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Journal of International Money and Finance journal homepage: www.elsevier.com/locate/jimf

Asset prices and twin crises Rajesh Singh* Department of Economics, 260 Heady Hall, Iowa State University, Ames, IA 50011, USA

a b s t r a c t JEL classiﬁcation: F3 F4 Keywords: Twin crises Currency crises Banking crises Asset prices Government bailout

Emerging market crises have been characterized by two key features: (i) banking crises generally precede currency crises, and (ii) asset prices decline in advance of currency crises. This paper argues that asset prices provide a key link between banking and currency crises. It is shown that a prospective currency crisis due to an unanticipated increase in the public debt triggers an asset price decline. Banks’ exposure to asset prices in turn deteriorates their balance sheets and precipitates a banking crisis. Under the assumption 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 the ratio 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 crises that occur together. In an extensive study of ‘twin’ crises, Kaminsky and Reinhart (1999) highlight that banking crises often begin before currency crises, and that asset prices decline in advance of currency crises. These facts combined with the evidence that banking sector problems generally begin from the asset-side and that banks carry substantial asset price exposures naturally raise the following questions. Do asset prices link banking and currency crises? In particular, what (if any) are the ‘fundamentals’ that activate the asset price link? Finally, if such a link is triggered, can it replicate the observed crises chronology? The crises literature is yet to address these issues despite their obvious

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

0261-5606/$ – see front matter 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jimonﬁn.2008.07.008

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27

importance for researchers as well as policymakers.1 This paper attempts to ﬁll this gap by providing an asset-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 sudden prospective increases in ﬁscal deﬁcits due to ﬁnancial sector meltdowns.2 Following this research, I ﬁrst study a currency crisis sparked by an unanticipated exogenous increase in ﬁscal liability. It is shown that a prospective currency crisis triggers an asset price decline, which in turn brings forth a banking crisis as banks are exposed to asset prices. Pursuing this further, under the assumption of government’s bailout of banks that generates an endogenous increase in ﬁscal liability, I show that twin crises are self-fulﬁlling. The analytical setup I utilize is particularly motivated by the following stylized facts relating to the East Asian crisis: (1) ﬁrms’ cash ﬂows were highly sensitive to interest rates; (2) banks were substantially exposed to asset prices; (3) equity prices declined in advance of currency crises leading to bank failures; and (4) eventually, governments bailed out insolvent banks.3 The model consists of a small open economy populated by households, ﬁrms, banks, and a government. Households derive utility by consuming and by holding deposits at banks. Banks take ` la Edwards and Ve´gh (1997).4 A fraction of deposits is held these deposits and advance loans to ﬁrms a at the central bank as required reserves which constitutes the monetary base. The demand for bank loans is motivated as follows. Firms produce output by combining their own capital with the capital rented from other ﬁrms and a fraction of the rental payments must be made through bank credit. As a result, the equilibrium return on capital and equity prices is sensitive to the changes in bank lending rates. Finally, banks also own equity in ﬁrms. To begin with, I assume, the economy is in a perfect foresight stationary equilibrium with a ﬁxed nominal exchange rate. Suppose now the government realizes an unanticipated increase in ﬁscal liability that is eventually expected to be ﬁnanced through seigniorage. Yet, the government sticks to the existing exchange rate peg until the public debt hits a publicly known upper bound, at which point the 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. An increase in nominal interest rate hikes up banks’ cost of holding fractional reserves. Although seigniorage revenue is directly raised from banks, indirectly, banks pass it on to households and ﬁrms by adjusting real interest rates on deposits and loans. Hence, a rise in the future nominal interest rate signals that ﬁrms’ cost of loans in future will rise and therefore its return on capital will decline. As asset prices reﬂect the present value of future earnings, they fall on impact and decline steadily thereafter. If the decline is sufﬁciently large, banks’ loss of capital may leave them with a negative net worth, which I deﬁne as banking crisis. Thus banks’ exposure to asset prices links a prospective currency crisis 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 government’s ﬁscal liability. Suppose instead the government has a bailout policy under which it assumes a ﬁxed percentage of banks’ capital loss if a crisis occurs. If agents believe that a currency crisis is imminent, ﬁrms’ equity prices fall; the increase in the stock of ﬁscal liability as a result of the government bailout policy then validates this belief. Here, the presence of bailout guarantee is necessary 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 of simultaneous 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 Ve´gh (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 when reserves 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.

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

forth a banking crisis, followed by a steady equity price decline that continues until the exchange rate peg 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 timing between a banking crisis and the eventual abandonment of ﬁxed exchange rate. This timing must be ﬁnite for crises to exist. The timing in turn is determined from the two policy parameters: (a) the bailout ratio, and (b) the upper bound on public debt at which the peg is abandoned. While the bailout creates additional ﬁscal liability, the upper bound on public debt determines the level of seigniorage raised after the exchange rate ﬂoats. The timing thus equilibrates the present value of seigniorage revenues 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 of central bank’s reserves that continues to fall until the exchange rate ﬂoats. In the model, the monetary base is proportional to the loans to ﬁrms that they need for paying rents on hired capital. As these rents also determine equity prices, their anticipated cumulative decline reﬂects in the initial fall in the equity price. In other words, the anticipated fall in the monetary base over time is proportional to the initial fall in the equity price. Thus, the total increase in the stock of ﬁscal liability (prospective loss of foreign reserves 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 that is eventually passed on to the ﬁrms. This share, given households’ preferences, uniquely depends on the 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 lower bailout ratio creates smaller increase in ﬁscal liability and thus requires a lower present value of seigniorage 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-fulﬁlling crises are ruled out if the bailout ratio falls below a lower bound. The households’ preferences affect the equilibrium bailout ratio in the following way. The more interest rate elastic is households’ demand for deposits, the less they are willing to contribute to seigniorage revenues. As a result, ﬁrms bear a larger burden 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’ demand for 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 increase in 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 higher debt 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 the demand for deposits is elastic, a relatively sharper decline in deposits causes a relatively higher loss of central bank reserves. In sum, the delaying effect of a higher public debt is more than offset by the combined effect of a higher equity price decline and loss of reserves. Hence, the exchange rate crisis is advanced. On the contrary, when the households’ demand for deposits is inelastic, the delaying effect of 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) ` la Diamond– currency mismatch models. Maturity mismatch models rely on self-fulﬁlling bank runs a Dybvig, in which the central bank’s role as a lender-of-last-resort links a run on deposits to a run on foreign reserves.9 In currency mismatch models, bailout guarantees induce banks to carry unhedged foreign currency liabilities that in the event of a devaluation cause bank failures. Then bailouts trigger and currency crises occur in self-fulﬁlling manner.10 Yet another set of currency mismatch papers, ` la Bernanke– although they do not study banking crisis explicitly, rely on ﬁnancial accelerator models a 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 presented in 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).

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occurs along with a fall in the economic activity.11 None of these models, however, address the intertemporal aspects of twin crises and both the banking (ﬁnancial) and the currency crisis occur simultaneously at a point in time. Moreover, any feasible extension of these models will not generate asset price dynamics that has been observed empirically. Thus, this paper makes the following contributions to the twin crises literature. First, it offers an asset price perspective of twin crises, which in the light of emerging market evidence competes well with maturity mismatch and/or currency mismatch explanations. Second, the model is better able to replicate empirically observed crises dynamics. In particular, banking crisis precedes currency crisis and asset prices decline in advance of currency crisis. Finally, the model generates a steady decline asset prices that closely mimics the data. The rest of the paper is organized as follows. Section 2 presents some evidence on the stylized facts relating to the East Asian crises. Section 3 sets up the basic model and derives the pre-crisis stationary equilibrium. 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 studies the existence and the properties of self-fulﬁlling twin crises equilibria under the presence of government bailout guarantees. Section 6 concludes. Algebraic tedious proofs are consigned to an Appendix. 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 substantially exposed to real estate and equity loans; (3) equity prices declined in advance of currency crises leading to major bank failures; (4) the depositors and the foreign bank creditors were often guaranteed by the governments, at least implicitly. I present some evidence on each of the above in the following paragraphs: (1) It has been argued that one of the main factors that left the East Asian countries vulnerable to a 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 earnings before interest, taxes, and depreciation (EBITDA) to interest payments, which indicate the adequacy 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/interest ratio, as at the end of 1996. It is fairly evident that the balance sheets of ﬁrms in Korea, Thailand, and Indonesia were vulnerable 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) and typically made loans to ﬁrms in the same group, thus, indirectly acquiring ownership stakes in related ﬁrms. Government directed lending to Chaebols in Korea, lending to related parties within large ﬁnancial (and non-ﬁnancial) conglomerates in Korea and Thailand, and ownership of weakly regulated banks by non-banks in Indonesia have been extensively documented in past studies.15 In addition, 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) report that ‘‘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). 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. 12

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(relative to tranquil times).’’ They further note that the equity prices are past their cyclical peak before the onset of banking crises. Fig. 1 shows the evolution of exchange rates and stock indexes in Thailand, 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 peak of 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 almost one third in Thailand. It is worth noting that Korea and Thailand suffered the earliest major bank failures 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 as a number of major conglomerates went bankrupt.19 By the ﬁrst week of September, six highly leveraged Chaebols had failed. In Thailand, many commercial banks reported a signiﬁcant increase in nonperforming loans in late 1996 and there were runs on the deposits of the Bangkok Bank of Commerce in May 1997. Less than a week before the baht was allowed to ﬂoat, the Bank of Thailand suspended 16 ﬁnance companies; another 42 were suspended on August 5.20 These developments were amply reﬂected by the stock market indices of the banking sector. In particular, the ﬁnancial sector stock indices 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 Thai baht came under attack already in November and December 1996. The Korean won was also under pressure in 1996, and had been allowed to depreciate in 1996 and early 1997. These events indicate that the markets were anticipating a currency crisis. Thus, the advance decline of equity prices, the timing of major bank failures, the continuing decline in equity prices until the peak of currency crises, and the recurrent speculative attacks on currencies provide evidence that the equity prices had been declining in anticipation of a currency crisis. The declines 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 also quantify the extent to which the implicit guarantees became explicit once the crises occurred. It has been reported that the cost of bank bailouts in Indonesia, Malaysia, South Korea, and Thailand has 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 and capital 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 when these 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 the onset of the regional crisis, banks and ﬁnance companies experienced a signiﬁcant decline in proﬁtability and asset quality deteriorated sharply. In Philippines, the deterioration in asset quality led to difﬁculties in some small banks, but most of the large commercial banks were relatively better capitalized and were able to withstand the increase in bankruptcies and debt restructuring. 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).

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Table 1 Corporate leverage. Country

Leverage

EBITDA/interest

Hong Kong Indonesia Korea Malaysia Philippines Singapore Thailand Germany USA

1.6 1.9 3.5 1.2 1.3 1.0 2.4 1.5 1.1

11.07 2.44 1.07 6.74 3.68 8.05 1.92 7.09 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. Each household owns a unit of labor which it supplies to the ﬁrms inelastically. There is a continuum of ﬁrms over a unit interval that produces consumption good. A ﬁrm is differentiated by the type of speciﬁc capital it owns. Each ﬁrm produces output by combining labor, its own capital, and capital rented from other ﬁrms. Rental payments between ﬁrms are made through bank credit and are denominated in home currency. A representative bank receives home currency deposits from households and lends to ﬁrms. Banks also hold equity in ﬁrms; banks are owned by households. All markets are perfectly competitive. Perfect integration in the goods market implies that the law of one price holds. Therefore, the home currency price of consumption is equal to the nominal exchange rate E, expressed in units of home currency per unit of world currency. Households, ﬁrms, banks, and the government can buy and sell an internationally traded bond at a constant world interest rate r. Perfect capital mobility implies that the interest rate parity holds. Therefore, the nominal interest rate is given by i ¼ r þ 3, where 3 is the devaluation rate. Notation. Unless otherwise indicated, subscripts denote partial derivatives. Time-dependent variables are denoted without time subscripts. x_ denotes dx/dt. Whenever necessary, time is indexed as an argument within small brackets. For example, x (t) denotes the value of x at time t. Finally, fjx¼a denotes the value of a function f(x) evaluated at x ¼ 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 logarithmic preferences, yield simple closed form expressions for the most of the results. However, separable preferences lead to the indeterminacy of crises equilibria as discussed in Section 5, and thus are treated only as special cases. For the main analysis, a general utility form is assumed, which yields results with a wider applicability.

3.1. Households A representative household’s lifetime utility is given by

Z

N

ebt uðc; dÞdt;

(1)

0

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 home currency deposits at banks. Deposits earn a nominal interest rate of id.

24

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

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

Korea

2500

2500

Thailand E

2000

2000 E

1500

1500 1000

1000 I

500

I

500

0

Malaysia

1600 1400 1200 1000 800 600 400 200 0

nJu 93 lJa 93 nJu 94 lJa 94 nJu 95 lJa 95 nJu 96 lJa 96 nJu 97 lJa 97 nJu 98 lJa 98 nJu 99 l-9 9

Ja

Ja

nJu 93 lJa 93 nJu 94 lJa 94 nJu 95 lJa 95 nJu 96 lJa 96 nJu 97 lJa 97 nJu 98 lJa 98 nJu 99 l-9 9

0

5000 E

4000 3000

I

Philippines E

I

2000 1000

Ja n M -95 ay Se -95 p Ja -95 n M -96 ay Se -96 p Ja -96 nM 97 ay Se -97 p Ja -97 n M -98 ay Se -98 p Ja -98 n M -99 ay Se -99 p99

Ja n M -95 ay Se -95 p Ja -95 n M -96 ay Se -96 p Ja -96 nM 97 ay Se -97 p Ja -97 n M -98 ay Se -98 p Ja -98 n M -99 ay Se -99 p99

0

Fig. 1. Stock indexes during East Asian currency crises. E: nominal exchange rate; I: stock index – Korea: KOSPI, Thailand: Bangkok S.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 whole one unit of labor for work. Then the household’s ﬂow budget constraint in terms of consumption goods can be expressed as

Dd ¼ Dbh if t˛G; a_ ¼ ra þ w þ Ub s I d d c if t;G;

(2)

where bh is the household’s net stock of international bonds; a h d þ bh is the total ﬁnancial wealth; s denotes lump-sum taxes paid to the government; and Ub is the dividends earned from banks.25 The household’s opportunity cost of holding deposits is given by Id h i id. G denotes a ﬁnite set of points in time when a discrete portfolio adjustment can occur. The household’s intertemporal budget constraint can be derived as (after imposing the transversality condition, limt/Na (t)ert ¼ 0)

að0Þ þ

Z 0

N

b

wþU

Z ert dt ¼

N

c þ I d d þ s ert dt:

(3)

0

Households maximize (1) subject to (3) taking the path of interest rates and income transfers {Id,r,s,Ub} as given. Assuming b ¼ 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 are held by households. This assumption helps avoid unnecessary algebra. All the results will go through, however, if households also held equity in ﬁrms directly.

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

Stock market peak

Major bank failures

Currency ﬂoat

% Decline in stock indexa

Thailand Korea

Jan 4, 1994 Nov 8, 1994

Jan–Aug, 1997 Jan–Sep, 1997

Jul 2, 1997 Nov 17, 1997

67.6 56.3

a Relative to the non-ﬁnancial sector, the stock market indices of the ﬁnancial sector in Thailand and Korea declined further by 30 and 34%, respectively, as of July 1, 1997: see Burnside et al. (2001, p. 1163, Table 3).

uc ¼ l;

(4a)

ud ¼ Id ; uc

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

3.2. Firms I derive below a rate of return schedule for ﬁrms’ capital that inversely varies with the interest rate on bank loans. For analytical convenience, I choose a set up in which ﬁrms’ input usage and consequently its output remain constant regardless of the variations in interest rates. To this end, I assume that there is a continuum of ﬁrms uniformly distributed over a unit interval, each owning a ﬁxed unit of differentiated capital. A ﬁrm ﬁrst combines its own capital with capital rented from other ﬁrms using a Dixit–Stiglitz aggregation technology. The aggregation technology is symmetric across ﬁrms. The aggregated 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 same amount of labor and capital, and their amount of input use, output, and the return on their capital will be identical. It is then convenient to describe ﬁrms’ decision rules in a representative manner. The details 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 other ﬁrms. In particular, ﬁrms need at least a fraction 4 of their rental liability as credit-in-advance. Formally,

z frk;

f˛ð0; 1;

(5) z

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

Fk ¼ r 1 þ fI z :

(6)

Similarly, a ﬁrm’s optimal labor employment equates the marginal product of labor with the market wage rate. Thus, w ¼ Fl.

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

In equilibrium, full one unit each of labor and capital are utilized since both wage and rental rate are positive. Hence, w ¼ Fl (1,1). Note further that a constant level of output y is given by

y ¼ Fð1; 1Þ ¼ w þ r 1 þ fI z ;

(7)

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

z ¼ fr:

(8)

Thus, after imposing the equilibrium capital and labor allocations, a ﬁrm’s ﬂow constraint can be consolidated as f a_ f ¼ raf þ y w frI z U ; f

(9) f

f

f

where U denotes ﬁrms’ dividend payments. Note that a h b z, where b denotes ﬁrms’ net stock of international bonds. I assume that ﬁrms hold bonds solely to accommodate discrete changes in the amount of loans. Otherwise, they do not accumulate ﬁnancial assets and af ¼ 0 for all t. Combining (6), (7) and (9) yields

Uf ¼ r ¼

9 1 þ fI z

;

(10)

where 9 h Fk (1,1) is the equilibrium marginal product of capital. Eq. (10) states that a ﬁrm’s dividends equal 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, and the nominal interest rate implied by the world interest parity. Additionally, banks equity transactions in the secondary market obtain a key equity price equation. I assume that the representative bank is a holding company that runs a depository bank and an investment bank. While depository banks intermediate funds between households and ﬁrms, investment banks raise funds by selling bonds and invest in ﬁrms’ equities. Depository banks take deposits from households and advance loans to ﬁrms. It is required that a fraction d of deposits be held as reserves with the central bank.26 These reserves solely constitute the monetary base denoted by m. Hence, m h dd. Banks costlessly convert deposits into loans after meeting the 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’ equities traded in perfectly competitive secondary markets. Using (11), the representative bank’s consolidated ﬂow constraint is given by

f a_ b ¼ rab þ I d þ ð1 dÞI z di d þ q_ rq þ U s;

(12)

where ab hqs bb , and where bb is the banks’ stock of debt in the international bond market, s and q denote 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 to increase this requirement during a currency crisis, especially if it is accompanied by bank failures. Then the results that follow will 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 ﬁxed cost on such conversions will obtain the same result.

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

35

imposing the transversality conditions, limt/N q(t) ert ¼ 0 and limt/N bb (t)ert ¼ 0, the present value of banks’ dividends can be derived as

Z

N

Ub ert dt ¼ bb ð0Þ þ qð0Þsð0Þ þ

0

Z

N h

i I d þ ð1 dÞI z di d þ q_ rq þ Uf s ert dt:

(13)

0

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

di Id ¼ Iz ð1 dÞ;

(14a)

rq ¼ q_ þ 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, while the right hand side represents its marginal revenue as a unit of deposit produces only 1 d unit of loans. A no-arbitrage equilibrium requires that both be equal. The intuition for Eq. (14b) is equally straightforward. While the right hand side represents net return from holding an equity as sum of capital 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 ¼

Z

N

rðxÞerðxtÞ dx;

(15)

t

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

Ub ert dt ¼ qð0Þsð0Þ bb ð0Þ;

(16)

0

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 s ¼ 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, or advanced as loans z to ﬁrms. Hence, banks’ equity capital, denoted by nwb, equals the difference between 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 in equity price, Dq, exceeds banks’ existing net worth, then nwb < 0 and the banking system becomes insolvent. Such an event is deﬁned as a banking crisis. As ﬁrms require bank credit in order to produce, I assume that the government bails out the banking system if a crisis occurs. In general, the level of government bailout will depend on the magnitude of stock market decline and the pre-crisis level of banks’ equity capital. If a market for banks’ equities were explicitly included in this model, then the percentage price decline of banks’ equities will exceed that of the ﬁrms. To see this, let qb denotes the banks’ equity price; clearly, qb ¼ nwb. As all ﬁrms’ equities are held by the banks, Dqb ¼ Dq.28 Then the ratio of the percentage equity price decline in the banking sector relative to that of the ﬁrms can be expressed as

Dqb Dq qb

=

q

¼

q bb ¼ 1 þ b: q qb

Notice from Table 3 that bb/qb equals banks’ leverage (debt/equity) ratio. A positive leverage then implies 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 continue to hold even if the assumption that banks are the sole owners of ﬁrms’ equities is relaxed.

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

Table 3 Banks’ balance sheet. Assets

Liabilities

1. Loans to ﬁrms (z) 2. Reserves (m)

1. Deposits (d) 2. International borrowing (bb) 3. Bank’s equity (nwb)

3. Stocks (q)

ﬁrms. Moreover, the higher the leverage, the higher will be the relative equity price declines in the banking 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;

_ þ 3m þ s if t;G; b_ ¼ rb þ m

(17)

bhbg

where 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 in the international bond market. The ﬁrst part of Eq. (17) states that discrete monetary changes are implemented by the central bank either through open market operations or by trading foreign reserves. Eq. (17) on the whole states that the government can discharge its debt only through taxes and/or seigniorage. For simplicity, I assume that the government does not hold nominal debt. Nominal debt offers an alternative 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 on the government’s unanticipated need for seigniorage. Allowing debt deﬂation, however, will only affect the results quantitatively. As long as seigniorage is inevitable, all the results will continue to hold qualitatively. Monetary policy. Monetary policy is set by the following rules. Under a ﬁxed exchange rate regime the government sets a devaluation rate 3, while under ﬂexible exchange rates it sets a money growth rate 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’s ﬂow constraint is given by

f_ ¼ rf þ y c h

f

(18)

b

where f ¼ b þ b b 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 given by (11), which then constitutes a constant fraction of ﬁrms’ returns, as given by (8). As a result, deposits relate to ﬁrms’ returns through

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

d ¼ qr;

37

(19)

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

m ¼ hr;

(20)

where h ¼ dq, and 1/h is the multiplier that relates money supply to ﬁrms’ returns. Thus, in equilibrium, for each unit of ﬁrms’ returns on capital, banks raise q units of deposits on which they earn qId from the households. On the other hand, for each unit of ﬁrms’ returns, banks are required to keep h units of reserves, which costs them hi in terms of lost interests. Banks’ zeroproﬁt condition (14a) implies that the total cost hi qI d ¼ fI z is in turn charged from ﬁrms for each unit of their returns. Thus, ﬁrms’ return on capital net of interest payments as given by (10) can be rewritten as

r¼

9

1 þ hi qI d

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

r¼

9 1 þ ðhi quudc

¼ rð i ; c Þ: þ

(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 simple interpretation. First, a higher nominal interest rate, ceteris paribus, raises marginal cost of loans depressing the demand for capital. As a result, the equilibrium rent on capital decreases with a rise in the nominal interest rate. Second, the higher the consumption, the higher is the demand for deposits that raises its price I d ¼ ud =uc . A lower marginal cost of raising deposits in turn lowers the interest rate that banks charge on loans. Thus, the equilibrium return on capital increases with consumption. Finally, the nominal interest rate i is determined from the government’s monetary policy and its ﬂow 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 ﬁrst characterize a stationary equilibrium under a ﬁxed exchange rate regime. This requires that the government set 3 ¼ 0 for all t. Assuming that the government runs a balanced budget, (17) can be rewritten as

s

b ¼ ; r

(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 3 ¼ 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 r

¼

rðr; cÞ r

;

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

(24)

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

4. Currency crisis and asset price decline This section ﬁrst speciﬁes the conditions and the policy rules under which a currency crisis is inevitable. Assuming that the pre-crisis economy follows an exchange rate peg, a currency crisis is deﬁned as an event when the peg is abandoned and the exchange rate ﬂoats. Then, equilibrium allocations and the time when the peg is abandoned are characterized. Finally, it is shown that the crisis brings 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, there is an unanticipated increase in the stock of government’s ﬁscal liability by an amount J > 0.29 Before t ¼ 0 the existing level of government debt is funded by the present value of taxes and (22) holds. The increase in debt J will now be funded either by discrete changes in money supply (Dm) or through _ þ 3mÞ as in (17).30 Hence, seigniorage ðm

J¼

Z

N 0

_ þ 3m ert dt þ Ss˛f0;Tg DmðsÞers : m

(25)

I assume that the government continues to peg the exchange rate at E. Note that as long as 3 ¼ 0, i ¼ r. From (17), (20) and (21), it then follows that there are no seigniorage revenues over t˛ð0; TÞ. Thus, the net public debt continues to grow. I assume that eventually when the net public debt hits an upper bound F, the government abandons the peg and the exchange rate ﬂoats. Henceforth, T will denote the time when this occurs. As will be shown below, discrete changes in base money can only occur at time 0 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 growth _ rate of money supply given by M=M ¼ m.31 It can be shown that a constant m is consistent only with a stationary equilibrium, which in turn implies that 3 ¼ m. Thus, the post-crisis (t > T) government’s budget constraint (17) can be rewritten as

r F ¼ 3m þ 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 r F s ¼ rðF bÞ is ﬁnanced by seigniorage after the exchange rate ﬂoats. Denoting the post-crisis debt service payment rðF bÞ by y, the rate of devaluation is given by

3¼

0; y

m;

t˛½0; TÞ; t˛½T; NÞ:

(27)

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

i ¼

r; t˛½0; TÞ; r þ my ; t˛½T; NÞ:

(28)

As shown below, the variables other than equity price q jump discretely at T but they have a ﬂat path otherwise. Therefore, an additional piece of notation will be helpful in what is to follow: the variables over 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 of unanticipated liabilities and instead consider its present value as a net stock addition to the current stock of liability. Explicitly including 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.

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

uc c1 ; qr1

¼ uc c2 ; qr2 :

39

(29)

Using (4b), (20) and (28) in (21) yields

r1 ¼ r r; c1 ;

r2 ¼ r r þ

y 2 ;c : hr2

(30)

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

c1 1 erT þ c2 erT 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 entails solving 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 an instant before the peg is abandoned. Lemma 1 and Proposition 1 verify that this result holds in the present setup. Lemma 1.

2

1

2

1

I d > I d ; I z > I z ; 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 of holding required reserves jumps at T. Observe that households’ and ﬁrms’ demand for deposits and loans, 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 is passed on to both households and ﬁrms, in terms of higher Id and Iz. Consequently, the rate of return 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) ¼ m2 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 fall at T. As the exchange rate is ﬁxed an instant before T, households and ﬁrms exchange their excess nominal balances at the central bank for bonds. While households buy bonds (Dd(T)) in exchange for their excess deposits, ﬁrms sell ((1 d)Dd(T)) bonds to the central bank to pay for their excess bank loans. 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. T ¼ 0.

J < JhF b hðr r~Þ, where r ¼ rðr þ hy~r; cÞ is the stationary return on capital if

~ ¼ r2 is then obtained from (30). The size of Suppose T ¼ 0. Then from (31) c2 ¼ c. By deﬁnition r ~ where m ~ ¼ h~ r. If the the run will be equal to the fall in base money, i.e., DhðT ¼ 0Þ ¼ m m, Assumption 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

40

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

~ ¼ p~r . Note that (29)–(31). In this case, the post-crisis stationary equity price will instantly adjust to q ~ ~ q < q, since r < r. It trivially follows that T ¼ 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 Assumption A.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

1 F b þ DmðTÞ : ln J Dmð0Þ r

(32)

Two features of (32) are noteworthy. First, the denominator J Dmð0Þ, which equals the increase in the 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 indeﬁnitely sustained. Second, as the Assumption A.1 holds, i.e., T 0, then F b þ DmðTÞ J Dmð0Þ, or equivalently F b þ J þ hðr r2 Þ.32 One can conjecture from Eq. (32) that the higher the unanticipated increase in the stock of ﬁscal liability J, the sooner the peg will be abandoned. Second, the larger the gap between the initial net public 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 in Appendix B.33 4.4. Asset price dynamics It is now shown that an unanticipated increase in ﬁscal liability not only precipitates an exchange rate 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 erT ert ;

t˛½0; TÞ; and q ¼

r2 r

;

t˛½T; NÞ:

(33)

Eq. (33) implies that at time 0

qð0Þ ¼

r2 1 erT þ erT : r r

r2

(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 value q(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, T ¼ 0, which violates the Assumption A.1. 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. 33

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

41

Fig. 2. Equity prices during currency crisis.

banks bear the whole brunt of additional ﬁscal liability. Banks, in turn, pass it on to households and ﬁrms by adjusting interest rates on deposits and loans, respectively. While the overall share of additional costs borne out by ﬁrms is determined from household preferences and other parameters of the model, the share nevertheless is positive. As a result, ﬁrms’ present value of dividends relative to its stationary level falls on impact, thus causing a drop in the equity price. The declining path of equity price is explained by the fact that the return on ﬁrms’ capital is relatively higher while the exchange rate is ﬁxed. Then the present value of ﬁrms’ dividends declines as the time of exchange rate crisis nears. Hence, the equity price declines over time. Logarithmic preferences. To get some idea of the share of additional liability borne by ﬁrms, it is useful to study a special case: u(c,d) ¼ ln c þ 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 away from ﬁrms. Intuitively, with logarithmic preferences households’ share of expenditures on consumption 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, the whole burden of seigniorage falls on ﬁrms. Hence, the equity price falls by J on impact and thereafter follows Eq. (35). 5. Government bailout and self-fulﬁlling twin crises Section 4 showed that an unanticipated increase in ﬁscal liability, in addition to generating a currency crisis, causes a decline in equity prices. In this section, I endogenize the increase in the stock of ﬁscal liability through a government bailout policy. Speciﬁcally, I propose that the government assumes a fraction g of the capital loss incurred by banks. Now, if agents believe that a currency crisis is imminent, equity price falls, and the increase in ﬁscal liability as a result of government bailout thus validates the belief. As a result, twin crises can be self-fulﬁlling. I ﬁrst establish the range of g over which 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 exchange rate crisis. Proposition 2 states that the equity price drops from q to q(0) under the anticipation of a future currency crisis. As a result, banks’ equity capital declines by q qð0Þ, which I assume to leaves banks

42

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

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

J ¼ gðq qð0ÞÞ:

(36)

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

ðF bÞerT ¼ gðq qð0ÞÞ þ hrðq qð0ÞÞ: The left hand side of above equation represents the present value of additional debt service that triggers from T onwards. The right hand side shows the present value of the increase in the stock of ﬁscal liabilities, while the exchange rate remains ﬁxed. The ﬁrst term represents the government bailout, while the second term denotes the present value of foreign reserves lost due to the exchange of nominal balances by the private sector. Note from (20) that the changes in base money are directly proportional to the changes in equilibrium returns on capital. Since q qð0Þ measures the present value of the changes in returns to capital, the second term denotes the present value of cumulative changes in base money. Alternatively, the above equation can be rewritten as

F ¼ xðq qð0ÞÞerT þ b;

(37)

where x h g þ hr. Note that under ﬁxed exchange rates the government can only sustain its pre-crisis debt 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

1 1 Fb : ln r x q qð0Þ

A casual glance at the above expression tells that the larger the upper bound on the net public debt, the longer the peg can be sustained. However, from (29)–(31) and (34) the fall in equity price depends on the anticipated rate of devaluation after T, which in turn depends on the net increase 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 exist for a given value of y.

5.1. Self-fulﬁlling equilibria Henceforth, for simplicity, I assume that uðc; dÞ ¼ ½ðc1ð1=sÞ þ cd1ð1=sÞ Þs=ðs1Þ 1ð1=3Þ =1 ð1=3Þ, where s and e are intratemporal and intertemporal elasticity of substitution, respectively. It is easy to verify that this CES form satisﬁes all assumptions as elaborated in Section 3.1. Note that when s ¼ e, utility is separable in c and d. As is well known, the relative intertemporal allocations crucially depend whether eXs, all three cases are analyzed in order to generalize the results. However, to avoid unnecessary algebra in derivations, it is assumed that e ¼ 1. Appendix D and Appendix F derive the relative 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 the economy 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’ equity capital.

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

43

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

g¼

F b rT hr: e q qð0Þ

(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 twin crises 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 T ¼ 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 later the peg will be abandoned. The result sounds intuitive. The explanation, however, is somewhat complicated. A lower bailout ratio implies a smaller increase in the stock of ﬁscal liability, had the fall in equity price remained constant. As additional stock of liability at t ¼ 0 must equal ðF bÞerT , it is implied that the peg can be sustained longer. But, the later the exchange rate ﬂoats, the later will be the decline in ﬁrms’ returns, and thus a smaller fall in equity price at time 0. From (38), it is then not obvious why g should move inversely with T. An inspection of (38) tells that the result in Proposition 3 can hold only if the rate at which the fall in equity price q qð0Þ decreases with respect to T is smaller than 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 dividends relative to the stationary economy is due to an increased cost of loans. The increase in costs in turn is driven by a combination of two factors. The ﬁrst is the increase in ﬁscal liability as a result of government bailout plus the loss of reserves due to cumulative change in base money. As discussed earlier, this component is a ﬁxed multiple x ¼ g þ hr of the fall in equity price. The second factor depends on the share of the increased liability borne by households. If households’ consumption and deposits are gross complements (s < e), they are willing to bear a larger share of seigniorage. Then ﬁrms’ burden is lessened. As a result, the fall in equity price is smaller than the increased stock of ﬁscal liability. On the other hand, if households’ consumption and deposits are gross substitutes (s > e), their deposits can fall signiﬁcantly if the interest rate on deposits falls. Now, in order to meet ﬁrms’ demand for loans, banks have to induce households to hold deposits by offering a higher interest rate. This additional cost of funds in turn further hikes the interest rate on loans. As a result, ﬁrms’ burden of ﬁnancing additional ﬁscal liability is further compounded. The adjustment of households’ deposit allocations, as T changes, works as follows. First, a constant level of wealth implies that consumption levels before and after T move in the same direction. For expositional convenience, I terms it as the ‘time’ effect. In general, the ‘time’ effect makes deposits commove 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 the changes 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 to changes in T is relatively milder. Note that for the CES utility form with s ¼ 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 invariant to the value of T.36 Therefore, as T changes, the present value of the households’ share of additional ﬁscal liability adjusts at a rate r. Note that the additional ﬁscal liability ðF bÞerT also adjusts at a rate r with a change in T. However, for sse, following the discussion in the preceding paragraph, the rate of

35 36

Except when s ¼ e ¼ 1; then the expenditure after T equals its level in the stationary economy. The case with s ¼ e is discussed below in more details.

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

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 s ¼ e be S. Consider now the case when consumption and deposits are gross complements. Then, for any given T, households in addition to S absorb an extra share S0 . Thus, 1 S S0 of the increased stock of ﬁscal liability xðq qð0ÞÞ is borne by ﬁrms. Since the share that is borne by ﬁrms is identically equal to the fall in equity price q qð0Þ, it must be the case that (1 S S0 )x ¼ 1. Suppose now T is increased. Then the new present value of additional ﬁscal liability is ðF bÞerT is smaller. However, as the rate of households’ deposits adjustment is higher than r, their share of ﬁscal burden decreases at a rate faster than the overall ﬁscal adjustment. In other words, S0 decreases. Then x must also decrease. Hence, a higher T is consistent with a lower bailout ratio. On the other hand, when consumption and deposits are gross substitutes, households’ deposit reallocation compounds ﬁrms’ burden by an extra share S00 . In equilibrium, (1 S þ S00 )x ¼ 1. Households’ deposit adjustment rate with respect to T is lower than r now. Therefore, for a higher T, the extra liability that households burden ﬁrms with decreases at a rate lower than r, the rate at which the ﬁscal liability 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 T ¼ 0, and g / g as T / N. Thus for a bailout ratio g g a self-fulﬁlling crisis can never occur.37 Appendix G computes the upper and lower bounds explicitly. Note that these results are applicable only if sse ¼ 1.38 On the other hand, when s ¼ e, the marginal utilities of consumption and deposits are independent of each other, and the intertemporal deposit allocations are independent of T, as can be seen from (30). This leads to indeterminacy of equilibria as discussed below.

5.2. Indeterminacy of crises equilibria When s ¼ e, the utility function takes the form u(c,d) ¼ y(c) þ w(d). Then, from (29)–(31), a constant marginal utility of consumption implies c1 ¼ c2 ¼ c. Further, r1 ¼ r. Then, using (34) in (38) yields

gjs¼e ¼

y hr: r r2

(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 up to T is identical to the pre-crisis equilibrium, as both the consumption and the nominal interest rate remain unchanged. Thus ﬁrms’ returns fall only at T, and the drop r r2 solely depends on the increase in debt service payments y, which is independent of T. The increase in the stock of ﬁscal liability at time 0 is a multiple x of the present value of the fall in ﬁrms’ returns. In equilibrium, this must equal the present value of the seigniorage y=r erT . Since both present values use the same discount rate, there is a 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 net public debt grows to its upper bound F. Then a question naturally arises: how T is affected by F? The following proposition provides an answer.

37 For g > , self-fulﬁlling crises will be consistent only if T < 0, which is impossible. All such cases imply T ¼ 0, which entails a discrete devaluation at 0. As discussed in Section 4, the analysis is restricted to T 0 cases as all other cases are ruled out by Assumption A.1. 38 It is conjectured that the result should hold for all e and s, such that sse. 39 When s ¼ e ¼ 1, i.e.,uðc; dÞ ¼ ln c þ c ln d, it is easy to verify that g ¼ 1.

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

45

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 cause a higher fall in equity prices. Then the bailout is larger, which advances T. When households’ deposits and consumption are gross complements, a higher post-crisis nominal interest rate, however, implies that agents tilt their consumption and deposits to a higher level while the exchange rate is ﬁxed, i.e., the nominal interest rate is low. As a result, there is a rise in households’ deposits relative to the pre-crisis level. The resulting increase in the foreign reserves of the central bank partially offsets the net increase in 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, households prefer 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 precrisis level at time 0. Now the delaying effect of a higher F yields to its advancing effect that stems jointly 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 key mechanism which links the twin crises hinges on the empirical facts that asset prices decline in advance of currency crises, and that banks in emerging markets carry large asset price exposure. The model presented in the paper generates an endogenous decline of asset prices when agents realize that a currency crisis is imminent. The decline in asset prices combined with the government bailout of banks thus spawns a self-fulﬁlling twin crises, in which banking crisis precedes currency crisis as has been observed empirically. The present analysis shows that the government bailout rule, whereby the government assumes a ﬁxed ratio of banks’ capital loss, crucially impacts the crises equilibria. In particular, the higher the bailout ratio, the steeper is the decline in asset prices and the sooner the government is forced to abandon its ﬁxed exchange rate policy. This result holds under a fairly general speciﬁcation of households’ preferences. However, the effect of a higher upper bound on the net public debt – the amount 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 currency crisis 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 critically hinge on the government’s dependence on seigniorage for ﬁnancing its unanticipated contingencies. In practice, debt deﬂation and ﬁscal reforms are equally (or even more) important sources of governments’ ﬁnance after crises (see Burnside et al., 2006). However, as long as seigniorage remains as one of the sources, including these alternatives will merely rescale the results. In particular, the smaller the share of seigniorage in the overall additional revenues, the longer a ﬁxed exchange rate regime can be 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 output is ﬁxed by construction. In a more realistic setting, declining asset prices can impinge on ﬁrms’ liquidity by either raising the cost of loans, or by constraining their access to ﬁnancial markets. I conjecture that adding these features will only amplify the present results and also generate substantial output and consumption 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 Ve´gh, and Joydeep Bhattacharya for helpful comments and suggestions. All remaining errors are mine.

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

Appendix.

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

yj ¼ F kj ; lj ; where F($,$) is homogeneous of degree one, and F1,F2 0. Further, a Z 1 a1 !a1 a kjl dl ;

kj ¼

a > 0;

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 R1 is the quantity of ﬁrm j’s capital used in its own production. Note that 0 klj dl ¼ 1; cj. Firm j’s creditin-advance constraint (5) can be written as

zj f

#

Z

rl kjl dl;

f˛ 0; 1 ;

½0;1yj

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 equilibrium, ﬁrm j’s ﬂow constraint is given by

Dbj ¼ Dzj ;

if t˛G;

Z a_ j ¼ raj þ F kj ; lj wlj

rl kjl 1 þ fIz dl þ rj

½0;1yj

Z

klj dl Uj ;

if t;G;

½0;1yj

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

Uj ¼ F kj ; lj wlj

Z

rl kjl 1 þ fIz dl þ rj ½0;1yj

Z

lj

½0;1yj

kj dl:

Integrating forward, the present value of dividends is

Z

N

j rt

Ue

dt ¼

Z

0

N

j

j

F k ;l

wl

0

j

Z ½0;1yj

l jl

z

r k 1 þ fI dl þ rj

Z

! lj

k dl ert dt:

½0;1yj

Given the path of interest rates, {iz,r,rl}, the shareholders of ﬁrm j maximize the present value of dividends by choosing the path of its capital use, {kjj,kjl}. The symmetric supply and demand schedules for all types of capital implies that their equilibrium rates of return are equal, i.e., rj ¼ r for all j. The ﬁrst order conditions require that the ratio of the marginal product of ﬁrm j’s use of its own capital, kjj, to its use of hired capital, kjl, be equal to the ratio of their respective opportunity costs, r and r(1 þ fIz). Hence,

a kjj ¼ 1 þ fI z ; kjl

cl˛½0; 1yj:

In equilibrium, ﬁrm j rents capital from all other ﬁrms in equal amounts, i.e., kjl ¼ kjn for all l and n s j.

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

47

The symmetry also implies an equal amount of mutual hiring, i.e., klj ¼ kjl. Moreover, each type of capital is hired in equal amounts by all ﬁrms, i.e., klj ¼ knj, for all l and n s j. Since ﬁrm j has a zero mass and the R1 use of its own capital, kjj, is ﬁnite, the aggregate equilibrium condition 0 klj dl ¼ 1 implies that klj ¼ 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) þ w(d). From (29) and (31) c1 ¼ c2 ¼ and r1 ¼ r. Then, (32) yields

T ¼

1 F b h r r2 ; ln J r

where from (30) r2 ¼ rðr þ ðy=hr2 Þ; cÞ. Clearly, r2 is independent of J and T. Hence, TJ < 0. On the other hand, r2 is decreasing in F b. Therefore, the effect of F and b on T is not obvious. As a further specialization, let u(c) ¼ ln c, and w(d) ¼ c ln d. Then,

T ¼

1 Fb ; ln r uJ

where u ¼ 1 þ 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 1 dr 1 udd uc ucd ud ¼ ¼ < 0; q dId quc ucc udd ucd udc dI d

(40)

where the last inequality follows from the properties of the u. Next, using (21) 1

Id ¼

2

Id ¼

1

u

q

1

q

u

9 ; r1

9y r2

(41)

where u ¼ 1 þ hr. Suppose I d < I d . Then, from (40) r1 < r2. But, from (41), I d > I d – a contradiction. Hence, 2

2

1

Id > Id ;

1

r1 > r2 :

2

1

(42)

The last result and (10) yield 2

1

IZ > IZ : , 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

dc ucd uc ¼ X0 iff ucd ,0: ucc udd ucd udc dI d Eqs. (4a), (4b) and (41) yield

(43)

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

ud c1 ; qr1 9 1 ¼ u ; q r1 uc c1 ; qr1 ud c2 ; qr2 9y 1 ¼ u ; q r1 uc c2 ; qr2 which imply c1 and c2 as functions of r1 and r2, respectively. Clearly,

dr1 ¼ dc1

udc uc ucc ud > 0; 1d 9 ucd ud udd uc ðu Þ2 1 c

f r

dr2 ¼ dc2

udc uc ucc ud > 0: 1d9y 2 ðu Þ ucd ud udd uc þ c 2

f

(44)

r

Case I ucd > 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 II ucd < 0. Using (31) with (43) yields c1 < c < c2 . Then from (42) and (44), r > r1 > r2 . Case III ucd ¼ 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 r1 r : 1 1 2 r r2 Fbh r r

(45)

Since ðJ hðr1 rÞÞ=ðF b hðr1 r2 ÞÞ > 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 F b r ~r 1þh ; J J r1 r where the second inequality follows from assumption that T 0. The above inequality in turn implies that

r r2 r ~r h : J r1 r

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

49

Recall from the discussion following Eq. (32) that J hðr1 rÞ. Hence, the above holds if r 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 ½ðc1ð1=sÞ þ cd1ð1=sÞ Þs=ðs1Þ 1ð1=eÞ =1 ð1=eÞ. It is easy to check that this satisﬁes all the assumptions on u ($,$) following Eq. (1). Furthermore, for s < e, ucd > 0 while for s > e, ucd < 0. For analytical convenience, I set e ¼ 1.

Case I

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

s > 1: Now, ucd < 0. Hence, c1 < c < c2 and r > r1 > r2 . Case III s ¼ 1: In this case, ucd ¼ 0. Hence, c1 ¼ c2 ¼ c and r ¼ r1 > r2 . Using (4b) and (41) obtain

9 s c1 ¼ kr1 u 1 ;

r

9y s ; c2 ¼ kr2 u 1

(46)

r

where k ¼ cs ð1 d=fÞs1 , Next, using (4a), (4b) and (41) yields

c1 þ ur1 9 ¼ c2 þ ur2 9 þ y :

(47)

For convenience, I rewrite (31) as

c1 T~ 1 þ c2 T~ 2 ¼ c;

(48)

where T~ 1 ¼ 1 erT , and T~ 2 ¼ erT . Observe that solving (46)–(48) obtains equilibrium allocations for all T 0. The following lemma is used in deriving some key results that follow. Lemma 2. Proof.

c1r1 Xc2r2 if and only if s,1.

Differentiating (46) yields

c1r1 c2r2

¼

u r91

u 9r2y

s1

u ð1 sÞr91

s1

u ð1 sÞ9r2y

1

2

Let x ¼ 9=ður1 Þ and y ¼ 9 y=ður2 Þ. Further, as I d < I d ; y < x. Hence,

c1r1 Xc2r2

iff

ð1 xÞ1s ð1 yÞ1s , iff s,1: ð1 1 sÞxÞ ð1 ð1 sÞyÞ

For future use, note that

50

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

92 9 s2 c1r1 r1 ¼ lsðs 1Þ 3 u 1 X0 iff sX1;

r

r1

9 y s2 ð9 yÞ2 X0 iff sX1: c2r2 r2 ¼ lsðs 1Þ 3 u 2

r2

r

(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) that d ððq qð0ÞÞerT Þ 0. In particular, I show that d ððq qð0ÞÞerT Þ > 0 if ss1, and d ððq qð0ÞÞerT Þ ¼ 0 if dT dT dT s ¼ 1. Next, the value of g for T ¼ 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, this is done at the end. The proof proceeds in the following ﬁve steps.

Step I Show that limT/N ðd=dTÞððq qð0ÞÞerT Þ ¼ 0; cs. First, from (34):

lim ðq qð0ÞÞ ¼ 0:

T/N

Applying L’Hospital’s rule yields

lim ðq qð0ÞÞerT ¼ lim

T/N

T/N

1 rT dqð0Þ e : r dT

(50)

Hence,

d dqð0Þ ðq qð0ÞÞerT ¼ lim rðq qð0ÞÞerT lim erT ¼ 0: dT T/N dT T/N T/N lim

Step II d ððq qð0ÞÞerT Þj d rT Show that dT T¼0 ¼ 0 for s ¼ 1, and dT ððq qð0ÞÞe ÞjT¼0 > 0 for ss1. Substituting (46) into (47), and differentiating along with (48) yields

dr1 dr2

2 3

2 2 ~2 1 4 cr2 þ u þ cr2 T 5 r erT c2 c1 dT; ¼ D c11 þ u þ c11 T~ 1 0 r r

where D ¼ c1r1 ðc2r2 þ uÞT~ 1 c2r2 ðc1r1 þ uÞT~ 2 . Hence,

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

51

dr1 1 < 0 if s < 1; c2 c1 T~ 2 ¼ r c2r2 þ u > 0 if s > 1; D dT

dr2 1 <0 ¼ r c1r1 þ u c2 c1 T~ 2 >0 D dT

if s < 1; if s > 1;

(51)

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

T~ 1 þ r21 T~ 2 d dqð0Þ r ; ðq qð0ÞÞerT ¼ rðq qð0ÞÞerT ¼ r r1 erT þ c1 c2 1 dT dT T~ c11 þ r21 T~ 2 c22 r

r2

where r1 ¼

ðc1

r þ

r

r

uÞ=ðc2

r þ uÞ. Observe that

d ððq qð0ÞÞerT Þ dT

0 iff

r r1 þ

c1 c

c2r2

T¼0

0;

(52)

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

c1 c > c1r1 > c2r2 : 1 T¼0 T¼0 r r T¼0 Similarly, for s > 1, (48), Lemma 2, and (49) imply

c1 c < c1r1 < c2r2 : 1 T¼0 T¼0 r r T¼0 Since for s ¼ 1, c1 ¼ c, r1 ¼ r, (52) holds for all values of s. In particular, it holds with strict inequality for ss1.

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

erT

for

T~ 1 þ r21 T~ 2 dqð0Þ r : ¼ r1 r2 þ c2 c1 1 dT T~ c11 þ r21 T~ 2 c22 r

r

s¼1

and

(53)

r

For s ¼ 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 second is negative. Note from Lemma 2 that c1r1 > c2r2 . Hence, erT ðdqð0Þ=dTÞ > 0 if ðc1 c2 Þ=ðr1 r2 Þ < c2r2 . From (46) c2 jr2 ¼r1 > c1 . Finally, using (49), c2r2 > ððc2 jr2 ¼r1 c1 Þ=ðr1 r2 ÞÞ > ðc1 c2 Þ=ðr2 r2 Þ. Hence, erT ðdqð0Þ=dTÞ > 0. For the next part, taking derivative of the RHS in (53) w.r.t. T, and after some algebra

3 r1 T~ 1 c1r1 r1 þ T~ 2 r2r1 c2r2 r2 ddT þ 2r2r1 c2r2 c1r1 rT~ 2 d dqð0Þ erT ¼ c2 c1 : 2 dT dT T~ 1 c1r1 þ r2r1 T~ 2 c2r2

52

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

For s ¼ 1, the RHS equals zero. Next, for s < 1, the RHS is clearly positive. For s > 1, using (51), the RHS is positive iff

2r2r1

c2r2 c1r1 c2 c1

3 T~ 1 c1r1 r1 þ T~ 2 r2r1 c2r2 r2 T~ 1 c1r1 þ T~ 2 r1r1 c2r2

;

which is easy to verify for T ¼ 0 and as T / N. By monotonicity, it holds for all T.

Step IV It has been shown that d=dtððq qð0ÞÞerT Þ 0 for T ¼ 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, then it will violate the result shown in Step I, i.e., limT/N ðd=dtÞððq qð0ÞÞerT Þ ¼ 0. Hence, ðd=dtÞððq qð0ÞÞerT Þ 0; cT. In particular,

d ðq qð0ÞÞerT ¼ 0; dt E d ðq qð0ÞÞerT 0; dt

s ¼ 1; ss1:

Step V Note from (38) that for T ¼ 0,

g¼

y df r ¼ g r r2 T¼0 1 d

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

g¼

y d fr ¼ g r r2 T/N þ 1 c2 T/N c 1 d c11 r

where

r2 j

c1 ¼c

2 2 1 T¼0 ; r jT/N ; c jT/N ; cr1 jr1 ¼r

are computed from (46)–(48).

,

H. Proof

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

vqð0Þ q qð0Þ þ dT y ¼ vy ; dy rðq qð0ÞÞ vqð0Þ vT

(54)

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

c1 c2r2 vqð0Þ 1 T~ 2 2 2 ~ 1 r1 r c ¼ 1 þ T 1 vy r T~ c11 þ r21 T~ 2 c22 y r u þ c2r2 r r r where

!

< 0;

(55)

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

53

9 y s1 c2y ¼ ks u 2 :

r

The denominator in (54) is positive as shown in Appendix G. However, the two terms in the numerator have opposite signs, and hence there sign is not so obvious. If the numerator has the same sign for both T ¼ 0 and as T / N, then from monotonicity it will have the same sign for all T. First, from (55) obtain

vqð0Þ vy

¼ T¼0

1 c2y ; r c2r2

Then using (46) and (54) obtain

r r2 1 s ry r u ð1 sÞ9y 2

dT ¼ dy T¼0

r

rðq qð0ÞÞ vqð0Þ vT

:

Notice that for s ¼ 1, the numerator is equal to zero. Hence, one needs to check the derivative of the numerator with respect to s around s ¼ 1:

9y y y d c ys þ r r2 u ð1 sÞ 2 ¼ ln 1 þ c c 0; k ds r þ9y þ9y k

k

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

9y dr2 c ln u r2 ¼ ; k u ds T¼0 9 dr c ln u r ¼ : k u ds Thus, I have shown that

dT X0 iff s,1: dy T¼0

(56)

Next, rewrite (54) as

vqð0Þ

q qð0Þ

erT dT y vy : ¼ dy rðq qð0ÞÞ vqð0Þ erT vT Note that obtain

þ

erT 0; cT, and limT/N ðrðq qð0ÞÞ ðvqð0Þ=vTÞÞerT ¼ 0. From (55) rðq qð0ÞÞ vqð0Þ vT

vqð0Þ erT vy

2 u þ c1 1 2 c 1 y 1 r þ cr1 cr2 ¼ : r T/N c1r1 u þ c2r2

Then (54) yields

c2y ðuþc11 Þþc11 c22 rT r r r limT/N ðqqð0ÞÞe 1r 1 u 2 y c 1 ð þc 2 Þ dT r r T/N : ¼ dy T/N erT limT/N rðq qð0ÞÞ vqð0Þ vT

Using (53), it can be shown that

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

lim

1 1 vqð0Þ 1 2 1 c c1 ¼ lim erT þ ¼ vT ry T/N r y c1r1 u

ðq qð0ÞÞerT

y

T/N

!

!

y þ : u

Hence,

!! 1 1 c2y u þ c1r1 þ c1r1 c2r2 vqð0Þ q qð0Þ rT 1 1 2 1 e ¼ c c1 lim þ þ þ : y vy ru ry r T/N c1r1 u c1r1 u þ c2r2

(57)

Again, note that for s ¼ 1, the above expression is equal to zero. For ss1, I take the derivative of the above expression with respect to s and evaluate around s ¼ 1. First, note that

9y c2s ¼ c2 ln u 2 ;

9 c1s ¼ c1 ln u 2 :

r

r

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

c2s

s¼1

c ¼ c ln k2 ;

c1s

r

s¼1

c dc2y k k 1 þ ln ; ¼ ds s¼1 r2

dc1r1 ds

c ¼ c ln k1 ;

r

dc2r2 ds

c 9y ¼ k u ln k2 þ 2 ; s¼1

r

r

c 9 ¼ k u ln k1 þ 1 ; s¼1

r

r

1 r d 1 c2 c1 c : ðr r2 Þ ¼ s 2s ¼ ln ds r2 u þ c r2 u þ c2r2 s¼1 Using the above relations in (57), and after some algebra, it can be shown that

y y d vqð0Þ q qð0Þ rT 1 c e ¼ ln 1 þ c c 0: lim þ y ds T/N vy ruy k kþ9y kþ9y Hence, for s close to 1, I have shown that

dT X0 iff s,1: dy T/0

(58)

It is conjectured that for all s40:

dT X0 iff s,1: dy ,

40

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

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

55

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