Solutions for Homework 6
Chapter 8: Problem 1
a. 7 units.
b. $28.
c. $32*7 = $224.
d. $14*7 = $98.
e. $224 – $98 = $126
f. ($28 – $32)*7 = – $28.
g. – $126, since its loss will equal its fixed costs.
h. Shut down.
Chapter 8: Problem 2
a. Set P = $80 = 8 + 4Q = MC to get Q = 18 units.
b. $80.
c. Tp = TR – TC = $80*18 – (40 + 8*18 + 2*182) = $608.
d. Entry will occur leading to lower market price and firm’s output. The long-run economic profits = 0.
Chapter 8: Problem 3
a. 7 units.
b. $130.
c. ($130 – $110)*7 = $140.
d. Demand will decrease over time as new firms enter the market. The long-run economic profits = 0.
Chapter 8: Problem 4
a. Setting MR = 200 – 4Q = 6Q = MC => Q = 20 and P = 200 – 2*20 = $160.
b. Tp = TR – TC = $160*20 – (2000 + 3*202) = 0.
c. Elastic.
d. Maximum TR when MR = 200 – 4Q = 0 => Q = 50 => P = 200 – 2(50) = $100.
e. Maximum R = $100*50 = $5,000.
f. Unitary.
Chapter 8: Problem 5
a. A perfectly competitive firm’s supply curve is its MC > minimum of its AVC curve.
MC = 50 – 8q + 3q2 = 50 – 4q + q2 = AVC => min AVC|q=2 = 50 – 4*2 + 22 = 46. The firm’s supply curve is P = 50 – 8q + 3q2 for P ³ 46, otherwise, zero units are produced.
b. A monopolist produces where MR = MC and has no supply curve.
c. A monopolistically competitive firm produces where MR = MC and has no supply curve.
Chapter 8: Problem 6
a. Q = 3 units; P = $70.
b. Q = 4 units; P = $60.
c. DWL = (70 – 40) / 2 = 15
Chapter 8: Problem 15
Solve system of linear equations: MR = 1,000 - 10Q = 10Q1 = MC1
MR = 1,000 - 10Q = 4Q2 = MC2
for Q1 = 200/9 = 22.22 and Q2 = 500/9 = 55.56 => P = f(Q1 + Q2) = 1,000 – 5*(700/9) = 611.11
Tp = TR – (TC1 + TC2) = $611.11*77.78 – (10,050 + 5*22.222 + 5,000 + 2*55.562) = $23,839.67.
Problem Set: Problem 25
a. True. The sum of the changes in TR as output changes necessarily equals TR.
b. False. The sum of the marginal costs equals TVC, not TC.
c. False. The sum of the marginal profit values equals total contribution profit above variable cost.
Problem Set: Problem 26
Profit is maximum where MR = P = 70 = 25 – 12Q + Q2 = MC => Q = 15.
Tp = TR – TC = $79*15 – (200 + 25*15 – 6*152 + 153/3) = $700 per week.
Problem Set: Problem 27
Problem Set: Problem 28
a. Price taker maximize profit where: MR = P = 40 = 4 + 2Q = MC => Q = 18
Firm's supply curve is its MC above its min AVC = 4 + Q (firm produces only if P ³ min AVC).
Derivative of AVC = 1 ¹ 0 => Q = 0 => min AVC|Q=0 = 4
Inverse supply curve is P = MC = 4 + 2Q, for P ³ $4, and supply curve is Qs = P/2 -2.
b. No effect on firm's output or short run supply curve, since TFC does not affect MC.
c. TpBefore = TR – TC = 40*18 – (100 + 4*18 + 182) = 224
TpAfter = TpBefore – (TFCAfter – TFCBefore) = 224 – (144 – 100) = 180
Since profit is positive in the short run, even following the more stringent environment regulation, in the long run more firms will enter and the price will be lower than $40/unit.
Problem Set: Problem 29
a. If tapes are given away P = 0, and thus Q = 1,600 – 200*0 = 1,600.
In absence of revenue, profit maximization becomes cost minimization:
TCA = 1,200 + 2(1,600) = 4,400 and TCB = 4(1,600) = 6,400 => buy from A.
b. To max profit MR = 8 – Q/100 = 2 = MCA => Q = 600 => P = 8 – 600/200 = 5
TpA = TR – TC = 5*1,600 – (1200 + 2(1600)] = 600
MR = 8 – Q/100 = 2 = MCB => Q = 400 => P = 8 – 400/200 = 6
TpB = 6*400 – 4*400 = 800
Thus, buy from B, and sell 400 tapes at $6 per tape.
Problem Set: Problem 30
a. BQ sets price to maximize 0.2(TR), which is the same as to maximize TR.
MR = 3 – Q/400 = 0 => Q = 1,200 => P = 3 – 1,200/800 = 1.5
BQ receives: 0.2(P*Q) = 0.2(1.5*1200) = 360
Franchise's profit: 0.8(P*Q) – AC*Q = 0.8*1.5*1200 – 0.8*1200 = 480
b. The franchise sets price to maximize its profit: 0.8(P*Q) – AC*Q = 0.8(3 – Q/800)Q – 0.8Q
Mp = 0.8(3 – Q/400) – 0.8 = 0 => Q = 800 => P = 3 – 800/800 = 2
BQ receives: 0.2(P*Q) = 0.2(2*800) = 320 < 360 from part a
Franchise receives: 0.8(P*Q) – AC*Q = 0.8*2*800 - 0.8*800 = 640 > 480 from part a
With conflict of interest: Part (a) Tp = 360 + 480 = $840 < Part (b) Tp = 320 + 640 = $960.
c. In profit sharing BQ and franchise maximize: t(P*Q – AC*Q) and (1-t) (P*Q – AC*Q), where t is the BQ’s share. Both objectives are equivalent to simple maximization of profits: (3 – Q/800)Q – 0.8Q.
Mp = 3 – Q/400 – 0.8 = 0 => Q = 880 => P = 3 – 880/800 = 1.9
BQ receives: 0.2(P*Q) = 0.2(1.9*880) = 334.4
Franchise receives: 0.8(P*Q) – AC*Q = 0.8*1.9*880 - 0.8*880 = 633.6
Without conflict of interests: Part (a) Tp = 840 Part (b) Tp = 960 < Part (b) Tp = 968.
d. Profit sharing is not widely practiced, since the cost of operating the franchise cannot be verified by BQ. There will be a tendency on the part of the franchise to misrepresent his operating costs.
Problem Set: Problem 31
a. MR = 20 – Q = 5 = MC => Q = 15 => P = 20 – 0.5*15 = 12.5
Tp = TR – TC = 12.5*15 – (30 + 5*15) = 82.5
b. TC = 30 + 8Q
MR = 20 – Q = 8 = MC => Q = 12 => P = 20 – 0.5*12 = 14
Tp = TR – TC = 14*12 – (30 + 8*12) = 42
Output price increase $1.5 is less input price increase $3, and hence profit decreases.
c. MR = 20 - Q = 5 = MC => Q = 15 => P = 20 - 0.5*15 = 12.5
after tax Tp = Tp – tax = (TR – TC) – tax = [12.5*15 – (30 + 5*15)] – 20 = 62.5
Change in property tax, which is a fixed cost, does not affect pricing.
d. MR = 20 – Q = 5 = MC => Q = 15 => P = 20 – 0.5*15 = 12.5
after tax Tp = (1 – t)p = 0.8(TR – TC) = 0.8[12.5*15 – (30 + 5*15)] = 66
Tax on profit does not affect pricing.
e. MR = .8(20 – Q) = 5 = MC => Q = 13.75 => P = 20 – 0.5*13.75 = 13.125
Tp = 0.8*TR – TC = 0.8(13.125*13.75) – (30 + 5*13.75) = 45.625
Sale tax affects MR and hence pricing.
Problem Set: Problem 32
The American Cracker Corporation should equate MR and MC to maximize profits.
Output in Cartons / Selling Price / TR / MR / Firm wide MC0
1
2
3
4
5
6
7
8
9
10
11
12 / .50
.48
.46
.44
.42
.40
.38
.36
.34
.32
.30
.28
.26 / ---
.48
.92
1.32
1.68
2.00
2.28
2.52
2.72
2.88
3.00
3.08
3.12 / .48
.44
.40
.36
.32
.28
.24
.20
.16
.12
.08
.04 / .10
.13
.13
.14
.14
.15
.16
.16
.16
.18
.19
.20
Nine cartons should be produced, of which 2 at Plant #1, 4 at Plant #2 and 3 at Plant #3.