Asset prices and twin crises

Journal of International Money and Finance 28 (2009) 26–55

Contents lists available at ScienceDirect

Journal of International Moneyand Finance

journal homepage: www.elsevier .com/locate/ j imf

Asset prices and twin crises

Rajesh Singh*

Department of Economics, 260 Heady Hall, Iowa State University, Ames, IA 50011, USA

JEL classification:F3F4

Keywords:Twin crisesCurrency crisesBanking crisesAsset pricesGovernment bailout

* Tel.: þ1 515 294 5213; fax: þ1 515 294 0221.E-mail address: [email protected]

0261-5606/$ – see front matter � 2008 Elsevier Ltdoi:10.1016/j.jimonfin.2008.07.008

a b s t r a c t

Emerging market crises have been characterized by two keyfeatures: (i) banking crises generally precede currency crises, and(ii) asset prices decline in advance of currency crises. This paperargues that asset prices provide a key link between banking andcurrency crises. It is shown that a prospective currency crisis dueto an unanticipated increase in the public debt triggers an assetprice decline. Banks’ exposure to asset prices in turn deterioratestheir balance sheets and precipitates a banking crisis. Under theassumption of government bailout of banks, it is shown that the‘twin’ crises are self-fulfilling and their time-line follows (i) and (ii)described above. The timing of currency crisis is decreasing in theratio of government’s bailout to banks’ loss of capital.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Emerging market crises in the past decade have been characterized by banking and currency crisesthat occur together. In an extensive study of ‘twin’ crises, Kaminsky and Reinhart (1999) highlight thatbanking crises often begin before currency crises, and that asset prices decline in advance of currencycrises. These facts combined with the evidence that banking sector problems generally begin from theasset-side and that banks carry substantial asset price exposures naturally raise the following ques-tions. Do asset prices link banking and currency crises? In particular, what (if any) are the ‘funda-mentals’ that activate the asset price link? Finally, if such a link is triggered, can it replicate theobserved crises chronology? The crises literature is yet to address these issues despite their obvious

d. All rights reserved.

mailto:[email protected]

www.sciencedirect.com/science/journal/02615606

www.elsevier.com/locate/jimf

R. Singh / Journal of International Money and Finance 28 (2009) 26–55 27

importance for researchers as well as policymakers.1 This paper attempts to fill this gap by providing anasset-side explanation of twin crises along with its observed chronology.

A number of researchers have argued that the East Asian currency crises were caused by suddenprospective increases in fiscal deficits due to financial sector meltdowns.2 Following this research, Ifirst study a currency crisis sparked by an unanticipated exogenous increase in fiscal liability. It isshown that a prospective currency crisis triggers an asset price decline, which in turn brings fortha banking crisis as banks are exposed to asset prices. Pursuing this further, under the assumption ofgovernment’s bailout of banks that generates an endogenous increase in fiscal liability, I show that twincrises are self-fulfilling.

The analytical setup I utilize is particularly motivated by the following stylized facts relating to theEast Asian crisis: (1) firms’ cash flows were highly sensitive to interest rates; (2) banks weresubstantially exposed to asset prices; (3) equity prices declined in advance of currency crises leading tobank failures; and (4) eventually, governments bailed out insolvent banks.3

The model consists of a small open economy populated by households, firms, banks, anda government. Households derive utility by consuming and by holding deposits at banks. Banks takethese deposits and advance loans to firms a la Edwards and Vegh (1997).4 A fraction of deposits is heldat the central bank as required reserves which constitutes the monetary base. The demand for bankloans is motivated as follows. Firms produce output by combining their own capital with the capitalrented from other firms and a fraction of the rental payments must be made through bank credit. Asa result, the equilibrium return on capital and equity prices is sensitive to the changes in bank lendingrates. Finally, banks also own equity in firms.

To begin with, I assume, the economy is in a perfect foresight stationary equilibrium with a fixednominal exchange rate. Suppose now the government realizes an unanticipated increase in fiscalliability that is eventually expected to be financed through seigniorage. Yet, the government sticks tothe existing exchange rate peg until the public debt hits a publicly known upper bound, at which pointthe peg is abandoned and the exchange rate floats. Thus an exchange rate crisis is now imminent.5

Agents anticipate a rise in nominal interest rate due to a rise in inflation after the peg is abandoned. Anincrease in nominal interest rate hikes up banks’ cost of holding fractional reserves. Althoughseigniorage revenue is directly raised from banks, indirectly, banks pass it on to households and firmsby adjusting real interest rates on deposits and loans. Hence, a rise in the future nominal interest ratesignals that firms’ cost of loans in future will rise and therefore its return on capital will decline. Asasset prices reflect the present value of future earnings, they fall on impact and decline steadilythereafter. If the decline is sufficiently large, banks’ loss of capital may leave them with a negative networth, which I define as banking crisis. Thus banks’ exposure to asset prices links a prospective currencycrisis to a current banking crisis. Furthermore, as a part of banks’ assets are purchased by issuing debt,the percentage decline in the net worth of the banking sector exceeds that of the firms.6

The fundamental that triggers the asset price channel above is the sudden increase in the gov-ernment’s fiscal liability. Suppose instead the government has a bailout policy under which it assumesa fixed percentage of banks’ capital loss if a crisis occurs. If agents believe that a currency crisis isimminent, firms’ equity prices fall; the increase in the stock of fiscal liability as a result of thegovernment bailout policy then validates this belief. Here, the presence of bailout guarantee isnecessary for self-fulfilling beliefs to trigger.7 Such crises begin with a fall in equity price that brings

1 Caballero and Krishnamurthy (2001) offer an explanation of emerging market crises, in which a crisis is an outcome ofsimultaneous asset price collapse along with binding collateral constraints.

2 See, for example, Krugman (1998), Corsetti et al. (1999) and Burnside et al. (2001).3 Section 2 presents evidence on each of the above.4 See also Lahiri and Vegh (2007).5 See, for example, Krugman (1979) and Flood and Garber (1984). In these models, an obvious inconsistency in the monetary

and fiscal policy leads to a secular loss of foreign reserves, eventually forcing the central bank to abandon the peg whenreserves hit a lower bound.

6 This is consistent with the evidence documented by Burnside et al. (2001).7 Government bailout guarantee as the fundamental behind a self-fulfilling twin crises has already been argued by Burnside

et al. (2004). See also Schneider and Tornell (2004) who study financial crises in a non-monetary setup.

R. Singh / Journal of International Money and Finance 28 (2009) 26–5528

forth a banking crisis, followed by a steady equity price decline that continues until the exchange ratepeg is abandoned. These results mimic the empirical evidence presented in Section 2.

If self-fulfilling twin crises exist then the equilibrium can be uniquely characterized by the timingbetween a banking crisis and the eventual abandonment of fixed exchange rate. This timing must befinite for crises to exist. The timing in turn is determined from the two policy parameters: (a) thebailout ratio, and (b) the upper bound on public debt at which the peg is abandoned. While the bailoutcreates additional fiscal liability, the upper bound on public debt determines the level of seigniorageraised after the exchange rate floats. The timing thus equilibrates the present value of seignioragerevenues with the stock of additional fiscal liability.

A bailout sparks off an increase in the stock of fiscal liability. This is further aggravated by the loss ofcentral bank’s reserves that continues to fall until the exchange rate floats. In the model, the monetarybase is proportional to the loans to firms that they need for paying rents on hired capital. As these rentsalso determine equity prices, their anticipated cumulative decline reflects in the initial fall in the equityprice. In other words, the anticipated fall in the monetary base over time is proportional to the initialfall in the equity price. Thus, the total increase in the stock of fiscal liability (prospective loss of foreignreserves plus bailout liability) depends on the initial fall in equity price and the bailout ratio.

On the other hand, the initial fall in the equity price equals the share of the government bailout thatis eventually passed on to the firms. This share, given households’ preferences, uniquely depends onthe timing of the exchange rate crisis. Hence, bailout ratio and the timing of crisis are uniquely related.It turns out that the lower the bailout ratio, the later is the exchange rate crisis. Intuitively, a lowerbailout ratio creates smaller increase in fiscal liability and thus requires a lower present value ofseigniorage revenues. Then an exchange rate peg can be sustained longer.8 Pursuing this logic further,the peg can be sustained indefinitely if the bailout ratio is sufficiently low. In other words, self-fulfillingcrises are ruled out if the bailout ratio falls below a lower bound. The households’ preferences affect theequilibrium bailout ratio in the following way. The more interest rate elastic is households’ demand fordeposits, the less they are willing to contribute to seigniorage revenues. As a result, firms bear a largerburden of additional fiscal liability. Thus for a given timing of crises, a relatively elastic (inelastic)demand for deposits calls for a low (high) bailout ratio.

The timing of crises in addition depends on the upper bound on public debt. If households’ demandfor deposits is interest rate elastic (inelastic), the higher the upper bound on public debt, the earlier(later) is the time of exchange rate crisis. Note that a higher upper bound implies that the net increasein public debt is eventually higher. The net increase in public debt has two mutually opposing effects.On the one hand, it allows the government to sustain the peg longer. On the other, it calls for a higherdebt service, a higher seigniorage, a higher interest rate, and thus a higher decline in equity price.A higher level of bailout in turn will then advance the time of exchange rate crisis. However, when thedemand for deposits is elastic, a relatively sharper decline in deposits causes a relatively higher loss ofcentral bank reserves. In sum, the delaying effect of a higher public debt is more than offset by thecombined effect of a higher equity price decline and loss of reserves. Hence, the exchange rate crisis isadvanced. On the contrary, when the households’ demand for deposits is inelastic, the delaying effectof a higher public debt dominates the other effects. Then, the exchange rate crisis is postponed.

The existing twin crises literature can be broadly classified into (a) maturity mismatch and (b)currency mismatch models. Maturity mismatch models rely on self-fulfilling bank runs a la Diamond–Dybvig, in which the central bank’s role as a lender-of-last-resort links a run on deposits to a run onforeign reserves.9 In currency mismatch models, bailout guarantees induce banks to carry unhedgedforeign currency liabilities that in the event of a devaluation cause bank failures. Then bailouts triggerand currency crises occur in self-fulfilling manner.10 Yet another set of currency mismatch papers,although they do not study banking crisis explicitly, rely on financial accelerator models a la Bernanke–Gertler. Here, the economies face multiple equilibria. In a crisis equilibrium, a currency devaluation

8 The logic behind this result is somewhat more complicated than the simple intuition given here. The details are presentedin Section 5.

9 See, for example, Radelet and Sachs (1998) and Chang and Velasco (2001).10 See, for example, McKinnon and Pill (1998) and Burnside et al. (2004).


occurs along with a fall in the economic activity.11 None of these models, however, address theintertemporal aspects of twin crises and both the banking (financial) and the currency crisis occursimultaneously at a point in time. Moreover, any feasible extension of these models will not generateasset price dynamics that has been observed empirically.

Thus, this paper makes the following contributions to the twin crises literature. First, it offers anasset price perspective of twin crises, which in the light of emerging market evidence competes wellwith maturity mismatch and/or currency mismatch explanations. Second, the model is better able toreplicate empirically observed crises dynamics. In particular, banking crisis precedes currency crisisand asset prices decline in advance of currency crisis. Finally, the model generates a steady declineasset prices that closely mimics the data.

The rest of the paper is organized as follows. Section 2 presents some evidence on the stylized factsrelating to the East Asian crises. Section 3 sets up the basic model and derives the pre-crisis stationaryequilibrium. In Section 4, a prospective currency crisis is shown to generate a decline in asset prices.This section also presents banks’ balance sheet and formally defines a banking crisis. Section 5 studiesthe existence and the properties of self-fulfilling twin crises equilibria under the presence ofgovernment bailout guarantees. Section 6 concludes. Algebraic tedious proofs are consigned to anAppendix.

2. Stylized facts

The following stylized facts have characterized emerging market, in particular the East Asian,12

crises: (1) the East Asian corporate leverage was exceptionally high; (2) banks were substantiallyexposed to real estate and equity loans; (3) equity prices declined in advance of currency crises leadingto major bank failures; (4) the depositors and the foreign bank creditors were often guaranteed by thegovernments, at least implicitly. I present some evidence on each of the above in the followingparagraphs:

(1) It has been argued that one of the main factors that left the East Asian countries vulnerable toa shift in market sentiment was exceptionally high leverage, defined as total debt/equity ratio.13

High leverage, in turn, implies high interest payments on debt. As a result, the ratios of earningsbefore interest, taxes, and depreciation (EBITDA) to interest payments, which indicate theadequacy of cash flows to service interest payments on outstanding debt, were also very high.Table 1 provides a cross-country comparison of corporate debt/equity ratio14 and EBITDA/interestratio, as at the end of 1996.

It is fairly evident that the balance sheets of firms in Korea, Thailand, and Indonesia werevulnerable to a prospective rise in the interest rates.

(2) It is well known that banks were often part of big conglomerates (e.g., Korean Chaebols) andtypically made loans to firms in the same group, thus, indirectly acquiring ownership stakes inrelated firms. Government directed lending to Chaebols in Korea, lending to related parties withinlarge financial (and non-financial) conglomerates in Korea and Thailand, and ownership of weaklyregulated banks by non-banks in Indonesia have been extensively documented in past studies.15 Inaddition, banks took substantial risks by lending heavily to the real estate sector and stock markets,and were often left with their diminished collateral.16

(3) In a study of 76 emerging market currency crises episodes, Kaminsky and Reinhart (1999) reportthat ‘‘during the 18 months prior to a BOP crisis, the equity market steadily underperforms

11 See, for example, Krugman (1999) and Aghion et al. (2004).12 Namely, Korea, Indonesia, Malaysia, Philippines, and Thailand.13 See, for example, IMF (1998a), pp. 34–35, and pp. 153–156, and Lane et al. (1999), p. 19. Pomerleano (1998) and Claessens

et al. (1998) hold the East Asian corporate leverage to be amongst the highest in the world.14 Also see Lane et al. (1999), p. 19, who report average leverage ratios of 3.95 and 4.5 for Korea and Thailand, respectively.15 See, for example, IMF (1997), p. 12, IMF (1998a), p. 35, 70, pp. 153–156, and IMF (1998b), p. 25.16 See, for example, Corsetti et al. (1998a), pp. 24–30, IMF (1997), pp. 12–13, IMF (1998a), p. 35, 37, pp. 153–156, IMF (1998b), p.

25, and Lane et al. (1999), p. 19, 28.


(relative to tranquil times).’’ They further note that the equity prices are past their cyclical peakbefore the onset of banking crises. Fig. 1 shows the evolution of exchange rates and stock indexes inThailand, Korea, Malaysia, and Philippines before, during, and after the crises.

The equity prices begin declining before the onset of currency crises, and recover only after the peakof currency crises.

Declining asset prices in Thailand and Korea provide the earliest signs of trouble in the region.During 1996, stock prices (in domestic currency terms) fell by more than 20% in Korea and by almostone third in Thailand. It is worth noting that Korea and Thailand suffered the earliest major bankfailures in the region.17,18 For these two countries, Table 2 presents the timing of stock market declines,bank failures, and the eventual collapse of the exchange rate pegs.

In Korea, the banking system exhibited increasing signs of stress during the first half of 1997 asa number of major conglomerates went bankrupt.19 By the first week of September, six highly lever-aged Chaebols had failed. In Thailand, many commercial banks reported a significant increase in non-performing loans in late 1996 and there were runs on the deposits of the Bangkok Bank of Commerce inMay 1997. Less than a week before the baht was allowed to float, the Bank of Thailand suspended 16finance companies; another 42 were suspended on August 5.20 These developments were amplyreflected by the stock market indices of the banking sector. In particular, the financial sector stockindices fell substantially more relative to that of the non-financial sector (see Burnside et al., 2001,p. 1163, Table 3).

While equity and property prices had been declining leading to a widespread banking sector crises,the currency crises did not come through without advance triggers. It is worth stressing that the Thaibaht came under attack already in November and December 1996. The Korean won was also underpressure in 1996, and had been allowed to depreciate in 1996 and early 1997. These events indicate thatthe markets were anticipating a currency crisis.

Thus, the advance decline of equity prices, the timing of major bank failures, the continuing declinein equity prices until the peak of currency crises, and the recurrent speculative attacks on currenciesprovide evidence that the equity prices had been declining in anticipation of a currency crisis. Thedeclines in stock prices and property prices eventually led to bankruptcies and bank failures.21

(4) Lastly, bailout guarantees have been extensively discussed in the literature.22 These studies alsoquantify the extent to which the implicit guarantees became explicit once the crises occurred. Ithas been reported that the cost of bank bailouts in Indonesia, Malaysia, South Korea, and Thailandhas been to the tune of 50%, 16%, 27%, and 50% of GDP, respectively.23

3. The model

Consider a small open economy perfectly integrated with the rest of the world in both goods andcapital markets. An infinitely lived representative household consumes a perishable good whose world

17 Although financial problems usually begin well before a bank is finally closed, major failures correspond to the time whenthese banks were publicly known to be insolvent.

18 In comparison, no major bank failures/closures were reported in Malaysia and Philippines. However, in Malaysia, with theonset of the regional crisis, banks and finance companies experienced a significant decline in profitability and asset qualitydeteriorated sharply. In Philippines, the deterioration in asset quality led to difficulties in some small banks, but most of thelarge commercial banks were relatively better capitalized and were able to withstand the increase in bankruptcies and debtrestructuring. See IMF (1998a,b).

19 See Corsetti et al. (1998b), pp. 4–8 for a detailed description of these events.20 See IMF (1998a), pp. 159–160.21 See IMF (1997), pp. 15–22, IMF (1998a), pp. 33–40, Lane et al. (1999), p. 28, and Corsetti et al. (1998a,b), pp. 24–30.22 See Burnside et al. (2001), Corsetti et al. (1998a), pp. 24–30, IMF (1998a), p. 35, IMF (1998b), p. 25, and Lane et al. (1999),p. 21.23 See Klingebiel and Laewen (2002).

Table 1Corporate leverage.

Country Leverage EBITDA/interest

Hong Kong 1.6 11.07Indonesia 1.9 2.44Korea 3.5 1.07Malaysia 1.2 6.74Philippines 1.3 3.68Singapore 1.0 8.05Thailand 2.4 1.92Germany 1.5 7.09USA 1.1 7.62

Leverage: total debt/equity ratio, source: Claessens et al. (1998).EBITDA: earnings before interest, taxes, and depreciation, source: Pomerleano (1998).


price is fixed at unity. Households in addition derive utility by holding deposits at banks. Eachhousehold owns a unit of labor which it supplies to the firms inelastically. There is a continuum of firmsover a unit interval that produces consumption good. A firm is differentiated by the type of specificcapital it owns. Each firm produces output by combining labor, its own capital, and capital rented fromother firms. Rental payments between firms are made through bank credit and are denominated inhome currency. A representative bank receives home currency deposits from households and lends tofirms. Banks also hold equity in firms; banks are owned by households. All markets are perfectlycompetitive.

Perfect integration in the goods market implies that the law of one price holds. Therefore, the homecurrency price of consumption is equal to the nominal exchange rate E, expressed in units of homecurrency per unit of world currency. Households, firms, banks, and the government can buy and sell aninternationally traded bond at a constant world interest rate r. Perfect capital mobility implies that theinterest rate parity holds. Therefore, the nominal interest rate is given by i¼ rþ 3, where 3 is thedevaluation rate.

Notation. Unless otherwise indicated, subscripts denote partial derivatives. Time-dependent vari-ables are denoted without time subscripts. _x denotes dx/dt. Whenever necessary, time is indexed as anargument within small brackets. For example, x (t) denotes the value of x at time t. Finally, fjx¼a denotesthe value of a function f(x) evaluated at x¼ a.

Household’s utility function. Some of the main results critically depend on the form of households’utility function. Preferences that are separable in consumption and deposits, particularly logarithmicpreferences, yield simple closed form expressions for the most of the results. However, separablepreferences lead to the indeterminacy of crises equilibria as discussed in Section 5, and thus are treatedonly as special cases. For the main analysis, a general utility form is assumed, which yields results witha wider applicability.

3.1. Households

A representative household’s lifetime utility is given by

Z N

0e�btuðc; dÞdt; (1)

where u($) has the following properties: uc> 0, ud> 0, ucc< 0, udd< 0, udcuc� uccud> 0,ucdud� udduc> 0, and uccudd� ucdudc> 0.24 b is the subjective discount factor. c denotes consumption,and d denotes deposits in terms of consumption goods. Thus d h D/E, where D denotes nominal homecurrency deposits at banks. Deposits earn a nominal interest rate of id.

24 These assumptions ensure that both c and d are normal goods.

Korea

0

500

1000

1500

2000

2500

Jan-93Jul-93

Jan-94Jul-94

Jan-95Jul-95

Jan-96Jul-96

Jan-97Jul-97

Jan-98Jul-98

Jan-99Jul-99

E

I

Thailand

0

500

1000

1500

2000

2500

Jan-93Jul-93

Jan-94Jul-94

Jan-95Jul-95

Jan-96Jul-96

Jan-97Jul-97

Jan-98Jul-98

Jan-99Jul-99

E

I

Malaysia

02004006008001000120014001600

Jan-95

May-95

Sep-95Jan-96

May-96

Sep-96

Jan-97

May-97

Sep-97

Jan-98

May-98

Sep-98

Jan-99

May-99

Sep-99

E

I

Philippines

0

1000

2000

3000

4000

5000

Jan-95

May-95

Sep-95

Jan-96

May-96

Sep-96

Jan-97

May-97

Sep-97

Jan-98

May-98

Sep-98

Jan-99

May-99

Sep-99

E

I

Fig. 1. Stock indexes during East Asian currency crises. E: nominal exchange rate; I: stock index – Korea: KOSPI, Thailand: BangkokS.E.T, Malaysia: Kuala Lumpur composite, Philippines: Philippines SE composite (source: datastream). Exchange rates for Thailand,Philippines, and Malaysia have been scaled up by a factor of 40, 100 and 300, respectively.


As the equilibrium wage rate w> 0 and as the household does not value leisure, it utilizes its wholeone unit of labor for work. Then the household’s flow budget constraint in terms of consumption goodscan be expressed as

Dd ¼ �Dbh if t˛G;

_a ¼ raþwþ Ub � s� Idd� c if t;G; (2)

where bh is the household’s net stock of international bonds; a h dþ bh is the total financial wealth; sdenotes lump-sum taxes paid to the government; and Ub is the dividends earned from banks.25 Thehousehold’s opportunity cost of holding deposits is given by Id h i� id. G denotes a finite set of points intime when a discrete portfolio adjustment can occur.

The household’s intertemporal budget constraint can be derived as (after imposing the trans-versality condition, limt/Na (t)e�rt¼ 0)

að0Þ þZ N

0

�wþ Ub

�e�rtdt ¼

Z N

0

�cþ Iddþ s

�e�rtdt: (3)

Households maximize (1) subject to (3) taking the path of interest rates and income transfers {Id,r,s,Ub}as given. Assuming b¼ r rules out trends. Then the first order conditions are

25 By assumption, firms’ equity is only held by banks. One can think of banks owning mutual funds, whose shares in turn areheld by households. This assumption helps avoid unnecessary algebra. All the results will go through, however, if householdsalso held equity in firms directly.

Table 2Major events/dates and % changes in the stock market indexes.

Country Stock market peak Major bank failures Currency float % Declinein stock indexa

Thailand Jan 4, 1994 Jan–Aug, 1997 Jul 2, 1997 67.6Korea Nov 8, 1994 Jan–Sep, 1997 Nov 17, 1997 56.3

a Relative to the non-financial sector, the stock market indices of the financial sector in Thailand and Korea declined further by30 and 34%, respectively, as of July 1, 1997: see Burnside et al. (2001, p. 1163, Table 3).


uc ¼ l; (4a)

ud

uc¼ Id; (4b)

where l is the Lagrangian multiplier for the intertemporal budget constraint (3).The first order conditions have the standard interpretations. Eq. (4a) states that since the inter-

temporal relative price of consumption is unity (as the market and the subjective discount rates areequal), households choose a constant marginal utility of consumption. Eq. (4b) states that the marginalutility of holding deposits equals its opportunity cost in terms of the lost marginal utility fromconsumption. Thus (4b) provides an inverse demand schedule that relates the quantity of real depositswith its opportunity cost, Id.

3.2. Firms

I derive below a rate of return schedule for firms’ capital that inversely varies with the interest rateon bank loans. For analytical convenience, I choose a set up in which firms’ input usage and conse-quently its output remain constant regardless of the variations in interest rates. To this end, I assumethat there is a continuum of firms uniformly distributed over a unit interval, each owning a fixed unit ofdifferentiated capital. A firm first combines its own capital with capital rented from other firms usinga Dixit–Stiglitz aggregation technology. The aggregation technology is symmetric across firms. Theaggregated capital, k, is then combined with labor, l, to produce the final good:

y ¼ Fðk; lÞ;

where F($,$) is homogeneous of degree one, and Fl, Fk� 0. In particular, Fl (1,1)> 0 and Fk (1,1)> 0.Observe that, in spite of the heterogeneity, the symmetry implies that each firm will employ the sameamount of labor and capital, and their amount of input use, output, and the return on their capital willbe identical. It is then convenient to describe firms’ decision rules in a representative manner. Thedetails of the derivation at an individual firm’s level is provided in Appendix A.

I assume that firms need a minimum line of credit from banks in order to rent capital from otherfirms. In particular, firms need at least a fraction 4 of their rental liability as credit-in-advance. Formally,

z � frk; f˛ð0;1�; (5)

where r is the rate of return on capital, and z is bank credit. Banks charge a nominal interest rate iz onloans implying an opportunity cost of Iz h iz� i. In equilibrium, Iz> 0, and hence (5) binds withequality. Thus, a firm’s effective cost of capital is r(1þ 4Iz) which at an optimum equals its marginalproduct:

Fk ¼ r�1þ fIz�: (6)

Similarly, a firm’s optimal labor employment equates the marginal product of labor with the marketwage rate. Thus, w¼ Fl.


In equilibrium, full one unit each of labor and capital are utilized since both wage and rental rate arepositive. Hence, w¼ Fl (1,1). Note further that a constant level of output y is given by

y ¼ Fð1;1Þ ¼ wþ r�1þ fIz�; (7)

where the latter equality follows from the homogeneity of F. Finally, each firm’s amount of bank creditis given by

z ¼ fr: (8)

Thus, after imposing the equilibrium capital and labor allocations, a firm’s flow constraint can beconsolidated as

_af ¼ raf þ y�w� frIz � Uf; (9)

where Uf denotes firms’ dividend payments. Note that af h bf� z, where bf denotes firms’ net stock ofinternational bonds. I assume that firms hold bonds solely to accommodate discrete changes in theamount of loans. Otherwise, they do not accumulate financial assets and af¼ 0 for all t. Combining (6),(7) and (9) yields

Uf ¼ r ¼ 9�1þ fIz

�; (10)

where 9 h Fk (1,1) is the equilibrium marginal product of capital. Eq. (10) states that a firm’s dividendsequal its return on capital net of interest payments.

3.3. Banking system

The main objective here is to derive a relationship between interest rates on deposits and loans, andthe nominal interest rate implied by the world interest parity. Additionally, banks equity transactionsin the secondary market obtain a key equity price equation. I assume that the representative bank isa holding company that runs a depository bank and an investment bank. While depository banksintermediate funds between households and firms, investment banks raise funds by selling bonds andinvest in firms’ equities.

Depository banks take deposits from households and advance loans to firms. It is required thata fraction d of deposits be held as reserves with the central bank.26 These reserves solely constitute themonetary base denoted by m. Hence, m h dd. Banks costlessly convert deposits into loans after meetingthe reserve requirement. Hence,27

z ¼ ð1� dÞd: (11)

Investment banks can sell bonds at the world interest rate r. Its assets consist solely of firms’ equitiestraded in perfectly competitive secondary markets. Using (11), the representative bank’s consolidatedflow constraint is given by

_ab ¼ rab þ�

Id þ ð1� dÞIz � di�

dþ�

_q� rqþ Uf�

s; (12)

where abhqs� bb, and where bb is the banks’ stock of debt in the international bond market, s and qdenote the amount and the price, respectively, of equity holdings in firms. Integrating (12) forward and

26 Although, for simplicity, the reserve requirement is assumed to remain fixed, it is not uncommon for central banks toincrease this requirement during a currency crisis, especially if it is accompanied by bank failures. Then the results that followwill be further strengthened.

27 It is critical for the results of the paper that banks’ deposit and lending rates respond to domestic monetary and fiscal policy.Therefore, it is assumed that foreign borrowings cannot be converted as loans to firms. Alternatively, a sufficiently large fixedcost on such conversions will obtain the same result.


imposing the transversality conditions, limt/N q(t) e�rt¼ 0 and limt/N bb (t)e�rt¼ 0, the present valueof banks’ dividends can be derived as

Z N

0Ube�rtdt ¼ �bbð0Þ þ qð0Þsð0Þ þ

Z N

0

h�Id þ ð1� dÞIz � di

�dþ

�_q� rqþUf

�sie�rtdt: (13)

Observe that the term on the left hand side represents bank’s net worth or equivalently its equitycapital, whereas the right hand side represents its assets net of liabilities. Maximizing net worth byshareholders leads to the following zero-profit and no-arbitrage conditions, respectively.

di� Id ¼ Izð1� dÞ; (14a)

rq ¼ _qþ Uf: (14b)

A unit of home currency deposit costs di in terms of the interest lost on reserves, whereas banks earn Id

for each unit. Thus, the left hand side of Eq. (14a) represents the marginal cost of a unit of deposit, whilethe right hand side represents its marginal revenue as a unit of deposit produces only 1� d unit ofloans. A no-arbitrage equilibrium requires that both be equal. The intuition for Eq. (14b) is equallystraightforward. While the right hand side represents net return from holding an equity as sum ofcapital gains and dividends, the left hand side represents its market opportunity cost.

Using (10) and integrating (14b) forward, firms’ equity price at any instant t is given by

qðtÞ ¼Z N

trðxÞe�rðx�tÞdx; (15)

Eq. (15) simply states that the value of a firm is equal to the present value of its return on capital. Finally,note that (13) together with (14a) and (14b) yields

Z N

0Ube�rtdt ¼ qð0Þsð0Þ � bbð0Þ; (16)

which states that banks’ net worth equals its current stock of assets net of liabilities.Banks’ balance sheet and banking crisis. After noting that in equilibrium s¼ 1, banks’ balance sheet is

shown in Table 3. Observe that deposits d are either kept as reserves m h dd with the central bank, oradvanced as loans z to firms. Hence, banks’ equity capital, denoted by nwb, equals the differencebetween the value of firms’ equity and banks’ bond market debt, as given by (16).

Clearly, a decline in firms’ equity price erodes banks’ equity capital. In particular, if a decline inequity price, Dq, exceeds banks’ existing net worth, then nwb< 0 and the banking system becomesinsolvent. Such an event is defined as a banking crisis. As firms require bank credit in order to produce, Iassume that the government bails out the banking system if a crisis occurs. In general, the level ofgovernment bailout will depend on the magnitude of stock market decline and the pre-crisis level ofbanks’ equity capital.

If a market for banks’ equities were explicitly included in this model, then the percentage pricedecline of banks’ equities will exceed that of the firms. To see this, let qb denotes the banks’ equityprice; clearly, qb¼ nwb. As all firms’ equities are held by the banks, Dqb¼Dq.28 Then the ratio of thepercentage equity price decline in the banking sector relative to that of the firms can be expressed as

Dqb

qb=

Dqq¼ q

qb¼ 1þ bb

qb:

Notice from Table 3 that bb/qb equals banks’ leverage (debt/equity) ratio. A positive leverage thenimplies that the percentage equity price decline in the banking sector will be larger relative to the

28 It bears emphasis that as long as banks’ exposure to asset prices and its leverage is sufficiently high the result will continueto hold even if the assumption that banks are the sole owners of firms’ equities is relaxed.

Table 3Banks’ balance sheet.

Assets Liabilities

1. Loans to firms (z) 1. Deposits (d)2. Reserves (m) 2. International

borrowing (bb)3. Stocks (q) 3. Bank’s equity (nwb)


firms. Moreover, the higher the leverage, the higher will be the relative equity price declines in thebanking sector. This is consistent with the evidence presented in Section 2 (see Table 2).

3.4. Government

The government includes the central bank. Their combined flow constraint is given by

�Db ¼ Dm if t˛G;

� _b ¼ �rbþ _mþ 3mþ s if t;G; (17)

where bhbg � h as the net public debt of the country; h denotes the stock of international bonds (i.e.,foreign reserves) held at the central bank, and bg denotes the stock of debt issued by the government inthe international bond market. The first part of Eq. (17) states that discrete monetary changes areimplemented by the central bank either through open market operations or by trading foreignreserves. Eq. (17) on the whole states that the government can discharge its debt only through taxesand/or seigniorage.

For simplicity, I assume that the government does not hold nominal debt. Nominal debt offers analternative way of financing unanticipated increases in fiscal liabilities through debt deflation. If so,seigniorage as a source of revenue becomes less important. The results in this paper critically hinge onthe government’s unanticipated need for seigniorage. Allowing debt deflation, however, will only affectthe results quantitatively. As long as seigniorage is inevitable, all the results will continue to holdqualitatively.

Monetary policy. Monetary policy is set by the following rules. Under a fixed exchange rate regimethe government sets a devaluation rate 3, while under flexible exchange rates it sets a money growthrate m.

3.5. Resource constraint

Combining the flow budget constraints of households, firms, banks, and the government as given by(2), (9), (12) and (17), respectively, and after substituting the equilibrium conditions, the economy’sflow constraint is given by

_f ¼ rf þ y� c (18)

where f¼ bhþ bf� bb� b is the economy’s net stock of international assets.

3.6. Monetary equilibrium

The monetary equilibrium involves the decisions of all agents, namely, households, firms, banks,and the government. Observe first that banks channel a fraction of deposits as loans to firms, as givenby (11), which then constitutes a constant fraction of firms’ returns, as given by (8). As a result, depositsrelate to firms’ returns through


d ¼ qr; (19)

where q h f/1� d, and 1/q is the deposits-to-returns multiplier. Similarly as banks’ fractional reservesdd solely constitute money supply, Eq. (19) yields

m ¼ hr; (20)

where h¼ dq, and 1/h is the multiplier that relates money supply to firms’ returns. Thus, inequilibrium, for each unit of firms’ returns on capital, banks raise q units of deposits on whichthey earn qId from the households. On the other hand, for each unit of firms’ returns, banks arerequired to keep h units of reserves, which costs them hi in terms of lost interests. Banks’ zero-profit condition (14a) implies that the total cost hi� qId ¼ fIz is in turn charged from firms foreach unit of their returns. Thus, firms’ return on capital net of interest payments as given by (10)can be rewritten as

r ¼ 9

1þ�

hi� qId�

substituting for Id from households’ demand for deposits (4b) yields

r ¼ 9

1þ ðhi� quduc

� ¼ rð i�; cþÞ: (21)

Eq. (21) implicitly defines r:r(i,c) as a function of i and c, as in equilibrium ud/uc is a function of r

and c. Given the assumptions on u, it is easy to show that ri< 0, and rc> 0. Eq. (21) has a simpleinterpretation. First, a higher nominal interest rate, ceteris paribus, raises marginal cost of loansdepressing the demand for capital. As a result, the equilibrium rent on capital decreases witha rise in the nominal interest rate. Second, the higher the consumption, the higher is the demandfor deposits that raises its price Id ¼ ud=uc. A lower marginal cost of raising deposits in turnlowers the interest rate that banks charge on loans. Thus, the equilibrium return on capitalincreases with consumption.

Finally, the nominal interest rate i is determined from the government’s monetary policy and itsflow constraint (17) as discussed below.

3.7. Pre-crisis stationary equilibrium

In what follows, x denotes a variable x’s value in the pre-crisis stationary economy. I firstcharacterize a stationary equilibrium under a fixed exchange rate regime. This requires that thegovernment set 3¼ 0 for all t. Assuming that the government runs a balanced budget, (17) can berewritten as

b ¼ sr; (22)

Eq. (22) states that the net public debt is financed by future tax revenues. The resource constraint (18)and the household’s optimality condition (4a) yields

c ¼ rf þ y; (23)

Eq. (23) simply states that the households consume the economy’s permanent income. Further, as 3¼ 0,i h r. Hence, the stationary level of return on capital r ¼ rðr; cÞ is obtained from (21). As a result, from(15), the equity price is given by

q ¼ r

r¼ rðr; cÞ

r; (24)

which reiterates that the equity price equals the present value of dividends.


4. Currency crisis and asset price decline

This section first specifies the conditions and the policy rules under which a currency crisis isinevitable. Assuming that the pre-crisis economy follows an exchange rate peg, a currency crisis isdefined as an event when the peg is abandoned and the exchange rate floats. Then, equilibriumallocations and the time when the peg is abandoned are characterized. Finally, it is shown that the crisisbrings forth a decline in asset prices as discussed in Section 2.

Suppose that the economy is in a stationary equilibrium as described in Section 3.7. At time 0, thereis an unanticipated increase in the stock of government’s fiscal liability by an amount J> 0.29 Beforet¼ 0 the existing level of government debt is funded by the present value of taxes and (22) holds. Theincrease in debt J will now be funded either by discrete changes in money supply (Dm) or throughseigniorage ð _mþ 3mÞ as in (17).30 Hence,

J ¼Z N

0

�_mþ 3m

�e�rtdt þ Ss˛f0;TgDmðsÞe�rs: (25)

I assume that the government continues to peg the exchange rate at E. Note that as long as 3¼ 0, i¼ r.From (17), (20) and (21), it then follows that there are no seigniorage revenues over t˛ð0; TÞ. Thus, thenet public debt continues to grow. I assume that eventually when the net public debt hits an upperbound F, the government abandons the peg and the exchange rate floats. Henceforth, T will denote thetime when this occurs. As will be shown below, discrete changes in base money can only occur at time0 and T. The second term in (25) takes this into account.

Finally, I assume that once the fixed exchange rate floats, the government sets a constant growthrate of money supply given by _M=M ¼ m.31 It can be shown that a constant m is consistent only witha stationary equilibrium, which in turn implies that 3¼ m. Thus, the post-crisis (t> T) government’sbudget constraint (17) can be rewritten as

rF ¼ 3mþ s; t˛½T;NÞ: (26)

In the pre-crisis economy, tax revenues support interest payments on the net public debt. Eq. (26)states that the net increase in debt service payments rF� s ¼ rðF� bÞ is financed by seigniorage afterthe exchange rate floats. Denoting the post-crisis debt service payment rðF� bÞ by y, the rate ofdevaluation is given by

3 ¼�

0; t˛½0; TÞ;ym; t˛½T;NÞ: (27)

As a result, the path of nominal interest rate is given by

i ¼�

r; t˛½0; TÞ;r þ y

m; t˛½T;NÞ: (28)

As shown below, the variables other than equity price q jump discretely at T but they have a flat pathotherwise. Therefore, an additional piece of notation will be helpful in what is to follow: the variablesover t˛½0; TÞ and t˛½T;NÞ are differentiated by using superscripts 1 and 2, respectively.

4.1. Crisis equilibrium

Using (19), the household’s first order conditions (4a) can now be rewritten as

29 The exercise follows Burnside et al. (2001) with one key difference. For simplicity, I abstract from the actual time-flow ofunanticipated liabilities and instead consider its present value as a net stock addition to the current stock of liability. Explicitlyincluding alternative time-flows will only complicate the analysis without adding any further insights.

30 Additionally, the government could raise revenues through explicit or implicit fiscal reforms (see Burnside et al., 2006).However, the results will continue to hold qualitatively as long as a non-zero amount of seigniorage is eventually raised.

31 Using more general rules will unnecessarily complicate the analysis without adding any further insights.


uc c1; qr1 ¼ uc c2; qr2 : (29)
� � � �
Using (4b), (20) and (28) in (21) yields

r1 ¼ r�

r; c1�;

r2 ¼ r

�r þ y

hr2; c2�: (30)

Finally, as the net asset level of the economy remains unchanged, the resource constraint (18)yields

c1�

1� e�rT�þ c2e�rT ¼ c: (31)

where c is as given in (23). Given T, {c1, c2, r1, r2} are easy to obtain from (29)–(31). Then {d1, d2, m1, m2}follow trivially from (19) and (20). However, the main variable of interest is T. Characterizing T entailssolving the aforementioned equations simultaneously with (22), (25) and (27), which is done below.

4.2. The run on reserves

A standard result in the ‘first-generation’ currency crisis models is that a run on reserves occurs aninstant before the peg is abandoned. Lemma 1 and Proposition 1 verify that this result holds in thepresent setup.

Lemma 1. Id2> Id1

; Iz2> Iz1

; and r1 > r2:

Proof see Appendix C.The result stated in Lemma 1 has a simple explanation. When the peg is abandoned at T,

inflation and nominal interest rates jump to a higher level. As a result, banks’ per unit cost ofholding required reserves jumps at T. Observe that households’ and firms’ demand for deposits andloans, respectively, are inversely related to their respective opportunity costs, Id and Iz. Moreover,loans are directly proportional to deposits in equilibrium. Hence, the higher cost of intermediation ispassed on to both households and firms, in terms of higher Id and Iz. Consequently, the rate ofreturn on capital drops at T.

Proposition 1. The exchange rate float is accompanied by a run on reserves.

Proof. Lemma 1 states that r1> r2. Then from (20) m1>m2. Hence, base money falls discretely at T,i.e., Dm(T)¼m2�m1<0. Then the combined goverment budget constraint (17) requires that

DbðTÞ ¼ �DmðTÞ ¼ h�

r1 � r2�> 0:

Again, this result has a simple intuition. Lemma 1 states that the demand for deposits and loans fallat T. As the exchange rate is fixed an instant before T, households and firms exchange their excessnominal balances at the central bank for bonds. While households buy bonds (Dd(T)) in exchange fortheir excess deposits, firms sell ((1� d)Dd(T)) bonds to the central bank to pay for their excess bankloans. Overall, base money and reserves fall by dDd(T)¼ h(r1� r2).

At this point, it is useful to make the following assumption that ensures that T� 0.

Assumption A.1. J < JhF� b� hðr� ~rÞ, where r ¼ rðr þ yh~r; cÞ is the stationary return on capital if

T¼ 0.

Suppose T¼ 0. Then from (31) c2 ¼ c. By definition ~r ¼ r2 is then obtained from (30). The size ofthe run will be equal to the fall in base money, i.e., DhðT ¼ 0Þ ¼ m� ~m, where ~m ¼ h~r. If theAssumption A.1 fails to hold, a run on reserves will leave agents with unexchanged nominal balances,and will require a discrete devaluation at 0. The resulting equilibrium can be trivially computed from


(29)–(31). In this case, the post-crisis stationary equity price will instantly adjust to ~q ¼ ~pr . Note that

~q < q, since ~r < r.It trivially follows that T¼ 0 for all J > J as T< 0 is impossible. In order to characterize functional

properties of T with respect to other variables of interest, it is henceforth assumed that the AssumptionA.1 always holds.

4.3. The time of exchange rate crisis

It is straightforward to obtain the time of exchange rate crisis from (22), (25) and (27) as

T ¼ 1rln

F� bþ DmðTÞJ� Dmð0Þ : (32)

Two features of (32) are noteworthy. First, the denominator J� Dmð0Þ, which equals the increase inthe stock of fiscal liability at time 0, must be positive if the peg is to be eventually abandoned.Otherwise, the net public debt will decline instead of growing and then the peg can be indefinitelysustained. Second, as the Assumption A.1 holds, i.e., T� 0, then F� bþ DmðTÞ � J� Dmð0Þ, orequivalently F � bþJþ hðr� r2Þ.32

One can conjecture from Eq. (32) that the higher the unanticipated increase in the stock of fiscalliability J, the sooner the peg will be abandoned. Second, the larger the gap between the initial netpublic debt b and its upper bound F, the longer the peg can be sustained. The results are intuitive. Note,however, that the changes in base money at 0 and T are functions of T and hence in turn of F, b, and J.Nevertheless, the conjecture stands verified with separable/logarithmic utility functions, as shown inAppendix B.33

4.4. Asset price dynamics

It is now shown that an unanticipated increase in fiscal liability not only precipitates an exchangerate crisis at T, but also triggers a fall in the equity price on impact followed by a declining path. Using(10), (15) and (30) the time path of equity price is given by

q ¼ r1 ��

r1 � r2�

e�rT ert ; t˛½0; TÞ; and q ¼ r2

r; t˛½T ;NÞ: (33)

Eq. (33) implies that at time 0

qð0Þ ¼ r2

r

�1� e�rT

�þ r2

re�rT : (34)

Proposition 2 presents the main result of this section.

Proposition 2. The equity price falls on impact of an unanticipated increase in the fiscal liability.Thereafter, it declines exponentially and eventually settles down to its post-crisis stationary valueq(T)¼ r2/r.

Proof. Appendix E shows that

qð0Þ < q:

The rest follows from Eq. (33). ,

Fig. 2 exhibits the equity price dynamics as stated in Proposition 2.The intuition behind these results is as follows. An unanticipated increase in the fiscal liability forces

the government to raise seigniorage eventually. As money supply comprises solely of banks’ reserves,

32 Suppose the inequality is violated. Then, as T < 0, T¼ 0, which violates the Assumption A.1.33 The result that T is increasing in F� b and decreasing in J holds with more general utility functions. We skip the required

analysis in order to focus on our main results.

Fig. 2. Equity prices during currency crisis.


banks bear the whole brunt of additional fiscal liability. Banks, in turn, pass it on to households andfirms by adjusting interest rates on deposits and loans, respectively. While the overall share of addi-tional costs borne out by firms is determined from household preferences and other parameters of themodel, the share nevertheless is positive. As a result, firms’ present value of dividends relative to itsstationary level falls on impact, thus causing a drop in the equity price. The declining path of equityprice is explained by the fact that the return on firms’ capital is relatively higher while the exchangerate is fixed. Then the present value of firms’ dividends declines as the time of exchange rate crisisnears. Hence, the equity price declines over time.

Logarithmic preferences. To get some idea of the share of additional liability borne by firms, it isuseful to study a special case: u(c,d)¼ ln cþ c ln d. Then, the time path of equity price is given by

q ¼ q�J ert ; t˛½0; TÞ;

q ¼ q�J ert ; t˛½T;NÞ: (35)

Note from Eq. (35) that qð0Þ ¼ q�J. Implicitly, the total increase in the fiscal liability is taxed awayfrom firms. Intuitively, with logarithmic preferences households’ share of expenditures on consump-tion and deposits remains fixed. Since the economy’s present value of wealth remains unchanged,households’ spending on consumption and deposits stays at its pre-crisis stationary level. Then, thewhole burden of seigniorage falls on firms. Hence, the equity price falls by J on impact and thereafterfollows Eq. (35).

5. Government bailout and self-fulfilling twin crises

Section 4 showed that an unanticipated increase in fiscal liability, in addition to generatinga currency crisis, causes a decline in equity prices. In this section, I endogenize the increase in the stockof fiscal liability through a government bailout policy. Specifically, I propose that the governmentassumes a fraction g of the capital loss incurred by banks. Now, if agents believe that a currency crisis isimminent, equity price falls, and the increase in fiscal liability as a result of government bailout thusvalidates the belief. As a result, twin crises can be self-fulfilling. I first establish the range of g overwhich self-fulfilling crises equilibria exist. Then I study how g affects the time of exchange rate crisis.Finally, I study the relationship between the upper bound on net public debt and the time of exchangerate crisis.

Proposition 2 states that the equity price drops from q to q(0) under the anticipation of a futurecurrency crisis. As a result, banks’ equity capital declines by q� qð0Þ, which I assume to leaves banks


with a negative net worth. Then, in order to keep banks financially viable, the government adopts thefollowing bailout rule34:

J ¼ gðq� qð0ÞÞ: (36)

As discussed in Section 4, an unanticipated increase in fiscal liability, in the absence of an alternativesource of revenue, implies that sooner or later the government will resort to seigniorage. Hence, a fixedexchange rate regime cannot be sustained indefinitely, and the net public debt will reach its upperbound F at a finite date T. As before, the rate of devaluation and the nominal interest rate will be givenby Eqs. (27) and (28), respectively. Further, for any T, the prices and allocations {c1, c2, r1, r2} can beobtained from (29)–(31). Finally, T will be pinned down from (25). Using Eqs. (27), (34) and (36) in Eq.(25) yields

ðF� bÞe�rT ¼ gðq� qð0ÞÞ þ hrðq� qð0ÞÞ:

The left hand side of above equation represents the present value of additional debt service thattriggers from T onwards. The right hand side shows the present value of the increase in the stock offiscal liabilities, while the exchange rate remains fixed. The first term represents the governmentbailout, while the second term denotes the present value of foreign reserves lost due to the exchange ofnominal balances by the private sector. Note from (20) that the changes in base money are directlyproportional to the changes in equilibrium returns on capital. Since q� qð0Þ measures the presentvalue of the changes in returns to capital, the second term denotes the present value of cumulativechanges in base money. Alternatively, the above equation can be rewritten as

F ¼ xðq� qð0ÞÞerT þ b; (37)

where x h gþ hr. Note that under fixed exchange rates the government can only sustain its pre-crisisdebt b through its tax revenues but the additional stock of liabilities keep growing up to T. As assumed,the two together equal F at T when the peg is abandoned. Eq. (37) directly yields

T ¼ 1rln

1x

F� bq� qð0Þ:

A casual glance at the above expression tells that the larger the upper bound on the net publicdebt, the longer the peg can be sustained. However, from (29)–(31) and (34) the fall in equityprice depends on the anticipated rate of devaluation after T, which in turn depends on the netincrease in the debt service y ¼ rðF� bÞ. Hence, Eq. (37) implicitly determines T as a function of y

and g. Thus, for any y and g, a self-fulfilling twin crises equilibrium exists only if (29)–(31) and(37) are satisfied. Section 5.1 characterizes the space of g over which self-fulfilling equilibria existfor a given value of y.

5.1. Self-fulfilling equilibria

Henceforth, for simplicity, I assume that uðc; dÞ ¼ ½ðc1�ð1=sÞ þ cd1�ð1=sÞÞs=ðs�1Þ�1�ð1=3Þ=1� ð1=3Þ,where s and e are intratemporal and intertemporal elasticity of substitution, respectively. It is easy toverify that this CES form satisfies all assumptions as elaborated in Section 3.1. Note that when s¼ e,utility is separable in c and d. As is well known, the relative intertemporal allocations crucially dependwhether eXs, all three cases are analyzed in order to generalize the results. However, to avoidunnecessary algebra in derivations, it is assumed that e¼ 1. Appendix D and Appendix F derive therelative magnitudes of {c1, c2, r1, r2} for all the three cases, sX1.

For verifying the existence of self-fulfilling equilibria I take the following approach. Let theeconomy be in a stationary equilibrium as described in Section 3.7. Fix T. Then, {c1, c2, r1, r2} are

34 This rule simplifies the analysis. However, the results will hold for any bailout rule that is linear in the loss of banks’ equitycapital.


obtained from (29)–(31). Next, q(0) is obtained from (34). Finally, the implied value of g is obtainedfrom (37) as

g ¼ F� bq� qð0Þe

�rT � hr: (38)

Proposition 3 establishes the existence of self-fulfilling twin crises equilibria over a range of g.

Proposition 3. For any pre-crisis stationary equilibrium, there exist g and g such that self-fulfilling twincrises exist only if g˛½g ;g�. Moreover, for sse, T is decreasing in g. Hence, g/ g as T / N and g ¼ g

when T¼ 0.

Proof see Appendix G.Proposition 3 states that self-fulfilling twin crises exist only if the bailout ratio falls in a certain

range. With non-separable preferences, it is further claimed that the lower the bailout ratio, the laterthe peg will be abandoned. The result sounds intuitive. The explanation, however, is somewhatcomplicated. A lower bailout ratio implies a smaller increase in the stock of fiscal liability, had the fall inequity price remained constant. As additional stock of liability at t¼ 0 must equal ðF� bÞe�rT , it isimplied that the peg can be sustained longer. But, the later the exchange rate floats, the later will be thedecline in firms’ returns, and thus a smaller fall in equity price at time 0. From (38), it is then notobvious why g should move inversely with T. An inspection of (38) tells that the result in Proposition 3can hold only if the rate at which the fall in equity price q� qð0Þ decreases with respect to T is smallerthan r. Appendix G proves that this is the case indeed.

A more intuitive explanation progresses in the following way. In a crisis, firms’ loss of dividendsrelative to the stationary economy is due to an increased cost of loans. The increase in costs in turn isdriven by a combination of two factors. The first is the increase in fiscal liability as a result ofgovernment bailout plus the loss of reserves due to cumulative change in base money. As discussedearlier, this component is a fixed multiple x¼ gþ hr of the fall in equity price. The second factordepends on the share of the increased liability borne by households. If households’ consumption anddeposits are gross complements (s< e), they are willing to bear a larger share of seigniorage. Thenfirms’ burden is lessened. As a result, the fall in equity price is smaller than the increased stock of fiscalliability. On the other hand, if households’ consumption and deposits are gross substitutes (s> e), theirdeposits can fall significantly if the interest rate on deposits falls. Now, in order to meet firms’ demandfor loans, banks have to induce households to hold deposits by offering a higher interest rate. Thisadditional cost of funds in turn further hikes the interest rate on loans. As a result, firms’ burden offinancing additional fiscal liability is further compounded.

The adjustment of households’ deposit allocations, as T changes, works as follows. First, a constantlevel of wealth implies that consumption levels before and after T move in the same direction. Forexpositional convenience, I terms it as the ‘time’ effect. In general, the ‘time’ effect makes depositscommove with consumption. When consumption and deposits are gross complements (s< e) the‘time’ effect and the substitution effect works in the same direction, and the response of deposits to thechanges in T is strong. On the other hand, when consumption and deposits are gross substitutes (s> e),the ‘time’ effect and the substitution effect oppose each other. Then, the response of deposits tochanges in T is relatively milder.

Note that for the CES utility form with s¼ e, utility is separable in consumption and deposits. Then,households’ expenditure on deposits up to T is the same as that in the pre-crisis stationary economy.Therefore, before T, households do not bear any share of additional fiscal liability. Although, T onwards,the expenditure on deposits generally differs from that in the stationary economy,35 its level is invariantto the value of T.36 Therefore, as T changes, the present value of the households’ share of additionalfiscal liability adjusts at a rate r. Note that the additional fiscal liability ðF� bÞe�rT also adjusts at a rate rwith a change in T. However, for sse, following the discussion in the preceding paragraph, the rate of

35 Except when s¼ e¼ 1; then the expenditure after T equals its level in the stationary economy.36 The case with s¼ e is discussed below in more details.


adjustment of deposits with respect to T is higher (lower) than the rate of overall fiscal adjustment r,when the consumption and deposits are gross complements (substitutes).

Let the share of additional fiscal liability that households partake when s¼ e be S. Consider now thecase when consumption and deposits are gross complements. Then, for any given T, households inaddition to S absorb an extra share S0. Thus, 1� S� S0 of the increased stock of fiscal liability xðq� qð0ÞÞis borne by firms. Since the share that is borne by firms is identically equal to the fall in equity priceq� qð0Þ, it must be the case that (1� S� S0)x¼ 1. Suppose now T is increased. Then the new presentvalue of additional fiscal liability is ðF� bÞe�rT is smaller. However, as the rate of households’ depositsadjustment is higher than r, their share of fiscal burden decreases at a rate faster than the overall fiscaladjustment. In other words, S0 decreases. Then x must also decrease. Hence, a higher T is consistentwith a lower bailout ratio.

On the other hand, when consumption and deposits are gross substitutes, households’ depositreallocation compounds firms’ burden by an extra share S00. In equilibrium, (1� Sþ S00)x¼ 1. House-holds’ deposit adjustment rate with respect to T is lower than r now. Therefore, for a higher T, the extraliability that households burden firms with decreases at a rate lower than r, the rate at which the fiscalliability decreases. In other words, S00 increases. Once again, x (and therefore g) must decrease with T.

Finally, as the bailout ratio g decreases with T, the upper bound g corresponds to T¼ 0, and g / g asT / N. Thus for a bailout ratio g� g a self-fulfilling crisis can never occur.37 Appendix G computes theupper and lower bounds explicitly.

Note that these results are applicable only if sse ¼ 1.38 On the other hand, when s ¼ e, themarginal utilities of consumption and deposits are independent of each other, and the intertemporaldeposit allocations are independent of T, as can be seen from (30). This leads to indeterminacy ofequilibria as discussed below.

5.2. Indeterminacy of crises equilibria

When s¼ e, the utility function takes the form u(c,d)¼ y(c)þw(d). Then, from (29)–(31), a constantmarginal utility of consumption implies c1¼ c2¼ c. Further, r1¼ r. Then, using (34) in (38) yields

gjs¼e¼y�

r� r2�� hr: (39)

Now g is independent of T, since r2 from (30) is independent of T. Thus, any T˛½0;NÞ satisfies (29)–(31)and (34), and is consistent with (38) if and only if g is given by (39).

The intuition behind the indeterminacy result is as follows. Notice first that the return on capital upto T is identical to the pre-crisis equilibrium, as both the consumption and the nominal interest rateremain unchanged. Thus firms’ returns fall only at T, and the drop r� r2 solely depends on the increasein debt service payments y, which is independent of T. The increase in the stock of fiscal liability at time0 is a multiple x of the present value of the fall in firms’ returns. In equilibrium, this must equal thepresent value of the seigniorage y=r e�rT . Since both present values use the same discount rate, there isa unique bailout ratio that balances the government’s budget. However, T is indeterminate.39

5.3. Upper bound on public debt and the time of exchange rate crisis

By assumption, the government’s exchange rate rule allows it to maintain the peg until the netpublic debt grows to its upper bound F. Then a question naturally arises: how T is affected by F? Thefollowing proposition provides an answer.

37 For g> , self-fulfilling crises will be consistent only if T< 0, which is impossible. All such cases imply T¼ 0, which entailsa discrete devaluation at 0. As discussed in Section 4, the analysis is restricted to T� 0 cases as all other cases are ruled out byAssumption A.1.

38 It is conjectured that the result should hold for all e and s, such that sse.39 When s¼ e¼ 1, i.e.,uðc; dÞ ¼ ln cþ c ln d, it is easy to verify that g¼ 1.


Proposition 4. When households’ deposits and consumption are gross complements (substitutes),a higher upper bound on net public debt causes the exchange rate float to be delayed (advanced).

Proof see Appendix H.The intuition behind this result is the following. A higher F (and hence F� b) has two mutually

opposing effects. First, a higher accumulation of debt allows the government to sustain the peg longer.On the other hand, the implied additional debt service then calls for a higher level of seigniorage,a higher rate of devaluation, and nominal interest rates. Higher nominal interest rates in turn causea higher fall in equity prices. Then the bailout is larger, which advances T. When households’ depositsand consumption are gross complements, a higher post-crisis nominal interest rate, however, impliesthat agents tilt their consumption and deposits to a higher level while the exchange rate is fixed, i.e., thenominal interest rate is low. As a result, there is a rise in households’ deposits relative to the pre-crisislevel. The resulting increase in the foreign reserves of the central bank partially offsets the net increasein the stock of fiscal liability due to the government bailout. Overall, it is the delaying effect of a higher F

that dominates. On the other hand, when consumption and deposits are gross substitutes, householdsprefer a higher consumption when the nominal interest rate is high, i.e., when the exchange rate floats.This implies a lower level of consumption while the peg exists. As a result, deposits drop from their pre-crisis level at time 0. Now the delaying effect of a higher F yields to its advancing effect that stemsjointly from the loss of foreign reserves and the bailout at time 0. Thus, T is advanced.

6. Conclusions

This paper shows that a prospective currency crisis can cause a banking crisis in advance. The keymechanism which links the twin crises hinges on the empirical facts that asset prices decline inadvance of currency crises, and that banks in emerging markets carry large asset price exposure. Themodel presented in the paper generates an endogenous decline of asset prices when agents realize thata currency crisis is imminent. The decline in asset prices combined with the government bailout ofbanks thus spawns a self-fulfilling twin crises, in which banking crisis precedes currency crisis as hasbeen observed empirically.

The present analysis shows that the government bailout rule, whereby the government assumesa fixed ratio of banks’ capital loss, crucially impacts the crises equilibria. In particular, the higher thebailout ratio, the steeper is the decline in asset prices and the sooner the government is forced toabandon its fixed exchange rate policy. This result holds under a fairly general specification ofhouseholds’ preferences. However, the effect of a higher upper bound on the net public debt – theamount of debt at which the government lets the exchange rate float – depends on households’preferences. If households’ nominal balances and consumption are gross complements, the currencycrisis is delayed. On the other hand, when they are gross substitutes, the crisis is advanced.

I conclude by discussing two obvious shortcomings of the present analysis. First, the results criti-cally hinge on the government’s dependence on seigniorage for financing its unanticipated contin-gencies. In practice, debt deflation and fiscal reforms are equally (or even more) important sources ofgovernments’ finance after crises (see Burnside et al., 2006). However, as long as seigniorage remains asone of the sources, including these alternatives will merely rescale the results. In particular, the smallerthe share of seigniorage in the overall additional revenues, the longer a fixed exchange rate regime canbe sustained. All the results are otherwise purely qualitative and will continue to hold.

Equally importantly, the model fails to generate any recessionary effects of twin crises as the outputis fixed by construction. In a more realistic setting, declining asset prices can impinge on firms’ liquidityby either raising the cost of loans, or by constraining their access to financial markets. I conjecture thatadding these features will only amplify the present results and also generate substantial output andconsumption declines as observed in the crises economies. This is left for future research.

Acknowledgments

I am thankful to an anonymous referee, Amartya Lahiri, Luisa Lamber-tini, Carlos Vegh, and JoydeepBhattacharya for helpful comments and suggestions. All remaining errors are mine.


Appendix.

A. Firm’s problem

There is a continuum of firms uniformly distributed over a unit interval, each owning a unit ofdifferentiated capital. A firm and its specific capital are indexed by j ˛ [0,1]. Firm j producesconsumption good with the following technology:

yj ¼ F�

kj; lj�;

where F($,$) is homogeneous of degree one, and F1,F2� 0. Further,

kj ¼ Z 1

0

�kjl�a�1

adl

! aa�1

; a > 0;

and where kjl (l ˛ [0,1], and l s j) denotes the quantity of type l capital hired by firm j from firm l, and kjj

is the quantity of firm j’s capital used in its own production. Note thatR 1

0 kljdl ¼ 1; cj. Firm j’s credit-in-advance constraint (5) can be written as

zj � f

Z½0;1�yj

rlkjldl; f˛

0;1

#;

where rl is the market rate of return on type l capital.Finally, firm j can hold international bonds, bj. Assuming that the credit constraint binds in equi-

librium, firm j’s flow constraint is given by

Dbj ¼ Dzj; if t˛G;

_aj ¼ raj þ F�

kj; lj��wlj �

Z½0;1�yj

rlkjl�1þ fIz�dlþ rjZ½0;1�yj

kljdl� Uj; if t;G;

where aj ¼ bj � zj, and Uj denotes firm j’s dividends payments. By assumption, bj¼ zj for all t. Then,firm j’s dividend payment at any instant t is given by

Uj ¼ F�

kj; lj��wlj �

Z½0;1�yj

rlkjl�1þ fIz�dlþ rjZ½0;1�yj

kljj dl:

Integrating forward, the present value of dividends is

Z N

0Uje�rtdt ¼

Z N

0

F�

kj; lj��wlj

Z½0;1�yj

rlkjl�1þ fIz�dlþ rj

Z½0;1�yj

kljdl

!e�rtdt:

Given the path of interest rates, {iz,r,rl}, the shareholders of firm j maximize the present value ofdividends by choosing the path of its capital use, {kjj,kjl}. The symmetric supply and demand schedulesfor all types of capital implies that their equilibrium rates of return are equal, i.e., rj¼ r for all j. The firstorder conditions require that the ratio of the marginal product of firm j’s use of its own capital, kjj, to itsuse of hired capital, kjl, be equal to the ratio of their respective opportunity costs, r and r(1þ fIz).Hence,

kjj

kjl¼�1þ fIz�a

; cl˛½0;1�yj:

In equilibrium, firm j rents capital from all other firms in equal amounts, i.e., kjl¼ kjn for all l and n s j.


The symmetry also implies an equal amount of mutual hiring, i.e., klj¼ kjl. Moreover, each type of capitalis hired in equal amounts by all firms, i.e., klj¼ knj, for all l and n s j. Since firm j has a zero mass and theuse of its own capital, kjj, is finite, the aggregate equilibrium condition

R 10 kljdl ¼ 1 implies that klj¼ 1 for

all l s j. Thus, the first order condition with respect to rented capital yields (6) in the main text.

B. Derivation of T with specific utility forms

Let u(c,d)¼ u(c)þw(d). From (29) and (31) c1¼ c2¼ and r1 ¼ r. Then, (32) yields

T ¼ 1rln

F� b� h�r� r2

�J

;

where from (30) r2 ¼ rðr þ ðy=hr2Þ; cÞ. Clearly, r2 is independent of J and T. Hence, TJ< 0. On theother hand, r2 is decreasing in F� b. Therefore, the effect of F and b on T is not obvious. As a furtherspecialization, let u(c)¼ ln c, and w(d)¼ c ln d. Then,

T ¼ 1rln

F� buJ

;

where u¼ 1þ hr. Thus TJ< 0, TF> 0, and Tb< 0.

C. Proof

Proof of Lemma 1. Using (4a) and (4b) it can be shown that

dr

dId¼ 1

q

dr

dId¼ 1

quc

udduc � ucdud

uccudd � ucdudc< 0; (40)

where the last inequality follows from the properties of the u. Next, using (21)

Id1 ¼ 1q

�u� 9

r1

�;

Id2 ¼ 1q

�u� 9� y

r2

�(41)

where u¼ 1þ hr. Suppose Id2< Id1

. Then, from (40) r1< r2. But, from (41), Id2> Id1

– a contradiction.Hence,

Id2> Id1

; r1 > r2: (42)

The last result and (10) yield

IZ2> IZ1

:

,

D. Allocations relative to pre-crisis levels

In this section, I retain a general utility form with assumptions uc, ud> 0, ucc, udd< 0,udcuc� uccud> 0, ucdud� udduc> 0, and uccudd� ucdudc> 0. Using (4a) and (4b), obtain

dcdId¼ � ucduc

uccudd � ucdudcX0 iff ucd,0: (43)

Eqs. (4a), (4b) and (41) yield


ud�c1; qr1

�� ¼ 1�

u� 9�

;
uc c1; qr1 q r1
ud�c2; qr2

�uc�c2; qr2

� ¼ 1q

�u� 9� y

r1

�;

which imply c1 and c2 as functions of r1 and r2, respectively. Clearly,

dr1

dc1 ¼udcuc � uccud

ucdud � udduc1� d

f

9

r1ðucÞ2> 0;

dr2

dc2 ¼udcuc � uccud

ucdud � udduc þ1� d

f

9� y

r2ðucÞ2

> 0: (44)

Case Iucd> 0. Using (31) with (43) yields c1 > c > c2. Observe from (4a), (4b) and (41) that if c2 ¼ c; r2 < r.Hence, from (42) and (44), r1 > r > r2.

Case IIucd< 0. Using (31) with (43) yields c1 < c < c2. Then from (42) and (44), r > r1 > r2.

Case IIIucd¼ 0. Using (4a) and (4b), with (31) yields c1 ¼ c2 ¼ c. Then from (42) and (44), r1 ¼ r > r2.

Note that in all cases, r1> r2. Hence, from (20), m1>m2.

E. Proof

Proof of Proposition 2. qð0Þ < q.Using (24), (32) and (34) obtain

q� qð0Þ ¼ r1 � r2

r

�J� h

�r1 � r

�F� b� h

�r1 � r2

��r1 � r

��r1 � r2

��: (45)

Since ðJ� hðr1 � rÞÞ=ðF� b� hðr1 � r2ÞÞ > 0, the proof for the case when ucd� 0 is trivial. For ucd> 0,observe from (45) that q� qð0Þ � 0 if and only if

r1 � r2

r1 � r� F� b

J� 1þ h

r� ~r

J;

where the second inequality follows from assumption that T� 0. The above inequality in turn impliesthat

r� r2

r1 � r� h

r� ~r

J:


Recall from the discussion following Eq. (32) that J � hðr1 � rÞ. Hence, the above holds ifr� r2 � r� ~r, or ~r � r2. On the other hand, since ucd> 0, c > c2. Following (44) then ~r � r2. ,

F. CES utility form

Let uðc;dÞ ¼ ½ðc1�ð1=sÞ þ cd1�ð1=sÞÞs=ðs�1Þ�1�ð1=eÞ=1� ð1=eÞ. It is easy to check that this satisfies allthe assumptions on u ($,$) following Eq. (1). Furthermore, for s < e, ucd > 0 while for s > e, ucd < 0. Foranalytical convenience, I set e¼ 1.

Case Is< 1: Here, ucd> 0. Hence, c1 > c > c2 and r1 > r > r2.

Case IIs> 1: Now, ucd< 0. Hence, c1 < c < c2 and r > r1 > r2.

Case IIIs¼ 1: In this case, ucd¼ 0. Hence, c1 ¼ c2 ¼ c and r ¼ r1 > r2.

Using (4b) and (41) obtain

c1 ¼ kr1�

u� 9

r1

�s

;

c2 ¼ kr2�

u� 9� y

r1

�s

; (46)

where k ¼ c�sð1� d=fÞs�1, Next, using (4a), (4b) and (41) yields

c1 þ�

ur1 � 9�¼ c2 þ

�ur2 � 9þ y

�: (47)

For convenience, I rewrite (31) as

c1~T1 þ c2~T2 ¼ c; (48)

where ~T1 ¼ 1� e�rT , and ~T2 ¼ e�rT . Observe that solving (46)–(48) obtains equilibrium allocationsfor all T� 0.

The following lemma is used in deriving some key results that follow.

Lemma 2. c1r1 Xc2

r2 if and only if s,1.

Proof. Differentiating (46) yields

c1r1

c2r2

¼

�u� 9

r1

�s�1�u� ð1� sÞ 9

r1

��

u� 9�yr2

�s�1�u� ð1� sÞ9�y

r2

�Let x ¼ 9=ður1Þ and y ¼ 9� y=ður2Þ. Further, as Id1

< Id2; y < x. Hence,

c1r1 Xc2

r2 iffð1� xÞ1�s

ð1� 1� sÞxÞ,ð1� yÞ1�s

ð1� ð1� sÞyÞ iff s,1:

For future use, note that


c1r1r1 ¼ lsðs� 1Þ 92�

r1�3

�u� 9

r1

�s�2

X0 iff sX1;

c2r2r2 ¼ lsðs� 1Þð9� yÞ2�

r2�3

�u� 9� y

r2

�s�2

X0 iff sX1: (49)

,

G. Proof

Proof of Proposition 3. Observe that q(0) is uniquely determined as a function of T from (46)–(48) and(34). Hence, to show that T is non-increasing in g, it suffices to show using (38) thatd

dTððq� qð0ÞÞerT Þ � 0. In particular, I show that ddTððq� qð0ÞÞerT Þ > 0 if ss1, and d

dTððq� qð0ÞÞerT Þ ¼ 0 ifs¼ 1.

Next, the value of g for T¼ 0, denoted as g , is easy to compute from (46)–(48), (34) and (38).However, computing the value of g as T / N, denoted as g, requires an intermediate result. Hence, thisis done at the end.

The proof proceeds in the following five steps.

Step IShow that limT/Nðd=dTÞððq� qð0ÞÞerT Þ ¼ 0; cs. First, from (34):

limT/N

ðq� qð0ÞÞ ¼ 0:

Applying L’Hospital’s rule yields

limT/N

ðq� qð0ÞÞerT ¼ limT/N

1r

erT dqð0ÞdT

: (50)

Hence,

limT/N

ddT

�ðq� qð0ÞÞerT

�¼ lim

T/Nrðq� qð0ÞÞerT � lim

T/NerT dqð0Þ

dT¼ 0:

Step IIShow that d

dTððq� qð0ÞÞerT ÞjT¼0 ¼ 0 for s¼ 1, and ddTððq� qð0ÞÞerT ÞjT¼0 > 0 for ss1. Substituting (46)

into (47), and differentiating along with (48) yields

dr1

dr2

¼ 1

D

24�

�c2

r2 þ u�þ c2

r2~T2

��

c1r1 þ u

�þ c1

r1~T1

35 r e�rT

�c2 � c1

�0

dT ;

where D ¼ �c1r1 ðc2

r2 þ uÞ~T1 � c2r2 ðc1

r1 þ uÞ~T2. Hence,


dr1¼ �1

r�

c2r2 þ u

� �c2 � c1

�~T2�< 0 if s < 1;

dT D > 0 if s > 1;

dr2

dT¼ �1

Dr�

c1r1 þ u

��c2 � c1

�~T2�< 0 if s < 1;> 0 if s > 1;

(51)

where the inequalities follow from the derivations in Section F and Lemma 2. Using (34) with (51)yields

ddTðq� qð0ÞÞerT ¼ rðq� qð0ÞÞerT dqð0Þ

dT¼�

r� r1�

erT þ�

c1 � c2� ~T1 þ r2

r1~T2

~T1c1r1 þ r2

r1~T2c2

r2

;

where r2r1 ¼ ðc1

r þ uÞ=ðc2r þ uÞ. Observe that

ddTððq� qð0ÞÞerT Þ

��T¼0� 0 iff

�r� r1

�þ�c1 � c

�c2

r2

� 0; (52)

since for T¼ 0, ~T1 ¼ 0 and ~T2 ¼ 0. Note that this holds with equality for s¼ 1. Then, for s< 1, (48),Lemma 2, and (49) imply that

c1 � cr1 � r

��T¼0

> c1r1

��T¼0

> c2r2

��T¼0

:

Similarly, for s> 1, (48), Lemma 2, and (49) imply

c1 � cr1 � r

��T¼0

< c1r1

��T¼0

< c2r2

��T¼0

:

Since for s¼ 1, c1 ¼ c, r1 ¼ r, (52) holds for all values of s. In particular, it holds with strict inequalityfor ss1.

Step IIIShow that erT ðdqð0Þ=dTÞ > 0; cs. Further d=dTðerT ðdqð0Þ=dTÞÞ ¼ 0; for s¼ 1 andd=dTðerT ðdqð0Þ=dTÞÞ > 0 for ss1. Using (34) with (51) yields

erT dqð0ÞdT

¼�

r1 � r2�þ�

c2 � c1� ~T1 þ r2

r1~T2

~T1c1r1 þ r2

r1~T2c2

r2

: (53)

For s¼ 1, the RHS equals r1� r2> 0. Further, for s> 1, both terms on the RHS are positive. Hence, (53)again holds with strict inequality. On the other hand, for s< 1, the first term is positive but the secondis negative. Note from Lemma 2 that c1

r1 > c2r2 . Hence, erT ðdqð0Þ=dTÞ > 0 if ðc1 � c2Þ=ðr1 � r2Þ < c2

r2 .From (46) c2jr2¼r1 > c1. Finally, using (49), c2

r2 > ððc2jr2¼r1 � c1Þ=ðr1 � r2ÞÞ > ðc1 � c2Þ=ðr2 � r2Þ. Hence,erT ðdqð0Þ=dTÞ > 0.

For the next part, taking derivative of the RHS in (53) w.r.t. T, and after some algebra

ddT

�erT dqð0Þ

dT

�¼��

~T1c1r1r1 þ ~T2

�r2

r1

�3c2

r2r2

�dr1

dT þ 2r2r1

�c2

r2 � c1r1

�r~T2

�~T1c1

r1 þ r2r1

~T2c2r2

�2

�c2 � c1

�:


For s¼ 1, the RHS equals zero. Next, for s< 1, the RHS is clearly positive. For s> 1, using (51), the RHS ispositive iff

2r2r1

�c2

r2 � c1r1

�c2 � c1 �

~T1c1r1r1 þ ~T2

�r2

r1

�3c2

r2r2

~T1c1r1 þ ~T2r1

r1 c2r2

;

which is easy to verify for T¼ 0 and as T / N. By monotonicity, it holds for all T.

Step IVIt has been shown that d=dtððq� qð0ÞÞerT Þ � 0 for T¼ 0 and as T / N. Further, d=dTðerT ðdqð0Þ=dTÞÞ �0 for all T. Thus d=dtððq� qð0ÞÞerT Þ ¼ rðq� qð0ÞÞerT � d=dtðerT ðdqð0Þ=dTÞÞ � 0 for all T. Suppose not.Then for some T ¼ T 0; ðd=dtÞððq� qð0ÞÞerT Þ < 0. Then, ðd=dtÞððq� qð0ÞÞerT Þ < 0 for all T > T 0. But, thenit will violate the result shown in Step I, i.e., limT/Nðd=dtÞððq� qð0ÞÞerT Þ ¼ 0. Hence,ðd=dtÞððq� qð0ÞÞerT Þ � 0; cT . In particular,

ddt


�¼ 0; s ¼ 1;

ddt


�E0; ss1:

Step VNote from (38) that for T¼ 0,

g ¼ y

r� r2��T¼0

� df

1� dr ¼ g

Further, using (50) and (53) in (38), as T/N

g ¼ y

r� r2��T/N

þ 1

c1r1

��c1¼c

�c2��T/N

�c�� d

1� dfr ¼ g

where r2jT¼0; r2jT/N; c

2jT/N; c1r1 jr1¼r are computed from (46)–(48). ,

H. Proof

Proof of Proposition 4. Note that y ¼ rðF� bÞ. Hence, the sign of dT/dF¼ sign of dT/dy. At this point,it is convenient to work with y. Differentiating (37) w.r.t. y yields

dTdy¼

vqð0Þvyþ q� qð0Þ

y

rðq� qð0ÞÞ � vqð0ÞvT

; (54)

where ðvqð0Þ=vTÞ is obtained from (53). Using (46)–(48), obtain

vqð0Þvy

¼ �1r

~T2

~T1c1r1 þ r2

r1~T2c2

r2

c2

y r2r1 þ ~T1

c1r1 � c2

r2

uþ c2r2

!< 0; (55)

where


c2y ¼ ks

�u� 9� y

r2

�s�1

:

The denominator in (54) is positive as shown in Appendix G. However, the two terms in the numeratorhave opposite signs, and hence there sign is not so obvious. If the numerator has the same sign for bothT¼ 0 and as T / N, then from monotonicity it will have the same sign for all T. First, from (55) obtain

vqð0Þvy

��T¼0¼ �1

rc2

y

c2r2

;

Then using (46) and (54) obtain

dTdy

��T¼0¼

r� r2

ry� 1

rs�

u� ð1� sÞ9�yr2

�rðq� qð0ÞÞ � vqð0Þ

vT

:

Notice that for s¼ 1, the numerator is equal to zero. Hence, one needs to check the derivative of thenumerator with respect to s around s¼ 1:

dds

�� ysþ

�r� r2

��u� ð1� sÞ9� y

r2

��¼ c

k

�ln�

1þ yckþ 9� y

�� y

ckþ 9� y

�� 0;

where, using (46), I have made use of the fact that

dr2

ds

��T¼0¼ �c

k

ln�

u� 9�yr2

�u

;

dr

ds¼ �c

k

ln�

u� 9r

�u

:

Thus, I have shown that

dTdy

��T¼0

X0 iff s,1: (56)

Next, rewrite (54) as

dTdy¼

�vqð0Þvyþ q� qð0Þ

y

�erT


vT

�erT

:

Note that�

rðq� qð0ÞÞ � vqð0ÞvT

�erT � 0; cT , and limT/Nðrðq� qð0ÞÞ � ðvqð0Þ=vTÞÞerT ¼ 0. From (55)

obtain

erT vqð0Þvy

��T/N

¼ �1r

c2y

�uþ c1

r1

�þ c1

r1 � c2r2

c1r1

�uþ c2

r2

� :

Then (54) yields

dTdy

��T/N

¼limT/N

ðq�qð0ÞÞerT

y � 1r

c2y ðuþc1

r1 Þþc1r1�c2

r2

c1r1 ðuþc2

r2 Þ

��T/N

limT/N


vT

�erT

:

Using (53), it can be shown that


limT/N

ðq� qð0ÞÞerT

y¼ lim

T/N

1ry

erT vqð0ÞvT

¼ 1ry

�c2 � c1

� 1c1

r1

þ 1u

!þ y

u

!:

Hence,

limT/N

�vqð0Þ

vyþq�qð0Þ

y

�erT ¼ 1

ruþ 1

ry

�c2� c1

� 1c1

r1

þ 1u

!!�1

r

c2y

�uþ c1

r1

�þ c1

r1 � c2r2

c1r1

�uþ c2

r2

� : (57)

Again, note that for s¼ 1, the above expression is equal to zero. For ss1, I take the derivative of theabove expression with respect to s and evaluate around s¼ 1. First, note that

c2s ¼ c2 ln

�u� 9� y

r2

�; c1

s ¼ c1 ln�

u� 9

r2

�:

Next, for s¼ 1, c1 ¼ c2 ¼ c, and r1 ¼ r. Hence,

c2s

��s¼1¼ c ln

� ck

r2

�; c1

s

��s¼1¼ c ln

� ck

r1

�;

dc2y

ds

��s¼1¼ k

�1þ ln

� ck

r2

��;

dc2r2

ds

��s¼1

¼ k

�u ln

� ck

r2

�þ 9� y

r2

�;

dc1r1

ds

��s¼1

¼ k

�u ln

� ck

r1

�þ 9

r1

�;

ddsðr1 � r2Þ

��s¼1¼ c2

s � c1s

uþ c2r2

¼ cuþ c2

r2

ln�

r1

r2

�:

Using the above relations in (57), and after some algebra, it can be shown that

dds

limT/N

�vqð0Þ

vyþ q� qð0Þ

y

�erT ¼ 1

ruy

ck

�ln�

1þ yckþ 9� y

�� y

ckþ 9� y

�� 0:

Hence, for s close to 1, I have shown that

dTdy

��T/0

X0 iff s,1: (58)

It is conjectured that for all s40:

dTdy

X0 iff s,1:

,

40 The conjecture is verified numerically. A formal proof is beyond the scope of the present analysis.


References

Aghion, P., Bachetta, P., Banerjee, A., 2004. A corporate balance-sheet approach to currency crises. Journal of Economic Theory119, 6–30.

Burnside, C., Eichenbaum, M., Rebelo, S., 2001. Prospective deficits and the Asian currency crisis. Journal of Political Economy109, 1155–1197.

Burnside, C., Eichenbaum, M., Rebelo, S., 2004. Government guarantees and self-fulfilling speculative attacks. Journal ofEconomic Theory 119, 31–63.

Burnside, C., Eichenbaum, M., Rebelo, S., 2006. Government finance in the wake of currency crises. Journal of MonetaryEconomics 53 (3), 401–440.

Caballero, R., Krishnamurthy, A., 2001. International and domestic collateral constraints in a model of emerging market crises.Journal of Monetary Economics 48, 513–548.

Chang, R., Velasco, A., 2001. A model of financial crises in emerging markets. Quarterly Journal of Economics 116 (2), 489–517.Claessens, S., Djankov, S., Lang, L., 1998. East Asian Corporates: Growth, Financing and Risks over the Last Decade. The World

Bank, Mimeo, pp. 1–23.Corsetti, G., Pesenti, P., Roubini, N., 1998a. What caused the Asian currency and financial crisis? Part I: A macroeconomic review.

NBER Working Paper no. 6833.Corsetti, G., Pesenti, P., Roubini, N., 1998b. What caused the Asian currency and financial crisis? Part II : The policy debate. NBER

Working Paper no. 6834.Corsetti, G., Pesenti, P., Roubini, N., 1999. Papers tigers? A model of the Asian currency crisis. European Economic Review 43 (7),

1211–1236.Edwards, S., Vegh, C., 1997. Banks and macroeconomic disturbances under predetermined exchange rates. Journal of Monetary

Economics 40, 239–278.Flood, R., Garber, P., 1984. Collapsing exchange rate regimes: some linear examples. Journal of International Economics 17, 1–13.International Monetary Fund, 1997. World economic outlook, Interim Assessment.International Monetary Fund, 1998a. International capital markets: developments, prospects, and key policy issues, world

economic and financial surveys, Washington.International Monetary Fund, 1998b. World economic outlook.Kaminsky, G., Reinhart, C., 1999. The twin crises: the causes of banking and BOP problems. American Economic Review 89,

473–500.Klingebiel, D., Laewen, L. (Eds.), 2002. Managing the Real and Fiscal Effects of Banking Crises. World Bank Discussion Paper no.

428.Krugman, P., 1979. A model of BOP crises. Journal of Money, Credit, and Banking 11, 311–325.Krugman, P., 1998. What Happened to Asia? MIT, Mimeo.Krugman, P., 1999. Analytical Afterthoughts on the Asian Crisis. MIT, Mimeo.Lahiri, A., Vegh, C., 2007. Output costs, currency crises, and interest rate defense of a peg. The Economic Journal 117, 216–239.Lane, T., Ghosh, A.R., Hamann, J., Phillips, M., Tsikata, T., 1999. IMF-supported programs in Indonesia, Korea and Thailand:

a preliminary assessment. IMF Working Paper, pp. 1–131.McKinnon, R., Pill, H., 1998. International overborrowing: a decomposition of credit and currency risks. World Development 26

(7), 1267–1282.Pomerleano, M., 1998. The East Asia crisis and corporate finances – the untold micro story. World Bank Working Paper, pp. 1–35.Radelet, S., Sachs, J., 1998. The East Asian financial crisis: diagnosis, remedies, prospects. NBER Working Paper no. 6680.Schneider, M., Tornell, A., 2004. Balance sheet effects, bailout guarantees and financial crises. Review of Economic Studies 71,

883–913.

Date post:	25-Oct-2016
Category:	Documents
Upload:	rajesh-singh
View:	220 times
Download:	0 times

Asset prices and twin crises

Documents