The
neoclassical economic growth model, in one form or another, encapsulates the conventional wisdom on
long-run economic growth. The model captures many of the issues thought to be important: productivity, capital formation, population growth, and technology. In this post, I will present the equations for the Solow-Swan model, convert the model into a causal path diagram and explore the model's assumptions, predictions, insights and shortcomings.
The heart of the Solow-Swan model is
Cobb-Douglas production function which models total production, Y, as a function of Total Factor Productivity Growth (TFPG) or technology,
A, capital,
K, and labor,
L. The exponent
alpha is simultaneously the elasticity of production, the capital-income ratio and the the shares of capital and labor in output. Capital investment (
I) comes from savings,
sY, and when compared to depreciation, (
1-d), determines growth of the capital stock. The growth of the labor force is a function of population growth,
N (the difference between births and deaths,
b-d), and the percent of population employed,
e. Technological change (TFPG), A, is exogenously given and grows at the rate
g.
Although the functional form used in the Solow-Swan model has appealing properties (constant elasticity of production, constant returns to scale, constant shares returned to capital and labor, diminishing marginal returns, etc.), there is really little theoretical justification for the equations as written (but see the
Kaya Impact Model below). Working with US data from 1927-1947,
Paul Douglas needed some functional form to use in regression analysis and the multiplicative model was suggested by the mathematician
Charles Cobb, without theoretical development.
Dropping the functional forms and mapping the causal connections among variables yields the directed graph above where total output,
Y, is replaced by gross domestic product,
Q. The causal model makes clear that as long as population and technological change are growing exponentially, gross domestic product will also grow exponentially, forever. Also, the idea that as long as technological change exceeds population growth, output-per-capita will increase is appealing to economists and addresses criticisms of the
Malthusian model where population growth exceeds productive growth.
The assumption of uncontrolled, unending exponential growth can be modified somewhat by assuming population growth is exogenous and specifying a regression model (
Wonnacott and Wonnacott, 1970: 93-95):
However, technological change (TFPG, the regression constant) still is assumed to grow exponentially. For reasonable long-run projections it can be assumed that, at some point in the future, technological progress will decline and eventually stop. For example, in the
DICE model:
"Assumptions about future growth trends are as followed: The rate of growth of population was assumed to decrease slowly, stabilizing at 10.5 billion people in the 22nd century. The rate of growth of total factor productivity [the growth rate of A(t) in Eq. 3] was calculated to be 1.3% per annum in the period from 1960 to 1989. This rate was assumed to decline slowly over the coming decades. We calibrate the model by fitting the solution for the first three decades to the actual data for 1965, 1975, and 1985 and then optimizing for capital accumulation and GHG emissions in the future..." page 1317 in
An Optimal Transition Path for Controlling Greenhouse Gases" William D. Nordhaus (1992).
There are many problems applying the neoclassical growth. For estimation, capital stock measures are not available or have questionable validity in many countries. Although the number of inputs (land, intermediate goods, etc.) are typically expanded when used for
growth accounting, as a transformation function the model does not make sense (neither labor or capital are not transformed into output through production). For cross-country comparisons, the assumption that one production function would cover all countries (Somalia would have the same production function as would Canada) seems unrealistic. Since the model is nonlinear, it cannot be aggregated from the individual level to the nation state or from the nation state to world system. Finally, the assumption of decreasing technological change in the future is only made because exponential growth forever is unreasonable not because there is any evidence for such an assumption.
A lot more can be said about the neoclassical production function, particularly that causality might run in the opposite direction from that assumed in the Cobb-Douglas production function. The other points are best made when considering other macro models, which I will do in future posts.
