Liquidity Shocks and Optimal Monetary and Exchange Rate Policies in a Small Open Economy?

Transcription

1 TBA manuscript No. (will be inserted by te editor) Liquidity Socks and Optimal Monetary and Excange Rate Policies in a Small Open Economy? Joydeep Battacarya, Rajes Sing 2 Iowa State University; joydeep@iastate.edu 2 Iowa State University; rsing@iastate.edu May 9, 2005(Very Preliminary Drat) Abstract Tis paper studies te potential or liquidity crises and teir impact on te course o monetary and excange rate policies in a microounded general equilibrium dynamic model in te tradition o Diamond and Dybvig (983) and Cang and Velasco (2000). We produce a small open economy pure excange overlapping generations model wit random relocation along te lines o Smit (2002). Te combination o random relocation and te assumed role o currency in interlocation trade creates random location - and country - speci c liquidity needs. Banks naturally arise to provide muc-desired insurance against tese liquidity socks. In tis setting, i witdrawal demand or te bank s deposits is ig enoug, te bank will exaust all its cas reserves and a liquidity sortage will occur. We provide a complete caracterization o optimal interest rate policies in tis setting. In a deterministic set up, we nd tat nominal interest rates tat are desirable rom a welare perspective may also lower te probability o a liquidity crisis. We go on to study te classic issue o te relative desirability o xed versus exible excange rate regimes by introducing time-varying random endowments into te above structure. Tis makes te banks portolio allocations dependent on te excange rate regime. Under a xed excange rate regime, by interest rate parity, te banks portolio coice is deterministic and constant over time and tis is supported by te injection/removal o nominal balances by te central bank. Under a exible excange rate regime, te money supply stays constant but a rate-o-return uncertainty emerges tat is in addition to te income uncertainty common to bot regimes. Our results sow tat a exible excange rate regime is superior in an ex-ante welare sense relative to te xed excange rate regime; owever, te ordering is reversed wen it comes to te likeliood o liquidity crisis.? Very preliminary drat prepared or submission to conerences. Te latest version can be downloaded rom ttp://

2 2 Joydeep Battacarya, Rajes Sing Introduction A bank s liabilities are typically sort-term deposits, wile most o its assets are usually eld in long-term, less liquid investments. As as been recognized at least since Diamond and Dybvig (983), wile te maturity transormation activity o banks produces major social bene ts, it also exposes te banking system to te possibility o liquidity socks. Suc socks ave long been understood to be an integral component o banking crises; indeed as early as Noyes (907), te de ning caracteristics o a banking panic ave been known to include te suspension o cas payments by banks to depositors, te depletion o cas reserves at banks, te emergence o a currency premium,... (Camp, Smit, and Williamson, 996). More recently, te importance o liquidity socks in te creation and propagation o nancial crises as been underscored by Cang and Velasco (998) wo identiy international illiquidity as a necessary and su cient condition or nancial crases especially in te context o te 997 Asian crisis. Tis paper studies te potential or liquidity crises and teir impact on te course o monetary and excange rate policies. Te principal ocus, as in Cang and Velasco (2000), is on te liquidity-provision and maturity transormation unctions o banks. In our setting, banks ace uncertain liquidity demand or a oreign currency rom teir depositors. 2 Wen suc demand is su ciently ig, banks run out o cas reserves in te sense o Noyes (907), and a liquidity crisis occurs. In an small open economy setting wit ree nancial ows, policy-induced canges in domestic as well as world interest rates determine te opportunity cost o liquidity, tereby in uencing te possibility o suc crises. Policy makers weig te social bene ts o liquidity provision and maturity transormation by banks against te potential costs o exposing te economy to a crusing liquidity crisis. Te kinds o monetary and excange rate policies tat a benevolent policy maker ougt to ollow in te wake o tis tension constitutes te principal subject matter o our paper. We produce a single good, pure excange overlapping generations, small open economy model, tat is a marriage o Smit (2002) wit Betts and Smit (997). In our set up, tere are two spatially separated locations in te domestic economy. Agents are initially assigned to a location witin te country. We start by studying a setting in wic tey receive a xed non-stocastic endowment only wen young and care only about teir consumption wen old. Near te end o a period, a xed raction o tese agents must move to te oter location witin te same country Banking crises, or more generally, substantial banking sector woes, are by no means rare. Lindgren et al. (996) ave reported tat over te period at least two-tirds o IMF member countries experienced signi cant banking sector problems. 2 Cang and Velasco (998) de ne a country s nancial system to be internationally illiquid i its potential sort term obligations in oreign currency exceed te amount o oreign currency it can ave access to on sort notice [pg. 2]

3 Title Suppressed Due to Excessive Lengt 3 (enceort ome movers ); additionally a random raction o agents will ave to relocate to te rest o te world (enceort oreign movers ). Te sole real asset is a storage tecnology wic, unlike in Smit (2002), cannot be prematurely liquidated. Te only asset tat individuals may transport across domestic locations is te domestic currency (enceort "pesos"); wen relocating to te rest o te world, individuals must carry te oreign currency (enceort "dollars") wit tem. Te domestic central bank issues its own at currency pesos wereas dollars can be obtained rom te rest o te world. Eac currency is dominated in its return by storage. Te combination o random relocation and te assumed unction o currency in interlocation excange creates random location and country speci c liquidity needs. Banks naturally arise to provide muc-desired insurance against te possibility o relocation. At te start o any period, tey take in deposits and divide teir portolio between pesos, dollars, and storage. Once te relocation sock is realized, tey provide payments to depositors tat are contingent on teir relocation status. As originally studied by Camp, Smit, and Williamson (996) and more recently by Smit (2002), suc a setting is conducive to te occurrence o liquidity crises. Indeed, i demand or dollars turns out to be too ig, te bank will exaust all its reserves o te dollars, and a liquidity crisis will appear. O course te banks portolio depends crucially on te returns to te domestic and oreign currencies, wic in turn, are determined by te money growt rates in te domestic country and te rest o te world. It is ere tat te international dimension added to Smit (2002) starts to sow some action. To x ideas, let us x our attention on te case tat we study extensively below, one were a xed known raction o agents in te economy move to oter witin-country locations but a random raction o agents in te ome country move to te rest o te world (or simplicity o exposition, assume no one rom te rest o te world moves to te ome country). In tis setting, wen te stationary nominal interest in te domestic economy (I ) is iger tan a cuto I p i.e., wen te peso as a su ciently low return relative to storage, we can sow tat banks use up all peso reserves to pay te ome movers. Clearly, non-movers get a iger return tan ome movers. Wen te realized raction o oreign movers is below an endogenous tresold, it is not e cient to exaust all dollar reserves on te ew oreign movers. Instead, it makes sense or te bank to equalize te ex-post returns to oreign movers and non-movers. Tis tey can acieve by selling some o its dollar reserves and paying te non-movers in goods in te ollowing period. Above te aorementioned tresold, all te dollar reserves o te banks are used up to pay te oreign movers. Te latter get a lower return tan te non-movers. As discussed in Smit (2002) and Camp, Smit, and Williamson (996), tis situation can be labelled a banking crisis or a liquidity crisis. As long as te ex-post rate o return is equalized between oreign movers

4 4 Joydeep Battacarya, Rajes Sing and non-movers, bot obtain a iger rate o return relative to ome movers. However, as te raction o oreign movers increases, te return obtained by oreign movers is lowered wile tat o non-movers is raised. I too many oreign movers are realized, ten te return obtained by oreign movers may even all below tat received by ome movers. For a uniorm distribution or te oreign relocation probability, we can analytically caracterize te portolio weigts or dollar and peso reserves. In particular, we can sow tat te iger is te oreign nominal interest rate (I ); te lower is te portolio weigt on dollars, and lower is te tresold oreign relocation probability at wic banks exaust all teir oreign cas reserves. In sort, te iger te opportunity cost o oreign unds, te iger is te probability o a liquidity crunc. Tis result is in line wit te empirical evidence on banking crises as presented in Eicengreen and Rose (998) were tey nd tat a one percent increase in te Nortern interest rates is associated wit an increase in te probability o Soutern banking crises o around tree percent. Wen I < I p obtains, matters are substantially di erent; indeed, tere are two endogenous tresolds or te oreign relocation socks tat we ave to contend wit. Below te rst tresold, te bank is able to provide complete insurance; non-movers, oreign and domestic movers all receive te same return ex-post. Tis requires banks to not use up all teir ome currency reserves to pay te domestic movers. Instead, or suc realizations o relocation socks, some pesos are eld back to buy goods or te non-movers. Similarly, not all te dollar reserves o te ome banks are used up to pay te oreign movers. Once te rst tresold is crossed, te banks can no longer protect te oreign movers even as tey end up exausting all teir dollar reserves. At tis point, te non-movers and te domestic movers continue to earn te same return, a return tat is iger tan wat te banks can o er te oreign movers. Ater te second tresold as been reaced, te return to te oreign movers continues to decline; te return to non-movers keeps increasing as te same amount o storage is now being divided among lesser and lesser number o non-movers. In turn now, it may not be necessary to transer pesos to non-movers in te same amount as beore. Banks exaust all pesos to pay te domestic movers wose return is now lower tan wat te non-movers get. We nd tat te portolio weigt attaced to pesos (dollars) alls (rises) wit te domestic nominal interest rate. Interestingly, te portolio weigt attaced to pesos is iger tan te known raction o ome movers implying tat banks optimally old more o te pesos tan could be justi ed by te purpose o paying te domestic movers. We can also sow tat te probability o a dollar liquidity crisis is decreasing in te domestic nominal interest rate. I te rest o te world ollows a zero dollar in ation policy, ex ante welare, in general equilibrium, is maximized wen te domestic nominal interest rate equals te rate o storage,

5 Title Suppressed Due to Excessive Lengt 5 i.e., te net peso in ation rate is zero. Tis result underscores a tension between welare and crisis probabilities rst recognized in Smit (2002): wile monetary policy can reduce te crisis probability by reducing te nominal interest rate (and approac te Friedman rule), suc policies may urt aggregate welare. We go on to study te classic issue o te relative desirability o xed versus exible excange rate regimes. To tat end, we introduce time-varying random endowments into te above structure. Tis makes te banks portolio allocations dependent on te excange rate regime. Under a xed excange rate regime, by interest rate parity, te banks portolio coice is deterministic and constant over time and is supported by te central bank injecting or witdrawing nominal balances. Tis in turn generates a novel inter-generational income redistribution. Under a exible excange rate regime, te money supply stays constant (ence no income redistribution) but a rate-o-return uncertainty emerges tat is in addition to te income uncertainty common to bot regimes. Our results sow tat a exible excange rate regime is welare superior relative to xed excange rate. However, te ordering is reversed wen it comes to dollar liquidity crisis. To get an intuitive albeit inormal sense o tis, we start by noting tat under te exible rate regime, te excange rate adjustments and consequent interest rate canges induce banks to tilt teir portolios towards te igest paying assets. Suppose or instance tat markets anticipate a low devaluation in any period. Banks will ten tilt teir portolio towards pesos, wic in turn will induce a current revaluation. As a result, te expected devaluation will now be iger. Tus, in equilibrium, excange rate adjustments will be muted due to canges in portolio demand. Tus, even toug te exible excange rate regime su ers rom te rate-o-return uncertainly, te price mecanism ensures tat te range o uctuations is diminised. Fixed excange rate regime, on te oter and, eliminates rate-o-return uncertainty, but te ine ciency tat stems rom income redistribution makes it worse. Essentially, te excange-rate adjustment mecanism along wit portolio coice implies tat banks will be allocating teir assets in te most e cient way. However, note tat, despite tese adjustments te domestic assets sometimes may become too attractive, and banks will put less weigt on dollar reserves under suc cases. As a result, te probability o crisis under exible excange rates is iger tan under xed excange rates. Our work is one in a line o papers tat ollow te spirit o te original modeling insigts o Diamond and Dybvig (983) and create microounded general equilibrium structures in wic banks arise naturally and are susceptible to crises. Te paper tat is closest in spirit to our work is Cang and Velasco (2000). Tere te autors study a small open Diamond-Dybvig economy wit tree periods were a subset o agents (te patient ) derive direct utility rom olding te domestic currency. We make progress by generalizing [teir] results to truly dynamic settings and to more satisactory speci cations o money demand. (Cang and Velasco, 2000).

6 6 Joydeep Battacarya, Rajes Sing Additionally, a major advantage o our analysis is tat we can quantiy te exact probability o a banking crisis; Cang and Velasco (2000) are only able to sow conditions under wic a banking crisis will occur. A point o contrast wit teir results is te ollowing. Cang and Velasco nd tat a exible rate system implements te social optimum and eliminates runs wile we nd tat te probability o crisis under exible excange rates is not zero and indeed is iger tan under xed excange rates. Te rest o te paper is organized as ollows. In Section 2, we extend Smit (2002) to an small open economy environment and ten caracterize banks equilibrium portolio allocations. Section 3 turns to studying optimal monetary policy in a deterministic environment. Section 4 introduces endowment uncertainty into our basic ramework and ten goes on to comparing xed and exible excange rate regimes. Some conclusions are o ered in Section 5. Proos o all major results are in te appendices. 2 Te Model 2. Te environment Consider a small open economy wit a single tradable good wose price is determined in te world markets in terms o te world currency. Te economy as two symmetric spatially separated locations. Eac location is te ome o an in nite sequence o two-period lived overlapping generations. At eac date t = ; 2; 3; :: a continuum o ex ante identical young agents o unit mass is born. A young agent receives an endowment o w; old agents receive no endowment. For te analysis in tis section we assume tat w is xed over time. Agents care only about teir second period consumption (c) ordered by ln (c). In wat ollows, variables superscripted wit () represent te rest o te world (domestic) variables. At te start o eac date, newly born agents are assigned to one o two locations in te ome country. Towards te end o te period, ater asset markets sut, some young agents will ave to move across locations domestically and some will ave to travel to te rest o te world. Te rest stay put (te non-movers ) in te location tey were born in. Speci cally, let be te raction o agents wo get relocated witin te country (enceort ome movers ) and denote te raction o agents wo get relocated to te rest o te world (enceort oreign movers ). We assume tat any agent s realization o te sock is public inormation. To keep analytics simple, we posit tat is known and constant. However, is assumed to be a random variable wit a known distribution (). Following Smit (2002), te only real asset is a (commonly available in te world) storage tecnology: F () =, i.e., unit invested in tis tecnology yields a sure gross real return o

7 Title Suppressed Due to Excessive Lengt 7 te ollowing period. Relocated agents cannot transport goods to teir new omes. Witin te country, agents can carry te domestic currency (ereater pesos ) to te oter location and subsequently trade teir currency against consumption goods. Relocations abroad require agents to carry wit tem te world currency (ereater dollars ). Unlike Smit (2002), we disallow investments in storage to be liquidated (scrapped) even at a cost. As will become clear below, tis assumption along wit te limited communication assumption implies tat towards te end o a period, any excess currency may be converted into goods to give non-movers in te same location, but any excess stored goods cannot be given to movers since tey are at a di erent location by ten. Tis asymmetry will ave important implications below. Te role o banks is motivated as ollows. At te start o a period, young agents at any location receive teir endowment; at tat point asset markets open. Agents potentially make decisions on ow muc currency and ow muc storage to old. Ater tis is done, asset markets close. Te relocation sock is subsequently revealed. At tis point, agents wo are relocated witin and across nations will need to acquire te appropriate currency. Only te currency o teir destination is valuable to tem; all teir oldings o oter assets, or lack o an open market or trade, is wortless to tem. In sort, agents in tis economy ace considerable private risks associated wit relocation. We assume competitive banks arise to supply insurance to tese risk averse agents. It will be evident tat all young agents will coose to deposit teir entire endowment wit a bank, i.e., all saving activity will be intermediated. As Nas competitors on bot te deposit and te asset sides, banks will make portolio coices about storage and te two currencies so tat tese coices maximize ex-ante expected utility o a representative depositor subject to balance seet constraints described below. Beore proceeding urter, some de nitions and notation are in order: m t M t p t ; m t = M t p t ; I t p t+ p t ; I t p t+ p ; () t were M denotes te supply o pesos, M denotes te amount o dollars eld te domestic economy; and were p and p denote te peso and dollar price o consumption good, respectively. It is assumed tat banks acquire dollars in eac period by selling goods, deposited by ouseolds, to te rest o te world at a price p. Assuming tat tere are no oter transactions costs, it is ten implied tat p = ep ; were e denotes te nominal excange rate denominated in units o domestic currency per unit o world currency. Using () wit purcasing power parity condition yields I t = I t p t+ p t p p t+ t = I t e t+ e t = I t ( + " t+ ) (2)

8 8 Joydeep Battacarya, Rajes Sing were " t+ denotes te (gross) rate o excange rate devaluation between t and t+. Henceort, we will ocus on situations were It ; I t olds, implying tat storage dominates te real return to olding eiter o te currencies. 2.2 Te bank s problem Te problem o portolio selection aced by banks is as ollows. As all agents are ex-ante identical, tey deposit teir endowment at banks wo in turn make portolio allocations between pesos, dollars, and storage as given by te per depositor resource constraint w + t m t + m t + s t ; (3) ere t = M t M t denotes te net government transer (seigniorage) to te young domestic p t agents, m and m denote te goods value o peso and dollar reserves respectively, and s denotes storage. No transers are made to te old. For uture reerence, de ne t m t ; t m t : (4) w + t w + t Ten ( ) represents te peso (dollar) reserve to deposit ratio o te ome banks. Notice tat = s w+ must old (tis is te portion invested in real storage). Banks, at te start o a period, and upon receiving deposits decide wat teir and sould be. At tis stage, banks know, but as not yet been realized. Ater asset markets close until te next period, te oreign relocation sock is revealed. At tat point, te banks will ave to pay domestic (oreign) movers in pesos (dollars). Muc will depend on te exact realization. In tis environment, unlike in Betts and Smit (997), banks announce a scedule o returns or te tree types o agents tat is contingent on. Let r () and r () denote te gross real returns promised to agents relocated domestically and abroad, respectively; let r n () denote te gross real return o ered to te non-movers. Finally let () denote te -contingent raction o dollar reserves (per depositor) tat is used to pay oreign movers; similarly, let () denote te -contingent raction o peso reserves (per depositor) tat is used to pay domestic movers. Tese returns must satisy (see Appendix 6.) ( ) r n t () ( t ()) t r t () t () t rt () t () t! p t p t+ pt p t+ p t p t+! + ( t ()) t ; (5a) ; and (5b) pt p t+ + t t : (5c)

9 Title Suppressed Due to Excessive Lengt 9 Tese constraints ave standard interpretations. Consider as an example, constraint (5a). Once is realized, te banks becomes aware tat it as to pay r () to eac o tese oreign movers. It uses up a raction o its per depositor dollar reserves or tis purpose. Foreign movers in total receive $ () p wic wen brougt to te rest o te world yields t per unit in te orm o goods next period. Te constraint (5b) is simpler. Banks know upront and so tey know in advance tat a raction o teir clientele will move domestically and will ave to be given r () : Tey can nance tis payout by (partially) using teir reserves o pesos, () : In te new location, tis cas is wort () pt p t+ p t+ in goods. Finally, note tat, ; () and () are cosen at te beginning o period, and can potentially be time-dependent; ence, we represent tem wit time subscripts. 3 On te oter and, is realized ater all coices ave been made. Hence, in order to economize on notation, we suppress time subscripts rom. As noted earlier, banks are Nas competitors in te deposit market, wic implies tat tey maximize te expected utility o a young agent. Once te deposits are made, banks coice problem eac period is identical, and we can drop time subscripts. Te banks problem can be stated as max ; ;; Z 0 ln r () + ln r () + ( ) ln r n () () d (6) subject to (3) - (5c) and non-negativity o ; ; ;and. Moreover, none o tese coices can exceed unity. For uture reerence, note tat in steady states, or stationary government policies, we can use () and (4) to rewrite constraints (5a) - (5c) more compactly as r n () = r () = () I ; r () = () I ; ( ()) I + ( ()) I + (7a) (7b) ; (7c) were, ater invoking optimality, it is assumed tat (5a)-(5c) will old wit equality. To reiterate, banks maximize (6) subject to (3) and (7a) - (7c). It is convenient to conceptualize te bank s problem as a two stage problem and work backwards: ) in te second stage, given and ; te banks coose () and () or eac, and 2) given te scedules () and (), tey coose and so as to maximize (6). Herein lies te seed o muc o wat is to come. Banks coose and by maximizing ex-ante expected utility. Since agents are risk-averse, te bank cooses and to try and provide insurance 3 As will be clear below, tese rules are time-invariant in te deterministic case wit stationary government policies. However, wen endowment uncertainty is introduced in 4, tese rules will depend on te endowment realization and ence will vary over time.

10 0 Joydeep Battacarya, Rajes Sing against te upcoming relocation sock. It aces a tension. On te one and, since te return on cas is lower tan tat on storage, te bank will want to economize on cas oldings. However, since storage cannot be liquidated, and insurance provision requires cas, te bank does not want too illiquid a portolio. Since asset markets close rigt ater tese coices ave been made, te bank is stuck wit its coices as it awaits te realization o : Ex post, te bank may or may not succeed in providing complete insurance. Too ig a witdrawal demand will compromise tis unction and precipitate a liquidity crisis. In particular, te bank may end up olding too many dollars or pesos wic cannot be converted into goods until te ollowing period (wen asset markets will open again). Limited communication implies tat any excess currency (converted next period into goods) can only be used to pay te non movers wo are te only depositors still present in te same location; te ome and oreign movers ave let and cannot be reaced any more. At te same time, i tere are excess stored goods, te bank cannot use tem to pay any o te relocated agents or te same reason. We start by ocusing on te second stage problem. Te stock o peso and dollar reserves is predetermined at tis point. It is easy to veriy tat te rst-order-condition or te coice o () is: r n () r () ; \ = i () < (8) Equation (8) states tat agents relocated abroad will get te same return as tose staying put (te bank provides complete insurance), only i tere is su cient dollar reserves in stock or te emigrants. O course tis does not mean tat in te event te (ex-post) realized is low, te bank would still distribute its entire dollar reserves equally among te oreign movers. In tat case, as we will see below, it may coose to convert some o its dollar reserves into goods and give to te non movers. I te ex-post is su ciently ig, eac oreign mover will get a low return and all dollar reserves will be exausted. Tis is because te number o non-movers is small and so teir per capita return (rom storage) will be ig relative to te oreign movers. However, it bears empasis tat since storage cannot be liquidated, banks cannot transer stored goods to agents wo ave already moved out. For uture reerence note tat using (), (2), (7a), and (7c), equation (8) can be rewritten as ( ()) ( ()) I + I + (), \ = i () < I (9) Similarly, te optimal coice o () ; i.e., ow muc o peso reserves to pay out, yields: r n () r () ; \ = i () < (0) Equation (0) as an interpretation similar to tat o (8): te bank provides complete insurance against te risk o domestic relocation, only i tere is su cient peso reserves in stock or te

11 Title Suppressed Due to Excessive Lengt ome movers. Again, using (), (2), (7b), (7c), equation (0) can be rewritten as ( ()) ( ()) I + I + (), \ = i () < I () Obviously, te decision to exaust reserves o any currency (as evident rom (9) or ()) partly depends on ow good one currency is relative to te oter. To oresadow, we nd tat tere is an (endogenously determined) nominal interest rate, I p ; tat serves as an important tresold rate. I te domestic interest rate, I ; is greater tan tis I p (i.e., te return to pesos is low enoug ) te ollowing are true and consistent in an equilibrium: a) oreign movers and non-movers receive a return equal to =I p i te realized is below a tresold, b) banks old a peso-reserve to deposit ratio equal to te raction o ome movers () ; and subsequently exaust teir peso reserves ( () = ; 8) or any realization o ; c) te ome movers get a return equal to I [see (7b)] wic is independent o I and any realization o. As I approaces I p and alls below it; te equilibrium canges; now te ome movers join te oter two types and get a common return (pooled return rom pesos, dollars, and storage) i te realized is below a tresold. In tis case, owever, te bank s coice o and () will also depend on I. 2.3 Te case o I > I p Suppose I > I. Ten pesos are te worst assets. Tere would be no gain in involving peso oldings into insuring movers or non-movers under varios scenarios. Given log preerences, ten banks would like to simply allocate a sare o current deposits to pesos. How about wen I < I? It turns out tat even wen I is smaller tan I ; but greater tan te endogenously determined common return on dollar oldings and storage, de ned as I p, banks still separate ome-movers portolio allocation problem rom te rest. In tis case =, and () = ; 8: Tese conjectures are veri ed in equilibrium below. Proposition Te ollowing constitute an equilibrium. De ne and Ten i I I p ; I p ; I ( ) + ( ) I : (2) 8 i < + I ; i ; () = ; (3) :, i >

12 2 Joydeep Battacarya, Rajes Sing () = ; 8; and te state contingent returns to te di erent types are given by 8 < ; i, r () = Ip : ; i >, I and Additionally, 8 < r n () = : r () = I p, i, ; i >. (4) (5) I < rn () = r () i (6) = ; and obtains. < and > Te state-contingent returns o ered by te banks are easily computed using (3) in (7a) and (7c). Using our conjecture tat = ; it is clear rom (7b) tat r () = ; wic does not I depend on I or on te realization o. Ten, using (4) - (6) in (6), along wit some rearrangement yields W = Z a 0 Z + ( a ) ln ( ) ln Z I () d + ln a I Z () d + ln 0 () d I () d (7) Te ex-ante coice o and is derived by maximizing te expression or stationary welare in (7). How is I p computed? Note tat as long as <, te rate o return obtained by oreign movers and non-movers are equal. Using (9) and (3) and = ; we get were I p implies tat I p I = + I I p is te common rate o return to oreign movers and non-movers or all. Tis i I.

13 Title Suppressed Due to Excessive Lengt 3 We are now in a position to veriy our conjecture, tat in equilibrium, te bank will exaust its peso reserves. Indeed () = or all is indeed true or all I > I. To see tis, note rom (9) and () tat () = or all i and only i () I or all () (8) I Furter, using (3) and te last part o Proposition, eq. (8) can be rewritten as I I (9) Tat is, () = or all i and only i (9) olds. Using te last part o Proposition, it ollows tat (9) trivially olds or all I I. In sum, te qualitative eatures o te equilibrium described in Proposition above, are as ollows. Wen te domestic nominal interest rate is greater tan I p (i.e., te return to pesos is low enoug ), we nd tat a) oreign movers and non-movers receive a return equal to =I p i te realized is below a tresold; oterwise te oreign movers receive muc less, b) banks old just enoug pesos to pay te ome movers and subsequently exaust teir peso reserves ( () = ; 8) or any realization o ; c) te ome movers get a return equal to I wic is insulated rom I and realizations o. [see (7b)] Intuitively, te ollowing issues are at te eart o te matter. Banks would like to provide insurance to te tree types, ome movers. oreign movers, and non movers, but only upto wat is allowable by te e ciency conditions. Holding too muc cas balances urts storage and olding too little urts te relocated agents. Te question central to all tis is: wen te return to pesos is low enoug, wy do banks old just te rigt amount o pesos and no more? in oter words, wy are ome movers kept insulated rom te insurance problem tat te bank is solving or te oter two types? In our setting, banks know te exact number o ome movers, but tey do not know ow many non movers and oreign movers will be realized. Consumption e ciency requires tem to provide insurance to te oreign movers. Te bank is aware tat tere is always a cance tat tere may be too many oreign movers. Since te oreign movers are risk averse, teir utility alls more rom a marginal loss in consumption tan it rises rom a marginal gain. As suc, te bank reserves more dollars tan wat te expected population sare o oreign movers would justiy. Tis comes at a cost; reserving extra dollars impedes production e ciency since it is associated wit too low investment in storage. Furtermore, given te aorediscussed asymmetric nature o te payouts, any excess dollars remaining (i ex post tere are too ew oreign movers) can only be used to pay te non movers but not te ome movers. In tis sense, te solution to te bank s insurance problem involves insulating te ome movers rom te relocation risk o oreign movers.

14 4 Joydeep Battacarya, Rajes Sing Overall, te bank s optimal sceme gives te same return (=I p ) to oreign movers and non movers and a lower return to ome movers =I i ew oreign movers are realized. I too many oreign movers are realized, te bank gives relatively more to non movers; te oreign movers su er and may even get a return lower tan te ome movers. As discussed above, i too ig a oreign relocation probability is realized, te bank cannot convert pesos to dollars (since te asset market does not reopen until te ollowing period by wic time, te oreign movers cannot be contacted any more). I a low enoug oreign relocation probability is realized, given te low return on te pesos, te oreign movers get a iger return anyway. As suc, under no circumstances can te bank use its peso reserves to o er additional consumption protection to te oreign movers. Consequently, it simply olds just enoug pesos to pay te ome movers and no more. 2.4 Te case o I < I p Wen te domestic currency yields a relatively superior return, i.e., I < I p ; banks may not always want to exaust teir reserves o domestic currency. Indeed, tere are two endogenous tresolds or te oreign relocation probability tat become relevant. Below te rst tresold, te bank is able to provide complete insurance; non-movers, oreign and domestic movers all receive te same return ex-post. Tis requires banks to not use up all teir peso reserves to pay te domestic movers. Instead, some pesos are converted into goods te ollowing period to pay te non-movers. Similarly, not all te dollar reserves o te banks are used up to pay te oreign movers. Once te rst tresold is crossed, te banks can no longer protect te oreign movers even as tey end up exausting all teir dollar (te weaker currency) reserves. At tis point, te non-movers and te ome movers continue to earn te same return, a return tat is iger tan wat te banks can o er te oreign movers. Ater te second tresold as been reaced, te return to te oreign movers gets lower and lower; te return to non-movers will keep increasing as te same amount o storage is now being divided among lesser and lesser number o non-movers. In turn now, it may not be necessary to transer peso reserves to non-movers in te same amount as beore. Banks exaust all peso reserves to pay te ome movers wose return is now lower tan wat te non-movers get. Formally, or all 2 [0; ]; (9) and () will old wit equality. Manipulating tem, one can rewrite + ( ()) I I + I ( ) (), \ = i () < (20a) ( ()) I I + + I ( ) (), \ = i () < (20b)

15 Title Suppressed Due to Excessive Lengt 5 Te optimal rules or use o currency reserves are collected in te next proposition Proposition 2 Wen I < I p ( )+ I ; is as obtained in Proposition were and 8 >< = >: ~ ( ) 8 < i ; = : i > ; i b ~ ; i 2 b ; ~ i ~ ; (2) i ; (22) I (23) I I I I + + ( ) I (24) I I + + I (25) Note tat te value o under I < I p is di erent rom tat in te case wen I > I p. But we continue to use te same notation; tis is due to te act tat any below gets a constant return tat equals tat o non-movers. Tis olds under bot cases. Figure exibits equations (2) and (22) grapically.

16 6 Joydeep Battacarya, Rajes Sing Figure For uture reerence note, r = r = r n = r = r = I I < r = r n = ~ I = I, or b b I ; or 2 ; ~ i < r = I < rn =, or 2 ~; i (26) We are now in a position to start investigating te optimal reserve oldings o te two currencies. Using (26), (6) can be rewritten as W = Z b 0 Z + + ~ Z ln I ( ) ln b ln Z ~ () d + I () d ( b ) ln! ~ I () d Z () d + ~ ln I () d We proceed to study ow te optimal reserve oldings respond to te domestic nominal interest rate, olding te oreign interest rate xed. Suppose we coose te ollowing parametric speci cation: = 0:3; = 0:6 < ; and I = :3; () is uniorm wit support [0; 0:4]and let I vary between.0 and.. Figure 2 plots and as I varies. Several items deserve mention. First, not surprisingly, and respond in exactly opposite ways; wile alls wit I ; increases wit it. As I increases wit I eld xed, and I stays below I ; banks decrease (increase) teir domestic (oreign) currency reserves. As I increases, banks want to economize on currency oldings; since te domestic opportunity cost is relatively increasing wit I, te bank reduces its peso oldings. Second, > or tis range o I and I implying tat banks optimally old more pesos tan tey need solely or te purposes o paying te ome movers. Finally, as a comparative static exercise, i I is raised to.35, te (down). locus sits up

17 Title Suppressed Due to Excessive Lengt 7 Figure 2 Discussion Contrast te two cases I > I p against I < I p. In general, banks would like to equate MRT wit MRS between te tree groups in expected terms. Wile tey know te number o ome movers, tey do not know ow many non-moverss and oreign movers are going to be out o te remaining. Tey would like to insure oreign movers in case too many o tem realize. Tis requires reserving more dollars tan teir expected population sare. Wy? So tat i too many come, tey don t end up wit too low a consumption. But, ten, reserving "extra" comes at te cost o sacri cing storage returns. Furtermore, given te asymmetric nature o payouts, any dollars let wen too ew oreign movers arrive can only be anded out to non-movers; but not to ome movers. In tis sense, te problem is jointly o oreign movers and non-movers. Te sceme gives same return to everyone i low is realized, gives more to non-movers wen is large. As long as te common return (or small ) is iger tan wat pesos get, tere is noting tat ome movers can do: even wen too many oreign movers arrive and teir return alls below ome movers, te latter are unable to elp tem. And o course, wit ew oreign movers, ome movers ave a lower return tan te rest anyway. Ten, te question is wat sare to reserve as pesos.? Here, one can argue tat, given te log utility, it is best to give to number o ome movers. However, i te ome movers return is ig enoug (i.e., I is low enoug) and is small, ome movers can surely elp by saring some pesos wit non-movers (and te latter can let

18 8 Joydeep Battacarya, Rajes Sing oreign movers ave more dollars in turn). But wat is te gain or te ome movers in tis sceme? Suppose peso sare is still, ten ome movers will be net losers. So, in order to get tem involved, wic ex-post makes sense, tey must be compensated by a peso sare iger tan. Tus, wen wen is ig, non-movers get ig returns, ome movers no longer need to pay (reinsure) tem, and tey can also enjoy a iger return (iger tan peso returns ( I > I ). Tus, a peso sare iger tan compensates omemovers under states o nature wit low. 3 General equilibrium, welare, and policy We allow te government to conduct monetary policy by canging te nominal stock o at currency at a xed non-stocastic gross rate > 0 per period, so tat M t = M t or all t. I te net money growt rate is positive ten te government uses te additional currency it issues to purcase goods, wic it gives to current young agents (at te start o a period) in te orm o lump-sum transers. I te net money growt rate is negative, ten te government collects lump-sum taxes rom te current young agents, wic it uses to retire some o te currency. Te tax (+) or transer ( or all t. In a stationary equilibrium, ten = ) is denoted t. Te budget constraint o te government is t = M t Mt p p = m t m t t t p t p t+ = and I = ; p t+ (27) m olds. For uture reerence, note tat maybe tougt o as te amount o seigniorage. We assume tat seigniorage is paid as lump sum transers to only young domestic agents. Since (w + ) = m; we ave = 2 4 w 3 5 ; and w + = w : Recall, owever, tat banks take as given and tereore teir equilibrium allocations ; are independent o te act tat equilibrium is a unction o. Fix I = :04; = :04; = 0:2; = 0:6. Te ollowing Figure 3 sows ow te welare o te economy varies wit te money growt rate.

19 Title Suppressed Due to Excessive Lengt 9 Figure 3 Proposition 3 Let te world rate o in ation be zero. Ten te optimal monetary policy is a xed money supply, i.e., ~ = : Te intuition beind tis result is simple. Given te act tat reserving domestic or oreign cas or movers is dominated by storage in te rate o return, te optimal allocation simply attempts to equate te marginal rate o substitution between consumption o di erent types o agents wit teir relative marginal rates o transormation. Any monetary policy oter tan a xed money supply introduces urter distortions troug transers rom ome-movers to te young o te next generation. Irrespective o weter tese transers are positive or negative, resulting allocations put a urter wedge in te planner s allocations and, tereore, are ine cient. How does beave as a unction o I, i.e., ow does te probability o a banking crisis cange wit te domestic monetary policy or, in turn, nominal interest rate? For te same parameter speci cations as in Figure 3, te ollowing Figure 4 presents (). Recall te iger te value o, te lower te probability o a crisis. Notice tat domestic interest rate can a ect only wen I < I p. Notice tat I p lies between :05 and :075:

20 20 Joydeep Battacarya, Rajes Sing Figure 4 Discussion It is clear tat te welare is maximized unambiguously by keeping te money supply or te price level constant, i.e., =. However, te probability o a dollar liquidity crisis may or may not be minimized at =. It as been sown tat as I increases, te crisis probability decreases. For simplicity, let s assume tat te monetary policy in te rest o te world is a constant money supply rule (it can be easily justi ed tat suc a rule is optimal or te rest o te world too). Ten, I =. Ten we ave te ollowing Lemma. Lemma Wen =, = minimizes crisis probability in addition to maximizing domestic welare. 4 Uncertainty and alternative excange rate regimes As observed above, wit te rest o te world ollowing an optimal monetary policy, te domestic optimal policy calls or a xed money supply. In our determinstic set up, tis also implies tat te price level or te nominal excange rate is also xed. Note, owever, tat in an uncertain world were agents ace exogenous uncertainty, eiter in terms o teir endowment or preerences, te excange rates will uctuate i a xed money supply policy is adopted. Ten, te question arises: wic excange rate regime, xed or exible, is welare superior? Does te regime tat obtains a iger welare also provides a better insurance against a dollar liquidity crisis? In order to meaningully analyze tese issues, in wat ollows, we introduce intrinsic uncertainty in te model.

21 Title Suppressed Due to Excessive Lengt 2 In particular, we propose tat young agents in eac period arrive wit wit uncertain endowment w. Speci cally w is i.i.d wit a known distribution over support [w; w]. Now tere are two sources o uncertainty. Wile w t is realized beore banks make portolio allocations, is realized aterwards. Hence, banks intial portolio allocation rules and as well as second stage reserve distribution rules and will be contingent on w. In addition, and will also be -contingent. As a result, uncertain endowments will induce a stocastic distribution o bot peso and dollar real balances. Below, we solve or stationary stocastic allocations as a xed point o a rational expectations equilibrium. As beore we continue to use witout time subscript since it is realized only ater all - contingent coices are made. However, all te rules will now depend on te current realization o w t. In order to denote teir w t dependence, we use time subscripts or all coices made at te beginning o t. Ater w is realized, te banks problem can be stated as Z Z max tt ; t ;t(); t ()g w t+ i ln rt () + ln r t () + ( ) ln rt n () () d subject to (7a) - (7c) wic are repeated below or convenience: r t () = t () t ; rt () = t () It ; " rt n () = ( t ()) t I t I t + ( t ()) I t # + t t ; (w t+ ) dw t+ Note tat w t+ denotes next period s endowment and (w t+ ) is its density unction. In addition, t ; t 0, t + t ; and t ; t 2 [0; ]. Notice tat portolio allocations and reserve distribution rules carry a subscript t wic denotes tat tey are w-contingent. As in te deterministic case, te problem is solved in two stages. In te second stage, ater is realized, t () and t () are optimally cosen given t state-contingent rules into account and determines t and t. (28) and t. Te rst stage takes tese optimal ex-post Witout loss o generality, we assume tat dollar in ation rate is zero; ten I =. Since and are now w-contingent, a constant money supply rule will imply tat te nominal excange rate will be stocastic. On te oter and, i te goverment sets an excange rate peg, te money supply will be stocastic. Potentially, te equilibrium allocations and welare under te two scenarios will be di erent, wic is te subject o study in te next section. A word on notation will be in order. Note tat all ex-ante ; and ; g rules will now be w-contingent: Tese rules will be cosen ater orming expectations o next period s

22 22 Joydeep Battacarya, Rajes Sing real balances tat in equilibrium, tat itsel will depend upon te aggregate rules ; and ; g. In turn, w-contingent rules will induce tese aggregate rules. In a stationary rational expectations equilibrium agents except tese rules to be ollowed next period and orm teir expectations o te aggregate variables on tis basis. It is in tis sense tat allocation rules and induced distributions constitute a rational expectations equilibrium. In equilibrium, current allocation rules coincide wit uture rules. Hence, given a realized value o w, eac period s problem is identical. Essentially, te solution entails nding a xed point o ex-ante portolio unctions (w), (w), (w; :) and (w; :). 4. Flexible excange rate regime Under a exible excange rate system, te money supply is kept constant, i.e., M t t. Ten, t = M t M t p t = 0. Hence, m t = t w t. Ten It = et+ e t banks rate-o-return constraints can be rewritten as = m t m t+ = M or all = t wt. Ten, t+wt+ r t () = t () t ; (30a) r () = () t m t+ m = t () w t+ t t+ ; (30b) w t r n () = ( ()) t + ( ()) w t+ t+ + t t ; (30c) w t Tus, banks maximize (28) subject to (30a) - (30c). Ater is realized, te rst-ordercondition or te coices o and are given by 8 < ( ) E t :( t ()) t + ( t ()) w t+ t+ w t + 8 < ( ) E t : w t+ t+ w t ( t ()) t + ( t ()) w t+ t+ w t + t t t t 9 = ; t () t 9 =, \ = i t ()(3) < (32) ; t (), \ = i t () < Banks ocs, (3) and (32), command tat te expected value o a marginal peso (dollar) be equal across ome (oreign) movers and non-movers, in terms o teir marginal utilities o consumption. Equation (3) states tat once is realized, te bank would like to equalize te expected marginal utilities o consumption across oreign movers and non-movers unless it as already exausted all its dollar reserves. Similar conditions exist between movers and non-movers as expressed by (32). Notice tat (3) and (32) can be more compactly written as ( ) c t E t c n ; \ = i t () < ; t c E t t c n, \ = i t () < ; t

23 Title Suppressed Due to Excessive Lengt 23 Below we sow tat muc depends on te realization o w t. I w t is su ciently ig, e t is low and expected devaluation or It is ig. Below, we sow tat te allocation rules ten coincide wit tose under te deterministic case. On te oter and, i w t is su ciently low te allocation rules may be signi cantly di erent. 4.. Te case o w t > w I w t is su ciently ig te expected return on pesos is su ciently low. We begin wit te conjecture tat ten banks separate ome-movers rom te oreign-movers insurance problem. As long as banks decide not to involve ome-movers into te insurance problem, te act tat ome-movers rate o return is uncertain is o no consequence. In order to compute equilibrium allocations, we rst look or cases were te expected rate o return on pesos is su ciently low so tat, as in te deterministic case, banks equals and () = n n or all. Recall tat te expected rate o return on pesos is E pt t = E mt + t m t o. De ne It e E t m t+ m t g as te opportunity cost o peso reserves relative to storage.4 Ten ollowing Lemma 2 veri es tat te above conjecture is indeed correct. Lemma 2 Let I e t > I p p t+ o ( ) were is computed as in Proposition. Ten te equilibrium allocations coincide wit tose in Proposition (a) t = ; t () = or all (b) were 8 i < + t t ; i t () = t ; t :, i >, ( ) t t ; t + t Te result ollows rom te act tat wit excange rate uncertainty it is only te omemovers tat ace a real rate o return uncertainty in case all pesos are disbursed to tem. Recall tat te realization o does not a ect te expected return o pesos. Since te expected rate o n o n o e return on pesos E t t e t+ = E mt+ t m t = I, it ollows tat as long as te expected rate o e return on pesos is less tan tan te "pooled" return on dollars and storage, I, it is optimal p or banks to coose = and () = or all. Note tat even te tresold value is now a unction o w t ; ence, it is denoted wit a time subscript. Note as t = or all It e > I p it ollows tat E mt+ t m t = Et( wt+) t+ w t < Ip : Since w is i.i.d. E t t+ w t+ = E wg. Furtermore, as I p is independent o w t, It e > I p olds i and only 4 In equilibrium m t = t wt. As w is i.i.d. and banks problem eac period, ater wt is realized, is otterwise identical, te portolio allocation rules t and t are stationary unctions o w t. Tus I e t = t w t Ewg :

24 24 Joydeep Battacarya, Rajes Sing i w t > w Ewg I. Intuitively, wen te current endowment is ig relative to te average, p te current price level (excange rate) is lower tan te average. Ten te peso is expected to depreciate. Once again, recall tat I p is determined endogenously wen ex-ante banks coose to separate. As expected return on pesos gets iger, suc tat t = may lead to peso returns exceed te pooled return on dollars and storage, banks will nd it ex-post e cient to not to disburse all pesos to omemovers i is su ciently low. In tis event, allocation rules will be substantially di erent as discussed in Section 2 and in Proposition 2. It is now obvious tat tis will appen i and only i w t < w: Tis is taken up next w < w Here w t < w. Ten te peso is expected to appreciate. It is ten not e cient to disburse all te pesos to ome movers wen is su ciently low because pooled dollar and storage return alls below expected peso returns. Banks ocs, (3) and (32), command tat te expected value o a marginal peso (dollar) be equal across ome (oreign) movers and non-movers, in terms o teir marginal utilities o consumption. O course, i is su ciently ig, te banks will disburse all te dollars to oreign movers. Similarly, i is too ig, a low number o non-movers obtain a ig return rom storage, and ten all pesos will be disbursed to ome-movers. In particular, wen < t, te pooled return rom dollar and storage is below tat expected on pesos. Ten banks allocation rules and ensure tat everyone gets te same return. Tis is done by letting linearly increase wit : At = t all te dollars are given to oreign movers, yet some pesos are retained or te non-movers. As gets larger a iger amount o storage goes to non-movers. Now banks would like to keep less o pesos or non-movers. Tus, begins to rise as rises beyond t. Finally, wen = ~ t, =. For all iger values, all pesos (dollars) are disbursed to ome (oreign) movers, and non-movers simply consume out o storage. Formally, or all 2 [0; t ]; it will be optimal or banks to equalize returns across all groups as some o te pesos can be carried over to provide goods to non-movers during te next period. Tus, as long as < t, te expected marginal utility across all tree groups is equalized and bot (3) and (32) old wit equality. Manipulating tem, t is now obtained as E t t ( t ( t)) w t+ t+ w +( t = ( ) t t t ) ; (33) + E t t ( t ( t)) w t+ t+ w +( t t t ) were t t is implicitly determined rom t t = + E t ( t ( t)) w t+ t+ w +( t t t ) E w t+ t+ w t t ( t ( t)) t+ w t+ w t +( t t )