Question

1. Let Q = 12(KL)² – K⁴.
            

1. What is the average product of capital?
2. Let L = 2. By plugging in values of K between 1 and 5, can you tell at what level of K the average product of capital is the highest?
3. Let L = 2. Using only the information you have been given so far, at what level K does the marginal product of capital equal the average product of capital?

Does this production function exhibit increasing, constant, or decreasing returns to scale?

Answer 1

a. The average product of capital will be total product divided by the total capital. Thus, the average product of capital= Q/K= (12(KL)^2-K^4)/K= 12K(L^2)-K^3

b. When L=2, Q= 12(K)^2(2*2)-K^4= 48K^2-K^4

The average product of capital= Q/K= (48K^2-K^4)/K= 48K-K^3

At K=1; average product of capital= 48(1)-(1)^3= 48-1= 47

At K=2; average product of capital= 48(2)-(2)^3= 96-8= 88

At K=3; average product of capital= 48(3)-(3)^3= 144-27= 117

At K=4; average product of capital= 48(4)-(4)^3= 192-64= 128

At K=5; average product of capital= 48(5)-(5)^3= 240-125= 115

The Average product of capital is the highest when K is 4, provided L is fixed at 2.

c. The average product of capital when L=2 is 48K-K^3

The marginal product of the capital = derivative of the total output with respect to K, at L=2 which is a derivative of 48K^2-K^4 with respect to K= 48*2(K)-4K^3= 96K-4K^3

Average product and marginal product are equal: 48K-K^3=96K-4K^3

4K^3-K^3=96K-48K

3K^3=48K

K^2= 48/3= 16

K= 4

To check the return to scale, we would increase both K and L by n and then compute the production function. If the new product is n times the old one, it implies a constant return to scale. If it is lesser than n times the old production then it signifies the decreasing returns and increasing return to scale when the new production function is more than n times the old production.

So, New production function= 12(nKnL)^2-(nK)^4= 12((n^2)^2(KL)^2)-(n^4)(K^4)= n^4 (12(KL)^2-(K)^4).

Since n^4 is greater than n, thus this production function is exhibiting the increasing returns to scale.