Financial Fragility and the Exchange Rate Regime

Journal of Economic Theory 92, 134 (2000) doi:10.1006jeth.1999.2621, available online at http:www.idealibrary.com on

Financial Fragility and the Exchange Rate Regime 1 Roberto Chang Federal Reserve Bank of Atlanta, Atlanta, Georgia 30303 roberto.changatl.frb.org

and Andres Velasco New York University, New York, New York 10003; University of Chile; and NBER velascofasecon.econ.nyu.edu Received April 23, 1999; final version received October 13, 1999

We study financial fragility, exchange rate crises, and monetary policy in a model of an open economy with DiamondDybvig banks. The banking system, the exchange rate regime, and central bank credit policy are seen as parts of a mechanism intended to maximize social welfare; if the mechanism fails, banking crises and speculative attacks on the currency become possible. We compare currency boards, fixed rates, and flexible rates, with and without a lender of last resort. A currency board cannot implement a social optimum; in addition, it allows bank runs to occur. A fixed exchange rate system may implement the social optimum but is more prone to bank runs and exchange rate crises than a currency board. A flexible rate system implements the social optimum and eliminates runs, provided that the exchange rate and credit policies of the central bank are appropriately designed. Journal of Economic Literature Classification Numbers: F3, E5, G2.  2000 Academic Press

1. INTRODUCTION Recent crises in emerging markets suggest that there is a tight link between the vulnerability of financial institutions and the occurrence of currency 1

Velasco is grateful for support provided by the C.V. Starr Center for Applied Economics at NYU and the Harvard Institute for International Development. We are indebted to an anonymous referee for very useful comments. We also thank Guillermo Calvo, Rudi Dornbusch, Allan Drazen, Scott Freeman, Javier Hamann, Paul Krugman, Robert Lawrence, Leonardo Leiderman, Mancur Olson, Maurice Obstfeld, Arvind Panagariya, Carmen Reinhart, Will Roberds, Dani Rodrik, Jeffrey Sachs, Bruce Smith, Shang-jin Wei, Richard Zeckhauser, and participants at various seminars for numerous suggestions. Of course, any errors and ours alone. The views expressed here are ours and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System.

1 0022-053100 35.00 Copyright  2000 by Academic Press All rights of reproduction in any form reserved.

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crashes. Both the analysis of particular episodes2 and formal econometric evidence 3 point to such a link. The proper choice of exchange rate regime, and the design of complementary monetary and fiscal policies, depend crucially on understanding this connection. It is often claimed, for instance, that fixed exchange rates limit a central bank's ability to act as a lender of last resort to the banking system. But there is strong disagreement on how to deal with this problem; suggestions include securing emergency lines of foreign credit, taxing capital inflows, or letting exchange rates float. This state of affairs has persisted, to a large extent, because there is little theoretical work linking financial fragility and exchange rate crises. In this paper we present a formal analysis of the subject. Our approach is novel in that it starts from the microfoundations of a country's financial system. More specifically, we study commercial banking, central bank policies, and exchange rate regimes as mechanisms intended to implement socially desirable allocations in an economy with incomplete financial markets. We examine when such mechanisms succeed, and also when they fail and result in a bank andor currency crisis. We study these issues in a monetary, open economy extension of the benchmark model by Diamond and Dybvig [10]. That model is a natural choice because it focuses squarely on the liquidity of banks, which has figured prominently in observed crises. Banks take liquid deposits and invest them partly in illiquid assets; such a maturity transformation can enhance welfare, but also leaves banks subject to self-fulfilling runs. Our analysis goes beyond that of Diamond and Dybvig in that we embed their banking story in a general equilibrium macroeconomic model which can operate under a variety of exchange rate and monetary arrangements or regimes. The resulting framework is rich enough to analyze currency boards, fixed exchange rates, and flexible exchange rates, with and without a lender of last resort. We also discuss the costs and benefits of precautionary measures, the implications of ``dollarization,'' and the impact of international capital flows. In all cases we investigate the efficiency of equilibria and whether self-fulfilling runs on the bank andor the currency can occur. The main results are: 1. A currency board, in which the exchange rate is fixed and the central bank issues no domestic credit, is vulnerable to self-fulfilling bank runs but (predictably) not to currency crises. In addition, a currency board cannot yield a socially optimal outcome. 2

In the context of the Mexican 1994 crisis, see [24]. For the recent Asian crisis, see [8]. The key reference is [18], which concludes that banking crises help predict exchange rate collapses. 3

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2. A less restrictive regime of fixed exchange rates, in which the central bank provides credit to commercial banks but does not act as a lender of last resort, has an equilibrium that yields the social optimum and, hence, may improve upon a currency board. Currency crises cannot occur. However, this regime is more prone to bank runs than a currency board. The intuition is that, with fixed exchange rates, the liabilities of banks are, implicitly, obligations in international currency. Hence bank runs are possible to the extent that the banking system's implicit liabilities exceed its internationally liquid assets. This gap turns out to be greater, relative to a currency board, when the central bank grants credit to the banking system. 3. If the central bank attempts to fix the exchange rate and to act as a lender of last resort, providing emergency credit to commercial banks in case of trouble, it precludes bank collapses only by using up its own international reserves when banks are run on; balance of payments crises may occur as a result. In fact, they may occur under the same conditions that make bank runs possible in the regime in which the central bank is not a lender of last resort. Hence the existence of crises depends just on the underlying international illiquidity of the economy; the lender of last resort policy determines only whether a crisis materializes as a generalized bank failure or as a collapse of the exchange rate peg. 4. Both a policy of high bank reserve requirements and one of large international reserves can eliminate equilibrium runs, but at the cost of inducing socially inefficient allocations. Hence, they both lower welfare relative to the good equilibrium without these policies. Moreover, the first policy unambiguously dominates the second one. 5. The combination of flexible exchange rates and a central bank that serves as lender of last resort dominates all other regimes: it can implement the social optimum while ruling out self-fulfilling runs. The intuition is simple. When the central bank acts as a lender of last resort, a run can occur only if each depositor expects the others will also run and deplete the international liquidity available to the central bank. In a run, depositors withdraw domestic currency from the commercial bank in order to buy foreign exchange from the central bank. This behavior is optimal for each depositor if, in a run, the central bank in fact uses up the available stock of foreign exchange, as would be the case in a fixed rate regime. But with flexible exchange rates the central bank is not compelled to sell all of its available reserves. Instead, those who run face a depreciation, while those who do not run know that there will still be foreign exchange available to them later. Hence, joining the run is no longer a best response, pessimistic expectations are not self-fulfilling, and a depreciation does not happen in equilibrium.

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6. Dollarization, in the form of foreign currency-denominated bank deposits. prevents the central bank from serving as a leader of last resort. This makes no essential difference for a currency board or a fixed exchange rate system, but eliminates the advantages of flexible rates. 7. An increase in the availability of international capital makes the country better off provided honest equilibria occur; it has ambiguous effects on the likelihood of run equilibria under fixed exchange rates.

Our paper is related to existing work along several dimensions, three of which deserve special comment. First, in attempting to provide a formal explanation of balance of payments crises we follow the huge literature initiated by [19]. 4 The role of a fragile financial system in such crises has been modeled only recently, however; notable contributions include [6, 7, 9, 16] and particularly [3]. Second, our analysis borrows heavily from the literature on banking crises, developed by [10] and others.5 Much remains to be done in adapting that literature to the study of exchange rate regimes. In particular, monetary versions of simple banking models such as [4, 26, 28] may prove useful in generalizing our results to truly dynamic settings and to more satisfactory specifications of money demand. Third, our paper contributes to the age-old debate on the optimal exchange rate regime. 6 The recent crises in emerging markets have made it clear that financial considerations are key to that choice; see [2], for example. Our paper is the first to present a formal model embodying that idea. 7 The paper is organized as follows. Section 2 describes the economic environment. Section 3 studies a currency board, while Section 4 analyzes the case of a fixed exchange rate with domestic credit. The effects of increasing reserve requirements and building a ``war chest'' of international reserves are considered in Section 5. Section 6 contains the analysis of flexible exchange rates. Section 7 considers dollarized bank deposits. Section 8 discusses the effects of international borrowing. Section 9 discusses some limitations of our analysis and concludes with suggestions for future research.

4

See [15] for a survey up to 1995. For more recent developments, see [12]. This literature is also enormous. See [13] for a recent summary. In addition to [10], we have greatly benefited from studying [27]. Like us, [16] discusses a version of the Diamond Dybvig model to discuss crises in open economies. 6 See [22, Chap. 9] for a useful summary of the different positions. 7 Also see [1] for a recent attempt at modeling links between banks and exchange rate issues. In contrast to our paper, however, [1] does not focus on crises. 5

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2. THE ECONOMIC ENVIRONMENT We will focus on a small open economy that lasts three periods indexed by t=0, 1, 2. There is a single consumption good whose price in the world market is fixed and normalized at one unit of a foreign currency (dollars). This good is freely traded at all times; hence we shall refer to units of consumption or dollars interchangeably. The economy under analysis is populated by a large number of ex ante identical agents, each endowed with e>0 dollars at t=0. This endowment can be invested in the world market, where each dollar invested at t=0 yields one dollar in either period 1 or period 2. Alternatively, it can be invested in a long term illiquid technology whose yield per dollar invested is r<1 units of consumption in period 1, and R>1 units of consumption in period 2. 8 At t=1 each agent discovers her type. With probability * she is impatient and derives utility only from period 1 consumption. With probability (1&*) she turns out to be patient. Patient agents derive utility from holding a domestic currency ( pesos) in period 1 and from consumption in period 2. Pesos are costlessly created andor destroyed by a Central Bank. Let x denote the typical agent's consumption in period 1 if she turns out to be impatient. Let M and y denote, respectively, the quantity of pesos acquired in period 1 and her period 2 consumption if she turns out to be patient. Finally, let E t the peso price of consumption in period t, which must equal the exchange rate. Then the expected utility of the representative agent can be represented by *u(x)+(1&*) u[/(ME 2 )+ y].

(2.1)

Note that real peso balances are measured by the quantity of pesos obtained in period 1 and deflated by the price of consumption in period 2. This is the natural assumption, since patient agents will hold such balances until period 2 and then use them to purchase consumption goods. The function u( . ) is smooth, strictly increasing, strictly concave, and satisfies Inada conditions. We assume also that /( . ) is smooth, strictly concave, and satisfies /(0)=0, /$(0)=, and /$(m )=0 for some m >0. This defines m as the satiation level of pesos. Our assumptions on peso demand are special in several ways. Impatient agents do not derive utility from holding pesos, and dollars do not enter utility functions; however, we argue in the last section of this paper that these aspects of the model can be relaxed at little cost. A more basic objection is that inserting pesos in utility functions is a very crude way to motivate 8

These assumptions on asset structure make our model similar to that in [17].

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peso demand. That objection is clearly warranted. On the other hand, the simplicity of our specification will allow us to obtain, as we shall see, a number of insights that may have been lost otherwise. Whether our results survive changes in the way we justify a demand for pesos is an open and promising question for future research; we speculate on the answer in the last section. Three further assumptions complete the description of the economic environment. As in [10], type realizations are i.i.d. across agents, and there is no aggregate uncertainty. The realization of each agent's type is private information to that agent. Finally, we assume until Section 8 that domestic residents (including the Central Bank) can invest but not borrow in the world market. This is mostly to simplify the exposition: Section 8 shows that the model can be amended to allow for international borrowing while preserving its main implications. Our basic assumptions on preferences and asset returns are similar to those in [10]; a main departure is that agents are assumed to demand domestic currency. Since only the domestic Central Bank can print pesos, the analysis of this model must include an assumption on the exchange ratemonetary regime, that is, a specification of the way pesos are issued and exchanged for dollars. The implications of alternative regimes are the subject of our ensuing discussion.

3. A CURRENCY BOARD This section analyzes regimes in which the Central Bank follows a very simple rule: it stands ready to exchange dollars for pesos at a fixed exchange rate and, in addition, it is committed not to create or destroy pesos in any other way. In other words, the Central Bank functions as a currency board. Under a currency board, the amount of base money in circulation is exactly equal to the foreign reserves of the Central Bank at all times. Hence there cannot be a balance of payments crisis. The private sector can be organized in various ways; we will examine one which resembles a banking system. Under plausible conditions, this system is Pareto superior to individual autarky. However, it may also be subject to bank runs. 3.1. A Banking System Because there is individual but not aggregate uncertainty, it is clear that domestic residents will find it advantageous to act collectively rather than in isolation. We shall assume that they in fact form such a coalition, which will be called a commercial bank (or simply ``bank'' when no confusion may arise) for reasons that will be clear shortly.

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The objective of the bank is to pool the resources of the economy in order to maximize the welfare of its representative member. Ideally this would require assigning consumption and peso holdings to each agent contingent on the realization of her type. In choosing such a contingent allocation, the bank is restricted not only by resource feasibility constraints but also by the fact that type realizations are private information. This implies that the bank must find some way of eliciting such information. Examining the bank's options is greatly simplified applying the Revelation Principle, which implies that attention can be restricted to feasible type contingent allocations that give no agent an incentive to misrepresent her type. To express the last condition correctly, it is necessary to make a more precise assumption about what the bank can and cannot observe. The simplest assumption, which will be maintained hereon, is that the commercial bank can observe each agent's transactions with the domestic banking system (the bank itself and the Central Bank), while consumption and world transactions cannot be monitored. Assume, without loss of generality, that the currency board guarantees that the exchange rate will be one peso for each dollar. Our discussion then implies that the bank will choose an allocation that maximizes (2.1), with E 2 =1, subject to k+be *x+(1&*) Mb+rl (1&*) y(1&*) M+R(k&l )

(3.1) (3.2) (3.3)

lk

(3.4)

k, b, x, y, M, l0

(3.5)

/(M)+ yx.

(3.6)

Quantities are denoted on a per depositor basis. In period 0, the bank must choose how much of the endowment e to invest in the world market, b, and in the long term asset, k; this is (3.1). Equation (3.2) is the feasibility constraint in period 1. The bank must assign x units of consumption to each impatient and M pesos to each patient; this is financed by liquidating the short term asset, b, and possibly some amount l of the long term asset. Note that (3.2) incorporates the assumption that the bank can acquire pesos from the Central Bank at an exchange rate of one. The feasibility constraint in period 2 is given by (3.3). To understand it, it is easiest to assume that the bank requires patient agents to return in period 2 the pesos they were given in period 1. The bank then resells the pesos to the Central

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Bank, 9 and in addition it liquidates the remainder of the long term investment. The proceeds from these operations are used to finance the period 2 consumption of the patient. The inequalities (3.4) and (3.5) are self explanatory. Finally, (3.6) is the incentive compatibility or truth telling constraint for patient agents. If a patient agent reports her type honestly she will receive M pesos in the first period and consume y in the second, whose value is u[/(M)+ y]. If she lies she will only be given x units of consumption in period 1; under our assumptions, the best she can do then is to exchange them for x dollars, which can be used in period 2 to buy x units of consumption. The value of lying is then u(x), which (3.6) requires not to be larger than the value of telling the truth. The solution to the above problem will be marked with tildes and called the currency board solution; its properties are easily derived. It can be shown that l =0that is, there is no liquidation in period 1 of the long term investment. This should be obvious, since the bank faces no aggregate uncertainty, and liquidating the long term asset in period 1 is costly. Given this fact, it is intuitive (and can be shown formally) that the currency board improves upon what each agent could achieve acting alone. The other features of the solution follow from marginal optimality conditions. 10 The optimal quantity of pesos to give patient agents is determined by /$(M )=R&1>0.

(3.7)

In words, the marginal utility of giving pesos to the patient must equal the marginal cost of pesos to the bank, which is R&1, and implies that M is less than m. Given this result, the choices of x and y must satisfy the following equation, which can be thought of as a transformation curve, R*x~ +(1&*) y~ =eR&(R&1)(1&*) M.

(3.8)

The final optimality condition is that the social indifference curve be tangent to the above transformation curve, *R *u$(x~ ) = (1&*) u$[/(M )+ y~ ] (1&*)

(3.9)

9 Note that in period 1 the bank sells (1&*) M dollars to the Central Bank in exchange for (1&*) M pesos; each patient agent withdraws M pesos, which she holds until period 2. The Central Bank then holds (1&*) M dollars as foreign exchange reserves between periods 1 and 2; these reserves are finally used to retire the outstanding amount of pesos. 10 The analysis of this section requires the RHS of (3.8) to be strictly positive. For this to be the case, it is sufficient that the satiation level of pesos be not too large relative to the wealth of the economy, that is, eR>(R&1)(1&*) m.

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or u$(x~ )=Ru$[/(M )+ y~ ]. (Note that, since R>1 and u is concave, (3.9) guarantees that the incentive constraint (3.6) does not bind.) Equations (3.7), (3.8), and (3.9) completely determine the optimal allocation of consumption and peso holdings by the two types. Next we discuss how this allocation can be implemented in a decentralized fashion. 3.2. Demand Deposits and Bank Runs The currency board solution is the best allocation that the bank can achieve, in principle, given the existence of a currency board. The next question is how that allocation will be implemented in practice and, more generally, how the banking system may try to accomplish any given objective. To answer that question, we shall assume that the commercial bank and its depositors establish a particular contract. Also, we shall assume that the Central Bank and the commercial bank agree on procedures about how dollars and pesos will be exchanged and, possibly, a lending-borrowing relationship. The contract between the bank and its depositors and the operating rules of the banking system, including Central Bank policy, is what we call a regime. We shall see that each regime induces a game played by the depositors. The analysis of a regime then reduces to the analysis of the equilibria of the induced game. One natural way in which the bank can implement the currency board solution is via demand deposits. Demand deposits are contracts that stipulate that each agent must turn over her endowment to the bank in period 0. The bank invests b in the world liquid asset and k in the long term technology. In return, the agent is promised some payments from the bank in periods 1 and 2, depending on her reported type. We shall impose some additional assumptions on the problem. First, we will assume until further notice that all payments from the bank to its depositors are denominated and are given in pesos. This implies that each agent is entitled to withdraw either x~ pesos in period 1, or M pesos in period 1 and y~ &M pesos in period 2, depending on the report of her type. Also, in order to buy consumption in the world market, the agent must visit the Central Bank to exchange pesos into dollars. The second assumption is that both the bank and the Central Bank must respect sequential service constraints. These constraints require, loosely speaking, that both the commercial bank and the Central Bank meet the requests of depositors on a first come-first served basis. The existence of sequential service constraints can be justified by more primitive features of the environment, as suggested by [27]. Accordingly, the sequence of events is as follows. At the start of period 1 depositors arrive to the bank in random order. Upon arrival, each agent reports her type realization and withdraws either x~ or M pesos, assuming the bank is still open. If the agent withdraws x~ pesos she must exchange

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them for dollars in period 1 at the Central Bank; to do this, she ``walks'' to the Central Bank to join a ``line'' waiting there. In contrast, if the agent withdraws M pesos she must hold them until period 2. The commercial bank services requested withdrawals sequentially, as long as it has assets to liquidate, in whose case it liquidates them as needed to obtain dollars that it sells for pesos to the Central Bank. If the bank exhausts its assets, it closes and disappears. After all depositors have visited the bank, the Central Bank starts selling dollars, at an exchange rate of unity, to the depositors that have withdrawn x~ pesos. We will assume that the Central Bank closes if it runs out of dollars (this cannot happen under a currency board, but will happen in other regimes). Finally, if the bank did not close in period 1, in period 2 the bank liquidates all of its remaining investments and sells the resulting dollar proceeds to the Central Bank. It pays y~ &M pesos plus any profits to agents that reported to be patient in period 1. These agents exchange the pesos thus accumulated at the Central Bank for dollars, which they subsequently use to buy consumption. The regime just described is, in short, a demand deposit system with a currency board. Given this regime, the consumers are engaged in an (anonymous) game. The outcomes of this regime are then given by the equilibria of the game. An equilibrium is a description of the strategies of each depositors and aggregate outcomes such that the aggregate outcomes are implied by the depositors' strategies and each depositor strategy is optimal for her given the aggregate outcomes. 11 The following can now be proven:

Proposition 3.1. There is an honest equilibrium. In this equilibrium, in period 1 impatient depositors retire x~ pesos and patient ones retire M. The bank does not fail, and pays y~ &M to patient depositors in the second period. The Central Bank ends period 1 with (1&*) M dollars, which it uses to redeem the money supply in period 2. The proof is standard and left to the reader. This proposition also clarifies the role of a banking system in a currency board regime (and, analogously, in other regimes). Here, the bank may implement the currency board solution which, as we observed before, improves upon what each agent can achieve in isolation. 11

This description is intentionally vague. This is because we have assumed a large number of depositors, who have measure zero. Given this, the equilibrium definition must ensure that depositors assume that their impact on aggregate outcomes is negligible. In such case the appropriate equilibrium concept is that of [25].

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However, the banking system may attain such an improvement only by holding less internationally liquid assets than its implicit liabilities. Consequently the banking system may be subject to a run. In particular, suppose that x~ >b +rk.

(3.10)

In such case, a run may emerge in equilibrium: Proposition 3.2. If (3.10) holds, there is an equilibrium in which all agents claim to be impatient and the bank fails in period 1. If (3.10) does not hold, there cannot be equilibrium bank runs. Proof. First we prove sufficiency. Suppose that every depositor claims to be impatient whatever her type. Then, (3.10) implies that the bank fails in period 1 after liquidating all investments. The Central Bank does not fail, but closes after selling b +rk dollars. No further payments will be given in period 2. Also, the Central Bank will hold no dollars in period 2, and hence the real value of pesos then will be zero. It follows that withdrawing M pesos in period 1 is worthless, and that attempting to withdraw x~ early is optimal for each depositor regardless of type. To prove necessity, suppose that a number *9 of depositors claim to be impatient in period 1. Clearly it is enough to restrict attention to the case in which *9 >*. We shall argue by contradiction and show that if (3.10) does not hold the bank is able to pay at least y~ to honest patient types in period 2, which implies that it cannot be optimal for the patient to lie. Let l9 denote the amount of liquidation of the long term asset; it is defined by *9 x~ +(1&*9 ) M =b +rl9 . Using this and *x~ +(1&*) M =b, one concludes that rl9 =(*9 &*)(x~ &M ). Now, it is easy to check that the bank will be able to pay y~ to the (1&*9 ) reportedly patient depositors if M +R(k &l9 ) (1&*9 ) y~. This inequality will hold, from the definition of y~ and the last equation for l9 , if rk (1&*)(x~ &M ). But the latter is equivalent to the failure of (3.10). The proof is complete. K The proposition states that (3.10) is not only sufficient but also necessary for the existence of a bank run. While (3.10) is a condition on the currency board solution, it can be easily translated into a condition on the fundamental parameters of the economy. A little manipulation shows that (3.10) is equivalent to x~ >

er+(1&r)(1&*) M #x. 1&(1&r) * 

(3.11)

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It can now be shown that a necessary and sufficient condition for (3.11) and, hence, for the existence of a bank run is: Proposition 3.3. Let y =(1&R) M +R(e&*x )(1&*). Then, a bank   run is possible if and only if u$(x )>Ru$(/(M )+ y ).   Corollary 3.4. regime.

(3.12)

Equilibrium bank runs may occur in a currency board

The proof is easy and left to the reader. 3.3. Summary Under a currency board, there can be no attacks on Central Bank reserves. A banking system operating with demand deposits can emerge to improve upon individual autarky. However, this arrangement may be prone to runs. In fact, since in a currency board all monetary holdings must be backed by ``real'' reserves (in this case, holdings of the liquid asset), this case is quite similar to the original DiamondDybvig setup in [10]. Both honest and run equilibria were possible in that context, and the same is true here. Notably, in a currency board runs on the financial system are not inconsistent with a successful defense of the exchange rate peg. This may help interpreting the recent ``success'' of Hong Kong and Argentina, where in recent years financial institutions came under intense pressure but currency boards survived.

4. FIXED EXCHANGE RATES AND CENTRAL BANK CREDIT Based on one of two possible conjectures, readers might guess that a currency board is inefficient. The first is that the rules of a currency board prevent the Central Bank from helping commercial banks in case of a run. Hence one might plausibly conjecture that welfare can be raised if the Central Bank is allowed to act as a lender of last resort. The second conjecture is more subtle. Under a currency board, each peso in circulation is backed by a dollar in the Central Bank's vault. This implies that in period 1 pesos are created at a positive cost, which is equal to the opportunity cost of the Central Bank's reserve holdings. The marginal utility of money holdings is equated to this positive cost in equilibrium. However, printing pesos is free for the Central Bank. This suggests that marginal utility is not equated to the ``true'' marginal cost of printing pesos, and therefore the currency board inefficiently prevents the Central

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Bank from providing an optimal amount of liquidity. Conceivably, a better allocation should be attainable. It is therefore important to discuss a fixed exchange rate system in which the Central Bank extends liquidity credit to the commercial bank; this is the objective of this section. We start by showing that such a regime may attain a better allocation than under a currency board; hence the second conjecture above is validated. This regime can be decentralized, as with the currency board, via a demand deposit system. As with a currency board, bank runs are possible. In addition, now balance of payments crises become possible. In both cases, the underlying cause is the international illiquidity of the banking system as a whole. The specific arrangements between the Central Bank and commercial banks only determine whether the illiquidity problem is manifested as a bank run or a balance of payments crisis, but this distinction has no economic significance. Hence the first conjecture turns out to be false if exchange rates are fixed. Finally, this section shows that the international liquidity of the banking system is less under a fixed exchange rate system with Central Bank credit than under a currency board. As a result, conventional fixed exchange rate systems are more prone to runs than are currency board systems. 4.1. The Optimal Supply of Central Bank Credit It is helpful at this point to study a social planning problem. Consider what the bank and the Central Bank can ideally achieve if they act together, assuming that the exchange rate will be fixed at one. The Revelation Principle applies to this problem, and the options open are given by all feasible incentive compatible allocations. Hence the solution, which will be called the social optimum, must maximize (2.1) (with E 2 =1) subject to (3.1) and *xb (1&*) yRk x, y, M, k, b0

(4.1) (4.2) (4.3)

and the incentive compatibility condition (3.6). In setting up the social planning problem, we have assumed that pesos can be provided at no cost to satisfy the peso demands of the patient types. We have also assumed that the exchange rate in period 2 is one. We shall see that both assumptions turn out to be justified, in the sense that the resulting allocation can be implemented. The social optimum will be marked with an overbar and can be intuitively understood as follows. First, since providing pesos is effectively gratis, the

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optimal allocation ensures that the real quantity of pesos is the satiation level, /$(M )=/$(m )=0

(4.4)

Second, the transformation curve is now given by R*x +(1&*) y =eR.

(4.5)

Note the difference with the transformation curve under a currency board, given by (3.8): in a fixed exchange rate system with optimal provision of pesos, the economy ``saves'' (R&1)(1&*) M, which is the opportunity cost of providing pesos for the patient types to hold between periods 1 and 2. The final optimality condition is that u$(x )=Ru$[/(M )+ y ]

(4.6)

which says that the social marginal rate of substitution must equal the slope of the transformation curve. The three conditions (4.4), (4.5), and (4.6) characterize the social optimum. Now we shall prove that the optimal allocation can be decentralized in a fixed exchange rate system with Central Bank credit. Suppose now that the commercial bank acts independently of the Central Bank and maximizes the welfare of the representative depositor. Assume also that pesos can be exchanged for dollars at a unity exchange rate. Finally, suppose that the Central Bank offers to lend pesos to the commercial bank in period 1, to be repaid without interest in period 2. The Revelation Principle applies once more and implies that the bank's problem is to maximize (2.1), with E 2 =1, subject to (3.1), and *x+(1&*) Mb+h (1&*) y(1&*) M+Rk&h x, y, M, k, b, h0

(4.7) (4.8) (4.9)

and the incentive compatibility constraint (3.6), where h denotes the bank's borrowing from the Central Bank. There is a difficulty with this argument as just stated, in that the commercial bank will choose to invest all of the economy's endowment in the long term asset and borrow from the Central Bank all it needs to service

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withdrawals in period 1. This is easily corrected, however, if we assume that the commercial bank must respect the following reserve requirement, 12 *xb.

(4.10)

With the reserve requirement, the bank's problem is to maximize (2.1) subject to (3.1), (4.7), (4.8), (4.9), (3.6), and (4.10). The solution of this problem can be easily seen to coincide with the social optimum described above. Some features of the social optimum are worth noting. It can be checked it results in a Pareto improvement over the currency board solution. This implies, in particular, that x >x~: impatient agents are entitled to more consumption in period 1 than in a currency board.13 Lastly, the optimal supply of credit from the Central Bank to the commercial banks is h = (1&*) M =(1&*) m. That is, the amount the Central Bank lends to the bank should be just enough to cover the demand for pesos of patient agents in period 1. Summarizing, in a fixed exchange rate system with a reserve requirement and optimal Central Bank credit the commercial bank will try to implement a socially optimal allocation. Next we shall study how this implementation can occur via a demand deposit system. 4.2. Demand Deposits and Bank Runs As with the currency board, a demand deposit in a fixed exchange rate regime is a contract that requires depositors to surrender their period 0 endowments to the bank. The bank invests b in the short term asset and k in the long term technology. In return, depositors get the promise of withdrawing, at their own discretion, either x pesos in period 1 or M pesos in period 1 and y &M pesos in period 2. Note that we maintain the assumption that deposits are denominated in pesos. We shall also maintain our assumptions on sequential service. The main difference with the currency board case is that now the existence of Central Bank credit makes it necessary to describe the credit arrangements between the Central Bank and the commercial bank. We shall examine two alternatives in turn, and argue that the main difference they make is whether bank runs or balance of payments crises can occur. 12

This condition may not look like a conventional reserve requirement. However, it is equivalent to suppose that the commercial bank is required to deposit at least *x pesos at the Central Bank between periods 0 and 1. To finance its reserve account, the commercial bank would have to sell at least *x dollars to the Central Bank in period 0. In period 1, the commercial bank would get the dollars back by liquidating its reserve account. 13 Whether y > y~ as well depends on the shape of the /( . ) function. The transformation curve has moved away from the origin, which would tend to increase y relative to y~. But /(M )>/(M ), which, ceteris paribus, would tend to reduce y relative to y~.

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4.3. A Regime with Central Bank Credit but No Lender of Last Resort Suppose that the Central Bank offers to lend h dollars to the commercial bank in period 1 at no interest, but restricts the commercial bank to use this line of credit only for withdrawals of the reportedly patient types. A rationale for this rule may be that the Central Bank wants to defend its reserves. An agent who reports that she is impatient will try to sell the pesos she withdraws within the period to the Central Bank. Hence, requiring the commercial bank to use the credit line only for patient withdrawals ensures that the Central Bank never runs out of dollars and that a balance of payments crisis is not possible. Given this arrangement, in period 1 the commercial bank services withdrawals of size M by borrowing from the Central Bank, and withdrawals of size x by liquidating b or k. The commercial bank is assumed to close if it liquidates all of its assets. In period 2, if the commercial bank did not close, it liquidates the rest of its assets. It uses the proceeds to repay the loan from the Central Bank: finally, it pays y &M plus any profits to each reportedly patient depositor. As in the currency board case, this modus operandi induces an anonymous game played by the depositors. Now it can be shown that: Proposition 4.1. There is an honest equilibrium whose outcome is the social optimum. In this equilibrium, neither the bank nor the Central Bank close. Proof. If all agents report their types honestly, in period 1 the bank services withdrawals of *x by liquidating the world asset and (1&*) M by borrowing from the Central Bank. This is feasible for the bank, given (4.10) and h =(1&*) M. Now, (4.8) ensures that the bank can repay the Central Bank's loan and service deposits in period 2. Hence the bank does not close. In period 1, the Central Bank receives b dollars from the bank, which are (just) enough to attend the demands of the impatient types. In period 2, the Central Bank receives Rk dollars from the bank, which is enough to satisfy the demand for dollars of the patient types. The rest of the proof is obvious and left to the reader. K Hence this system may implement the optimal allocation. However, Proposition 4.2.

Assume that x >b +rk.

(4.11)

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Then there is an equilibrium in which all depositors claim to be impatient and the bank fails in period 1. Conversely, if (4.11) does not hold there cannot be an equilibrium with a bank run. Proof. First we prove the sufficiency of (4.11). Suppose all consumers attempt to withdraw x. Then, given the assumption that the bank cannot use the Central Bank credit line to service impatient withdrawals, (4.11) implies that the bank fails in period 1. Under the same conditions, there is no Central Bank lending in period 1. In period 1, the Central Bank receives b +rk dollars from the bank. This is just enough to sell dollars to the depositors that hold pesos. Hence the Central Bank does not close in period 1. The rest of the proof is similar to previous proofs and left to the reader. K The condition (4.11) is obviously analogous to (3.10). Now, it turns out that: Proposition 4.3. true.

If (3.10) is satisfied, so is (4.11). The converse is not

Proof. Using (3.1), (4.11) can be rewritten as x >er(1&(1&r) *). Using the definition of x given in (3.11), this is equivalent to x +(1&r)(1&*) M (1&*(1&r))>x. Since x >x~ the proposition follows immediately. K  Hence, if runs are possible in a currency board they are also possible in a fixed exchange rate regime, but the converse is not true. There is a sense, therefore, in which a fixed exchange rate regime is more conducive to bank runs than is a currency board. On the other hand, the fixed exchange rate regime can implement an allocation (if the ``good'' equilibrium obtains) that improves upon the best a currency board can attain. The question of which regime is to be preferred is subtle and depends on how an equilibrium is selected when there are many of them. Unfortunately, our analysis is not informative on this respect, and supplementing it with a theory of equilibrium selection goes beyond the scope of this paper. 4.4. Fixed Exchange Rates with a Lender of Last Resort In the preceding subsection the Central Bank did not act as a lender of last resort in the sense that, if there was a bank run, the Central Bank did not extend credit to the commercial bank. We shall now examine the consequences of dropping that assumption. However, we shall still assume that the Central Bank tries to maintain a fixed exchange rate regime in the sense that it will defend the peso until it is impossible for the economy to obtain more dollars.

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Such a regime is best formalized as follows. In addition to the line of credit of h, the Central Bank offers to lend an unlimited quantity of pesos to the commercial bank in case more than * customers claim to be impatient. If such emergency credit is used, we plausibly assume that the Central Bank obtains control over the long term asset in period 1, and liquidates the asset as needed to sell dollars to agents claiming impatience. In period 1, the commercial bank pays reportedly patient types by borrowing from the ``normal'' line of credit, and it pays withdrawals of the impatient by liquidating the world asset first, and then drawing on the emergency credit line. Because emergency credit is unlimited, the bank does not close. The Central Bank sells dollars at an exchange rate of one, using first the dollars obtained from the bank, and then liquidating the long term asset. If the long term asset is completely liquidated in period 1, the Central Bank is forced to stop selling dollars; if this occurs when there are still some depositors trying to exchange pesos for dollars, we say that there is a ``balance of payments crisis.'' Under these assumptions one can show that if all depositors act honestly the social optimum is implemented: Proposition 4.4. The fixed exchange rate system with a lender of last resort has an honest equilibrium. The proof is similar to previous ones and left to the reader. Now, the same international illiquidity condition under which there was a bank run in the previous regime implies that with a lender of last resort there is a balance of payments crisis: Proposition 4.5. If (4.11) holds, there is an equilibrium in which all depositors claim to be impatient and the Central Bank is unable to service all dollar demands in period 1. Proof. If all depositors claim to be impatient, they all arrive to the Central Bank with x pesos in period 1. The Central Bank obtains b dollars from the bank, and rk from liquidating the long term asset. Given (4.11), the Central Bank runs out of dollars and closes. Clearly, the bank does not close in period 1. Since all deposits are paid in the first period, the bank has no further liabilities vis-a-vis depositors in period 2. 14 It remains to check that each depositor is acting optimally. This is easy and left to the reader. K 14

The bank has no resources to repay its debt to the Central Bank in period 2but this is not important, since by then the Central Bank has become bankrupt. Alternatively, one can assume that the debt is cancelled when the Central Bank fails to return the long term asset to the commercial bank.

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Note that in this case there is still a bank run in the sense that all agents declare themselves to be impatient. However, there is no bank failure, since the Central Bank stands ready to rescue the commercial bank. If agents run, it is not because they want to get to the commercial bank in time, but because they want to get to the Central Bank in time with the pesos they withdraw from the commercial bank. One conclusion from the analysis is that, with fixed exchange rates, whether international illiquidity leads to bank crises or to balance of payments crises depends crucially on the credit arrangements between the Central Bank and the commercial bank. However, the root source of financial fragility is the same. The socially optimal allocation implies that investment in the world liquid asset is less than the implicit short term dollar liabilities of the banking system. This has been identified before as a potential problem under fixed rates. However, our analysis differs from previous ones in showing that international illiquidity has benefits as well as costs: the fixed rates regime of this section exacerbates international illiquidity but has a good equilibrium that improves upon a currency board. The analysis of this subsection assumes that the Central Bank fights devaluation to the bitter end. This follows, in particular, from the assumption that the Central Bank stops selling dollars in the first period only after the long term asset has been liquidated. One may ask what would happen with a less committed Central Bank; for example, the Central Bank procedure could call for stopping the sale of dollars before the long term asset has been exhausted. Such a rule, in fact, would amount to a rule for when to devalue the peso and, hence, is best delayed until flexible exchange rates are considered. Before that, we shall study some policy measures that could be used to prevent runs under fixed rates.

5. NARROW BANKING VERSUS WAR CHESTS 5.1. Narrow Banking Given the possibility of bank runs and balance of payments crises with fixed rates, some observers have suggested that commercial banks should be required to hold enough liquidity to cover all of their demandable debt. We now show that this policy, which has been termed narrow banking, can in fact eliminate equilibrium runs, but at the cost of implementing an inefficient allocation. To analyze narrow banking, suppose that the Central Bank fixes the exchange rate at one. Also, commercial banks are allowed to borrow pesos in period 1 to be repaid in period 2 at no interest. Finally, in order to ensure

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the ``stability'' of the financial system, commercial banks are constrained by xb

(5.1)

which ensures that the bank's total amount of demandable debt is completely backed by the world liquid asset. This restriction is the crucial difference between the fixed exchange rate system with Central Bank credit and a narrow banking system. The analysis of the bank's problem is a standard exercise. Here we will only discuss the relevant aspects of the solution, which will be identified with asterisks. First, under narrow banking the bank provides the satiation level of pesos to patient agents, that is, M*=m. This is as expected, because in period 1 the marginal cost of pesos to the bank is zero. However, it can also be shown that Central Bank credit is not necessary. 15 The reserve requirement 5.1 is strictly binding at the solution. This and the bank's feasibility constraints now imply that [R&(1&*)] x*+(1&*) y*=eR

(5.2)

which should be thought of as the transformation curve in a narrow banking regime. Comparing (5.2) against (4.5) makes clear why narrow banking is costly: the ``price'' of x under narrow banking is R&(1&*), which is larger than *R, the price of x when narrow banking is not imposed. Intuitively, this occurs because, under narrow banking, an increase of x of, say, dx, requires increasing b by more than *dx, the difference being an increase in required reserves. The last optimality condition is the natural tangency one, R&(1&*) *u$(x*) = . (1&*) u$[/(m )+ y*] 1&*

(5.3)

Comparisons with the social optimum are now straightforward. It can be shown that x*
These facts do not contradict each other because it turns out that, at the optimum, the bank's holdings of the world liquid asset are more than sufficient to finance all period 1 withdrawals. In other words, in period 1 the bank holds ``excess liquidity,'' which is of course a consequence of the imposition of narrow banking.

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21

However, this comes as a cost, as the resulting allocation is not socially optimal. 16 5.2. War Chests It has also been suggested that, to prevent crises, governments should implement a tax-transfer scheme to finance an ``emergency'' dollar fund to be used in the event of bank runs. This is allegedly equivalent to using ``fiscal,'' as opposed to ``monetary'' policy, to ensure the stability of the banking system. It turns out such a war chest policy may eliminate runs, just like a narrow banking approach, but is also inefficient. In fact, narrow banking turns out to dominate the war chest approach. To support our assertions, suppose that the Central Bank builds a ``war chest'' of dollars to be used in case of a bank run. To do this, it imposes a lump sum tax of T dollars on banks in period 0. If there is no run, it returns the T dollars to the banks in period 2 as a lump sum transfer. The statement of the commercial bank's problem is straightforward and we only discuss the relevant aspects of the solution, which will be identified by hats. First, the bank provides the satiation level of pesos to patient agents in period 1that is, M =m. This should be obvious since the bank's cost of pesos between periods 1 and 2 is zero. The tangency condition now is that the slope of the bank's objective function be equated to the slope of its perceived transformation curve, R* *u$(x^ ) = . (1&*) u$[/(m )+ y^ ] 1&*

(5.4)

Note that the relative prices perceived by the bank are the same as with a fixed exchange rate with no taxes. Finally, the bank's problem clearly depends on the size of the ``war chest'' T. To permit comparison with the narrow banking case, consider the case in which b+T=x^

(5.5)

so that the economy's liquid assets are sufficient to back the total amount of demandable debt. In that case, the solution must lie on the following social transformation curve, [R&(1&*)] x^ +(1&*) y^ =eR.

(5.6)

16 This follows from the fact that the bank is now solving the same problem as in the fixed rates case studied above, but with the tighter constraint xb instead of x*b.

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Clearly, if (5.5) holds, the resulting allocation can be decentralized in a demand deposit system that cannot be subject to equilibrium runs. In that sense, narrow banking and war chests are similar. But they are not equivalent. Comparing (5.2) and (5.6) implies that the economy faces the same set of options between x and y as with narrow banking. However, (5.4) must also hold. This implies that, with a war chest, the social marginal rate of substitution between x and y is not equated to the social marginal rate of transformation at (x^, y^ ). The intuition is clear: because taxes and transfers are lump sum, the bank does not internalize the implicit cost of building the war chest T. This cost exists because the Central Bank keeps all of T in a liquid form, instead of investing T optimally. The divergence between the social and private returns to investment creates a distortion that does not exist under narrow banking. 5.3. Observations on Deposit Insurance Instead of a buildup of foreign exchange reserves at the Central Bank, the war chest policy just discussed could be interpreted as a deposit insurance institution. Under such an interpretation, the tax T would be seen as an ``insurance premium'' paid by commercial banks and held by the insurance agency. The analysis of the preceding subsection would then mean that deposit insurance, if designed appropriately, can prevent crises. However, the analysis also implies that deposit insurance can eliminate crises only if at least two conditions are met. First, the size of the insurance fund, plus the bank's liquid assets, must be enough to cover the potential liabilities of the banking system, as given by (5.5). Second, the financing of deposit insurance is crucial: here, the insurance agency must collect the premium T in period 0 and invest it in the liquid asset. In practice, deposit insurance systems may fail on both counts. Insurance schemes are often designed to cover only the idiosyncratic liquidity shortages of individual banks; as a result, their size is smaller than needed in the event of an economy-wide crisis. Such an ``underfunded'' insurance scheme cannot be effective in preventing crises; this is clearly the case in our model if T fails to satisfy (5.5). Also, it may be the case that the way deposit insurance schemes are financed is inconsistent with other policy objectives, particularly a fixed exchange rate. Suppose, for instance, that the deposit insurance institution is funded not by collecting the premium T from the commercial banks, but with a credit from the Central Bank. It is easy to see that the analysis is essentially the same as when the Central Bank acts as a lender of last resort: even if T is large enough to prevent the commercial banks from defaulting on its nominal obligations, there may be a panic equilibrium in which all depositors withdraw early and attack the Central Bank.

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23

The fact that recent crises have occurred in spite of the existence of explicit or implicit deposit insurance may indicate that insurance schemes are poorly designed, in the sense just discussed. Finally, it bears reemphasizing that, in our model deposit insurance can be effective only by giving up implementing the social optimum.

6. FLEXIBLE EXCHANGE RATES Until now we have assumed that the Central Bank has the policy of defending the exchange rate until it is no longer feasible. In this section we drop that assumption and analyze a flexible rate system. Our main result is that flexible rates result in the implementation of the social optimum. Moreover, the possibility of a devaluation implies that equilibrium runs are eliminated if policy is chosen appropriately. 6.1. Flexible Rates with a Lender of Last Resort We focus on the question of how the social optimum can be implemented by a demand deposit system. To examine this issue, we retain all of the assumptions of the case of fixed rates with a lender of last resort, except that the Central Bank is no longer committed to selling dollars to depositors at a fixed exchange rate of one for as long as this is possible. Instead, we assume that the Central Bank allows exchange rates to change according to financial events. Consider the following mechanism. The commercial bank and depositors agree on the same contract as under fixed rates. When period 1 begins, depositors arrive to the commercial bank in sequence, and each withdraws either x or M pesos. The commercial bank services withdrawals of M pesos by borrowing from the Central Bank; withdrawals of x pesos are financed by first liquidating the world liquid asset and selling the dollar proceeds to the Central Bank at an exchange rate of one, and then by borrowing from the Central Bank. After all depositors have visited the commercial bank, the Central Bank sells dollars to depositors at an exchange rate equal to E 1 =Max[(* r*), 1] where * r denotes the fraction of agents reporting impatience. Hence, if * r >* there will be a devaluation. This devaluation rule can be interpreted in one of two equivalent ways. The first is that the Central Bank, after observing the actual pattern of withdrawals at the commercial bank, announces a devaluation. This rule is consistent with the existence of a sequential service constraint at the Central Bank, since it will be seen that the devaluation is such that the Central Bank will not run out of dollars. A second interpretation, perhaps more appealing, is that the Central Bank does not ``fix'' the exchange rate, but rather there is no sequential service at the Central Bank. Instead, the

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exchange rate is determined by an auction. In the auction, the Central Bank offers the dollars obtained from the commercial bank, and each of the * r depositors that reported impatience offers the x pesos withdrawn from the commercial bank. The devaluation rule implies that each agent that reported impatience receives xE 1 dollars in the first period, which can be used for purchasing consumption in the world market. In period 2 the bank liquidates the long term asset and sells the resulting Rk dollars to the Central Bank at an exchange rate of Q pesos per dollar. Here Q=1+[(* r &*) x ]Rk and, therefore, it may depend on what happened in period 1; we will discuss shortly the reason for this devaluation rule. The commercial bank then repays its debt to the Central Bank and distributes its remaining pesos to the (1&* r ) agents that reported patience in period 1. Finally, these agents use their pesos to purchase dollars from the Central Bank at an exchange rate E 2 determined in an auction, and in turn they use those dollars to purchase consumption. Given this procedure, it can be proven that the optimal allocation is an equilibrium: Proposition 6.1. There is an honest equilibrium whose outcome is the social optimum. In this equilibrium the exchange rate is one at all times. Proof. Suppose all agents report their types honestly. Then * r =*, and hence E 1 =Q=1. It can be shown that in period 2 the commercial bank distributes exactly Rk pesos to the patient depositors. Since in period 2 the Central Bank sells the Rk dollars obtained from the bank, the exchange rate at which those dollars are sold must also be one. The rest of the proof follows previous arguments and is left to the reader. K The Proposition makes it clear that this regime may implement the social optimum. In addition, if this equilibrium prevails, the outcome is the same as in the honest equilibria with fixed rates. In spite of the equivalence of their honest outcomes, flexible rates and fixed rates are quite different. Under the proposed flexible rate regime, the possibility of a peso devaluation eliminates the incentives for any runs: Proposition 6.2.

Runs cannot occur in equilibrium.

Proof. Clearly, if a run is to occur, * r must be greater than *. We will show that this implies that lying is not optimal for patient depositors. Given the rules in this regime, neither the bank nor the Central Bank fails in any period. Now, Q has been defined so that E 2 is exactly one. To see this, note that in period 2 the debt of the commercial bank to the Central

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Bank is (* r &*) x +(1&* r ) M. Hence the commercial bank distributes QRk &(* r &*) x &(1&* r ) M to (1&* r ) depositors, each of which arrives to period 2 with M pesos. As a consequence, depositors go to the Central Bank with a total of QRk &(* r &*) x pesos to buy Rk dollars. The definition of Q ensures that these two quantities are equal, and hence E 2 =1. If a patient depositor lies, she gets xE 1 =*x* r *, Rk(1&* r )>Rk(1&*)= y. Since *x * r
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real peso holdings close to the satiation level. 17 This is necessary to ensure that patient agents do not profit from participating in a run. The devaluation policy described in the previous subsection is one of the many possibilities that accomplish that goal. 18 Finally, it should be noted that, in equilibrium, a devaluation does not occur. 19 In other words, our flexible rate regime imply that the exchange rate remains fixed at one. This result is clearly special and hinges on the absence of aggregate uncertainty. However, it underscores that distinguishing between flexible and fixed rate regimes may be difficult in practice; in particular, the variance of the nominal exchange rate may be a misleading criterion.

7. ON DOLLARIZATION So far we have maintained the assumption that bank deposits must be paid in domestic currency. But bank accounts denominated in foreign currency (e.g., dollars) are observed on many countries, and there is heated debate about their desirability. Our framework can be easily amended to shed light on such dollarization questions. Consider first the currency board regime of Section 3, but assuming that deposits are paid in dollars instead of pesos. The main change is that, in period 1, the bank pays x~ dollars to depositors reporting impatience and M dollars to those reporting patience. However, since the latter need pesos and not dollars, they instantaneously exchange the M dollars for M pesos at the bank. The bank finances such an exchange, in turn, by selling the M dollars to the currency board. With this system, there is not need for depositors to visit the Central Bank. The other details should be obvious. Clearly, the regime just described works as if the bank paid patient withdrawals in period 1 in pesos and all others in dollars. It should also be clear that this is only a mechanical difference, and the outcomes are exactly the same as those of a non-dollarized currency board. Consider next happens in a dollarized fixed exchange rate regime. If the Central Bank extends liquidity but not emergency credit to the commercial 17

This is not an issue for the devaluation policy that we consider, which implies that the price level (and the exchange rate) is always one in period 2. 18 Here is an alternative devaluation policy, assuming that y x In period 2, the Central Bank sells pesos to the bank at an exchange rate of one, and later places only Rk &(* r &*) x dollars for sale to reportedly patient depositors. Since the latter will arrive to the auction with Rk &(* r &*) x pesos, E 2 will be one. It can also be shown that, if y x, each patient depositor will consume at least y. Hence patient types will not lie. 19 In this sense, flexible rates are similar to suspension of payments in [10].

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27

bank, as in Subsection 4.3, the analysis is exactly analogous to that of the dollarized currency board, except that the M pesos paid to patient types in period 1 are financed by Central Bank credit. It should be easy to see that the outcomes of this regime are the same as those of its non-dollarized counterpart. Dollarization, however, makes a big difference when we consider the Central Bank's potential role as lender of last resort. The Central Bank can print pesos but not the dollars needed to honor the dollar claims of impatient depositors. Nor can it acquire more dollars than those already available to the commercial bank by liquidating b and k. In short, the Central Bank cannot serve as a lender of last resort in a dollarized economy. The last implication of dollarization now emerges naturally: flexible exchange rates cannot help prevent bank runs. This is because it was not flexible exchange rates alone that were able to rule out runs in Section 6 above, but the coupling of exchange rate flexibility with a lender of last resort. In short, dollarization makes no essential difference in a currency board or a fixed exchange rate system. But it prevents the Central Bank from serving as a lender of last resort, and consequently it eliminates the potential benefits of flexible rates. In this sense, our analysis gives credence to concerns that dollarization can be destabilizing. 20

8. THE ROLE OF INTERNATIONAL BORROWING The assumption that domestic agents cannot, either individually or collectively, borrow in the international market is very strong; fortunately, it is also much stronger than we need. To show this, in this section we sketch how the model can be amended to allow for international borrowing. Such an extension is also useful to discuss the effects of an increase in capital inflows, such as that experienced by emerging markets in the first half of the 1990s. To allow for inflows of international capital, suppose that the economy is the same as before, except that each domestic agent can borrow in the world market up to a strictly positive credit ceiling f, at the world interest rate of zero. The ceiling f is assumed to be finite, but it can be arbitrarily large. Hence this extension represents a considerable relaxation of our previous assumption. On the other hand, the existence of the credit ceiling will be taken as exogenously given. While it deserves further justification, we delay discussion of this issue to the next section. With our new assumption, the analysis of the model is formally very similar to that of its previous version. For concreteness, and to show the 20

See, in particular, [23].

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necessary modifications, we now sketch the analysis of a fixed exchange rate regime. Consider first the socially optimal allocation. The country's planning problem is now to maximize (2.1) subject to (4.1), (4.3), (3.6), (3.5), d0, and k=d+e (1&*) y+(d+b)Rk d+b f.

(8.1) (8.2) (8.3)

In this problem, d denotes the amount borrowed abroad in period 0. The rest of the notation is the same as before, with an important exception: b now denotes foreign borrowing in period 1. With these changes, the problem is formally the same as in the previous version of the model, except that (4.2) and (3.1) are replaced by (8.2) and (8.3), respectively. The resource constraints have a slightly different interpretation. To limit the number of cases to be considered, we assume that the credit ceiling f is sufficiently large relative to the initial endowment e so that in period 0 illiquid investments take up the whole of e plus some foreign borrowing, as shown by (8.1). Constraint (4.1) now says that the consumption of impatient agents must be financed by additional borrowing abroad in period 1. Expression (8.3) restricts total foreign borrowing not to exceed the credit ceiling; note that the ceiling applies to the outstanding debt in either of the first two periods. Finally, (8.2) requires that the return on the long term investment be sufficient to pay for the consumption of the patient and the service of accumulated external debt. The optimal allocation will be denoted by x + and so on, and must satisfy (4.4), (4.6), and R*x + +(1&*) y + =eR+ f (R&1)

(8.4)

which is the social transformation curve in this case. Comparing (8.4) against (4.5) it is clear that foreign borrowing increases the economy's wealth by allowing it to take advantage of the profitable cross-border return differential. This advantage is growing in the size of the credit ceiling f. The analysis of the implementation of the social optimum via demand deposits is essentially the same as that of the earlier version of the model. One minor change is that the demand deposit contract requires individuals not only to turn their endowments over to the bank, but also to allow the bank, effectively, to borrow abroad on their behalf. Alternatively, the bank

FINANCIAL FRAGILITY AND EXCHANGE RATES

29

contract could require individuals to borrow abroad d + dollars in period 0 and b + dollars in period 1, and deposit those dollars at the bank. An additional assumption is required because the commercial bank now has an additional set of claimants: foreign creditors. The way these claimants are treated, and the possibility of international default, may in fact affect the analysis of crises in a substantial way. The simplest possibility is that external debts are repaid under all circumstances; we shall focus on this case here, leaving an informal discussion of the others for the next section. The repayment of foreign debt can be guaranteed if the liquidation of the long term asset is limited to at most 21 l+=

Rk + & f . R

(8.5)

Under these assumptions, it is easy to show that the results are similar to those of the model with no international borrowing. There is an honest equilibrium that implements the social optimum, and run equilibria exist if and only if x + >b + +rl + .

(8.6)

In this setup we can ask: what are the effects of an increase in international capital mobility, as captured by an increase in the credit ceiling f ? Clearly, a larger f makes it possible for the country to enjoy an improvement in welfare if the honest equilibrium occurs. On the other hand, whether it makes crisis condition (8.6) more likely to hold may depend on the third derivative of u. 22 Hence little can be said, at this level of generality, about how increased international mobility affects the possibility of crises. Despite the ambiguity just noted, our analysis clarifies conditions under which the abundance of international capital may contribute to financial fragility. An increase in the credit ceiling f makes it possible for a country to achieve better allocations. But this potential may not be realized, since 21

This limit is derived from the condition that, even if that much of the long term investment is liquidated, the yield on the remainder be sufficient to repay the foreign debt, or d + +b + = f R(k + &l + ). Hence, if the Central Bank does not provide last resort funding, the commercial bank always services the foreign debt. If it does, our assumption is that, in a run, the Central Bank also respects the limit l + in liquidating the long term asset and takes over the service of foreign debt. 22 To see this, note that (8.5) and the constraints on the planning problem imply that + x &(b+ +rl + )=(1&*) x + &rl + =[(1&*)(Rx + &ry + )]R. Hence x + &(b + +rl + ) increases in f if Rx + &ry+ does. But, using the Implicit Function Theorem in (8.4) and (4.6), one arrives at R x +f &r y + f B R 2u"[/(M + )+ y + ]u"(x + )&r, which can be positive or negative.

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the country's international liquidity position may deteriorate. In our model. this occurs when a larger f increases the discrepancy between the potential short run obligations of the banking system and the amount of short run funds it can tap. Should policy discourage borrowing from abroad? An analogy with the case of the precautionary measures considered in Section 5 may be illuminating. Those measures implied setting aside some resources in liquid form, which could eliminate run equilibria at the cost of inducing inefficient outcomes. Here, in the case of limited borrowing ability on the part of domestic agents, some of that borrowing capacity could be set aside to be used in the event of a runbut, for now obvious reasons, that would be inefficient. By contrast, if domestic agents use the credit line efficiently, devoting part of it to the long term investment, they can improve welfare by shifting out the social transformation curve; yet at the same time, they can make the economy less liquid and render a run equilibrium more likely.

9. DISCUSSION AND FINAL REMARKS We have provided a detailed and formal account of the possible interactions between bank fragility and the exchange rate and monetary regimes. We have shown, in a world in which banks play a well defined microeconomic role, that different nominal regimes induce different real allocations, and also that they imply varying degrees of financial fragility. Some of the assumptions we have made are particularly strong and may call into question the generality of our results. Consequently, in this section we elaborate on these assumptions, discuss how they could be weakened, and speculate on the consequences of doing so. We noted in Section 2 that our way of modeling the demand for pesos was overly restrictive. To be sure, some aspects of our specification can be changed without altering the essence of the analysis. In particular, the assumption that impatient agents do not derive utility from holding pesos is only one of convenience; assuming that their utility, like that of patient types, depends on real peso balances can be accommodated easily. Suppose, for instance, that the utility of impatient agents is not u(x) but u[x+/(NE 1 )], where N denotes pesos held between periods 1 and 2. The first order conditions for the social planning problem of Subsection 4.1 would then include (4.5), /$(N)=0, and u$[x+/(N)]=Ru$[ y+/(M)]. Amending the rest of our analysis to deal with this case is a straightforward matter. In a demand deposit system, impatient agents would have the right to withdraw x+N pesos in period 1, and to exchange x pesos for dollars at the Central Bank; the illiquidity condition (4.11) would still be the crucial condition for the

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existence of crises, although the values of x, b, and k would be those given by the new planning problem. One noteworthy difference, which the interested reader can verify, is that the illiquidity condition (4.11) would be less likely to hold with this specification. The reason is that, given that impatient agents gain from holding pesos, the planning solution would reduce x relative to the previous case. Likewise, it would be straightforward to allow for utility functions that include dollar balances. For example, one may suppose that the utility of patient agents is given by u[ y+/(ME 2 )]+&(D), where D denotes dollars held between periods 1 and 2 and & is an increasing, concave function. The planning solution would then assign some dollars to patient agents in period 1. The interested reader is invited to supply the remaining details. Clearly, it would be efficient for D to be financed by increasing the bank's liquid investment. Hence this extension, like the previous one, would make illiquidity less likely. While some features of our model can be generalized easily, there is the more fundamental objection that introducing a demand for pesos by inserting them into utility functions is a brute-force method. We decided to maintain that assumption for tractability, and also because assuming money in the utility function has been shown, in some contexts, to be a valid approximation to other, more natural ways to justify a demand for fiat money. But further exploration is needed to identify how such an approximation affects our analysis. This may involve explicitly modeling the fundamental frictions that may give money a role, as in [1, 4, 26, 28], and other recent monetary versions of the DiamondDybvig model. One result that does seem sensitive to the way we introduce pesos is the welfare difference between currency boards and fixed exchange rates. That difference depends crucially on the fact that a currency board imposes a positive opportunity cost, associated with the foreign exchange reserves backing peso holdings between periods. That opportunity cost would disappear, for example, if agents (somewhat implausibly) did not demand pesos between periods; this would occur under some assumptions on transactions patterns. In such a case, currency boards and fixed exchange rates would differ only in that the former does not allow the Central Bank to be the lender of last resort. We believe that, with the qualifications just mentioned, our other results are robust to changes in the way the demand for pesos is introduced. This is because our analysis is largely based on the relationships between depositors, the commercial bank, and the Central Bank; the precise source of the demand for pesos does not make much of a difference. A second assumption which deserves elaboration is that foreign borrowing has a limit, even if that limit can be arbitrarily large. We are not alone in making this assumption: the huge literature spawned by [19] uniformly

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assumes that domestic central banks cannot borrow international reserves, and some limit on international borrowing would seem to be needed if we are to study currency crises. 23 Rationalizing some credit limit should not be, by itself, a hard enterprise: one may appeal to the many theories of informational imperfections or frictions associated with sovereign debt. However, the rationale one chooses may affect the particular form of the credit limit. Our analysis has postulated a very simple specification of the credit limit; our results may depend on that specification. This is, in our view, a promising question for future research. A final comment is warranted on the assumption, made in the previous section, that the liquidation of the long term asset is limited so as to always guarantee the repayment of foreign debt. This assumption was made mostly for the sake of tractability. Dropping it would not eliminate the results of this paper; however, it would introduce new possibilities, derived from the potential default on foreign debt. In particular, the possibility of foreign creditors' panics, and their interaction with domestic bank runs, would have to be examined. We did not tackle this issue here in order to keep our focus on exchange rate regimes and monetary policy; however, we have made some progress in [5]. Our analysis can and should be extended in many other directions. In studying the choice of exchange rate regime, one should compare the advantages of flexible rate systems identified here against some well understood advantages of fixed ratessuch as their beneficial impact on policy credibilitythat we have abstracted from. Another important extension would focus on the role of international rescue packages and, in particular, the need for an international lender of last resort. Many of the problems with fixed rates identified here follow from the Central Bank's ability to print pesos but not dollars. Hence, the Federal Reserve System and international institutions such as the IMF, which do have access to dollars in times of trouble, can potentially play a crucial role in the design of appropriate policy to prevent and manage crises. Finally, our focus has been on self fulfilling runs, but this is of course not the only possible explanation of currency crises. In particular, the Asian collapse has given rise to a number of models that emphasize moral hazard in the presence of government guarantees to private borrowing. 24 The question of whether actual crises are best explained by such incentive problems or by self-fulfilling panics remains a fertile area for research. 23 Such a crisis is by definition a situation of extreme scarcity of foreign exchange, which would not occur if solvent domestic residents could borrow abroad all they wanted at the world rate of interest. This is why all prominent theories of currency crises, including [19], impose some restriction on foreign borrowing. 24 See, in particular, [11, 20, 21].

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