The Cobb-Douglas Production Function

The Cobb-Douglas Production Function

Introduction

A production function is a function which relates the factors of production, capital and labour, to output given the available technology. Capital here refers to things workers use in producing goods and services such as computers. Labour refers to the time that people spend working. The production function can thus be written as: Y = F(K, L) where Y denotes output, K denotes capital, L denotes labour and F is the relationship between the factors of production and output. Knowing the specific form of the production is important as it can explain the incomes of capital and labour, i.e. how much of output is compensated to workers and how much is compensated to owners of capital. This may seem a daunting task given modern economies’ vast array of industries which use labour and capital differently but there is a particular form which has proven to be resilient since its discovery.

In 1927, Paul Douglas, a professor of economics in the United States (U.S.), discovered a pattern in data he had constructed from 1899 to 1922. Labour share of total income was relatively constant over this long period. This meant that even as the economy grew, total income of capital owners and workers grew at almost exactly the same rate. He consulted Charles Cobb, a mathematician and colleague, on what sort of production function would yield constant factor shares. Cobb showed that the function with this property was: Y = F(K, L) = AKαL1-α, where α < 1. A is the residual which explains any output growth not explained by changes in capital or labour, and is also known as total factor productivity. Besides capital and labour, technology is another major determinant of output and A can be taken to be the effect of technology on output.

Derivations, Proof And Assumptions

Some economics and mathematics is required to show that the Cobb-Douglas production function exhibits constant factor shares. Firstly, assume firms are profit-maximising. In this way, firms will wish to rent capital and hire workers as long as the benefit accrued exceeds the cost of doing so. Thus, firms will rent capital and hire workers until the marginal benefit and marginal cost of each factor are equal. Now, it may seem odd how analysis done thus far assumed capital was rented when firms often own the capital they use in the real world. However, the opportunity cost is the real rental price whether capital is rented or hired. Let us look at labour first and then capital.

Define MPL as the marginal product of labour or the additional output produced from hiring 1 more unit of labour. Thus, MPL = F(K, L+1) – F(K, L). From this, it is clear that MPL is the partial derivative of Y with respect to L. Hence, MPL = AKαL-α(1 – α) = (Y/L)(1 – α). The cost of hiring 1 more unit of labour is the nominal wage W. However, MPL is expressed in real terms, in terms of how much output. Thus, the marginal cost must be expressed in real terms, i.e. how much output can be bought with W. This marginal cost is then the real wage or W/P where P is the general price level. Thus, MPL = (Y/L)(1 – α) = W/P. Labour share of output = Real Wage x No. of Workers = MPL x L = Y(1 – α).

Define MPK as the marginal product of capital or the additional output produced from renting 1 more unit of capital. Thus, MPK = F(K+1, L) – F(K, L). From this, it is clear that MPK is the partial derivative of Y with respect to K. Hence, MPK = AK-αL1-α(α) = α(Y/K). The cost of renting 1 more unit of capital is the nominal rent R. However, MPK is expressed in real terms, in terms of output. Thus, the marginal cost must be expressed in real terms, i.e. how much output must be paid with R. This marginal cost is then the real rental price or R/P where P is the general price level. Thus, MPK = α(Y/K) = R/P. Capital share of output = Real Rental Price x Units of Capital = MPK x K = αY.

Empirical Evidence For The Cobb-Douglas Production Function

As seen from the simple derivations done above, capital share of output and labour share of output are constant proportions of output even if technology (A) changes. Now that the model has been proven to be mathematically robust, it is time to see if it still applies in the contemporary context.

Figure 1: U.S. Labour Share of Total Income/Output, 1960 – 2009

Table 1: Wage, Profit and Tax Shares of Gross Domestic Product (GDP), 1980 – 2009

Period Wage Share (%) Profit Share (%) Tax Share (%)

1980 – 1989 41.8 50.7 7.5

1990 – 1999 41.9 48.8 9.3

2000 – 2009 42.5 50.4 7.1

As