Based on Lecture Notes from Dr. Naish at CSUF (Topics in Economic Analysis, Fall 2009)
In contemporary macroeconomic textbooks the assumption of constant returns to scale and perfect competition are the building blocks of much of the economic growth models and theories. The models imply that the only method for increasing per capita income is via technological progress. The following calculations demonstrate that with Increasing Returns to Scale the growth of the population can also be a factor even if there is no technological progress.
Basic Equations and Assumptions:
MACROECONOMIC COBB-DOUGLAS PRODUCTION FUNCTION WITH INCREASING RETURNS TO SCALE
The increasing returns to scale are symbolized by the fact that alpha plus beta is greater than one. It can be shown that this inequality implies that the produciton function, Y, exhibits increasing returns to scale. The increasing returns to scale imply that there are cost benefits to operating at a larger scale whether this is achieved through increasing the size of firms, opening up to international trade, or simply by increasing the population of a country is another question that should be handled in the context of the resource constraints and the overall economic goals of a nation. This derivation is to simply demonstrate that there can be advantages to operating at larger scales even without technological progress which is something most macroeconomic textbooks ignore.
There are 2 other variables which need a more rigorous definition and those are the rate of population growth and the growth rate of capital since both are elements in the production function above.
RATE OF GROWTH OF CAPITAL STOCK
The rate of growth of the capital stock is equal to I, which is investment per period. Investment is equal to the marginal propensity to save multiplied by total GDP since most savings is in on way or another an investment. Money saved in the bank become mortgages, loans, and bond purchases by the banks which essentially means that, roughly speaking, investments equal the proportion of savings in an economy.
LABOR AS A FUNCTION OF TIME AND GROWTH RATE
The formula for population growth in this model is an exponential growth model. The population grows starting from an initial population and grows exponentially as a function of the time multiplied by the growth rate n. This model of population growth is well suited for developing countries where populations increase at high rates. A Logistic Model for population growth incorporates both the exponential growth pattern of developing nations as well as the increasing at decreasing rates of growth in industrialized countries. The exponential model will suit this example well and the marginal benefit of a more accurate population model is less than the cost of using the logistic growth model for most countries in the world.
GROWTH IN THE LABOR FORCE
The following calculation take advantage of the previous derivations of population and capital growth rates to examine how increasing returns to scale can have a higher growth rate in the capital to labor ratio which in turn increase productivity. This increase in productivity has nothing to do with increases in technological progress, but comes strictly from the benefits of operating at larger scales.
In equilibrium the rate of growth in the capital per workers should be constant since any increase would cause an increase in output and thus by definition the system would not be in equilibrium. This yields the final equation for growth in the capital to labor ration with no technological progress, but with increasing returns to scale:
THIS FORMULA DEMONSTRATES THE IMPORTANCE OF ECONOMIES OF SCALE IN THE GROWTH OF CAPITAL PER WORKER AND THUS WAGES
Technological progress is not the only way to increase the capital to labor ratio. The fact that the rate of population growth is included in the rate of capital per worker above in such a manner that demonstrates a positive relationship between increasing rates of population growth and increasing capital to labor ratios is apparent in a country which experiences large increasing returns to scale. Countries with higher capital to labor ratios have higher GDP per capita and thus higher income. High income countries tend to have better educated and healthier people so the importance of economies of scale cannot be downplayed much like it has in many books on macroeconomic growth.