Question

There are two kinds of factors of production, labor L and capital K, which are only available in non-negative quantities. There are two firms that make phones, Apple and Banana. To make qA phones, Apple’s input requirement of (L,K) is given by production function f(L,K) = L0.6K0.2. To make qB phones, Banana’s input requirement of (L,K) is given by production function g(L,K) = L0.75K0.25.
(a) (Time: 3 minutes) How many phones can Apple make with factor bundle (L1,K1) = (1,1)? And how many phones can Banana make with this factor bundle? How many phones can Apple make with factor bundle (L2,K2) = (2,2)? And how many phones can Banana make with this factor bundle?

(b) (Time: 2 minutes) More generally, for any factor bundle (L,K) where L > 1 and L > 1, would you agree that Banana can make more phones than Apple? Explain your answer.

In the case of Apple and Banana, suppose that they both face the same input prices for labor and capital, w and r, respectively, and that the price at which they can sell their output is the same as well, equal to p. Also, assume they are price takers in factor markets as well as the output market. Suppose w = 3 and r = 1.
(c) (Time: 6 minutes) From your answer above, it should be clear that Apple could produce 1 unit of output by using factor bundle (L1,K1). Show why this input bundle is actually cost-minimizing, i.e. the cheapest possible input bundle that ensures an output of 1.

(d) (Time: 2 minutes) Explain why, at output level of 1, Apple’s marginal cost is 5.

Whether or not an output level of 1 maximizes profit will depend upon the price of output. Continue to assume w = 3 and r = 1, and now also suppose p = 5.
(e) (Time: 2 minutes) Explain why output level qA = 1 is profit-maximizing for Apple.

Answer 1

Part (a)

Part (b)

if inputs are doubled

Doubling of inputs leads to doubling of output for banana phones. but less than proportionate increase in output is observed for apple phones,

production function for apple phones shows decreasing returns to scale whereas production function for banana phones shows constant returns to scale

Part (c) cost is minimized at point where above condition is fulfilled

we have to find cost minimizing input bundle for an output of 1 unit of apple phones

Part (d)

we know from the cost minimizing condition

Part (e)

given price of an apple phone is 5

maximizing profit by equating first derivative with respect to output to 0