Cobb-Douglas Production Function

• The Cobb-Douglas functional form of production functions is widely used to represent the relationship of an output to inputs.

• In 1928 Charles Cobb and Paul Douglas published a study in which they modeled the growth of the American economy during the period 1899 - 1922.

INTRODUCTION

• 1. If either labor or capital vanishes, then so will production.

• 2. The marginal productivity of labor is proportional to the amount of production per unit of labor.

• 3. The marginal productivity of capital is proportional to the amount of production per unit of capital.

Assumptions

• The cobb-douglass production function in its stochastic from may be expressed as

Where Y= output

X2= labor input X3= capital input u = stochastic disturbance term e = base of natural logarithum

FORMULA

Yi=β1 X2i β2 X3i β3 eui

• The relationship between output and the two inputs is nonlinear.• However, if we log-transform this model, we obtain:

• Thus written the model is liner in the parameters β0 β2 and β3 is therefore a linear regression model. Notice though it is nonlinear in the variables y and x but linear in the logs of these variables.

• In short is a log -log, double -log or log linear model the multiple regression counterparts of the two variable log linear models

In Yi= Inβ1+β2Ln X2i+β3ln X3i+ui =β0+β2ln X2i+β3ln X3i+ui

• β2Is the elasticity of output with respect to the labour input that is tit measures the percentage change in output for say a i percent change in the labour input holding the capital input constant

• Likewise, β3 is the elasticity of output with respect to the capital input holding the labour input constant.

• The sum ( β2+β3) gives information about the returns to scale that is the response of output to a proportionate change in the inputs.

properties

• If this sum is 1 then there are constant returns to scale, that is doubling the inputs will double the output tripling the inputs will triple the output and so on.

• If the sum is less than 1 there are decreasing returns to scale- doubling the inputs will less than double the output.

• Finally. If the sum is greater than 1, there are increasing return to scale-doubling the inputs will more than double the output.

• To illusatre the cobb-Douglas production function we obtained the data shown in table these date are for the agricultural sector of Taiwan for 1958- 1972.

• Assuming that the model satisfies the assumptions of the classical linear regression model we obtained the following regression by the OLS method .

Example

WINDOWS

NEW

EXCEL SHEET

DATA

LOG

LOG Y,

LOG Y, X2,X3

DATA-DATA ANALYSIS

Input Y, X

result

result

• In yi =-2.80027 +1.514626 * In X2 + 0.429014 (1.899659 ) (0.416656 ) (0.081337 ) *= 5% level of significant R Square =0.918563 Adjusted R Square =0.903757

Result

• From we see that in the; Taiwanese agricultural sector for the period 1958-1972 the output elasticises of labour and capital were and respectively

• In other words over the period of study holding the capital input constant 1percent increase in the labour input led on the average t o about a 1.5 percentage increase in the output.

• Similarly holding the labour input constant, a 1 percent incerease in the capital input led on the average to about 0.5 percent increase the output.

• Adding the two output elasticises we obtain 1.9887, which gives the value of the returns t scale parameter.

•

Result

• As i s evident over the period of the study the Taiwanese agriculture sector ws characterized by increasing returns to scale.

• From a purely statistical viewpoint the estimated regression line fits the data quite well. The value of means that about 91percent of the variations in the output is explained by the labour and capital.

• We shall see how the estimated standard errors can be used to test hypotheses about the true values of the parameters of the cobb - Douglas production function for the Taiwanese economy.

Result

Thank u