Columbia University. Department of Economics Discussion Paper Series. Optimal Exchange Rate Policy Under Collateral Constraints and Wage Rigidity

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1 Columbia Universiy Deparmen of Economics Discussion Paper Series Opimal Exchange Rae Policy Under Collaeral Consrains and Wage Rigidiy Pablo Oonello Discussion Paper No.: Deparmen of Economics Columbia Universiy New York, NY 127 June 213 Updaed February 214

2 Opimal Exchange-Rae Policy Under Collaeral Consrains and Wage Rigidiy Pablo Oonello February 24, 214 Firs Draf: April 16, 212 Absrac his paper conducs a quaniaive sudy of he opimal exchange-rae policy in a small open economy ha faces he credi access unemploymen rade-off: In he presence of nominal wage rigidiy, exchange-rae depreciaion reduces unemploymen; in he presence of collaeral consrains linking exernal deb o he value of income, exchange-rae depreciaion ighens he collaeral consrain and leads o higher consumpion adjusmens. I is shown ha he opimal policy during financial crises generally feaures large currency depreciaion, since welfare coss relaed o higher unemploymen and lower consumpion ypically ouweigh welfare coss associaed wih ineremporal misallocaion of consumpion. he opimal policy also implies a lower currency depreciaion han ha necessary o achieve full employmen, which is consisen wih a managed-floaing exchange-rae policy, frequenly observed during financial crises in emerging marke economies. Sudden sops (or large curren-accoun adjusmens) are par of he endogenous response o large negaive shocks under he opimal exchange-rae policy. JEL Classificaion: E52, F41 Keywords: Exchange-rae policy, financial crises, deb deflaion, sudden sops, downward wage rigidiy, unemploymen Deparmen of Economics, Columbia Universiy. po2171@columbia.edu. I would like o hank my advisors Marin Uribe, Sephanie Schmi-Grohe, Guillermo Calvo, and Michael Woodford for guidance and suppor. I would also like o hank, for very useful commens and suggesions Javier Bianchi, Saki Bigio, Emi Nakamura, Jaromir Nosal, Diego Perez, Alessandro Rebucci, Ricardo Reis, Bernard Salanie, Jon Seinson, Joseph Sigliz, Erneso alvi, Valenina Duque, Mariana Garcia, Kyle Jurado, Joan Monras, Sebasian Rondeau, Dmiriy Sergeyev, and seminar paricipans a Columbia Universiy, he 212 EconCon held a Princeon Universiy, he 212 Midwes Fall Macro Conference held a he Universiy Colorado Boulder, he 213 CEPR European Summer Symposium in Inernaional Macroeconomics (ES- SIM), held a Izmir, urkey, he 213 Workshop on Macroeconomics, Financial Fricions and Asse Prices, held a Pavia, Ialy, and he 213 IEA-BCU Roundable held a Monevideo, Uruguay. 1

3 1 Inroducion During an exernal crises, exchange-rae policy in emerging marke economies (EMs) seem o leave policymakers beween a rock and a hard place: Prevening currency depreciaion could bring more unemploymen, bu if liabiliies are denominaed in foreign currency, currency depreciaion could increase deb in erms of domesic income, leading o financial desabilizaion, and compromising credi access. he poenial conflic for exchange-rae policy beween hese wo welfare concerns, credi access and unemploymen, is ofen a cenral elemen of he policy debae, as was observed during he Eas Asian and Lain American crises in he lae 199s (Fischer, 1998; Calvo, 21; Sigliz, 22) and during he peripheral European crises ha sared in 28 (see, for example, Krugman, 21; Feldsein, 211). his paper conducs a quaniaive analysis of he opimal exchange-rae policy when facing his credi access unemploymen rade-off. I consrucs an environmen ha provides a heoreical jusificaion for his rade-off, combining wo fricions ha have been largely sudied in he lieraure: a downward nominal wage rigidiy (as in Schmi- Grohe and Uribe, 211), and a financial fricion by which exernal borrowing is denominaed in he inernaional uni of accoun and limied by he value of collaeral in he form of radable and nonradable income (as in Mendoza, 22). In his framework, crediaccess and unemploymen are wo conflicing facors affecing welfare: Devaluaions are associaed wih a welfare gain by decreasing he real value of wages hey reduce involunary unemploymen bu are also associaed wih a welfare cos by increasing he value of exernal deb in erms of domesic income, hey ighen he collaeral consrain and can rigger an endogenous sudden sop. 1 he main finding, in calibraed versions of he model, is ha wo feaures characerize he opimal exchange-rae policy during financial crises (defined as episodes of binding credi consrains). Firs, he opimal allocaion generally implies a large real exchange rae depreciaion (beween a 17 and 4 percen fall on average in he relaive price of nonradables), which is achieved by allowing for nominal currency depreciaion. he reason is ha, he welfare coss relaed o higher unemploymen and lower consumpion are ypically higher han he welfare coss relaed o ineremporal misallocaion of consumpion. Second, opimal currency depreciaion is generally lower han ha associaed wih full employmen. hus, he opimal policy is consisen wih a managed-floaing exchangerae policy, frequenly observed in EMs during financial crises (see, for example, Calvo and Reinhar, 22). Moreover, he real exchange-rae depreciaion and curren accoun adjusmen under he opimal exchange rae policy during episodes of binding collaeral consrains is in line wih he dynamics observed in he daa during sudden sops. he 1 Calvo (1998) labeled sudden sops episodes of large and abrup reversals in exernal credi flows ha characerize EMs. For a review of he Fisherian deb-deflaion approach o sudden sops, including he form of collaeral consrain used in his paper, see Korinek and Mendoza (213). 2

4 paper shows ha he naure of he shocks and he srucural characerisics of he economy are key deerminans for he opimal degree of fear of floaing during financial crises: Higher exernal ineres raes, a larger ineremporal or inraemporal elasiciy of subsiuion, or a large mobiliy of labor across secors, call for a smaller unemploymen; a more elasic labor supply, or a higher share of income ha can be used as collaeral, call for more conained currency depreciaion. Welfare under he opimal exchange-rae policy is compared o ha under a fullemploymen and a fixed exchange-rae regimes. he full-employmen exchange-rae regime is cosly in erms of welfare for making consumpion adjus more han wha is opimal during periods of binding collaeral consrains. he fixed exchange-rae regime is cosly for inducing an inefficien adjusmen o negaive shocks wih involunary unemploymen, in periods of boh nonbinding and binding collaeral consrains. his differen naure of welfare coss generally makes he fixed exchange-rae regime more cosly, in erms of welfare, han he full-employmen exchange-rae regime. he welfare cos of he full-employmen and fixed exchange-rae regimes, wih respec o he opimal exchangerae policy, are larger in regions of he sae-space where he collaeral consrain binds, wih an average welfare cos during periods of binding collaeral consrains of.6 percen and 1.8 percen of consumpion per period, respecively. his is he firs paper ha conducs a quaniaive sudy of nominal exchange-rae policy under a collaeral consrain by which deb is limied by he value of collaeral in he form of radable and nonradable income. Inroduced in Mendoza (22), his form of financial fricion has been widely used o capure he main sylized facs abou sudden sops in EMs. 2 his form of collaeral consrain causes endogenous sudden sops hrough Fisher s (1933) deb-deflaion mechanism: Binding consrains lead o deleveraging, which leads o a fall in he price of nonradables, which furher ighens he collaeral consrain. Previous sudies using his form of financial fricion have considered real models, in which he policy insrumen during periods of binding collaeral consrains is a ax (subsidy) on nonradable or radable goods (see, for example, Benigno e al., 212a). he presen paper expands his lieraure by considering a nominal model and a moneary insrumen, which presen he policymaker a differen rade-off: While subsidizing nonradable goods leads simulaneously o increased employmen and higher prices of nonradable goods (relaxing he credi consrain), currency depreciaion leads o an increase in employmen and a decrease in he price of nonradable goods (ighening he credi consrain). 3 2 See, for example, Mendoza (25), Durdu, Mendoza, and errones (29), Korinek (211), Bianchi (211), Benigno e al. (211,212a,b,c). 3 In paricular, i can be shown ha in he model economy presened in Secion 2, using axes on nonradable or radable consumpion ogeher wih he capial-conrol ax, a Ramsey planner can achieve an allocaion characerized by full employmen and nonbinding collaeral consrain in all saes. Benigno e al. (212b) for a similar resul in an economy wihou wage rigidiy. See 3

5 he paper is relaed o he large body of lieraure ha sudies nominal exchange-rae policy in small open economies during financial crises. A key difference wih respec o his lieraure is he form of financial fricion sudied in he presen paper, which in urn leads o differen policy implicaions. For insance, in a large subse of his lieraure, borrowing access is linked o asse prices: Cespedes, Chang, and Velasco (24), Devereux, Lane, and Xu (26), Curdia (27), and Gerler, Gilchris, and Naalucci (27) sudy economies feauring he financial acceleraor mechanism (Bernanke, Gerler, and Gilchris, 1999). In a recen relaed paper, Fornaro (213) sudy an economy wih a collaeral consrain ha limis exernal deb o a fracion of he marke value of asse holdings (as in Bianchi and Mendoza, 211). In hese frameworks, currency depreciaions have a posiive effec on oupu, ha leads o higher asse prices and improved credi access. As a consequence, conrary o he presen paper s resul, in hese papers flexible exchange raes lead o more financial sabiliy during crises han fixed exchange raes. In he presen seup, borrowing access is linked o goods prices: Deb denominaed in a foreign currency is limied by he marke value of radable and nonradable income. he combinaion of a nonradable secor and liabiliy dollarizaion creaes a currency mismach ha makes currency depreciaion financially desabilizing (in line wih he radiional original sin argumen; see Eichengreen, Hausmann, and Panizza, 25; Calvo, 1999). he hypohesis of currency depreciaions being financially desabilizing has been previously formalized, for insance, in Aghion, Bacchea, and Banerjee (21), and in Braggion, Chrisiano, and Roldos (29) using credi consrains on firms. In hese papers, however, currency depreciaions are financially desabilizing because hey cause oupu conracion. In he presen paper, currency depreciaions are no conracionary (hey reduce unemploymen), bu he associaed currency mismach reduces he value of income, leading o a large consumpion adjusmen under binding credi consrains and enailing a welfare cos. For hese reasons, he form of financial fricion considered in his paper gives rise o a rade-off beween credi access and unemploymen ha has no been formally sudied in he lieraure of exchange-rae policy in small open economies during financial crises. he policy choice under he rade-off sudied in his paper has engendered a longsanding and sill lively policy debae. Keynes, for insance, was firs acively opposed o he reurn of Briain o he gold sandard afer World War I, arguing ha i would be associaed wih high unemploymen (Keynes, 1925). However, when he Grea Depression sared, Keynes recommended agains devaluaion, claiming ha now he coss in erms of deb revaluaion and financial desabilizaion would ouweigh he benefis (Irwin, 211). In he same line, Diaz-Alejandro (1965), analyzing Argenina s exchange-rae policy in he 195s, highlighed he possibiliy ha devaluaions would lead o negaive wealh effecs and adjusmen in consumpion from income disribuion and balance-shee effecs. his policy debae was riggered again by he crisis in peripheral Europe ha sared 4

6 in 28, in which here are, simulaneously, high unemploymen and high deb levels denominaed in euros. Moreover, he empirical lieraure suggess ha boh sides of he debae are suppored by evidence. Cross-counry regressions for EMs end o show boh ha fixing he exchange rae during financial crisis episodes is associaed wih larger oupu conracions (see, for example, Oriz e al., 29), and ha currency mismach plays a key role in deermining he access o inernaional credi markes (see, for example, Calvo, Izquierdo, and Mejia, 28). Finally, i is worh noing ha while mos of he above-menioned lieraure on nominal exchange rae policy in small open economies during financial crises compare differen (possibly nonopimal) exchange-rae regimes, he presen paper derives he fully opimal exchange-rae policy. 4 he paper shows ha he opimal allocaion is a nonmonoonic funcion of he saes; herefore, considering he opimal policy, insead of comparing exchange-rae regimes, is relevan. he res of he paper is organized as follows. Secion 2 presens he model economy. Secion 3 defines hree possible exchange-rae regimes in his seup (opimal, fullemploymen, and fixed exchange-rae policies) and provides analyical resuls describing he exchange-rae policy rade-off ha emerges in his economy. Secion 4 presens he quaniaive analysis comparing he aggregae dynamics and welfare under he hree exchange-rae regimes. Secion 5 examines he sensiiviy of resuls o differen calibraions and changes in he baseline model s assumpions. Secion 6 concludes. 2 he Model Economy his secion describes he model economy used o conduc exchange-rae policy analysis. I exends he wo-secor (radable and nonradable), dynamic, sochasic, small open economy model wih a downward nominal wage rigidiy from Schmi-Grohe and Uribe (211), o include a collaeral consrain in he form of radable and nonradable income. he economy only has access o a one-period, non-sae-coningen deb insrumen, denominaed in unis of radable goods, capuring liabiliy dollarizaion. he model hen feaures a nominal rigidiy and wo financial fricions ha will inerac o deermine he exchange-rae policy rade-off. radables are endowed o he economy, and heir price is deermined by he law of one price. Nonradables are produced by he economy, and heir price is deermined by domesic demand and supply. Flucuaions in he small open economy are driven by 4 Opimal moneary policy has been largely sudied in open economies wih complee asse markes, and in open economies in which he financial fricion is ha financial markes are incomplee; see, for example, Clarida, Gali, and Gerler (21), Schmi-Grohe and Uribe (21), Devereux and Engel (23), Corsei and Peseni (25), Gali and Monacelli (25), De Paoli (29), Corsei, Dedola, and Leduc (21), Schmi-Grohe and Uribe (211). he presen paper consiues a conribuion in his direcion for a small open economy in which financial fricions include an imperfec access o credi markes, wih he presence of occasionally binding collaeral consrains. 5

7 exogenous shocks o he value of he radable endowmen (which can be inerpreed as shocks o erms of rade or o produciviy in he radable secor) and o he ineres rae on exernal deb, wo sources of business-cycle flucuaions ha have been widely sudied in EMs (Mendoza, 1995; Neumeyer and Perri, 25; Uribe and Yue, 26). 2.1 Households Households preferences over consumpion are described by he expeced uiliy funcion: E β U (c ), (1) = where c denoes consumpion in period ; he funcion U ( ) is assumed o be coninuous, wice differeniable, sricly increasing, and concave; he subjecive discoun facor β (, 1), and E denoes expecaion condiional on he informaion se available a ime. he consumpion good is assumed o be a composie of radable and nonradable goods, wih a CES aggregaion echnology: c = A ( c, c N [ ) = a ( c ) (1 a) ( c N ] ) 1 1 1, (2) where c denoes radable consumpion and c N denoes nonradable consumpion. Each period, households receive a sochasic endowmen (y ) and profis from he ownership of firms producing nonradable goods (Π ). hey inelasically supply h hours of work o he labor marke. (Secion 5.3 relaxes hese assumpions sudying producion in he radable secor and an elasic labor supply.) Due o he presence of he wage rigidiy (discussed in deail in he nex secions), households will only be able o sell h h hours in he labor marke. he level of acual hours worked (h ) is deermined by firms and is aken as given by he households. Households have access o a one-period, non-sae-coningen bond denominaed in unis of radable goods ha can be raded inernaionally paying an exogenous and sochasic gross ineres rae R. he model herefore assumes full liabiliy dollarizaion. I is assumed ha he vecor of exogenous saes, s X [ y ], R, follows a firs-order Markov process. Deb acquired in period is axed a rae τ d. Households sequenial budge consrain is herefore given by d ( ) +1 1 τ d = d + c + p c N ( y ) + w h + Π, (3) R where d +1 denoes he level of deb assumed in period and due in period + 1, p P N P denoes he relaive price of nonradables in erms of radables, w denoes he wage rae in erms of radable goods, and denoes a lump sum ransfer in period. I is assumed ha households face a collaeral consrain by which exernal deb canno exceed a fracion κ of income: d +1 κ ( y + w h + Π ), (4) 6

8 where κ >. his form of collaeral consrain, inroduced in Mendoza (22), has been used exensively in he lieraure on small open economies o capure he effec of currency mismach on exernal credi-marke access: While collaeral includes income from boh radable and nonradable secors, exernal deb is fully denominaed in unis of radables. he credi-marke fricions from which his consrain arises are no modeled here explicily, bu his form of collaeral consrain can be seen as describing an environmen in which lenders manage defaul risk by imposing a deb limi linked o households curren income, as is ypically he case of lending crieria in morgage or consumer credi markes. 5 Empirical evidence suggess ha curren income is a significan deerminan of credi marke access (Jappelli, 199). In addiion, households are assumed o face a no-ponzi game consrain of he form d +1 d N, (5) where d N denoes he naural deb limi. As in Aiyagari (1994), his is defined as he maximum value of exernal deb ha he household can repay almos surely saring from ha period, assuming ha is radable consumpion is zero forever. Formally, denoing y as he minimum possible level of radable endowmen and R as he maximum possible level of exernal ineres rae, he naural deb limi is defined as d N R R 1 y. Since he collaeral value in he credi limi (4) depends on relaive prices which can be affeced by policy variables, consrain (5) is imposed in addiion o (4) is in order o preven Ponzi schemes induced by he policymaker (Mendoza, 25; Benigno e al., 212b). he household problem is o choose sae-coningen plans for c, c, c N, and d +1 ha maximize he expeced uiliy (1) subjec o he consumpion aggregaion echnology (2), he sequenial budge consrain (3), he collaeral consrain (4), and he no-ponzi game consrain (5), for a given iniial deb level, d ; for he given sequence of prices, w and p ; for he given sequence of hours worked, h, profis, Π, sochasic radable endowmen, y, and ineres rae, R ; and for he given sequence of policies, τ d and. Denoing by λ he Lagrange muliplier associaed wih he budge consrain (3) and by µ he Lagrange muliplier associaed wih he collaeral consrain (4), he opimaliy condiions (provided d +1 < d N ) are (2), (3), and (4), wih he firs-order condiions ( ) λ R 1 1 τ d = βe λ +1 + µ, (6) U c A ( c, c N ) = λ, (7) ( ) ( 1 a c and he complemenary slackness condiions a c N ) 1 P ( c, c N ) = p, (8) µ, µ ( κ ( y + w h + Π ) d+1 ) =. (9) 5 Korinek (211) shows ha his form of he collaeral consrain can be raionalized as a renegoiaionproof form of deb conrac in an imperfec credi marke in which households can renegoiae exernal deb and lenders can exrac a mos a fracion of borrowers curren income if deb is renegoiaed. 7

9 2.2 Firms Each period, operaing in compeiive labor and produc markes, firms hire labor o produce he nonradable good, y N. Profis each period are given by Π = p F (h ) w h, where he producion funcion, F ( ), is assumed o be increasing and concave. he firms problem is o choose h o maximize profis given prices p and w. he firs-order condiion of his problem is p F (h ) = w. (1) his condiion implicily defines he firms demand for labor. 2.3 he Labor Marke Nominal wages (W ) are assumed o be downwardly rigid as in Schmi-Grohe and Uribe (211): 6 for γ >. W γw 1, I is assumed ha he law of one price holds for radable goods, implying ha P = E P, where E is he nominal exchange rae and P is he foreign currency price of radable goods. Assuming ha P is consan and normalized o one, wages in erms of radable goods (w ) can be expressed as w = W E. From his, he wage rigidiy can be expressed as w γ w 1 ɛ, (11) where ɛ is he gross depreciaion rae of he nominal exchange rae: ɛ E E 1. However, acual hours worked canno exceed he inelasically supplied level of hours: h h. (12) When he nominal wage rigidiy binds, he labor marke can exhibi involunary unemploymen, given by h h. his implies a slackness condiion mus hold a all daes and saes: ( w γ w ) 1 (h ) h =. (13) ɛ 6 he assumpion of an asymmeric nominal wage rigidiy is consisen wih empirical evidence using microeconomic daa (e.g., Goschalk, 25; Baraieri, Basu and Goschalk, 21; Daly, Hobijn, and Lucking, 212). 8

10 his condiion means ha when he nominal wage rigidiy is no binding, he labor marke mus exhibi full employmen, and if i exhibis unemploymen, i mus be he case ha he nominal wage rigidiy is binding. 2.4 he Governmen he governmen deermines he exchange-rae depreciaion, ɛ, and imposes a proporional ax (subsidy) on deb τ d, which is rebaed lump sum o households ( ), o balance is budge each period: d +1 R τ d =. (14) Secion 3 defines differen exchange-rae regimes and how he capial-conrol ax is deermined. 2.5 General Equilibrium Dynamics he marke for nonradable goods clears a all imes: c N = F (h ). (15) Combining he equilibrium price equaion, (8), wih condiion (15), he firms opimaliy condiion, (1), can be expressed as ( ) 1 a (c ) 1 w = F (h ) 1 F (h ). (16) a Combining condiion (15) wih households budge consrain, (3), he definiion of firms profis, and he governmen s budge consrain, (14), he resource consrain of he economy becomes d +1 R = d + c y. (17) Using he definiion of firms profis, he equilibrium price equaion, (8), and he marke clearing condiion for nonradables, (15), he collaeral consrain, (4), can be reexpressed as ( d +1 κ y + ( 1 a a ) ) (c ) 1 F (h ) 1 d ( h, c, y ). (18) he general equilibrium dynamics are hen given by sochasic processes {c N, c, h, p, w, d +1, λ, µ, } = saisfying he se of equaions (GE): 9

11 (5): d +1 d N, (6): λ R 1 ( ) 1 τ d = βe λ +1 + µ, ( (7): U c c, c N ) ( A c, c N ) = λ, (8): p = P ( c, c N ), ( ( (9): µ, µ κ y + p F (h ) ) ) d +1 =, (11): w γ w 1 ɛ, (12): h h, ( ) (13): w γ w 1 (h ) ɛ h =, (14): = τ d d +1 R 1, (15): c N = F (h ), (16): w = ( ) ( ) 1 1 a a c F (h ) 1 F (h ), (17): d +1 R 1 = d + c y, ( (18): d +1 κ y + ( ) ( ) 1 ) 1 a a c F (h ) 1, given an exchange-rae policy {ɛ } =, a capial-conrol ax policy {τ d } =, iniial condiions w 1 and d, and exogenous sochasic processes {y, R } =. 3 Exchange-Rae Regimes: Definiions and Analyical Resuls his secion formally defines he opimal exchange-rae policy, and discusses he rade-off beween credi access and unemploymen ha exchange-rae policy can face in he model economy presened in he previous secion. Analyical resuls relaing credi access and unemploymen are esablished, providing a framework for undersanding he quaniaive characerizaion of he opimal exchange-rae policy o be presened in he nex secion. wo addiional exchange-rae regimes are also defined in his secion full-employmen and fixed exchange-rae policy o provide sandard benchmarks for he sudy of he opimal exchange-rae policy. 3.1 Definiion of Exchange-Rae Regimes his secion defines hree possible exchange-rae regimes: opimal, full-employmen, and fixed exchange-rae policy. Exchange-rae regimes are defined condiional on an opimal capial-conrol ax policy (τ d ). he reason for using his capial-conrol ax is wofold. Firs, previous lieraure has shown ha boh he credi consrain and he downward wage rigidiy considered in his paper embody a pecuniary exernaliy ha may induce inefficien exernal borrowing (Bianchi, 211; Beningo e al., 212a; and Schmi-Grohe and Uribe, 213). 7 he opimal capial-conrol ax policy eliminaes any borrowing inef- 7 Inefficien borrowing arises when he social coss of borrowing differ from he privae coss of borrowing. Bianchi (211) shows ha in an endowmen economy, he collaeral consrain in he form of radable 1

12 ficiency, and allows for a comparison across exchange-rae regimes isolaing he effec of exchange-rae policy from his disorion. Second, wihou he opimal capial-conrol ax, he se of resricions for he opimal policy includes a forward-looking consrain (namely, he household s ineremporal borrowing decision (6)). As shown in Kydland and Presco (1977), Bellman s (1957) principle of opimaliy fails in his conex, and sandard dynamic programming echniques canno be applied. Using an opimal capial-conrol ax echnically simplifies he problem, allowing for he use of sandard dynamic programming echniques. Neverheless, Secion 5.4 sudies he sensiiviy of he opimal exchange-rae policy o he assumpion of opimal capial-conrol axes by resricing he Ramsey planner s se of available insrumens o he nominal exchange rae. In his conex, ime-invarian opimal policies under commimen are obained using he recursive saddle-poin mehod developed in Marce and Marimon (211) Opimal Exchange-Rae Policy Definiion 1 he opimal exchange-rae policy wih opimal capial-conrol axes is he se of processes { ɛ, τ d } ha maximize households expeced lifeime uiliy (1) subjec o he se of equaions describing he general equilibrium dynamics (GE). o characerize he allocaion under he opimal exchange-rae policy wih opimal capial-conrol axes, I se up he Ramsey problem dropping consrains (6) (9), (11), and (13) (16). Appendix A shows ha any { d +1, c, h } ha saisfy (5), (12), (17), and (18) also saisfy (GE). he Ramsey problem is hen o maximize (1) wih respec o { d +1, c, h }, subjec o (5), (12), (17), and (18). he dynamics under he opimal exchange-rae policy wih opimal capial-conrol axes can be hus expressed wih he Bellman equaion, V OP ( s X, d ) [ ( ( = max U A c, F (h) )) + βe s X V OP ( s X, d )] (19) d,c,h s.. d R = d + c y, ( ( ) 1 a (c d κ y + ) ) 1 F (h) 1, a d d N, h h, and nonradable income induces overborrowing, in he sense ha he social coss of borrowing exceed he privae coss of borrowing; in his seup, he consrained social planner borrows less han he compeiive equilibrium. Beningo e al. (211, 212a) define overborrowing (underborrowing) as a siuaion in which a consrained social planner would ake on less (more) deb han decenralized agens; in his sense, he auhors find ha wheher an economy wih his form of collaeral consrain feaures overborrowing or underborrowing depends on he srucure of he economy (e.g., endowmen or producion), and on he calibraion. Schmi-Grohe and Uribe (213) show ha he downward wage rigidiy, combined wih a fixed exchange-rae policy, induces overborrowing. 11

13 where ime subscrips for variables daed in period have been dropped, and a prime is used o indicae variables daed in period +1; V OP ( s X, d ) denoes he value funcion for households under opimal exchange-rae and capial-conrol ax policies. his formulaion will be used in he quaniaive analysis Full-Employmen Exchange-Rae Policy For his regime, consider an exchange-rae policy aimed a mainaining full employmen a all saes and daes: Under he full-employmen policy, h = h,. (2) Definiion 2 he full-employmen exchange-rae policy wih opimal capial-conrol axes is he se of processes { ɛ, τ d } ha maximize households expeced lifeime uiliy (1) subjec o he se of equaions describing he general equilibrium dynamics (GE), and he full-employmen consrain (2). o characerize he opimal allocaion under he full-employmen policy, I follow he same sraegy as for he opimal exchange-rae policy and drop consrains (6) (9) and (11) (16). Appendix A shows ha any { d +1, c, h } ha saisfy (5), (17), (18), and (2), also saisfy (GE) and (2). herefore, he dynamics under he full-employmen exchangerae policy wih opimal capial-conrol axes can be expressed wih he Bellman equaion, V F E ( s X, d ) = max d,c [ U ( A ( c, F ( h ))) + βe s X V F E ( s X, d )] (21) s.. d R = d + c y, ( ( ) 1 a (c d κ y + ) 1 F ( h ) ) 1, a d d N, where V F E ( s X, d ) denoes he value funcion for households under he full-employmen exchange-rae policy wih opimal capial-conrol axes Fixed Exchange-Rae Policy Finally, consider a policy aimed a keeping he exchange rae fixed a all saes and daes: Under he fixed exchange-rae policy or currency peg, ɛ = 1,. (22) Definiion 3 he fixed exchange-rae policy wih opimal capial-conrol axes is he se of processes { ɛ, τ d } ha maximize households expeced lifeime uiliy (1) subjec o he se of equaions describing he general equilibrium dynamics (GE), and currency peg consrain (22). 12

14 o characerize he allocaion under he currency peg wih opimal capial-conrol axes, I follow a similar sraegy o ha of he opimal exchange-rae policy and drop consrains (6) (9) and (14) (15). Appendix A shows ha any { d +1, c, h, w, ɛ } ha saisfy (5), (11) (13), (16) (18), and (22), also saisfy (GE) and (22). hus, he dynamics under he currency peg wih opimal capial-conrol ax policy can be expressed wih he Bellman equaion, V P EG ( s X, d, w 1 ) = s.. d max [ ( ( U A c, F (h) )) + βe s X V P EG ( s X, d, w )] (23) d,c,h,w R = d + c y, ( ( ) 1 a (c d κ y + ) ) 1 F (h) 1, a d d N, w γw 1, h h, (w γw 1 ) ( h h ) =, ( ) 1 a (c w = ) 1 F (h) 1 F (h), a where V P EG ( s X, d, w 1 ) denoes he value funcion for households under he currency peg and opimal capial-conrol axes and he subscrip 1 is used o indicae variables daed in period Opimal Exchange-Rae Policy, Unemploymen and Credi Limi: Analyical Resuls his secion sudies he relaionship beween unemploymen and he credi limi under he opimal exchange-rae policy. Alhough, given he complexiy of he model, a numerical soluion is required for a full characerizaion, some analyical resuls can be obained o show he rade-off involved in exchange-rae policy. hese resuls will be relevan o undersanding he nex secion s numerical soluion for he dynamics of he economy under he opimal exchange-rae policy. Proposiion 1 characerizes he allocaion under he opimal exchange-rae policy defined in he previous secion. Proposiion 1. Under he opimal exchange-rae policy wih opimal capial-conrol axes (Definiion 1) he following condiions hold a all daes and saes: ( ) ( ( If < 1, h h d h, c, y ) ) d+1 =. If 1, h = h. Proof. See Appendix B. wo conclusions follow from his resul. Firs, he allocaion under he opimal exchangerae policy and capial-conrol axes feaures no unemploymen under no binding collaeral 13

15 consrains. Given ha he capial-conrol ax eliminaes any inefficien borrowing, eliminaing unemploymen when he credi consrain does no bind leads o a welfare gain (a higher consumpion of nonradables), wihou any welfare cos. 8 Second, if he inraemporal elasiciy of subsiuion is less han one, a slackness condiion is esablished beween unemploymen and he collaeral consrain under he opimal exchange-rae policy: If he collaeral consrain is no binding, he labor marke mus exhibi full employmen, and if here is unemploymen, he collaeral consrain mus be binding. As discussed a he end of his secion, empirical evidence from EMs provides wide suppor for he inraemporal elasiciy of subsiuion being less han one. o undersand he role of he inraemporal elasiciy of subsiuion and he ineracion beween unemploymen and he collaeral consrain under he opimal exchange-rae policy, a discussion is in order regarding he rade-off facing exchange-rae policy in his economy he Credi-Access Unemploymen rade-off Parallel o he radiional inflaion unemploymen rade-off in he New Keynesian lieraure, he exchange-rae policy in his economy may face a credi-access unemploymen rade-off. Under binding nominal downward wage rigidiy, a depreciaion of he nominal exchange rae decreases real wages and, hus, helps reduce unemploymen. Bu i is also associaed wih a real exchange-rae depreciaion, which decreases he value of nonradable oupu in radable unis. Recall ha he collaeral in his economy is given by he value, in radable unis, of radable and nonradable oupu. Accordingly, if he price effec (real exchange-rae depreciaion) dominaes he quaniy effec (employmen increase), an exchange-rae depreciaion can decrease he collaeral value and ighen he credi limi. he price effec dominaes he quaniy effec if he inraemporal elasiciy of subsiuion beween radables and nonradables is less han one ( < 1). As discussed in he nex secion, his assumpion is widely suppored by empirical evidence from EMs. Under his assumpion, he following proposiion can be esablished: Proposiion 2. If < 1, given an iniial sae ( s X ), d, for any deb level d +1 wih associaed radable consumpion c = ( d +1 R 1 d + y ) > for which d +1 > d ( h, c, y ), here exiss a level of employmen h (, h ) for which d +1 = d ( h, c, y ). Proof. See Appendix B. his resul shows ha for any deb level ha does no saisfy he credi limi under full employmen, here exiss a level of employmen below full employmen for which he real exchange rae is sufficienly appreciaed o ensure he credi limi is saisfied for 8 Noe ha while he presence of incomplee financial markes leads o inefficien consumpion flucuaions relaive o an economy wih complee asse markes, eliminaing unemploymen does no lead o more inefficien consumpion flucuaions. 14

16 ha deb level. his resul sems from he fac ha if he inraemporal elasiciy of subsiuion is less han one ( < 1), he collaeral consrain is decreasing in he level of employmen. 9 his provides a heoreical jusificaion for he exisence of he exchangerae policy debae, ypically observed during financial crises in EMs, ha weighs he wo policy objecives. he opimal choice under his rade-off can be characerized using he firs-order condiions of he opimal policy problem (19): Remark 1 If < 1, in an allocaion under he opimal exchange-rae policy wih opimal capial-conrol axes (Definiion 1) in which, a ime, h < h, he following condiions hold: ( U c A N c, F (h ) ) ( ) =φ µ 1 κp ( c, F (h ) ), (24) ( ) φ µ φ F = βe φ F +1, (25) R where φ µ, and φf denoe he nonnegaive mulipliers associaed wih he collaeral consrain (18), and he resource consrain (17), respecively, in he Ramsey problem of opimal exchange-rae policy wih opimal capial-conrol axes. Proof. See Appendix B. Equaion (24) shows ha in any opimal allocaion in which here is unemploymen, he Ramsey planner equaes he marginal benefi of increasing employmen, given by he marginal uiliy of nonradable consumpion, o is marginal cos in erms of ighening he collaeral consrain. Equaion (25) shows ha he shadow price of relaxing he credi consrain for he Ramsey planner, φ µ, is he wedge beween he curren shadow value of wealh for he Ramsey planner and he expeced value of reallocaing wealh o he nex period. his shows a relevan aspec of he rade-off involved in exchange-rae policy: While he coss of exchange-rae depreciaions are associaed wih ineremporal misallocaion of consumpion, heir benefis are relaed o a higher level of consumpion Empirical Evidence on he Inraemporal Elasiciy of Subsiuion As shown in Proposiions 1 and 2, a ension exiss beween credi access and unemploymen only if he elasiciy of subsiuion beween radable and nonradable goods is less han one ( < 1). If his is he case, radable and nonradable goods are gross complemens, and he price effec (real exchange-rae depreciaion) associaed wih increasing employmen dominaes he quaniy effec (employmen increase). As a resul, 9 If he inraemporal elasiciy of subsiuion is greaer han or equal o one ( 1), he credi access unemploymen rade-off vanishes, as implied by Proposiion 1. In paricular, if he inraemporal elasiciy of subsiuion is equal o one ( = 1), employmen does no influence he collaeral consrain. If he inraemporal elasiciy of subsiuion is greaer han one ( > 1), he credi-access unemploymen rade-off overurns, and a decrease in unemploymen also helps relax he collaeral consrain. 15

17 exchange-rae depreciaion can decrease he collaeral value and make he credi limi igher. here is wide suppor from he empirical lieraure for he inraemporal elasiciy of subsiuion being less han one. In a sample of developed and emerging marke economies, Sockman and esar (1995) esimae a value of he elasiciy of subsiuion of.44. Separaing he samples of developed and emerging economies, Mendoza (1995) finds values of he elasiciy of.74 and.43, respecively. In sudies for EMs, Gonzalez- Rozada e al. (24) found esimaes in he range beween.4 and.48 for Argenina and Lorenzo, Aboal, and Osimani (25), found esimaes in a range beween.46 and.75 for Uruguay. 1 Moreover, following his empirical lieraure, he sudies referenced in he presen paper ha calibrae a wo-secor, small open economy model generally use a parameer value of he elasiciy of subsiuion in he range beween.44 and Quaniaive Analysis he objecive of his secion is o quaniaively characerize he aggregae dynamics of he model economy under he opimal exchange-rae policy and o compare is performance, in erms of welfare, o ha under he full-employmen and fixed exchange-rae policies, boh during periods of financial crises and under regular business-cycle flucuaions. 4.1 Calibraion and Compuaion o characerize he aggregae dynamics under he differen exchange-rae regimes, calibraed versions of he funcional equaions (19), (21) and (23) are solved numerically. Due o he presence of occasionally binding consrains, I resor o he mehod of valuefuncion ieraion over a discreized sae-space o compue he numerical soluions. As menioned in Secion 2, he consumpion aggregaor is assumed o be a CES aggregaor. I also assume a CRRA period uiliy funcion and an isoelasic form for he producion funcion: U(c) = c1 σ 1 1 σ, F (h) = h αn. (26) he model is calibraed a he annual frequency, o mach Argeninean daa. Argenina is used as a benchmark o conduc his exercise as an EM counry whose exchangerae regimes and financial crises have been widely sudied, paricularly in he wo branches of he lieraure his paper combines. 1 Osry and Reinhar (1992) found evidence inconclusive in his respec wih esimaes beween.66 and 1.44, depending on he EM region and he insrumenal variable considered. For a survey on he mehodologies used o esimae he elasiciy of subsiuion beween radable and nonradable goods, see Akinci (211). 16

18 able 1. Baseline Parameer Values Parameer Value Descripion σ 2 Inverse of ineremporal elasiciy of subsiuion.44 Inraemporal elasiciy of subsiuion a.295 Share of radables β.8 Annual subjecive discoun facor κ.263 Share of income used as collaeral α N.75 Labor share in nonradable secor γ.96 Degree of downward nominal rigidiy All parameer values used in he baseline calibraion are shown in able 1. inverse of he ineremporal elasiciy of subsiuion is se o σ = 2, a sandard value in he business-cycle lieraure for small open economies (see, for example, Mendoza 1991). he inraemporal elasiciy of subsiuion is se o =.44, using he esimaes of Gonzalez-Rozada e al. (24) for Argenina (see Secion 3.2 for a review of he lieraure on his parameer). I se α N =.75, following he evidence in Uribe (1997) on he labor share in he nonradable secor in Argenina, and γ =.96, following he evidence in Schmi-Grohe and Uribe (211) on downward nominal wage rigidiy. he mean level of radable oupu and he labor endowmen (h) are normalized o one. he parameers {β, a, κ} are used o mach hree key momens in he ergodic disribuions of he model under he opimal exchange-rae policy o he ones observed in hisorical Argeninean daa (for he period ). he hree daa momens considered are ypically argeed in he relaed lieraure (following Bianchi, 211): an average level of exernal deb-o-gdp raio of 21 percen, a share of radable oupu in GDP of 32.9 percen, and a frequency of sudden sops of 5.5 percen. A sudden sop in he model is defined as a period in which he economy exhibis a change in he curren accoun larger han one sandard deviaion, following Eichengreen, Gupa and Mody (26), from which he frequency of sudden sops is obained for a sample of EMs. 11 he (See he Daa appendix for furher deails on daa sources, and on he consrucion of he series). he parameer values obained from his calibraion are β =.8, a =.295 and κ =.263. Secion 5.1 sudies he sensiiviy of he opimal policy o his calibraion. I is assumed ha he wo exogenous driving forces, he radable endowmen and he ineres rae, follow a firs-order VAR of he form [ ( )] ln y ln ( R ) = Φ ln ( y ) [ ] ( 1 ) ε y ln R 1 +, (27) R R ε R where [ ε y ε R ] i.i.d. N (, Ω) and R denoes he mean ineres rae level. 11 he frequency for EMs is similar o he frequency in Argenina during his period, and o oher empirical esimaes, such as Calvo e al. (28). 17

19 he parameers of his sochasic process are esimaed using Argeninean daa since radable endowmen is measured wih he cyclical componen of value added in agriculure and manufacuring. Ineres raes on exernal deb are measured as he sum of EMBI spreads and he reasury-bill rae. (Secion 5.2 sudies an economy wih ineres rae shocks calibraed o hose of he risk-free rae). Since he daa on EMBI spreads for Argenina is available since 1994, he series were exended back o 1983, using he Neumeyer and Perri (25) daase, which uses a measure similar o he one considered here. he ineres rae series is hen deflaed wih a measure of expeced dollar inflaion. (See he Daa appendix for furher deails on daa sources, and on he consrucion of radable endowmen and ineres raes.) he years 22 25, in which Argenina defauled and was excluded from inernaional markes (Cruces and rebesch, 213), are no included in he esimaion. he following OLS esimaes are obained. [ ] [ ] Φ =, Ω =, R = his process is approximaed wih a Markov chain, seing a grid of 15 equally spaced poins for boh ln ( y ) ( and ln R ) R, yielding 225 exogenous saes. o esimae he ransiion-probabiliy marix, I use he mehod proposed by erry and Knoeck (211) exending auchen (1986). 12 Finally, o approximae he aggregae dynamics of he economy under he opimal and he full-employmen policies, I discreize he endogenous sae space (d ) using 1,1 equally spaced poins. o approximae he dynamics under a currency peg, I use 251 equally spaced poins for deb (d ) and 25 equally spaced poins for he log of previous period wage (w 1 ). he nex secions presen he resuls of he quaniaive analysis. 4.2 Policy Funcions his secion analyzes he policy funcions under he opimal exchange-rae policy and compares hem o hose under he wo benchmark exchange-rae regimes: full-employmen and fixed exchange rae. Figure 1 shows decision rules for he nominal devaluaion rae, he real exchange rae, unemploymen, and nex-period deb as a funcion of he sae variables: curren deb, radable endowmen, and he exernal ineres rae. In each panel, only one sae variable varies (on he horizonal axis), and he remaining sae variables are fixed a heir uncondiional means (under he opimal policy, if he sae is endogenous). In each panel, a shaded region depics he sae-space in which he collaeral consrain binds under he opimal policy. he panels on he righ do no have a shaded region since varying he 12 I am graeful o Sephen J. erry and Edward S. Knoek II for sharing he code for he Markovchain approximaions of vecor auoregressions, which were used in his paper o esimae he ransiionprobabiliy marix of he sochasic process. 18

20 ineres rae while keeping he res of he saes fixed a heir respecive means is no sufficien o make he collaeral consrain bind. he decision rules for he nominal devaluaion rae and real exchange rae under he opimal policy are nonmonoonic, in sharp conras wih he decision rules under he full-employmen or fixed exchange-rae policies. he change of he sign in he slope under he opimal policy occurs a he poin a which a higher iniial level of deb or a lower radable endowmen enails a binding credi consrain. In he region of nonbinding collaeral consrain, he decision rules of opimal and full-employmen policies coincide, as implied by Proposiion 1. In his region, currency depreciaion is increasing in he iniial deb level and he ineres rae, and decreasing in he level of radable endowmen. In he region of binding collaeral consrain, while currency depreciaion in he fullemploymen policy coninues o be increasing in he iniial deb level and decreasing in he level of radable endowmen, currency depreciaion under he opimal exchange-rae policy becomes decreasing in he iniial deb level and increasing in he level of radable endowmen. Posiive unemploymen emerges under he opimal exchange-rae policy in he region of binding collaeral consrain, increasing in he iniial level of deb and decreasing in he level of radable endowmen. he decision rule for nex-period deb under he opimal policy is monoonic, as i would be wihou an endogenous collaeral consrain. Again in sharp conras, under he full-employmen policy, he decision rule for nex-period deb is nonmonoonic, wih a change in he slope a he poin a which a higher iniial level of deb or a lower radable endowmen implies a binding credi consrain (for a similar resul in an endowmen economy, see Bianchi, 211). Hence, consisen wih Proposiion 2, he opimal policy resores he monooniciy in he policy funcions of deb by making he decision rule of he real exchange rae nonmonoonic. In oher words, he opimal choice under he credi-access unemploymen rade-off implies no corner soluion: he opimal policy is willing o choose unemploymen in he region of binding collaeral consrain o allow for a higher nex-period deb. he decision rules for he currency peg show ha his regime, in conras o he opimal and full-employmen policies, implies posiive unemploymen in he sae-space regions in which he collaeral consrain does no bind. In hese regions, he fixed exchange-rae regime makes he downward rae rigidiy binding. Consisen wih his, he decision rule of he real exchange rae under he currency peg displays less sensiiviy han ha of he oher wo exchange-rae regimes. 4.3 Opimal Exchange-Rae Policy during Financial Crises Under no binding collaeral consrains, he opimal exchange-rae policy always consiss of depreciaing he nominal exchange rae in response o negaive shocks o achieve full employmen, as implied by Proposiion 1. his secion characerizes he opimal 19

21 1.5.8 Devaluaion Rae ε 1 p d d ε 1 p log(y ) Real Exchange Rae.1.5 log(y ) Unemploymen Rae.1 ε 1 p R 1, in percen R 1, in percen.6 1 h h.5 1 h d log(y ) Nex Period Deb R 1, in percen d d d log(y ) Opimal Policy Full Employmen Policy Currency Peg d R 1, in percen Figure 1. Policy Funcions. Noe: he real exchange rae is expressed in log deviaions from is sample mean. Devaluaion rae, unemploymen rae and nex-period deb are expressed in levels. In each panel, only one sae variable varies (on he horizonal axis), and he remaining sae variables are fixed a heir uncondiional means (under he opimal policy, if he sae is endogenous). Shaded regions denoe regions of he sae-space in which he collaeral consrain binds under he opimal policy. 2

22 exchange-rae policy under periods of binding collaeral consrains, or financial crises, and compares he dynamics of he economy under he differen exchange-rae regimes. o do his, he calibraed version of he model is simulaed for 2 million quarers, idenifying periods in which he collaeral consrain is binding under he opimal exchangerae policy. he beginning of a financial crisis episode ( = ) is defined as he firs period in which he collaeral consrain binds. he responses of he variables, during all episodes of financial crisis, are hen averaged. Figure 2 depics he average exernal shock during a financial-crisis episode. In he wo years ha precede such an episode, radable endowmen conracs and ineres raes increase. A he crisis rough ( = ), radable oupu is 1 percen below is mean, and he annual ineres rae is 16 percen, 4 percenage poins above is mean. In he hree years following he rough, boh radable oupu and he ineres rae recover heir precrisis levels. radable Endowmen 18 Ineres Rae log(y ), in percen R 1, in percen Figure 2. Financial Crises Exogenous Variables. he average responses of he nominal exchange rae and endogenous variables under he differen exchange-rae regimes are shown in Figure 3. Opimal and full-employmen exchange-rae policies display sriking similariies and offer a sharp conras o he response under a currency peg. Even under binding collaeral consrains ( = ), he opimal exchange-rae policy does no fix bu i subsanially depreciaes he nominal exchange rae, 52 percen on average. his depreciaion is less han ha under he fullemploymen policy (71 percen). As a resul, some involunary unemploymen emerges under binding collaeral consrains (1.6 percen on average a he crisis rough). How- 21

23 ever, unemploymen under he opimal exchange-rae policy is significanly lower han ha observed under he currency peg (6.2 percen on average a he crisis rough). ε Devaluaion Rae p Real Exchange Rae w Real Wage Exernal Deb h Unemploymen Rae Curren Accoun d +1.1 (d +1 d ) Opimal Policy Full Employmen Policy Currency Peg Figure 3. Financial Crises Endogenous Variables. Noe: Real exchange rae, real wage, and exernal deb expressed in log deviaions from heir sample means. Curren accoun expressed in deviaions of is sample mean. Devaluaion rae and unemploymen rae expressed in levels. In periods of binding collaeral consrains, he large real-exchange-rae depreciaion under he opimal exchange-rae policy (he relaive price of nonradable goods being 39 percen below is mean a he crisis rough), implies a large adjusmen of exernal deb and radable consumpion. Under he opimal policy, he conracion of radable 22