Game Theory with Applications to Finance and Marketing, I

Transcription

1 Game Theory with Applications to Finance and Marketing, I Solutions to Homework 1. (A Strategic Role of Futures Contracts) Consider example 1 in Lecture 1, part I, where firms 1 and can costlessly produce a product and engage in Cournot competition with the inverse demand being, in the relevant range, P (q 1 + q ) = 1 q 1 q. This problem is a modification of the above Cournot game. (i) Assume that there are two dates. The two firms will compete at date 1, but at date 0, both firms can correctly expect the date-1 inverse demand function, which is the P ( ) defined above. At date 0, the futures market opens for the product produced by the two firms. There are price-competitive investors in the futures market, who, just like the two firms, are risk neutral without time preferences (that is, there will be no discounting for anyone). The extensive game is as follows. At date 0, (only) firm 1 can sign a futures contract with the competitive investors. In the futures contract, firm 1 promises to deliver f 1 units of the product at date 1 to one of the investors (say, Mr. A), and Mr. A promises to pay the price F (referred to as the date-0 futures price of the product). We assume that firm 1 announces f 1, and the competitive investors then determine the futures price F. Assume that investors have ratinal expectations; that is, upon seeing f 1, they can use backward induction to anticipate the date-1 price of the product (called the date-1 spot price of the product), and to rule out arbitrage opportunities, in the date- 0 equilibrium, F must equal the anticipated date-1 price P (q 1, q ) so that Mr. A would get zero profits from futures trading. At date 1, upon seeing firm 1 s date-0 futures contract (f 1, F ), the two firms choose q 1 and q simultaneously. Then, after firms set q 1 and q, firm 1 must deliver f 1 units of the product to Mr. A, and Mr. A must pay firm 1 F f 1 dollars. 1

2 Then, consumers arrive, and they purchase f 1 units of the product from Mr. A, q 1 f 1 units from firm 1, and q units from firm. Since consumers purchase q 1 + q units in total, the date-1 spot transaction price is P (q 1, q ). Mr. A s profit is then [P (q 1, q ) F ]f 1. Firm 1 s profit as a function of q 1, q is Π 1 (q 1, q ; f 1 ) = [1 q 1 q ][q 1 f 1 ] + F f 1. Firm s profit function is still Π (q 1, q ) = [1 q 1 q ]q. Find the SPNE of this extensive game. Explain why firm 1 may benefit from futures trading. 1 (ii) Now, suppose that both firms can engage in futures trading at date 0, with f 1 and f units sold respectively at the futures price F determined at date 0. Again, assume that all investors in the futures market have rational expectations when they compete in price to determine F. Re-derive the SPNE. Explain why the two firms might be hurt by the availability of futures trading. 1 Hint: Use backward induction. First consider the date-1 subgame with f 1 given. This is just a Cournot game with the two firms profit functions being Π 1 and Π specified above. Let the subgame equilibrium be (q 1(f 1 ), q (f 1 )), which depends on f 1. Now move backwards to consider firm 1 s date-0 choice of f 1. Remember that the investors in the futures market can rationally expect the date-1 spot price of the product, which is P ((q 1(f 1 ), q (f 1 )), and given f 1, they will compete in price so that in the date-0 futures market equilibrium, F = P ((q 1(f 1 ), q (f 1 )). Given that F = P ((q 1(f 1 ), q (f 1 )), find firm 1 s optimal f 1. Hint: Again, consider the date-1 subgame with f 1, f given. Now for i = 1,,, firm i s profit function becomes Π i (q i, q j ; f i ) = [1 q i q j ][q i f i ] + F f i. Find the Nash equilibrium (q1(f 1, f ), q(f 1, f ))for this subgame. Now return to the date-0 futures market, where the two firms must simultaneously choose f 1 and f. For each pair (f 1, f ) announced, the investors can correctly expect the date-1 spot price, which must be P ((q1(f 1, f ), q(f 1, f )). Knowing that the futures price will be such that F = P ((q1(f 1, f ), q(f 1, f )), the two firms choices (f 1, f ) must form a Nash equilibrium at date 0.

3 Solution. Consider part (i). It is straightforward to show that the two firms date-1 reaction functions are r 1 1(q ; f 1 ) = 1 + f 1 q Hence we have the subgame equilibrium, r(q 1 1 ) = 1 q. q 1(f 1 ) = f 1, q (f 1 ) = f 1. Now consider firm 1 s date-0 choice of f 1. Since F = P ((q 1(f 1 ), q (f 1 )) (a no-arbitrage condition!), at date 0 firm 1 seeks to max f 1 P ((q 1(f 1 ), q (f 1 ))q 1(f 1 ) = 1 3 (1 f 1)( f 1), for which the necessary and sufficient first-order condition gives implying that, in equilibrium, f 1 = 1 4, F = P = 1 4, q 1 = 1, q = 1 4, Π 1 = 1 8, Π = Next consider part (ii). Given (f 1, f ), now the subgame equilibrium becomes q 1(f 1, f ) = f f, q (f 1, f ) = f 1 3 f 1, P (f 1, f ) P (q 1(f 1, f ), q (f 1, f )) = 1 3 (1 f 1 f ). Now consider the date-0 futures market equilibrium. Firm i s problem is to, given the conjectured f j, max P (qi (f i, f j ), qj (f i, f j ))qi (f i, f j ) = 1 f i 3 (1 f i f j )( f i 1 3 f j). 3

4 The necessary and sufficient first-order condition gives firm i s date-0 reaction function Thus the date-0 equilibrium is implying that ri 0 (f j ) = 1 f j, i, j = 1,, i j. 4 f 1 = f = 1 5, q 1 = q = 5, F = P = 1 5, Π 1 = Π = 5. Remark. In part (i), firm 1 is better off with futures trading. The reason is that after commtting to sell f 1 units at a fixed price F, which will not fall when firm 1 expands output at date 1, firm 1 has an incentive to choose a higher total output at date 1. This fact results in firm lowering output accordingly (because output choices are strategic substitutes). In essense, firm 1 s selling futures contracts serves as a commitment that tells its rival that its reaction function is now shifted upwards. Consequently, firm 1 benefits from futures trading, which hurts firm at the same time. Compared to the Cournot equilibrium profit, however, both firms are worse off in part (ii). The reason is that, as in the game of prisoner s dilemma, here each firm intends to hold a short position in the futures contract as an attempt to force its rival to produce less. With the short positions in the futures contract, both firms are faced with a residual inverse demand with lower elasticity to their output expansion. Consequently, both firms choose to produce more in the subgame where futures contracts have been signed, leading to a lower spot and futures price for the product, and lower profit for each firm. 3. (A Strategic Role of Option Contracts) This exercise can be applied to joint ventures, but we shall consider a simpler interpretation. 3 This exercise is adapted from Biaise Allaz and Jean-Luc Vila, 1993, Cournot Competition, Forward Markets and Efficiency, Journal of Economic Theory, 59,

5 There are two players in this sequential game, a landlord (L) and a tenant (T). The landlord can first spend a [0, 1] to build a house, and then after the tenant moves in, the tenant can spend b [0, 1] to make improvements on the house. The resale value of the house is v(a, b) = a f + b h, where the constants f, h (0, 1). (Of course the landlord charges a rent from the tenant, say r, for renting the house for a given period, say a year, but this rental transaction has nothing to do with our main analysis and so we shall forget about it at this moment.) Let us call the social benefit, and the solution S(a, b) v(a, b) a b (a, b ) = arg max S(a, b) a,b [0,1] will be called the first-best investments. We shall assume that a, b can only be observed by the landlord and the tenant but not by the court of law (i.e., they are non-verifiable variables), and hence cannot be put into a legally binding contract. Moreover, S(a, b) is not verifiable either. 4 What L and T can do is to sign a contract to decide who owns the house. The timing of the game is as follows. The two first sign 4 We claim that if instead S(a, b) is verifiable, then there exists a simple sharing rule that gives L and T respectively the payoffs αs(a, b ) and (1 α)s(a, b ) for some α [0, 1], and that contract induces L and T to choose respectively a and b. To see this, recall that v(a, b) is verifiable, and if S(a, b) is verifiable also, then a + b must also be verifiable. Consider the following contract: If S(a, b) = S(a, b ), and if a + b = a + b also, then T would get a fraction (1 λ) of the proceeds v(a, b ) from selling the house, where λ satisfies both L s and T s individual rationality conditions; and in any other event regarding (a, b), both L and T would get nothing from the proceeds of selling the house (the entire v(a, b) would be donated to charity). Now, given λ, define α associated with this λ as such that (1 α)s(a, b ) + b = (1 λ)v(a, b ). Now, if L chooses any a a, then by the uniqueness of (a, b ), it is in T s interest to choose b = 0 rather than any b > 0 such that S(a, b ) = S(a, b ), as there is not other pair (a, b ) satisfying both a + b = a + b and S(a, b ) = S(a, b ). Thus by choosing some a a, S would get the payoff a. On the other hand, if L chooses a = a, then it is obviously in T s interest to choose b = b. Thus with the above contract, in equilibrium L and T get respectively αs(a, b ) and (1 α)s(a, b ). 5

6 an ownership contract, and then given the contract L first chooses a, and upon seeing a, T must choose b. Then the house is sold after the rental period, and the two people share the proceeds according to the ownership contract. (i) Compute a, b. Suppose first that a, b are contractible. Show that if L and T are both rational, they will put a = a, b = b in the contract. From now on, return to our initial assumption that a, b cannot be verified in the court of law, and hence L and T can only try to implement efficient a, b by choosing a smart ownership contract. (ii) Suppose that L owns the house exclusively (so that T cannot share a cent when the house is sold), determine the a, b and v(a, b) by backward induction. (iii) Suppose that before building the house, L sells the house to T by making a take-it-or-leave-it offering price q (so that L cannot share a cent when the house is sold). Determine the a, b and v(a, b) by backward induction. Find q. (iv) Suppose that before building the house, T agrees to pay L some money z to jointly own the house with L, and L and T will subsequently receive respectively λv(a, b) and (1 λ)v(a, b) when selling the house (where λ is exogenously given). Determine the a, b and v(a, b) by backward induction. Find z, assuming that L has all the bargaining power in determining z. (v) Finally, consider the following contingent ownership contract: L owns the house initially, and he gives an option for free (why for free?) to T, and the option allows T to buy the house at the exercise price p = v(a, b ) b after L chooses a but before T chooses b. Find the SPNE by backward induction. Determine the equilibrium a, b and 6

7 v(a, b). (vi) Explain why the contingent ownership contract attains the firstbest efficiency, while the other ownership contracts do not. (vii) Now suppose instead that after L chooses a but before T decides to or not to exercise the option, L can offer a new contract to T. (We call this re-contracting event a renegotiation. ) This new contract will replace the existing option contract if and only if both L and T agree to do so. The new contract states a (probably) different exercise price p that allows T to pay p to L and get the house before T chooses b. Find the equilibrium a and b chosen by L and T respectively. Solution. Consider part (i). The first-best investment levels (a, b ) must solve the following maximization problem max S(a, b) = v(a, b) a b = a f + b h a b. a,b The necessary and sufficient first-order conditions yield a = f 1/(1 f) and b = h 1/(1 h). Since rational people must sign a Pareto efficient contract, these will be L and T s choices if they can sign complete contracts. Consider part (ii). Obviously, T will choose b = 0 since he cannot share the proceeds from selling the house. Thus L seeks to max v(a, 0) a = a f a. a The solution is a = a. Hence when L owns the house exclusively, v(a, b) = v(a, 0) and L s payoff is S(a, 0) < S(a, b ). Consider part (iii). Suppose that T has already paid q to L before L chooses a. Then L will choose a = 0. Thus T seeks to max v(0, b) b = b h b, b 7

8 yielding b = b. Thus, the proceeds from selling the house will be v(0, b ). For T to be willing to pay q for the house in the first place, it must be that q v(0, b ) b. Thus L optimaly chooses q = v(0, b ) b. It follows that L s payoff is S(0, b ) < S(a, b ). Consider part (iv). Consider the subgame where T has already paid z to L for the right of jointly owning the house. Given that L has chosen a, T seeks to max b (1 λ)v(a, b) b = (1 λ)(a f + b h ) b. Thus T optimally chooses b = [(1 λ)h] 1/(1 h) b(λ). Rationally expecting T s behavior, in choosing a, L seeks to max a λv(a, b(λ)) a = λ{a f + [b(λ)] h } a. The solution is a = (λf) 1/(1 f) a(λ). The proceeds from selling the house will thus be v(a(λ), b(λ)). Thus T will accept z if and only if z (1 λ)v(a(λ), b(λ)) b(λ). Consequently, L will choose z = (1 λ)v(a(λ), b(λ)) b(λ), which yields for L the payoff S(a(λ), b(λ)). It is easy to see that S(a(λ), b(λ)) < S(a, b ). Consider part (v). If T does not exercise the option, then he must choose b = 0 because he does not get to share the proceeds from selling the house. If T exercises the option, then given any a he will choose b to max v(a, b) b = a f + b h b, b since he exclusively owns the house. Thus T will choose b = b after he exercises the option. Should T exercise the option? T knows that he will choose b = b if he exercises the option, and hence he chooses to exercise the option if and only if v(a, b ) b p = v(a, b ) b [v(a, b ) b ] 0 a a. The result is not surprising. The house value depends not only on b but also on a. From T s perspective, given the strike price, the house 8

9 is worth buying only if a is large enough. Indeed, the higher the strike price chosen by L, the higher a must be in order to induce T to exercise the option. By wisely setting p = v(a, b ) b, L knows that T will exercise the option if and only if L chooses some a a. Now, what is L s optimal choice about a? If L chooses some a < a, then T will not exercise the option, and T will subsequently choose b = 0, leading to the payoff S(a, 0) for L. If L chooses some a a, then T will exercise the option and L s payoff would become S(a, b ). Thus L s optimal choice is a = a, which generates for L the first-best payoff S(a, b ). An interesting question here is why L offers the option for free? In fact, regardless of the strike price chosen by L, T will refuse to pay anything for the option. Why? Note that after T obtains the option, L will choose some a that makes T feel indifferent between to and not to excercise the option. In other words, L will choose some a that ensures that T makes zero profits by exercising the option. Therefore, for any strike price chosen by L, T will attach zero value to the option. Consider part (vi). The above discussion shows that the first-best efficiency is attained in part (v) but not in parts (ii), (iii), or (iv). There is a free-rider problem in parts (ii), (iii) and (iv), which prevents the first-best efficiency from prevailing. On the other hand, in part (v), T s incentive to choose b can be ensured by making T the sole owner at the time the house is sold (or equivalently, making T the sole residual claimant). For L, on the other hand, by wisely choosing the strike price v(a, b ) b for the option, L can be induced to choose a = a. This explains how the first best efficiency is attained in part (v). Finally, consider part (vii). Note that in part (v), given the existing option contract T will not exercise the option if a < a, which is not efficient because b = 0 rather than b will then be chosen by T. We have assumed in part (v) that the existing option contract cannot be renegotiated, even though such inefficiency may exist. What if L and T can renegotiate the existing option contract? Does the opportunity of renegotiating an inefficient old contract undermine our result that option contracts can help attain the first-best efficiency? 9

10 Recall from part (v) that T will exercise the option if and only if v(a, b ) b p = v(a, b ) b [v(a, b ) b ] 0 a a. Now, if a new contract specifies a strike price p > p, T will never agree to replace the old contract p by this new contract p. Thus if L wants to offer a new contract to T, he must choose some p p. Suppose that L has already spent some a a. Since T is willing to exercise the option under old contract, L will optimally choose p = p in this case, so that contract renegotiation does not arise in this case. What if L has spent some a < a? To induce T to agree to replace the old contract p by this new contract p, it is necessary and sufficient that the new strike price p satisfies v(a, b ) b p 0. Hence from L s perspective the optimal p = v(a, b ) b. Therefore, if L has chosen some a < a, he will offer a new contract that yields for L the payoff v(a, b ) b a = S(a, b ). It follows that L should optimally choose a = a! Our conclusion is that, allowing renegotiation does not change our main result that option contracts can help resolve the free-rider problem and attain the first-best efficiency. Remark. As we explained in part (vi), the free-rider problem in parts (ii), (iii) and (iv) that prevents the first-best efficiency from prevailing is removed in part (v), where T has the correct incentive to choose b because T is the residual claimant) when choosing b and the wisely chosen strike price v(a, b ) b induces L to optimally choose a = a. The problem with this wisely designed contract is that it leads to b = 0 even if a is only slightly lower than a, an outcome which is not productive efficient. Thus subgame perfection implies that L and T may wish to replace this contract by a new one as a remedy, if it did happen that somehow L has chosen a < a. By giving L full bargaining power in contract renegotiation, we show that (a, b ) are still the two players equilibrium choices, even if they are allowed to replace an old contract by a new one after a is chosen. The idea here is that L knows that he will get all the surplus (and T 10

11 will get zero surplus) generated from the replacement of p by p, given that he has all the bargaining power against T in the regenotiation subgame, and for this reason L should choose a to maximize the social benefit (given that, by backward induction, T will always choose b after T agrees to exercise the new option under the price p ). Although we have assumed in this exercise a special functional form for v(a, b), the above results stand valid rather generally even if v is not additively separable in a and b (Borrowing Bank Loan or Issuing Corporate Bond?) An entrepreneur needs to invest 1 dollar to build a firm at date 0, while he has only w < 1 dollars. There are two types of investors in the financial market: households and commerical banks. The entrepreneur and all investors are risk neutral without time preferences; that is, they all seek to maximize expected profits and future cash flows are never discounted. The difference between the two types of investors is that banks have committed to spend a cost c > 0 to monitor each borrowing firm s operations, but households do not have the expertise that is required to oversee the firm s operations. Because banks will have to spend on monitoring, either the entrepreneur chooses to borrow 1 w from households only, or he must borrow (at least partially) from banks, and in the latter case, he needs to raise 1 w + c, instead of 1 w. After the entrepreneur gets the funding, he can incur a private cost ϕ 0 to determine the quality p of the firm s investment project at date 1. For simplicity, suppose that a project of quality p may generate Y dollars with probability p and nothing with probability 1 p at date 3, where the constant Y > 1. The personal cost ϕ is a function of p, and let us assume that, for some constant K > Y, ϕ(p) = K p, p [0, 1]. If a bank lends to the firm at date 0, then it can see p after spending 5 This exercise is adapted from George Nöldeke and Klaus M. Schmidt, 1998, Sequential Investments and Options to Own, Rand Journal of Economics, 9,

12 c > 0 before date. The firm can be liquidated at date, and its (nonnegative) liquidation value is L < 1. Let p be the first-best project quality; that is, 6 p = arg max py ϕ(p). p [0,1] Assume that py ϕ(p) > 1. (A) First suppose that banks do not exist. In this case, the entrepreneur can first decide to or not to offer a financial contract to a household, and in case he does, the household can either accept or reject it. A financial contract, or a corporate bond, is defined as (I, Q, R), such that, according to this contract, (i) the household needs to give the entrepreneur I dollars at date 0; (ii) the household (or, the bondholder) can choose to or not to liquidate the firm at date ; (iii) the household will get Q [0, L] in the event that the firm is liquidated at date ; and (iv) the household will get R [0, Y ] in the event that the firm is not liquidated at date, and it generates Y at date 3. Show that there exists w (0, 1) such that in equilibrium the entrepreneur chooses to borrow from a household and the household is willing to lend I = 1 w if and only if w w. From now on, assume the following numerical values: K = 7, Y = 13, L = 3 5, w = 1. 6 Verify that p = Y K and hence the assumption py ϕ(p) > 1 reduces to Y > K. Given Y, this last inequality holds for some K if Y >. 1

13 Let π E be the entrepreneur s expected wealth and p the equilibrium project quality. Show that under the optimal corporate bond (I, Q, R), 7 p = 1 8, R = 4, π E = (B) Next suppose that the entrepreneur can only borrow from banks. In this case, the entrepreneur can first decide to or not to offer a financial contract to a bank, and in case he does, the bank can either accept or reject it. A financial contract, or a bank loan, is defined as (i, q, r), such that, according to this contract, (i) the bank needs to give the entrepreneur i dollars at date 0; (ii) the bank can choose to or not to liquidate the firm at date, after it sees the firm s choice of p at date 1; (iii) the bank will get q [0, L] in the event that the firm is liquidated at date ; and 7 Hint: Since the household must decide to or not to liquidate the firm before knowing the entrepreneur s equilibrium choice p, the game between the entrepreneur and the household is a simultaneous game. Given (I, Q, R), and given p (which the household did not observe but can conjecture correctly in equilibrium), the household should liquidate the firm if and only if Q p R. On the other hand, the entrepreneur will choose p = 0 if he believes that the household will subsequently liquidate the firm, and he will choose p =argmax p [0,1] p(y R) ϕ(p) if he expects no liquidation. Note that there may exist multiple Nash equilibria for the subgame where the household has already lent to the entrepreneur. Verify that for this subgame it is an equilibrium where the household always liquidates the firm and the entrepreneur always picks p = 0. However, the household would not have lent to the entrepreneur in the first place if they expect this bad subgame equilibrium to subsequently prevail. In other words, bond financing is feasible only if the household expects the entrepreneur to pick p > 0 and only if the household can at least break even (that is, p R 1 w). 13

14 (iv) the bank will get r [0, Y ] in the event that the firm is not liquidated at date, and it generates Y at date 3. Show that the first-best p can be attained in equilibrium if L 1 w + c. 8 Show that, with the above specified numerical values, under the optimal bank loan contract (i, q, r), p = p = 13 7, π E = c, so that the entrepreneur prefers borrowing bank debt to issuing a corporate bond if c < 1 9. Solution. Consider part (A). Consider the subgame where the household has already lent 1 w to the firm, and the entrepreneur is about to choose p. Assuming that the household will not liquidate the firm subsequently (forward induction!), the entrepreneur seeks to max p(y R) ϕ(p), p so that p = Y R K. At the time the household lends to the entrepreneur, the household rationally expects the entrepreneur to choose p = Y R, given R. Hence K the entrepreneur should offer the household the contract (I, Q, R) that satisfies 8 Hint: The bank can see the entrepreneur s choice p before it decides to or not to liquidate the firm. Thus the game between the entrepreneur and the lending bank is a sequential game. The bank gets q L by liquidating the firm, and the bank gets pr by letting the firm continue its operations. If the contract (i, q, r) is such that 1 w + c = q = pr, then the bank will liquidate the firm if and only if p < p. Facing this credible threat, the entrepreneur s best response is p = p, fulfilling the productive efficiency. The only problem is q must not exceed L, and hence such a wisely designed contract is feasible if and only if L 1 w + c. 14

15 This implies that either Rp = R( Y R K ) = 1 w. R = Y + Y 4K(1 w) or R = Y Y 4K(1 w). It also follows that either p = Y + Y 4K(1 w) K or p = Y Y 4K(1 w). K Apparently, the entrepreneur prefers to implement a higher p. Thus we conclude that with bond financing we must have p = Y + Y 4K(1 w) K R = Y Y 4K(1 w). In equilibrium, by lending 1 w at date 0, the household expects to receive p R = Y [Y 4K(1 w)] = 1 w, 4K so that the household does break even. Note that we have assumed that Y 4K(1 w) is well-defined, or equivalently, w w 1 Y 4K > 1 Y 4. 15

16 If w < w, then for all R [0, Y ], Rp = R( Y R K ) < 1 w, so that the household can never break even from lending to the entrepreneur. Note also that, for p to be a legitimate probability, p must lie between zero and one, which is true: 0 < p = Y + Y 4K(1 w) K < Y + Y K = Y K 1. With the specified numerical values, we have p = 1 8, R = 4, implying that π E = p (Y R ) ϕ(p ) = Now, consider part (B). Under the optimal (i, q, r), the first-best quality p = 13 will be implemented, so that the total social surplus is 7 py ϕ(p) = Thus we have π E = (1 w) c = c, which is greater than 9 16 if and only if c 1 9. Remark. Forward induction tells us that either bond financing does not work, or following bond financing the bondholders would never force the firm to liquidate early. Thus the entrepreneur is free to choose any p that he likes. Since an overly low p would not allow bondholders 16

17 to break even, and since the entrepreneur would choose a high p only if the returns generated by the project will mostly be accrued to the entrepreneur, bond financing is feasible if and only if the entrepreneur himself can finance a large portion of the project; that is, w must be sufficiently close to 1. When bond financing is feasible, the chosen p is always lower than its first-best level p. Bank financing has the merit that the lending bank can force early liquidation after seeing the p chosen by the entrepreneur. The lending bank may have an incentive to let the firm continue (because continuation may generate for the bank a higher payoff than liquidation), but it is generally possible to design a contract that induces the bank to choose liquidation over contiuation whenever the entrepreneur chooses an overly low p. Given such a bank-loan contract, the entrepreneur knows that choosing an overly low p will result in the firm being liquidated, and he will not get much in the latter event. Thus bank financing can force the entrepreneur to choose a high p. Indeed, we have seen that it is sometimes (i.e. when L 1 w + c) possible to force the entrepreneur to implement p using bank financing. However, this productive efficiency comes at a cost: the bank must spend c to do auditing, which could be spared if the project were instead financed by issuing a bond. Given w, when c is small and L large, bank financing is apparently better than bond financing because it ensures the first-best productive efficiency. This exercise offers an explanation regarding why most firms are bankfinanced in a developing economy like Taiwan in the period. During that time most Taiwanese firms were small in size (and hence relatively easy to monitor, implying a small c) and possessed mostly general-purpose tangible assets (like factories and physical equipments that are valuable in most traditional business lines), which tend to have higher liquidation values L than intangible assets (like human capital or goodwill) or industry-specific tangible assets possessed by, say, a hightech company. As bank financing tends to dominate bond financing for a developing economy, it also renders an explanation regarding the late development of a bond market in a developing economy. 9 9 This exercise is adapted from Repullo, R., and J. Suarez, 1998, Monitoring, Liquida- 17

18 4. (Currency Attack.) This exercise concerns the strategic interactions between the government of a nation and a group of speculators in the foreign exchange market. 10 The state of the economy is denoted by θ, which is uniformly distributed over the unit interval [0, 1]. The exchange rate in the absence of government intervention is f(θ), with f ( ) > 0, so that a higher realization of θ represents a stronger state of the economy. The exchange rate is initially pegged by the government at e. Facing the government is a continuum of identical speculators located at the unit interval: for each a [0, 1], there is exactly one speculator located at a. Thus the total population of speculators is one. Each speculator may either attack the currency by selling short one unit of the currency, or refrain from trading. Short-selling costs t to the speculator. If a speculator short-sells the currency and the government abandons the exchange rate peg, then the speculator s payoff is e f(θ) t; but if the government defends the peg, then the speculator gets t. Assume that 11 e f(1) > e t = f(θ) > f(0). If the speculator chooses not to attack the currency, his payoff is zero. tion, and Security Design, Review of Financial Studies, 11, The following story can be found on the internet: The Quantum Group of Funds are privately owned hedge funds based in Curacao (Netherlands Antilles) and Cayman Islands. They are currently advised by George Soros through his company Soros Fund Management. Soros started the fund in 1973 in partnership with Jim Rogers. The shareholders of the funds are not publicly disclosed although it is known that the Rothschild family and other wealthy Europeans put $6 million into the funds in In 199, the lead fund, Soros Quantum Fund, became famous for breaking the Bank of England, forcing it to devalue the pound. Soros had bet his entire fund in a short sale on the ultimately fulfilled prediction that the British currency would drop in value, a coup that netted him a profit of $1 billion. In 1997, Soros was blamed for forcing sharp devaluations in Southeast Asian currencies. In July 011, the Quantum Fund announced they would be ending the fund, and will be returning all outside money by the end of 011. The fund will now exclusively manage Soros family money. 11 That is, given that the government would defend the peg, attacking the currency is worthwhile to a speculator if and only if the state of economy is weaker than or equal to θ. 18

19 The government s payoff to abandoning the exchange rate is zero, while the payoff to defending the exchange rate is v c(α, θ), where α is the population of speculators who choose to attack the currency, v is a positive constant representing the value of maintaining the peg, and c(, ) is continuously differentiable and such that c α > 0 > c θ ;1 c(1, 1) > v; 13 c(0, θ) < c(0, θ) = v < c(0, 0). 14 The game proceeds as follows. All speculators and the government first observe the realization of θ. Then, all speculators simultaneously decide whether to spend c and attack the currency. Then the government learns the population α of speculators that have chosen to attack the currency. The government then decides to or not to abandon the peg. The game ends after the government makes the decision. Assume that a speculator will not attack the currency if he feels indifferent about attacking or not attacking, and that the government will abandon the peg if it feels indifferent about abandoning or not abandoning the peg. (i) Suppose that θ [0, θ]. Does the government abandon the peg in equilibrium? Determine the equilibrium α. 1 That is, it is less costly for the government to defend the currency if the economy is in a stronger state or if there are fewer speculators choosing to attack the currency. 13 That is, when all speculators choose to attack the currency, the government s cost of defending the currency falls short of its value, regardless of the state of the economy. 14 That is, when the economy is in the weakest state, it is never worthwhile for the government to defend the peg; and when no speculators choose to attack the currency, there exists a benchmark state θ < θ of the economy such that the government would defend the peg if and only if θ θ. 19

20 (ii) Suppose that θ (θ, θ). Does the government abandon the peg in equilibrium? Determine the equilibrium α. 15 (iii) Suppose that θ [θ, 1]. Does the government abandon the peg in equilibrium? Determine the equilibrium α. Solution. that Consider part (i). For θ [0, θ], we have by c α > 0 > c θ v = c(0, θ) c(0, θ) c(α, θ), α [0, 1], and hence it is the government s dominant strategy to not defend the peg. Rationally expecting that the government will not defend the peg, a speculator can get e t f(θ) = f(θ) f(θ) > 0 by attacking the currency, since f ( ) > 0 and θ θ < θ. Thus in equilibrium we have α = 1, and the government will abandon the peg with probability one. Next, consider part (iii). Given that θ [θ, 1], a speculator would get e t f(θ) = f(θ) f(θ) 0 by attacking the currency, and hence no speculator chooses to attack the currency. Thus we have α = 0 in equilibrium. It follows from c > 0 > c and α θ θ θ > θ 15 Unlike in part (i) and part (iii), there may be multiple equilibria in part (ii). 0

21 that v = c(0, θ) > c(0, θ) c(0, θ), and hence the government chooses to defend the peg in equilibrium. Finally, consider part (ii). We claim that the equilibria obtained in respectively part (i) and part (iii) are both equilibria in part (ii). Indeed, if a single speculator expects all his fellow speculators will attack the currency, so that α = 1, then since v < c(1, 1) < c(1, θ), θ (θ, θ), the government will abandon the peg upon seeing α = 1. It follows that by joining his fellow speculators to attack the currency, that single speculator would get e t f(θ) = f(θ) f(θ) > 0, which implies that the single speculator had better follow suit. This proves that there does exist an equilibrium with α = 1. On the other hand, if a single speculator expects all his fellow speculators will not attack the currency, so that α = 0, 16 then since v = c(0, θ) > c(0, θ), θ (θ, θ), the government will defend the peg upon seeing α = 0, which implies that the single speculator had better follow suit and not attack the currency either. This proves that there does exist an equilibrium with α = With a continuum of speculators (represented by the unit interval), each single speculator is of zero Lebesgue measure. In other words, the equilibrium α is independent of the equilibrium behavior of a single speculator. 1

22 If we remove the assumption that the government will abandon the peg if it feels indifferent about abandoning or not abandoning the peg, and if we assume instead that the government may adopt a mixed strategy in the latter situation, then there also exists an equilibrium for part (ii) where the government randomizes between defending and abandoning the peg. Given θ (θ, θ), we can define two constants a(θ) and π(θ) as follows. By the monotonicity and continuity of c(, θ), and by the intermediate value theorem, there must exist a unique a(θ) (0, 1) such that c(1, θ) > v = c(a(θ), θ) > c(0, θ), since v < c(1, 1) < c(1, θ), θ (θ, θ), and v = c(0, θ) > c(0, θ), θ (θ, θ). Now, define π(θ) (0, 1) as the solution to the following equation (1 π)[f(θ) f(θ)] πt = 0 π(θ) = f(θ) f(θ) f(θ) f(θ) + t. Then, for each θ (θ, θ), there exists an equilibrium where (A) all speculators feel indifferent about attacking or not attacking the currency, and in equilibrium a group of speculators with population a(θ) choose to attack the currency; and (B) upon seeing α = a(θ), the government feels indifferent about defending or abandoning the peg, and in equilibrium the government may defend the peg with probability π(θ).

23 Remarks. This exercise is taken from Morris and Shin (1998, Unique Equilibrium in a Model of Self-Fulfilling Currency Attacks, American Economic Review). The multiplicity of equilibrium in part (ii) is disturbing, because the government cannot develop effective anti-attack strategies without knowing which equilibrium will prevail. The authors show that if the speculators do not observe θ directly but instead each receives a private signal about θ, then the problem of equilibrium multiplicity disappears. For this reason, the government has reasons to withhold some detailed information about the true state of the economy from the public investors. We shalll come back to this point after we talk about games with incomplete information. Observe that the likelihood of currency attacks decreases when the economy grows stronger; in particular, as shown in part (iii), when θ θ, we have α = 0 in equilibrium. In this sense, high real productivity guards financial stability. 5. (Price Competition in a Hotelling Model.) Consider a linear city where consumers uniformly reside on the unit interval [0, 1], and two firms A and B are located respectively at 0 and at 1 (the left and the right endpoints of the unit interval). Firm j produces a single product j costlessly, j {A, B}, and each consumer may either buy 1 unit of product A, or buy 1 unit of product B, or buy nothing. At each point t [0, 1], there exists exactly one consumer, whom we shall refer to as consumer t. Consumer t would obtain a surplus v p A tc if he buys from firm A, and a surplus v p B (1 t)c if he buys from firm B, where v > 0 is the gross utility that consumer t derives from consuming 1 unit of product A or product B, c > 0 represents the unit transportation cost facing the consumer, and p j is the retail price chosen by firm j. A consumer gets zero surplus if he buys nothing. The game proceeds as follows. The two firms must simultaneously announce retail prices, and upon (costlessly) learning about the two retail prices, consumers must simultaneously decide whether to visit one retailer and make a purchase, or to stay home and buy nothing. We shall concentrate on the pure-strategy Nash equilibria for this game. 3

24 (i) Suppose that (p A, p B) is a pure-strategy equilibrium. with this equilibrium, define for j = A, B, Associated T j {t [0, 1] : Buying product j is an equilibrium best response for consumer t}. (i-1) Show that for j {A, B}, either T j = or T j is a closed interval, where in the latter case, for some t a, t b [0, 1], T A = [0, t a ] and T B = [t b, 1]. (i-) Show that either T A TB = or there exists a unique t [0, 1] such that T A TB = {t }. 17 (i-3) Show that T A T B. (i-4) Show that if in a pure-strategy NE, T A TB =, then t a < t b, and consumer t a gets zero surplus from buying product A and consumer t b gets zero surplus from buying product B. (i-5) Suppose that in the pure-strategy NE (p A, p B), T A TB =. Show that, given any tiny ϵ, if given p B firm A deviates and prices at p A + ϵ instead, then firm A s payoff would become L(ϵ) (p A + ϵ) 1 c (v p A ϵ), so that in equilibrium we must have L (0) = 0, proving that p A = v. Likewise, show that p B = v. Show that (p A, p B) = ( v, v ) is indeed a pure-strategy NE if and only if 0 < v < c. 17 Note that in the former case, some consumers are left unserved in equilibrium. 4

25 (i-6) Suppose that in the pure-strategy NE (p A, p B), T A TB = {t }, where 0 t 1. Show that this implies that consumer t can obtain a non-negative surplus from buying either product, which in turn implies that c max(p A p B, p B p A). Show that given p B, firm A s payoff as a function of p A is p A ( p B p A c + 1 ); and given p A, firm B s payoff as a function of p B is p B ( p A p B c + 1 ). Show that in such an equilibrium, necessarily (p A, p B) = (c, c), and this pure-strategy NE exists if and only if v 3c. (ii) Now, on top of the above extensive game, let us add an earlier stage where firms must decide simultaneously whether to provide a presale service. Providing such a service will cost k j to firm j, where we assume that k B = k > k A = 0, and with firm j providing this service, a consumer s gross utility from buying product j rises from v to v + s. From now on, we shall assume that the unit transportation cost is c = 1, and that v is very large (so that T A TB is always non-empty). Suppose further that 0 < k < s 3 s 9, s < 3. Does firm B provide the presale service in the unique SPNE? Does the provision of presale services enhance the profits of the two firms? 5

26 Solution. Consider (i-1). We shall prove the assertion for the case j = A, and the case j = B can be analogously proved. If T A is nonempty, then either T A = {0}, which is a closed interval, or T A contains t > 0, and in the latter case, we claim that for all non-negative t < t, t T A also. Indeed, note that t T A if and only if and (IR) v p A tc 0; (IC) v p A tc v p B (1 t)c. If t > 0 satisfies both of these inequalities, then so does any t with 0 t < t. Hence T A must be an interval taking the form of [0, t a ) or [0, t a ]. Note that the two functions and f(t) v p A tc g(t) [v p A tc] [v p B (1 t)c] are both continuous in t, and that t T A if and only if f(t) 0 and g(t) 0. This implies that T A must be a closed set: let {t n } be a sequence in T A such that lim n t n = t a, then we have f(t a ) = f(lim t n ) = lim f(t n ) 0, g(t a ) = g(lim t n ) = lim g(t n ) 0, proving that t a T A also. Thus T A must take the form of [0, t a ]. Consider (i-). If T A TB contains t and t with t < t, then by (IC) we have v p A tc = v p B (1 t)c and which yields a contradiction. v p A t c = v p B (1 t )c, 6

27 Consider (i-3). We shall prove the assertion for the case j = A, and the case j = B can be analogously proved. Suppose that T A =. Note that this can happen only if either or (IC ) p A + tc p B + (1 t)c, t [0, 1], (IR ) p A + tc > v, t [0, 1]. By (IC ) and the fact that p B 0, we have implying that By (IR ), we have implying that p A (1 t)c, t [0, 1], p A c. p A v tc, t [0, 1], p A v. Note that firm A s equilibrium payoff is zero, and yet given p B, firm A can deviate and price at, for example, p = 1 min(v, c), and obtain a strictly positive payoff. Indeed, when firm A announces p A = p, even if p B = 0, consumer t would strictly prefer buying product A to buying product B if t max( v c, v c 1, 1 v 4c, 1 4 ), implying for firm A a deviation payoff of at least p > 0, which contradicts the assumption that p A is firm A s equilibrium best response 4 against p B. Hence in a pure-strategy NE, T A must be non-empty. Consider (i-4). In a pure-strategy NE where T A TB is empty, apparently we must have t a < t b, which follows directly from (i-1). Now, suppose that consumer t a gets a strictly positive equilibrium payoff from buying product A; i.e., v p A t a c > 0. 7

28 Then for e > 0 such that e < t b t a, e < 1 c (v p A) t a, obviously consumer t a + e should also be an element of T A, which contradicts the definition of t a! The same argument applies to consumer t b. Consider (i-5). By (i-4), we know that for ϵ R such that ϵ is sufficiently small, we have t a + ϵ < t c b, so that consumer t a + ϵ would c get a negative surplus if he accepts p B and buys product B. Thus consumer t a + ϵ would either purchase product A or buy nothing. By c changing its price from p A to p A ϵ, therefore, firm A s sales volume would change from t a to t a + ϵ, so that its payoff would then change c from its equilibrium payoff L(0) to L( ϵ). Since ϵ = 0 is an interior optimum for firm A s problem of maximizing L in a small neighborhood around ϵ = 0, we must have L (0) = 0 (the first-order condition), which gives rise to p A = v. The same analysis can be conducted on the part of firm B. Hence, necessarily, (p A, p B) = ( v, v ) in such an equilibrium. For such an NE to actually prevail, we must also require that t a = v < t c b = 1 v, or equivalently that c > v. c Consider (i-6). Note that buying either product is an equilibrium best response for consumer t, as t T A TB, and hence v p A t c = v p B (1 t )c, implying that t = p B p A c and for t to lie in [0, 1], we must have + 1, c max(p A p B, p B p A). Given p B, firm A s payoff, by changing p A slightly from p A, would be p A ( p B p A c + 1 ), 8

29 so that its best response p A must satisfy p A = p B + c. The same analysis on the part of firm B gives rise to p B = p A + c. Thus, necessarily (p A, p B) = (c, c), implying that t = 1. For consumer t to obtain a non-negative surplus, we must have v 3c. Finally, consider part (ii). By backward induction, we must consider 4 subgames when analyzing the two firms price competition (as in part (i)): (1) the subgame where both firms have spent on service provision; () the subgame where neither has chosen to provide the presale service; (3) the subgame where only firm A has spent on service provision; and (4) the subgame where only firm B has spent on service provision. Now we summarize the subgame equilibria. In the subgame where both firms have spent for service provision, p A = p B = c = 1, and the firms equilibrium payoffs are Π A = 1 4 and Π B = 1 4 k. In the subgame where both firms have chosen not to spend for service provision, p A = p B = c = 1, and the firms equilibrium payoffs are Π A = 1 and Π 4 B = 1. 4 In the subgame where only firm A has chosen to spend for service provision, p A = 1 + s, p 3 B = 1 s, and the firms equilibrium 3 payoffs are Π A = [ 1 + s 3 ] and Π B = [ 1 s 3 ]. In the subgame where only firm B has chosen to spend for service provision, p A = 1 s, p 3 B = 1 + s, and the firms equilibrium 3 payoffs are Π A = [ 1 s 3 ] and Π B = [ 1 + s 3 ] k. 9

30 By assumption, we have k < s s, and hence in equilibrium even firm 3 9 B chooses to spend for service provision. This equilibrium is Pareto inefficient for the firms (but not from the perspective of the entire society, because consumers benefit from service provision obviously), since the two firms would be better off if they could coordinate and commit to not spending on service provision. What happens here is that when v is very large, in the symmetric pricing equilibrium the firms prices depend only on c but not on v, and hence raising v by providing a presale service is totally wasteful from the firms perspective. In equilibrium, however, out of the fear that it might lose too big a market share to firm A if it did not provide the service, firm B chooses to spend on service provision also. 6. (Duopolistic Firms Submitting Supply Curves.) Recall the Cournot game in Example 1 of Lecture 1, Part I. Assume that c = F = 0 and the inverse demand in the relevant range is P (Q) = 1 Q, 0 Q = q 1 + q 1. Now consider a similar but different game, where the two firms must simultaneously choose supply curves instead of supply quantities. 18 That is, the two firms first choose supply functions at the same time, where for i = 1,, firm i s supply curve is denoted by S i (P ), and after they submit their supply functions, the product price P is determined via the following market-clearing condition S 1 (P ) + S (P ) = D(P ), where, from the inverse demand function, the demand curve is D(P ) = 1 P. 18 If we interpret the firms as financial institutions selling a stock, and consumers as market makers absorbing the financial institutions sell orders, then we are assuming in the Cournot game that financial institutions can only submit market orders, whereas in the current example, the financial institutions can submit either market orders or any number of limit orders. This exercise thus shows that large strategic traders in the stock market can manipulate the stock price by submitting limit orders. 30

31 (i) A linear Nash equilibrium is one in which both firms submit linear supply functions; that is, S i (P ) = a i + b i P, i = 1,, for some constants a 1, a R and b 1, b 0. In such an equilibrium, expecting firm j to submit the supply curve S j (P ) = a j + b j P, firm i optimally submits the supply curve S i (P ) = a i + b i P in equilibrium. Now, look for a symmetric linear Nash equilibrium, where symmetry means a 1 = a = a and b 1 = b = b 0. (ii) Determine if firms are better off in equilibrium because they can choose supply curves rather than supply quantities. (iii) We have confined attention to equilibria where b 1, b 0. What 31

32 would your answer to part (ii) become if we allow b 1, b < 0? 19 (iv) Show that, in fact, when the firms can submit any linear supply curves (with a 1, a, b 1, b R), every non-negative output pair (q 1, q ) with q 1 + q 1 can be sustained in some Nash equilibrium. Solution. Consider part (i). In order to solve for the Nash equilibrium, observe that given firm j s supply curve S j ( ), firm i becomes a monopoly facing the following residual demand curve D i (P ) D(P ) S j (P ), and all firm i needs to do is to find a point on this residual demand at which its profit is maximized. After such a point is found, any supply curve S i ( ) that passes through that point is one optimal supply curve 19 Hint: In order to solve the Nash equilibrium, observe that given firm j s supply curve S j ( ), firm i becomes a monopoly facing the following residual demand curve D(P ) S j (P ), and all firm i needs to do is to find a point on this residual demand at which its profit is maximized. After such a point is found, any supply curve S i ( ) that passes through that point is one optimal supply curve for firm i! However, we cannot just use any supply curve that passes through that point; we need to find for firm i a linear supply curve passing through that point and inducing firm j to optimally adopt S j ( ) in the first place. In particular, if firm i believes that firm j will submit S j (P ) = a j + b j P, show that firm i should select the point P (a j, b j ) = 1 a j + b j on its residual demand curve. Any linear S i ( ) passing through this point on firm i s residual demand curve is one best response for firm i, and such a supply curve must satisfy a i + b i P (a j, b j ) = S i (P (a j, b j )) = 1 a j (1 + b j )P (a j, b j ), so that after you impose symmetry (a i = a j = a and b i = b j = b), you will get a continuum of equilibria, where for each b 0, a is determined by a(b) = b 1 3. For part (ii), notice that when b = 0, this equilibrium coincides with Cournot equilibrium. 3