Microeconomics: Pricing 31E11100
Fall 2016
Juuso Välimäki
TA: Julia Salmi
Hints for Problem Set 1
1. True or false:
(a) All Nash equilibria are Pareto-efficient.
Solution. False. E.g. prisoner’s dilemma.
(b) Price discrimination is good for sellers and therefore bad for buyers.
Solution. False. First degree price discrimination is bad for consumers since the seller
gets the whole surplus. However, third degree price discrimination is obviously good
for those consumers who are eligible for discounts. Second degree price discrimination
has ambiguous effects on consumers’ surplus.
(c) My incentives to acquire additional price information are at the highest when very few
other buyers are informed.
Solution. True. Recall the story about search externalities in Lecture 2: If the others
search, the firms are forced to price lower, and that reduces my incentives to search.
Though, notice that if no one searches, all firms charge the monopoly price and there
no incentives to search.
2. Consider a setting where the buyers know the distribution of prices in the market, but must
pay a cost of c > 0 to get a quote from any individual seller. The buyers are all identical
with the same downward sloping demand curve. Assume that even at the highest price in
the price distribution, consumer surplus is higher than c.
(a) Suppose the buyer must choose the number of price quotes n at the cost nc. Assume
that each price quote is an independent draw from the same distribution of prices.
Discuss (in words) how to determine the optimal number of price quotes in this simultaneous quote setting.
Solution. The (expected) marginal utility of each extra quote should be larger than
marginal cost c. Notice that the marginal utility is decreasing in the number of quotes:
quotes are identically distributed and more there are existing quotes, less likely it is
that a new quote is lower than the lowest existing one and even if it is, the difference
is more likely small.
(b) Suppose next that the buyer can get the quotes sequentially. In other words, the buyer
gets an additional quote by paying c. How does this setting differ from the previous
one?
Solution. Assume that the market is large so that a consumer can make arbitrarily
many draws from the distribution.
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Now, the marginal utility depends on the lowest existing quote only. The lower the
existing offer, the less likely it is to get a much better offer. The buyer uses a cutoff
rule: draw new quotes as long as the lowest existing one is above the cutoff.
The difference between a) and b) arises from the possibility to observe the outside
option (no quote: take the lowest existing quote) in b). In a), the buyer forms expectations of the outside option. The expectation depends on the number of quotes, but
the number of quotes does not matter ones the lowest quote is known.
(c) Can you think of any practical reasons why one might choose the simultaneous quote
procedure over the sequential.
Solution. Will be discussed in the class.
(d) (Harder) Can you give a mathematical formulation for the two optimization problems
and their optimality condition. Let F (p) denote the distribution of prices and let p(q)
denote the inverse demand curve of the buyer and q(p) be the demand curve so that
the consumer surplus at price p is
ˆ
q(p)
(p(q) − p)dq.
CS(p) =
0
If you want, you can consider the case of unit demand where the willingness to pay for
the buyer is v and the price distribution is on [0, pmax ] for some pmax < v.
Solution. Sequential case.
It is profitable to draw a new quote if the lowest existing quote is p̂ iff
P r(p ≤ p̂) E(CS(p)|p ≤ p̂) − CS(p̂) ≥ c
ˆ p̂
CS(p)f (p)dp − F (p̂)CS(p̂) ≥ c.
⇔
0
The left hand side is increasing in p̂. Therefore, there exists a p̄ such that the above
condition holds for all p̂ ≥ p̄ and never for p̂ < p̄. The buyer stops, i.e. does not pay
for new quotes, when she gets a quote less or equal to the cutoff p̄.
Simultaneous case.
Now the buyer has to form expectations of the lowest quote when taking n draws from
the distribution. The larger is the expected lowest quote, the higher is the expected
gain from an extra draw.
Expected gain from n + 1:th quote:
!
G(n + 1) = E P r(p ≤ p̂) E(CS(p)|p ≤ p̂) − CS(p̂) |p̂ lowest of n
ˆ
pmax
ˆ
!
p̂
CS(p)f (p) − F (p̂)CS(p̂)dp f 1 (p̂)dp̂,
0
0
where f 1 is the density function for the lowest p:
f 1 (p) = P r(p lowest of n)f (p) = n(1 − F (p))n−1 f (p).
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The buyer chooses n∗ = min{n ∈ N : G(n + 1) < c}.
(Advanced: In order to make the reasoning complete, one should show that the buyer
does not want to take m additional draws if she does not want to take one. This
follows from that G(n+1) is decreasing in n. Intuitively, G(n+1) is decreasing because
the lowest existing quote is expected to be lower if there were more draws. To show
this formally, notice that the cumulative distribution function of the lowest quote
is 1 − (1 − F (p))n , which is increasing in n for all p. Hence, smaller n first order
stochastically dominates larger n and hence there is more probability mass on large
values of p̂ when n is small.)
3. A monopoly phone company is offering telephone services for both businesses and families.
The demand curve for businesses is given by qB = 100 − pB whereas families’ demand is
given by qF = 15 − p2F . The cost of providing the services is (qB + qF ) (i.e. all calls have a
marginal cost of one).
(a) Suppose the company can set different prices for the two markets and find the optimal
monopoly prices.
Solution. The firm maximizes profits:
max q(p)(p − c),
where c is marginal cost, here c = 1.
To businesses:
max(100 − pB )(pB − 1)
⇔ pB = 50.5.
To families:
max(15 − 1/2pF )(pF − 1)
⇔ pF = 15.5.
(b) Suppose next that a start up company figures out a way to buy family services and sell
those services to the business market at zero marginal cost. How would the monopoly
company react to this (in its pricing decisions)?
Solution. Now there is one price only. The combined demand is
(
115 − 32 p if p ≤ 30
q(p) =
100 − p otherwise.
argmax q(p)(p − 1) = 50.5.
The firm sells to the businesses only.
(c) Compare the welfare (profit plus consumer surplus) in the market across the two cases.
Solution. Case (a) leads to higher welfare.
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4. Two sellers contemplate making a price offer to a single buyer. The goods that the sellers
have are identical, and the buyer buys the good from the seller that makes the lower offer
as long as the offer is below her willingness to pay 100. At equal prices, the buyer buys from
each firm with probability 12 It costs each firm 10 to make the offer regardless of whether it
is accepted. The sellers have no other costs.
(a) The sellers decide simultaneously their offers. Let p1 be the price offer of firm 1 and p2
be the offer of firm 2. If the firm does not make the offer, then we denote the offer by
n. What are the payoffs to the two firms in the game where they choose their offers
simultaneously?
Solution. Firms can deviate to n and hence an equilibrium payoff must be nonnegative. Both firms must play p = 10 with positive probability. If not, say the smallest
price is some p̂, the other firm sets p̂ − and guarantees that the consumer buys from
her. But then the first firm makes losses and has a profitable deviation to n.
Since firms are indifferent between playing 10, which gives them at most profit 0, the
expected payoffs cannot be strictly positive. Combining this with non-negativity gives
that the expected payoffs are 0.
(b) Show that the game has no pure strategy Nash equilibria.
Solution. Use the same reasoning as in (a) to rule out any constant price above 10.
Playing always 10 cannot be an equilibrium strategy. The best response to p = 10 is
to play n, but the best response to n is 100, not 10.
(c) (Harder) Those familiar with mixed Nash equilibria should compute the symmetric
mixed strategy Nash equilibrium for the game. This mixed strategy equilibrium leads to
price dispersion since the firms make offers of different prices with positive probability.
Solution. Let the mixed strategy follow a cumulative distribution F , conditional on
making an offer. As argued in (a), p = 10 must be part of the support. Let the support
be n ∪ [10, 100]. (You should convince yourself that there cannot be gaps or atoms in
the support. Pricing at 100 must be part of the support because otherwise it would
be a profitable deviation to offer 100 instead of the largest price in the support.) The
indifference condition gives:
((1 − P r(n))(1 − F (p)) + P r(n))p = 10
1
P r(n)100 = 10 ⇔ P r(n) =
10
10 − 100 p1
⇒ F (p) =
.
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(d) Discuss the economic reasons for price dispersion in this example.
Solution. A firm does not know if the other firm is in the market (analogous to the
case where buyers may not see other prices) and hence faces a trade-off between getting
a higher price and selling with a higher probability.
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