Question

3. Suppose that you have a Cobb-Douglas production function of the following form: Y = 0.25K0.24L 0.40D 0.10 (1) where Y is output, K is capital stock, L is labour, and D is land. (a) What is the interpretation of the individual exponents on K, L and D respectively? (b) What is the interpretation of the sum of these coefficients (i.e., which represents the degree of homogeneity for this function)? Is this function subject to constant, decreasing or increasing returns to scale? Explain your reasoning.

Answer 1

a) individual coefficients represent the elasticity of output with respect to the change in particular input. for instance, here, the coefficient on capital is 0.24, which means output elasticity with respect to change in the capital is 0.24. it implies if the capital increases by 10%, the output will be increased by 24%. similarly, with respect to labour, the output elasticity is 0.40, implying an increase in output by 40% with a 10% increase in labour. and finally, the output elasticity with respect to land is 0.10, implying an increase in output by 10% with 10% increase in land.

b) the sum of the coefficients on all the three inputs reflect the return to scale. it implies if all the inputs increase proportionally, then to what extent output will increase. this is determined by the sum of the coefficients. if the sum is greater than 1, it implies increasing returns to scale, i.e. output will increase more than proportional increase inputs. if sum is less than 1, it implies decreasing returns to scale, i.e. output will increase less than proportional increase in inputs, and finally, if sum is exactly equal to 1, it implies constant returns to scale, i,e. output increases in the same proportion as of the increase in inputs.

here, in this example, the sum of all coefficients, i.e. 0.24+0.40+0.10= 0.74. it is less than 1, thereby this example reflects decreasing return to scale, it means the increase in output will be less than proportional increase in inputs. here i am attaching the mathematical proof of decreasing returns to scale.