CERGE-EIFall 2010 Industrial Organization I Practice Problems Problem 1: Consider a monopolist that faces a downward-sloping demand curve, and produces at constant marginal cost, $c per unit. Suppose that the monopolist produces an output Q > 0 when facing a per-unit tax of $t. Show that there exists an ad valorem (percentage of sales revenue) tax that induces the monopolist to produce the same output, Q. Which tax raises more revenue? Explain. [Let the producer price under the ad valorem tax be p (1 − 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 ). When the firm has to pay a unit tax its problem is max π Q = p(Q)Q − (c + t)Q (1) FOC: p(Q) + p (Q)Q = c + t. With an ad valorem tax, though, we have max π Q = (1 − τ )p(Q)Q − cQ (2) FOC: p(Q) + p (Q)Q = c/(1 − τ ). 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 − τ ) τ = t/c. We now need to compare the revenue in each case. In the first case this is t simply Qt, whereas in the second case the revenue is Qp c . These expressions 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 > M R = c, so the ad valorem tax revenue is larger. The intuitive reason for this is depicted in Figure 1 (the taxed demand is in gray). In the ad valorem tax case the monopolist faces a steeper (less elastic) demand and, therefore, charges a higher price. For the same amount of output that corresponds to a higher tax revenue. 1 p demand p demand MC MC MR }t MR 0 Q 0 Q Figure 1: Monopolist with unit (left) and ad valorem tax (right). Problem 2: Show that if indirect demand p(Q) is concave, then p(Q) is logconcave which in turn implies that 1/p(Q) is convex. Also show that if p(Q) is log-concave or if 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(Q) is convex. If we take the second derivative of 1/p(Q) we get an expression that is positive if p(Q) ≥ 0 and p (Q) ≤ 0. Hence, 1/p(Q) is convex. 2p (Q)2 − p (Q)p(Q) ∂2 1 = . ∂Q2 p(Q) p(Q)3 Now, if we take the second derivative of log p(Q) we get an expression that is negative if p(Q) ≥ 0 and p (Q) ≤ 0 ∂ 2 log p(Q) p (Q)p(Q) − p (Q)2 = . ∂Q2 p(Q)2 A counterexample to the concavity of a log-concave function is p(Q) = Q2 . This function is convex but log p(Q) = 2 log 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 − 200p 2 (p measured in cents) and Firm B’s is qB = 5000 − 100p. The marginal cost of providing a phone call is 6 cents. Assuming the phone company has to charge the same monthly rental fee and unit price to all its customers (i.e. single two-part 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 tariff confirms this assumption. In this case the phone company can set the rental fee (fixed 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 CSA (p) = (14 − p)(2800 − 200p) (14 − p)qA (p) = 2 2 where 14 is Firm A’s maximum willingness to pay for a unit of service (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 = 2CSA (p) + (p − 6)[qA (p) + qB (p)]. p Substituting and taking the first order condition we find that the maximizing price is p∗ = 20 > 14, which is a contradiction to our starting assumption. both 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 CSB (p) = (50 − p)(5000 − 100p) (50 − p)qB (p) = . 2 2 In the same way as before, we can now write the phone company’s problem max π = CSB (p) + (p − 6)qB (p). p The solution is the two-part tariff p∗ = 6 and F = 96800. The 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 discrimination? The monopolist 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. Find the prices and quantities for the low-demand and the highdemand market. CS1 = 49/2. Compute the inverse demand for the integrated market with two consumer groups. Find price. q2 4 . it can be expressed as p(Q) = 12 − Q 10 − Q/2 0≤Q≤4 4 < Q ≤ 20. The marginal revenue is ∂p(Q)Q/∂Q for each segment of inverse demand. Solution: Because of the kink in (inverse) demand. Suppose that the monopolist charges a uniform price on the integrated market and that his marginal cost is M C = c = 0. Solution: In this case the monopolist will maximize for each market separately. Formally. the best way to solve this problem is by using a graph. so in our case. The total welfare therefore is W = 79. The problems are max π1 = (12 − q1 )q1 q1 max π2 = (8 − q2 )q2 . It is depicted in Figure 2 in solid black. When demand is linear. b. and CS2 = 9/2. third degree price discrimination will be harmful when all markets are served under uniform pricing. quantity sold. Plot it with p on the vertical and Q = q1 + q2 on the horizontal axis. Now assume that the monopolist can price discriminate between the market segments. Solving two monopoly maximizations is straightforward. Show that welfare goes down. Figure 2 shows both the demand from part (a) and the discontinuous marginal revenue curve (solid red). Formally M R(Q) = 12 − 2Q 10 − Q 0≤Q≤4 4 < Q ≤ 20. the condition for maximum profit is M R = M C. consumer surplus and monopolist’s profit. each valid in the respective interval. Its y-intercept is at p = 12 and it has a kink at p = 8. a. p = 8 − q2 . Suppose there are two groups of consumers: highdemand (1) and low-demand (2). Compute the social welfare (sum of consumer surpluses and profits). c. From here it is easy to determine the profit and total surplus: π ∗ = 50. Find the monopolist’s profit and total consumer surplus. As usual. Inverse demands are: p = 12 − q1 . Solution: The integrated inverse demand is the horizontal sum of the individual inverse demands because we are summing up quantities at each price. Q∗ = 10 and p∗ = 5. 5.375. for a total welfare of W = 78. The profits and 2 1 ∗ ∗ consumer surpluses for each market are: π1 = 36. Unsurprisingly. Solution: Again.5 and q2 = 0. q2 ∗ ∗ The solutions are: q1 = 2. now the monopolist has to solve two disjoint problems max π1 = (12 − q1 )q1 − 7q1 q1 max π2 = (8 − q2 )q2 − 7q2 . Show that now all markets are served and that welfare goes up relative to the uniform pricing (the case in part d).75. As expected. and they yield 1 2 a social welfare of W = 9. which means only market 1 will be served. Like before.5 and p = 9.5 4 10 20 Q Figure 2: Monopolist with two demands. Now assume that demands are the same but marginal cost is higher: M C = c = 7. d. Compute the social welfare. this is lower than the welfare without discrimination we found in part (b). p∗ = 7.5.5.5. we use the graph in Figure 2 and check where the M C = 7 line crosses the M R curve.p 12 10 7 5 MC p(Q) MR(Q) 0 2. p∗ = 6 and q2 = 4. ∗ ∗ The solutions are: q1 = 6. CS2 = 8. CS1 = 18 and π2 = 16. the social welfare is straightforward to compute: W = 9. This happens at q = 2. Show that under the uniform pricing the low-demand group does not buy. Solution: Just like in part (c). e. this is larger than what we found in part (d). 5 . p∗ = 9. p∗ = 4. Compute prices and quantities with M C = c = 7. Allow for third degree price discrimination. 6 . because it is selling below production cost. b. We will refer to Figure 3 during our analysis. But it would be making positive profits on the sale of goods by C in market 1 and C + D in market two. The left graph shows the effect of an infinitesimally small decrease in p below M C.Problem 5: Show in a diagram that if two consumers have linear demands. This loss is the size of segment A for market 1 and segment A + B for market 2. 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. 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. however. We can conclude that decreasing the price below p = M C will lower the profit. with one consumer demanding more than the other at any price. p p<MC p p>MC A B p(q1) MC p(q2) C D p(q1) MC p(q2) 0 Q 0 Q Figure 3: Single two-part tariff. To prove that the monopolist charges p > M C we will be using the graph on the right hand side of Figure 3. a. 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. Consider an infinitesimally small increase in p above M C. It would be making a loss. Hence it is profitable to increase the price above p = M C by a positive amount. The monopolist will be able to charge a slightly higher fixed fee by the size of segment A to each market. Our argument will be very similar. Hence the net change of this deviation would bring a loss. The monopolist will have to charge a slightly smaller fixed fee by the amount of the length of the segment marked C. 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. By charging p1 > M C the monopolist will forgo K from the fixed fee in market 1. p p(q2) p A G B C F D p(q1) H E p2>p1 p1=MC p2<p1 J K M L N p1>MC p2=MC p(q2) p(q1) 0 Q 0 Q Figure 4: Separate two-part tariffs. it ends up losing H. Nevertheless. 7 . Consider what happens if p2 > p1 . If the monopolist sets p2 = M C it can extract the additional surplus (J + K + L + N ) using F2 . it now has to compensate the customers in this market for the drop in consumer surplus by lowering F + H from the fixed fee. In this case. The left graph depicts the situation when the monopolist charges p1 = M C to market 1. It is possible to argue that the best the monopolist can do is to offer the same tariff to market 2. While the monopolist makes a profit of F on sales in market 2. it has to provide at least as much surplus for the high demand as they would get under tariff 1. Now that we established that the monopolist does not gain from charging at cost in market 1. for p2 < p1 we can show that the monopolist will make a loss of E. The analysis that follows refers to Figure 4. For this analysis we will be referring to the right hand side graph of Figure 4. this will also make the first market tariff much less attractive to high demand consumers. Similarly. Hence. we can proceed to show that it will gain from charging a higher price in this market. That means that while it can still charge fixed fee F1 = CS1 . Their surplus from choosing that tariff has dropped to only M . market 2 consumers can get a surplus of F + G + H by choosing this tariff. Then it can ask for F1 = A + B in that market. ¯ Let’s call this level of utility Ui . a line with slope pi and a y-intercept ∂i . the utility from having no insurance (T = Y = 0). What is the optimal contract if the insurance company cannot price discriminate? Solution: Since we don’t know the precise expression for v(·) but only its concave shape. ∂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. Hence we are dealing with a concave function. where pL < pH . we can still determine its shape using differential calculus. The individual may be a careful or a careless person. We can also say that the indifference curve that starts from the origin represents the outside option of the consumer. 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 . The individual clearly knows this but the insurance company does not. a. c) = v(w). the individual bears a total loss of L if his car is stolen. Y ) − Ui = 0 ∂F/∂Y pi v (w0 − T + Y − L) ∂T =− = . When we draw it in our usual (T. we will use graphs to illustrate a general answer to this problem. Without insurance. the insurance company compensates the individual with a total of Y . 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 utility of the individual is given by u(w. where v (w) > 0 and v (w) < 0. Most important. Y ) is given by Ui = (1 − pi )v(w0 − T ) + pi v(w0 − T − Y − L). Y )-space the function of the isoprofit line is Ti = πi + pi Y . that is. The probability of his auto being stolen is pL if the individual is careful and is pH if she is careless. (3) While it is not possible to get a formula for the indifference curve. In case of theft.Problem 6: Suppose that an individual who has an initial wealth of w = wo is contemplating buying insurance against auto theft. The insurance company’s belief is that the individual is careful with a probability µ. The utility of the consumer with theft probability pi from an insurance contract (T. we can find the slope if we consider (3) as an implicit function F (T. Let T be the payment from the individual to the insurance company if he chooses to buy insurance. and that ∂T /∂Y > pi before this point (Y < L) and ∂T /∂Y < pi after it (Y > L). The indifference curves are a bit more complicated to derive. 8 . that is. If it offers point 2. If µ is small enough or pH − pL is large enough.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. 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. it will be making a profit of µπL + (1 − µ)˜H . it will be selling only to the high-risk consumers for a profit of (1 − µ)πH . nondiscriminating (left) and discriminating (right). Formally. Since the high-risk type is always willing to pay more for the same amount of coverage Y . the insurer will only sell to high-risk types. we can already say that only the individual rationality constraint (participation) for the low-risk consumers. the problem of the 9 . Solution: The monopolist cannot simply offer points 1 and 2 that we mentioned in part (a) as contracts. Hence. The left graph on Figure 5 shows this point for both types. the profit from selling low-risk insurance to high-risk consumers will only ˜ be positive if the difference in pi between the types is not large. If it offers point 1. We already know that at this point L = Y . How do they depend on the parameters of the model. he is analogous to the high demand type that we encounter in other incarnations of this problem. Figure 5: Insurance seller. The non-discriminating monopolist will offer one of these points. in particular on µ? Give an intuitive explanation for your answers. Note that π πH . and the incentive compatibility constraint for the high-risk types will hold in equilibrium. b. because the high-risk types would prefer the low-risk contract to their own. C. the low-risk insurance coverage must be lower than L. whereas the high-risk types will earn some information ∗ ¯ rent (UH > UH ).insurance company (writing only the binding constraints) is the following TL . θH and θL . Derive his optimal pricing policy. which is only satisfied at YH = L. 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. Show that he serves both classes of consumers if either θL or λ is “large enough. and its cost of production per unit is c > 0. T ) = θv(x) − T . p). The only constraint for the monopolist is the participation constraint max(pi − c)xi xi .YL . (4) s. 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.” Solution: The pricing policy is a single couple (x. Problem 7: MWG 14. v(x) = a.YH max µ(TL − pL YL ) + (1 − µ)(TH − pH YH ) ¯ (1 − pL )v(w0 − TL ) + pL v(w0 − TL + YL − L) ≥ UL (1 − pH )v(w0 − TH ) + pH v(w0 − TH + YH − L) ≥ (1 − pH )v(w0 − TL ) + pH v(w0 − TL + YL − L).: If we take the first order conditions for TH and YH we get the equation v (w0 − TH + YH − L) = (1 − pH )v (w0 ) + pH v (w0 − TH + YH − L). This information rent will be increasing in µ. We will start by deriving the best pricing policy for a consumer type θi . For this type of contract to be incentive compatible. A type θ’s utility when consuming an amount x of the good and paying a total of T for it is u(x. as shown in the right graph of Figure 5.7 Assume that there are two types of consumers for a firm’s product. In fact. Consider a nondiscriminating monopolist. if the fraction of low-risk consumers falls. as we expect in these kind of problems. 2 The firm is the sole producer of this good.t. As usual. the high-risk types will get a socially optimal insurance coverage. because.t. Hence. 10 (5) .TH .: θi v(xi ) − pi xi ≥ 0. the low-risk types will be left with a utility equal to their ¯ outside option (UL ). The proportion of type θL consumers is λ. however.pi s. where 1 − (1 − x)2 . He will serve both types if πL ≥ (1 − λ)πH . and price according to θL . pricing according to θH . pi ) = θi v(xi ) − pi xi . The problem is max θL v(xL ) − pxL + (p − c)[λxL + (1 − λ)xH ]. For a any price pi we can derive the demand of θi by maximizing their utility max ui (xi . We know this charge will equal the utility of the low types in equilibrium. Using this as a demand for the market and maximizing profits with respect to price we get: p∗ = (θi + c)/θi and i xi = 1/2 − c/(2θi ). Characterize his optimal prices. xi which gives us xi (pi ) = (1 − pi /θi ). so we can derive demands and solve the profit maximization problem. Solution: Now we have two separate markets and the utilities of their respective customers. p (6) After some tedious algebra we find the optimal price p∗ = c[(1 − λ)θL + λθH ] . (θi + c)/2]. Interpret.The maximizing pricing policy is (x∗ . 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 . πi = + 2 2θ A monopolist can choose whether it wants to serve both types. the monopolist will prefer to serve the whole market for a high enough λ and θL . that yields i i a profit of c2 θ − c. or only the high types. 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). c. 2(1 − λ)θL − (1 − 2λ)θH 11 . b. 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. p∗ ) = [1 − c/θi . Hence. and we can plug the demands we derived in part (b) xi = (1 − p/θi ). Consider a monopolist who can distinguish the two types (by some characteristic) but can only charge a simple price pi to each type θi . Suppose the monopolist cannot distinguish the types. a).TH max λ(TL − cxL ) + (1 − λ)(TH − cxH ) θL v(xL ) − TL ≥ 0 θH v(xH ) − TH ≥ θH v(xL ) − TL . 12 . Just like we expected. though. When we solve the problem with these substitutions in place. tourists and business.xH . By the same reasoning H we can see that the low demand type consumes less than the optimal amount because the demand corresponds to pL > c. the high demand type consumes an optimal amount because x∗ corresponds to the demand when pH = c. There are two types of passengers. we can directly write the problem with the binding constraints only xL . so we can solve them to get both Ti in terms of pi to simplify the problem TL = θL v(xL ) TH = θH v(xH ) − v(xL )(θH − θL ).: We already know that the constraints will hold with equality. The two types do differ.) More specifically.t. 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. The airline. We can also derive F ∗ = θL v(1−p∗ /θL )−p∗ (1−p∗ /θL ). (Passengers do not like to commit themselves to traveling at a particular time.It is possible to show that p∗ > c. Problem 8: MWG 14. (7) s. however. for any given amount of time W prior to the flight that the ticket is purchased are given by Business : T ourist : v − θB P − W. the utility levels of each of the two types net of the price of the ticket. Compute the fully optimal nonlinear tariff. Business travelers are willing to pay more than tourists. v − θT P − W. P . which confirms our intuition (see problem 5.8 Air Shangri-la is the only airline to fly between the islands of Shangri-la and Nirvana. d.C. we get the optimal xi x∗ = 1 − λc/[θL − (1 − λ)θH ] L ∗ xH = 1 − c/θH . in how much they are willing to pay to avoid having to purchase their tickets in advance. cannot tell directly whether a ticket purchaser is a tourist or a business traveler. The monopolist will serve both types if the demand for the low type at p∗ is positive.TL . Draw the airline’s isoprofit curves.] Solution: The indifference and isoprofit curves are depicted in Figure 6.: .PT . W v ^ ^ uT(P. if it charged a negative price. (Note that for any given level of W . 13 s.where 0 < θB < θT . W )-space. WT . it would sell an infinite number of tickets at this price. W) 0 v/θ T v/θ B P Figure 6: The problem of Air Shangri-la. Draw the indifference curves of the two types in (P. Formally.WT max λPT + (1 − λ)PB (i) (ii) (iii) (iv) (v) θT PT + WT ≤ v θB PB + WB ≤ v θT PT + WT ≤ θT PB + WB θB PB + WB ≤ θB PT + WT PT . Also. [Hint: Impose nonnegativity of prices as a constraint since. Air Shangri-la solves the following problem PB . PB . a. the business traveler is willing to pay more for his ticket. WB ≥ 0.) 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. Now formulate the optimal (profit-maximizing) price discrimination problem mathematically that Air Shangri-la would want to solve. the business traveler is willing to pay more for any given reduction in W .WB .t. W) isoprofit ^ W ^ ^ uB(P. Assume that {(PT . That means that constraint (i) is binding (satisfied with equality). Describe fully the optimal price discrimination scheme under the assumption that they sell to both types. In numbers: they have to lower PT by /θT in order to increase WT by . This means tourists will not get any surplus above their outside option. 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 < . and it can do this by increasing WT . Solution: We can prove by contradiction that business traveler will not have to book in advance. constraints (i) and (iv) imply that constraint (ii) is satisfied with strict inequality and can be ignored. WB )} is a solution to the firm’s problem and that WB > 0. Solution: Knowing that θB < θT . On the other hand. however. it must couple this with cuts in PT . This new solution solves the incentive compatibility constraint for tourists (iii) and provides a higher profit. if business travelers strictly prefer their own ticket then it would be possible to increase PB without violating their incentive compatibility constraint (iv). This trade-off does not depend in the level of prices or WT so it θT θB holds anywhere (it is linear). in the optimal solution they must be indifferent between the two types of tickets. Show that in the optimal solution. c. θB . Then the airline can change this to WB = 0. To keep the tourists in the market. while increasing PB by WB /θB to keep their utility unchanged.b. Show that in the optimal solution. This contradicts our starting assumption. otherwise Air Shangrila could increase both prices by a small amount and still all constraints would hold. therefore PB cannot be higher than 0. This will enable them to raise PB by (θT −θB )) . θT and c? Solution: The airline faces the following problem: It must make the tourist ticket unattractive to business travelers despite the higher PB . How does it depend on the underlying parameters λ. business travelers never buy their ticket prior to the flight and are just indifferent between doing this and buying when tourists buy. Hence. d. (8) 1−λ θB 14 . (PB . tourists are indifferent between buying a ticket and not going at all. WT ). In this case the optimal contract is {(0. 0)}. If c > θB then Air Shangri-la cannot operate at all. Plugging this in the problem and solving the first order 15 . so the ticket will be (P. where 2 < θL < 10. They will be constrained by the willingness to pay of tourists. and it can only offer a single two-part tariff in the form (p. Under what circumstances will Air Shangri-la choose to serve only business travelers? Solution: Aside from the conditions mentioned in part (d). 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. W ) = (v/θT . (v/θB . and low-types (L) with a demand function qL (p) = θL − p. The proportion of L-types in the population is α ∈ (0. T ). 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. the firm may also decide to drop its tourist customers if the pooling ticket does not justify its cost. A j-type consumer accepts this contract as long as T + pqj (p) does not exceed her surplus from consuming qj (p) units. Derive the optimal two-part tariff under the assumption that the monopolist serves both types of consumers. The monopolist is risk-neutral.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). 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. 1). v). – 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. There are two types of consumers for the monopolist’s product: high-types (H) with a demand function qH (p) = 10 − p. if v/θT < c < v/θB . 0). a. Solution: Assuming that the monopolist serves both types of customers. that is. When will the monopolist serve only the H-types? Interpret. that is: T (p) = qL (p)2 /2. Suppose the monopolist cannot distinguish the types. 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. A j-type consumer accepts (q. and it can only offer a single two-part tariff in the form (q. T (q)) as long as T (q) does not exceed her surplus from consuming q units. T (q)) gives her the highest utility among all the contracts. T (q)).condition. T (q)) as long as T (q) does not exceed her surplus from consuming q units. b. Its profit is maximized at q ∗ = 8 and it is equal to 32(1 − α). Only in the case when the monopolist would serve only H-types under the two-part tariff is the monopolist indifferent between the two. The problem is max T (q) − 2q q and it is maximized at q ∗ = θL − 2. The maximized profit is (θL − 2)2 /2. 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). In general the monopolist prefers the two-part tariff to the (T. because it can collect all of their surplus using either contract. So. If. who consume more. we have p∗ = 2 + (1 − α)(10 − θL ). and (q. Solution: Assuming it serves both types. the monopolist will serve only type H if (θL − 2)2 < 64(1 − α). q) contract because charging a unit price will ensure the H-types. the monopolist chooses to serve only type H. Derive the optimal two-part tariff under the assumption that the monopolist serves both types of consumers. Suppose the monopolist cannot distinguish the types. also pay more and cover their production costs. This will not happen if α or θL are small enough. Now suppose the monopolist can discriminate across the two groups of consumers and is free to offer any contract in the form (q. c. The monopolist will serve both types if the demand of type L at p∗ is positive θL > p∗ = 2 + (1 − α)(10 − θL ). A j-type consumer accepts (q. that is if the demand of type L can be sacrificed either because of their small numbers or small willingness to pay. Clearly state the monopolist’s maximization problem and derive 16 . T (q)). then T is constrained by T (q) = q(10 − q/2). instead. The buyer accepts a particular contract (˜. that is. However. ∗ The monopolist will decide to serve to only H-types if qL < 0∗. constraint (i) gives us TL = qL (θL − qL /2) and constraint (iv) gives us TH = qH (10 − qH /2) − 2qL . There is a single buyer who wishes to consume this good. T (˜)) as q q long as θ˜ ≥ T (˜). Problem 10: Consider a market for a single good. θ is uncertain to the monopolist. A risk-neutral monopolist supplies the good and the cost of production is assumed to be zero. if αθL < 0. we are only concerned that each quantity of production does not exceed capacity. but the same consumer in two states of the world. and the respective Ti . the 17 . 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.TH max α(TL − 2qL ) + (1 − α)(TH − 2qH ) (i) (ii) (iii) (iv) qL (θL − qL /2) − TL ≥ 0 qH (10 − qH /2) − TH ≥ 0 qL (θL − qL /2) − TL ≥ qH (θL − qH /2) − TH qH (10 − qH /2) − TH ≥ qL (10 − qL /2) − TL . where T (q) is the total amount to be paid by the buyer for q units.t. Which q q constraints will be binding in the optimal solution? Derive the optimal menu of contracts. The buyer’s willingness-to-pay is θq for q units consumed. Solution: In this problem. and recalling that 2 < α < 10. it is equally likely to be 1 or 2. Since we are not dealing with two consumers. T (q)). s. Formally. The buyer has a large income. Does the monopolist ever prefer to serve only the H-types? Explain why or why not.the optimal non-linear contract. 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 qL .qH . it can supply at most 10 units of the good. aside from the usual conditions on the optimal contracts we also have a constraint on capacity. whereas the L-type consume below this. Suppose the monopolist can offer a menu of contracts in the form (q. Thus. the monopolist is capacity-constrained. As expected.TL . we can see that both types will end up consuming more under the discriminating monopolist. Assuming both types are served in both cases. the H-type consumer a socially optimal amount.: 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. Clearly state the monopolist’s maximization problem. qH ≥ 0. The inverse demand function in the first period is summarized by p = 100 − Q. either in the first period or in the second period). We are left only with binding conditions (i) and (iv) which give us TL = qL and TH = 2qH − qL . TL ). which implies that (iii) does not bind. with all constraints included. Hence. It is assumed for simplicity that production is costless. Problem 11: Shy 5. if (iv) binds then we can write qL − TL = qH − TH + (qH − qL ). on the other hand. 1 pS the second-period sale price and pR the per-period rental price. Let pS be the first-period sale price. that is.TH max (1/2)TL + (1/2)TH (i) (ii) (iii) (iv) (v) qL − TL ≥ 0 2qH − TH ≥ 0 qL − TL ≥ qH − TH 2qH − TH ≥ 2qL − TL qL . we have a continuum of equilibria given by ∗ ∗ ∗ ∗ {(qL . any level ∗ of qL ∈ [0. ∗ The first order condition for qH is always positive. s. we will filter out the two redundant conditions. x). 20 − x)} for all x ∈ [0. a consumer can use it for one period only if she leases it. 18 . she will have it for her entire life (i.monopolist’s problem. If the monopolist leases the product for a single product..t.t: qH . 10]. which implies that qH = 10. then (ii) holds with inequality.e.qH max qH s. because the product is durable. Substituting into the objective function we have qL . (qH .1 A monopoly produces a durable product that lasts for two periods. Less obvious is that the first order condition for qL is always zero. They have different valuations for the product. 10] is optimal. qL = 10.: As usual. The respective payments Ti∗ can be derived from (i) and (iv). otherwise we could increase the profit by raising TH without violating any constraint. If (i) and (iv) hold. then (iv) must hold with equality in equilibrium. The monopoly 2 maximizes the sum of profits from the sales in the two periods.TL . (10. TH )} = {(x.qH . If the monopolist offers the product for sale and a consumer purchases the product. and there is no discounting the future. Given this. is the following qL .5. consumers buy it once in their life. Futhermore. There is a continuum of consumers who live for two periods and desire this product for the two periods of their lives. and everyone with a smaller valuation will decide to buy later. pS and pS . so now we know all of the consequences of setting a price in the first period. Now that we know what is the best we can do in the second period. we can move to the first. if the monopoly only rents this product. Thus we can find where this split in the demand will occur solving 2(100 − qi ) − p1 = (100 − q1 ) − (50 − q1 /2). 19 . the monopolist maximizes its profit by solving max π2 = p2 (100 − q1 − p2 ). 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 . Hence. if the monopoly 2 1 only sells this product. Solution: This is a simple monopoly problem played twice. It is important to realize that here buyers have a choice of waiting to buy in the second period. where q1 is the number of buyers who bought the good in the first period. there will be one marginal buyer who will determine the demand and who is indifferent between buying in this period and waiting. q1 = 40. The solution is q1 = 150 − 2p1 /3. b. pS = 30 1 2 S and q2 = 30 which give the monopolist a total profit of π S = 4500 < 5000. Taking this demand for granted. p1 s Taking the first derivative and solving we find: pS = 90. Solution: In this case the monopolist will solve the game using backward induction. Find the equilibrium per-period rental price. Everyone with a valuation higher than his will prefer to buy in the first period. and a total revenue of 2πi = 2 · 50(100 − 50) = 5000. his residual demand is q2 = 100 − q1 − p2 . Each period the maximization is R max πi = (100 − pR )pR pR and it is solved at pR = 50. at any price in this period. and we can solve the maximization problem max π1 + π2 = p1 q1 (p1 ) + (50 − q1 (p1 )/2)2 . p2 S which gives pS = 50 − q1 /2 and sell q2 = 50 − q1 /2 units for a profit of 2 S 2 π2 = (50 − q1 /2) . pR .a. Starting with the second period. the monopolist will be facing a residual demand of the (100 − q1 ) buyers with the lowest valuations. Characterize the equilibrium sale price per period. Moreover. b.5. where pS is the first period sale price. 20 . L. They both desire this product for the two periods of their lives. a. the monopolist prefers renting. and pS is the second period sale price. It is assumed for simplicity that production is costless. We are given that V H > 2V L . if the monopoly only rents this product. Solution: Just like in problem 11. so we can build up a strategy on that. either in the first period or in the second period. are given by: if i buys in period 1 2V i − pS 1 i S V − p2 if i buys in period 2 Ui = 0 if i does not buy. 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. or rent at V L to both. They 1 2 may also rent the product in each period if the monopoly offers renting. and there is no discounting the future. In this case. which confirms the Coase conjecture. Characterize the equilibrium sale price per period. The utility functions of the consumers. There are only two consumers living for two periods. It is easy to see that these are the best responses H if nobody bought in period 1 V S L V if only H bought in period 1 p2 = 0 if both bought in period 1. The monopolist already knows whether someone already bought the durable good in the first period. for i = H. where V H > 2V L > 0. pS and pS . 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. Solution: Now we must deal with both periods jointly. but they differ in their willingness to pay for it.2 A monopoly produces a durable product that lasts for two periods. where pR is the per-period rental price. Find the equilibrium per-period rental price. so we start from the second using backward induction.a. if the monopoly 1 2 only sells this product. consumers buy it once in their life. their per-period utilities are V i − pR . pR . Problem 12: Shy 5.c. hence the monopolist will rent at V H . this is simply the same game played twice. Any other prices are dominated by one of these two. which yields a total revenue of π R = 2V H over both periods. The monopoly maximizes the sum of profits from the sales in the two periods. Because the product is durable. Does the monopolist prefer renting or selling? Solution: As shown above. instead.1 We can assume that he buys. firms can distinguish consumers at different locations and therefore can price discriminate by choosing a delivered price for each particular consumer. The lines extending from each position show the lowest price a firm can charge to deliver the product at that location. c. Without loss of generality we can assume that x2 ≥ x1 . Problem 14: Consider the standard Hotelling model with two firms. We can no move on to the first period. Solution: Self-explanatory. We will denote the position of firm i by xi . and still leaves him the possibility to sell in the second period at V L to consumer L. This example goes to show that the Coase conjecture may fail when we deal with discrete demand. we were dealing with a continuum of consumers of measure 1. We have ignored the production costs c because they do will not affect our analysis and will only clutter notation. Recall that the utilities of each consumer net of the price are 2V i because they will enjoy the good in both periods. This is because he can use the two periods to differentiate between the two customers. if he waits the pS will be V H 2 and not V L . In the first case. first analyzing the optimal pricing decisions given locations. but now suppose that firms deliver the product.It is impossible only for consumer L to buy. in this case the monopolist makes a higher profit by selling rather than renting. The monopolist can charge pS = 2V H aimed only at consumer H or pS = 2V L aimed at both. We focus on the firm 1’s decisions. consumer H is indifferent between buying now and waiting. In particular. Compare the outcome with question 11. 1 21 . Think what would happen if. Why is it different? Solution: Unlike problem 11. 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. So. all exercises in chapter 1 (answers are in the book!). In the 1 1 second case the monopolist does not make any revenue in the second period and his total revenue is 2V L . Argue how this may change the findings of Hotelling. This is clearly the best strategy for the monopolist yielding π S = 2V H + V L . which earns the monopolist a first period revenue of 2V H . We can only claim this because we are dealing with a single consumer H who bears the whole consequence of his action. Problem 13: Tirole. Figure 7 shows the market with given firm locations. and transportation cost is proportional to the distance between the firm and the consumer served by that firm. because. The production costs are c per unit for each firm. and then moving to the choice of location. Figure 7: Price competition with given locations. Assume transportation costs t per unit. Firm 1 solves max π1 (x1 . 2 2 2 2 2 C1 = tx1 /2 + t(q1 − x1 ) /2 = tx1 /2 + t(x2 − x1 )2 /8. 2 1 x1 whose first order condition gives us the response function x∗ = x2 /3. Problem 15: Consider a linear city of length 1. x2 ) = R1 − C1 = (t/4)(x2 − 3x2 + 2x1 x2 ). We need to define everything in terms of locations. The 1 2 response functions are linear. There is a single buyer who wishes to consume this good. That is. if we switch the reference point to the other end of the segment. The monopolist can produce any quantity at zero cost. of course. Thus firm 1 will sell to all the market from 0 to q1 . knows her 22 . to get the equilibrium: x∗ = 1/4 and x∗ = 3/4. We can do so by constructing firm 1’s best response to firm 2’s location. 1 − x2 = x1 . and the resulting profits. so that we can use the results in the first stage. There is a risk-neutral monopolist located at point 0 who supplies a good. By symmetry. it will charge (slightly under) the lowest price firm 2 can afford. for a profit of π1 (shaded area). 1 the same is valid for firm 2. if we start with the assumption 1 2 that x2 ≤ x1 . The buyer’s location is uncertain to the monopolist (the buyer. There is another (tivial) equilibrium where x∗ = 3/4 and x∗ = 1/4. The revenue and costs of firm 1 are given by the areas R1 = tx2 /2 − t(x2 − q1 )2 /2 = tx2 /2 − t(x2 − x1 )2 /8. Firm 1 will charge the highest price it can without risking to be undercut by firm 2. q1 = (x2 + x1 )/2 because it is in the middle. The prices that lead to a positive profit (above firm 1’s own break-even line) are highlighted. Knowing the optimal pricing decision of the last period. we can move to the first period and examine the choice of location. so they cross only at this point. Hence. Hence. : Not all these constraints are binding in equilibrium. the maximization problem for the monopolist is qL .own location). we will call the buyer sitting at (1/3) high type and the buyer sitting at (2/3) low type. That tells us that (i) must be binding. x) ≥ 0. Suppose the monopolist can offer a menu of contracts in the form (q. Hence. otherwise we could increase profits by raising both TL and TH by some small amount. a.TH max (1/2)TL + (1/2)TH (i) (ii) (iii) (iv) U (qL . Substituting into the objective function and solving the first order conditions ∗ ∗ we find that qL = 0 and qH = 2/3. TL . b. TL . TH .t. From the monopolist’s point of view.qH . 1/3) ≥ U (qL . The buyer accepts a particular contract (˜. Which constraints will be binding in the optimal solution? Explain. by subtracting qH from each side. That also tells us that (iv) must bind.TL . U (q. the buyer is equally likely to be located at 1/3 and 2/3. in equilibrium only (i) and (iv) hold with equality. we see that (iii) does not bind. By this notation. x) = 2 where T is the transfer paid by the buyer for q units. it doesn’t bind. then (ii) must hold with inequality. Note that this is an internal solution that 23 . otherwise we could increase profits by raising TH . s. and x is the buyer’s location (so. then. TH . Solution: At any q the willingness to pay of the nearest buyer is greater than the furthest one because of the lower transportation cost. Derive the optimal menu of contracts. 1/3) ≥ 0 U (qL . The buyer’s net utility from consuming q units is given by 1 − (1 − q)2 − xq − T. if (i) and (iv) hold. T (q)). Solution: From constraints (i) and (iv) we can write TL = [1 − (1 − qL )2 ]/2 − 2qL /3 TH = [1 − (1 − qH )2 ]/2 − qH /3 − qL /3. Hence. 2/3) ≥ U (qH . T. per-unit transportation cost is 1 for each unit of distance travelled). 2/3) U (qH . TL . though. If (iv) holds with equality. T (˜)) as long as U (˜. that is. TH . 1/3)L. T (˜). q q q q Clearly state the monopolist’s maximization problem. First. 2/3) ≥ 0 U (qH . would the buyer’s purchase decision reveal her location? What about the second period? Discuss. this is also the socially optimal location. If we plug these solutions and their respective Ti∗ into the objective function and take the first derivative with respect to x we will find it is positive. whereas the distance from the furthest buyer is 2/3−x. The maximizing contracts are ∗ ∗ ∗ ∗ {(qL . 24 . 2/3 − x) ≥ 0 U (qH . 0). Therefore. because here the monopolist will not price discriminate. if we change the denomination such that the distance is measured from the end of the segment. Suppose the monopolist can change its location prior to offering the contracts.t. x − 1/3) ≥ U (qL . is only valid for x ≤ 1/2 because of our definitions of high and low type. thus no surplus will be lost in order to impose incentive compatibility. In fact.coincides with the corner solution of selling only to the nearest customer. (2/3. Suppose the monopolist is located at 0 again. 2/9)}. the monopolist always gain from moving towards the furthest buyer. Omitting the nonbinding constraints. (qH . s. the problem is symmetric. Incidentally. TH . TH )} = {(0.TH max (1/2)TL + (1/2)TH U (qL .: ∗ We can now solve the problem just like we did in part (b) to find qL = 3x−2/3 ∗ and qH = 4/3 − x.qH . Here. This means the monopolist gains by moving towards the furtherst customer. the monopolist’s problem then becomes qL . Solution: It is not difficult to argue that the monopolist will decide to choose somewhere between the two buyers. TL . the current analysis holds with the buyer at (2/3) as the high type. but now suppose the game is repeated twice. TL ). 2/3] interval will only increase the cost of both buyers and reduce demand. so that the customer at 1/3 is still the one with the highest demand. What is the optimal location? Is it socially optimal? Explain. the monopolist maximizes its profit. at an equal distance to each consumer. 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. The only point where this improvement is not possible only at x = 1/2. x − 1/3)L. however. because being outside the [1/3. Suppose also that the monopolist chooses a position x ∈ [1/3. c. This specification. The distance from this buyer is now x − 1/3. TL .TL . d. 1/2]. in this scenario. If the monopolist learns from the first period the location of the buyer based on the contract he chooses. They pay transportation costs of $1 per unit of distance. Find all the symmetric equilibria in which firms charge price p at their outer outlets. cannot be an equilibrium. as in Figure 8: the upper locations F1 . however. This. then in the second period there will be no information surplus. 1. see Figure 8. you can undercut a seller by infinitesimally lowering your price and gaining a chunk of the market. The outlets are symmetrically distributed so that the distance between any two neighboring outlets is 1/6. Figure 8: The circular city. TL ). so that he will reveal his identity and get no surplus in period 2. F3 belong to the first firm. Each firm has variable costs c per unit and aims to maximize the sum of the profits of the 3 outlets. Two firms locate on the circle. while the lower locations f1 . 0).Solution: The addition of another period changes the problem because of the revelation of information involved. because. F2 . 3. and price q at their inner outlets. we can only use calculus 25 . Therefore. the high type will choose (qL . to signal that he is a low type and take a more profitable deal in the second period. that is. whose length is 1. It is only possible to have an equilibrium where customer sitting at 1/3 is offered a higher surplus in period 1. each with 3 outlets. in this case (0. outlets no. f3 belong to the second firm. the monopolist would put no faith in the signal. outlets no. 2. Problem 16: Consumers are uniformly distributed (with density 1) on the circumference of a circle. If offered the contracts ∗ ∗ derived in part (b). f2 . Solution: Linear transportation costs in location models cause discontinuities in the strategic variable. given that the problem is symmetric. The most obvious scenario is when both inner and outer outlets are working. Now we can write the firm’s problem as max π(p. We started with the assumption that all outlets work.in the intervals when these discontinuities do not occur and make sure to cover all intervals. p) = (p − q)/2 + 1/12. which means that no outlets are undercut. First. i. We now consider what happens if firms decide to use only their inner outlets f2 and F2 . The left graph on Figure 10 shows the competition between these 26 . The consumer sitting at x will be indifferent between buying from f1 for p + x and buying from F1 from p + (1/6 − x). Figure 9: Both inner and outer outlets working. q. That gives us x as a function of the surrounding outlet prices as x(p. We have a symmetric equilibrium where p = p = p∗ . In this case. let’s define the demands. let x denote the demand between f1 and F1 that accrues to f1 and let z denote the demand between f1 and f2 that accrues to f2 . p ) = (q − c)z + (p − c)(1/6 − z + x). We will be working only on optimizing this section. We must now check for what values of c (our only parameter) is this equilibrium valid. Our firm takes p as given and sets its own strategic variables p and q that maximize its profit. By the same token z(q. p. Figure 9 shows half of the market of the lower-case firm linearized for convenience. The condition for this can be written as p∗ − 1/6 < q ∗ < p∗ + 1/6. This condition is satisfied for all values of c. p ) = (p − p)/2 + 1/12.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. ii. The profit of the deviant is π d = 25/144 > 1/8 which means that we do not have an equilibrium. Given that the competitor sells for q ∗ = c + 1/2 at F2 what is the highest profit our firm can achieve by setting pd and q d ? In this case. And now for the last case. q ) = (q − c)y. when only the outer outlets are working. which is again satisfied for all values of c. The disputed demand for 27 . The deviant’s problem is max π(p. This scenario is shown in the left graph of Figure 11. q whose symmetric equilibrium is q = q = q ∗ = c + 1/2 with a (half) profit of π = c/4 + 1/8. p. This deviation is depicted in the right graph of Figure 10. iii. q)d = (q − c)y1 + (p − c)(1/6 − y1 + y2 ). Figure 11: Only outer outlets working. q ) = (q − q)/2 + 1/4 in the same way we did in part (i). 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. We can define the demand of outlet f2 as y(q. demand y1 = (p − q)/2 − 1/12 and demand y2 = (q ∗ − p)/2 − 1/6. firms.q which is maximized at pd = c+7/12 and q d = c+2/3. Our firm’s problem is given by max π(q. For this deviation to be valid (no undercutting) we must have that pd − 1/6 < q d < pd + 1/6. Most important. for our equilibrium to exist this deviation must not be profitable.Figure 10: Only inner outlets working. and the problem to solve is max π(p. The seller aims to maximize his profit. because the firm take p∗ = c + 1/3 as given. 28 s. this is not an equilibrium. Thus we again have a profitable deviation. p The problem is solved at p = p = p∗ = c + 1/3 for a profit of π = 1/18. it is CQ − P . therefore. p ) = (p − c)(w + 1/6).: . The deviant’s problem is max π(p. 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. Again. This deviation is shown on the right graph of Figure 11. The deviant could start selling from outlet f2 and charge prices pd and q d .t. There are 3 consumers. Which of those are equalities? (There is no need to solve for the optimal plan). we check whether there are profitable deviations. p ) = (p − p)/2 + 1/12. characterized by the three constants A > B > C > 0. p. For consumer 2. it is BQ − P .Qi QA (PA − K) + QB (PB − K) + QB (PB − K) (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) AQA − PA ≥ 0 BQB − PB ≥ 0 CQC − PC ≥ 0 AQA − PA ≥ AQB − PB AQA − PA ≥ AQC − PC BQB − PB ≥ BQA − PA BQB − PB ≥ BQC − PC CQC − PC ≥ CQA − PA CQC − PC ≥ CQB − PB . If consumer 1 buys quantity Q and pays P for it. Problem 17: A seller sells rice which he produces at variable cost K. Solution: Using our usual notation. the problem of the rice seller is the following max Pi . and for consumer 3. Write the inequalities defining the seller’s optimal plan.q which is maximized at pd = c + 5/12 and q d = c + 1/2 for a profit of π d = 13/144 > 1/18. The demands are w1 = (p−q)/2+1/12 and w2 = (p∗ − p)/2 + 1/12. his utility is: AQ − P .our firm is w(p. q)d = (q − c)w1 + (p − c)(1/6 − w1 + w2 ). assuming that he is selling to all three buyers. Figure 12 shows the constraints in the familiar (P. Therefore (iv) must bind. then (v) must not. otherwise we could increase PB without violating any constraints. Therefore. based on the previous point (vii) must bind.Figure 12: Indifference curves of the three buyers. • Either (vi) or (vii) must be binding. • By a similar reasoning. • Assume (iv) binds. Q)-space and will help to keep track of the arguments that follow: • If (iii) and (v) hold then (i) is not binding. then (vi) must not bind because AQA − PA = BQB − PB + (A − B)QB . • (iii) must be binding. given (i) and (ii) don’t bind either. • If (iii) and (vii) hold then (ii) is not binding. but in that case CQA − PA = CQC − PC + (A − C)(QA − QC ) contradicts (viii). • If (iv) does not bind. otherwise. because AQA − PA = AQC − PC + (A − B)QC . binding (vii) and (iv) also imply non-binding (viii) and (ix). • Either (iv) or (v) must be binding. • If (iv) and (vii) bind. then (v) must. otherwise we could increase PA without violating any constraints. 29 . we could increase all prices by the same amount without violating any constraints. there exists a resale market in which the good bought in the first period may change hands in the second period). Their per-period valuations for the good are distributed uniformly along the unit line [0. we need to clarify the role of second hand sales. This marginal buyer will be indifferent between buying in the first period and waiting. In fact. there cannot be trade between those who choose to buy first and those who decide to wait. The production is costless. If he waits.Problem 18: A monopolist produces a durable good that lasts for two periods (no depreciation). Now that we have established this. Since the q1 who value the good most have already purchased it before. the implied initial total demand is q = 1 − p). because the former always value the good more. we go ahead to solve the game without bothering ourselves with the second hand sales. Keep in mind that 1 − q1 is his value from owning the good for one period. This buyer is indifferent between buying now at p1 to get a surplus of 3(1 − q1 )/2 − p1 and waiting. we will determine the first period demand q1 by finding who is the buyer who values the good least among them. If a consumer purchases the good in the first period. Hence. or may sell it in the second period (hence. that is. As usual we begin from the last period. he will get a surplus of (1 − q1 − p2 )/2. the monopolist can only commit to charging the price that maximizes the profit in this period. The monopoly maximizes the sum of profits and the common discount rate is 1/2. the sales in the second period from those buyers who bought in the first period. respectively. Let p1 and p2 be the first. 30 . a. Hence. 2 Moving on to the first period. we must determine q1 by finding the marginal buyer. which has been discounted to reflect the fact that the purchase occurs later. The monopolist offers the good for sale in both periods. to which we have added the discounted second period value. 1] (thus. Characterize the equilibrium sale price in each period. only those who value it most will decide to buy it rather than wait for a lower price later. p2 ∗ which is maximized at p∗ = (1 − q1 )/2 for a profit of π2 = (1 − q1 )2 /4. she may hold on to it for two periods. There is a continuum of consumers each of whom lives for two periods and desires the good in each period. in the second period the residual demand is q2 = 1−q1 −p2 . Solution: To begin with.and the second-period prices. the monopolist’s problem is max p2 (1 − q1 − p2 ). When the monopolist charges a price in the first period. Given the residual demand. From here we can find all other variables: 1 ∗ p∗ = 5/18 and q1 = 4/9. 2 From here we can directly write the maximization problem for both periods max p1 . Problem 19: A monopolist supplies rides at an amusement park. Normally. what would be the optimal prices? Solution: Now the problem is somewhat different. his demand in the first period is given by an equality similar to part (a): 1 3 (1 − q1 ) − p1 = [1 − q1 − p2 ] 2 2 p2 q 1 = 1 − p1 + . 2 b.Hence. though. p1 which is maximized at p∗ = 25/36. 1 2 Note that given the residual demand q2 = 1/2 − p2 nobody will buy in the second period at p∗ = 1/2. we constrained the monopolist to the only p2 he could commit. 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). q1 can be identified by the equality 1 1 − q1 3 (1 − q1 ) − p1 = 1 − q1 − 2 2 2 4 q 1 = 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 . Before. by restriction of subgame perfection. The demand for rides of each consumer is shown in 31 . it would not be possible for the 2 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. If he could credibly choose another price. If the monopolist could credibly commit in the first period to a price schedule. If the monopolist can commit to any p2 . Taking both first order conditions and solving them simultaneously we get ∗ to the prices p∗ = 3/4 and p∗ = 1/2. p2 π1 + π2 /2 = p1 q1 + (1/2)p2 (1 − q1 − p2 ). the price that maximized that period’s profit. as well as first period demand q1 = 1/2. There is a large pool of identical consumers. The advantage of this policy is that the monopolist does not incur the costs of printing and collecting tickets. what is the entrance fee. per-unit charge. the marginal cost for each ride is c + d. which will be equal to area A. Since we are dealing with a single demand. The number of rides they will demand is denoted by Q2 . so this must be the per-unit charge. because the price of each ride only pays for its cost. The monopolist is considering two pricing policies: Policy 1: An entrance fee and a per-unit charge for each ride. Policy 2: Just an entrance fee. The number of rides demanded under this policy is denoted by Q1 on Figure 13.the figure below. due to the much larger consumer surplus. and d. the park is able to extract all the consumer surplus using the entry fee. The profit under this policy is equal to Π1 = A. The fee will be equal to the sum of areas A + B + D + E + F + G. Hence. the monopolist will make a loss on the rides. Each per-unit cost is independent of the number of rides offered at the park. the cost of printing and collecting tickets at each ride. but will be able to charge a higher entrance fee. The capital letters in the figure represent the areas. 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. 32 . Under policy 2. the cost of supplying the equipment. Using the information given in the graph. The per-unit cost of supplying rides has two components: c. the profit under this policy is Π2 = A + B + D − H. total number of rides supplied per consumer. but running the park will cost E + F + G + H. p2 33 . Problem 20: Consumers are uniformly distributed on an interval of length 1. The marginal consumer will be indifferent between p1 + y/2 = p2 + (1 − y) y = 2(p2 − p1 + 1)/3. Two firms producing this homogeneous good are located at the two end points of the interval. This length will be defined by the marginal consumer who is indifferent between buying at each firm. They simultaneously choose prices for their product.5 per kilometer (the cost of reaching firm 2 remains $1 per km). which is true if B + D > H. This gives Firm 1 a profit of π1 = 1/2. We will denote by x the demand for Firm 1’s product when the transportation costs are equal for both firms’ customers. 1 2 Now for the case when it only costs consumers 1/2 per unit of transport to reach Firm 1. Firm 1’s problem then is max p1 (p2 − p1 + 1)/2. Let y be the demand of Firm 1 in this case. Each wants to buy one unit of a homogeneous good. 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. p1 and it gives the same response function as before p∗ (p2 ) = (p2 + 1)/2. Firm 1’s problem is max 2p1 (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. Since our equilibrium is 1 ∗ symmetric we have p∗ = p∗ = 1.Policy 2 will be more profitable than policy 1 if Π2 > Π1 . p1 which gives us the response function p∗ (p2 ) = (p2 + 1)/2. They pay transportation costs of $1 per kilometer. Calculating these profits is straightforward. Let pi be the price charged by firm i: p1 + x = p2 + (1 − x) x = (p2 − p1 + 1)/2. Firm 2’s 1 problem is given by max p2 [1 − 2(p2 − p1 + 1)/3]. 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. he is indifferent between not buying and (qL . The higher demand is not sufficient to offset the effect of the lower price. Under the new conditions Firm 1 makes a lower profit than under symmetric costs. he is indifferent between (qL . under these conditions. TL ) and (qH . TL ) if θL < θ < θH . All other things equal. where q is the amount offered and T (q) is the payment requested for q units. Assume that there are no production costs. Suppose that the monopolist can offer at most 2 non-linear contracts in the form (q. Therefore the resulting lower p2 will force Firm 1 to charge less in equilibrium. Which constraints bind in equilibrium? Characterize (or discuss if you cannot) the equilibrium. Solution: The consumer will buy iff 10θq ≥ pq ⇔ p ≤ 10θ. 10 b. and not just the myopic choice of an investment to lower transportation costs. So. in an 34 . Solving both response functions 2 simultaneously we arrive at the result: p∗ = 5/6 and p∗ = 2/3. q ≤ 1. So. a. Write down the profit maximization problem. When θ = θL . Suppose the monopolist has to charge a single linear price. 1]. which yields a profit for Firm 1 of π1 = 25/54 < 1/2.and it gives the response function p∗ = (2p1 +1)/4. TL ). The consumer has a maximum willingnes-to-pay equal to 10θq for every q units bought. The consumer buys if her net surplus is nonnegative. That is. Prices are strategic complements in game theory. T (q)). Solution: Consider (qL . Plugging into the 1 2 ∗ demand we find that y ∗ = 5/9. TH ). buys (qL . Problem 21: Consider a monopolist that faces a consumer with unit demand. TL ) and (qH . meaning a lower price from one firm induces a response by a lower price from the other firm (the response functions have positive slopes). TH ) if θ > θH . Alternatively: 1 max p p/10 pdθ = p(1 − p ) ⇒ p∗ = 5. However. This apparently puzzling result can be explained if we consider the strategic interactions between firms. a lower transportation cost for Firm 1 means more customers and a higher profit. and when θ = θH . the monopolist would optimally set p∗ = E[10θ] = 5. and is privately informed about θ. TH ) such that the consumer does not to buy either if θ < θL . The monopolist’s belief for θ is described by a uniform density function over [0. and buys (qH . Find the profitmaximizing price. that is why a sensible investment under naive consideration is not profitable if strategic interactions are taken into account. Firm 2 will decide to charge a lower price. the consumer buys 1 unit if p ≤ 10θ. Two firms are located at the two end points. The maximand is linear in q. 5). T ∗ ) = (1. The solution will be the same as before.. Integrating gives T (q(θ)) = 10θq(θ) + c. Solution: Let the non-linear (strictly increasing) function T (q) describe the contracts the monopolist offers. c = 0. 10(qH − qL ) Furthermore. so the monopolist would want to set q = 1 for everyone. i. Transportation costs are linear and cost t per unit of distance. A customer with income d who buys one unit of a good of quality q has the utility qd. An individual located at point d (distance d from the point 0) has income d. because the type θ = 0 has zero willingness-to-pay. Let θ > 0 be the indifferent consumer type. of the interval. Assume that t > q0 − q1 > 0 and 35 . From part (a).e. it must be (q ∗ . This consumer type will have zero surplus ˜ ˜ in equilibrium. (1 − qL )2 qL But this is possible only if θH = θL . 0 and 1. Which constraints bind in equilibrium? Carefully explain. it is straightforward that qL < qH = 1. T (q(θ)) = 10θq(θ). Hence. i. which means that the monopolist offers only one contract.e. their products differ in quality: q0 > q1 . 5). Write down the profit maximization problem. so.TH 2 (TH − TL )2 TL [qL ] : − + 2 = 0. c. T (q) must be continuous (why?). the monopolist will offer only one contract (q ∗ . T ∗ ) = (1. Problem 22: Customers are uniformly distributed on a Hotelling interval [0. Hence. Firms have no production costs.equilibrium. suppose that the monopolist is unconstrained in the number of contracts it can offer.TL . 10qL TH − TL . The reason is the linearity of the consumer’s utility function. the maximization problem is: max [(1 − θH )TH + (θH − θL )TL ] = TH − T2 (TH − TL )2 + L 10(1 − qL ) 10qL qL . The consumer chooses: q(θ) = arg maxq (10θq − T (q)). we must have: 10θH qH − TH = 10θH qL − TL ⇒ θH = 10θL qL − TL = 0 ⇒ θL = TL . The monopolist never sells to all types of the ˜ consumer. so T (q(θ)) = 10θ. 1]. Now. the maximization problem is: 1 1 max T (q) ˜ θ T (q(θ))dθ ≡ max q(θ) ˜ θ 10θq(θ)dθ. 36 .find the prices that the two firms charge in equilibrium. Undercutting may be the cause of non existence of equilibrium if there is a discontinuity of payoffs. Write the profit functions of the two firms and derive the Nash Equilibrium. Note that customers close to 0 DO NOT buy. Solution: For given prices find the marginal consumer.Here there is no discontinuity: The payoff of a firm gradually and continuously shrinks as the price of the other falls. Discuss the possibility of undercutting in this model. The market share of the firm located at 0 should be adjusted and trimmed.