Douglas Production Function
Many economists have studied actual production functions and have used statistical methods to find out relations between changes in physical inputs and physical outputs. A most familiar empirical production function found out by statistical methods is the cobb-Douglas production function. Originally cob- Douglas production function was applied not to the production process of an individual firm but to the whale of the manufacturing industry. Output in this function was thus manufacturing production. Two factors Cobb-Douglas production function takes the following math metical form:
Q = AL K
Where Q is the manufacturing output L is the quantity of employed K is the quantity of capital employed and an ad and b are parameters of the function.
Roughly speaking Cobb-Douglas production function found that about 75% of the increase in manufacturing production was due to the labour input and the remaining 25 % was due to the capital input. Cobb-Douglas production can be estimated by regression analysis by first converting it into the following log form.
Log Q = log A + a log L + b log K
Cobb - Douglas production function in log form is a linear function.
Cobb-Douglas production function is used in empirical studies to estimate returns to scale n various industries as to whether they are increasing consent or decreasing. Further Cobb-Douglas production function is also frequently used to estimate output elasticizes of labour and capital. Output elasticity of a factor shows the percentage change in output as results of a given percentage change in the quantity of a factor.
Cobb- Douglas production has the following useful properties:
1. The sum of the exponents of factors in Cobb-Douglas production function that is a + b measure returns to scale.
If a = b = 1, returns to scale are constant
If a = b > 1, returns to scale are increasing
If a = b < 1, returns to scale are decreasing
2. In a linear homogeneous Cobb-Douglas production function Q = AL K average and marginal products of a factor depend on ratio of factors used in produiotn and is independent of the absolute quantities of the factors used. In the linear homogeneous Cobb-Douglas production function.
Q = AL K where a =+1 - a = 1
Average product of labour can be obtained from dividing the production function by the amount of labour L
Average product of labour = AL K / L = AK / L = A (K/L)
Since A and a are constants average product of labour will depend on the ratio of factors ( K/L) of the factors used and will not depend upon their absolute quantities.
Like the average product of a factor the marginal product of a factor of a linear homogeneous Cobb-Douglas production function also depends upon the ratio of the factors used and is independent of the absolute quantities of the factors used. Note that marginal product of a factor say labour is first derivative of the production function with respect to labour. The marginal product of labour of Cobb- Douglas production can be obtained as under
Q = AL K
Marginal product of labour dQ / dL = AdL K
= AaL K/L
= AaL K / L = AaK / L
= Aa (K/L)
Since A and a are constants marginal product of labour will depend on capital - labour ratio (k/L) that is capital per worker and is independent of the absolute quantities of the factors employed.
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