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Advanced Industrial Organization: Problem Set 2—Answerkey
CRN 25965 - ECON 485 B Pascal Courty University of Victoria
Exercise 1 Damaged-good strategy (25%, 40 words per answer on average)
A firm sells a product in a market where there are two types of consumers,
high and low-valuation consumers. There are equally many of the two types of
consumers, and the total number of consumers is normalized to 1. The product
has value 3 to the high-valuation consumers and value 1 to the low-valuation
consumers. All consumers have unit demand, i.e., they buy either one unit or
do not participate. The product is produced at constant marginal cost equal to
0.
1. Find the profit maximizing price and calculate the firm’s profit.
The seller may either sell to 1/2 of the consumers at 3 or to all at 1. Since
3/2 > 1 the former strategy dominates the later. Profits are 3/2.
2. The firm considers introducing a damaged version of the product. The
damaged version is produced at constant marginal cost equal to 1/10. It
results in a utility of 5/10 to the low-valuation consumers and of 6/10
to the high valuation consumers. Find the optimal price of the normal
and of the damaged version of the product. Should the firm introduce the
damaged version? What are the welfare consequences of the introduction
of the damaged version?
The firm will charge 5/10 to the low valuation consumers for the damaged version and bind the incentive compatibility constraint of the high
valuation consumers 3 − p = 6/10 − 5/10. The price of the non-damaged
product is 29/10 and profits are 1/2(29/10)+1/2(5/10-1/10)=33/20. Welfare increases by 1/2(5/10-1/10)=1/5. Welfare is not maximized because
the low types get a damaged good that is inefficiently produced.
Exercise 2 Bundling (25%, 30 words per answer on average)
Suppose that a monopolist produces two products, product 1 and product
2. There is a mass 1 of consumers. A share λ of consumers are heterogeneous
among each other and are described by their type θ. This type is distributed
uniformly on the unit interval. The willingness-to-pay for product 1 and 2 is
assumed to be r1 = θ and r2 = 1 − θ. A share (1 − λ)/2 of consumers has
willingness to pay r1 = 2/3 and r2 = 0. The remaining share (1 − λ)/2 of
consumers has willingness to pay r1 = 0 and r2 = 2/3. Firms can sell products
1 and 2 independently at prices p1 and p2 , respectively. Alternatively, it may
only sell a bundle at price p. This is a situation referred to as pure bundling. A
third possibility is that the firm sells the bundle and the independent products,
a situation referred to as mixed bundling.
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1. Suppose λ = 1. Determine whether independent selling, pure or mixed
bundling are profit maximizing. Calculate associated prices and profits.
Under independent selling, the firm charges 1/2 for each good and earns
2(1/4)=1/2. Under pure bundling, the firm charges 1 for the bundle and
earns 1. Under mixed bundling, the firm cannot do better than pure
bundling.
2. Suppose that λ > 0 and characterize the solution under independent selling for all λ > 0.
You should draw the inverse demand curve for each product. It starts
at (0,1), decreases to (λ/3, 2/3), remains flat up to ((3 − λ)/6, 2/3), and
finaly decreases again to ((1 + λ)/2, 0). In maximizing profits, the firm
will have to watch out for corner solutions. The firm can either charge 2/3
per product. Alternatively, the firm can charge less than
and earn 3−λ
9
2
(1+λ)
1+λ
2/3 and earn p( 1+λ
2 − λp). The FOC gives p = 4λ and profits
16λ
per product. This value for profits is correct only for p < 2/3 which is
2
> 3−λ
equivalent to λ > 3/5. One can show that (1+λ)
16λ
9 . Therefore, the
1+λ
firm will charge 2/3 if λ < 3/5 and p = 4λ otherwise.
3. Suppose that λ = 4/5. Characterize the profit-maximizing solution under
independent selling, pure bundling, and mixed bundling. Show which of
the selling strategies is profit-maximizing. Discuss your result.
Under pure bundling, you can derive the inverse demand curve as a step
function. The firm can either charge 1 and earn λ or charge 2/3 and earn
2/3.
Under mixed bundling, the firm will charge 2/3 for the individual products
(there is not reason to charge less since the θ types will buy the bundle).
The issue is whether the seller wants to charge less than 1 for the bundle.
The problem with charging 1 is that some of the θ types will prefer to
buy only one of the individual products. Let p12 denote the price of the
bundle. Very low θ types and very high ones will prefer to buy the separate
good and types in the middle will prefer the bundle. The indifferent types
are such that 1 − p12 = θ − 2/3 and 1 − p12 = 1 − θ − 2/3. These
two threshold types determines how the θ types sort across products.
Collecting sales from the different consumers give profits p12 (7/3−2p12 )λ+
(4/3)λ(p12 − 2/3) + (2/3)(1 + λ) where the first term captures the sales
from the bundle to intermediate types, the second term captures the sales
of a single product to extreme types, and the third term captures sales of
a single product to the types that are willing to pay 2/3 for one product.
Taking FOC gives the optimal price p12 = 11/12 which is very close to 1.
Plugging this expression back gives profits λ/8 + 2/3 and using λ = 4/5
gives profits of 23/30 which is less thant 4/5. We conclude that mixed
bundling is dominated. The intuition is that since λ is high, it is costly to
offer separate sales. The monopolist prefers to sell the bundle only at 1.
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4. Suppose that λ = 2/3. Characterize the profit-maximizing solution under
independent selling, pure bundling, and mixed bundling. Show which of
the selling strategies is profit-maximizing. Discuss your result.
The analysis is the same as above but plug the value λ = 2/3 in the profits
formula to obtain 3/4 which dominates the profits under pure bunding
which are 2/3. Since λ is not so high, the monopolist should sell to all
consumers and mixed bundling is optimal.
Exercise 3 Research Article (50%, 40 words per answer on average)
Timothy Bresnahan and Peter Reiss. Entry and Competition in Concentrated
Markets. Journal of Political Economy, 1991. You should read sections I and
II and only what is necessary to understand Figures 3 and 4 in section III (you
can skip IIIC and IV).
A local market is populated by S consumers. Each consumer has demand
d(p) = 100 − p for a homogenous product (such as dental care, plumbing).
Firms have to invest fixed cost F to enter the market and then pay c(q) = q 2
to produce q units.
1. (a) Compute the inverse demand curve p(Q) in a local market. (b) Plot
p(Q) for S = 1, S = 2 and S = 100. What happens to p(Q) when the
number of consumers in the market increases?
The inverse demand curve is p(Q) = 100−Q/S. It becomes flatter (rotates
upward around the point (0,100)) as S increases. For S = 100 it is almost
flat relative to S = 2.
2. (a) What is the smallest population size S such that a monopolist earns
non-negative profits? Denote that value sM . (b) What is the smallest
population size such each member of a cartel formed of n firms earns
non-negative profits?
(a) The monopolist maximizes q(100−q/S)−q 2 −F . The optimal quantity
S
50S
. Profits 502 S+1
− F are positive if S > 502F−F . (b) The cartel
is q = S+1
will behave as a monopolist but the difference is that each member of the
cartel has to recover its fixed cost. If the are n members in the cartel,
market size must be at least n 502F−F .
3. Consider a large local market (S large) with many firms who are price
taker and earn zero profits. How many consumers does each firm serve?
Denote that value s∞ .
2
Marginal cost is 2q, average cost is F +q
competitive quantity
q . At the √
√
√
MC=AC
so
q
=
F
and
p
=
M
C
gives
S(100
−
2
F ) = F or s∞ =
√
F√
.
100−2 F
4. (a) Show that sM and s∞ are positive if and only if F < 502 . (b) Assume
this condition holds and show that sM < s∞ . Interpret.
√
We have
F√
100−2 F
>
F
502 −F
iff F < 502 .
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5. Consider a local market where there are n identical firms. Define firm
size sn as the observed number of consumers served by each firm. Provide
informal arguments: (a) Why would you expect firm size to increase with
the number of firms in a market (sn+1 > sn )? (b) Why should the ratio
sn+1 /sn converge to one as n increases?
(a) As market size increases, more firms enter, competition increases, firms
earn lower profits, so they need to sell more to cover their fixed cost.
Firm size increases with the number of firms in the market. (b) A n
increases, prices converge to marginal cost and firms are close to break
even. Additional entry has a small impact on prices. Firm size converges
to s∞ and sn does not increase much with n.
6. How would you derive sM , s2 and s∞ from Figure 3?
S M is the threshod that allows at least one firm to enter. It is above 500
since the majority of towns with less than 500 people have no dentist.
It is less than 2000 since the majority of towns larger than 2000 have
2 dentists. So sM is in between 500 and 2000. s2 is around 2000 (the
boundaries are not precisely defined because towns of similar sizes can
have 1 or 2 denstist. s∞ is the increment in town size such that beyond a
given town size a dentist enters each time size increases by that increment.
It is difficult to read what this increment is on the figure but it seems that
by increasing town size by 2000, the average number of dentists increases
by one. Consider the move from 3K-4K to 5K-6K or from 4K-5K to 5K6K. This is a bit subjective, however.
7. From looking at Figure 4, which markets would you think are the least
competitive? What could explain differences in plots across markets?
A market is competitive if the ratio sn+1 /sn quickly reaches 1. Druggist and tire dealers seem the least competitive. This could be due to
regulation, ability to collude, or supplier power that reduces competition
downstream.
8. Would the approach presented in the paper work for other markets such
as retailing, food industry, restaurants?
What is driving the model is the fixed cost of entry that prevents new
entrants from breaking even. Markets need to be independent so that entry decision in one market does not depend on entry decision in another.
In particular, the fixed cost of entry in one market must be independent
of what firms do in other markets. Retailers and restaurant chains do
not meet that condition. Entry costs depend on what firms do in other
markets, because for example, distance to distribution centers (cost increases with distance and this explains along with other factors concentric
growth). The same holds for the food industry. Food manufacturers offer
their products in markets where they can find a local distributor. Again,
independent fixed entry costs in each market is not what is driving the
entry decision.
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