Question

Given the following Cobb-Douglas production functions:
F(L,K) = LK^2
F(L,K) = L^3/4K^1/4
F(L,K) = L^1/2K^1/4
1. Determine the returns to scale for each function.

2. For the rest of this exercise assume that the price of labor, w, and the price of capital, r,
equal 1: w = r = 1. Find the conditional input demand functions of labor and capital (the
cost-minimizing combinations of labor and capital).

3. Now find the cost functions for each of the production functions.

4. For each of the above production functions, functions, plot the cost function on the same
graph with Q on the horizontal axis and total cost on the vertical axis.

5. Find and plot the average and marginal cost functions with Q on the horizontal axis and
average cost on the vertical axis. Related these graphs with your answers to the returns to
scale of the production function you found in subquestion 1.

Answer 1

(1)

A production function shows increasing (decreasing) returns to scale if, when both inputs are doubled, output increases by more than (less than) double. A production function shows constant returns to scale if, when both inputs are doubled, output increases by exactly double.

(a) Q = F(L, K) = LK²

When both inputs are doubled, new production function is:

Q* = (2L)(2K)² = 2 x L x (2)² x K² =2 x 4 x L x K² = 8 x Q

Q*/Q = 8 > 2

Since doubling both inputs has more than doubled the output, there is increasing return to scale.

(b) Q = F(L, K) = L^3/4K^1/4

When both inputs are doubled, new production function is:

Q* = (2L)^3/4(2K)^1/4 = (2)^3/4 x (L)^3/4 x (2)^1/4 x (K)^1/4 = 2 x L^3/4K^1/4 = 2 x Q

Q*/Q = 2

Since doubling both inputs has exactly doubled the output, there is constant return to scale.

(c) Q = F(L, K) = L^1/2K^1/4

When both inputs are doubled, new production function is:

Q* = (2L)^1/2(2K)^1/4 = (2)^1/2 x (L)^1/2 x (2)^1/4 x (K)^1/4 = 2^3/4 x L^1/2K^1/4 = 2^3/4 x Q

Q*/Q = 2^3/4 < 2

Since doubling both inputs has less than doubled the output, there is decreasing return to scale.

NOTE: As per Answering Policy, 1st part with multiple sub-part is answered.