Skip to: [Printable Version (PDF)]

Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 848, This Draft April 2006 --- Screen Reader Version*

Fighting Against Currency Depreciation, Macroeconomic Instability and Sudden Stops¹

Luis-Felipe Zanna²

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.

Abstract:

In this paper we show that, in the aftermath of a currency crisis, a government that adjusts the nominal interest rate in response to domestic currency depreciation can induce aggregate instability in the economy by generating self-fulfilling endogenous cycles. We find that, if a government raises the interest rate proportionally more than an increase in currency depreciation, then it induces self-fulfilling cycles that, driven by people's expectations about depreciation, replicate several of the salient stylized facts of the "Sudden Stop" phenomenon. These facts include a decline in domestic production and aggregate demand, a collapse in asset prices, a sharp correction in the price of traded goods relative to non-traded goods, an improvement in the current account deficit, a moderately higher CPI-inflation, more rapid currency depreciation, and higher nominal interest rates. In this sense, an interest rate policy that responds to depreciation may have contributed to generating the dynamic cycles experienced by some economies in the aftermath of a currency crisis.

Keywords: Small Open Economy, Interest Rate Policies, Currency Depreciation, Self-fulfilling Cycles, Sudden Stops, Collateral Constraints.

JEL classification: E32, E52, E58, F41.

1  Introduction

One of the policy debates that emerged out of the Asian Crisis of 1997 was that of the appropriate interest rate policy to fight domestic currency depreciation. On one hand, the IMF advocated for higher interest rates to prevent excessive depreciation. It claimed that such a policy would reduce capital outflows by raising the cost of currency speculation and induce capital inflows by making domestic assets more attractive.³ On the other hand, some critics of the IMF policy prescription recommended lowering interest rates in the face of depreciation.⁴ These critics argued that lower rates would alleviate the subsequent recessions in the Asian economies, and improve the banks' and corporations' balance sheets, which had weakened significantly as a result of the negative effects of rapid currency depreciation on their dollarized liabilities.

These two views had something in common. They both advocated for changes in interest rates in response to some macroeconomic indicators, in this case the currency depreciation rate. They both implied a feedback mechanism, positive or negative, from the current depreciation rate to the nominal interest rate.⁵ In this paper, we study some of the possible macroeconomic consequences of these interest rate feedback policies in countries that have been hit by a currency crisis. Our main result is that in the aftermath of a crisis, an interest rate policy that calls for changing the nominal interest rate, positively or negatively, in response to currency depreciation can induce aggregate instability in the economy by generating self-fulfilling endogenous cycles. That is, this policy can cause cycles in the economy that are driven exclusively by people's self-fulfilling beliefs rather than fundamentals, and regardless of whether the response to depreciation is positive or negative.

We find that if a government raises the interest rate proportionally more than an increase in currency depreciation, then it induces self-fulfilling cycles that replicate several of the salient empirical regularities of emerging market crises. In particular, we build a model in which these cycles, which are driven exclusively by people's expectations about currency depreciation, replicate many of the stylized facts following a currency crisis. These facts, labeled by Calvo (1998) as the "Sudden Stop" phenomenon, include a decline in domestic production and aggregate demand, a collapse in asset prices, a sharp correction in the price of traded goods relative to non-traded goods, an improvement in the current account deficit, a moderately higher CPI-inflation, more rapid currency depreciation and higher nominal interest rates. In this sense, interest rate policies that respond to depreciation may have contributed to generating the dynamic cycles experienced by some economies in the aftermath of a currency crisis.

As a building block to our full model, we first construct a simple environment of a small open economy in which we are able to derive analytical results. We show that interest rate policies that respond only to currency depreciation can induce real indeterminacy or multiple equilibria, which in turn opens the possibility of self-fulfilling equilibria.⁶ The intuition for this result is the following. Suppose that in response to a "sunspot," agents in the economy expect a higher inflation rate. Because the interest rate policy responds only to currency depreciation - and not to inflation - the real interest rate can fall, boosting aggregate demand. As firms see aggregate demand rising, they raise prices, thus validating the original expectations of a higher inflation.⁷

Our full model is a sticky-price small open economy with traded and non-traded goods with two important features: an interest rate policy that responds to currency depreciation, and a collateral constraint which captures the fact that international loans must be guaranteed by physical assets such as capital. Loosely speaking, the crisis is modelled as a time when this constraint is unexpectedly binding.⁸ The model also includes other features that have become very useful to match quantitatively some of the stylized facts of the aftermath of a crisis. In particular, we consider non-traded distribution services, and domestic and international loan requirements to hire labor and to purchase an imported intermediate input.⁹

In this full model, an interest rate policy that responds only to depreciation also induces multiple equilibria. However, the interaction of this policy with the collateral constraint induces self-fulfilling cycles. This result is related to, but differs from, the seminal work by Kiyotaki and Moore (1997). They show that a binding collateral constraint induces credit cycles that may amplify business cycles driven by fundamentals. In our work this constraint creates cycles, but the interest rate policy makes them self-fulfilling to the extent that they are driven by people's expectations about the economy.

We think our results have at least three important implications. First, they do not support any of the views of the previously mentioned debate. In fact we unveil a peril that may be present in both policy recommendations. What seems crucial is not whether the government increases or decreases the interest rate, but instead the feedback response of the nominal interest rate to currency depreciation that most of the previous studies have ignored.

Second, our results provide a possible explanation of why the empirical literature has not been able to obtain conclusive evidence about whether higher interest rates can cause nominal currency depreciation or, instead, appreciation in the aftermath of a crisis.¹⁰ This literature has tried to control for the variables that influence the nominal exchange rate. But our results suggest that there can be potential influences that may depend on "sunspots," which in turn can induce self-fulfilling cycles in the nominal exchange rate (or the nominal depreciation rate) as well as in other variables. Clearly these influences do not depend on fundamentals, and their effect should be taken into account by this literature.

Third, since interest rate policies that respond to currency depreciation can induce expectations-driven fluctuations, then they can destabilize the economy. Therefore they can be costly in terms of macroeconomic instability and welfare.¹¹ This has not been studied in the previous literature and deserves further research. For instance, Lahiri and Vegh (2002, 2003) and Flood and Jeanne (2000) focus on the fiscal and output costs of higher interest rates before and after a crisis. In addition Lahiri and Vegh (2003) claim that there is a non-monotonic relationship between welfare and the increase in interest rates. Christiano, Gust and Roldos (2004) explore conditions under which a cut (rise) in the interest rate in the midst of a crisis will stimulate output and improve welfare. Aghion, Bacchetta and Banerjee (2000) find that it might not be optimal to raise interest rates either when the proportion of foreign currency debt is not too large, or when credit provision, domestic investment and production are highly sensitive to changes in nominal interest rates; whereas Braggion, Christiano and Roldos (2005) build a model where, in response to a financial crisis, it is optimal to raise the interest rate immediately, and then reduce it sharply.

The remainder of this paper is organized as follows. In Section 2, we consider a simple economy in "good" times and characterize analytically the equilibrium under interest rate policies that respond only to currency depreciation. In Section 3, we build our full model, which includes, among other things, a collateral constraint. Here we study the determinacy of equilibrium analysis through numerical simulations. In addition, we show that even if the interest rate policy responds to past CPI-inflation, it can still induce multiple equilibria as long as it responds to currency depreciation.¹² In Section 4, we use our full model to construct a self-fulfilling equilibrium that captures the stylized facts of a "Sudden Stop." Section 5 concludes.

2  The Simple Model

In this section we develop a simple infinite-horizon small open economy model. The economy is populated by a continuum of identical household-firm units and a government who are blessed with perfect foresight. Before we describe in detail the behavior of these agents we state a few general assumptions and definitions.

There are two consumption goods: a traded good and a composite non-traded good whose prices are denoted by $P_{t}^{T}$ and $P_{t}^{N}$ respectively. We assume that the law of one price holds for the traded good. Then $P_{t} ^{T}=\mathcal{E}_{t}P_{t}^{T\ast}$ where $\mathcal{E}_{t}$ is the nominal exchange rate, and $P_{t}^{T\ast}$ is the foreign price of the traded good. Below we will relax this assumption. We also set $P_{t}^{T\ast}=1$ implying that $P_{t}^{T}=\mathcal{E}_{t}$.

The real exchange rate ($e_{t}$) is defined as the ratio between the price of traded goods and the aggregate price of non-traded goods, $e_{t} =\mathcal{E}_{t}/P_{t}^{N}$. From this definition we deduce that

$$e_{t}=e_{t-1}\left( \frac{\epsilon_{t}}{\pi_{t}^{N}}\right) ,$$ (1)

where $\epsilon_{t}=\frac{\mathcal{E}_{t}}{\mathcal{E}_{t-1}}$ is the gross nominal depreciation, and $\pi_{t}^{N}=\frac{P_{t}^{N}}{P_{t-1}^{N}}$ is the gross non-traded goods inflation$.$

3  The Government

The government issues two nominal liabilities: money, $M_{t}^{g}$, and a domestic bond, $B_{t}^{g},$ that pays a gross nominal interest rate $R_{t}$. It does not have access to foreign debt and makes lump-sum transfers to the household-firm units, $\tau_{t},$ pays interest on its domestic debt, ( $R_{t}-1)B_{t}^{g}$, and receives revenues from seigniorage$.$ Its budget constraint is described by $m_{t}^{g}+b_{t} ^{g}=\frac{m_{t-1}^{g}}{\epsilon_{t}}+\tau_{t}+\frac{R_{t-1}b_{t-1}^{g} }{\epsilon_{t}}$ where $m_{t}^{g}=\frac{M_{t}^{g}}{\mathcal{E}_{t}}$ and $b_{t}^{g}=\frac{B_{t}^{g}}{\mathcal{E}_{t}}.$ It follows a Ricardian fiscal policy by setting the lump-sum transfers in order to satisfy the intertemporal version of its budget constraint and its transversality condition $\underset{t\rightarrow\infty}{\lim}\frac{b_{t}^{g}}{\underset{s=0} {\overset{t-1}{\prod}}\left( \frac{R_{s}}{\epsilon_{s+1}}\right) }=0.$

The description of monetary policy is motivated by the debate that emerged after the Asian crisis about the appropriate interest rate policy to fight against currency depreciation. We assume the government implements an interest rate policy that responds to the deviation of the current depreciation rate from a depreciation target.¹³ That is,

$$R_{t}=\bar{R}\rho\left( \frac{\epsilon_{t}}{\bar{\epsilon}}\right) ,$$ (2)

where $\rho(.)$ is a continuous, differentiable, and strictly positive function in its argument with $\rho\left( 1\right) =1$ and $\rho_{\epsilon }\equiv\rho^{\prime}\left( 1\right) \neq0$; and $\bar{R}$ and $\bar {\epsilon}$ are the targets of the nominal interest rate and the nominal depreciation rate that the government wants to achieve.¹⁴

This policy can respond positively to the deviation of the nominal depreciation rate from its target, $\rho_{\epsilon}>0$, capturing the policy recommendation of the IMF; or it can react negatively, $\rho_{\epsilon}<0$, describing, to some extent, the policy recommendation of the opposite view. Nevertheless we exclude the cases $\rho_{\epsilon}=-1,1.$ ¹⁵ In other words, the interest rate policy corresponds to (2) with $\rho_{\epsilon}\equiv\rho^{\prime}\left( 1\right) \neq0$ and either $\left\vert \rho_{\epsilon}\right\vert >1$ or $\left\vert \rho_{\epsilon}\right\vert <1.$

2.2  The Household-Firm Unit

There is a large number of identical household-firm units. They have perfect foresight, live infinitely and derive utility from consuming, not working, and liquidity services of money. The intertemporal utility function of the representative unit is described by

$$\overset{\infty}{\sum_{t=0}}\beta^{t}\left[ U(c_{t}^{T})+V(c_{t}^{N} )+H(h_{t}^{T})+L(h_{t}^{N})+J(m_{t})\right] ,$$ (3)

where $\beta\in(0,1)$ corresponds to the subjective discount rate, $c_{t}^{T}$ and $c_{t}^{N}$ denote the consumption of traded and non-traded goods respectively, $h_{t}^{T}$ and $\ h_{t}^{N}$ are the labor allocated to the production of the traded good and the non-traded good, and $m_{t}$ refers to real money holdings measured with respect to foreign currency. The specification in (3) assumes separability in the single period utility function among consumption, labor, and real money balances. Hence there are no distortionary effects of transactions money demand.¹⁶ Moreover the utility function is separable in $c_{t}^{T},$ $c_{t}^{N},$ $h_{t}^{T},$ and $h_{t}^{N}.$ This allows us to derive analytical results in the determinacy of equilibrium analysis. To complete the characterization we also make the following assumption.

Assumption 1. a)$\ U(.),$ $V(.),$ $H(.),$ $L(.)$ and $J(.)$ are twice continuously differentiable; and b) $U(.),$ $V(.),$ and $J(.)$ are strictly increasing ( $U_{T} \equiv\frac{dU}{dc_{t}^{T}}>0,$ $V_{N}>0,$$J_{m}>0$) and strictly concave ($U_{TT}<0,$ $V_{NN}<0,$$J_{mm}<0$), whereas $H(.)$ and $L(.)$ are strictly decreasing ( $H_{T}\equiv\frac{dH}{dh_{t}^{T}}<0,$ $L_{N}<0$) and strictly concave ($H_{TT}<0,$ $L_{NN}<0$).

The household-firm unit produces a flexible-price traded good and a sticky-price non-traded intermediate differentiated good by employing labor from perfectly competitive markets. For simplicity we assume that there is zero labor mobility.¹⁷ The technologies are described by

$$y_{t}^{T}=F\left( \breve{h}_{t}^{T}\right)$$   and$$\qquad\tilde {y}_{t}^{N}=G\left( \tilde{h}_{t}^{N}\right) ,$$

where $\breve{h}_{t}^{T}$ and $\tilde{h}_{t}^{N}$ denote the labor hired by the household-firm unit for the production of the traded good and the non-traded good, respectively. The technologies satisfy the following assumption.

Assumption 2. $F\left( .\right)$ and $G\left( .\right)$ are twice continuously differentiable, strictly increasing ( $F_{T}\equiv\frac{dF}{dh_{t}^{T}}>0,$ $G_{N}>0$), and strictly concave ($F_{TT}<0,$ $G_{NN}<0$ )$.$

Consumption of the non-traded good, $c_{t}^{N},$ is a composite good made of a continuum of intermediate differentiated goods. The aggregator function is of the Dixit-Stiglitz type. Each household-firm unit is the monopolistic producer of one variety of non-traded intermediate goods. The demand for the intermediate good is of the form $C_{t}^{N}d\left( \frac{\tilde{P}_{t}^{N} }{P_{t}^{N}}\right)$ satisfying $d\left( 1\right) =1$ and $d^{\prime }\left( 1\right) =-\mu,$ with $\mu>1,$ where $C_{t}^{N}$ denotes the level of aggregate demand for the non-traded good, $\tilde{P}_{t}^{N}$ is the nominal price of the intermediate non-traded good produced by the household-firm unit, and $P_{t}^{N}$ is the price of the composite non-traded good. The unit sets the price of the good it supplies, $\tilde{P}_{t}^{N}$, taking the level of aggregate demand for the good as given. Specifically, the monopolist is constrained to satisfy demand at that price. That is,

$$G\left( \tilde{h}_{t}^{N}\right) \geq C_{t}^{N}d\left( \frac{\tilde{P} _{t}^{N}}{P_{t}^{N}}\right) .$$ (4)

Following Rotemberg (1982), we introduce nominal price rigidities for the intermediate non-traded good. The household-firm unit faces a resource cost of the type $\frac{\gamma}{2}\left( \frac{\tilde{P}_{t}^{N}}{\tilde{P}_{t-1} ^{N}}-\bar{\pi}^{N}\right) ^{2},$ where $\bar{\pi}^{N}$ is the steady-state level of the gross non-traded goods inflation.

There are incomplete markets. The representative household-firm unit has access to the following two risk free bonds: a government domestic bond, $B_{t}$, that pays a gross nominal interest rate $R_{t},$ and a foreign bond, $b_{t}^{\ast},$ that pays a gross foreign interest rate $R_{t}^{\ast}.$ In addition, the unit receives income from working, $W_{t}^{T}h_{t}^{T}+W_{t} ^{N}h_{t}^{N}$, transfers from the government, $\tau_{t}$, and dividends. Then the flow budget constraint in units of the traded good can be written as

$$m_{t}+b_{t}\leq\frac{m_{t-1}}{\epsilon_{t}}+\frac{R_{t-1}b_{t-1}}{\epsilon _{t}}+w_{t}^{T}h_{t}^{T}+w_{t}^{N}h_{t}^{N}+\tau_{t}+\Omega_{t}-c_{t} ^{T}-\frac{c_{t}^{N}}{e_{t}},$$ (5)

where $b_{t}=\frac{B_{t}}{\mathcal{E}_{t}},$ $w_{t}^{i}=\frac{W_{t}^{i} }{\mathcal{E}_{t}}$ with $i=T,N,$ and

$$\Omega_{t}=\left[ F\left( \check{h}_{t}^{T}\right) -w_{t}^{T}\check{h} _{t}^{T}\right] -\frac{1}{e_{t}}\left[ \frac{\tilde{P}_{t}^{N}}{P_{t}^{N} }C_{t}^{N}d\left( \frac{\tilde{P}_{t}^{N}}{P_{t}^{N}}\right) -e_{t}w_{t} ^{N}\tilde{h}_{t}^{N}-\frac{\gamma}{2}\left( \frac{\tilde{P}_{t}^{N}} {\tilde{P}_{t-1}^{N}}-\bar{\pi}\right) ^{2}\right] -R_{t-1}^{\ast} b_{t-1}^{\ast}+b_{t}^{\ast}.$$ (6)

Equation (5) says that the end-of-period real financial domestic assets (money plus domestic bond) can be worth no more than the real value of financial domestic wealth, brought into the period, plus the sum of wage income, transfers, and dividends ( $\Omega_{t}$) net of consumption. The dividends described in (6) correspond to the difference between sale revenues and costs, taking into account that through the firm-side the representative unit can hold foreign debt, $b_{t}^{\ast}$. For holdings of foreign debt the unit pays interests ( $R_{t-1}^{\ast}-1)b_{t-1}^{\ast}.$ Moreover the unit is subject to a Non-Ponzi game condition

$$\underset{t\rightarrow\infty}{\lim}\frac{n_{t}}{\underset{s=0}{\overset {t-1}{\prod}}R_{s}^{\ast}}\geq0,$$ (7)

where $n_{t}=b_{t}+m_{t}-b_{t}^{\ast}.$

The problem of the household-firm unit consists of choosing the set of sequences $\{c_{t}^{T},$ $c_{t}^{N},$ $h_{t}^{T},$ $h_{t}^{N},$ $\check{h} _{t}^{T},$ $\tilde{h}_{t}^{N},$ $\tilde{P}_{t}^{N},$ $b_{t}^{\ast},$ $b_{t},$ $m_{t}\}_{t=0}^{\infty}$ in order to maximize (3) subject to (4), (5), (6) and (7), given the initial condition $n_{-1}$ and the set of sequences { $R_{t}^{\ast}$, $R_{t}$, $\ \epsilon_{t},$ $e_{t},$ $P_{t}^{N},$ $w_{t}^{T},$ $w_{t}^{N},$, $\tau _{t}$, $C_{t}^{N}\}.$ Note that since the utility function specified in (3) implies that the preferences of the agent display non-sasiation, then both constraints (5) and (7) hold with equality. The Appendix contains a detailed derivation of the necessary conditions for optimization. Imposing these conditions along with the market clearing conditions in the labor markets, the equilibrium symmetry ( $\tilde{P}_{t} ^{N}=P_{t}^{N}$ and $\tilde{h}_{t}^{N}=h_{t}^{N}),$ the market clearing condition for the non-traded good

$$G\left( h_{t}^{N}\right) =c_{t}^{N}+\frac{\gamma}{2}\left( \pi_{t}^{N} -\bar{\pi}^{N}\right) ^{2},$$ (8)

and the definitions $\pi_{t}^{N}=P_{t}^{N}/P_{t-1}^{N},$ $d(1)=1$ and $d^{\prime}(1)=-\mu,$ we obtain

$$R_{t}^{\ast}=\frac{R_{t}}{\epsilon_{t+1}},$$ (9)

$$-\frac{H_{T}(h_{t}^{T})}{U_{T}(c_{t}^{T})}=F_{T}\left( h_{t}^{T}\right) ,$$ (10)

$$U_{T}(c_{t}^{T})=\beta R_{t}^{\ast}U_{T}(c_{t+1}^{T}),$$ (11)

$$V_{N}(c_{t}^{N})=\frac{\beta R_{t}}{\pi_{t+1}^{N}}V_{N}(c_{t+1}^{N}),$$ (12)

$$\frac{V_{N}(c_{t+1}^{N})\left( \pi_{t+1}^{N}-\bar{\pi}^{N}\right) \pi _{t+1}^{N}}{V_{N}(c_{t}^{N})}=\frac{\left( \pi_{t}^{N}-\bar{\pi}^{N}\right) \pi_{t}^{N}}{\beta}+\frac{\mu c_{t}^{N}}{\beta\gamma}\left( \frac{\mu-1}{\mu }-mc_{t}\right) ,$$ (13)

where $mc_{t}=-\frac{L_{N}(h_{t}^{N})}{V_{N}(c_{t}^{N})G_{N}\left( h_{t} ^{N}\right) }$ corresponds to the marginal cost of producing the non-traded good. In addition equilibrium in the traded good market implies that

$$b_{t}^{\ast}-b_{t-1}^{\ast}=\left( R_{t-1}^{\ast}-1\right) b_{t-1}^{\ast }+c_{t}^{T}-F\left( h_{t}^{T}\right) .$$ (14)

The interpretation of these equations is straightforward. Condition (9) corresponds to an Uncovered Interest Parity condition (UIP) that equalizes the returns of the foreign and domestic bonds. Equation (10) makes the marginal rate of substitution between labor, assigned to the production of the traded good, and consumption of the traded good equal to the marginal product of labor in the production of the traded good. Equations (11) and (12) are the standard Euler equations for consumption of the traded good and consumption of the non-traded good. Equation (13) corresponds to the augmented Phillips curve for the sticky-price non-traded goods inflation. And (14) describes the dynamics of the current account deficit.

2.3  Capital Markets

We introduce imperfect international capital markets using the following ad-hoc supply curve of funds

$$R_{t}^{\ast}=R^{\ast}f\left( \frac{b_{t}^{\ast}}{\bar{b}^{\ast}}\right) , f^{\prime}\left( \frac{b_{t}^{\ast}}{\bar{b}^{\ast} }\right) >0, f\left( 1\right) =1, f^{\prime}\left( 1\right) =\psi>0,$$ (15)

and where $f\left( \frac{b_{t}^{\ast}}{\bar{b}^{\ast}}\right)$ corresponds to the country-specific risk premium, and $R^{\ast}$ is the risk free international interest rate. This specification captures the idea that the small borrowing economy faces a world interest rate, $R_{t}^{\ast}$, that increases when the stock of foreign debt, $b_{t}^{\ast}$, is above its long run level, $\bar{b}^{\ast}$. Then as the external debt grows, so does the risk of default, and in order to compensate the lenders for this risk, the economy has to pay them a premium over the risk free international interest rate. We also assume that the long-run level of foreign stock of debt is positive.

Assumption 3. The long-run level of the foreign stock of debt is positive: $\bar{b}^{\ast}>0.$

By introducing (15), we "close the small open economy" and avoid the unit root problem, as discussed in Schmitt-Grohé and Uribe (2003). This will allow us to pursue a meaningful analysis of the dynamics of the non-linear equations that describe the economy, using a log-linear approximation.¹⁸

2.4  A Perfect Foresight Equilibrium

We are ready to provide a definition of a perfect foresight equilibrium in this economy.

Definition 1   Given the initial condition $b_{-1}^{\ast},$ the steady-state level of foreign debt $\bar{b}^{\ast},$ and the depreciation target $\bar{\epsilon},$ a symmetric perfect foresight equilibrium is defined as a set of sequences $\{c_{t}^{T},$ $c_{t}^{N},$ $h_{t}^{T},$ $h_{t}^{N},$ $b_{t}^{\ast},$ $\epsilon_{t},$ $\pi_{t}^{N},\ R_{t},$ $R_{t}^{\ast}\}_{t=0}^{\infty}$ satisfying: a) the market clearing conditions for the non-traded and traded goods, (8) and (14), b) the UIP condition (9), c) the intratemporal efficient condition (10), d) the Euler equations for consumption of traded and non-traded goods, (11) and (12 ), e) the augmented Phillips curve, (13), f) the interest rate policy (2) and g) the ad-hoc upward-sloping supply curve of foreign funds (15).

Since fiscal policy is Ricardian, this definition ignores the budget constraint of the government and its transversality condition. The definition also ignores real money balances. This is because monetary policy is described as an interest rate policy in (2),and real balances enter in the utility function in a separable way. In addition, note that once we solve for $\{c_{t}^{T},$ $c_{t}^{N},$ $h_{t}^{T},$ $h_{t}^{N},$ $b_{t}^{\ast },$ $\epsilon_{t},$ $\pi_{t}^{N},\ R_{t},$ $R_{t}^{\ast}\}_{t=0}^{\infty},$ it is possible to retrieve the set of sequences $\{\lambda_{t},$ $e_{t},$ $m_{t},$ $b_{t},$ $w_{t}^{T},$ $w_{t}^{N},$ $mc_{t}\}_{t=0}^{\infty}$ using (1), (5), and equations (41)-(44), (46), and (48) that are presented in the Appendix.

2.5  The Determinacy of Equilibrium Analysis

In order to pursue the determinacy of equilibrium analysis, we log-linearize the system of equations that describe the dynamics of this economy around a steady state $\{\bar{c}^{T},$ $\bar{c}^{N},$ $\bar{h}^{T},$ $\bar{h}^{N},$ $\bar{b}^{\ast},$ $\bar{\epsilon},$ $\bar{\pi}^{N},\ \bar{R},$ $\bar{R}^{\ast}\}$. In the Appendix we characterize this steady state.

Log-linearizing the equations of Definition 1 around the steady-state yields

$$\hat{R}_{t}=\rho_{\epsilon}\hat{\epsilon}_{t} \rho_{\epsilon}\neq0 \left\vert \rho_{\epsilon}\right\vert >1 \left\vert \rho_{\epsilon}\right\vert <1,$$ (16)

$$\hat{R}_{t}=\hat{R}_{t}^{\ast}+\hat{\epsilon}_{t+1}, \hat{R}_{t}^{\ast}=\psi\hat{b}_{t}^{\ast},$$ (17)

$$\hat{c}_{t}^{N} =\hat{c}_{t+1}^{N}- \xi^{N}\left( \hat{R}_{t}-\hat {\pi}_{t+1}^{N}\right) ,$$ (18)

$$\hat{\pi}_{t}^{N}=\beta\hat{\pi}_{t+1}^{N}+\beta\varphi \hat{c}_{t} ^{N},$$ (19)

$$\hat{b}_{t}^{\ast}=\left( \frac{1+\psi}{\beta}\right) \hat{b}_{t-1}^{\ast }+\kappa \hat{c}_{t}^{T},$$ (20)

$$\hat{c}_{t}^{T} =\hat{c}_{t+1}^{T} -\xi^{T}\left( \hat{R}_{t}-\xi ^{T}\hat{\epsilon}_{t+1}\right) ,$$ (21)

where $\hat{x}_{t}=\log\left( \frac{x_{t}}{\bar{x}}\right)$ and

$$\xi^{T}=-\frac{U_{T}}{U_{TT}\bar{c}^{T}}>0,\qquad\xi^{N}=-\frac{V_{N}} {V_{NN}\bar{c}^{N}}>0,\qquad\sigma^{T}=\frac{H_{T}}{H_{TT}\bar{h}^{T} }>0,\qquad\sigma^{N}=\frac{L_{N}}{L_{NN}\bar{h}^{N}}>0,$$

$$\omega^{T}=-\frac{F_{T}}{F_{TT}\bar{h}^{T}}>0,\qquad\omega^{N}=-\frac{G_{N} }{G_{NN}\bar{h}^{N}}>0,$$ (22)

$$\kappa=\frac{1}{\bar{b}^{\ast}}\left[ \bar{c}^{T}+\frac{F_{T}\bar{h} ^{T}\sigma^{T}\omega^{T}}{\left( \sigma^{T}+\omega^{T}\right) \xi^{T} }\right] >0,\text{ \ and \ \ }\varphi=\left[ \frac{(\mu-1)\bar{c}^{N}} {\beta\gamma\bar{\epsilon}^{2}}\right] \left[ \frac{\bar{c}^{N}(\sigma ^{N}+\omega^{N})}{G_{N}\bar{h}^{N}\sigma^{N}\omega^{N}}+\frac{1}{\xi^{N} }\right] >0,$$

whose signs are derived using Assumptions 1, 2 and 3. Equations (16 )-(21) correspond to the reduced log-linear representations of the interest rate policy, the UIP condition, the Euler equation for consumption of the non-traded good, the augmented Phillips curve, the current account equation, and the Euler equation for consumption of the traded good, respectively.

Our main goal is to show that the policy in (16) is prone to induce multiple equilibria in the economy described by equations (17 )-(21). Proving this implies that this policy can cause fluctuations in the economy that are driven by people's self-fulfilling beliefs rather than fundamentals. In fact, before we provide a formal proof, it is worth developing a simple intuition. To do so, it is sufficient to concentrate on equations (16)-(19). Note that given the international interest rate, $\hat{R}_{t}^{\ast},$ then the policy (16) and the UIP condition (17) determine the dynamics of the depreciation rate, $\hat{\epsilon}_{t},$ and the nominal interest rate, $\hat{R}_{t}.$ Moreover, the nominal interest rate, $\hat{R}_{t},$ is not affected by either the non-traded good inflation, $\hat{\pi}_{t}^{N},\$or the consumption of the non-traded good, $c_{t}^{N}$. Taking this into account, we can construct the following self-fulfilling equilibrium.

Assume that agents, in response to a "sunspot," expect a higher non-traded good inflation in the next period. Since the interest rate policy does not react to these expectations, then the real interest rate measured with respect to the expected non-traded good inflation, $\hat{R}_{t}-\hat{\pi}_{t+1}^{N},$ declines.¹⁹ In response, households increase consumption of non-traded goods - see (18) - which leads firms to raise their prices, inducing a higher non-traded inflation - see (19). But by doing this, firms end validating the original non-traded inflation expectations.

This simple intuition is appealing but incomplete, unless we show that all the equilibrium conditions (16)-(21) are satisfied on the entire equilibrium path. In other words, we need to characterize formally the equilibrium of this economy. To accomplish this, we start by writing equations (16)-(21) as

$$\left( \begin{array}[c]{c} \hat{\epsilon}_{t+1}\\ \hat{b}_{t}^{\ast}\\ \text{$\hat{c}_{t+1}^{T}$}\\ \hat{\pi}_{t+1}^{N}\\ \text{$\hat{c}_{t+1}^{N}$} \end{array} \right) =\underset{J^{c}}{\underbrace{\left( \begin{array}[c]{ccccc} \rho_{\epsilon} & -\frac{\psi\left( 1+\psi\right) }{\beta} & -\psi\kappa & 0 & 0\\ 0 & \frac{1+\psi}{\beta} & \kappa & 0 & 0\\ 0 & \frac{\psi\left( 1+\psi\right) \xi^{T}}{\beta} & \left( 1+\psi\kappa \xi^{T}\right) & 0 & 0\\ 0 & 0 & 0 & \frac{1}{\beta} & -\varphi\\ \rho_{\epsilon}\xi^{N} & 0 & 0 & -\frac{\xi^{N}}{\beta} & \left( 1+\varphi \xi^{N}\right) \end{array} \right) }}\left( \begin{array}[c]{c} \hat{\epsilon}_{t}\\ \hat{b}_{t-1}^{\ast}\\ \text{$\hat{c}_{t}^{T}$}\\ \hat{\pi}_{t}^{N}\\ \text{$\hat{c}_{t}^{N}$} \end{array} \right) .$$ (23)

Then, we use this system to find and to compare the dimension of its unstable subspace with the number of non-predetermined variables.²⁰ If the dimension of this subspace is smaller than the number of non-predetermined variables, then, from the results by Blanchard and Kahn (1980), we can infer that there exist multiple perfect foresight equilibria. This forms the basis for the existence of self-fulfilling fluctuations.

The following Proposition states the main result of the determinacy of equilibrium analysis: an interest rate policy that raises or lowers the nominal interest rate in response to current currency depreciation leads to multiple perfect foresight equilibria or, equivalently, to real indeterminacy.

Proposition 1   If the government follows an interest rate policy that responds to currency depreciation such as $\hat{R}_{t}=\rho_{\epsilon}\hat{\epsilon }_{t}$ with $\rho_{\epsilon}\neq0$ and either $\left\vert \rho_{\epsilon }\right\vert >1$ or $\left\vert \rho_{\epsilon}\right\vert <1,$ then there exists a continuum of perfect foresight equilibria (indeterminacy) in which the sequences $\{\hat{\epsilon}_{t},$ $\hat{b}_{t}^{\ast},\ \hat{c}_{t}^{T},$ $\hat{\pi}_{t}^{N},\ \hat{c}_{t}^{N}\}_{t=0}^{\infty}$ converge asymptotically to the steady state. In addition,

a) if $\left\vert \rho_{\epsilon}\right\vert >1$ then the degree of indeterminacy is of order 1.²¹

b) if $\left\vert \rho_{\epsilon}\right\vert <1$ then the degree of indeterminacy is of order 2.

Proof. The eigenvalues of the matrix $J^{c}$ in (23) correspond to the roots of the characteristic equation $\mathcal{P}^{c}(v)=\left\vert J^{c}-vI\right\vert =0.$ Using the definition of $J^{c}$ in (23), this equation can be written as

$$\mathcal{P}^{c}(v)=\left( v-\rho_{\epsilon}\right) \mathcal{P}^{f}(v)=0,$$ (24)

where

$$\mathcal{P}^{f}(v)=\left[ v^{2}-\left( 1+\frac{1+\psi}{\beta}+\psi\kappa \xi^{T}\right) v+\frac{1+\psi}{\beta}\right] \left[ v^{2}-\left( 1+\frac{1}{\beta}+\varphi\xi^{N}\right) v+\frac{1}{\beta}\right] .$$

By Lemma 4 in the Appendix, $\mathcal{P}^{f}(v)=0$ has real roots satisfying $\left\vert v_{1}\right\vert <1,$ $\left\vert v_{2}\right\vert >1$, $\left\vert v_{3}\right\vert <1,$ and $\left\vert v_{4}\right\vert >1.$ The fifth root of $\mathcal{P}^{c}(v)=0$ is $v_{5}=\rho_{\epsilon}.$ Clearly, if $\left\vert \rho_{\epsilon}\right\vert >1$ then $\left\vert v_{5}\right\vert >1,$ whereas if $\left\vert \rho_{\epsilon}\right\vert <1$ then $\left\vert v_{5}\right\vert <1.$ Therefore, using this, the characterization of the roots of $\mathcal{P}^{f}(v)=0,$ and (24), we can conclude the following. If $\left\vert \rho_{\epsilon}\right\vert >1$ then $\mathcal{P} ^{c}(v)=0$ has three explosive roots, namely $\left\vert v_{2}\right\vert >1$, $\left\vert v_{4}\right\vert >1$ and $\left\vert v_{5}\right\vert >1.$ While if $\left\vert \rho_{\epsilon}\right\vert <1$ then $\mathcal{P}^{c}(v)=0$ has two explosive roots, namely $\left\vert v_{2}\right\vert >1$ and $\left\vert v_{4}\right\vert >1.$ Hence, regardless of whether $\left\vert \rho_{\epsilon }\right\vert >1$ or $\left\vert \rho_{\epsilon}\right\vert <1,$ there are at most three explosive roots. Given that there are four non-predetermined variables, $\hat{\epsilon}_{t}$, $\hat{c}_{t}^{T}$, $\hat{\pi}_{t}^{N}$ and $\hat{c}_{t}^{N}$, then the number of non-predetermined variables is greater than the number of explosive roots. Applying the results of Blanchard and Kahn (1980), it follows that there exists an infinite number of perfect foresight equilibria converging to the steady state. Finally, parts a) and b) follow from the difference between the number of non-predetermined variables and the number of explosive roots, when $\left\vert \rho_{\epsilon}\right\vert >1$ and $\left\vert \rho_{\epsilon}\right\vert <1,$ respectively. $\qedsymbol$

Proposition 1 has two important implications. First, provided that the fiscal policy is Ricardian, then the interest rate policy induces not only real indeterminacy but also nominal indeterminacy. That is, the nominal exchange rate is not pinned-down.²² This follows from the definition of the nominal depreciation rate, and the fact that the nominal exchange rate is also a non-predetermined variable. Second, and in contrast to the intuition provided above, Proposition 1 suggests that it is possible to construct self-fulfilling equilibria that are based on expectations of a different variable from the non-traded inflation. For instance, we can construct a self-fulfilling equilibrium driven by people's beliefs about currency depreciation. We will pursue this exercise in Section 4.

Table 1. Determinacy of Equilibrium Analysis - Degree of Responsiveness of Interest Rate Policy

	The Simple Model - $\left\vert \rho_{\epsilon}\right\vert <1$	The Simple Model - $\left\vert \rho _{\epsilon}\right\vert >1$	The Full Model - $\left\vert \rho _{\epsilon}\right\vert <1$	The Full Model - $\left\vert \rho_{\epsilon }\right\vert >1$
Forward-Looking $\hat{R}_{t}=\rho_{\epsilon}\hat{\epsilon}_{t+1}$with $\rho_{\epsilon}\neq0$	M	M	M	M
Contemporaneous $\hat{R}_{t}=\rho_{\epsilon}\hat{\epsilon}_{t}$ with $\rho _{\epsilon}\neq0$	M	M	M	M
Backward-Looking $\hat{R}_{t}=\rho_{\epsilon}\hat{\epsilon}_{t-1}$ with $\rho_{\epsilon}\neq0$	M	U	M	M or U

Note: $M$refers to multiple equilibria and $U$refers to a unique equilibrium.

The results of Proposition 1 also pose the question of whether policies that respond exclusively to either the future depreciation rate, a forward-looking policy $\hat{R}_{t}=\rho_{\epsilon}\hat{\epsilon}_{t+1},$ or to the past depreciation rate, a backward-looking policy $\hat{R}_{t} =\rho_{\epsilon}\hat{\epsilon}_{t-1}$, can induce multiple equilibria. The answer is yes and the analysis is provided in the Appendix. In particular, we find that forward-looking policies can still lead to multiple equilibria when the interest rate response coefficient to future depreciation satisfies either $\left\vert \rho_{\epsilon}\right\vert >1$ or $\left\vert \rho_{\epsilon }\right\vert <1.$ On the contrary, backward-looking policies that are very aggressive with respect to past depreciation, and satisfy $\left\vert \rho_{\epsilon}\right\vert >1,$ will guarantee a unique equilibrium; whereas timid policies that satisfy $\left\vert \rho_{\epsilon}\right\vert <1$ can lead to real indeterminacy. These results, as well as the results from Proposition 1, are presented in Table 1 in the columns labeled as "The Simple Model." ²³

Note that the assumption of zero labor mobility is not crucial for the results of Proposition 1. In fact, it is possible to obtain similar analytical results under perfect labor mobility and some extra assumptions about the disutility of working.²⁴ Then, what are the features of the model that drive these results? The crucial features are the following: the description of monetary policy as an interest rate feedback rule, non-traded goods price-stickiness, and the exclusive dependence of the policy on currency depreciation. By Sargent and Wallace (1975), we know that the first feature by itself leads to nominal indeterminacy of the exchange rate level (price level), in a flexible price model under a Ricardian fiscal policy. The second characteristic together with the policy elucidate why nominal indeterminacy turns into real indeterminacy. And finally, the first two features, in tandem with the exclusive response of the policy to depreciation, are what explains why multiple equilibria arise, to some extent, regardless of the degree of responsiveness of the policy.²⁵

This simple model suffers from at least two drawbacks. On one hand, there is no feature that captures the fact that the economy is in a crisis. On the other hand, the dynamics of non-traded consumption and inflation, that are supported as a self-fulfilling equilibrium, are completely at odds with the stylized facts of a "Sudden Stop." In particular, the self-fulfilling equilibrium that we constructed above implies that non-traded consumption and inflation are positively correlated. In contrast, the stylized facts suggest that there was a strong decline in consumption accompanied by an increase in inflation. To improve the model, in the next section, we will add some extra features.

3  The Full Model

In this section we enrich the simple model in several dimensions. First, in accord with Burnstein, Eichenbaum, and Rebelo (2005a,b), we introduce non-traded distribution services. This together with price stickiness are crucial to explain the large movements in real exchange rates after large devaluations. Second, following Lahiri and Vegh (2002) and Christiano, Gust, and Roldos (2004) we assume that the household-firm unit requires domestic and international loans to hire labor and to purchase an imported intermediate input, respectively. This is important to obtain a decline in output and demand in the midst of the crisis, when interest rates rise. Third, as in the new literature about currency crises, we introduce a collateral constraint. That is, international loans must be guaranteed by physical assets such as capital. This will help to provide a definition of a crisis. Fourth, we consider a utility function that is not separable in the two types of consumption. We proceed to explain how we introduce these features and their influence in the previous equations.

3.1  The Additional Features

As in Burnstein et al. (2003), we assume that the traded good needs to be combined with some non-traded distribution services before it is consumed. In order to consume one unit of the traded good, it is required $\eta$ units of the basket of differentiated non-traded goods. Let $\breve {P}_{t}^{T},$ $P_{t}^{T}$ and $P_{t}^{N}$ be the producer's price of the traded good, the consumer's price of the traded good, and the general price level of the basket of differentiated non-traded goods, respectively. All of them are expressed in domestic currency. Hence the consumer's price is simply $P_{t}^{T}=\breve{P}_{t}^{T}+\eta P_{t}^{N}.$ And since PPP holds at the production level of the traded good ( $\breve{P}_{t}^{T}=\mathcal{E}_{t} \breve{P}_{t}^{T\ast}$), and the foreign price of the traded good is normalized to one ( $\breve{P}_{t}^{T\ast}=1$), we have that $P_{t} ^{T}=\mathcal{E}_{t}+\eta P_{t}^{N}.$

The production of the differentiated non-traded good is still demand determined by

$$G\left( \tilde{h}_{t}^{N},K^{N}\right) \geq C_{t}^{N}d\left( \frac {\tilde{P}_{t}^{N}}{P_{t}^{N}}\right) +\eta C_{t}^{T}d\left( \frac{\tilde {P}_{t}^{N}}{P_{t}^{N}}\right) ,$$ (25)

where $d\left( 1\right) =1,$ $d^{\prime}\left( 1\right) =-\mu,$ $C_{t} ^{N}$ denotes the level of aggregate demand for the non-traded good, $\tilde{P}_{t}^{N}$ is the price of the intermediate non-traded good set by the household-firm unit, and $C_{t}^{T}$ corresponds to the level of aggregate consumption of the traded good. But now the demand requirements come from two sources.²⁶They come from consumption of non-traded goods $C_{t}^{N}d\left( \frac {\tilde{P}_{t}^{N}}{P_{t}^{N}}\right)$ that provide utility, and from non-traded distribution services $\eta C_{t}^{T}d\left( \frac{\tilde{P} _{t}^{N}}{P_{t}^{N}}\right)$ that are necessary to bring one unit of the traded good to the household-firm unit. Note that there is no difference between non-traded consumption goods and non-traded distribution services. As a consequence, in equilibrium the basket of non-traded goods required to distribute traded goods will have the same composition as the non-traded basket consumed by the household-firm unit.

The introduction of the loan requirements and the collateral constraint in the model follows Christiano et al. (2004). The household-firm unit requires domestic loans to hire labor ( $\check{h}_{t}^{T}$ and $\tilde{h}_{t}^{N}$), and international loans to buy an imported input ($I_{t}$), that will be used in the production of the traded good. These loans are obtained at the beginning of the period and repaid at the end of the period. In this sense, they represent short-term debt and differ from long-term foreign debt $b_{t-1}^{\ast}.$ We do not model, however, the financial institutions that provide these loans. Instead, we assume that the domestic loans are provided by the government, whereas foreign loans are supplied by foreign creditors.²⁷ For these loans the unit pays interests $(R_{t}-1)W_{t} (\check{h}_{t}^{T}+\tilde{h}_{t}^{N})$ and ( $R_{t}^{\ast}-1)\tilde{P}_{t} ^{T}I_{t},$ that are accrued between periods, where $R_{t}$ is the domestic nominal interest rate, and $R^{\ast}$ is the international interest rate. The latter is assumed to be constant and equal to $\frac{1}{\beta}$.

The production technology of the traded good uses labor ( $\breve{h}_{t}^{T}$), an imported input ($I_{t}$), and capital ($K^{T}$); whereas the technology for the non-traded intermediate differentiated goods only requires labor ( $\tilde{h}_{t}^{N}$) and capital ($K^{N}$). That is,

$$y_{t}^{T}=F\left( \breve{h}_{t}^{T},I_{t},K^{T}\right)$$     and  $$\tilde{y}_{t}^{N}=G\left( \tilde{h}_{t}^{N},K^{N}\right) .$$

Furthermore, as in Christiano et al. (2004) and Mendoza and Smith (2002), among others, capital is assumed to be time-invariant, does not depreciate, and there is no technology to making it bigger. Under these new features, the dividends that the household-firm unit receives can be written as

$$\Omega_{t} =F\left( \breve{h}_{t}^{T},I_{t},K^{T}\right) +\frac{1} {e_{t}}\left[ \frac{\tilde{P}_{t}^{N}}{P_{t}^{N}}G\left( \tilde{h}_{t} ^{N},K^{N}\right) -\frac{\gamma}{2}\left( \frac{\tilde{P}_{t}^{N}}{\tilde {P}_{t-1}^{N}}-\bar{\pi}\right) ^{2}\right]$$ (26)

$$-w_{t}^{T}R_{t}\check{h}_{t}^{T}-w_{t}^{N}R_{t}\tilde{h}_{t}^{N}-R^{\ast }I_{t}-R^{\ast}b_{t-1}^{\ast}+b_{t}^{\ast},$$

where $w_{t}^{i}=\frac{W_{t}^{i}}{\mathcal{E}_{t}}$ with $i=T,N,$ and $e_{t}=\frac{\mathcal{E}_{t}}{P_{t}^{N}}.$

To model the crisis, we follow Christiano et al. (2004) by imposing the following collateral constraint:

$$R^{\ast}b_{t-1}^{\ast}+R^{\ast}I_{t}+R_{t}\left( w_{t}^{T}\check{h}_{t} ^{T}+w_{t}^{N}\tilde{h}_{t}^{N}\right) \leq\phi\left( q_{t}^{N}K^{N} +q_{t}^{T}K^{T}\right) ,$$ (27)

where $q_{t}^{N}$ and $q_{t}^{T}$ represent the real value, in units of foreign currency, of one unit of capital for the production of the non-traded and traded goods, respectively, and $\phi$ is the fraction of these stocks that foreign creditors accept as collateral. The constraint (27) says that the total value of foreign and domestic debt, that the representative household-firm unit has to pay to completely eliminate the debt of the firm by the end of period $t,$ cannot exceed the value of the collateral. The crisis makes this constraint unexpectedly binding in every period henceforth without the possibility of being removed.

Finally, we relax the assumption in (3) of a separable utility function between the two types of consumption, $U(c_{t}^{T})+V(c_{t}^{N})$. But we will still assume separability among consumption, labor, and real money balances. We proceed to study how all these features influence the previous optimal conditions.

3.2  The New Equilibrium Conditions

The problem that the household firm unit has to solve is similar to the one presented in the simple model. The agent chooses the set of sequences $\{c_{t}^{T},$ $c_{t}^{N},$ $h_{t}^{T},$ $h_{t}^{N},$ $\check{h} _{t}^{T},$ $\tilde{h}_{t}^{N},$ $I_{t},$ $\tilde{P}_{t}^{N},$ $b_{t}^{\ast},$ $b_{t},$ $m_{t}\}_{t=0}^{\infty}$ in order to maximize

$$\overset{\infty}{\sum_{t=0}}\beta^{t}\left[ \tilde{U}(c_{t}^{T},c_{t} ^{N})+H(h_{t}^{T})+L(h_{t}^{N})+J(m_{t})\right]$$

subject to the budget constraint

$$m_{t}+b_{t}\leq\frac{m_{t-1}}{\epsilon_{t}}+\frac{R_{t-1}b_{t-1}}{\epsilon _{t}}+w_{t}^{T}h_{t}^{T}+w_{t}^{N}h_{t}^{N}+\tau_{t}+\Omega_{t}-\left( 1+\frac{\eta}{e_{t}}\right) c_{t}^{T}-\frac{c_{t}^{N}}{e_{t}}$$

and the constraints (7), (25), (26), and (27), given the initial conditions $b_{-1}^{\ast},$ $b_{-1},$ and $m_{-1}$ and the set of sequences { $R_{t}^{\ast}$, $R_{t}$, $\ \epsilon _{t},$ $e_{t},$ $P_{t}^{N},$ $w_{t}^{T},$ $w_{t}^{N},$ $\tau_{t}$, $C_{t} ^{N},$ $C_{t}^{T}\}.$ The optimization conditions together with symmetry and market clearing conditions can be used to find the laws of motion of the economy. They correspond to (1), (2), (27) with equality,

$$G\left( h_{t}^{N},K^{N}\right) =c_{t}^{N}+\frac{\gamma}{2}\left( \pi _{t}^{N}-\bar{\pi}\right) ^{2}+\eta c_{t}^{T},$$ (28)

$$R_{t}=R^{\ast}(1+\zeta_{t+1})\epsilon_{t+1},$$ (29)

$$-\frac{H_{T}(h_{t}^{T})}{\tilde{U}_{T}(c_{t}^{T},c_{t}^{N})}=\frac{w_{t}^{T} }{1+\frac{\eta}{e_{t}}},$$ (30)

$$\tilde{U}_{T}(c_{t}^{T},c_{t}^{N})=\frac{\beta R_{t}}{\pi_{t+1}^{T}}\tilde {U}_{T}(c_{t+1}^{T},c_{t+1}^{N}), \pi_{t+1}^{T} =\epsilon_{t+1}\frac{\left( 1+\frac{\eta}{e_{t+1}}\right) }{\left( 1+\frac{\eta}{e_{t}}\right) },$$ (31)

$$\tilde{U}_{N}(c_{t}^{T},c_{t}^{N})=\frac{\beta R_{t}}{\pi_{t+1}^{N}}\tilde {U}_{N}(c_{t+1}^{T},c_{t+1}^{N}),$$ (32)

$$\frac{\tilde{U}_{N}(c_{t+1}^{T},c_{t+1}^{N})(\pi_{t+1}^{N}-\bar{\pi}^{N} )\pi_{t+1}^{N}}{\tilde{U}_{N}(c_{t}^{T},c_{t}^{N})}=\frac{(\pi_{t}^{N} -\bar{\pi}^{N})\pi_{t}^{N}}{\beta}+\frac{\mu\left( c_{t}^{N}+\eta c_{t} ^{T}\right) }{\beta\gamma}\left( \frac{\mu-1}{\mu}-mc_{t}\right) ,$$ (33)

$$b_{t}^{\ast}-b_{t-1}^{\ast}=(R^{\ast}-1)b_{t-1}^{\ast}+R^{\ast}I_{t}+c_{t} ^{T}-F\left( h_{t}^{T},I_{t},K^{T}\right) ,$$ (34)

$$F_{T}\left( h_{t}^{T},K^{T},I_{t}\right) =w_{t}^{T}(1+\zeta_{t})R_{t},$$ (35)

$$mc_{t}=\frac{w_{t}^{N}e_{t}(1+\zeta_{t})R_{t}}{G_{N}\left( h_{t}^{N} ,K^{N}\right) },$$ (36)

$$F_{I}\left( h_{t}^{T},K^{T},I_{t}\right) =(1+\zeta_{t})R^{\ast},$$ (37)

$$-\frac{L_{N}(h_{t}^{N})}{\tilde{U}_{N}(c_{t}^{T},c_{t}^{N})}=w_{t}^{N}e_{t},$$ (38)

where $\lambda_{t}\zeta_{t}$ and $\lambda_{t}$ are the Lagrange multipliers of the collateral constraint and the budget constraint. The latter multiplier evolves according to the asset pricing equation

$$\lambda_{t}=\beta R^{\ast}(1+\zeta_{t+1})\lambda_{t+1}.$$ (39)

Equations (28)-(34) are equivalent to equations (8)-(14) in the simple model.²⁸ Therefore they have a similar interpretation. Equations (35)-(37) correspond to the optimal conditions that determine the demands for labor (for the production of the traded and non-traded good) and for the imported input; and (38) is the intratemporal condition that makes the marginal rate of substitution between labor, assigned to the production of the non-traded good, and consumption of the non-traded good equal to the real salary measured in units of the non-traded good.

The new laws of motion reveal that the additional features have some important consequences. On one hand, distribution services affect the relative price of the traded good at the consumer level with respect to the nominal exchange rate. In the simple model this relative price was equal to one. In the full model this price is $1+\frac{\eta}{e_{t}},$ which depends on the distribution costs parameter $\eta.$ From (30) and (31), it is clear that through this price, distribution costs affect, in equilibrium, the optimal intratemporal decisions between labor and consumption of the traded good, as well as the optimal intertemporal choices for consumption of the traded good. On the other hand, distribution services generate an extra demand of non-traded goods, as is captured by the last term, $\eta c_{t}^{T}$, of the right hand side of (28). This extra demand also influences the dynamics of non-traded goods inflation, as can be seen in (33).

The binding collateral constraint generates an endogenous risk premium, as reflected by the modified UIP condition in (29). In fact, because of this constraint, the effective international interest rate that domestic agents pay becomes $(1+\zeta_{t+1})R^{\ast}.$ Thus, raising external debt $b_{t}^{\ast}$ not only requires the payment of interests ( $R^{\ast} b_{t}^{\ast}$), but also tightens the binding constraint ( $\zeta_{t+1}>0$). This generates an additional interest cost.²⁹

The requirement of loans to hire labor and the binding collateral constraint affect the labor demand decisions, as can be inferred from (35) and (36). Keeping the rest constant, these two features and the fact that in the short run $\zeta_{t}>0$ imply the following. Increases in the effective interest rate, $(1+\zeta_{t})R_{t},$ push the cost of hiring labor up, which in turn discourages the demands for labor for the production of the traded and non-traded goods.

The optimal condition for demand of the imported input, equation (37), equalizes the marginal product of this input to the effective cost of foreign working capital, $(1+\zeta_{t})R^{\ast},$ that is necessary to import it. As the constraint tightens and $\zeta_{t}$ goes up, the effective cost raises and the demand for the imported input decreases. Furthermore, the purchases of this input influence the current account equation (34). A decrease in the imported input can cause an improvement in the current account deficit. This improvement is almost immediate, given the assumption that external short term loans to finance the intermediate input have to be repaid at the end of the period, and not at the beginning of next period.

Finally, it is possible to derive the equilibrium value of the prices of capital. The equilibrium value of these asset prices are described by

$$q_{t}^{T}=\frac{F_{K}+\left( \frac{R_{t}}{\epsilon_{t+1}}\right) ^{-1}q_{t+1}^{T}}{(1-\phi\zeta_{t})} \qquad q_{t}^{N} =\frac{(\frac{mc_{t}}{e_{t}})G_{K}+\left( \frac{R_{t}}{\epsilon_{t+1} }\right) ^{-1}q_{t+1}^{N}}{(1-\phi\zeta_{t})},$$ (40)

where $F_{K}$ and $G_{K}$ are the marginal products of capital in the production of the traded and non-traded goods, respectively.

3.3  The Determinacy of Equilibrium Analysis

Using this set-up, the definition of equilibrium under a currency crisis is the following.

Definition 2   Given $b_{-1}^{\ast},$ $R^{\ast}$, $K^{N},$ $K^{T}$ and the depreciation target $\bar{\epsilon},$ a perfect foresight equilibrium is defined as a set of sequences $\{c_{t}^{T},$ $c_{t}^{N},$ $\zeta_{t},$ $h_{t}^{T},$ $h_{t}^{N},$ $I_{t}$, $b_{t}^{\ast},$ $mc_{t},$ $e_{t},$ $q_{t}^{T},$ $q_{t}^{N},$ $w_{t}^{T},$ $w_{t}^{N}$, $\epsilon_{t},$ $\pi_{t}^{N},\ R_{t}\}_{t=0}^{\infty}$ satisfying equations (1), (2), the collateral constraint (27) as an equality, (28)-(37) and (40).

Note that, as in Christiano et al. (2004) among others, we model the crisis as a collateral constraint that binds in every period. Nevertheless, at the steady state, the shadow price of the collateral constraint is equal to zero. To see this use (39) in tandem with $\beta=\frac{1}{R^{\ast}}$ to obtain $\bar{\zeta}=0.$ This implies that the collateral constraint is marginally not binding at the steady state. Hence the credit restrictions disappear marginally. In contrast, in the short run the shadow price of the collateral constraint may vary as $\zeta_{t}$ changes. When $\zeta_{t}>0$ is high then the collateral constraint tightens.

As before, to pursue the determinacy of equilibrium analysis, we log-linearize the system of equations around the perfect-foresight steady state. Then we characterize the dimension of the unstable subspace of the system and compare it to the number of non-predetermined variables. By log-linearizing the system, we are following the same approach that Kiyotaki and Moore (1997) adopt to solve for an equilibrium of a model with a binding collateral constraint. This precludes the possibility of exploring non-linear equilibrium dynamics.³⁰Moreover, since it is not possible to derive analytical results, we have to rely on numerical simulations. We use the following functional forms. For consumption and labor pref