2 b. 2 d 4,7 2, 1 2,3 10,3

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1 Problem Set 3 Do problem 4a or 4b or 4c, not all three (1.) Backwards Induction and Subgame Perfection: Find all subgameperfect equilibria of the following games. Explicitly write down the behavior strategies that support a subgame-perfect equilibrium (don t just point out the equilibrium path through the game). Are there any Nash equilibria that aren t sub-game perfect? (a.) a L b 1 R c d 10,3, 1,3 4,7 (i.) First, let s rewrite it as a simultaneous-move game. First, how many behavioral strategies does have? He can use a against L and c against R, b against L and c against R, a against L and d against R, b against L and d against R: ac ad bc bd L 10,3 10,3, 1, 1 1 R,3 4,7,3 4,7 So this game ends up having three Nash equilibria: (L,ac), (L,ad), and (R,bd). But this is equivalent to saying, if both players chose strategies simultaneously and independently, what are possible candidate equilibria? Now, let s use our information about the timing to find the Subgame Perfect equilibria. Start at the bottom of the tree and work backward. If one played L, two is choosing between a and a payoff of 3, or b and a pay-off of 1, so that two chooses a. If one played R, two gets to pick from c and 3 or d and 7; so two chooses d. Anticipating all this, player one gets to choose from L and a pay-off of 10, or R and a pay-off of 4; so one chooses L. Then our sub-game perfect equilibrium is (L, ad) with an equilibrium path prediction of one playing L followed by two playing a. You can see that if we said the equilibrium strategies are (L,ac), this is not subgame perfect. In particular, if player two ended up at the node where she had to choose between c and d, she would never choose c, since d provides a strictly better pay-off. That is why we throw this Nash equilibrium out. Likewise, if we said the equilibrium strategies are (R,bd), it hinges on a non-credible threat: Player two is saying she will use strategy b if one plays L. That would give her a pay-off of -1 instead of the 3 for using a, so two would never actually play b. The only reason this equilibrium is a mutual best response in a simultaneous move game is because we threw out the information about timing. Now

2 that one knows two would never use b when a is available, we can safely rule it out. In other words, one can force two to choose between a and b, knowing two prefers a. (b.) a L b 1 R c d 0,10 3, 3,1 6, Again, the strategic form is: ac ad bc bd L 0,10 0,10 3, 3, 1 R 3,1 6, 3,1 6, We get multiple Nash equilibria: (R,ac), (R,bc), (L,ad). Solving by backwards induction, So only (R,ac) is subgame perfect. The other two are Nash in the sense that if player two could commit to playing irrationally, going along with the equilibrium profile would be a best-response for player one, but believing such non-credible threats is ridiculous. (R, bc) has the same equilibrium path as (R, ac), but involves player two claiming she would use b against L, for a pay-off of instead of 10 so, it isn t a subgame perfect equilibrium, since it involves sub-optimal play in one of the subgames. Likewise, (L,ad) relies on player two threatening to use d in response to R; this would give a pay-off of - for d instead of 1 for playing c so, this is irrational in one of the sub-games, so it isn t sub-game perfect. (c.) a L b 1 R c d 10,3 1,3,1 5,4

3 Repeat the same analysis from part i. Strategic form: ac ad bc bd L 10,3 10,3 1,3 1,3 1 R,1 5,4,1 5,4 Now, we have three Nash equilibria. However, player two is now indifferent about her pay-off in the subgame after one plays L. Let s try backwards induction: If one plays R, two certainly plays d, since 4 > 1. If one plays L, two might play a or b; there s no way to throw either possibility out. Then and specify two sub-game perfect equilibria. In particular, we throw out (L,ac), because it hinges on player one believing that player two would use c against R. You can see this is irrational; if one plays R, two only hurts herself by using c when d is available, since 4 > 1. Again, we ve used backwards induction and subgame perfection to throw out an unreasonable Nash equilibrium. (). Suppose there is a single manufacturer, M and two retailers R 1 and R. First, the manufacturer posts a price h for its good; Second, the two retailers observe h and simultaneously place orders q 1 and q for the manufacturer s good(the two retailers participate in a simultaneous Cournot market, where their marginal cost is h). The market price of the good is p(q 1,q ) = A (q 1 +q ). The manufacturer s total costs of production are C(q 1 +q ) = c (q 1 +q ). (a). Solve for a subgame perfect Nash equilibrium of the game. What are the firms profits? What is the market price and market quantity? We use backwards induction. In the final period, the two retailers are behaving as Cournot duopolists, so we need to solve the Cournot game. (1) The firm s payoffs are π 1 = (A q 1 q )q 1 hq 1 π = (A q 1 q )q hq () Maximizing gives best-response functions π r1 = A q 1 q h = 0 q 1 = A q h q 1 π r = A q q 1 h = 0 q = A q 1 h q

4 (3) Solving the best-response functions simultaneously yields the equilibrium q 1 = q = A h 3 So the total order received by the manufacturer is Q(h) = q 1 +q = 3 (A h) Now we take a step back. The manufacturer can predict the consequences of choosing different values of h, and solves or Maximizing gives maxq(h)h cq(h) = max Q(h)(h c) h h π m = max h 3 (A h)(h c) π m h = A h h+c = 0 h = A+c So the subgame perfect Nash equilibrium is that the manufacturer chooses h = (A+c)/, and the retailers then each order (A h)/3. The price, then is p = A (/3)(A h ) = (A+c)/3. Then each retailer orders a quantity q (h) = (A c)/6, for Q (h) = (A c)/3 Each retailer s profits, then, are π r = p q i h q i = (p h )q i = (A c) 36 The manufacturer s profits are π m = A c (h c) = (A c) 3 6 (b). Suppose the two retailers merge into a single firm, or horizontally integrate, so that there is a single manufacturer and a single retailer. Is this profitable for the retailers (in the sense that the joint profits are higher after the merger)? If the retailers merge, they are a single firm deciding how much to order as the last move in the game: max q (A q)q hq 0 = A q h q (h) = A h

5 Then the manufacturer, knowing how the retailer will behave, maximizes max h q (h)(h c) = max h A h (h c) This leads to the same price, h = (A + c)/. The SPNE then is that the manufacturer posts a price of h = (A+c)/, and the retailer orders q (h) = (A h)/ units. The retailer s profits are ( A c (A q (h ) h )q (h ) = 4 Since(A c) /16 ((A c) /36) = (A c) /18, themergerisprofitable for the retailers. This improves their bargaining position against the retailer, in particular. The price in the product market is p = (3A + c)/4, and the total quantity is Q = A c 4 ) (c). Suppose the manufacturer decided to vertically integrate by buying one of the retailers and refusing to sell its good to the other retailer. Is this profitable for the manufacturer and integrated retailer, in the sense that the vertically integrated monopolist generates more profits than the manufacturer and a single retailer make in the presence of the other firm? (Or, can the manufacturer and retailer make more money by cutting the other retailer out of the loop?) Now, the integrated firm is a monopolist, so it maximizes max q {(A q)q hq}+{hq cq} = max(a q)q cq Maximizing gives the quantity q = (A c)/ and price p = (A+c)/. Is this more profitable than part (a)? Well the total profits for the monopoly firm here are (A c) /4. The profits for a monopolist and one of the retailers in part (a) are (A c) 36 + (A c) 6 q = 7(A c) 36 And since 7/ <.5, the integrated firm makes more profits than a monopolist and one retailer. Why does this matter? Because if the retailer and monopolist merge and cut out the other retailer, this move has to profitable for both parties. If profits don t increase enough to be able to compensate both parties, then they won t want to merge. (d). In class, we looked at the case with a single retailer and concluded

6 that welfare for consumers will improve from vertical integration, since price goes down and quantity goes up. Are the same conclusions true in b and c here? When does consumers welfare go up or down? Briefly explain your results. So let s summarize... P A A+c 3 B 3A+c 4 C A+c Q A c 3 A c 4 A c Now, we can order the quantities and prices. For consumers, the industry in part (c) wins, since it has the highest quantity and lowest price, then part (a), then part (b). So the monopolist is unambiguously the best, followed by the monopolist/two retailers market, followed by the monopolist/one retailer market. (3). There is a firm that produces specialized machines, which require a substantial investment in learning the details of the client s business; think of industrial sewing machines or equipment for physics laboratories. First, the firm invests effort e in producing the machine, at cost c e. The client s profits are increasing in the effort put into the machine, with π(e) = ae. Second, the client proposes a trading price p. If the firm refuses to sell the machine at the proposed price, the firm can sell the machine on the market for a payoff be, b < a. If the client has no machine, the client gets a pay-off of r, with r small. (a.) Draw an extensive form representing this game and solve for a subgame perfect Nash equilibrium. We sketched an extensive form for a game like this in class. Ask me if you want to see it drawn out again. We solve using backwards induction: In the final period, the firm should accept if p c e be c e or p be. The firm should reject otherwise. When deciding what offer to make, the client is trying to maximize maxae p subject to p be. Since the client s payoff is decreasing in p, it should be made as small as possible, or p = be.

7 Then the firm, when trying to decide how much effort to exert, is maximizing π(e) = maxbe c e e The condition for maximization is Then the SPNE is dπ(e) de The firm chooses e = b/c The client offers a payment p = be = b ce = 0 e = b c The firm accepts if p be and rejects otherwise. (b.) Solve the social planner s problem: max(ae p)+(p c e e ) and thereby find the efficient level of effort. Is it greater or less than the subgame perfect Nash level? Explain your results. The efficient level of effort is a/c. This is strictly greater than b/c because a > b, so there is a suboptimal level of investment in the equilibrium of the dynamic game. Thereason is that not all of the returnsto the effort accrue to the firm: A large portion go to the client, since the client s payoff is (a b)e = (a b)b/c. This reduces the firm s incentive to exert effort, and an inefficient investment results. (d.) Does the equilibrium choice of effort, e, depend on the client s outside option, r? Is this realistic? Briefly explain. It doesn t depend on r as long as r is sufficiently small, or r < (a b)b/c. Is this realistic? Yes: The players are bargaining over the surplus generated by providing the machine to the client, so the client s outside option should only matter to the extent that the trade is worthwhile for both parties. At that point, it is merely a question of the two parties bargaining power against each other, with the firm potentially walking away with the specialized machine. (4a). Imagine an alternating-offer bargain game, but instead of discounting future payoffs by δ and keeping the size of the pie fixed at 1, assume that in each period the value of the pie is reduced by one In the first

8 period, v 1 = V, in the second v = V 1, in the third v 3 = V, etc., and there is no discounting. (a). Find the subgame perfect Nash equilibrium for V = 3 and V = 4. If V = 4, there are only four times at which trade can happen before the pie vanishes. In the fourth offer period, there is once slice left, and player is making an offer to player 1. Player should offer player 1 nothing, and keep the whole pie. Since player 1 is indifferent between accepting and rejecting, he accepts, and the game ends. In the third offer period, there are two slices left, and player 1 is making an offer to player. Player 1 should offer player one slice, since we know that player can get a slice by rejecting now and waiting to make an offer next period. So player 1 gets a slice and player gets a slice. Otherwise, player rejects. In the second offer period, there are three slices left, and player is making an offer to player 1. Player should offer player 1 one slice, since we know that player 1 can get a slice by rejecting now and waiting to make an offer next period. So player gets two slices and player 1 gets one slice. Otherwise, player 1 rejects. In the first offer period, there are four slices left, and player 1 is making an offer to player. Player 1 should offer player two slices, since we know that player can get two slices by rejecting now and waiting to make an offer next period. So player 1 gets two slices and player gets two slices. Otherwise, player rejects. If V = 3, the work is the same as the final three offer periods above, but switching the names of player 1 and player. (b). Solve for the subgame perfect Nash equilibrium for arbitrary V (you can guess at the solution, then explain why you think it is true). If V is even, t 1 = V and if V is odd, t 1 = V +1 The reason is that as V grows, the players are basically tacking on an extra period to the front of the case studied in (a). If we add a slice to an odd period, the pie becomes even, and the players should split it equally because delays lead to lost slices of pie. If we add a slice to

9 an even period, we get an odd pie, and there is essentially once slice up for grabs. If players split it equally in even periods, a rejections of t 1 = V/+1 leads to having V/ pie in the next period for the rejector, which is the same as if he had accepted today. (If you really wanted, you can prove this formally by mathematical induction on V = 3,4,...: Start with V = 3 as the odd base case and V = 4 as the even base case Suppose that it is a SPNE to offer t 1 = V/ if V is even and t = V/ + 1 if V is odd. Then show it is an SPNE to offer t 1 = V/+1 if V is even and t 1 = V/ if V is odd. This is easy to do, since we just do one step of the kind of work we did in part (a). This proves that the induction hypothesis holds. Since the claim holds for V = 3,4, and we can show the claim holds for any V +1 starting from any V, then this is an equilibrium for all V.) (4b). As a regulator, you might be worried about predatory investment: When an incumbent firm over-invests to keep entrants out of the market. But what does predatory investment even mean? In our model, we have an incumbent firm i and an entrant e. First, the incumbent chooses either to invest in cost-reducing research that reduces its marginal costs to zero at a cost of 5, or no investment in cost-reducing research, in which case its marginal cost is. Second, the entrant observes the investment choice and decides whether to pay a fixed cost 1/ to start its business and enter the market with a marginal cost of, or not. Third, if entry occurs, they become Cournot duopolists. If entry does not occur, the entrant gets a payoff of zero, and the incumbent becomes a monopolist. The market price is p(q e,q i ) = 5 q e q i. The total costs for the incumbent if it invested is 5; the total costs for the incumbent if it didn t invest are q i. The total costs of production for the entrant if entry occurs are q e +1/. (a). Solve for the firms equilibrium choice of quantity in the Cournot Duopoly, given that the entrant chose to enter (note that this depends the investment level of the incumbent). Solve for the incumbent s monopoly strategy, given that the entrant chose not to enter (note that this depends the investment level as well). Compute the profits of the firms in each scenario. The incumbent s profits from no investment and entry are π i ne = (5 q e q i )q i q i

10 and the entrant s profits from no investment and entry are πne e = (5 q e q i )q e q e 1/ The equilibrium strategies are qne i = qe ne = 1 The firms profits are πne i = 1,πe ne = 1/ The incumbent s profits from investment and entry are πie i = (5 q e q i )q i 5 and the entrant s profits from no investment and entry are πie e = (5 q e q i )q e q e 1/ The equilibrium strategies are The firms profits now are q i ie = 7 3,qe ie = 1 3 π i ie =.4444,π e ie =.967 The incumbent s profits as a monopolist with no investment and no entry π i nn = (5 q i )q i q i The equilibrium strategy requires The incumbent s profits are q i nn = 3 π i nn = 9 4 =.5 The incumbent s profits as a monopolist from investment and no entry are π i in = (5 q i )q i 5 The equilibrium strategy is The incumbent s profits are q i in = 5 π i in = 5 4 = 1.5

11 (b). Solve for a subgame perfect equilibrium of the entry game, using the payoffs you computed in part (a). What happens in equilibrium? Here s an extensive form: Inv Ent Enter Don t Inc No Inv Ent Enter Don t.44, , 0 1,.5.5, 0 The unique SPNE is that the incumbent Invests, and the Entrant (i) Doesn t enter if the incumbent invested and (ii) Does enter if the incumbent doesn t invest. So by investing in cost-reduction, the incumbent can keep the entrant out of the market. (c). Consider the following definition: Investment is predatory if (i) the incumbent chooses a level of investment in the presence of the potential entrant that results in the entrant staying out of the market and (ii) in the absence of the potential entrant, the incumbent would have chosen a lower level of investment. Does predatory investment occur in our model? Is this a good definition of predatory investment? Identify one practical or theoretical problem with using it, and briefly explain either how your problem can be solved or why it is impossible to fix the definition. Should predatory investment be illegal? Briefly explain. Predatory investment does occur in our model. This is a good definition because it captures the idea that in the absence of the entrant, the incumbent would have chosen a much lower level of investment, so the only explanation for the investment is that it frightens away potential competition. This doesn t necessarily benefit consumers, because competition may have lead to lower prices and higher quantities than the over-investment monopolist chooses; in these cases, firms should be prosecuted. However, it is difficult to tell both (i) whether the incumbent would have invested less in the absence of an entrant, (ii) whether a particular entrant is serious about going into a market, since you don t want to encourage spuriouslaw suits, (iii) whether or not the firm would have been more profitable without making the investment. These are all very hard questions, and the best we can usually do is formulate a model like the one above, estimate the parameters, and ask what if questions. These are called structural models, and are sometimes used in court cases. For example, one of my advisors testified in a court case where two firms were accused of colluding by reneging on plans to build

12 new factories to keep themselves weak, and thereby able to charge high prices to the plaintiff. So these kinds of issues do come up in real life, and game theory is used to analyze them. (d). Briefly explain how investment in this model can behave as a commitment device for the incumbent to be more aggressive in dealing with any potential entrants. Because we have a game of strategic substitutes (Cournot), investing lowers one firm s marginal cost, which means that in the new equilibrium, the firm that invested will produce more and the firm that did not will produce less (the entrant). Since the entrant has a fixed start-up cost, if the incumbent can invest and drive down the entrant s market share enough, it will no longer be profitable to enter. In this sense, the investment acts as a commitment device because it shifts the market shares and allows the incumbent to credibly threaten to play hard ball with the entrant. (4c). We mentioned that if the buyer privately knew his value of the good, the seller would have a more difficult time capturing all the value as happens in the dictator game. This question explores this idea. Thereisasinglebuyerandasingleseller. Thebuyer svalueisdistributed uniformly between zero and 1, so Pr[v b < x] = F(x) = x and Pr[v b > x] = 1 F(x) = 1 x. The buyer knows her value, but the seller does not. The seller has zero value for the good. The buyer s utility is u b = δ k (v t) if they trade at price t at date k, and zero otherwise. The seller s utility is u s = δ k t if they trade at price t at date k, and zero otherwise. There are potentially two trading dates: k = 0,1. (a). If the seller gets to make a single offer to the buyer announces a t which the buyer can accept or reject what should it be? What is the probability that trade occurs? Trade occurs only if v t, so the probability trade occurs is Pr[v t] = 1 t. Then the seller s expected profits are max t Pr[v t]t = max(1 t)t t so t = 1/. Then probability trade occurs is 1/ as well. (b). If the seller gets to make two offers t 0 and t 1 and both players discount the second trade by δ as described above, what should the offers be? Think of it like this: There is a range of values [ˆv,1] for which the buyer accepts right away, and a range [ṽ,ˆv] where the buyer accepts in the second

13 period, and a range [0,ṽ] for which the buyer rejects both offers (why?). So, there is an agent ˆv for whom ˆv t 0 = δ(ˆv t 1 ), and an agent ṽ for whom δ(ṽ t 1 ) = 0. Solve backwards. and or Solving for ˆv and ṽ yields ṽ = t 1 ˆv = t 0 δt 1 1 δ You could just write out the seller s discounted profits and maximize: max t 0,t 1 (1 ˆv)t 0 +δ(ˆv ṽ)t 0 ( max 1 t ) ( 0 δt 1 t0 δt 1 t 0 +δ t 1 )t 0 t 0,t 1 1 δ 1 δ But this is actually pretty ugly to solve. If you solve by backwards induction, the seller in the second period is solving ( t0 δt 1 max t 1 )t 1 t 1 1 δ where t 0 is a constant that was already determined. Maximizing yields the optimal t 1 : t 1 = t 0/ I m not kidding, it s really that simple. Then in period one, the seller is maximizing ( max 1 t ) ( 0 δt 0 / t0 δt 0 / t 0 +δ t 0 1 δ 1 δ Maximizing yields the optimal t 0 : t 0 = 1 δ δ ) t 0 / t 0 This implies that t 1 = 1 δ ( δ) (c). How do your equilibrium t 0 and t 1 depend on δ? As δ 1, what happens to t 0 and t 1?

14 As δ 1, t 0 = 0/ and t 1 = 0/. The price goes to zero in both periods (or marginal cost, since marginal cost is zero here). (d). Does the seller make more profits from making two offers as δ 1, or a single offer? Coase conjectured that as the time between offers went to zero, a monopolist would be unable to maintain high prices, and would end up with low profits. What do you observe here? That s what we see here. Since the monopolist is competing against his future self, letting δ 1 erodes all of his market power. It becomes like a Bertrand game where we have firm 1 at date 0 and firm at date 0 + (1 δ), and as δ 1, they re essentially competing at the same time. This is a funny result because you d think as δ 1, it would approximate the monopoly outcome with only one period of trade, but instead, you get intra-self/inter-temporal competition that leads to the perfectly competitive outcome. Crazy, right?