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Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 924, March 2008--- Screen Reader Version*

A Solution to the Default Risk-Business Cycle Disconnect^*

Enrique G. Mendoza, University of Maryland and NBER
Vivian Z. Yue, New York University

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publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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:

Models of business cycles in emerging economies explain the negative correlation between country spreads and output by modeling default risk as an exogenous interest rate on working capital. Models of strategic default explain the cyclical properties of sovereign spreads by assuming an exogenous output cost of default with special features, and they underestimate debt-output ratios by a wide margin. This paper proposes a solution to this default risk-business cycle disconnect based on a model of sovereign default with endogenous output dynamics. The model replicates observed V-shaped output dynamics around default episodes, countercyclical sovereign spreads, and high debt ratios, and it also matches the variability of consumption and the countercyclical fluctuations of net exports. Three features of the model are key for these results: (1) working capital loans pay for imported inputs; (2) imported inputs support more efficient factor allocations than when these inputs are produced internally; and (3) default on the foreign obligations of firms and the government occurs simultaneously.

Keywords: Business cycles, sovereign default, emerging economies

JEL classification: E32, E44, F32, F34

1  Introduction

Three key empirical regularities characterize the relationship between sovereign debt and economic activity in emerging economies:

(1) Output displays V-shaped dynamics around default episodes. Recent default episodes have been associated with deep recessions. Arellano (2007) shows that GDP deviations from trend in the quarter in which default occurred were -14 percent in Argentina, -13 percent in Russia and -7 percent in Ecuador. Using quarterly data for 39 developing countries over the 1970-2005 period, Levy-Yeyati and Panizza (2006) show that the recessions associated with defaults tend to begin prior to the defaults and generally hit bottom when the defaults take place. Tomz and Wright's (2007) study of the history of defaults for industrial and developing countries during the period 1820-2004 reports that the frequency of defaults is at its maximum when output is at least 7 percent below trend. They also found, however, that some defaults occurred with less severe recessions, or when output is not below trend in annual data.

(2) Interest rates on sovereign debt and domestic output are negatively correlated. Neumeyer and Perri (2005) report that the cyclical correlations between these interest rates and GDP range from -0.38 to -0.7 in five emerging economies, with an average correlation of -0.55. Uribe and Yue (2006) report correlations for seven emerging economies ranging from zero to -0.8, with an average of -0.42.¹

(3) External debt as a share of GDP is high on average, and high when countries default. Foreign debt was about a third of GDP on average over the 1998-2005 period for the entire group of emerging and developing countries as defined in IMF (2006). Within this group, the highly indebted poor countries had the highest average debt ratio at about 100 percent of GDP, followed by the Eastern European and Western Hemisphere countries, with averages of about 50 and 40 percent of GDP respectively. Reinhart et al. (2003) report that the external debt ratio during default episodes averaged 71 percent of GDP for all developing countries that defaulted at least once in the 1824-1999 period. The default episodes of recent years are in line with this estimate: Argentina defaulted in 2001 with a 64 percent debt ratio, and Ecuador and Russia defaulted in 1998 with debt ratios of 85 and 66 percent of GDP respectively.

These empirical regularities have proven difficult to explain. On one hand, quantitative business cycle models can account for the negative correlation between country interest rates and output if the interest rate on sovereign debt is introduced as the exogenous interest rate faced by a small open economy in which firms require working capital to pay the wages bill.² On the other hand, quantitative models of sovereign default based on the classic setup of Eaton and Gersovitz (1981) can generate countercyclical sovereign spreads if the sovereign country faces stochastic shocks to an exogenous output endowment.³ These models require exogenous output costs of default with special features in order to support non-trivial levels of debt together with observed default frequencies, but even with these costs they either produce mean debt ratios under 10 percent of GDP or underestimate default probabilities by a wide margin.⁴ Thus, there is a crucial disconnect between business cycle models and sovereign default models: the former lack an explanation of the default risk premia that drive their findings, while the latter lack an explanation of the business cycle dynamics that are critical for their results.

The country risk-business cycle disconnect raises three important questions: Would a business cycle model with endogenous default risk still be able to explain the stylized facts that models with exogenous country risk have explained? Can a model of sovereign default with endogenous output dynamics produce the large output declines needed to support high ratios of defaultable debt as an equilibrium outcome? Would a model that endogenizes both country risk and output dynamics be able to mimic the V-shaped dynamics of output associated with defaults, and the countercyclical behavior of default risk?

This paper aims to answer these questions by studying the quantitative implications of a model of sovereign default with endogenous output fluctuations. The model borrows from the sovereign default literature the workhorse Eaton-Gersovitz recursive formulation of strategic default in which a sovereign borrower makes optimal default choices by comparing the payoffs of repayment and default. In addition, the model borrows from the business cycle literature a transmission mechanism that links default risk with economic activity via the financing cost of working capital. We extend the two classes of models (sovereign debt and business cycle models) by developing a framework in which the equilibrium dynamics of output and default risk are determined jointly, and influence each other via the interaction between foreign lenders, the domestic sovereign borrower, and domestic firms. In particular, a fall in productivity in our setup increases the likelihood of default and hence sovereign spreads, and this in turn increases the firms' financing costs leading to a further fall in output, which in turn feeds back into default incentives and sovereign spreads.

We demonstrate via numerical analysis that the model can explain the three key empirical regularities of sovereign debt mentioned earlier: The model mimics the V-shaped pattern of output dynamics around defaults with large recessions that hit bottom during defaults, yields countercyclical interest rates on sovereign debt, and supports high debt-GDP ratios on average and in default episodes. These results are obtained requiring only a small fraction of firms' factor costs to be paid with working capital (only 10 percent of the cost of imported inputs). Moreover, the model matches key business cycle features like the variability of consumption and the countercyclical behavior of net exports.

These results hinge on three key assumptions of the model: First, producers of final goods obtain working capital loans from abroad to finance purchases of imported intermediate goods. Second, these producers can choose optimally to employ domestic intermediate goods instead of imported inputs, but this shift entails an efficiency loss. Third, the government can divert the firms' repayment of working capital loans when it defaults on its own debt, so that both agents default on their foreign obligations at the same time, and the interest rates they face are equal at equilibrium.

The transmission mechanism that connects country risk and business cycles in our model operates as follows: Final goods producers maximize profits and choose optimally whether to use imported inputs or inputs produced in the domestic economy. These two inputs are perfect substitutes in the production technology, but imported inputs have a higher financing cost because they need to be paid in advance using working capital, while domestic inputs require costly reallocation of labor away from final goods production into intermediate goods production. Thus, a shift from imported to domestic inputs causes an efficiency loss in production of final goods due to the reallocation of labor.⁵

The choice of imported v. domestic inputs by final goods producers depends on the country interest rate (inclusive of default risk), which drives the financing cost of working capital, and on the state of total factor productivity (TFP). When the country has access to world financial markets, they choose imported intermediate goods if the country interest rate is low enough and/or TFP is high enough for the efficiency loss from using domestic inputs to exceed the higher financial cost of imported inputs. That is, final goods producers trade off the higher financing cost of imported inputs for the enhanced efficiency in the use of labor services (which are fully allocated to final goods production). In this situation, fluctuations in default risk affect the cost of working capital and thus induce fluctuations in factor demands and output. Conversely, above (below) a threshold value of the interest rate (TFP) firms choose to use domestic inputs because the financing cost of imported inputs exceeds the efficiency loss due to domestic labor reallocation, with labor services now being allocated to both final and intermediate goods production.

When the economy defaults, both the government and firms are excluded from world credit markets for some time, with an exogenous probability of re-entry as is common in the recent quantitative studies of sovereign default. Since the probability of default depends on whether the country's value of default is higher than that of repayment, there is feedback between the economic fluctuations induced by changes in interest rate premia, default probabilities, and country risk. In particular, rising country risk in the periods leading to a default causes a decline in economic activity as the firms' financing cost increases. In turn, the expectation of lower output at higher levels of country risk alters repayment incentives for the sovereign, affecting the equilibrium determination of default risk premia.

The transmission mechanism linking country risk and business cycles generates an endogenous output cost of default that is larger in " better" states of nature (i.e., increasing in the state of TFP). This result follows from the efficiency loss caused by the optimal shift from imported to domestic inputs when default takes place. Since default yields an effective financing cost of working capital loans that is too high for firms to employ foreign inputs, firms always use domestic inputs when the country is in financial autarky. Before default, however, if the interest rate is low enough and/or TFP is high enough, firms operate with imported inputs, and therefore final goods production is higher than in the default scenario, in which final goods producers shift to domestic inputs. Hence, the decline in GDP at the moment of default is higher the higher TFP was just before default, and the fraction of output loss caused by a default increases with TFP. This increasing output cost of default is a key feature of the model because it implies that the option to default brings more " state contingency" into the asset market, allowing the model to produce equilibria that support significantly higher mean debt ratios than those obtained in existing models of sovereign default.

The increasing output cost of default also implies that output can fall sharply when the economy defaults, and that, because this output drop is driven by an efficiency loss due to sectoral labor reallocation, part of the output collapse will appear as a drop in the Solow residual (i.e. the fraction of aggregate GDP not accounted for by capital and labor). This is consistent with the data of emerging economies in crisis showing that a large fraction of the output collapse is attributed to the Solow residual (see Meza and Quintin (2006) and Mendoza (2007)). Moreover, Benjamin and Meza (2007) show that in Korea's 1997 crisis, the productivity drop did follow from a sectoral reallocation of labor from more to less productive sectors.

Our treatment of the financing cost of working capital differs from the treatment in Neumeyer and Perri (2005) and Uribe and Yue (2006), both of which treat the interest rate on working capital as an exogenous variable set to match the interest rate on sovereign debt. In contrast, in our setup both interest rates are driven by endogenous sovereign risk. In addition, in the Neumeyer-Perri and Uribe-Yue models, working capital loans pay the wages bill in full, while in our model firms use working capital to pay only for a fraction of imported intermediate goods. This lower working capital requirement is desirable because, at standard labor income shares, working capital loans would need to be about 2/3rds of GDP to cover the wages bill, and this is difficult to reconcile with observed ratios of bank credit to the private sector as a share of output in emerging economies, which hover around 50 percent (including all credit to households and firms at all maturities, not just short-term revolving loans to firms).

The rest of the paper proceeds as follows: Section 2 presents the model. Section 3 explores the model's quantitative implications for a benchmark calibration. Section 4 conducts sensitivity analysis. Section 4 concludes.

2  A Model of Sovereign Default and Business Cycles

We study a dynamic stochastic general equilibrium model of sovereign default and business cycles. There are four groups of agents in the model, three in the "domestic" small open economy (households, firms, and the sovereign government) and one abroad (foreign lenders).

2.1  Households

Households derive utility from consumption and disutility from labor. Their preferences are given by a standard time-separable utility function $E\left[ {\textstyle\sum\nolimits_{t=0}^{\infty}} \beta^{t}u\left( c_{t}-h(L_{t})\right) \right] ,$ where $0<\beta<1$ is the discount factor, and $c_{t}$ and $L_{t}$ denote consumption and " composite" labor effort supplied in period $t$ respectively. $u(\cdot)$ is the period utility function, which is continuous, strictly increasing, strictly concave, and satisfies the Inada conditions. Following Greenwood, Hercowitz and Huffman (1988), we remove the wealth effect on labor supply by specifying period utility as a function of consumption net of the disutility of labor $h(L_{t})$, where $h(\cdot)$ is increasing, continuously differentiable and convex. This formulation of preferences has been shown to play an important role in allowing international real business cycle models to explain observed business cycle facts, and it also simplifies the supply-side of the model by removing intertemporal considerations from the labor supply choice.

Households choose consumption and sectoral allocations of labor offered to producers of final goods and intermediate goods ( $L_{t}^{f}$ and $L_{t}^{m}$ respectively). These sectoral labor supply allocations aggregate into a composite amount of labor effort represented by a labor transformation curve $\Psi\left( L_{t}^{f},L_{t}^{m}\right)$, where $\Psi(\cdot)$ is a CES aggregator. $L^{f}$ and $L^{m}$ can thus be viewed as efficiency units of labor that households allocate across the two sectors out of a given amount of labor effort $L.$

Households take as given the sectoral wage rates $\left( w_{t}^{f},w_{t} ^{m}\right)$, the profits paid by firms $\left( \pi_{t}^{f},\pi_{t} ^{m}\right)$ and government transfers $\left( T_{t}\right)$. Households do not borrow directly from abroad, but they are still able to smooth consumption because the government borrows, pays transfers, and makes default decisions internalizing their utility function. This assumption implies that the households' optimization problem reduces to the following static problem:

$$\max_{c_{t},L_{t}^{m},L_{t}^{f},L_{t}}E\left[ \sum\beta^{t}u\left( c_{t}-h\left( L_{t}\right) \right) \right]$$ (1)

$$s.t. c_{t} =w_{t}^{f}L_{t}^{f}+w_{t}^{m}L_{t}^{m}+\pi_{t} ^{f}+\pi_{t}^{m}+T_{t}$$ (2)

$$L_{t} =\Psi\left( L_{t}^{m},L_{t}^{f}\right)$$ (3)

The optimality conditions for labor supply are:

$$w_{t}^{f} =h^{\prime}\left( L_{t}\right) \Psi_{L_{f}}^{\prime}\left( L_{t}^{f},L_{t}^{m}\right)$$ (4)

$$w_{t}^{m} =h^{\prime}\left( L_{t}\right) \Psi_{L_{m}}^{\prime}\left( L_{t}^{f},L_{t}^{m}\right)$$ (5)

Hence, optimal sectoral allocations of labor are obtained when the relative wage rates equal the sectoral marginal rate of transformation:

$$\frac{w_{t}^{f}}{w_{t}^{m}}=\frac{\Psi_{L^{f}}^{\prime}(L_{t}^{f},L_{t}^{m} )}{\Psi_{L^{m}}^{\prime}(L_{t}^{f},L_{t}^{m})}$$ (6)

The labor disutility function is defined in isoelastic form $h(L)=\frac {L^{\omega}}{\omega}$ with $\omega>1.$ The period utility function takes the standard constant-relative-risk-aversion form $u\left( c,L\right) =\frac{\left( c-\frac{L^{\omega}}{\omega}\right) ^{1-\sigma}-1}{1-\sigma}$ with $\sigma>0$. The labor transformation curve is given by $\Psi\left( L_{t}^{f},L_{t}^{m}\right) =[(L_{t}^{f})^{\upsilon}+(L_{t}^{m})^{\upsilon }]^{1/\upsilon}$ with $0\leq\upsilon\leq1$. $\upsilon=1$ implies costless reallocation of homogenous labor, $L_{t}=L_{t}^{f}+L_{t}^{m},$ and $\upsilon=0$ implies that the cost of reallocating labor across sectors is infinite. With these functional forms, the optimality condition for sectoral labor supply allocations reduces to:

$$\frac{w_{t}^{f}}{w_{t}^{m}}=\left( \frac{L_{t}^{f}}{L_{t}^{m}}\right) ^{\upsilon-1}$$ (7)

Hence, the elasticity of substitution between $L_{t}^{f}$ and $L_{t}^{m}$ is equal to $1/(\upsilon-1)$.

2.2  Final Goods Producers

Firms are divided into a sector $f$ of final goods producers and a sector $m$ of producers of intermediate goods, both of which maximize profits. Firms in the $f$ sector use labor and intermediate goods, and face Markov TFP shocks $\varepsilon_{t},$ with transition probability distribution function $\mu\left( \varepsilon_{t}\vert\varepsilon_{t-1}\right)$. The production function of the $f$ sector is Cobb-Douglas:

$$y_{t}=\varepsilon_{t}(m_{t})^{\alpha_{m}}(L_{t}^{f})^{\alpha_{L}}k^{\alpha _{k}}$$ (8)

with $0<\alpha_{L},\alpha_{m},\alpha_{k}<1$ and $\alpha_{L}+\alpha_{m} +\alpha_{k}=1$.

The $f$ sector chooses optimally whether to import intermediate goods from abroad or buy them from the $m$ sector at home. Imported inputs are sold in a competitive world market at the exogenous relative price $p_{m}^{\ast}$.⁶ A fraction $\theta$ of the cost of these imported inputs needs to be paid in advance using working capital loans $\kappa_{t}\,$, which are intraperiod loans repaid at the end of the period that are offered by foreign creditors at the interest rate $r_{t}$. This interest rate is linked to the sovereign interest rate at equilibrium, as shown in the next section. Working capital loans satisfy the standard payment-in-advance condition:

$$\frac{\kappa_{t}}{1+r_{t}}\geq\theta p_{m}^{\ast}m_{t}$$ (9)

Profit-maximizing firms choose $\kappa_{t}$ so that this condition holds with equality.

The profits of final goods producers when they use imported inputs are:

$$\pi_{t}^{\ast}=\varepsilon_{t}(m_{t})^{\alpha_{m}}(L_{t}^{f})^{\alpha_{L} }k^{\alpha_{k}}-p_{m}^{\ast}(1+\theta r_{t})m_{t}-w_{t}^{f}L_{t}^{f}$$ (10)

Alternatively, when they use domestic intermediate goods, their profits are given by:

$$\pi_{t}^{d}=\varepsilon_{t}(m_{t})^{\alpha_{m}}(L_{t}^{f})^{\alpha_{L} }k^{\alpha_{k}}-p_{m}m_{t}-w_{t}^{f}L_{t}^{f}$$ (11)

where $p_{m}$ is the endogenous price of intermediate goods produced at home. As noted earlier, domestic inputs do not require working capital financing. This assumption is just for simplicity, the key element for the analysis is that at high levels of country risk (including periods without access to foreign credit markets) the financing cost of foreign inputs is higher than that of domestic inputs.

Final goods producers maximize profits taking the sectoral wage rate, the interest rate, and intermediate goods prices as given, and choosing whether to use domestic or imported intermediate goods and the optimal amount of intermediate goods and labor to buy in each case. This is equivalent to first evaluating the profit-maximizing plans under each alternative (domestic v. imported inputs) and then choosing the one that yields higher profits:

$$\pi_{t}^{f}=\max\left[ \max_{m_{t},L_{t}^{f}}(\pi_{t}^{\ast}),\max _{m_{t},L_{t}^{f}}(\pi_{t}^{d})\right]$$ (12)

When imported intermediate goods are used, the optimality conditions are

$$\alpha_{m}\varepsilon_{t}(m_{t})^{\alpha_{m}-1}(L_{t}^{f})^{\alpha_{L} }k^{\alpha_{k}} =p_{m}^{\ast}(1+\theta r_{t})$$ (13)

$$\alpha_{L}\varepsilon_{t}(m_{t})^{\alpha_{m}}(L_{t}^{f})^{\alpha_{L-1} }k^{\alpha_{k}} =w_{t}^{f}$$ (14)

Alternatively, when domestic inputs are used, the optimality conditions are:

$$\alpha_{m}\varepsilon_{t}(m_{t})^{\alpha_{m}-1}(L_{t}^{f})^{\alpha_{L} }k^{\alpha_{k}} =p_{t}^{_{m}}$$ (15)

$$\alpha_{L}\varepsilon_{t}(m_{t})^{\alpha_{m}}(L_{t}^{f})^{\alpha_{L-1} }k^{\alpha_{k}} =w_{t}^{f}$$ (16)

These two sets of optimality conditions are standard: Marginal products of factors of production equal the corresponding marginal costs.

2.3  Intermediate Goods Producers

Domestic inputs do not require advance payment, but in order to produce them labor has to be reallocated from the $f$ sector to the $m$ sector. At equilibrium, the $m$ sector operates only if the market price of its output is positive, which occurs only if the $f$ sector chooses to use domestic inputs.

Producers in the $m$ sector operate with a production function given by $m=A(L_{t}^{m})^{\gamma}$, with $A>0$ and $0\leq\gamma\leq1.$ Given the domestic price of intermediate goods and the sectoral wage rate, they choose labor demand so as to solve the following profit maximization problem:

$$\max_{L_{t}^{m}}\ \pi_{t}^{m}=p_{t}^{_{m}}A(L_{t}^{m})^{\gamma}-w_{t}^{m} L_{t}^{m}$$ (17)

If sector $f$ producers find it optimal to use imported inputs, the demand for domestic intermediate goods is zero, and hence $p_{t}^{_{m}}$ and $L_{t}^{m}$ are zero and the $m$ sector is idle. If final goods producers demand domestic intermediate goods, optimal labor demand by producers of intermediate goods satisfies

$$\gamma p_{t}^{_{m}}A(L_{t}^{m})^{\gamma-1}=w_{t}^{m}$$ (18)

2.4  Competitive Equilibrium of the Private Sector

Definition 1   A competitive equilibrium for the private sector of the economy is given by sequences of allocations $\left[ c_{t},L_{t},L_{t}^{f},L_{t}^{m},m_{t} ,\kappa_{t}\right] _{t=0}^{\infty}$ and prices $\left[ w_{t}^{f},w_{t} ^{m},p_{t}^{m},\pi_{t}^{f},\pi_{t}^{m}\right] _{t=0}^{\infty}$ such that:

1. The allocations $\left[ c_{t},L_{t},L_{t}^{f},L_{t}^{m}\right] _{t=0}^{\infty}$ solve the households' utility maximization problem$.$

2. The allocations $\left[ L_{t}^{f},m_{t},\kappa_{t}\right] _{t=0}^{\infty }$ solve the profit maximization problem of sector $f$ producers.

3. The allocations $\left[ L_{t}^{m}\right] _{t=0}^{\infty}$ solve the profit maximization problem of sector $m$ producers.

4. The labor market-clearing conditions hold.

Standard national income accounting implies that the economy's GDP is equal to either: (a) the gross output of the $f$ sector net of the cost of imported inputs if final goods producers use imported inputs, or (b) the gross output of the $f$ sector if final goods producers use domestic inputs. In the first case, the $m$ sector is not operating and GDP at factor costs follows from the definition of profits of the $f$ sector: $w_{t}L_{t}^{f}+\pi_{t} ^{f}=\varepsilon(m_{t})^{\alpha_{m}}(L_{t}^{f})^{\alpha_{L}}k^{\alpha_{k} }-p_{m}^{\ast}(1+\theta r_{t})m_{t}=\left( 1-\alpha_{m}\right) \varepsilon(m_{t})^{\alpha_{m}}(L_{t}^{f})^{\alpha_{L}}k^{\alpha_{k}}$ . This excludes $(1-\alpha_{m})$ of gross output of final goods because imports of intermediate goods are factor payments to foreigners. In the second case, the definitions of profits of the $f$ and $m$ sectors yield: $w_{t}^{f}L_{t} ^{f}+w_{t}^{m}L_{t}^{m}+\pi_{t}^{f}+\pi_{t}^{m}=\varepsilon(m_{t})^{\alpha _{m}}(L_{t}^{f})^{\alpha_{L}}k^{\alpha_{k}}.$

A key constraint on the problem of the sovereign borrower making the default decision will be that the private-sector allocations must be a competitive equilibrium. Since the sovereign government's problem and the equilibrium of the credit market will be characterized in recursive form, it is useful to also characterize the allocations of the above competitive equilibrium in recursive form (i.e. as functions defined in the state space domain). This is done by first expressing the optimal allocations of labor and intermediate goods when sector $f$ uses imported inputs as the following functions of $r$ and $\varepsilon$:

$$m^{\ast}(r,\varepsilon) =\left[ \alpha_{L}^{\alpha_{L}}\left( \varepsilon k^{\alpha_{k}}\right) ^{\omega}\left( \frac{\alpha_{m}}{p_{m}^{\ast }(1+\theta r)}\right) ^{\omega-\alpha_{L}}\right] ^{\frac{1}{\omega\left( 1-\alpha_{m}\right) -\alpha_{L}}}$$ (19)

$$L^{f\ast}(r,\varepsilon) =\left[ \alpha_{L}\left( \varepsilon k^{\alpha_{k}}\right) ^{\frac{1}{1-\alpha_{m}}}\left( \frac{\alpha_{m} }{p_{m}^{\ast}(1+\theta r)}\right) ^{\frac{\alpha_{m}}{1-\alpha_{m}}}\right] ^{\frac{1-\alpha_{m}}{\omega\left( 1-\alpha_{m}\right) -\alpha_{L}} }$$ (20)

If sector $f$ uses domestic inputs instead, the optimal allocations of factors of production in the $f$ and $m$ sectors are:

$$L^{d}\left( \varepsilon\right) =\left[ (\alpha_{L}+\gamma\alpha _{m})\varepsilon k^{\alpha_{k}}A^{\alpha_{m}}\left( z_{L_{m}}\right) ^{\alpha_{m}\gamma}\left( z_{L_{f}}\right) ^{\alpha_{L}}\right] ^{1/\left( \omega-\alpha_{L}-\alpha_{m}\gamma\right) }$$ (21)

$$L^{f,d}\left( \varepsilon\right) =z_{L_{f}}L^{d}\left( \varepsilon \right)$$ (22)

$$L^{m,d}\left( \varepsilon\right) =z_{L_{m}}L^{d}\left( \varepsilon \right)$$ (23)

$$m^{d}\left( \varepsilon\right) =A\left( L^{m,d}\left( \varepsilon \right) \right) ^{\gamma}$$ (24)

where $z_{L_{m}}=\left( \frac{\gamma\alpha_{m}}{\gamma\alpha_{m}+\alpha_{L} }\right) ^{1/\nu}$ and $z_{L_{f}}=$ $\left( \frac{\alpha_{L}}{\gamma \alpha_{m}+\alpha_{L}}\right) ^{1/\nu}$ . Note also that the equilibrium price of the domestic intermediate goods is $p_{m}\left( \varepsilon\right) =\alpha_{m}\varepsilon\left( m^{d}\left( \varepsilon\right) \right) ^{\alpha_{m}-1}$ $\left( L^{f,d}\left( \varepsilon\right) \right) ^{\alpha_{L}}k^{\alpha_{k}}.$

It follows from the above solutions that final goods production is not affected by foreign interest rates when firms use domestic intermediate goods, because sector $f$ is not borrowing from abroad in this case. In contrast, when producers of final goods use imported inputs, their demand for these inputs and labor decreases with $r$. Thus, in this situation, sovereign risk affects the actions of sector $f$ firms. Because, as we show later, the interest rate on foreign working capital loans is driven by the sovereign interest rate, these firms face higher financing costs when default risk rises, and so their factor demands and output fall. One special case of this situation is the state when default occurs, in which the country has no access to working capital because effectively $r$ has gone to infinity. In this case, firms cannot import inputs from abroad and switch to use domestic substitutes. Note, however, that the interest rate does not need to rise to infinity for the switch to occur. Firms switch to domestic inputs at a finite interest rate that is high enough for $\pi^{d}>\pi^{\ast}$.

Next we define the indicator function $\Phi(r,\varepsilon)$ to identify whether the $f$ sector is using domestic or imported inputs at the current state of interest rates and TFP. In particular, $\Phi(r,\varepsilon)=1$ if $\pi^{f}=\max(\pi^{\ast})$ and $\Phi(r,\varepsilon)=0$ if $\pi^{f}=\max (\pi^{d})$ for a given $(r,\varepsilon)$ pair. Hence, firms use imported (domestic) inputs when $\Phi(r,\varepsilon)=1$ ( $\Phi(r,\varepsilon)=0$). The competitive equilibrium allocations of factor demands and working capital can now be expressed as functions of $r$ and $\varepsilon$ as follows:

$$\kappa(r,\varepsilon) =\Phi(r,\varepsilon)\theta p_{m}^{\ast}m^{\ast }(r,\varepsilon)+\left( 1-\Phi(r,\varepsilon)\right) \cdot0$$ (25)

$$m(r,\varepsilon) =\Phi(r,\varepsilon)m^{\ast}(r,\varepsilon)+\left( 1-\Phi(r,\varepsilon)\right) m^{d}(\varepsilon)$$ (26)

$$L(r,\varepsilon) =\Phi(r,\varepsilon)L^{f\ast}(r,\varepsilon)+\left( 1-\Phi(r,\varepsilon)\right) L^{d}\left( \varepsilon\right)$$ (27)

$$L^{f}(r,\varepsilon) =\Phi(r,\varepsilon)L^{f\ast}(r,\varepsilon)+\left( 1-\Phi(r,\varepsilon)\right) L^{f,d}(\varepsilon)$$ (28)

$$L^{m}(r,\varepsilon) =\Phi(r,\varepsilon)\cdot0+\left( 1-\Phi (r,\varepsilon)\right) L^{m,d}(\varepsilon)$$ (29)

2.5  Endogenous Output Cost of Default

The decision by firms in the $f$ sector to shift between foreign and domestic inputs depends on the states of the interest rate and TFP. The mechanism that drives this shift can be illustrated by examining the $f$ sector firms' optimal choice of intermediate goods using Figure 1. For simplicity, we draw this figure assuming that total labor effort $L$ is inelastic. The demand for intermediate goods is determined by the marginal product of $m$. The corresponding marginal productivity curve when foreign (domestic) inputs are used is labeled $\varepsilon f_{m\ast}$ ( $\varepsilon f_{md}$). The marginal productivity of intermediate goods employed in final goods production is always lower when domestic inputs are used, because of the reallocation of labor from final goods production to production of intermediate goods. Given the Cobb-Douglas production function for $f$, the lower labor input available to the $f$ sector when it uses domestic inputs reduces the marginal product of intermediate goods in production of final goods.⁷ Moreover, because the reallocation of labor is costly, one unit of labor taken away from the $f$ sector yields less than one unit of labor in the $m$ sector, and the higher this reallocation cost the lower the marginal product of domestic intermediate goods relative to that of imported intermediate goods (i.e. the larger the gap between $\varepsilon f_{m\ast}$ and $\varepsilon f_{md}$).

Figure 1:  The Intermediate Goods Market

The supply of imported inputs is infinitely elastic at an exogenous price $p_{m}^{\ast}(1+\theta r)$. In contrast, the supply of domestic inputs ( $m_{d}$ in Figure 1) is determined by the production plans of the $m$ sector. This supply function is given by $m^{d}\left( .\right) =A\left( \frac{\gamma Ap^{m}}{w^{m}}\right) ^{\frac{\gamma}{1-\gamma}}$ .

If the interest rate is sufficiently low, the firms' optimal plans call for using imported inputs up to the amount at which the marginal product of $m$ equals the marginal cost $p_{m}^{\ast}(1+\theta r)$. This is point A in Figure 1. Around point A, output fluctuates as a result of changes in $r$ and $\varepsilon$. Consider first the interest rate. Given that marginal products are decreasing and continuously differentiable, it follows that as $r$ rises the demand for imported inputs and the profits of final goods producers decline, until we reach a threshold value $r^{\prime}$ at which $\pi^{\ast }=\pi^{d}.$ $r^{\prime}$ is an interest rate high enough for these producers to find it optimal to switch to the domestic inputs, because $r>r^{\prime}$ yields $\pi^{d}>\pi^{\ast}$. This threshold point is shown as point C in Figure 1.

When the interest rate reaches $r^{\prime}$, final goods producers switch to domestic inputs and the equilibrium price and quantity of intermediate goods are determined at point B. Notice that, because imported inputs have higher marginal product, when interest rates are high (but not yet at $r^{\prime}$) it can be optimal for firms to use quantities of imported inputs that are lower than what they use if they operate with domestic inputs ($m_{d}$). This is because in this situation firms still make more profits with the foreign inputs than by switching to domestic inputs.

Around point B, fluctuations in output are driven by changes in $\varepsilon,$ but output is no longer affected by the interest rate. This has two important implications. First, since in principle $r^{\prime}$ can be reached before the country defaults, high interest rates can trigger a switch to domestic inputs even before default occurs. Second, since $r^{\prime}$ is well defined and at default $r\rightarrow\infty$, firms always use domestic inputs when the economy defaults.

Productivity shocks can also cause the switch from imported to domestic inputs, even if $r$ remains constant. As with the interest rate, there is a threshold TFP level at which final goods producers are indifferent between using imported or domestic inputs because $\pi^{\ast}=$ $\pi^{d}$. For TFP shocks below this threshold, these producers opt for domestic inputs. The reason is that a low $\varepsilon$ lowers the marginal product of imported inputs but firms still pay the extra marginal cost due to the cost of working capital. Hence, firms choose to use domestic inputs (and bear the efficiency loss) rather than paying this financing cost.

The switch from imported to domestic inputs that occurs at high interest rates has important implications for the output cost of default. In particular, it makes the cost of default an increasing function of the state of TFP. This property of the default cost can be illustrated by studying how productivity shocks affect the fraction of GDP lost in a default $1-(Y^{d}/Y^{\ast}),$ where $Y^{\ast}$ and $Y^{d}$ represent GDP when the economy has access to credit markets and when the economy defaults respectively (both given by the fraction $\left( 1-\Phi(r_{t},\varepsilon_{t})\alpha_{m}\right)$ of final goods production.

Figure 2 shows how $Y^{\ast},$ $Y^{d}$ and the output loss at default change with TFP shocks for a given $r$. If the country defaults, exclusion from world credit markets prevents final goods producers from accessing working capital loans and forces them to switch to domestic inputs, so along the $Y^{d}$ line firms always operate with domestic inputs. If the country has access to world credit markets, final goods producers choose optimally whether to use imported or domestic inputs. Hence, $Y^{\ast}$ is produced with imported inputs as long as $\varepsilon$ is above the threshold at which final goods producers switch to domestic inputs, and $Y^{\ast}=Y^{d}$ otherwise.

Figure 2:  Output and the Output Cost of Default as Functions of TFP

Data for Figure 2

e shock	GDP	GDP in default	GDP_loss
-0.1500	0.1638	0.1647	0.2960
-0.1342	0.1694	0.1686	0.3031
-0.1184	0.1752	0.1727	0.3102
-0.1026	0.1812	0.1768	0.3171
-0.0868	0.1875	0.1810	0.3240
-0.0711	0.1939	0.1854	0.3309
-0.0553	0.2006	0.1898	0.3376
-0.0395	0.2075	0.1944	0.3443
-0.0237	0.2147	0.1990	0.3509
-0.0079	0.2220	0.2038	0.3575
0.0079	0.2297	0.2087	0.3640
0.0237	0.2376	0.2137	0.3704
0.0395	0.2458	0.2188	0.3768
0.0553	0.2542	0.2241	0.3831
0.0711	0.2630	0.2294	0.3893
0.0868	0.2720	0.2349	0.3955
0.1026	0.2814	0.2406	0.4016
0.1184	0.2911	0.2463	0.4076
0.1342	0.3011	0.2522	0.4136
0.1500	0.3114	0.2583	0.4195

As Figure 2 shows, the output cost of default increases with the size of the TFP shock, because default is accompanied by a switch from $Y^{\ast}$ to $Y^{d},$ so default is more painful at higher levels of TFP. This property of the output cost of default is key for the model's ability to support high debt levels together with observed default frequencies, because it makes the default option more attractive to the country at lower states of productivity, and works as a desirable implicit hedging mechanism given the incompleteness of asset markets.

This finding is in line with Arellano's (2007) result showing that an exogenous default cost with similar features can allow the Eaton-Gersovitz model to support non-trivial levels of debt together with observed default frequencies. In particular, she proposed an exogenous default cost function such that below a threshold level of an output endowment default does not entail an output cost, but above that threshold default reduces the endowment to a state-invariant fraction of the long-run average of GDP. In this second range, the size of the output loss is increasing in the output realization at the time of default. Still, the mean debt ratio in her baseline calibration was only about 6 percent of GDP (assuming output at default is 3 percent below mean output), while we show later that our model with an endogenous output cost of default yields a mean debt ratio about four times larger.

2.6  The Sovereign Government

The sovereign government trades with foreign lenders one-period, zero-coupon discount bonds, so markets of contingent claims are incomplete. The face value of these bonds specifies the amount to be repaid next period and is denoted as $b_{t+1}$. When the country purchases bonds $b_{t+1}>0$, and when it borrows $b_{t+1}<0$. The set of bond face values is $B=\left[ b_{\min},b_{\max }\right] \subset R$, where $b_{\min}\leq0\leq b_{\max}$. We set the lower bound $b_{\min}<-\frac{\overline{y}}{r}$, which is the largest debt that the country could repay with full commitment. The upper bound $b_{\max}$ is the highest level of assets that the country may accumulate.⁸

The sovereign cannot commit to repay its debt. As in the Eaton-Gersovitz model, we assume that when the country defaults it does not repay at date $t$ and the punishment is exclusion from the world credit market in the same period. The country re-enters the credit market with an exogenous probability $\eta$, and when it does it starts with a fresh record and zero debt.⁹ Also as in the Eaton-Gersovitz setup, the country cannot hold positive international assets during the exclusion period, otherwise the model cannot support equilibria with debt.

We add to the Eaton-Gersovitz setup an explicit link between default risk and private financing costs. This is done by assuming that a defaulting sovereign can divert the repayment of the firms' working capital loans to foreign lenders. Hence, both firms and government default together. This is perhaps an extreme formulation of the link between private and public borrowing costs, but we provide later some evidence in favor of this view.

The sovereign government solves a problem akin to a Ramsey problem.¹⁰ It chooses a debt policy (amounts and default) that maximizes the households' welfare subject to the constraints that: (a) the private sector allocations must be a competitive equilibrium; and (b) the government budget constraint must hold. The state variables are the initial foreign asset position, working capital loans as of the end of last period, and the state of TFP, denoted by the triplet $\left( b_{t},\kappa _{t-1},\varepsilon_{t}\right)$. The price of sovereign bonds is given by the bond pricing function $q_{t}\left( b_{t+1},\varepsilon_{t}\right)$. Since at equilibrium the default risk premium on sovereign debt will be the same as on working capital loans, it follows that the interest rate on working capital is a function of $q_{t}\left( b_{t+1},\varepsilon_{t}\right)$. Hence, the recursive expressions that represent the competitive equilibrium of the private sector derived earlier can be expressed as as $\kappa\left( q_{t}\left( b_{t+1},\varepsilon_{t}\right) ,\varepsilon_{t}\right)$ , $m\left( q_{t}\left( b_{t+1},\varepsilon_{t}\right) ,\varepsilon _{t}\right) ,$ $L^{f}\left( q_{t}\left( b_{t+1},\varepsilon_{t}\right) ,\varepsilon_{t}\right) ,$ $L^{m}\left( q_{t}\left( b_{t+1},\varepsilon _{t}\right) ,\varepsilon_{t}\right)$ , $L\left( q_{t}\left( b_{t+1} ,\varepsilon_{t}\right) ,\varepsilon_{t}\right)$ , and $\Phi(q_{t}\left( b_{t+1},\varepsilon_{t}\right) ,\varepsilon_{t})$ .

The recursive optimization problem of the government is summarized by the following value function:

$$V\left( b_{t},\kappa_{t-1},\varepsilon_{t}\right) =\left\{ \begin{array}[c]{ll} \max\left\{ v^{nd}\left( b_{t},\varepsilon_{t}\right) ,v^{d}\left( \kappa_{t-1,}\varepsilon_{t}\right) \right\} & \text{for }b_{t}<0\\ v^{nd}\left( b_{t},\varepsilon_{t}\right) & \text{for }b_{t}\geq0 \end{array} \right.$$ (30)

If the country has access to the world credit market at date $t$, the value function is the maximum of the value of continuing in the credit relationship with foreign lenders (i.e., repayment or "no default"), $v^{nd}\left( b_{t},\varepsilon_{t}\right)$, and the value of default, $v^{d}\left( \kappa_{t-1,}\varepsilon_{t}\right)$. If the economy holds a non-negative net foreign asset position, the value function is simply the continuation value because in this case the economy is using the credit market to save, receiving a return equal to the world's risk free rate $r^{\ast}$.

The continuation value $v^{nd}\left( b_{t},\varepsilon_{t}\right)$ is defined as follows:

$$v^{nd}\left( b_{t},\varepsilon_{t}\right) =\underset{c_{t},b_{t+1}}{\max }\left\{ \begin{array}[c]{c} u\left( c_{t}-h(L\left( q_{t}\left( b_{t+1},\varepsilon_{t}\right) ,\varepsilon_{t}\right) )\right) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ \;\;\;\;\;\;\;+\beta E\left[ V\left( b_{t+1},\kappa\left( q_{t}\left( b_{t+1},\varepsilon_{t}\right) ,\varepsilon_{t}\right) ,\varepsilon _{t+1}\right) \right] \end{array} \right\}$$ (31)

subject to

$$c_{t}+q_{t}\left( b_{t+1},\varepsilon_{t}\right) b_{t+1}-b_{t} \leq \ \ \ \ \ \ \ \ \ \ \ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$

$$\left[ 1-\Phi(q_{t}\left( b_{t+1},\varepsilon_{t}\right) ,\varepsilon _{t})\alpha_{m}\right] \varepsilon_{t}f\left( m\left( q_{t}\left( b_{t+1},\varepsilon_{t}\right) ,\varepsilon_{t}\right) ,L^{f}\left( q_{t}\left( b_{t+1},\varepsilon_{t}\right) ,\varepsilon_{t}\right) ,k\right)$$ (32)

The constraint of this problem is the resource constraint of the economy at a competitive equilibrium. The left-hand-side is the sum of consumption and net exports, and the right-hand-side is GDP. This constraint is obtained by combining the households' budget constraint (2) with the government budget constraint, $T_{t}=b_{t}-q_{t}\left( b_{t+1},\varepsilon_{t}\right) b_{t+1}$ , and noting that the firms' optimality conditions imply that total domestic factor payments, $w_{t}^{f}L_{t}^{f}+w_{t}^{m}L_{t}^{m}+\pi_{t}^{f}+\pi _{t}^{m}$ , equal the fraction $(1-\Phi(r,\varepsilon)\alpha_{m})$ of gross output of final goods $\varepsilon f(m,L^{f},k).$

The resource constraint captures three important features of the model: First, the government internalizes how interest rates affect the competitive equilibrium allocations of output and factor demands. Second, the households cannot borrow from abroad, but the government internalizes their desire to smooth consumption and transfers to them an amount equal to the negative of the balance of trade (i.e. it gives the private sector the flow of resources it needs to finance the gap between GDP and consumption). Third, the working capital loans $\kappa_{t-1}$ and $\kappa_{t}$ do not enter explicitly in the continuation value or in the resource constraint, because working capital payments are included in the fraction of gross output allocated to payments of intermediate goods, $\alpha_{m}f(m,L^{f},k)$. Still, we need to keep track of the state variable $\kappa_{t}$ because the amount of working capital loans taken by final goods producers at date $t$ affects the sovereign's incentive to default at $t+1,$ as explained below.

The value of default $v^{d}\left( \kappa_{t-1},\varepsilon_{t}\right)$ is:

$$v^{d}\left( \kappa_{t-1},\varepsilon_{t}\right) =\max_{c_{t}}\left\{ \begin{array}[c]{c} u\left( c_{t}-h(L(\varepsilon_{t}))\right) \;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ \;\ \;\;\;\;\;\;\;\;\ \ \ \ \ \ \ \ +\beta\left( 1-\eta\right) Ev^{d}\left( 0,\varepsilon_{t+1}\right) +\beta\eta EV\left( 0,0,\varepsilon_{t+1}\right) \end{array} \right\}$$ (33)

subject to:

$$c_{t}=\varepsilon_{t}f\left( m^{d}(\varepsilon_{t}),L^{f,d}(\varepsilon _{t}),k\right) +\kappa_{t-1}$$ (34)

Note that $v^{d}\left( \kappa_{t-1},\varepsilon_{t}\right)$ takes into account the fact that in case of default at date $t,~$the country has no access to financial markets this period, and hence the country consumes the total income given by the resource constraint in the default scenario. In this case, since firms cannot borrow to finance purchases of imported inputs, $m^{d}(\cdot)$, $L(\cdot)$ and $L^{f,d}\left( \cdot\right)$ are the competitive equilibrium allocations that correspond to the case when the $f$ sector operates with domestic inputs. Moreover, because the defaulting government diverts the repayment of last period's working capital loans, total household income includes government transfers equal to the appropriated repayment for the amount $\kappa_{t-1}$ (i.e., on the date of default, the government budget constraint is $T_{t}=\kappa_{t-1}$). The value of default at $t$ also takes into account that at $t+1$ the economy may re-enter world capital markets with probability $\eta$ and associated value $V\left( 0,0,\varepsilon_{t+1}\right)$, or remain in financial autarky with probability $1-\eta$ and associated value $v^{d}\left( 0,\varepsilon _{t+1}\right)$.

For a debt position $b_{t}<0$ and given a level of working capital $\kappa_{t-1}$, default is optimal for the set of realizations of the TFP shock for which $v^{d}\left( \kappa_{t-1},\varepsilon_{t}\right)$ is at least as high as $v^{nd}\left( b_{t},\varepsilon_{t}\right)$:

$$D\left( b_{t},\kappa_{t-1}\right) =\left\{ \varepsilon_{t}:v^{nd}\left( b_{t},\varepsilon_{t}\right) \leq v^{d}\left( \kappa_{t-1},\varepsilon _{t}\right) \right\}$$ (35)

It is critical to note that this default set has a different specification than in the typical Eaton-Gersovitz model of sovereign default (see Arellano (2007)), because the state of working capital affects the gap between the values of default and repayment. This results in a two-dimensional default set that depends on $b_{t}$ and $\kappa_{t-1}$, instead of just $b_{t}.$

Despite the fact that the default set depends on $\kappa_{t-1}$, the probability of default remains a function of $b_{t+1}$ and $\varepsilon_{t}$ only. This is because the $f$ sector's optimality conditions imply that the next period's working capital loan $\kappa_{t}$ depends on $\varepsilon_{t}$ and the interest rate, which is a function of $b_{t+1}$ and $\varepsilon_{t}$. Thus the probability of default at $t+1$ perceived as of date $t$ for a country with a productivity $\varepsilon_{t}$ and debt $b_{t+1}$, $p_{t}\left( b_{t+1},\varepsilon_{t}\right)$, can be induced from the default set, the decision rule for working capital, and the transition probability function of productivity shocks $\mu\left( \varepsilon _{t+1}\vert\varepsilon_{t}\right)$ as follows:

$$p_{t}\left( b_{t+1},\varepsilon_{t}\right) =\int_{D\left( b_{t+1} ,\kappa_{t}\right) }d\mu\left( \varepsilon_{t+1}\vert\varepsilon_{t}\right)$$ (36)

$$\kappa_{t} =\kappa\left( q_{t}\left( b_{t+1},\varepsilon _{t}\right) ,\varepsilon_{t}\right)$$ (37)

The economy is considered to be in financial autarky when it has been in default for at least one period and remains without access to world credit markets as of date $t.$ As noted above, the economy can exit this exclusion stage at date $t+1$ with probability $\eta$. We assume that during the exclusion stage the economy cannot build up its own stock of savings to supply working capital loans to firms, which could be used to purchase imported inputs.¹¹ This assumption ensures that, as long as the economy remains in financial autarky, the optimization problem of the sovereign is the same as the problem in the default period but evaluated at $\kappa_{t-1}=0$ (i.e. $v^{d}\left( \varepsilon_{t},0\right)$).

We also studied an alternative setup in which we allowed for a domestic financial market to operate during the exclusion stage. In this case, households make saving plans to offer working capital loans to firms at a market-determined interest rate, and firms demand these loans if the endogenous domestic interest rate is low enough to make productions plans using foreign inputs more profitable than with domestic inputs, despite the higher financing cost of the former. In this case, domestic loans are included as an additional state variable and their interest rate is determined as an equilibrium outcome. We found, however, that for parameter values around our baseline calibration this domestic financial market is not viable: The interest rate at which households would find it optimal to accumulate savings is too high for firms to optimally choose to obtain domestic working capital loans to purchase imported inputs, instead of just using domestic inputs. Hence, the equilibrium for the model with the domestic financial market operating during the exclusion stage is the same as that for the model that simply assumes that firms operate with domestic inputs whenever they cannot access world credit markets.

The model preserves a standard feature of the Eaton-Gersovitz model: Given $\varepsilon_{t}$, the value of defaulting is independent of the level of debt, while the value of not defaulting increases with $b_{t+1}$, and consequently the default set and the equilibrium default probability grow with the country's debt. The following theorem formalizes this result:

Theorem 1   Given a productivity shock $\varepsilon$ and level of working capital loan $\kappa$, for $b^{0}<b^{1}\leq0$, if default is optimal for $b^{1}$, then default is also optimal for $b^{0}$. That is $D\left( b^{1},\kappa\right) \subseteq D\left( b^{0},\kappa\right)$ . The country agent's probability of default in equilibrium satisfies $p^{\ast}\left( b^{0},\varepsilon\right) \geq p^{\ast}\left( b^{1},\varepsilon\right)$ .
Proof. See Appendix. $\qedsymbol$

2.7  Foreign Lenders

International creditors are risk-neutral and have complete information. They invest in sovereign bonds and in private working capital loans. Foreign lenders behave competitively and face an opportunity cost of funds equal to the world risk-free interest rate. Competition implies that they expect zero profits at equilibrium, and that the returns on sovereign debt and the world's risk-free asset are fully arbitraged:

$$q_{t}\left( b_{t+1},\varepsilon_{t}\right) =\left\{ \begin{array}[c]{cc} \frac{1}{1+r^{\ast}} & \text{if }b_{t+1}\geq0\\ \frac{\left[ 1-p_{t}\left( b_{t+1},\varepsilon_{t}\right) \right] }{1+r^{\ast}} & \text{if }b_{t+1}<0 \end{array} \right.$$ (38)

This condition implies that at equilibrium bond prices depend on the risk of default. For a high level of debt, the default probability is higher. Therefore, equilibrium bond prices decrease with indebtedness. This result, formalized in Theorem 2 below, is consistent with the empirical evidence documented by Edwards (1984).

Theorem 2   Given a productivity shock $\varepsilon$ and level of working capital loan $\kappa$, for $b^{0}<b^{1}\leq0$, the equilibrium bond price satisfies $q^{\ast}\left( b^{0},\varepsilon\right) \leq q^{\ast}\left( b^{1} ,\varepsilon\right)$
Proof. See Appendix. $\qedsymbol$

The returns on sovereign bonds and working capital loans are also fully arbitraged. Because the sovereign government diverts the repayment of working capital loans when it defaults, foreign lenders assign the same risk of default to private working capital loans as to sovereign debt, and hence the no- arbitrage condition between sovereign lending and working capital loans implies:

$$r_{t}\left( b_{t+1},\varepsilon_{t}\right) =\frac{1}{q_{t}\left( b_{t+1},\varepsilon_{t}\right) }-1 b_{t+1}<0 \kappa _{t}>0$$ (39)

2.8  Country Risk & Private Interest Rates: Some Empirical Evidence

The result that the interest rates on sovereign debt and private working capital loans are the same raises a key empirical question: Are sovereign interest rates and the rates of interest faced by private firms closely related in emerging economies?

Providing a complete answer to this question is beyond the scope of this paper, but we do provide empirical evidence suggesting that indeed interest rates on loans to private firms and on sovereign bonds move together. To study this issue, we constructed country estimates of firms' financing costs that aggregate measures derived from firm-level data. We constructed a measure of firm-level effective interest rates as the ratio of a firm's total debt service divided by its total debt obligations using the Worldscope database, which provides the main lines of balance-sheet and cash-flow statements of publicly listed corporations. We then constructed the corresponding aggregate country measure as the median across firms.

Table 1:  Sovereign Interest Rates and Firm Financing Cost

Country	Sovereign Interest Rates	Median Firm Interest Rates	Correlation
Argentina	13.32	10.66	0.87
Brazil	12.67	24.60	0.14
Chile	5.81	7.95	0.72
China	6.11	5.89	0.52
Colombia	9.48	19.27	0.86
Egypt	5.94	8.62	0.58
Malaysia	5.16	6.56	0.96
Mexico	9.40	11.84	0.74
Morocco	9.78	13.66	0.32
Pakistan	9.71	12.13	0.84
Peru	9.23	11.42	0.72
Philippines	8.78	9.27	0.34
Poland	7.10	24.27	0.62
Russia	15.69	11.86	-0.21
South Africa	5.34	15.19	0.68
Thailand	6.15	7.30	0.94
Turkey	9.80	29.26	0.88
Venezuela	14.05	19.64	0.16

The comparison of this measure of interest rates faced by private firms with the standard EMBI+ measure of interest rates on sovereign debt shows two striking facts (see Table 1): First, the two interest rates are positively correlated in most countries, with a median correlation of 0.7, and in some countries the relationship is very strong (see Figure 3).¹² Second, the effective financing cost of firms is generally higher than the sovereign interest rates. This fact indicates that the common conjecture that firms (particularly the large corporations covered in our data) may pay lower rates than governments with default risk is incorrect.

The study by Arteta and Hale (2007) provides further and more systematic evidence on the strong effects of sovereign debt on the terms of private-sector debt contracts of emerging economies. In particular, they show strong, systematic negative effects on private corporate bond issuance during and after default episodes.

Figure 3:  Sovereign Bond Interest Rates and Median Firm Financing Costs

________  Sovereign Bond Interest Rates
-----------  Median Firm Financing Cost

Data for Figure 3

Year	Argentina: Firm	Argentina: Sovereign	Chile: Firm	Chile: Sovereign	Malaysia: Firm	Malaysia: Sovereign	Mexico: Firm	Mexico: Sovereign	Peru: Firm	Peru: Sovereign	Thailand: Firm	Thailand: Sovereign
1994	8.05	12.87	-	-	-	-	7.78	10.84	-	-	-	-
1995	7.18	18.24	-	-	-	-	17.43	17.13	-	-	-	-
1996	9.33	13.81	-	-	-	-	14.64	13.61	-	-	-	-
1997	9.08	10.13	-	-	6.58	6.92	11.28	10.08	9.99	10.03	7.64	7.86
1998	9.02	11.07	-	-	9	10.04	12.3	10.92	9.5	10.74	9.29	12.65
1999	10.55	12.57	8.67	7.33	8.29	9.38	13.1	11.83	11.29	11.51	7.78	10.2
2000	10.92	12.26	8.8	8.05	6.91	8.25	12.93	9.66	11.3	11.5	7.75	8.17
2001	13.76	15.64	8.18	6.53	6.87	7	11.83	8.42	10.7	11.15	6.15	7.17
2002	20.01	61.44	6.79	5.65	6.19	5.69	9.14	7.06	9.36	9.88	4.83	5.49
2003	-	-	6.14	4.38	5.73	4.67	9.79	5.63	7.09	7.78	3.97	4.41
2004	-	-	5.79	4.29	5.33	4.54	10.13	5.37	7.53	7	4.15	3.83
2005	-	-	6.17	4.67	5.32	4.9	11.72	5.65	9.74	6.05	4.6	4.32

There is also evidence suggesting that our assumption that the government can divert the repayment of the firms' foreign obligations is realistic. In particular, it is not uncommon for the government to take over the foreign obligations of the corporate sector in actual default episodes. The following quote by the IMF historian explains how this was done in Mexico's 1982-83 default, and notes that arrangements of this type have been commonly used since then:

"A simmering concern among Mexico's commercial bank creditors was the handling of private sector debts, a substantial portion of which was in arrears...the banks and some official agencies had pressured the Mexican government to assume these debts...Known as the FICORCA scheme, this program provided for firms to pay dollar-denominated commercial debts in pesos to the central bank. The creditor was required to reschedule the debts over several years, and the central bank would then guarantee to pay the creditor in dollars. Between March and November 1983, close to $12 billion in private sector debts were rescheduled under this program... FICORCA then became the prototype for similar schemes elsewhere."
(Boughton (2001), Ch. 9, pp. 360-361)

2.9   Recursive equilibrium

Definition 2   The model's recursive equilibrium is given by (i) a decision rule $b_{t+1}$ $\left( b_{t},\kappa_{t-1},\varepsilon_{t}\right)$ for the sovereign government with associated value function $V\left( b_{t},\kappa _{t-1},\varepsilon_{t}\right)$, consumption and transfers rules $c\left( b_{t},\kappa_{t-1},\varepsilon_{t}\right)$ and $T\left( b_{t},\kappa _{t-1},\varepsilon_{t}\right) ,$ default set $D\left( b_{t},\kappa _{t-1}\right)$ and default probabilities $p^{\ast}\left( b_{t+1} ,\varepsilon_{t}\right)$; and (ii) an equilibrium pricing function for sovereign bonds $q^{\ast}\left( b_{t+1},\varepsilon_{t}\right)$ such that:

1. Given $q^{\ast}\left( b_{t+1},\varepsilon_{t}\right)$, the decision rule $b_{t+1}\left( b_{t},\kappa_{t-1},\varepsilon_{t}\right)$ solves the recursive maximization problem of the sovereign government (30).

2. The consumption plan $c\left( b_{t},\kappa_{t-1},\varepsilon_{t}\right)$ satisfies the resource constraint of the economy

3. The transfers policy $T\left( b_{t},\kappa_{t-1},\varepsilon_{t}\right)$ satisfies the government budget constraint.

4. Given $D\left( b_{t},\kappa_{t-1}\right)$ and $p^{\ast}\left( b_{t+1},\varepsilon_{t}\right) ,$ the bond pricing function $q^{\ast}\left( b_{t+1},\varepsilon_{t}\right)$ satisfies the arbitrage condition of foreign lenders (38).

Condition 1 requires that the sovereign government's default and saving/borrowing decisions be optimal given the interest rates on sovereign debt. Condition 2 requires that the private consumption allocations implied by these optimal borrowing and default choices be both feasible and consistent with a competitive equilibrium (recall that the resource constraint of the sovereign's optimization problem considers only private-sector allocations that are competitive equilibria). Condition 3 requires that the decision rule for government transfers shifts the appropriate amount of resources between the government and the private sector (i.e. an amount equivalent to net exports when the country has access to world credit markets, or the diverted repayment of working capital loans when a default occurs, or zero when the economy is in financial autarky beyond the date of default). Notice also that given conditions 2 and 3, the consumption plan satisfies the households' budget constraint. Finally, Condition 4 requires the equilibrium bond prices that determine country risk premia to be consistent with optimal lender behavior.

A solution for the above recursive equilibrium includes solutions for $\kappa\left( q^{\ast}\left( b_{t+1},\varepsilon_{t}\right) ,\varepsilon _{t}\right)$ ,