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^M 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

At Equilibrium: MPL/w=MPK/r
MPL=(Y)'=2 and MPK=(Y)'=5

as w=2, r=3
MPL/2=2/2
MPK/3=5/3
Since 5/3 is highest value - thus capital gives higher product per dollar spent - so producer should choose only capital in production and no labor at all

L=0, y=2L+5K = 5K
K=y/5=
TC=2*0 + 3*y/5

Total Cost Function: (y+1)^2 + 0.6y = y2 +1 +2y + 0.6y

Variable Cost: 2y +2.6

Average cost : y + 1/y + 2.6

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