Question 1 Given the production function for good X is Q = 2K1/4L1/2 In the shor

Question

Question 1 Given the production function for good X is Q = 2K1/4L1/2 In the short run, the amount of capital is fixed at K-16. The capital rental rate is r = $2.5, and wage rate is w = $10. 1) Derive the short-run total cost function. 2) Derive the short-run average cost (AC), average fixed cost (AFC), and average variable cost AVC). Show them on the same graph. Derive short-run marginal cost curve. Show on the same graph as 2). Derive the short-run supply function of good x. Suppose now the firm is selling good X at a fixed price $15, will the firm making any profit at this price? If so, how much profit does it make? What will be the price level if the firm does not earn any profit? 3) 4) 5) 6)

Explanation / Answer

a) Production function is Q = 2K^0.25L^0.5. Fixed cost = 16*2.5 = $40. This implies Q = 2*(16^0.25)*L^0.5 = 4L^0.5

This gives

L = (0.25Q)^2 = 0.0625Q^2

Short run cost function is C = 10*0.0625Q^2 + 40 = 0.625Q^2 + 40

b) AC = 0.625Q + 40/Q. AFC = 40/Q and AVC = 0.625Q

c) MC = dTC/dQ = 1.25Q

d) Let the price of good be P. Then it implies VMPL = wage

P x d4L^0.5/dL = 10

P x 2 x L^-0.5 = 10

Here L =  0.0625Q^2

0.2P =  (0.0625Q^2)^0.5

0.2P = 0.25Q

P = 1.25Q

Note that is the same function as MC so we believe that supply function is rising portion of MC.

e) Price is 15. Hencem Q = 12. Total cost at Q = 12 is 0.625*(12^2) + 40 = 130. Profit = 12*15 - 130 = $50

f) Break even price has TR = TC

1.25Q^2 = 0.625Q^2 + 40

This gives Q* = 8. At this the price is 1.25*8 = $10. So the breakeven price is 10.

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