ANSWERS, HOMEWORK 2
Team “Micro Math” Homework Problems.
Homework Assignment II
Due Date, July 16, 2002
1. With TC = 20,000 + 4Q + .5Q2, find
a. AFC b. AVC c. AC (=ATC), and d. MC
Answer: la. AVC = TVC/Q = (4Q + .5 Q2)/Q = 4 + .5Q
b. ATC = TC/Q = (20,000 + 4Q + .5 Q2)/Q = 20,000/Q + 4 + .5Q
c. MC = dTC/dQ = dTVC/dQ = 4 + Q. And, of course, AFC = 20,000/Q
2. With a Total Fixed Cost of 10 and an AVC of 26 - 5Q + .5Q2, find
a. TVC, b. ATC, And c. MC.
Answer: 2a. TVC = AVC(Q) = (26- 5Q + .5 Q2) Q = 26Q- 5 Q2 + .5 Q3
b. ATC = TC/Q = (10 + 26Q- 5Q2 + .5 Q3)/Q = lO/Q + 26- 5Q + 0.5 Q2
c. MC = dTC/dQ = 26- l0Q + 1.5 Q2.
3. By way of reminder, the formula for price elasticity of demand is
E = dQ/dP(P/Q). At Q = 10 and P = 1000 + 3Q - 4Q2, find E.
Answer: 3. At Q = 10, P = 1000 + 3(10)- 4(10)2 = 630
dP/dQ = 3- 8Q = 3- 8(10) = -77; dQ/dP = - 1/77
E = dQ/dP(P/Q) = - 1/77(630/10) = - (630/770) = -.82
4. A firm in a purely competitive industry faces a fixed price of $70 for its product. With a TC function
TC = 150 + 25Q - 6Q2 + 1/3Q3.
Show how this firm maximizes profit. (Hint: Equate MC and MR. When you encounter an expression with a Q2, factor it, getting two roots. The larger one is the profit-maximizing output.) Show what short-run profits will be.
Answer. 4. Note that where P is constant, P = MR, so MR = 70
MC = dTC/dQ= 25- 12Q + Q2
Equating MC and MR and solving for the profit-maximizing Q gives:
25 - 12Q + Q2 = 70
-45 - 12Q + Q2 = 0
Now, we factor the expression on the left.
(-15+q)(3+q)=0
Q = 15,-3
The largest root, 15, is the profit-maximizing output rate. At Q = 15, TR = $1050 and TC = 300, so short-run profits are $750 per week.
5. Determine the profit-maximizing price and quantity for a firm with demand function,
P = 1625 - 6Q and MC = 25 - 12Q + Q2.
Answer. 5. TR = PQ = (1625-6Q)Q = 1625Q-6Q2
MR = dTR/dQ = 1625- 12Q
MC = dTC/dQ = 25- 12Q + Q2
Equate MC and MR, then solve for Q:
1625- 12Q = 25- 12Q + Q2
1600 = Q2
Q = 40. At Q = 40, P = 1625- 6(40) = $1385.