Question

A firm produces a product with labor and capital. Its production function is described by Q = min(L, K). Let w and r be the prices of labor and capital, respectively.

a) Find the equation for the firm’s long-run total cost curve as a function of quantity Q and input prices, w and r.

b) Find the solution to the firm’s short-run cost minimization problem when capital is fixed at a quantity of 5 units (i.e., K = 5). Derive the equation for the firm’s short- run total cost curve as a function of quantity Q. Graph this curve together with the long-run total cost curve for w = 1 and r = 1.

c) How do the graphs of the long-run and short-run total cost curves change when w = 1 and r = 2?

d) How do the graphs of the long-run and short-run total cost curves change when w = 2 and r = 1?

Answer 1

Part A

suppose the firm have complementary inputs, then cost minimization firm can uses the same proportion of inputs. So we have,

Part B

Given, , then

It means that in the short run they can't produce more than 5 goods. It is because not be greater than 5. So, we need to produce then we can set with , thus,

Here 10 is the fixed cost of visible input and 5 is the fixed cost. Below given the diagram,

Part--C

If and then we have,

and Here Q is not greater than 5, Then

Part--D

if and , then we have,

and , here Q not greater than 5,