1. The demand function for a firm’s product is Q(P) = 50-P/10. The firm’s cost o

Question

1. The demand function for a firm’s product is Q(P) = 50-P/10. The firm’s cost of production is C(Q) = Q3-20Q2+125Q. The firm’s problem is to choose the value of Q>=0 that maximizes its profit. You may occasionally find an irrational number and in those cases simplify your answer as much as possible.

(a) Calculate the firm’s inverse demand function.

(b) Calculate the firm’s marginal and average cost functions.

(c) Over what range of Q does the firm have economies to scale? Over what range of Q does it have diseconomies to scale? What is the firm’s lowest possible average cost of production?

(d) Does the firm’s profit-maximization problem satisfy the global SOC?

(e) Find all values of Q (if any) that satisfy the first-order condition for the firm’s problem.

(f) Calculate the firm’s profit-maximizing price and quantity. Justify your answer carefully.

(g) Calculate the firm’s maximized profit, and the revenue and cost that produce that profit.

(h) Calculate the elasticity of demand at the profit-maximizing point.

(i) What is the firm’s markup at the profit-maximizing point? Confirm that this markup has the expected relationship to the elasticity of demand calculated in part (h).

(j) Calculate the price(s) that would cause the firm to break even, meaning: earn exactly zero profit.

(k) For this part only, change the demand function by assuming that demand (at any given price) is half of what it was before. In this new situation, calculate the firm’s inverse demand function, profit-maximizing point, and maximized profit.

(l) For this part only, suppose that the problem is to maximize revenue instead of profit. Does this problem satisfy the global SOC? Find all points (if any) that satisfy the FOC. Calculate the revenue-maximizing price and quantity. (Justify your answer carefully.) Calculate the firm’s maximized revenue. How much profit does the firm sacrifice by choosing to maximize revenue instead of profit?

[The remaining parts of the original assignment are postponed to HW #10.]

2. The demand function for a firm’s product is Q = P-3 . The firm’s marginal cost of production is constant at MC(Q) = 12.

(a) Calculate the elasticity of demand, as a function of Q.

(b) Does the firm’s profit maximization problem satisfy the global SOC?

(c) Using your answers to (a) and (b), what is the firm’s profit-maximizing markup? (Justify your answer carefully. Do not forget about the possibility of a boundary solution.)

(d) Based on your answer to (c), what is the firm’s profit-maximizing price?

(e) Based on your answer to (d), what is the firm’s profit-maximizing quantity?

Explanation / Answer

Ans a)

P= 500- 10 Q

INVERSE DEMAND FUNCTION IS PRICE EXPRESSED IN TERMS OF QUAQNTITY JUST BY REARRANGING

ANS B)

AR = P= 500-10Q

TR= AR*Q = 500Q-10Q2

MR= dTR/dQ= 500 – 20 Q

ANS C)

C(Q) = Q3-20Q2+125Q.

AC(Q) = C/Q = Q2 -20 Q +125

FOR MINIMUM AC. FIRST ORDER DERIVATIVE OF AC SHOULD BE = 0

dAC/dQ = 2 Q -20 = 0

   Q= 20/2= 10

SO AT Q= 10 . AC IS MINIMUM . BEFORE Q= 10, ECONOMIES OF SCALE OPERATES AND AFTER THIS Q DISECONOMIES OF SCALE OPERATES

AT Q= 10, MIN AC= 102- 20(10)+125 = 25

ANS e)

Profit = TR-TC =500Q-10Q2 - Q3+20Q2-125Q

PROFIT = 375Q +10Q2 - Q3

1 ST ORDER CONDITION

dPROFIT/dQ = 375 +20Q – 3 Q2 =0

           Q= 15

        Q=-25/3( NEGATIVE QUANTITY NOT POSSIBLE)

SO Q=15

AND D)

d2profit/d2Q = 20 - 6Q = 20 – 6(15) =-70<0 SO Q=15 IS MAXIMISING PROFIT ACCORDING TO SOC

ANS F) Q AND P AT MAXIMUM PROFIT

Q= 15

P =500-10Q = 350

ANS G)

PROFIT, REVENUE AND COST AT PROFIT MAXIMISING POINT

TR= 500Q-10Q2 = 7500-2250 =5250

C = TC= Q3-20Q2+125Q = 3375 – 20 (225) + 1875 =3375 -4500+1875 =750

PROFIT= TR- TC =5250 -750 =4500

ANS H)

Q(P) = 50-P/10

ELASTCITY Ep = -(dQ/dP )* (P/Q)= - (-1/10) *(350/15) = 2.33

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