Recitation 7

Krzysztof Makarski

Auction

1. At Toivo’s auction house in Ishpemming, Michigan, a beautiful stuffed moosehead is

being sold by auction. There are 5 bidders in attendance: Aino, Erkki, Hannu, Juha,

and Matti. The moosehead is worth $100 to Aino, $20 to Erkki, and $5 to each of the

others. The bidders do not collude and they don’t know each others’ valuations.

(a) If the auctioneer sells it in an English auction, who would get the moosehead and

approximately how much would the buyer pay?

Answer:

Aino would get it for $20.

(b) If the auctioneer sells it in a sealed-bid, second-price auction and if no bidder

knows the others’ values for the moosehead, how much should everyone bid, whowould get the moosehead and how much would he pay?

Answer:

Aino $100, Erkki $20, and the others $5.Aino would get it for $20.

2. Charlie Plopp sells used construction equipment in a quiet Oklahoma town. He hasrun short of cash and needs to raise money quickly by selling an old bulldozer. If he

doesn’t sell his bulldozer to a customer today, he will have to sell it to a wholesaler for

$1,000. Two kinds of people are interested in buying bulldozers. These are professional

bulldozer operators and people who use bulldozers only for recreational purposes onweekends. Charlie knows that a professional bulldozer operator would be willing to

pay $6, 000 for his bulldozer but no more, while a weekend recreational user would be

willing to pay $4, 500 but no more. Charlie puts a sign in his window. “Bulldozer SaleToday.” Charlie is disappointed to discover that only two potential buyers have come

to his auction. These two buyers evidently don’t know each other. Charlie believes

that the probability that either is a professional bulldozer operator is independent of

the other’s type and he believes that each of them has a probability of 1/2 of being aprofessional bulldozer operator and a probability of 1/2 of being a recreational user.

Charlie considers the following three ways of selling the bulldozer:

• Method 1. Post a price of $6,000, and if nobody takes the bulldozer at that price,sell it to the wholesaler.

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• Method 2. Post a price equal to a recreational bulldozer user’s buyer value andsell it to anyone who offers that price.

• Method 3. Run a sealed-bid auction and sell the bulldozer to the high bidder atthe second highest bid (if there is a tie, choose one of the high bidders at random

and sell the bulldozer to this bidder at the price bid by both bidders.)

(a) What is the probability that both potential buyers are professional bulldozer

operators? What is the probability that both are recreational bulldozer users?

What is the probability that one of them is of each type?

Answer.

1/4, 1/4 and 1/2.

(b) If Charlie sells by method 1, what is the probability that he will be able to sell

the bulldozer to one of the two buyers? What is the probability that he will have

to sell the bulldozer to the wholesaler? What is his expected revenue?

Answer.34, 14, 34× $6, 000 + 1

4× $1, 000 = $4, 750.

(c) If Charlie sells by method 2, how much will he receive for his bulldozer?

Answer.

$4,500.

(d) Suppose that Charlie sells by method 3 and that both potential buyers bid ra-

tionally. What will be Charlie’s expected revenue from selling the bulldozer by

method 3?

Answer.14× $6, 000 + 3

4× $4, 500 = $4, 750.

(e) Which of the three methods will give Charlie the highest expected revenue?

Answer.

Method 3.

(f) Suppose that a weekend recreational user would be willing to pay $3, 500 (com-

paring to $4, 500 before). How would your answer to (d) and (e) change?

3. Late in the day at an antique rug auction there are only two bidders left, April andBart. The last rug is brought out and each bidder takes a look at it. The seller says

that she will accept sealed bids from each bidder and will sell the rug to the highest

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bidder at the highest bidder’s bid. Each bidder believes that the other is equally likely

to value the rug at any amount between 0 and $1,000. Therefore for any number xbetween 0 and 1,000, each bidder believes that the probability that the other bidder

values the rug at less than x is x/1, 000. The rug is actually worth $800 to April. If

she gets the rug, her profit will be the difference between $800 and what she pays for

it, and if she doesn’t get the rug, her profit will be zero. She wants to make her bid insuch a way as to maximize her expected profit.

(a) Suppose that Bart will bid exactly what the rug is worth to him. If April bids $xfor the rug, what is the probability that she will get the rug? What is her profit

if she gets the rug for $x? What is her expected profit if she bids $x?

Answer.x

1000, $(800− x), $(800− x) · x

1000

(b) Find the bid x that maximizes her expected profit. (Hint: Take a derivative.)

Answer.

x = 400.

(c) Now let us go a little further toward finding a general answer. Suppose that thevalue of the rug to April is $V and she believes that Bart will bid exactly what

the rug is worth to him. Write a formula that expresses her expected profit in

terms of the variables V and x if she bids $x.

Answer.

$(V − x)( x1000

)

(d) Now calculate the bid $x that will maximize her expected profit. (Same hint:

Take a derivative.)

Answer.

x = V/2.

4. If you did the previous problem correctly, you found that if April believes that Bart

will bid exactly as much as the rug is worth to him, then she will bid only half as much

as the rug is worth to her. If this is the case, it doesn’t seem reasonable for April tobelieve that Bart will bid his full value. Let’s see what would the best thing for April

to do if she believed that Bart would bid only half as much as the rug is worth to him

(a) If Bart always bids half of what the rug is worth to him, what is the highest

amount that Bart would ever bid?

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Answer.

$500.

(b) Suppose that the rug is worth $800 to April and she bids $x for it. April will onlyget the rug if the value of the rug to Bart is less than $2x. What is the probability

that she will get the rug if she bids $x for it? What is her expected profit if she

bids $x? What should she bid?

Answer.2x1000

, $(800− x) 2x1000

, $400

(c) Suppose that April values the rug at $V and she believes that Bart will bid half

of his true value. What should April bid?

Answer.

Maximize (V − x) 2x1000

, to get x = V/2.

(d) Suppose that April believes that Bart will bid half of his actual value and Bart

believes that April will bid half of her actual value. Suppose also that they both

act to maximize their expected profit given these beliefs. Will these beliefs beself-confirming in the sense that given these beliefs, each will take the action that

the other expects?

Answer.

Yes.

Monopoly

1. Professor Bong has just written the first textbook in Punk Economics. It is called

Up Your Isoquant. Market research suggests that the demand curve for this book will

be Q = 2, 000 − 100P , where P is its price. It will cost $1, 000 to set the book in

type. This setup cost is necessary before any copies can be printed. In addition to thesetup cost, there is a marginal cost of $4 per book for every book printed. Find his

optimal output, price and profits. Use two different methods (profit maximization and

MR =MC approach). Show his profits on a graph.

Answer.

Q∗ = 800.

2. A monopolist has an inverse demand curve given by p(y) = 12 − y and a cost curve

given by c(y) = y2.

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(a) What will be its profit-maximizing level of output?

Answer.

3

(b) What will be the Pareto optimal level of output?

Answer.

4

(c) Compute and draw the deadweight loss.

Answer.32

(d) Suppose this monopolist could operate as a perfectly discriminating monopolist

and sell each unit of output at the highest price it would fetch. What would the

output and the deadweight loss be in this case?

Answer.

4 and 0

(e) Suppose the government decides to put a tax on this monopolist so that for each

unit it sells it has to pay the government $2. What will be its output under thisform of taxation?

Answer.

2.5.

(f) If you wanted to choose a price ceiling for this monopolist so as to maximizeconsumer plus producer surplus, what price ceiling should you choose? How

much output will the monopolist produce at this price ceiling?

Answer.

8, 4.

(g) Suppose now that the government puts a lump sum tax of $10 on the profits of

the monopolist. What will be its output?

Answer.

3.

Monopoly behavior

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1. Ferdinand Sludge has just written a disgusting new book, Orgy in the Piggery. His

publisher, Graw McSwill, estimates that the demand for this book in the United Statesis Q1 = 50, 000− 2, 000P1, where P1 is the price in the U.S. measured in U.S. dollars.The demand for Sludge’s opus in England is Q2 = 10, 000 − 500P2, where P2 is itsprice in England measured in US dollars. His publisher has a cost function C(Q) =

$50, 000 + $2Q, where Q is the total number of copies of Orgy that it produces.

(a) If McSwill must charge the same price in both countries, how many copies should

it sell? What price should it charge to maximize its profits? How much will thoseprofits be?

27, 500, $13, $252, 500.

(b) If McSwill can charge a different price in each country and wants to maximize

profits, how many copies should it sell in the United States? What price should itcharge in the United States? How many copies should it sell in England? What

price should it charge in England? How much will its total profits be?

Answer.

US: 23, 000, $13.5.

England: 4, 500, $11

Total profits = $255, 000.

2. Consider a monopoly producing two goods A and B. The monopoly has the following

cost structure: C(yA) = 0 and C(yB) = 0. There are two types of consumers called

1 and 2 with unit demand functions for each good. For simplicity each type consistsonly of one consumer. The following are reservations prices (or valuations) of goods A

and B by consumers: v1A = 80 and v1B = 80, v2A = 120 and v2B = 10.

(a) What are the optimal prices of both goods, pA and pB without any bundling?

Compute profits πa associated with this pricing strategy.

Answer.

pA = 80 and pB = 80, which implies profits π = 3 ∗ 80 = 240(b) What is optimal price of the bundle pbundle using pure bundling pricing (i.e. only

bundles are sold). Compute profits πb associated with this pricing strategy.

Answer.

pbundle = 130, which implies profits π = 2 ∗ 130 = 260

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(c) What are optimal prices of the bundle pbundle and goods pA and pB using mixed

bundling pricing (i.e. bundles as well as separate goods are sold). Compute profitsπc associated with this pricing strategy.

Answer.

pbundle = 160, pA = 120 and pB > 40, which implies profits π = 1 ∗160+1∗ 120 =280

3. A phone company ABC monopolized the telecommunications market in the country X.

It is not an usual market, as there are only two types of customers in this market: type1 (residential customer with low demand - low type) and 2 (business customer with

high demand). More, there is only one customer of each type. After extensive market

research the monopolist ABC estimated that demands of different types of customersare: D1(P1) = 5− P1 for type 1, and D2(P2) = 10− 2P2 for type 2. Let’s assume forsimplicity that cost of ABC are zero, i.e. C(Q) = 0.

(a) Consider a monopolist that doesn’t engage in price discrimination by any means

(in other words, it uses linear pricing). What are monopolist’s profits in this case

Answer.

Problem maxP

P (15− 3P ) gives P ∗ = 2.5 and π = 18.75

(b) Consider a monopolist that wants to charge 2 different two-part tariffs: T1(Q1) =

f1 + P1Q1 customized to type 1 consumers and T2(Q2) = f2 + P2Q2 customized

to type 2 customers. Formulate now the problem of profit maximizing monop-

olist ABC subject to participation and incentive compatibility (self selection)constraints for each type. Denote a net surplus of type 1 at two part tariff

T (Q) = f + PQ as CS1(P ) − f, and by analogy a net surplus of type 2 at

two part tariff T (Q) = f + PQ as CS2(P )− f.

Answer.

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Problem

maxf1,f2,P1,P2

f1 + f2 + P1D1(P1) + P2D2(P2)

sb. to

CS1(P1)− f1 ≥ 0CS2(P2)− f2 ≥ 0

CS1(P1)− f1 ≥ CS1(P2)− f2

CS2(P2)− f2 ≥ CS2(P1)− f1

(c) At optimal solution the participation constraint for type 1 (low type) and incentive

compatibility constraint for type 2 (high type) holds with equality. The other

constraints are unimportant and can be neglected at optimal solution (check at

the end if they are satisfied). Using this info rewrite the profit maximizationproblem of ABC from b) to get unconstrained problem with only P1 and P2 as

decision variables. Solve first the for optimal P1, P2, and then for associated f1, f2and profits Π∗.

Answer.

maxP1,P2

CS1(P1) + CS2(P2)−CS2(P1) + CS1(P1) + P1D1(P1) + P2D2(P2)

ormaxP1,P2

2CS1(P1) + CS2(P2)−CS2(P1) + P1D1(P1) + P2D2(P2)

Note that CS1(P ) = 0.5(5− P )2 and CS2(P ) = 0.5(5− P )(10− 2P ) = (5− P )2

(recall consumer surplus is the area below demand curve and above the price).

Now plugging all function of prices into objective problem we get:

maxP1,P2

(5− P2)2 + P1(5− P1) + P2(10− 2P2)

FOCs:

With respect to P1 :5− P1 − P1 = 0

P ∗1 = 2.5

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With respect to P2 :

2(5− P2)(−1) + 10− 2P2 − 2P2 = 0

P ∗2 = 0

Now we can compute f1 and f2 :

f1 = CS1(5) = 0.5(5− 2.5)2 = 3.125f2 = CS2(0)−CS2(2.5) + CS1(2.5) = (5− 0)2 − (5− 2.5)2 + 0.5(5− 2.5)2

= 4 ∗ 2.52 − 2.52 + 0.5 ∗ 2.52 = 3.5 ∗ 2.52 = 21.875

and profits Π∗ = 3.125+ 21.875+0(10− 2 ∗ 0)+ 2.5(5− 2.5) = 25+ 6.25 = 31.25(d) What is a percentage increase in profits of a monopolist in point c) as compared

to point a)?

Answer.

Percentage increase in profits = (31.2518.75 − 1) ∗ 100 = 66.667%

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