Back Matter
  • 1 https://isni.org/isni/0000000404811396, International Monetary Fund

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Appendix 1: The Firms’ Dynamic Optimization Problem with Working Capital

We review here the optimization problem faced by a representative firm that needs working capital to pay for a fraction of the cost of imported inputs. The setup is based on a similar derivation in Uribe and Yue (2006). For simplicity, we characterize the problem in partial equilibrium, assuming that all intermediate goods are imported, and that there is a single homogeneous foreign input. We are interested in particular in the following two results:

Consider a representative firm in a small open economy that produces output by means of a production function that uses imported intermediate goods and labor as inputs,

yt=F(mt,Lt),(31)

where the function F is homogeneous of degree one, increasing in both arguments, and concave. Firms buy their inputs from perfectly competitive markets.

Production is subject to a working capital constraint that requires firms to pay in advance for a fraction θ of the cost of imported inputs, which have a world-determined relative price p. The working capital constraint is:

ktRtθpmt;θ0,

where kt denotes the amount of working capital held by the representative firm in period t.

The above formulation of the working capital constraint corresponds to a timing of transactions akin to a “cash-in- advance constraint,” by which firms must hold non-interest-bearing foreign assets by an amount equal to the fraction θ of the cost of imported inputs. There is also an alternative formulation known as the “shopping time” formulation, according to which firms need to have the working capital θpmt at the end of the period (see Uribe and Yue 2006 and Oviedo 2005). The two differ only in that the former increases the cost of inputs by θ (Rt – 1), as we show below, and the latter by θ (Rt – 1) / Rt, but in both cases the relevant interest rate is determined by the same interest rate as for one-period debt Rt.

The debt position of the firm, denoted by dt, evolves according to the following period-by-period budget constraint:

dt=Rt-1dt-1-F(mt,Lt)+wtLt+pmt+πt-kt-1+kt,

where πt denotes profits in period t, and Rt is the interest rate in one-period bonds. Thus, there is no assumption requiring that the interest rate on between period debt be the same as that on working capital loans.

Define the firm’s total net liabilities at the end of period t as atRtdt – Kf Then, we can rewrite the budget constraint as:

atRt=at-1-F(mt,Lt)+wtLt+pmt+πt+(Rt-1Rt)kt.

Assume the interest rate is positive at all times. This implies that the working capital constraint always binds, or otherwise the firm would incur in unnecessary financial costs, which would be suboptimal. Since the working capital constraint holds with equality, we can eliminate kt from the above expression to get:

atRt=at-1-F(mt,Lt)+pmt[1+θ(Rt-1)]+wtLt+πt.(32)

The first result we highlighted earlier is evident in this expression: The working capital constraint increases the unit cost of imported inputs by the amount θ(Rt – 1), which is increasing in the interest rate Rt, where Rt is the same interest rate as the one charged on one-period debt.

The firm’s problem is to maximize the present discounted value of the stream of profits. In the paper the owners are domestic residents, so firms discount at the households’ stochastic discount factor βtλtλo, where λt = β RtEtλt+1 is the Euler equation for bond holdings. Alternatively, firms can be assumed to discount profits at the world interest rates Rt. For the results we show here this does not matter. Under the first alternative, the firm’s problem is,

maxE0Σt=0βtλtλ0πt.

Using constraint (32) to eliminate πt from the firm’s objective function, the firm’s problem can be stated as choosing at, mt, and Lt so as to maximize

E0Σt=0βtλtλ0{atRt-at-1+F(mt,Lt)-pmt[1+θ(Rt-1)]-wtLt},

subject to the no-Ponzi-game constraint limjEtat+jπs=0jRt+s<0.

The first-order conditions associated with this problem are the Euler equation for net liabilities, λt = β RtEtλt+1, the no-Ponzi-game constraint holding with equality, and

Fm(mt,Lt)=p[1+θ(Rt-1)](33)
FL(mt,Lt)=wt.(34)

It is clear from condition (33) that the working capital requirement drives a wedge between the marginal product of imported inputs and their world relative price p. This wedge is larger the larger the financing cost of working capital, (Rt – 1), or the higher the fraction of the cost of imported inputs that needs to be paid with credit, θ.

It is critical to note that, since total net liabilities are irrelevant for the optimal choices of labor and imported inputs and the payoff function of the firm is linear with respect to net liabilities (and all the terms in λt = βRtEt λt+1 are exogenous to the firm’s choices), any process at satisfying equation (32) and the firm’s no-Ponzi-game constraint is optimal. Hence, if firms start out with zero net liabilities, then an optimal plan consists in holding no liabilities at all times (at = 0 for all t ≥ 0), with distributed profits given by

πt=F(kt,ht)-pmt[1+θ(Rt-1)]-wtLt

This implies that firms do not accumulate precautionary holdings of assets, regardless of the input prices they face. Their choices of labor and imported inputs follow from the optimality conditions (33)-(34), which depend only on current values of factor prices, the rate of interest and TFP. These are not assumptions attached to the working capital requirement, but a result that follows from the linear nature of the firm’s payoff: If firms maximize the present value of profits, with discount rates independent of the firm’s choices and net liabilities entering linearly in profits, there is no incentive for precautionary asset holdings by firms. One can of course propose alternative formulations that deviate from these conditions and would produce precautionary asset demand by firms, but these are not conditions assumed in the setup of the model.

Appendix 2: Decentralized Equilibrium

The social planner’s problem studied in Section 2 can be decentralized by formulating the problem of the sovereign as an optimal policy problem akin to a Ramsey problem.39 The government chooses a debt policy (amounts and default or repayment) that maximizes the households’ welfare given a bond pricing function qt (bt+1, εt) and subject to the constraints that: (a) the private sector allocations must be a competitive equilibrium; and (b) the government budget constraint must hold. The first constraint is dealt with by using conditions (14)-(20) to write down recursive functions that represent the competitive equilibria of factor allocations and factor prices as functions of bond prices and TFP: k (qt(bt+1, εt), M (qt (bt+1, εt), εt), m* (qt (bt+1t), εt), md (qt(bt+1t), Lf (qt(bt+1t), εt, Lm (qt, (bt+1, εt), εt) and L(qt (bt+1, εt), εt), recalling that there is a one-to-one mapping between qt (bt+1, εt) and rt. These functions are then entered into the government’s optimization problem below.

This decentralization also requires an assumption about how the government deals with the diverted repayment of working capital loans. This diverted repayment can be treated as a cost of default, in which case we obtain the same allocations and prices in the decentralized equilibrium as the planner’s problem of Section 2. Alternatively, if the government rebates the repayment as a lump-sum transfer to households, the stock of working capital becomes an extra state variable, because of the income effect on households resulting from the transfer of the working capital repayment when the country defaults. To consider both cases, in what follows we define the state variables of the government’s problem as (bt, kt-1, εt), but kt–1 is only a relevant state if we assume that diverted repayments of working capital are rebated to households. Note, however, that since kt-1 is small in the calibration, our quantitative results change very little regardless of whether we assume that these repayments are rebated or not.

The recursive optimization problem of the government is:

V(bt,kt-1,εt)=max{vnd(bt,εt),vd(kt-1,εt)}.(35)

The continuation value is defined as follows:

vnd(bt,εt)=maxct,bt+1{u(ct-g(L(qt(bt+1,εt),εt)))+βE[V(bt+1,k(qt(bt+1,εt),εt),εt+1)]},(36)

subject to

ct+qt(bt+1,εt)bt+1-bt<εtf(M(qt(bt+1,εt),Lf(qt(bt+1,εt),εt),k)-m*(qt(bt+1,εt),εtPm*(1qt(bt+1,εt)-1),(37)

where f()=MαM(Ltf)αLkαk Note that the constraint of this problem is again the resource constraint of the economy at a competitive equilibrium.

The working capital loans kt-1 and kt do not enter explicitly in the continuation value or in the resource constraint but they are relevant state variables, 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. In particular, the value of default is:

vnd(kt-1,εt)=maxct{u(ct-g(L˜(εt)))+β(1-ϕ)Evd(0,εt+1)+βϕEV(0,0,εt+1)]},(38)

subject to:

ct=εtf(M˜(εt),L˜f(εt),k)-m*(εt)P˜*+kt-1.(39)

Note that νd (kt-1t) takes into account the fact that in case of default at date t, the country has no access to financial markets that 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 the subset Ω of imported inputs, M˜(ε),L˜(ε) and L˜f(ε) are competitive equilibrium allocations that correspond to the case when the f sector operates without those inputs, and P˜*=[θ1(pj*)vv1dj]v1v. Moreover, because the defaulting government diverts the repayment of last period’s working capital loans, household income includes government transfers equal to the appropriated repayment kt-1 (i.e., on the date of default, the government budget constraint is Tt = kt-1).

The default set is now defined as:

D(bt,kt-1)={εt:vnd(bt,εt)<vd(kt-1,εt)}.(40)

This default set has a different specification than in the typical Eaton-Gersovitz model (see Arellano (2008)), 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 bt and kt-1, instead of just bt. Despite of this, however, the probability of default remains a function of bt+1 and εt only. This is because the f sector’s optimality conditions imply that the next period’s working capital loan kt depends on εt and the interest rate, which is a function of bt+1 and εt. Thus the probability of default at t + 1 perceived as of date t for a country with a productivity εt and debt bt+1, pt (bt+1, εt) can be induced from the default set, the decision rule for working capital, and the transition probability function of productivity shocks μ(εt+1t) as follows:

pt(bt+1,εt)=D(bt+1,kt)dμ(εt),where kt=k(qt(bt+1,εt).(41)

To show the equivalence between the decentralized equilibrium when diverted working capital repayments are not rebated and the planner’s problem of Section 2, consider first that, without that rebate, kt-1 is removed from the set of state variables in the government’s problem, and from the right-hand-side of the resource constraint when the country defaults. Next, consider the fact that the recursive functions defined above to characterize factor allocations and prices as functions of qt (bt+1, εt), εt imply that conditions (14)-(20) hold. These are identical to the first-order conditions that set factor allocations for the social planner in Section 2, and the corresponding shadow prices of that planner’s problem determine the same wage rate and price of domestic inputs. From these results it follows that, for given qt (bt+1, εt), bt+1 and bt, the resource constraints in the continuation and default branches are also identical in the two problems, which therefore means that the value functions and optimal bond decision rules, assuming these are well-defined, are also identical.

Appendix 3: Theorem Proofs

Proof of Theorem 1

Given a productivity shock ε, the utility from defaulting νd (ε′) is independent of b. We can also show that the utility from not defaulting νnd (b, ε′) is increasing in 6t+1. Therefore, if V (b1ε′, = νd ε′, then it must be the case that V (b0,ε′) = νd (ε′). Hence, any ε′, that belongs in D (b1, ε) must also belong in D (b0, (ε).

Let d*(b, ε′) be the equilibrium default decision rule. The equilibrium default probability is then given by

p(b,ε)=d*(b,ε)(ε|ε).

From D(b1ε′) ⊆D(b0, ε′), if d* b1ε′) = 1, then d* (b0, ε′) = 1. Therefore,

p(bo,ε)p(b1,ε).

Proof of Theorem 2

From Theorem 1, given a productivity shock b0< b1 ≤ 0, p* (b0ε) ≥p* (b1ε). The equilibrium bond price is given by

q(b,ε)=1p(b,ε)1+r.

Hence, using Theorem 1, we obtain that:

q(b0,ε)<q(b1,ε).

Appendix 4: Data Definition and Data Source

Table A1 describes the variables and data sources of our data set for cross-country event studies. Table A2 summarizes the list of countries, default episodes, and the available variables in the analysis.

Table A1:

Variables and Sources

article image
Table A2:

List of countries and variables included in the event analysis.

article image

The default episodes are based on Yeyati and Panizza (2011), Benjamin and Wright (2010), and Standard and Poors report. Because we need the measure of economic variables which typically reflect the impact of default with some lag, we use the quarter after the default announcement date in the event analysis. This treatment is the same as Yeyati and Panizza who study the drop of GDP in the post-default quarters.

GDP, consumption, and trade balance/GDP are from International Financial Statistics and Yeyati and Panizza (2011). Yeyati and Panizza (2011) compiled the real GDP for countries from national sources for periods when IFS does not record their GDP. GDP, consumption, and trade balance/GDP are H-P detrended.

The imported intermediate inputs are sum of categories for intermediate goods based on the classification of Broad Economic Categories (BEC). The categories for intermediate goods are: (111*) Food and beverages, primary, mainly for industry, (121*) Food and beverages, processed, mainly for industry, (21*) Industrial supplies not elsewhere specified, primary, (22*) Industrial supplies not elsewhere specified, processed, (31*) Fuels and lubricants, primary, (322*) Fuels and lubricants, processed (other than motor spirit), (42*) Parts and accessories of capital goods (except transport equipment), (53*) Parts and accessories of transport equipment.

Intermediate goods are from United Nation, National Accounts Official Country Data. The data is taken from Table 4.1 Total Economy (S.1), I. Production account - Uses Intermediate consumption, at purchaser’s prices.

Labor data is the total paid employment data from LABORSTA dataset collected by International Labor Organization.

Because of the serious data limitation for imported inputs, total intermediate goods, and labor, we cannot use HP filter. Imported inputs and total intermediate goods are log-linearly detrended. Labor data is indexed so that the employment 4 years before default is 1.

*

The previous version of this paper was entitled † solution to the Disconnect between Country Risk and Business Cycles Theories.† We thank Cristina Arellano, Mark Aguiar, Andy Atkeson, Fernando Broner, Jonathan Eaton, Gita Gopinath, Jonathan Heathcote, Olivier Jeanne, Pat Kehoe, Tim Kehoe, Narayana Kocherlakota, Guido Lorenzoni, Andy Neumeyer, Fabrizio Perri, Victor Rios-Rull, Thomas Sargent, Stephanie Schmitt-Grohe, Martin Eribe, Mark Wright, and Jing Zhang for helpful comments and suggestions. We also acknowledge comments by participants at various seminars and conferences.

This version of the paper was prepared while Enrique Mendoza was a visiting scholar with the Research Department, and he is grateful for Department’s hospitality and support.

1

Neumeyer and Perri used data for Argentina, Brazil, Korea, Mexico and the Philippines. Uribe and Yue added Ecuador, Peru and South Africa, but excluded Korea.

3

See, for example, Aguiar and Gopinath (2006), Arellano (2008), Bai and Zhang (2005) and Yue (2010).

4

As a result, part of the output drop that occurs when the economy defaults shows as a fall in the Solow residual (i.e. the fraction of aggregate GDP not accounted for by capital and labor). This is consistent with the data from emerging markets crises showing that a large fraction of the observed output collapse is attributed to the Solow residual (Meza and Quintin (2006), Mendoza (2010)). Moreover, Benjamin and Meza (2007) show that in Korea’s 1997 crisis, the productivity drop followed in part from a sectoral reallocation of labor.

5

Arellano (2008) obtained a mean debt-output ratio of 6 percent using her asymmetric cost. Aguiar and Gopinath (2006) obtained a mean debt ratio of 19 percent using the fixed percent cost, but at a default frequency of only 0.23 percent. Yue (2010) used the same cost in a model with renegotiation calibrated to observed default frequencies, and obtained a mean debt ratio of 9.7 percent. Studies that have obtained higher debt ratios with modifications of the Eaton-Gersovitz environment, but still assuming exogenous endowments, include: Cuadra and Sapriza (2008), D’Erasmo (2008), Bi (2008a) and (2008b), Chatterjee and Eyigungor (2008), Benjamin and Wright (2008), and Lizarazo (2005).

6

If the inputs are perfect substitutes there is no output cost of default, because firms can shift inputs without affecting production and costs. If they are complements, production is either zero (with unitary elasticity of substitution) or not defined (with less-than-unitary elasticity) when the economy defaults and cannot access imported inputs.

7

Removing the wealth effect on labor supply is useful because otherwise the wealth effect pushes labor to display a counterfactual rise when TFP falls or when consumption drops sharply, as is the case in default episodes.

8

This assumption is very common in the Eaton-Gersovitz class of models but it is not innocuous, because whether private foreign debt contracts are allowed, and whether they are enforceable vis-a-vis government external debt, affects the efficiency of the credit market equilibrium (see Wright (2006)).

9

Sovereign debt models generally abstract from capital accumulation for simplicity. Adding capital makes the recursive contract with default option significantly harder to solve because it adds an additional endogenous state variable. Moreover, changes in the capital stock have been estimated to play a small role in output dynamics around financial crises (see Meza and Quintin (2006) and Mendoza (2007)).

10

This structure of aggregation of imported and domestic inputs is similar to those used in the empirical work of Gopinath and Neiman (2010) and Halpern, Koren and Szeidl (2009).

11

We assume that the entire cost of purchasing the varieties in Ω needs to be paid in advance. Hence, θ determines the “intensitity” of the working capital friction in a similar way as the standard working capital models use θ to define the fraction of the cost of a single input that is paid in advance (e.g. Neumeyer and Perri (2005) and Uribe and Yue (2006). We could also introduce an extra parameter so that the varieties in Ω require that only a fraction of their cost be paid in advance, but lowering this fraction would have similar effects as keeping the fraction at 100 percent and lowering Ω instead.

12

We use P * = 1, which follows from the fact that pj*=1 for all j ∈ [0.1] and assuming a zero real interest rate in the base year. Note, however, that changes in our quantitative results are negligible if we use the equilibirum price index P * (rt) instead, because default is a low frequency event, and outside default episodes interest rates display very small fluctuations.

13

b max exists when the interest rates on a country’s saving are suffciently small compared to the discount factor, which is satisfied in our paper since (1 +r *) β < 1.

14

We asbtract from debt renegotiation. See Yue (2010), Bi (2008b) and Benjamin and Wright (2008) for quantitative studies of sovereign default with renegotiation.

15

Notice that existing models of emerging markets business cycles with working capital (e.g. Neumeyer and Perri (2005) and Uribe and Yue (2006)) already assume that the sovereign interest rates and priviate financing costs are equal. Here we endogenize interest rates and the two rates are equalized as an equilibrium outcome.

16

We show in Appendix 2 that this planner’s problem yields the same equilibrium as a decentralized Ramseylike equilibrium in which the government maximizes households’ utility subject to the resource constraints, the government budget constraint, the constraint that factor allocations need to be consistent with private equilibrium conditions, and the assumption that the diverted working capital repayments are not rebated to households (i.e. they are an extra default cost or a tax used to finance unproductive government purchases). We also discuss in Appendix 2 the decentralized equilibrium in the alternative case in which these repayments are rebated.

17

Arellano and Kocherlakota (2007) and Agca and Celasun (2009) provide further empirical evidence of the positive relationship between private domestic lending rates and sovereign spreads. Corsetti, Kuester, Meier and Muller (2010) show that this feature is also present in the data of OECD countries.

18

In addition, since factor allocations satisfy conditions (14)-(20), these allocations are also consistent with a competitive equilibrium in factor markets.

19

Note that the threshold would be at the unitary elasticity of substitution if labor supply were inelastic.

20

This is the case in turn because of the “strong” convexity of Cobb-Douglas marginal products. Consider for simplicity the case in which production εF (m) requires a single input m. In this case, “strong convexity” means that F (m) satisfies F ′′′(m) > (F ′′(m))2/F′(m), which holds in the Cobb-Douglas case.

21

We also found that adjusting A has qualitatively similar effects as changing ω.

22

In Figure 6, we hold constant pm for simplicity. At equilibrium, the relative price of domestic inputs changes, and this alters the value of the marginal product of Ld, and hence labor demand by the m sector. The results of our numerical analysis do take this into account and still are roughly in line with the intuition derived from Figure 6.

23

The last effect hinges on the fact that the gap between LmD and LD widens as the wage falls. This is a property of factor demands with Cobb-Douglas production.

24

Mendoza (2010) reports a very similar share for Mexico, and Gopinath, Itskhoki, and Rigobon (2010) show shares in the 40-45 percent range for several countries.

25

The two studies use different definitions of re-entry. Gelos et al. use actual external bond issuance of public debt. Dias and Richmond define rentry when either the private or public sectors can borrow again, and they also distinguish partial reaccess from full reaccess (with the latter defined as positive net debt flows larger than 1.5 percent of GDP). Gelos et al. estimate an avearge exclusion of 5.4 years in the 1980s and nearly 1 year in the 1990s.

26

Several countries have input expenditure ratios similar to Mexico’s, but the ratios can vary widely. Goldberg and Campa (2008) report ratios of imported inputs to total intermediate goods for 17 countries that vary from 14 to 49 percent, with a median of 23 percent. This implies ratios of imported to domestic inputs in the 16-94 percent range, with a median of 30 percent.

27

The prediction that net exports go to zero when the country is excluded from credit markets is not particular to our model. All existing quantitative models of sovereign default in the Eaton-Gersovitz class have the same feature, because the only way to finance a trade imbalance in this class of models is with foreign credit.

28

Only 23 countries have defaulted on the IMF since it was created in 1945 (see Aylward and Thorned (1998)), and these are low income countries or countries in armed conflicts without access to private lenders (e.g. Liberia, Somalia, Congo, Sudan, Afghanistan, Iraq). In all the sovereign defaults included in the event analysis of Figure 1, payments to the IMF continued even after countries defaulted on private lenders, except in the case of Peru in the 1980s.

29

A is useful for targeting the output drop at default because, as mentioned in Section 2, changes in A have similar effects as changes in ω. In particular, lower values of A yield larger output drops at default without altering the slope of the relationship between TFP and these output drops.

30

Argentina declared default in the last week of December in 2001, but it is reasonable to assume that, in quarterly data, the brunt of the real effects of the debt crisis were felt in the first quarter of 2002. Arellano (2008) also follows this convention to date the default as of the first quarter of 2002. She estimated the output cost at 14 percent, measured as a deviation from a linear trend.

31

Note that β is relatively low compared to typical RBC calibrations, but is in the range of values used in sovereign default models (e.g. β in Aguiar and Gopinath (2006), Arellano (2008), and Yue (2010) ranges from 0.8 to 0.953). These lower discount factors are often justified by arguing that political economy incentives lead government decision-makers to display higher rates of time preference.

32

We exclude financial autarky periods in computing correlations with spreads because in the model the default spread goes to infinity when the economy defaults, and hence correlations with the country interest rate are undefined.

33

We provide both the cross-country medians and the observations for Argentina’s default because we aim to illustrate how well the model can match both the behavior across the default events in our cross-country dataset and the data for Argentina. The former is harder because the model’s calibration is based on Argentine data, and thus misses cross-country variation in the model’s key parameters.

34

As explained earlier, we do not show interest rates for t ≥ 0 because in the default state the default risk is infinite.

35

To make debt ratios comparable across data and model during periods of exclusion, we adjusted the model’s measure to match the practice followed in the World Bank dataset, which includes defaulted debt and the corresponding interest in arrears in the debt estimates. Thus, the mean debt ratio for the model after default in the event plot is the average of the pre-default debt ratio and the debt ratio chosen in the case of re-entry.

36

Mendoza (2010) proposed an alternative model of endogenous Sudden Stops based on collateral constraints and Irving Fisher’s debt-deflation mechanism instead of sovereign default risk.

37

We also generated plots with the default event dynamics of output comparable to Figure 7. The quantitative differences are small and qualitatively they all have the same pattern, so we decided not to put them in the paper. We also conducted the full sensitivity analysis without the exogenous component of capital flows xt. Most of the results are similar to those reported in Table 4, which again verifies that xt plays a minor role, except for enabling the model to produce the trade surplus after default.

38

Note that the calibrated proportional drop in TFP is less than 1 percent in this experiment. If we set a 2 percent drop as in the literature, the output drop in default becomes 15 percent, the mean debt ratio is 24 percent, and the average spread falls to 0.01 percent, which are results in line with the literature (e.g. Aguiar and Gopinath (2006)).

39

See Cuadra and Sapriza (2007) for an analysis of optimal fiscal policy as a Ramsey problem in the presence of sovereign default in an endowment economy.

A General Equilibrium Model of Sovereign Default and Business Cycles
Author: Miss Zhanwei Z. Yue and Mr. Enrique G. Mendoza