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Econ 210, Microeconomic Theory HW 8 - Answers Auctions and Price Discrimination Professor Guse November 12, 2015 1. First-price sealed bid auction. Assume there are n bidders. Each bidder i has a true value, vi distributed uniformly on [0, 1]. Each bidder knows her own value but does not know the value of the other bidders in the game. (a) When there are only two bidders, show that there is an equilibrium in linear strategies. In other words, show that if player 2 employs a strategy with the form b2 (v2 ) = a2 v2 , that bidder 1’s best response is to employ a similar strategy. ANSWER: Player 1’s problem is to maximize expected consumer surplus max(v1 − b1 )P r(b1 > b2 ) b1 substituting in the assumed form for player 2’s strategy for choosing b2 we can rewrite this as max(v1 − b1 )P r(b1 > a2 v2 ) b1 ⇔ max(v1 − b1 ) b1 b1 a2 Using the product rule to differentiate with respect to b1 , we get a first order condition for an optimal choice of b1 : (v1 − b1 ) ⇔b1 = 1 v1 2 1 b1 − =0 a2 a2 (b) Solve for the linear equilibrium in the n = 2 case and calculate the expected revenue generated by the auction in that equilibrium. ANSWER. As we just saw, there is a linear equilibrium in which both players bid half of their true value for the object. To calculate the expected revenue we need to calculate E(Rev) = E(max{ v1 v2 1 , }) = E(max{v1 , v2 }) 2 2 2 In other words, since, in equlibrium, both player bid half their true value, the highest bid will be half of the highest true value. So how do we calculate the expected value of the maximum of a pair of random uniformly distributed variables? To answer this question, we need to review what an expectation is. By definition the expected value of a random variable x whose density is described by f (x) is given by E(x) = Z ∞ xf (x)dx x=−∞ We want E(max{v1 , v2 }) = = Z 1 1 Z max{v1 , v2 }dv1 dv2 v1 =0 v2 =0 Z 1 Z 1 v1 =0 v2 =v1 v2 dv1 dv2 + Z 1 v1 =0 Z v1 v1 dv1 dv2 v2 =0 Note that in the last step we just decomposed the integral into the lower and upper triangles of the square. The first term represents all the cases where v2 > v1 and the second case represents all the cases where v2 < v1 . Note also that even though they look a little different they have to evaluate to the same value since by symmetry the expected value of v1 when its bigger than v2 cannot be different that the expected value of v2 when its bigger than v1 . Therefore we really need only compute one of the terms and then multiply it by two for our answer. Nevertheless I will explicitly evaluate both terms. 2 E(max{v1 , v2 }) = Z 1 v1 =0 Z 1 Z 1 v2 dv1 dv2 + v2 =v1 1 (1 − v12 )dv1 + E(max{v1 , v2 }) = 2 v1 =0 1 1 1 = (1 − ) + 2 3 3 2 = 3 Z Z 1 v1 =0 Z v1 v1 dv1 dv2 v2 =0 1 v12 dv1 dv2 v1 =0 Since the bid will be half of the max value, the expected revenue will be n = 2. 1 3 for (c) Repeat the above two questions for the general case of n bidders. ANSWER Again consider Player 1’s problem when facing off against linear strategies b2 (v2 ) = av2 , b3 (v3 ) = av3 , ... (Note that we assume that Players 2, 3, ... n following the same strategy.) max(v1 − b1 )P r(b1 > b2 &b1 > b3 ...) b1 Assuming independence and plugging in the other player’s strategic forms we get P r(b1 > b2 &b1 > b3 ) = P r(b1 > av2 ) × P r(b1 > av3 ) × ... = b1n−1 an−1 Plugging this into 1’s objective function, differentiating it w.r.t. b1 and setting it equal to 0, we get the FOC: − Dividing thru by b1n−2 an−1 b1n−1 (n − 1)b1n−2 + (v − b ) =0 1 1 an−1 an−1 we get −b1 + (v1 − b1 )(n − 1) = 0 (n − 1) v1 b1 = n 3 So, 1’s best response when everyone is playing the av strategy is to play a linear strategy and specifically one where she bids n−1 n times her true value. Therefore everyone employing such a strategy consititutes an equilibrium. To calculate n the expected revenue, one needs to recognize that E(max{vi }ni=1 ) = n+1 . (This follows by using the logic of integration we applied for the case of n = 2 above). Therefore the expected revenue will be n−1 n n−1 = n n+1 n+1 Note, therefore that expected revenue increases toward 1 as n increases for two reasons. First, as the auction becomes more competetive, bidders bid closer to their true value. Second, the max true value drawn will on average be higher the more true values are drawn. 2. Second Degree Price Discrimination. Suppose there are two types of consumers ,L and H, with demands for donuts given by the following inverse demand equations. P L (y) = 100 − y P H (y) = 150 − y Assume that there is one firm that serves this market and that the marginal cost of making donuts is zero. (a) If the firm pursues a strategy of setting one price, what price should it set? How much revenue will it earn? ANSWER Since cost is zero, this is a revenue maximization problem. In order to solve it, we need to derive a revenue function. But first we need an aggregate demand function. Horizontally adding the two demand curves above (note they are presented in their vertical forms), we have y Agg (p) = y L (p) + y H (p) = max{100 − p, 0} + max{150 − p, 0} = max{250 − 2p, 150 − p, 0} 250 − 2p if p < 100 150 − p if 100 < p < 150 = 0 if p > 150 Therefore revenue as a funciton of price is 4 R(p) = y Agg (p) × p 250p − 2p2 if p < 100 150p − p2 if 100 < p < 150 = 0 if p > 150 Marginal revenue is M R(p) = y Agg (p) × p 250 − 4p if p < 100 150 − 2p if 100 < p < 150 = 0 if p > 150 Normally the first order condition for maximizing revenue would be M R = 0. However we must be cautious when working with a discontinuous function. One thing we can say is that M R = 0 in the lower segment of the price ranges at p = 250 4 = 62.5, so we can say that this is the best price to set of all the prices between 0 and 100. The firm’s revenue at that price would be R(62.5) = 7812.5 However we need to check whether we can achieve a higher level of revenue in the upper range - between 100 and 150. Marginal revenue is negative at 100 and just gets more negative at higher prices. Therefore in that price range the solution is a corner; it’s 100 and at that price, revenue would be 5000. Therefore the revenue maximizing price is 62.5. (b) If the firm pursues a strategy of marketing two different sized donut packs, what would be the optimal sized donuts packs and the prices for them? Would this strategy make more revenue than the single-price approach?ANSWER Following the example in Varian, the optimal size of the the large package would be 150 donuts (where the MWTP is zero for the high demand group), while the optimal size for the smaller package is the quantity at which the MWTP of the High demand group is exactly twice the MWTP of the Low demand group: P H (y) = 2P L (y) ⇔150 − y = 2(100 − y) ⇔y = 50 Again following the example in Varian, the menu would look as follows 5 size 50 150 price 3750 8750 where 3750 is the area under the L-type’s MWTP curve beteen quantities of 0 and 50 and 8750 is that same area plus the area under the H-type’s MWTP curve between quantities of 50 and 150. 3. Suppose that demand for water by each industrial users (I) is M W T PI (QI ) = .5 − .00005QI While demand for water by each residential user (R) is M W T PR (QR ) = 1 − .01QR There are 1 industrial user and 100 residential users. The fixed cost to build a water supply system to supply both groups is 5000, the marginal cost of supplying the water, once the system is built is $.01 per gallon. If each case below, say what water consumption levels for each group would be as well as profit for the water company. (a) If the water company were allowed to charge a single per-unit price. What would it charge to maximize profit? ANSWER. First find aggregate demand curve by re-writing each demand function according to its horizontal interpretation. (i.e. in terms of quantity)... .5 − P = 10000 − 20000P .00005 1−P = 100 − 100P QR (P ) = .01 QI (P ) = Now add these together multiplying the residential by 100 (since there are 100 of them) to get aggregate demand. Q(P ) = QI (P ) + 100QR (P ) = 20000 − 30000P Invert this to get aggregate marginal willingness to pay. 6 ∀P ≤ .5 (1) P (Q) = 20000 − Q 30000 (2) Total revenue is price × quantity and marginal revenue is the first derivative of TR w.r.t Q Q(20000 − Q) 30000 20000 − 2Q ⇒M R(Q) = 30000 T R(Q) = Setting M R equal to M C we get the profit maximizing quantity to provide for a sigle price monopoly which is about 10,000 gallons. 20000 − 2Q = .01 30000 ⇒20000 − 2Q = 300 19700 ∼ 10000 ⇒Q = 2 Plus back into equation ??, the inverse of the aggregate demand curve, to find the price that would be set so that this quantity is demanded Psp (Q = 10000) = ⇒Psp = 20000 − 10000 30000 1 3 (b) If the water company could charge different per-unit price to industrial users and residential users, what would these prices be? ANSWER. Since marginal cost is constant we can simply set each individual marginal revenue equal to the marginal cost. Notice that since we have linear demand, the MR curves are just the individual MWTP curve with twice the vertical slope. First do industrial customers M W T PI (QI ) = .5 − .00005QI ⇒M RI = .5 − .0001QI Setting M RI equal to M C, we get 7 .5 − .0001QI = .01 .5 − .01 ⇒QI = = 4900 .0001 ⇒PI = .5 − .00005(4900) = .255 while for residental M W T PR (QR ) = 1 − .01QR ⇒M R = 1 − .02QR .99 = 49.5 .02 ⇒ PR = 1 − .01(49.5) ≃ .50 ⇒ QR = So under a single price monopoly, both industrial and residential customers would pay about 33 cents per gallong, but if the monopoly can discriminate between residential and industrial then it would charge the industrial about .25 per gallon and residential customers about .50. (c) Suppose the local community will not agree to the new water system unless, the firm charges only the marginal cost for each unit of water. What does the company tell the community? ANSWER At .01 per gallong total demand according to equation ?? woudl be Q(P = .01) = 20000 − 30000(.01) = 19700 At a price of .01, total revenue would be $197. However costs would be 5000 + .01(19700) = 5197, hence the firm would be losing 5000 - its entire fixed cost. Therefore at that price, this firm would exit in the long run (or never enter to begin with). (d) Suppose that the community does not like what it hears. Instead it decides to let the water company charge a single price which is just high enough that it will not require a subsidy. What price do they dictate? ANSWER. First calculate average cost , AC... AC = Now set AC = MWTP. 8 5000 + .01Q Q (3) ⇒ ⇒ ⇒ ⇒ 5000 + .01Q 20000 − Q = Q 30000 150, 000, 000 + 300Q = 20000Q − Q2 Q2 − 19700Q + 150, 000, 000 = 0 p 19700 ± 197002 − 600, 000, 000 Q= 2 p 19700 ± 197002 − 600, 000, 000 Q= 2 It turns out that that there is no real-valued solution since the expression inside the root sign is negative. This can be interpretted as, “ the demand curve never intersects the average cost curve”. In fact, it turns out that this company makes zero profit in parts (a) and (b) as well. The only way for this water company to make a positive profit would be to charge fixed “service” charges either instead of or in addition to a per-unit charge of .01. For example charging the industrial customer a $1000 “hook-up” fee and each residential customer a $40 “hook-up” fee in addition to charging everyone $0.01 per gallon. Doing this, the company would break even and each customer would willingly hook-up and still derive some consumer surplus. (e) Come up with a pricing plan that lowers prices to consumers and still allows the water company to break even. ANSWER. For example, one could increase the hook-up fee charge to the industrial customer. 9