Question

? = ??^(1/3) L^(2/3)

Assume that total factor productivity, A, equals 2 in all parts of this question.

(a) Compute the level of output produced when K=10 and L=20.

(b) Does this production function exhibit essentiality? Prove this.

(c) Does this production function exhibit constant returns to scale? Explain why it does or does not and prove that your answer is correct by computing the effect of simultaneously tripling the quantities of K and L from their levels in (a).  

(d) Does this production function exhibit positive but diminishing productivity of each factor of production? Explain what we mean by this property, and prove that your answer is correct by computing the effect for output of doubling the quantity of K repeatedly (do this at least twice, relative to the level of K in (a)) while holding L constant, and the effect for output of doubling the quantity of L repeatedly while holding K constant (do this at least twice, relative to the level of L in (a)).

(e) Re-write the production function as a relation between output per worker and capital per worker. Compute the effects for output per worker of repeatedly doubling the amount of capital per worker. Draw a diagram illustrating the relationship between capital per worker and output per worker.

(f) Does this production function satisfy the Inada conditions? Explain your answer.  

Answer 1

The production function is given to be or .

(a) For the given K and L, the production would be or untis.

(b) For K=L=0, we have . Hence, the production function exhibits essentiality.

(c) Increasing the input by a times, we have or or . Hence, increasing the input by a times, the output increases by a times, meaning that the production function does exhibits constant returns to scale.

Tripling the inputs in part-a, we have or or units. As can be seen, this output is three times the output in part-a, as 31.748*3=95.244 units.

(d) The marginal products are and , meaning that the marginal products are indeed positive. But, we have and , meaning that as the input increases, the marginal product increases at a decreasing rate (meaning that they are diminishing).

To check for K (taking L=1), we have from part a, and doubling it twice, we have and . As can be seen, increasing K by 10 units, the output increased by 1.1199 ( = 5.4288-4.3089). Again increasing it by 20 units, the output increased by 1.4111 ( = 6.8399-5.4288). While the output should have been increased by 1.1199 times 2, since the labor is increased by that times, we may say that there is diminishing marginal product of labor.

To check for L (taking K=1), we have from part a, and doubling it twice, we have and . As can be seen, increasing L by 20 units, the output increased by 8.656. Again increasing it by 40 units, the output increased by 13.7406. While the output should have been increased by 8.656 times 2, since the capital is increased by that times, we may say that there is diminishing marginal product of capital.