In economics, theory of Economic Growth was initially developed at a purely theoretical level, with just a few empirical implications that could attract the interest of researchers. These were mainly related to the implications of exogenous growth models regarding the rate of growth of an economy as well as the convergence in income per capita among a set of countries that were soon put to test through regression analysis. The Neoclassical Exogenous Growth model with a constant savings rate, introduced in the seminal papers of Solow and Swan incorporated a constant returns to scale assumption in the production of the final good, which was shown to imply zero long-run growth for per-capita variables. This model is able to explain positive long-term growth in per-capita variables only through some type of exogenous growth in productivity. Only that way could the model be made consistent with some regularity observed in actual data.
► Exogenous growth refers such growth that is not being produced by either the decisions made by private economic agents, or by any policy intervention .
► Endogenous growth refers to growth being influenced on the decisions made by private agents as well as on policy choices.
The Solow Neoclassical Growth Model: Origin of Exogenous Growth Model.
The first growth model was published almost simultaneously by R. Solow and S. Swan in two different papers in 1956. In fact, as we will see, the assumptions embedded in this model imply that, in the long run, and in the absence of technological growth , economies do not grow in per-capita terms. As economics papers put it, the possibility of aggregate growth arises only from either population growth or growth in factor productivity. Since neither factor is supposed to depend on the decision of economic agents, this is known as an Exogenous Growth Model .
Per capita income, the most obvious indicator of the state of a given economy, displays two different characteristics in most developed countries:
(a) it increases over time, and
(b) it experiences cyclical fluctuations around its long-term trend over relatively short periods of time.
The model focuses on explaining the first characteristic, long-term growth . A stochastic version of growth models is needed if we want the model to re-produce the statistical characteristics of business cyclical fluctuations in actual economies. We will also consider a stochastic version of the Solow–Swan growth model. This model describes the time evolution of an economy in which there is growth from some initial, known conditions.
ASSUMPTIONS OF THE MODEL
1. Continuous time.
2. Single good produced with a constant technology.
3. No government or international trade.
4. All factors of production are fully employed.
5. Labor force grows at constant rate n such that
6. Initial values for capital, K 0 and labor, L 0 given.
FORMULATION OF THE MODEL
The Solow-Swan model assumes that GDP is produced according to an aggregate production function technology. It is worth flagging that in macroeconomics research papers most of the key results for Solow’s model can be obtained using any of the standard production functions that you see in microeconomic production theory. However, a case in which the production function takes the Cobb-Douglas form is adopted in this article:
Growth rate of gross domestic product is stated as:
We are interested in is growth rates of series so a production function taking the Cobb-Douglas form is stated thus:
In addition to the production function, the model has four other equations.
- Capital accumulates according to:
In other words, the addition to the capital stock each period depends positively on savings and negatively on depreciation, which is assumed to take place at rate
2. Labour input grows at rate n :
3. Technological progress grows at rate g :
4. A fraction of output is saved each period
Two assumptions underlying the Cobb-Douglas production function:
►Constant returns to scale: A doubling of inputs leads to a doubling of outputs.
►Decreasing marginal returns to factor accumulation (adding extra capital while holding labour input fixed yields ever-smaller increases in output).
If a firm acquires an extra unit of capital, it should raise its output. But if the firm keeps piling on extra capital without raising the number of workers available to use this capital, the increases in output will gradually lessen. In the Cobb-Douglas case, the parameter α dictates the pace at which output gradually lessens. The use of logarithms is very helpful when dealing with dynamic models.
As specified in equation II: Y t = A t K t α L t 1-α applying the properties of logarithm which are:
After taking the derivative with respect to time we find the required formula describing exogenous growth theory.
is the rate of change in gross domestic product
is change in technological progress (productive efficiency)
is the rate of change in capital consumption
is the rate of change in labour consumption
The Solow model helps to picture what the economy looks like along a path on which output growth is constant. In macroeconomics research papers, reference is made to such constant growth paths as steady-state growth paths. Given constant growth rates for technology and labour input, all variations in output growth are due to variations in the growth rate of capital input:
For output growth to be constant, we must also have capital growth being constant. To achieve this we restated the capital accumulation equation using equations (3) and (5) to arrive at
Dividing across by K t on both sides
With this result in mind, we see that the steady-state growth rate must satisfy
Subtracting from both sides of equation 17, we get
Therefore, an economy will experience steady-state growth at the rate of such that;
This model is expressed in explicit econometric linear equation form as:
Technological progress is an exogenous component of the solow model which is unobservable. The parameter alpha-subscript-0 replaced technological progress in the linear representation of the model as the y-intercept . The greek letter epsilon is the regression residual or disturbance term.
This states the theoretical predictions of the signs of the parameters employed in the model. Specifically, it explains what the theory predicted about the relationship between explanatory variables: capital, labour and the explained variable gross national product. From the model specified above, a positive relationship exists in theory between capital and output. A positive relationship also exists in theory between labour and output.
The stochastic version of the solow growth model is needed if we want the model to reproduce the statistical characteristics of business cyclical fluctuations in actual economies.
This approach to growth modelling in economics has had a tremendous impact on the way we think about the analysis of effects of the different exogenous shocks in an economy. We are able to analyze which among the possible shocks is more likely to produce a given statistical characteristic of the solution, or which one is more useful in order for a model to replicate a given statistical regularity observed in actual time series data.
Similarly, we can characterize how economic policy influences the dynamics of relevant variables, as well as the co-movements between them. So, it is not surprising that it is in the normative analysis of economic policy where stochastic, dynamic models with microeconomic foundations have become standard.
Later on, several other growth models were built to show that the rate of saving of households s can be endogenous. It was such growth models that were termed Endogenous Growth Models .
Robert Solow and Trevor Swan independently developed the exogenous growth model in 1956. They noted that the possibility of aggregate growth arises only from either population growth or growth in factor productivity since neither factor of production say, labour and capital depended on the decisions of economic agents. The research method adopted in this theory is macro research. Hence, macroeconomics research papers ought to be written in line with clear objectives and variables.
Alfonso Novales et. al. (2009): “Economic Growth- Theory and Numerical Solution Methods”, Springer-Verlag Berlin Heidelberg.
Gujarati, D. N. (2003): Basic Econometrics, 4 th ed., McGraw-Hill Higher education, Newyork.