A competitive firm has a production function of the form Y = 2L + 5K. Suppose th

Question

A competitive firm has a production function of the form Y = 2L + 5K. Suppose that the price of L is w = 2 and the price of K is r = 3. What will be the minimum cost of producing y units of output? Due to some regulations on this output product, there is an additional cost (y +1)^2 when producing y units of output. Now, what is the total cost function (C)? What is the total variable cost function (VC)? (In what follows below, use this total cost function) Compute the average cost function (AC), average variable cost function (AVC), average fixed cost function (AFC). Compute the marginal cost function (MC), marginal variable cost function (MVC), and marginal fixed cost function (MFC). Is it true that Integral_0^y MC(x)dx = C(y)? Is it also true that Integral_0^y MC(x)x = VC(y)? Now, consider a general cost function C(y). Prove that AC'(y) > 0 if AC(y)

Explanation / Answer

Multiple questions asked.

First 4 are answered below.

Production function: Y=2L+5K

Cost constraint: 2L+3K

1)

To find the cost minimizing point, setup the optimization problem as follows:

Minimize 2L+3K

Such that y=2L+5K

Since the production function represents perfect substitutes, equate 2L=5K

Use this relation to substitute in the cost function.

This makes the cost function as: 2L+3K = 5K+3K = 8K or, 3.2L

2)

New total cost function becomes: 2L+3K+(y+1)2 = 2L+3K+(2L+5K+1)2 = 100K2+28K+1

Variable cost: dependent on K = 100K2+28K

3)

AVC = TVC/K = 100k+28

AFC = TFC/K = 1/k

4)

MC = dTC/dK = 200K+28

MVC = dTVC/dK = 200K+28

MFC = dTFC/dK = 0

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