.823(Ct - Ct-1)

(11.259) (1.638)

This would suggest that the parameteric values of the accelerator coefficient is 0.823, although the relatively low t- statistic value of 1.638 suggests that we cannot place too much confidence on this parameter.

·Therefore we can state: g = 0.714

a = 0.823.

We also know from Samuelson's model that if ag<1 then the model is stable. From the value above we see that ag = 0.587 < 1. Hence, the U.S. economy is a stable economy that experiences cyclical fluctuations.

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