The Cobb-Douglas production function models the relationship between an output and inputs like labor and capital. It assumes outputs increase with inputs but at a decreasing rate. The formula relates the natural log of output to the natural log of inputs with elasticity coefficients representing the percentage change in output from a 1% change in an input. If the coefficients sum to 1 there are constant returns to scale, less than 1 is decreasing returns, and more than 1 is increasing returns. An example using Taiwan agricultural data from 1958-1972 estimated elasticities of 1.5 for labor and 0.4 for capital, indicating increasing returns to scale.

1. Cobb-Douglas Production Function

2. • 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

3. • 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

4. • 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

5. • 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

6. • 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

7. • 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.

8. • 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

9. WINDOWS

10. NEW
EXCEL SHEET

11. DATA

12. LOG

13. LOG
Y,

14. LOG
Y, X2,X3

15. DATA-DATA ANALYSIS

16. Input Y, X

17. result

18. result

19. • 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

20. • 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

21. • 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

22. Thank u