CERGE-EI
Fall 2010
Industrial Organization I Practice Problems Consider a monopolist monopolist that faces a down downwardward-slopi sloping ng demand curve, Problem 1: Consider and produces at constant marginal cost, $c $c per unit. Suppose that the monopolist produces an output Q > 0 when facing a per-unit tax of $t $t. Show that there exists an ad valorem (percentage of sales revenue) tax that induces the monopolist to produce the same output, Q. Which Which tax raises raises more more revenue? revenue? Explai Explain. n. [Let [Let the producer price under the ad valorem tax be p (1 − τ )] τ )] Solution: We start by writing the condition for profit maximization in both cases,
with a unit tax t and with an ad valorem tax (1 − τ ). τ ). When the firm has to pay a unit tax its problem is max π
=
Q
p(Q)Q − (c + t)Q
FOC: p(Q) + p + p (Q)Q = c + t.
(1)
With an ad valorem tax, though, we have max π Q
=
(1 − τ ) τ ) p( p(Q)Q − cQ
FOC: p(Q) + p + p (Q)Q = c/(1 c/(1 − τ ) τ ).
(2)
Since Q is the same in both cases, then from equations (1) and (2) we have the condition for equivalence of the tax rates c + t = c/(1 c/(1 − τ ) τ ) τ = t/c. We now now need need to compare compare the reven revenue ue in each each case. case. In the first first case this is simply Qt, Thes esee express expressio ions ns Qt, whereas in the second case the revenue is Qp ct . Th would be equal if instead of c in the denominator of the ad valorem tax revenue we had p. We also know that for a monopoly p > MR = c, so the ad valorem tax revenu revenuee is larger. larger. The intuiti intuitive ve reason reason for this is depicted depicted in Figure Figure 1 (the taxed demand is in gray). In the ad valorem tax case the monopolist faces a steeper (less elastic elastic)) demand demand and, therefor therefore, e, charge chargess a higher higher price. For the same amount amount of output that corresponds to a higher tax revenue. 1
Figure 1: Monopolist with unit (left) and ad valorem tax (right). Problem Problem 2: Show that if indirect demand p(Q) is concave, then p(Q) is log-
concave which in turn implies that 1/p 1/p((Q) is conv convex. ex. Also Also show show that that if p if p(Q) is log-concave or if 1/p 1/p((Q) is convex, p(Q) is not necessarily concave (give counterexamples). Solution: There are several definitions for the concavity of a function, but for
this problem we will assume that the functions are twice continuously differentiable (C 2 ) and will use the p ≤ 0 definition. First, we can establish that if a function p(Q) is concave then 1/p 1/p((Q) is convex. If we take the second derivative of 1/p 1/p((Q) we get an expression that is positive if 1/p((Q) is convex. p( p(Q) ≥ 0 and p (Q) ≤ 0. Hence, 1/p
2 p (Q)2 − p (Q) p( ∂ 2 1 p(Q) = . ∂Q 2 p( p(Q) p( p(Q)3
Now, if we take the second derivative of log p log p((Q) we get an expression that is negative if p(Q) ≥ 0 and p (Q) ≤ 0
log p((Q) ∂ 2 log p p (Q) p( p(Q) − p (Q)2 = . ∂Q 2 p( p(Q)2
A counterexample to the concavity of a log-concave function is p(Q) = Q2 . This function is convex but log p log p((Q) = 2log Q is clearly concave. Problem 3: An upstart phone company has only two potential large customers,
Firm A and Firm B. Firm A’s monthly demand for phone calls is qA = 2800 − 200 p 200 p 2
( p measured in cents) and Firm B’s is qB = 5000 − 100 p 100 p.. Th Thee margin marginal al cost cost of providing a phone call is 6 cents. Assuming the phone company has to charge the same same monthl monthly y rental rental fee and unit price price to all its custome customers rs (i.e. (i.e. single single two-pa two-part rt tariff same for both firms), at what level should it set these charges? Solution: Let us start by assuming that the phone company chooses to serve
both firms. We will later check whether the optimal two-part two-part tariff confirms this assum assumpti ption. on. In this this case case the phone phone compa company ny can set set the rental rental fee (fixed (fixed part part of the tariff) to equal Firm A’s consumer surplus, because Firm A has a lower demand at any given price (the usual low type). Since we are dealing with a linear demand, the consumer surplus is the triangle given by C S A ( p) p) =
(14 − p) (14 − p)(2800 200 p)) p)qA ( p) p) p)(2800 − 200 p = 2 2
where 14 is Firm A’s maximum willingness to pay for a unit of service (q ( qA (14) = 0). Knowing this will give us the fixed part of the tariff we can write the phone company’s maximization problem max π = 2CS A ( p) ( p − 6)[q 6)[qA ( p) p) + ( p p) + qB ( p)] p)].. p
Substituting and taking the first order condition we find that the maximizing price is pboth = 20 > 14, which is a contradiction to our starting assumption. Hence, it is clear that the phone company will only serve Firm B. In this case, the constraint for the rental fee becomes Firm B’s consumer surplus. Similar to the previous constraint, we have ∗
C S B ( p) p) =
(50 − p) (50 − p)(5000 100 p)) p)qB ( p) p) p)(5000 − 100 p = . 2 2
In the same way as before, we can now write the phone company’s problem max π = C S B ( p) ( p − 6)q 6)qB ( p) p) + ( p p). p
The solution is the two-part tariff p = 6 and F = 96800. 96800. Th Thee result result is not surprising because this is a case of a simple two part tariff with only one demand. It is always optimal for the firm to price and marginal cost and scoop up all the potential surplus with the fixed fee. ∗
Problem 4: Should we allow a monopolist to implement the third degree price
discriminatio discrimination? n? The monopolist will will clearly be no worse off than under the uniform pricing (because he still has the choice of setting prices equal across all market segments). But what about social welfare? Third degree price discrimination will 3
be beneficial if some markets are not served under uniform pricing. When demand is linear, third degree price discrimination will be harmful when all markets are served under uniform pricing. Suppose there are two groups of consumers: highdemand (1) and low-demand (2). Inverse demands are: p = 12 − q1 , p = 8 − q2 . a. Compute the inverse demand for the integrated market with two consumer
groups. Plot it with p on the vertical and Q = q1 + q2 on the horizontal axis. Solution: The integrated inverse demand is the horizontal sum of the indi-
vidual inverse demands because we are summing up quantities at each price. Its y-intercept is at p = 12 and it has a kink at p = 8. It is depicted in Figure 2 in solid black. Formally, it can be expressed as p(Q) =
12 − Q
10 − Q/2
0≤Q≤4 4 < Q ≤ 20.
b. Suppose that the monopolist charges a uniform price on the integrated mar-
ket and that his marginal cost is MC = c = 0. Find price, quantity sold, consumer surplus and monopolist’s profit. Compute the social welfare (sum of consumer surpluses and profits). Solution: Because of the kink in (inverse) demand, the best way to solve
this problem is by using a graph. Figure 2 shows both the demand from part (a) and the discontinuous marginal revenue curve (solid red). The marginal revenue is ∂p(Q)Q/∂Q for each segment of inverse demand, each valid in the respective interval. Formally MR(Q) =
12 − 2Q 10 − Q
0≤Q≤4 4 < Q ≤ 20.
As usual, the condition for maximum profit is MR = MC , so in our case, Q = 10 and p = 5. From here it is easy to determine the profit and total surplus: π = 50, CS 1 = 49/2, and CS 2 = 9/2. The total welfare therefore is W = 79. ∗
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c. Now assume that the monopolist can price discriminate between the market
segments. Find the prices and quantities for the low-demand and the highdemand market. Find the monopolist’s profit and total consumer surplus. Show that welfare goes down. Solution: In this case the monopolist will maximize for each market sepa-
rately. Solving two monopoly maximizations is straightforward. The problems are max π1 = (12 − q1 )q1 q1
max π2 = (8 − q2 )q2 . q2
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Figure 2: Monopolist with two demands. The solutions are: q1 = 6, p1 = 6 and q2 = 4, p2 = 4. The profits and consumer surpluses for each market are: π1 = 36, CS 1 = 18 and π2 = 16, CS 2 = 8, for a total welfare of W = 78. As expected, this is lower than the welfare without discrimination we found in part (b). ∗
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d. Now assume that demands are the same but marginal cost is higher: MC =
c = 7. Show that under the uniform pricing the low-demand group does not buy. Compute the social welfare. Solution: Again, we use the graph in Figure 2 and check where the MC = 7
line crosses the MR curve. This happens at q = 2.5 and p = 9.5, which means only market 1 will be served. Like before, the social welfare is straightforward to compute: W = 9.375. e. Allow for third degree price discrimination. Compute prices and quantities
with MC = c = 7. Show that now all markets are served and that welfare goes up relative to the uniform pricing (the case in part d). Solution: Just like in part (c), now the monopolist has to solve two disjoint
problems max π1 = (12 − q1 )q1 − 7q1 q1
max π2 = (8 − q2 )q2 − 7q2 . q2
The solutions are: q1 = 2.5, p1 = 9.5 and q2 = 0.5, p2 = 7.5, and they yield a social welfare of W = 9.75. Unsurprisingly, this is larger than what we found in part (d). ∗
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Problem 5: Show in a diagram that if two consumers have linear demands, with
one consumer demanding more than the other at any price, a. It is never optimal for a monopolist that charges both consumers the same
two-part tariff to set the unit price equal to or less than marginal cost. Solution: When a monopolist charges both consumers we know that his
fixed fee for each market is the consumer surplus of the market with lower demand. We will refer to Figure 3 during our analysis. The left graph shows the effect of an infinitesimally small decrease in p below M C . The monopolist will be able to charge a slightly higher fixed fee by the size of segment A to each market. It would be making a loss, however, because it is selling below production cost. This loss is the size of segment A for market 1 and segment A + B for market 2. Hence the net change of this deviation would bring a loss. We can conclude that decreasing the price below p = MC will lower the profit.
Figure 3: Single two-part tariff. To prove that the monopolist charges p > MC we will be using the graph on the right hand side of Figure 3. Our argument will be very similar. Consider an infinitesimally small increase in p above MC . The monopolist will have to charge a slightly smaller fixed fee by the amount of the length of the segment marked C . But it would be making positive profits on the sale of goods by C in market 1 and C + D in market two. Hence it is profitable to increase the price above p = M C by a positive amount. b. It is never optimal for a monopoly that charges each consumer a separate
two-part tariff to set the unit price of the low-type equal to OR less than marginal cost. 6
Solution: Now the monopoly has to satisfy the incentive compatibility
constraint for the high demand consumers in order to prevent them from choosing the tariff meant for the low demand. That means that while it can still charge fixed fee F 1 = CS 1 , it has to provide at least as much surplus for the high demand as they would get under tariff 1.
Figure 4: Separate two-part tariffs. The analysis that follows refers to Figure 4. The left graph depicts the situation when the monopolist charges p1 = MC to market 1. Then it can ask for F 1 = A + B in that market. In this case, market 2 consumers can get a surplus of F +G + H by choosing this tariff. It is possible to argue that the best the monopolist can do is to offer the same tariff to market 2. Consider what happens if p2 > p1 . While the monopolist makes a profit of F on sales in market 2, it now has to compensate the customers in this market for the drop in consumer surplus by lowering F + H from the fixed fee. Hence, it ends up losing H . Similarly, for p2 < p1 we can show that the monopolist will make a loss of E . Now that we established that the monopolist does not gain from charging at cost in market 1, we can proceed to show that it will gain from charging a higher price in this market. For this analysis we will be referring to the right hand side graph of Figure 4. By charging p1 > M C the monopolist will forgo K from the fixed fee in market 1. Nevertheless, this will also make the first market tariff much less attractive to high demand consumers. Their surplus from choosing that tariff has dropped to only M . If the monopolist sets p2 = MC it can extract the additional surplus (J + K + L + N ) using F 2 .
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Problem 6: Suppose that an individual who has an initial wealth of w = wo
is contemplating buying insurance against auto theft. Without insurance, the individual bears a total loss of L if his car is stolen. The individual may be a careful or a careless person. The individual clearly knows this but the insurance company does not. The insurance company’s belief is that the individual is careful with a probability µ. The probability of his auto being stolen is pL if the individual is careful and is pH if she is careless, where pL < pH . Let T be the payment from the individual to the insurance company if he chooses to buy insurance. In case of theft, the insurance company compensates the individual with a total of Y . The utility of the individual is given by u(w, c) = v(w), where v (w) > 0 and v (w) < 0.
a. What is the optimal contract if the insurance company cannot price discrim-
inate? Solution: Since we don’t know the precise expression for v(·) but only
its concave shape, we will use graphs to illustrate a general answer to this problem. Let’s characterize the isocost lines for the firm and the indifference curves for the consumer as sharply as we can. The isoprofit lines are easy. The profit of the firm from offering an insurance compensation of Y in case of theft that happens with probability pi for a payment of T is πi = T − i − pi Y . When we draw it in our usual (T, Y )-space the function of the isoprofit line is T i = πi + pi Y , that is, a line with slope pi and a y-intercept ∂ i . The indifference curves are a bit more complicated to derive. The utility of the consumer with theft probability pi from an insurance contract (T, Y ) is given by (3) U i = (1 − pi )v(w0 − T ) + pi v(w0 − T − Y − L). While it is not possible to get a formula for the indifference curve, we can still determine its shape using differential calculus. Most important, we can find the slope if we consider (3) as an implicit function F (T, Y ) − U i = 0 ∂T ∂F/∂Y pi v (w0 − T + Y − L) =− = . ∂Y ∂F/∂T (1 − pi )v (w0 − T ) + pi v (w0 − T + Y − L))
It is important to notice that the slope equals ∂T/∂Y = pi at Y = L, and that ∂T/∂Y > pi before this point (Y < L) and ∂T/∂Y < pi after it (Y > L). Hence we are dealing with a concave function. We can also say that the indifference curve that starts from the origin represents the outside option of the consumer, that is, the utility from having no insurance (T = Y = 0). ¯i . Let’s call this level of utility U
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From these two lines we can determine that the insurance company maximizes its profit at the point of tangency between this outside option indifference curve and an isoprofit line. We already know that at this point L = Y . The left graph on Figure 5 shows this point for both types. The non-discriminating monopolist will offer one of these points. If it offers point 2, it will be selling only to the high-risk consumers for a profit of (1 − µ)πH . If it offers point 1, it will be making a profit of µπL + (1 − µ)˜πH . Note that ˜H , the profit from selling low-risk insurance to high-risk consumers will only π be positive if the difference in pi between the types is not large. If µ is small enough or pH − pL is large enough, the insurer will only sell to high-risk types.
Figure 5: Insurance seller, nondiscriminating (left) and discriminating (right). b. What is the optimal separating contract if the insurance company can price
discriminate? Find the surplus that accrues to each type of the individual in the optimal solution. How do they depend on the parameters of the model, in particular on µ? Give an intuitive explanation for your answers. Solution: The monopolist cannot simply offer points 1 and 2 that we men-
tioned in part (a) as contracts, because the high-risk types would prefer the low-risk contract to their own. Since the high-risk type is always willing to pay more for the same amount of coverage Y , he is analogous to the high demand type that we encounter in other incarnations of this problem. Hence, we can already say that only the individual rationality constraint (participation) for the low-risk consumers, and the incentive compatibility constraint for the high-risk types will hold in equilibrium. Formally, the problem of the 9
insurance company (writing only the binding constraints) is the following max
T L ,Y L ,T H ,Y H
s.t.:
µ(T L − pL Y L ) + (1 − µ)(T H − pH Y H )
(4)
¯L (1 − pL )v(w0 − T L ) + pL v(w0 − T L + Y L − L) ≥ U (1 − pH )v(w0 − T H ) + pH v(w0 − T H + Y H − L) ≥ (1 − pH )v(w0 − T L ) + pH v(w0 − T L + Y L − L).
If we take the first order conditions for T H and Y H we get the equation
v (w0 − T H + Y H − L) = (1 − pH )v (w0) + pH v (w0 − T H + Y H − L), which is only satisfied at Y H = L. Hence, as we expect in these kind of problems, the high-risk types will get a socially optimal insurance coverage. For this type of contract to be incentive compatible, however, the low-risk insurance coverage must be lower than L, as shown in the right graph of Figure 5. As usual, the low-risk types will be left with a utility equal to their ¯L ), whereas the high-risk types will earn some information outside option (U ¯H ). This information rent will be increasing in µ, because, if rent (U H > U the fraction of low-risk consumers falls, the monopolist will forgo more of the profit gained from the contract at point 3 in order to make deviation less appealing to the high-risk consumers. In fact, for a low enough µ the monopolist will choose to ignore low-risk types altogether and will offer only the contract at point 2 to high-risk consumers. ∗
Problem 7: MWG 14.C.7 Assume that there are two types of consumers for a
firm’s product, θH and θL . The proportion of type θL consumers is λ. A type θ’s utility when consuming an amount x of the good and paying a total of T for it is u(x, T ) = θv(x) − T , where 1 − (1 − x)2 v(x) = . 2 The firm is the sole producer of this good, and its cost of production per unit is c > 0. a. Consider a nondiscriminating monopolist. Derive his optimal pricing policy.
Show that he serves both classes of consumers if either θL or λ is “large enough.” Solution: The pricing policy is a single couple (x, p). We will start by
deriving the best pricing policy for a consumer type θi . The only constraint for the monopolist is the participation constraint max( pi − c)xi xi ,pi
s.t.: θi v(xi ) − pi xi ≥ 0. 10
(5)
The maximizing pricing policy is (xi , pi ) = [1 − c/θi , (θi + c)/2], that yields a profit of θ c2 − c. πi = + 2 2θ ∗
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A monopolist can choose whether it wants to serve both types, and price according to θL , or only the high types, pricing according to θH . He will serve both types if πL ≥ (1 − λ)πH . If we take the derivative of profits with respect to type we will find it is positive if θi > c (a condition which must hold if there will be a market). Hence, the monopolist will prefer to serve the whole market for a high enough λ and θL . b. Consider a monopolist who can distinguish the two types (by some charac-
teristic) but can only charge a simple price pi to each type θi . Characterize his optimal prices. Solution: Now we have two separate markets and the utilities of their
respective customers, so we can derive demands and solve the profit maximization problem. For a any price pi we can derive the demand of θi by maximizing their utility max ui (xi , pi ) = θi v(xi ) − pi xi , xi
which gives us xi ( pi ) = (1 − pi /θi ). Using this as a demand for the market and maximizing profits with respect to price we get: pi = (θi + c)/θi and xi = 1/2 − c/(2θi ). ∗
c. Suppose the monopolist cannot distinguish the types. Derive the optimal
two-part tariff (a pricing policy consisting of a lump-sum charge F plus a linear price per unit purchased of p) under the assumption that the monopolist serves both types. Interpret. When will the monopolist serve both types? Solution: We already have the demand functions and the utility function we
need to determine the lump-sum charge F . We know this charge will equal the utility of the low types in equilibrium, and we can plug the demands we derived in part (b) xi = (1 − p/θi ). The problem is max θL v(xL ) − pxL + ( p − c)[λxL + (1 − λ)xH ]. p
After some tedious algebra we find the optimal price ∗
p =
c[(1 − λ)θL + λθH ] . 2(1 − λ)θL − (1 − 2λ)θH 11
(6)
It is possible to show that p > c, which confirms our intuition (see problem 5.a). We can also derive F = θL v(1 − p /θL ) − p (1 − p /θL ). The monopolist will serve both types if the demand for the low type at p is positive. ∗
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d. Compute the fully optimal nonlinear tariff. How do the quantities purchased
by the two types compare with the levels in (a) to (c)? Solution: This is a standard problem of adverse selection, we can directly
write the problem with the binding constraints only max
xL ,T L ,xH ,T H
s.t.:
λ(T L − cxL ) + (1 − λ)(T H − cxH )
(7)
θL v(xL ) − T L ≥ 0 θH v(xH ) − T H ≥ θH v(xL ) − T L .
We already know that the constraints will hold with equality, so we can solve them to get both T i in terms of pi to simplify the problem T L = θL v(xL ) T H = θH v(xH ) − v(xL )(θH − θL ). When we solve the problem with these substitutions in place, we get the optimal xi ∗
xL = 1 − λc/[θL − (1 − λ)θH ] xH = 1 − c/θH . ∗
Just like we expected, the high demand type consumes an optimal amount because xH corresponds to the demand when pH = c. By the same reasoning we can see that the low demand type consumes less than the optimal amount because the demand corresponds to pL > c. ∗
Problem 8: MWG 14.C.8 Air Shangri-la is the only airline to fly between the
islands of Shangri-la and Nirvana. There are two types of passengers, tourists and business. Business travelers are willing to pay more than tourists. The airline, however, cannot tell directly whether a ticket purchaser is a tourist or a business traveler. The two types do differ, though, in how much they are willing to pay to avoid having to purchase their tickets in advance. (Passengers do not like to commit themselves to traveling at a particular time.) More specifically, the utility levels of each of the two types net of the price of the ticket, P , for any given amount of time W prior to the flight that the ticket is purchased are given by Business : Tourist :
v − θB P − W, v − θT P − W, 12
where 0 < θB < θT . (Note that for any given level of W , the business traveler is willing to pay more for his ticket. Also, the business traveler is willing to pay more for any given reduction in W .) The proportion of travelers who are tourists is λ. Assume that the cost of transporting a passenger is c. Assume in (a) to (d) that Air Shangri-la wants to carry both types of passengers. a. Draw the indifference curves of the two types in (P, W )-space. Draw the
airline’s isoprofit curves. Now formulate the optimal (profit-maximizing) price discrimination problem mathematically that Air Shangri-la would want to solve. [Hint: Impose nonnegativity of prices as a constraint since, if it charged a negative price, it would sell an infinite number of tickets at this price.] Solution: The indifference and isoprofit curves are depicted in Figure 6.
Figure 6: The problem of Air Shangri-la. Formally, Air Shangri-la solves the following problem λP T + (1 − λ)P B
max
P B ,W B ,P T ,W T
s.t.:
(i) (ii) (iii) (iv) (v)
θT P T + W T ≤ v θB P B + W B ≤ v θT P T + W T ≤ θT P B + W B θB P B + W B ≤ θB P T + W T P T , W T , P B , W B ≥ 0. 13
b. Show that in the optimal solution, tourists are indifferent between buying a
ticket and not going at all. Solution: Knowing that θB < θT , constraints (i) and (iv) imply that con-
straint (ii) is satisfied with strict inequality and can be ignored. That means that constraint (i) is binding (satisfied with equality), otherwise Air Shangrila could increase both prices by a small amount and still all constraints would hold. This means tourists will not get any surplus above their outside option. c. Show that in the optimal solution, business travelers never buy their ticket
prior to the flight and are just indifferent between doing this and buying when tourists buy. Solution: We can prove by contradiction that business traveler will not
have to book in advance. Assume that {(P T , W T ), (P B , W B )} is a solution to the firm’s problem and that W B > 0. Then the airline can change this to W B = 0, while increasing P B by W B /θB to keep their utility unchanged. This new solution solves the incentive compatibility constraint for tourists (iii) and provides a higher profit. This contradicts our starting assumption, therefore P B cannot be higher than 0. On the other hand, if business travelers strictly prefer their own ticket then it would be possible to increase P B without violating their incentive compatibility constraint (iv). Hence, in the optimal solution they must be indifferent between the two types of tickets. d. Describe fully the optimal price discrimination scheme under the assump-
tion that they sell to both types. How does it depend on the underlying parameters λ, θB , θT and c? Solution: The airline faces the following problem: It must make the tourist
ticket unattractive to business travelers despite the higher P B , and it can do this by increasing W T . To keep the tourists in the market, however, it must couple this with cuts in P T . In numbers: they have to lower P T by /θT in order to increase W T by . This will enable them to raise P B by (θ θ )) . This trade-off does not depend in the level of prices or W T so it θ θ holds anywhere (it is linear). This will be profitable if and only if the loss in revenue from tourists is compensated by a higher increase in revenue from businesses (θT − θB ) λ < (1 − λ) θT θT θB which is equivalent to λ θT − θB (8) < . 1−λ θB T − B T
B
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Assuming that the airline makes a profit (costs are low enough) then two scenarios can arise – if (8) holds then only business travellers will be served (it makes sense
to trade off all of the tourist revenue). In this case the optimal contract is {(0, v), (v/θ B , 0)}. – if (8) doesn’t hold then the airline will serve both types the same ticket
because it does not make sense for them to introduce waiting times in order to differentiate among customers. They will be constrained by the willingness to pay of tourists, so the ticket will be (P, W ) = (v/θ T , 0). From here it is easy to see that if λ is small enough or (θT − θB ) is big enough the firm will choose to serve only businesses. e. Under what circumstances will Air Shangri-la choose to serve only business
travelers? Solution: Aside from the conditions mentioned in part (d), the firm may
also decide to drop its tourist customers if the pooling ticket does not justify its cost, that is, if v/θT < c < v/θB . If c > θB then Air Shangri-la cannot operate at all. Problem 9: Consider a monopolist who can produce a quantity q of a product
at a constant marginal cost of c = 2 and no fixed costs. There are two types of consumers for the monopolist’s product: high-types (H ) with a demand function qH ( p) = 10 − p, and low-types (L) with a demand function qL ( p) = θL − p, where 2 < θL < 10. The proportion of L-types in the population is α ∈ (0, 1). The monopolist is risk-neutral. a. Suppose the monopolist cannot distinguish the types, and it can only offer
a single two-part tariff in the form ( p,T ). A j-type consumer accepts this contract as long as T + pq j ( p) does not exceed her surplus from consuming q j ( p) units. Derive the optimal two-part tariff under the assumption that the monopolist serves both types of consumers. When will the monopolist serve only the H -types? Interpret. Solution: Assuming that the monopolist serves both types of customers,
his problem is max T ( p) + ( p − 2)[αqL ( p) + (1 − α)qH ( p)], p
where T ( p) is constrained by the consumer surplus of the low type, that is: T ( p) = qL ( p)2 /2. Plugging this in the problem and solving the first order 15
condition, we have ∗
p = 2 + (1 − α)(10 − θL ). The monopolist will serve both types if the demand of type L at p is positive ∗
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θL > p = 2 + (1 − α)(10 − θL ). This will not happen if α or θL are small enough, that is if the demand of type L can be sacrificed either because of their small numbers or small willingness to pay. b. Suppose the monopolist cannot distinguish the types, and it can only offer
a single two-part tariff in the form (q, T (q)). A j-type consumer accepts (q, T (q)) as long as T (q) does not exceed her surplus from consuming q units. Derive the optimal two-part tariff under the assumption that the monopolist serves both types of consumers. When will the monopolist serve only the H -types? When does the monopolist choose this form of two-part tariff over the one in part (a)? Explain. Solution: Assuming it serves both types, the monopolist will charge a fixed
fee that is constrained by the L-type consumer surplus at consuming a quanq tity q: T (q) = 0 (θL − y)dy = q(θL − q/2). The problem is
max T (q) − 2q q
and it is maximized at q = θL − 2. The maximized profit is (θL − 2)2 /2. ∗
If, instead, the monopolist chooses to serve only type H , then T is constrained by T (q) = q(10 − q/2). Its profit is maximized at q = 8 and it is equal to 32(1 − α). So, the monopolist will serve only type H if (θL − 2)2 < 64(1 − α). ∗
In general the monopolist prefers the two-part tariff to the (T, q) contract because charging a unit price will ensure the H -types, who consume more, also pay more and cover their production costs. Only in the case when the monopolist would serve only H -types under the two-part tariff is the monopolist indifferent between the two, because it can collect all of their surplus using either contract. c. Now suppose the monopolist can discriminate across the two groups of con-
sumers and is free to offer any contract in the form (q, T (q)). A j-type consumer accepts (q, T (q)) as long as T (q) does not exceed her surplus from consuming q units, and (q, T (q)) gives her the highest utility among all the contracts. Clearly state the monopolist’s maximization problem and derive 16
the optimal non-linear contract. Does the monopolist ever prefer to serve only the H -types? Explain why or why not. How do the quantities purchased by the two types compare with the levels in part (a)? Solution: In this case the problem of the firm becomes
α(T L − 2qL ) + (1 − α)(T H − 2qH )
max
qL ,T L ,qH ,T H
s.t.:
(i) (ii) (iii) (iv)
qL (θL − qL /2) − T L ≥ 0 qH (10 − qH /2) − T H ≥ 0 qL (θL − qL /2) − T L ≥ qH (θL − qH /2) − T H qH (10 − qH /2) − T H ≥ qL (10 − qL /2) − T L .
We know that in this type of problem at the optimal contracts only constraints (i) and (iv) are binding and they are satisfied with equality. Thus, constraint (i) gives us T L = qL (θL − qL /2) and constraint (iv) gives us T H = qH (10 − qH /2) − 2qL . Plugging these into the objective function and solving the first order conditions for qL and qH we get qL = θL − 2/α and qH = 8, and the respective T i . As expected, the H -type consumer a socially optimal amount, whereas the L-type consume below this. ∗
∗
The monopolist will decide to serve to only H -types if qL < 0∗, that is, if αθL < 0. Assuming both types are served in both cases, and recalling that 2 < α < 10, we can see that both types will end up consuming more under the discriminating monopolist. ∗
Problem 10: Consider a market for a single good. A risk-neutral monopolist
supplies the good and the cost of production is assumed to be zero. There is a single buyer who wishes to consume this good. The buyer’s willingness-to-pay is θq for q units consumed. θ is uncertain to the monopolist; it is equally likely to be 1 or 2. The buyer has a large income. However, the monopolist is capacity-constrained; it can supply at most 10 units of the good. Suppose the monopolist can offer a menu of contracts in the form (q, T (q)), where T (q) is the total amount to be paid by the buyer for q units. The buyer accepts a particular contract (˜ q, T (˜ q )) as long as θ˜ q ≥ T (˜ q). Clearly state the monopolist’s maximization problem. Which constraints will be binding in the optimal solution? Derive the optimal menu of contracts. Solution: In this problem, aside from the usual conditions on the optimal con-
tracts we also have a constraint on capacity. Since we are not dealing with two consumers, but the same consumer in two states of the world, we are only concerned that each quantity of production does not exceed capacity. Formally, the 17
monopolist’s problem, with all constraints included, is the following max
(1/2)T L + (1/2)T H
qL ,T L ,qH ,T H
s.t.:
(i) (ii) (iii) (iv) (v)
qL − T L ≥ 0 2qH − T H ≥ 0 qL − T L ≥ qH − T H 2qH − T H ≥ 2qL − T L qL , qH ≥ 0.
As usual, we will filter out the two redundant conditions. If (i) and (iv) hold, then (ii) holds with inequality. Given this, then (iv) must hold with equality in equilibrium, otherwise we could increase the profit by raising T H without violating any constraint. Futhermore, if (iv) binds then we can write qL − T L = qH − T H + (qH − qL ), which implies that (iii) does not bind. We are left only with binding conditions (i) and (iv) which give us T L = qL and T H = 2qH − qL . Substituting into the objective function we have max qH s.t: qH , qL = 10.
qL ,qH
The first order condition for qH is always positive, which implies that qH = 10. Less obvious is that the first order condition for qL is always zero, that is, any level of qL ∈ [0, 10] is optimal. The respective payments T i can be derived from (i) and (iv). Hence, we have a continuum of equilibria given by ∗
∗
∗
∗
∗
∗
∗
{(qL , T L ), (qH , T H )} = {(x, x), (10, 20 − x)} for all x ∈ [0, 10]. Problem 11: Shy 5.5.1 A monopoly produces a durable product that lasts for
two periods. It is assumed for simplicity that production is costless. There is a continuum of consumers who live for two periods and desire this product for the two periods of their lives. They have different valuations for the product. The inverse demand function in the first period is summarized by p = 100 − Q. If the monopolist offers the product for sale and a consumer purchases the product, she will have it for her entire life (i.e., because the product is durable, consumers buy it once in their life, either in the first period or in the second period). If the monopolist leases the product for a single product, on the other hand, a consumer can use it for one period only if she leases it. Let pS 1 be the first-period sale price, S R p2 the second-period sale price and p the per-period rental price. The monopoly maximizes the sum of profits from the sales in the two periods, and there is no discounting the future. 18
a. Find the equilibrium per-period rental price, pR , if the monopoly only rents
this product. Solution: This is a simple monopoly problem played twice. Each period
the maximization is max πiR = (100 − pR ) pR pR
and it is solved at pR = 50, and a total revenue of 2πi = 2 · 50(100 − 50) = 5000. S b. Characterize the equilibrium sale price per period, pS 1 and p2 , if the monopoly
only sells this product. Solution: In this case the monopolist will solve the game using backward
induction. Starting with the second period, the monopolist will be facing a residual demand of the (100 − q1 ) buyers with the lowest valuations, where q1 is the number of buyers who bought the good in the first period. Hence, his residual demand is q2 = 100 − q1 − p2 . Taking this demand for granted, the monopolist maximizes its profit by solving max π2 = p2 (100 − q1 − p2 ), p2
S which gives pS 2 = 50 − q1 /2 and sell q2 = 50 − q1 /2 units for a profit of π2S = (50 − q1 /2)2 .
Now that we know what is the best we can do in the second period, we can move to the first. It is important to realize that here buyers have a choice of waiting to buy in the second period. Moreover, at any price in this period, there will be one marginal buyer who will determine the demand and who is indifferent between buying in this period and waiting. Everyone with a valuation higher than his will prefer to buy in the first period, and everyone with a smaller valuation will decide to buy later. This marginal buyer is indifferent between buying now to get 2(100 − qi ) − p1 and waiting to buy next period to get (100 − q1) − p2 . Thus we can find where this split in the demand will occur solving 2(100 − qi ) − p1 = (100 − q1 ) − (50 − q1/2). The solution is q1 = 150 − 2 p1 /3, so now we know all of the consequences of setting a price in the first period, and we can solve the maximization problem max π1 + π2 = p1 q1 ( p1 ) + (50 − q1 ( p1 )/2)2 . p1
s S Taking the first derivative and solving we find: pS 1 = 90, q1 = 40, p2 = 30 and q2S = 30 which give the monopolist a total profit of πS = 4500 < 5000.
19
c. Does the monopolist prefer renting or selling? Solution: As shown above, the monopolist prefers renting, which confirms
the Coase conjecture. Problem 12: Shy 5.5.2 A monopoly produces a durable product that lasts for
two periods. It is assumed for simplicity that production is costless. There are only two consumers living for two periods. They both desire this product for the two periods of their lives, but they differ in their willingness to pay for it. Consumer H is willing to pay up to V H and consumer L is willing to pay up to V L for use of this product in each period, where V H > 2V L > 0. Because the product is durable, consumers buy it once in their life, either in the first period or in the second period. The utility functions of the consumers, for i = H, L, are given by:
2V − p U = V 0 − p i
i
i
S 1 S 2
if i buys in period 1 if i buys in period 2 if i does not buy,
S where pS 1 is the first period sale price, and p2 is the second period sale price. They may also rent the product in each period if the monopoly offers renting. In this case, their per-period utilities are V i − pR , where pR is the per-period rental price. The monopoly maximizes the sum of profits from the sales in the two periods, and there is no discounting the future.
a. Find the equilibrium per-period rental price, pR , if the monopoly only rents
this product. Solution: Just like in problem 11.a, this is simply the same game played
twice, because the decision in one period does not affect the outcome in the other. Hence, the monopolist faces this dilemma: rent at V H only to consumer H , or rent at V L to both. Any other prices are dominated by one of these two. We are given that V H > 2V L , hence the monopolist will rent at V H , which yields a total revenue of π R = 2V H over both periods. S b. Characterize the equilibrium sale price per period, pS 1 and p2 , if the monopoly
only sells this product. Solution: Now we must deal with both periods jointly, so we start from the
second using backward induction. The monopolist already knows whether someone already bought the durable good in the first period, so we can build up a strategy on that. It is easy to see that these are the best responses pS 2
V = V 0
H L
if nobody bought in period 1 if only H bought in period 1 if both bought in period 1. 20
It is impossible only for consumer L to buy. We can no move on to the first period. Recall that the utilities of each consumer net of the price are 2V i because they will enjoy the good in both periods. The monopolist can charge H L aimed only at consumer H or pS pS 1 = 2V 1 = 2V aimed at both. In the second case the monopolist does not make any revenue in the second period and his total revenue is 2V L . In the first case, consumer H is indifferent H between buying now and waiting, because, if he waits the pS 2 will be V and not V L .1 We can assume that he buys, which earns the monopolist a first period revenue of 2V H , and still leaves him the possibility to sell in the second period at V L to consumer L. This is clearly the best strategy for the monopolist yielding πS = 2V H + V L . c. Compare the outcome with question 11. Why is it different? Solution: Unlike problem 11, in this case the monopolist makes a higher
profit by selling rather than renting. This is because he can use the two periods to differentiate between the two customers. This example goes to show that the Coase conjecture may fail when we deal with discrete demand. Problem 13: Tirole, all exercises in chapter 1 (answers are in the book!). Solution: Self-explanatory. Problem 14: Consider the standard Hotelling model with two firms, but now
suppose that firms deliver the product. So, firms can distinguish consumers at different locations and therefore can price discriminate by choosing a delivered price for each particular consumer. The production costs are c per unit for each firm, and transportation cost is proportional to the distance between the firm and the consumer served by that firm. Argue how this may change the findings of Hotelling. In particular, does a SPNE in pure strategies exist in which firms choose locations first and then their (delivered) price schedules? Solution: We will start by backward induction; first analyzing the optimal pricing
decisions given locations, and then moving to the choice of location. We will denote the position of firm i by xi . Figure 7 shows the market with given firm locations. We have ignored the production costs c because they do will not affect our analysis and will only clutter notation. The lines extending from each position show the lowest price a firm can charge to deliver the product at that location. Without loss of generality we can assume that x2 ≥ x1 . We focus on the firm 1’s decisions. 1
We can only claim this because we are dealing with a single consumer H who bears the whole consequence of his action. Think what would happen if, instead, we were dealing with a continuum of consumers of measure 1.
21
Figure 7: Price competition with given locations. Firm 1 will charge the highest price it can without risking to be undercut by firm 2. That is, it will charge (slightly under) the lowest price firm 2 can afford. The prices that lead to a positive profit (above firm 1’s own break-even line) are highlighted. Thus firm 1 will sell to all the market from 0 to q1, for a profit of π1 (shaded area). Assume transportation costs t per unit. We need to define everything in terms of locations, so that we can use the results in the first stage. Hence, q1 = (x2 + x1 )/2 because it is in the middle. The revenue and costs of firm 1 are given by the areas R1 = tx22 /2 − t(x2 − q1 )2/2 = tx22 /2 − t(x2 − x1 )2 /8, C 1 = tx21 /2 + t(q1 − x1 )2 /2 = tx21 /2 + t(x2 − x1 )2 /8. Knowing the optimal pricing decision of the last period, and the resulting profits, we can move to the first period and examine the choice of location. We can do so by constructing firm 1’s best response to firm 2’s location. Firm 1 solves max π1 (x1 , x2 ) = R1 − C 1 = (t/4)(x22 − 3x21 + 2x1 x2 ), x1
whose first order condition gives us the response function x1 = x2 /3. By symmetry, the same is valid for firm 2, if we switch the reference point to the other end of the segment. Hence, 1 − x2 = x1 , to get the equilibrium: x1 = 1/4 and x2 = 3/4. The response functions are linear, so they cross only at this point. There is another (tivial) equilibrium where x1 = 3/4 and x2 = 1/4, if we start with the assumption that x2 ≤ x1. ∗
∗
∗
∗
∗
Problem 15: Consider a linear city of length 1. There is a risk-neutral monopolist
located at point 0 who supplies a good. The monopolist can produce any quantity at zero cost. There is a single buyer who wishes to consume this good. The buyer’s location is uncertain to the monopolist (the buyer, of course, knows her 22
own location). From the monopolist’s point of view, the buyer is equally likely to be located at 1/3 and 2/3. The buyer’s net utility from consuming q units is given by 1 − (1 − q)2 − xq − T. U (q , T , x) = 2 where T is the transfer paid by the buyer for q units, and x is the buyer’s location (so, per-unit transportation cost is 1 for each unit of distance travelled). a. Suppose the monopolist can offer a menu of contracts in the form (q, T (q)).
The buyer accepts a particular contract (˜ q, T (˜ q )) as long as U (˜ q, T (˜ q), x) ≥ 0. Clearly state the monopolist’s maximization problem. Which constraints will be binding in the optimal solution? Explain. Solution: At any q the willingness to pay of the nearest buyer is greater
than the furthest one because of the lower transportation cost. Hence, we will call the buyer sitting at (1/3) high type and the buyer sitting at (2/3) low type. By this notation, the maximization problem for the monopolist is max
(1/2)T L + (1/2)T H
qL ,T L ,qH ,T H
s.t.:
(i) (ii) (iii) (iv)
U (qL , T L , 2/3) ≥ 0 U (qH , T H , 1/3) ≥ 0 U (qL , T L , 2/3) ≥ U (qH , T H , 2/3) U (qH , T H , 1/3) ≥ U (qL , T L , 1/3)L.
Not all these constraints are binding in equilibrium, though. First, if (i) and (iv) hold, then (ii) must hold with inequality, that is, it doesn’t bind. That tells us that (i) must be binding, otherwise we could increase profits by raising both T L and T H by some small amount. That also tells us that (iv) must bind, otherwise we could increase profits by raising T H . If (iv) holds with equality, then, by subtracting qH from each side, we see that (iii) does not bind. Hence, in equilibrium only (i) and (iv) hold with equality. b. Derive the optimal menu of contracts. Solution: From constraints (i) and (iv) we can write
T L = [1 − (1 − qL )2 ]/2 − 2qL /3 T H = [1 − (1 − qH )2 ]/2 − qH /3 − qL /3. Substituting into the ob jective function and solving the first order conditions we find that qL = 0 and qH = 2/3. Note that this is an internal solution that ∗
∗
23
coincides with the corner solution of selling only to the nearest customer. The maximizing contracts are ∗
∗
∗
∗
{(qL , T L ), (qH , T H )} = {(0, 0), (2/3, 2/9)}. c. Suppose the monopolist can change its location prior to offering the con-
tracts. What is the optimal location? Is it socially optimal? Explain. Solution: It is not difficult to argue that the monopolist will decide to
choose somewhere between the two buyers, because being outside the [1/3, 2/3] interval will only increase the cost of both buyers and reduce demand. Suppose also that the monopolist chooses a position x ∈ [1/3, 1/2], so that the customer at 1/3 is still the one with the highest demand. The distance from this buyer is now x − 1/3, whereas the distance from the furthest buyer is 2/3 − x. Omitting the nonbinding constraints, the monopolist’s problem then becomes max
qL ,T L ,qH ,T H
s.t.:
(1/2)T L + (1/2)T H U (qL , T L , 2/3 − x) ≥ 0 U (qH , T H , x − 1/3) ≥ U (qL , T L , x − 1/3)L.
We can now solve the problem just like we did in part (b) to find qL = 3x−2/3 and qH = 4/3 − x. If we plug these solutions and their respective T i into the objective function and take the first derivative with respect to x we will find it is positive. This means the monopolist gains by moving towards the furtherst customer. ∗
∗
∗
This specification, however, is only valid for x ≤ 1/2 because of our definitions of high and low type. In fact, the problem is symmetric, if we change the denomination such that the distance is measured from the end of the segment, the current analysis holds with the buyer at (2/3) as the high type. Therefore, the monopolist always gain from moving towards the furthest buyer. The only point where this improvement is not possible only at x = 1/2. Here, at an equal distance to each consumer, the monopolist maximizes its profit. Incidentally, this is also the socially optimal location, because here the monopolist will not price discriminate, thus no surplus will be lost in order to impose incentive compatibility. d. Suppose the monopolist is located at 0 again, but now suppose the game is
repeated twice. How would this change the monopolist’s strategy? Would the monopolist still offer a menu of (possibly non-linear) contracts in the first period? If it does, would the buyer’s purchase decision reveal her location? What about the second period? Discuss. 24
Solution: The addition of another period changes the problem because of
the revelation of information involved. If the monopolist learns from the first period the location of the buyer based on the contract he chooses, then in the second period there will be no information surplus. If offered the contracts derived in part (b), the high type will choose (qL , T L ), in this case (0, 0), to signal that he is a low type and take a more profitable deal in the second period. This, however, cannot be an equilibrium, because, in this scenario, the monopolist would put no faith in the signal. It is only possible to have an equilibrium where customer sitting at 1/3 is offered a higher surplus in period 1, so that he will reveal his identity and get no surplus in period 2. ∗
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Problem 16: Consumers are uniformly distributed (with density 1) on the cir-
cumference of a circle, whose length is 1. They pay transportation costs of $1 per unit of distance. Two firms locate on the circle, each with 3 outlets, as in Figure 8: the upper locations F 1, F 2 , F 3 belong to the first firm, while the lower locations f 1 , f 2 , f 3 belong to the second firm. Each firm has variable costs c per unit and aims to maximize the sum of the profits of the 3 outlets. The outlets are symmetrically distributed so that the distance between any two neighboring outlets is 1/6.
Figure 8: The circular city. Find all the symmetric equilibria in which firms charge price p at their outer outlets, outlets no. 1, 3, and price q at their inner outlets, outlets no. 2, see Figure 8. Solution: Linear transportation costs in location models cause discontinuities in
the strategic variable, that is, you can undercut a seller by infinitesimally lowering your price and gaining a chunk of the market. Therefore, we can only use calculus 25
in the intervals when these discontinuities do not occur and make sure to cover all intervals.
Figure 9: Both inner and outer outlets working. i. The most obvious scenario is when both inner and outer outlets are working.
In this case, let x denote the demand between f 1 and F 1 that accrues to f 1 and let z denote the demand between f 1 and f 2 that accrues to f 2 . Figure 9 shows half of the market of the lower-case firm linearized for convenience. We will be working only on optimizing this section, given that the problem is symmetric. Our firm takes p as given and sets its own strategic variables p and q that maximize its profit. We have a symmetric equilibrium where p = p = p .
∗
First, let’s define the demands. The consumer sitting at x will be indifferent between buying from f 1 for p + x and buying from F 1 from p + (1/6 − x). That gives us x as a function of the surrounding outlet prices as
x( p,p ) = ( p − p)/2 + 1/12. By the same token z(q, p) = ( p − q)/2 + 1/12. Now we can write the firm’s problem as
max π( p,q,p ) = (q − c)z + ( p − c)(1/6 − z + x). p,q
Taking the first order conditions and then substituting the condition for symmetric equilibrium p = p = p we get p = c + 1/2 and q = c + 7/12. We must now check for what values of c (our only parameter) is this equilibrium valid. We started with the assumption that all outlets work, which means that no outlets are undercut. The condition for this can be written as p − 1/6 < q < p + 1/6. This condition is satisfied for all values of c.
∗
∗
∗
∗
∗
∗
ii. We now consider what happens if firms decide to use only their inner outlets
f 2 and F 2 . The left graph on Figure 10 shows the competition between these 26
Figure 10: Only inner outlets working. firms. We can define the demand of outlet f 2 as y(q, q ) = (q − q)/2 + 1/4 in the same way we did in part (i). Our firm’s problem is given by
max π(q, q ) = (q − c)y, q
whose symmetric equilibrium is q = q = q = c + 1/2 with a (half) profit of π = c/4 + 1/8. But in this case we need to check that there is no profitable deviation for one of the firms by selling at its outer outlets too. This deviation is depicted in the right graph of Figure 10. Given that the competitor sells for q = c + 1/2 at F 2 what is the highest profit our firm can achieve by setting pd and q d ? In this case, demand y1 = ( p − q)/2 − 1/12 and demand y2 = (q − p)/2 − 1/6. The deviant’s problem is
∗
∗
∗
max π( p,q)d = (q − c)y1 + ( p − c)(1/6 − y1 + y2 ), p,q
which is maximized at pd = c+7/12 and q d = c+2/3. For this deviation to be valid (no undercutting) we must have that pd − 1/6 < q d < pd + 1/6, which is again satisfied for all values of c. Most important, for our equilibrium to exist this deviation must not be profitable. The profit of the deviant is πd = 25/144 > 1/8 which means that we do not have an equilibrium.
Figure 11: Only outer outlets working. iii. And now for the last case, when only the outer outlets are working. This
scenario is shown in the left graph of Figure 11. The disputed demand for 27
our firm is w( p,p ) = ( p − p)/2 + 1/12, and the problem to solve is
max π( p,p ) = ( p − c)(w + 1/6). p
The problem is solved at p = p = p = c + 1/3 for a profit of π = 1/18. Again, we check whether there are profitable deviations. The deviant could start selling from outlet f 2 and charge prices pd and qd . This deviation is shown on the right graph of Figure 11. The demands are w1 = ( p−q)/2+1/12 and w2 = ( p − p)/2 + 1/12, because the firm take p = c + 1/3 as given. The deviant’s problem is
∗
∗
∗
max π( p,q)d = (q − c)w1 + ( p − c)(1/6 − w1 + w2 ), p,q
which is maximized at pd = c + 5/12 and qd = c + 1/2 for a profit of πd = 13/144 > 1/18. Thus we again have a profitable deviation, therefore, this is not an equilibrium. Note that in this deviation we did not have undercutting because pd − 1/6 < q d < pd + 1/6 and p − 1/6 < q d < p + 1/6 are both satisfied for all values of c. ∗
∗
Problem 17: A seller sells rice which he produces at variable cost K . There are
3 consumers, characterized by the three constants A > B > C > 0. If consumer 1 buys quantity Q and pays P for it, his utility is: AQ − P . For consumer 2, it is BQ − P , and for consumer 3, it is CQ − P . The seller aims to maximize his profit. Write the inequalities defining the seller’s optimal plan. Which of those are equalities? (There is no need to solve for the optimal plan). Solution: Using our usual notation, assuming that he is selling to all three buyers,
the problem of the rice seller is the following QA (P A − K ) + QB (P B − K ) + QB (P B − K )
max P i ,Qi
s.t.:
(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)
AQA − P A ≥ 0 BQ B − P B ≥ 0 CQ C − P C ≥ 0 AQA − P A ≥ AQB − P B AQA − P A ≥ AQC − P C BQ B − P B ≥ BQ A − P A BQ B − P B ≥ BQ C − P C CQ C − P C ≥ CQA − P A CQ C − P C ≥ CQB − P B . 28
Figure 12: Indifference curves of the three buyers. Figure 12 shows the constraints in the familiar (P, Q)-space and will help to keep track of the arguments that follow:
• If (iii) and (v) hold then (i) is not binding. • If (iii) and (vii) hold then (ii) is not binding. • (iii) must be binding, otherwise, given (i) and (ii) don’t bind either, we could increase all prices by the same amount without violating any constraints. • Either (iv) or (v) must be binding, otherwise we could increase P A without violating any constraints. • Either (vi) or (vii) must be binding, otherwise we could increase P B without violating any constraints. • Assume (iv) binds, then (vi) must not bind because AQA − P A = BQ B − P B + (A − B)QB . Therefore, based on the previous point (vii) must bind. • If (iv) and (vii) bind, then (v) must not, because AQA − P A = AQC − P C + (A − B)QC . • By a similar reasoning, binding (vii) and (iv) also imply non-binding (viii) and (ix). • If (iv) does not bind, then (v) must, but in that case CQA − P A = CQC − P C + (A − C )(QA − QC ) contradicts (viii). Therefore (iv) must bind.
29
Problem 18: A monopolist produces a durable good that lasts for two periods
(no depreciation). The production is costless. There is a continuum of consumers each of whom lives for two periods and desires the good in each period. Their per-period valuations for the good are distributed uniformly along the unit line [0, 1] (thus, the implied initial total demand is q = 1 − p). The monopolist offers the good for sale in both periods. Let p1 and p2 be the first- and the second-period prices, respectively. If a consumer purchases the good in the first period, she may hold on to it for two periods, or may sell it in the second period (hence, there exists a resale market in which the good bought in the first period may change hands in the second period). The monopoly maximizes the sum of profits and the common discount rate is 1/2. a. Characterize the equilibrium sale price in each period. Solution: To begin with, we need to clarify the role of second hand sales,
that is, the sales in the second period from those buyers who bought in the first period. When the monopolist charges a price in the first period, only those who value it most will decide to buy it rather than wait for a lower price later. In fact, we will determine the first period demand q1 by finding who is the buyer who values the good least among them. This marginal buyer will be indifferent between buying in the first period and waiting. Hence, there cannot be trade between those who choose to buy first and those who decide to wait, because the former always value the good more. Now that we have established this, we go ahead to solve the game without bothering ourselves with the second hand sales. As usual we begin from the last period. Given the residual demand, the monopolist can only commit to charging the price that maximizes the profit in this period. Since the q1 who value the good most have already purchased it before, in the second period the residual demand is q2 = 1 − q1 − p2 . Hence, the monopolist’s problem is max p2 (1 − q1 − p2 ), p2
which is maximized at p2 = (1 − q1 )/2 for a profit of π2 = (1 − q1 )2 /4. ∗
∗
Moving on to the first period, we must determine q1 by finding the marginal buyer. This buyer is indifferent between buying now at p1 to get a surplus of 3(1 − q1 )/2 − p1 and waiting. Keep in mind that 1 − q1 is his value from owning the good for one period, to which we have added the discounted second period value. If he waits, he will get a surplus of (1 − q1 − p2 )/2, which has been discounted to reflect the fact that the purchase occurs later.
30
Hence, q1 can be identified by the equality
3 1 1 − q1 (1 − q1) − p1 = 1 − q1 − 2 2 2 4 q1 = 1 − p1 . 5
We now have all the necessary components needed to solve the first period maximization problem max π1 + π2 /2 = p1 q1 + (1/8)(1 − q1)2 , p1
which is maximized at p1 = 25/36. From here we can find all other variables: p2 = 5/18 and q1 = 4/9. ∗
∗
∗
b. If the monopolist could credibly commit in the first period to a price schedule,
what would be the optimal prices? Solution: Now the problem is somewhat different. Before, by restriction of
subgame perfection, we constrained the monopolist to the only p2 he could commit, the price that maximized that period’s profit. If he could credibly choose another price, though, he would probably want a higher price to force more buyers into purchasing in the first period (where they would be willing to pay more). If the monopolist can commit to any p2, his demand in the first period is given by an equality similar to part (a): 3 1 (1 − q1 ) − p1 = [1 − q1 − p2 ] 2 2 p2 q1 = 1 − p1 + . 2 From here we can directly write the maximization problem for both periods max p1 , p2 π1 + π2/2 = p1 q1 + (1/2) p2 (1 − q1 − p2 ). Taking both first order conditions and solving them simultaneously we get to the prices p1 = 3/4 and p2 = 1/2, as well as first period demand q1 = 1/2. Note that given the residual demand q2 = 1/2 − p2 nobody will buy in the second period at p2 = 1/2. Normally, it would not be possible for the monopolist to commit to this price because he would be tempted to lower it in order to earn something from the second period as well. ∗
∗
∗
∗
Problem 19: A monopolist supplies rides at an amusement park. There is a large
pool of identical consumers. The demand for rides of each consumer is shown in 31
the figure below. The capital letters in the figure represent the areas. The per-unit cost of supplying rides has two components: c, the cost of supplying the equipment, and d, the cost of printing and collecting tickets at each ride. Each per-unit cost is independent of the number of rides offered at the park. The monopolist is considering two pricing policies: Policy 1: Policy 2:
An entrance fee and a per-unit charge for each ride. Just an entrance fee. The advantage of this policy is that the monopolist does not incur the costs of printing and collecting tickets.
Using the information given in the graph, what is the entrance fee, per-unit charge, total number of rides supplied per consumer, and total profits under each policy? Under what conditions will profits under policy 2 exceed those under policy 1?
Figure 13: Demand for the amusement park rides. Solution: Under policy 1, the marginal cost for each ride is c + d, so this must be
the per-unit charge. The number of rides demanded under this policy is denoted by Q1 on Figure 13. Since we are dealing with a single demand, the park is able to extract all the consumer surplus using the entry fee, which will be equal to area A. The profit under this policy is equal to Π1 = A, because the price of each ride only pays for its cost. Under policy 2, the monopolist will make a loss on the rides, but will be able to charge a higher entrance fee, due to the much larger consumer surplus. The number of rides they will demand is denoted by Q2 . The fee will be equal to the sum of areas A+ B +D +E +F +G, but running the park will cost E + F + G+H . Hence, the profit under this policy is Π2 = A + B + D − H . 32
Policy 2 will be more profitable than policy 1 if Π2 > Π1 , which is true if B + D > H . Problem 20: Consumers are uniformly distributed on an interval of length 1.
They pay transportation costs of $1 per kilometer. Each wants to buy one unit of a homogeneous good. Two firms producing this homogeneous good are located at the two end points of the interval. They simultaneously choose prices for their product. Firm 1 has the option of paying a bus company a lump sum $L so that customers can reach firm 1 with transportation costs $0.5 per kilometer (the cost of reaching firm 2 remains $1 per km). How much is firm 1 willing to pay the bus company for this privilege? Solution: The willingness to pay of Firm 1 will be determined by the difference in
profit between the symmetric case and the case where its customers transportation costs are halved. Calculating these profits is straightforward. We will denote by x the demand for Firm 1’s product when the transportation costs are equal for both firms’ customers. This length will be defined by the marginal consumer who is indifferent between buying at each firm. Let pi be the price charged by firm i: p1 + x = p2 + (1 − x) x = ( p2 − p1 + 1)/2. Firm 1’s problem then is max p1 ( p2 − p1 + 1)/2, p1
which gives us the response function p1 ( p2 ) = ( p2 + 1)/2. Since our equilibrium is symmetric we have p1 = p2 = 1. This gives Firm 1 a profit of π1 = 1/2. Now for the case when it only costs consumers 1/2 per unit of transport to reach Firm 1. Let y be the demand of Firm 1 in this case. The marginal consumer will be indifferent between ∗
∗
∗
∗
p1 + y/2 = p2 + (1 − y) y = 2( p2 − p1 + 1)/3. We now have to solve both firms’ problems to get their response functions because we cannot rely on the property of symmetry anymore. Firm 1’s problem is max 2 p1 ( p2 − p1 + 1)/3, p1
and it gives the same response function as before p1 ( p2 ) = ( p2 + 1)/2. Firm 2’s problem is given by max p2 [1 − 2( p2 − p1 + 1)/3], ∗
p2
33
and it gives the response function p2 = (2 p1 +1)/4. Solving both response functions simultaneously we arrive at the result: p1 = 5/6 and p2 = 2/3. Plugging into the demand we find that y = 5/9, which yields a profit for Firm 1 of π1 = 25/54 < 1/2. Under the new conditions Firm 1 makes a lower profit than under symmetric costs. This apparently puzzling result can be explained if we consider the strategic interactions between firms, and not just the myopic choice of an investment to lower transportation costs. All other things equal, a lower transportation cost for Firm 1 means more customers and a higher profit. However, under these conditions, Firm 2 will decide to charge a lower price. Prices are strategic complements in game theory, meaning a lower price from one firm induces a response by a lower price from the other firm (the response functions have positive slopes). Therefore the resulting lower p2 will force Firm 1 to charge less in equilibrium. The higher demand is not sufficient to offset the effect of the lower price, that is why a sensible investment under naive consideration is not profitable if strategic interactions are taken into account. ∗
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∗
∗
∗
Problem 21: Consider a monopolist that faces a consumer with unit demand.
The consumer has a maximum willingnes-to-pay equal to 10θq for every q units bought, q ≤ 1, and is privately informed about θ. The monopolist’s belief for θ is described by a uniform density function over [0, 1]. The consumer buys if her net surplus is nonnegative. Assume that there are no production costs. a. Suppose the monopolist has to charge a single linear price. Find the profit-
maximizing price. Solution: The consumer will buy iff 10θq ≥ pq ⇔ p ≤ 10θ. That is, the
consumer buys 1 unit if p ≤ 10θ. So, the monopolist would optimally set p = E [10θ] = 5. Alternatively: ∗
1
max p
pdθ = p(1 −
p/10
p ) ⇒ p = 5. 10 ∗
b. Suppose that the monopolist can offer at most 2 non-linear contracts in
the form (q, T (q)), where q is the amount offered and T (q) is the payment requested for q units. Write down the profit maximization problem. Which constraints bind in equilibrium? Characterize (or discuss if you cannot) the equilibrium. Solution: Consider (qL , T L ) and (qH , T H ) such that the consumer does not
to buy either if θ < θL , buys (qL , T L ) if θL < θ < θH , and buys (qH , T H ) if θ > θH . When θ = θL , he is indifferent between not buying and (qL , T L ), and when θ = θH , he is indifferent between (qL , T L ) and (qH , T H ). So, in an 34
equilibrium, we must have: 10θH qH − T H = 10θH qL − T L ⇒ θH = 10θL qL − T L = 0 ⇒ θL =
T H − T L , 10(qH − qL )
T L . 10qL
Furthermore, it is straightforward that qL < qH = 1. Hence, the maximization problem is: (T H − T L )2 T L2 max [(1 − θH )T H + (θH − θL )T L ] = T H − + q ,T ,T 10(1 − qL ) 10qL (T H − T L )2 T L2 [qL ] : − + 2 = 0. (1 − qL )2 qL L
L
H
But this is possible only if θH = θL , which means that the monopolist offers only one contract. From part (a), it must be (q , T ) = (1, 5). ∗
∗
c. Now, suppose that the monopolist is unconstrained in the number of con-
tracts it can offer. Write down the profit maximization problem. Which constraints bind in equilibrium? Carefully explain. Solution: Let the non-linear (strictly increasing) function T (q) describe the
contracts the monopolist offers. T (q) must be continuous (why?). The consumer chooses: q(θ) = arg maxq (10θq − T (q)), so T (q(θ)) = 10θ. Integrating gives T (q(θ)) = 10θq(θ) + c. The monopolist never sells to all types of the consumer, because the type θ = 0 has zero willingness-to-pay. Let θ˜ > 0 be the indifferent consumer type. This consumer type will have zero surplus ˜ = 10θq(θ), ˜ so, c = 0. Hence, the maximization in equilibrium, i.e. T (q(θ)) problem is:
1
max T (q)
θ˜
1
T (q(θ))dθ ≡ max q(θ)
θ˜
10θq(θ)dθ.
The maximand is linear in q, so the monopolist would want to set q = 1 for everyone. The solution will be the same as before, i.e., the monopolist will offer only one contract (q , T ) = (1, 5). The reason is the linearity of the consumer’s utility function. ∗
∗
Problem 22: Customers are uniformly distributed on a Hotelling interval [0, 1].
An individual located at point d (distance d from the point 0) has income d. Two firms are located at the two end points, 0 and 1, of the interval, their products differ in quality: q0 > q1 . A customer with income d who buys one unit of a good of quality q has the utility qd. Transportation costs are linear and cost t per unit of distance. Firms have no production costs. Assume that t > q0 − q1 > 0 and 35
