The Exchange Rate Mismatching in a General Equilibrium Approach Ilan Goldfajn Marcos Antonio Silveira Central Bank of Brazil PUC-RJ1 PUC-RJ June, 2001 1 PontiÞcal Catholic University of Rio de Janeiro Abstract A general equilibrium model is built to explain how exchange rate volatility smoothing policies may bring a Pareto-improvement for a small open economy. The model shows that this may occur when the home economy is paying a large spread on the default risk-free world interest rate and market imperfections, such as nontradable goods and imperfect information, prevent home economy’s Þrms from internalizing all beneÞts and costs of the exchange rate risk realignment into their allocative decisions. The reason is that the wealth volatility of an individual Þrm impacts on both its foreign credit’s supply and demand curves and then on the interest rate it pays on its foreign liabilities. 1 Introduction A general equilibrium model is built to explain how market imperfections, such as nontradable goods and information asymmetry between foreign creditors and home borrowers, allow exchange rate volatility smoothing (ERVS) policies to bring a Pareto-improvement for a small open economy. Fundamentally, given the balanced government budget constraint, these policies amount to a realignment of the exchange rate risk exposure across the home economy. However, if they are efficient, in the sense that they bring a Pareto-improvement for the home economy, why don’t competitive markets signal the correct incentives to the risk reallocation? This question is mainly relevant for many emerging markets economies with a well developed Þnancial market, for which market incompleteness can not be used as a ground for policy justiÞcation. With full information and perfect markets, the risk inherent to any source of uncertainty must be efficiently reallocated across market participants. As a consequence, Pareto-improvement is possible only if there is some positive externality underlying the risk redistribution across the home economy which is not efficiently allocated by the market. More precisely, the welfare gain provided by ERVS policies must be enough large to pay the sectors with a broader exchange rate risk exposure. The model shows that this may arise when the home economy is paying a spread over the default risk- free world interest, either because reputational costs are not strong enough to induce repayment or because foreign creditors are overpessimistic about the home economy’s performance. In this case, as the foreign debt burden on the tradable sector falls as a result of ERVS policies, the home economy must export less to Þnance the capital account’s deÞcit, increasing in this way the supply of tradable goods for the home market. As a result, not only the tradable sector wealth and welfare increase, but also the nontradable sector is beneÞted by a higher relative price for its output. It is important to note that exchange rate mismatching is observed in both sectors. Unless the tradable goods’s weight in the price index is very small, even the tradable sector’s wealth is not fully immunized to exchange rate shocks. Assuming the law of one price, the effect of a higher exchange rate on the tradable goods’s relative price is partially neutralized by a higher general level of prices. It rests then a question: how can ERVS policies to affect the foreign debt cost? The answer is that both 1 the foreign loans’s demand and supply curves faced by each home resident depend on its wealth volatility, which impacts on the cost of its debt directly, as it changes its default probability, and indirectly, as it changes its incentives for production. Therefore, if home residents fail to internalize all beneÞts and costs of the exchange rate risk realignment into their allocative decisions, there is a scope for ERVS policies. The model examines two reasons why this may occur, both related to market failures commonly referred in international economics. Firstly, foreign creditors might be imperfectly informed about the home Þrms’s economic and Þnancial standing. As an example, they could observe only the average level of production and wealth volatility of each home economy’s sector. Therefore, as each individual Þrm is able to free ride on the rest of its sector, not accepting a larger exchange rate risk exposure turns into a strictly dominant strategy, even if a lower foreign debt cost makes the net welfare effect of this action positive. Secondly, even if foreign creditors are fully informed, it is impossible for nontradable Þrms to prevent the rest of its sector from taking advantage of a higher relative price. The model also explores in some detail the different channels through which ERVS policies may affect the interest rate that the home economy pays on its foreign debt. These effects turn out to be very ambiguous, so that it is important to understand their determinants. As an example, reducing the tradable sector’s wealth volatility may or not promote exports and then increase the foreign credit supply. Even so, credit demand also increases with wealth in order to smooth consumption over time. Moreover, ERVS policies have also an ambiguous and direct effect on the foreign credit supply for the home economy as it changes the wealth volatility of their sectors. To a certain extent, the paper goes along the lines of the literature on the determinants of the optimal currency composition of the foreign debt. The model does not explain why the foreign debt is denominated mostly in foreign currency. Rather, we take this fact as an assumption, ”the original sin” by Eichengreen (1999). Although we model a nonmonetary economy, home and foreign shocks to exchange rate can be proxied by a productivity shock on the home economy’s tradable sector. 2 2 Description of the Model This section describes the central aspects of the economy that we model to explain the main issues presented in the previous section. 2.1 World economy: basics Consider a non-monetary, small open economy, which lasts for two periods: t = 0, 1. We call this economy and the rest of the world as home country and foreign country respectively, indexed by j = H, F . The home country comprises a tradable and a nontradable sector, indexed by i = T, N T . Each sector produces a single good. The home country is competitive: there is a large number of identical individuals in each sector. Then, we can assume a representative agent for each sector, which realizes that its individual actions have no effect on the market prices. For sake of simplicity, foreign country’s residents are risk- neutral, whereas home country’s residents are risk averse. We assume rational expectations and that home country’s sectors share the same information set. There are no barriers to the international ßow of goods and capital. The subscript t indicates that a variable is known from period t on. 2.2 Technology All the goods supplied for the home market at period 1 must be produced only with labor. For this, each sector has access to a technology, described by the production function i i ¡ i¢ ki 1 ¡ i ¢1+λi y1 = y1 l 0 ≡ i l0 , (1) 1+λ where y1 is the sector i’s output at period 1, l0 is the sector i’s labor supply at period 0 and ki is a i i 1 productivity shock, which is the only source of uncertainty in the model. The parameter λi is the labor- elasticity of the sector i’s output. We assume constant or decreasing returns to scale by imposing λi ≤ 0. The production of both goods takes one period. An important assumption is the information asymmetry with respect to the home country’s performance at period 1: in the country j’s beliefs, formed at period 3 ¯ 0, ki is a random variable uniformly distributed between ki and ki . More formally, 1 ¯j j £ ¤ ¯ ki ∼ U ki , ki , (2) 1 j j ¯ conditioned on all information available for country j at period 0, where ki ≡ µi + η i − 1 ; (3) ¯j j j ¯ ki ≡ µi − ηi + 1 ; (4) j j j µi ≥ 1 ; 1 > η i > 0 . j j (5) It follows from (3)-(5) that ¯ ki > ki > 0 ; (6) j ¯j j £ i¤ E0 k1 = µi ;j (7) £ i¤ 1¡ ¢2 V ARj k1 = 0 1 − ηi . j (8) 3 The parameters µi and η i determine the mean and the volatility of the shocks on the sector i’s productivity. j j In addition, we deÞne the parameters αi and ρi as µi F αi ≡ i ; (9) µH ηi F ρi ≡ i , (10) ηH so that they measure, respectively, how much divergent the home and the foreign countries’s beliefs are with respect to the mean and the volatility of the home sectors’s productivity: the greater they are, the deeper the divergence is. The nontradable sector is endowed with a positive amount of the nontradable good at period 0, which NT is denoted by y0 . There is no exogenous endowment of the tradable good, so that its supply for the home market at period 0 is provided only by importation. This is a technical assumption that forces the tradable sector to be a net foreign debtor, which is well appropriate to the purpose of the paper. 4 2.3 International capital market International capital market is competitive. The only available asset for intertemporal wealth transference between the home and foreign countries is an one-period bond denominated in tradable goods. Both home country’s sectors may have incentive to default on the foreign debt. As the default prob- ability may differ across sectors, the interest rate they pay for the loans are not necessarily equal. The reputational costs implied by default lead to a positive loss of utility, denoted by ²i . Nevertheless, the H foreign creditors may have a different belief of this loss, which in turn is denoted by ²i . This difference F may result from an information asymmetry as to the nature and size of the costs incurred by the sector i in case of default. By assumption, foreign debtors never default when the home country is lending to the foreign country. We deÞne the parameter φj (j = T, N T ) as ¡ ¢ j 1 − exp −²j F φ ≡ , (11) 1 − exp (−²i ) H which determines how much divergent the home and foreign country’s beliefs are with respect to the utility loss caused by default: the smaller φj , the greater this divergence is. The foreign creditors are capable of monitoring only the home country’s aggregate labor supply. The labor supplied by a particular Þrm in not observed directly, but only deducted indirectly from the aggregate level and from the fact that sector T ’s producers are identical and then supply the same amount of labor. As we will see later, this assumption is crucial for determining the incentives that the tradable sector has to increase its production in order to improve the credit terms on its foreign debt. In addition, we make the somewhat strong assumption that the foreign creditors realize that the nontradable producers have much less incentive to repay loans than the tradable producers. A theoretical justiÞcation is that the reputational costs could result mostly from the loss or reduction of foreign trade credit, which is the main source of funding to export. For sake of simplicity, we suppose that φNT = 0: foreign creditors are so pessimistic about the sector NT ’s willingness to repay loans that it has no access to the international capital market. In addition, we assume that foreign creditors can monitor the Þnancial accounts of the tradable producers, so that the home capital market can not be used to transfer foreign funds to the nontradable sector. 5 2.4 Preferences We assume that the home country’s sectors consume both goods in each period and that increasing labor supply reduces welfare.Then, the sector i’s preferences can be represented by the lifetime utility function ¡ ¢ £ ¡ ¢¤ ¡ i¢ u0 ci + βE0 u1 δ i , ci − v i l0 , 1 > β > 0 , 0 1 (12) such as ¡ ¢ ¡ ¢ u0 ci = ln ci ; 0 0 (13) ¡ i i¢ ¡ i¢ u1 δ , c1 = ln c1 − δ i ²i ;j (14) h iθ h i1−θ ci = c (T )i t t c (NT )i t , 0< θ<1; (15) ¡ i¢ 1 ¡ i ¢2 vi l0 = l , (16) 2 0 where β is the subjective temporal discount factor, θ is the preference parameter, c (T )i and c (NT )i are t t the sector i’s demand for the tradable and nontradable good at period t respectively, ci is a composite t consumption index for sector i at period t and δ i is an indicator function deÞned by ½ i 1 ; if sector i defaults ; δ = . (17) 0 ; if sector i does not default. The period 1-utility in the equation (14) depends on the consumption and on whether the sector i defaults or not. The labor desutility function in the equation (16) is strictly increasing and convex. 2.5 Relative prices The tradable good is the numeraire of the home country. The sector i’s total expenditure at period t, denoted by ei , is deÞned by the function t ¡ ¢ ei = ei pT , pNT , ci ≡ t t t t t min pT c (T )i + pNT c (NT )i t t t t (18) c(T )i t , c(NT )i t h iθ h i1−θ i i s.a. ci t = c (T )t c (N T )t , 6 where pT and pNT are the prices of the tradable and the nontradable goods at period t respectively. Solving t t the optimization problem (18), we have that · ¸1−θ θ NT c (T )i = p ci ; (19) t 1−θ t t · ¸−θ θ NT c (NT )i = p ci , (20) t 1−θ t t whereas the total expenditure of both sectors can be written as ¡ ¢ ¡ ¢1−θ i ei = ei pNT , ci = ϕ pN T t t t t t ct , (21) where ϕ ≡ θ−θ (1 − θ)θ−1 . Note that pT was omitted as argument of the function in (21) because, by t assumption, pT = 1. Finally, the aggregate price level, denoted by pt , is deÞned as t ¡ ¢ ¡ ¢1−θ pt = ei pN T , 1 = ϕ pNT t t t , (22) Note that pt can be seen as a consumption-based price index: it is the minimal total expenditure to have ci = 1. . t 2.6 Policy instrument Aiming to implement a reallocation of the home country’s exposure to the productivity shocks across sectors, the government transfers for only one sector, at period 0, a given amount of a Þnancial asset that yields, at period 1, a pay-off (per unit) given by ¡ T ¢ T £ T¤ k1 − µT ¡ T ¢1+λT H y1 − E0 y1 = − l0 . (23) 1 + λT There is no disbursement at period 0: the asset can be seen as a derivative similar to a future contract. The asset works as a policy instrument to smooth the home country’s wealth volatility: the pay-off is negative (positive) when the tradable sector’s output is above (below) its expected level. From now on, we call this asset as the smoothing security. The amount supplied of this security and the recipient sector are determined by the size and the sign of the variable h0 , which summarize all the information on the 7 volatility smoothing policy: when h0 > 0 (h0 < 0), the tradable (nontradable) sector is endowed with |h0 | units of the security. This variable is exogenously determined by the government and should be regarded as an economic policy parameter. The fact that only one sector’s wealth volatility can be effectively reduced follows directly from the balanced government budget constraint. The reason is that this volatility smoothing policy works as a channel of transmission of the effect of productivity shocks on the wealth of the home country’s sectors. By changing the tax burden on the sector not holding the smoothing security, the government is able to transfer to this sector the effect of a shock on the wealth of the sector holding the security. We also assume that only the aggregate supply of the smoothing security can be observed by foreign creditors. The amount that each particular producer has in its portfolio can only be deducted indirectly from the aggregate level and from the fact that identical individuals have the same incentives. This assumption is crucial for determining the willingness of each sector in accepting or not the public supply of the security. 2.7 Consumer-producer behavior Both sectors maximize the lifetime utility, subject to an intertemporal constraint, given by p0 ci = pi y0 + di , 0 0 i 0 (24) ¡ T ¢ ¡ ¢ T k1 − µT ¡ T ¢1+λT H p1 cT = y1 − 1 + g0 d0 − 1 T T l0 h0 ; (25) 1 + λT ¡ ¢ NT NT NT ¡ NT ¢ NT kT − µT ¡ T ¢1+λT 1 H p1 c1 = p1 y1 − 1 + g0 d0 + l0 h0 , (26) 1 + λT where di is the sector i’s net foreign debt and g0 is the interest rate on this debt. Note that, by assumption, 0 i y0 = 0 and pT = 1. T 0 8 2.8 Market equilibrium All the home country’s markets clear in both periods, so that yt − xT = c(T )T + c(T )NT ; T t t t (27) NT yt = c(NT )T + c(NT )NT ; t t (28) where xT is the tradable good’s net exports from the home country to the foreign country at period t. t 3 General Equilibrium This section derives and interprets the general equilibrium solution for the world economy. First, we derive equations for exports, prices and consumption as functions of the vector z0 ≡ (di , g0 , l0 )i=T,NT , which 0 i i comprises the net foreign debt, the interest rate on this debt and the labor supply for both sectors. Next, we Þnd a general equilibrium solution for these variables. 3.1 Exports, prices and consumption It follows from the equations (21)-(22), (24)-(26) and (27)-(28) that the home country’s balance of payments is given by ¡ ¢ xT + dT + dNT = 0 ; 0 0 0 (29) ¡ ¢ ¡ ¢ xT − 1 + g0 dT − 1 + g0 dN T = 0 . 1 T 0 NT 0 (30) Then, the tradable good’s exports is given by ¡ ¢ xT = xT (z0 ) = − dT + dNT ; 0 0 0 0 (31) ¡ ¢ ¡ ¢ xT = xT (z0 ) = 1 + g0 dT + 1 + g0 dNT . 1 1 T 0 NT 0 (32) 9 Substituting (19)-(20) and (31)-(32) into (27)-(28), we have that the prices of the goods are given by pT = pT (z0 ) = 1 ; t t (33) 1 − θ dT + dNT 0 0 pNT = pNT (z0 ) = 0 0 NT ; (34) θ y0 ¡ ¢ ¡ ¢ 1 − θ y1 − 1 + g0 dT − 1 + g0 T dN T T T 0 N 0 pNT = pNT (z0 ) = 1 1 NT , (35) θ y1 whereas substituting (34)-(35) into (22), we have that the price index is given by µ ¶1−θ 1 dT + dNT 0 0 p0 = p0 (z0 ) = NT ; (36) θ y0 à ¡ ¢ ¡ ¢ !1−θ 1 y1 − 1 + g0 dT − 1 + g0 dNT T T 0 NT 0 p1 = p1 (z0 ) = NT . (37) θ y1 Finally, it follows from the equations (1) and (24)-(26) that the consumption indices are given by pi i 1 ci 0 = ci 0(z0 ) = 0 y0 + di , i = T, N T ; (38) p0 p0 0 ¡ ¢ 1 kT 1 ¡ T ¢1+λT 1 ¡ ¢ T 1 kT − µT ¡ T ¢1+λT 1 cT 1 = cT (z0 ) = 1 l − T 1 + g0 d0 − l0 h0 ; (39) p1 1 + λT 0 p1 p1 1 + λT ¡ NT ¢1+λNT ¡ ¢ ¡ T ¢1+λT pNT kNT l0 1 1 1 ¡ ¢ NT 1 kT − µT l0 1 h0 cNT 1 NT = c1 (z0 ) = NT − NT 1 + g0 d0 + T , (40) p1 1+λ p1 p1 1+λ whereas the consumption levels of both goods are given by the equations (21)-(22). 3.2 Condition for default As we want to derive the default probability of the home country’s sectors, we assume throughout this section that di > 0 for i = T, NT . Lifetime utility maximization implies that sector i repays its debt only 0 when the utility gain with default, denoted by χi , is smaller than the utility loss from reputational costs. Therefore, in the country j’s belief, the sector i defaults if and only if χi > ²T , j (41) 10 where à ¡ ¢ ! 1 kT − kT − µT h0 ¡ T ¢1+λT 1 1 H χT ≡ ln T l0 (42) p1 1+λ à ¡ ¢ ! 1 kT − kT − µT h0 ¡ T ¢1+λT 1 1 H 1 ¡ T ¢ T − ln T l0 − 1 + g0 d0 p1 1+λ p1 and à ¡ ¢ ! NT pNT kNT ¡ NT ¢1+λNT 1 1 1 kT − µT h0 ¡ T ¢1+λT 1 H χ ≡ ln l + l0 (43) p1 1 + λNT 0 p1 1 + λT à ¡ ¢ ! pNT kNT ¡ NT ¢1+λNT 1 1 1 ¡ NT ¢ NT 1 kT − µT h0 ¡ T ¢1+λT 1 H − ln NT l0 − 1 + g0 d0 + T l0 > ²NT , j p1 1 + λ p1 p1 1+λ By noting (39)-(40), we can see that the expressions into the Þrst and the second brackets in (42)-(43) are, respectively, the sector i’s consumption indices when it default and doesn’t: the only difference is the term with the debt. Note also that the period 1-price index in these expressions is the same because we assume that the home country’s producers realize that their individual actions, such as default on loans, do not affect the market prices. Otherwise, as we can infer from (36)-(37), the period 1-price index would be higher when loans are not repaid since default does increase the amount of tradable goods supplied to the home country. The conditions (41)-(43) can be rewritten as kT < bi , 1 j (44) such that ¡ ¢¡ ¢ 1 + λT 1 + g0 dT T 0 h0 µTH bT j = bT (z0 , h0 ; Ωj ) ≡ j £ ¡ T ¢¤ T 1+λT − ; (45) (1 − h0 ) 1 − exp −²j (l0 ) (1 − h0 ) " #¡ ¢¡ ¢ 1 1 − θ 1 + λT 1 + g0 dNT NT 0 bN T j NT = bj (z0 , h0 ; Ωj ) ≡ ¡ NT ¢ + £ 1−θ ¤ T 1+λT (46) 1 − exp −²j θ + h0 (l0 ) θ ¡ T ¢¡ T ¢ T T 1−θ 1+λ 1 + g0 d0 h0 µH + £ 1−θ ¤ T 1+λT + 1−θ , θ + h0 (l ) θ + h0 θ 0 11 ¡ ¢ where Ωj ≡ β, θ, λT , λNT , µT , ²T , ²NT 1 . Therefore, we have that the country j’s belief on the sector i ’s H j j default probability, denoted by π i , is given by the function j £ ¤ £ ¤ π i = π i (z0 , h0 ; Φj ) ≡ Prj δ i = 1 | z0 , h0 ; Φj = Prj kT < bi (z0 , h0 ; Ωj ) | Φj , j j 1 1 j (47) ¡ ¢ where Φj ≡ µT , µNT , η T , ηNT , Ωj . Given the probability distribution for the productivity shock in (2)- j j j j (5), we have from (47) that " # bi − kT j j 1 b i − µT j j π i = ¯T ¯ T = ¯ + 1 , if kT < bi < kT , (48) j kj − kj 2 1 − ηT j ¯j j j ¯ and ½ 0, if bi ≤ kT j πi j = ¯j . (49) ¯ 1, if bj ≥ kT i j Comparative Statistics for π i j Comparative statistics for π i are important to understand what deter- j mines the foreign credit supply to the home’s country, which will be examined in the next subsection. As it will be clear in subsection (3.4), we can focus our analysis on the sector T . To better understand the results below, note in (42)-(43) that, for any kT , the wealth’s marginal utility without default is greater 1 than with default. Hence, χT decreases monotonically with kT . 1 The effect of a change in z0 and h0 It follows from (48) that ∂π i j 1 ∂bi = ¡ ¢ j ; (50) ∂z0 2 1 − η T ∂z0 j ∂π i j 1 ∂bi = ¡ ¢ j , (51) ∂h0 2 1 − η T ∂h0 j 1 As µT is part of the hedge contract’s clauses, it is a parameter observed by both countries and then included in ΩH and H ΩF . 12 ¯ if kT < bi 0 (52) ∂d0 ∂g0 ∂l0 It is easy to see that, for any kT , χT increases with g0 or dT , making π T higher. In addition, as the wealth’s 1 T 0 j marginal utility is decreasing, χT decreases with l0 , pushing π T down. T j Differently, the effect of a change in h0 is not so obvious. As to the sector T , it follows from (45) that ∂bT j bT − µT j H = . (53) ∂h0 1 − h0 Hence, we have that ∂π T ∂bT T 0 ⇐⇒ T 0 ⇐⇒ bT T µT . j j j H (54) ∂h0 ∂h0 To understand the result (54), consider an increase in h0 . In this case, χT increases (decreases) for kT > µT 1 H ¡ T ¢ k1 < µT and remains the same for kT = µT . Again, this occurs because decreasing wealth’s marginal H 1 H utility implies that the size of the utility change without default is higher than with default. Therefore, π T increases (decreases) with h0 when bT > µT (bT < µT ). Note also that the size of the change in χT , j j H j H and consequently also in πT , increases with the size of the difference between bT and µT . j j H The effect of a change in µT , η T and ²T F F F Now, we derive the effect of changes in the country F ’s beliefs, which are given by ∂π i F 1 ∂biF = <0; (55) ∂²T F 2 (1 − η T ) ∂²T F F ∂π i F 1 = − <0; (56) ∂µTF 2 (1 − η T ) F ∂π i F 1 bi − µi F F = , (57) T ∂ηF T 2 2 (1 − ηF ) ¯ if kT < bi bi . F F 3.3 Foreign credit supply for the tradable sector In this section, we derive the equilibrium foreign credit supply for the tradable sector, denoted by dS,T , as 0 a function of all the variables observed by the foreign creditors at period 0, which are given by the vector i i w0 ≡ (l0 , g0 )i=T,NT , the policy parameter h0 and the vector ΦF . Note that, as the international capital market is competitive, foreign creditors take g0 (i = T, N T ) as given. By deÞnition, dS,T gives the amount i 0 of foreign credit supplied for the tradable sector such that • (C1) all foreign creditors lending to the home country are maximizing proÞts; • (C2) no other foreign saver has incentive to lend to the sector T . Firstly, note that, as the foreign creditors are risk-neutral, the equilibrium conditions (C1)-(C2) imply that dS,T , if positive, must satisfy the equation 0 ³ ´ h i¡ S,T ¡ T ¢ T S,T T ¢ 1 − πF 1 + g0 = PrF k1 ≥ bF | ΦF 1 + g0 = 1 + r0 , (58) such that ³ ´ S,T z0 ≡ dS,T , dNT , w0 ; 0 0 (59) ³ ´ bS,T F F S,T ≡ bT z0 , h0 ; ΩF ; (60) ³ ´ π S,T F F S,T ≡ π T z0 , h0 ; ΦF , (61) where we make use of (47). The condition (58) deÞnes implicitly dS,T as a function of w0 , h0 and ΦF : dS,T is 0 0 the net amount of foreign credit for the sector T that make the expected rate of return on the loans to T equal the default risk-free interest rate. When g0 ≥ r0 , it follows from (48)-(49) that ˆ bS,T = kT , (62) F 14 ˆ where kT is deÞned as ¡ ¢ ¯F (1 + r0 ) kT − kT ˆ kT ¯ ≡ kT − ¯F (63) F T 1 + g0 · ¸ T ¡ T ¢ 1 + r0 T = µF + 1 − η F 1 − 2 T , if g0 > r0 ; (64) 1 + g0 ˆ kT ≡ τ kT ≤ kT , if g0 = r0 , T (65) ¯F ¯F for any 0 ≤ τ ≤ 1. By using (45), we have from (59)-(60) and (62) that dS,T can be explicitly deÞned as 0 h i£ ¡ ¢¤ ˆ (1 − h0 ) kT + h0 µT 1 − exp −²T ¡ ¢1+λT H F dS,T (w0 , h0 ; ΦF ) = 0 ¡ ¢ T l0 , (66) (1 + g0 ) 1 + λT T When g0 < r0 , it is easy to see that the condition (58) is not met for any positive dT . Then, we have that T 0 dS,T (w0 , h0 ; ΦF ) = 0 , if g0 < r0 . 0 T (67) Comparative Statistics for dS,T 0 Next, we get comparative statistics results for the credit supply when T T g0 > r0 . For g0 = r0 , changes in w0 , h0 or ΦF can be accommodated by a change in τ , deÞned in (65). Starting from an equilibrium solution for the foreign country, a change in w0 or h0 that reduces π T or F T increases g0 makes the expected rate of return to get above r0 . ProÞt maximizer foreign creditors are so encouraged to supply more credit to the sector T , which in turn pushes π T up. As a result, dS,T increases F 0 up to the level at which a new equilibrium solution is reached. The effect of a change in ²T , µT and η T F F F To better understand the effect of a change in w0 , it is helpful to derive Þrstly the effects of a change in the country F ’s beliefs, which are given, respectively, by h i ¡ ¢ S,T ˆ (1 − h0 ) kT + h0 µT exp −²T ¡ ¢1+λT ∂d0 H F T = ¡ ¢ l0 >0 (68) ∂²TF T (1 + g0 ) 1 + λ T £ ¡ ¢¤ ∂dS,T 0 (1 − h0 ) 1 − exp −²T ¡ T ¢1+λT F = ¡ ¢ l0 >0; (69) ∂µTF (1 + g0 ) 1 + λT T · ¸ £ ¡ ¢¤ ∂dS,T 0 2 (1 + r0 ) (1 − h0 ) 1 − exp −²T ¡ T ¢1+λT F = −1 ¡ ¢ l0 . (70) ∂η T F 1 + g0T (1 + g0 ) 1 + λT T 15 The intuition behind these results follows directly from (55)-(57): a change in one of these parameters causes an increase in dS,T if and only if it leads to a reduction in π S,T . As to last derivative, we have from 0 F (57) and (64) that π T decreases with η T if and only if bT < µT , which in turn occurs if and only if the F F F F term into brackets in (70) is positive. T T The effect of a change in l0 Deriving (66) with respect to l0 , we have that h i£ ¡ ¢¤ S,T ˆ (1 − h0 ) kT + h0 µT 1 − exp −²T ¡ ¢λT ∂d0 H F T T = T l0 . (71) ∂l0 1 + g0 For h0 < 1, this derivative is strictly positive. The intuition follows from (42) : for any kT , the higher the 1 sector T ’s wealth at period 1, which increases with l0 , the lower χT . Hence, dS,T must increase for π S,T to T 0 F get unaltered. Now, we examine how the country F ’s beliefs affect the size of this derivative. As we can see from (66), in equilibrium, the sector T ’s foreign liabilities is a constant fraction of its period 1-wealth ˆ when kT =kT . Then, everything else constant, the positive effect of l0 on dS,T increases with dS,T . As a T 1 0 0 result, the effect of ²T , µT and η T on the derivative in (71) depends only on how they affect dS,T . In this F F F 0 sense, we have in (68)-(70) that dS,T always increases with µT and ²T , whereas it increases with η T if and 0 F F F T only if 1 + g0 < 2 (1 + r0 ). The effect of a change in hT 0 Now, deriving (66) with respect to hT we have that 0 ³ ´£ ¡ ¢¤ ˆ kT − µT 1 − exp −²T ¡ ¢1+λT ∂dS,T 0 (w0 , h0 ; ΦF ) H F T =− T ¡ T ¢ l0 . (72) ∂h0 (1 − h0 ) (1 + g0 ) 1 + λ ˆ Note that the sign and the size of this derivative depends on the difference between kT and µT . As we saw H ˆ in (51)-(53), when kT > (<) µT , π T increases (decreases) with this difference. The intuition behind this H F result shed light on the role played by the country’s F beliefs in the effect of h0 . The derivative in (72) ˆ always decreases with µT and increases (decreases) with ²T if and only with kT < (>) µT . Finally, note F F H from (63) that it increases with ηT if only if 1 + g0 > 2 (1 + r0 ). F T 16 T T The effect of a change in g0 The most interesting result is the effect of a change in g0 . Differently T from l0 and h0 , this variable affects not only the default probability, but also the contractual credit cost. T Deriving (66) with respect to g0 , we have that ¡ ¢ ∂dS,T 0 1 − exp −²TF ¡ T ¢1+λT T = K¡ ¢ 2 l0 (73) ∂g0 1 + λT (1 + g0 ) T where ¡ T ¢ K = K g0 , h0 ; ΦF ¡ ¢ £ ¯ ¤ 2 (1 − h0 ) (1 + r0 ) kT − kT ¯ T T ≡ − (1 − h0 ) kF + h0 µH + F ¯ F (74) T 1 + g0 · µ ¶ ¸ T ¡ T ¢ 1 + r0 T = − (1 − h0 ) µF + (1 − h0 ) 1 − η F 1 − 4 T + h0 µH 1 + g0 As we see in (45) and (58), an increase in g0 has two reverse effects on dS,T . On a hand, a higher g0 T 0 T implies that foreign creditors make more proÞts on the loans they will be actually repaid. Then, dS,T 0 must increase to push π T up. On the other hand, a higher g0 implies that π T increases. Then, dS,T must F T F 0 decrease to push π T down. It follows from (74) that the relative strength of these effects depends on the F parameters µT and η T . As dS,T increases with µT , the higher this parameter, the larger the increase in the F F 0 F interest expenses caused by a higher g0 and then the larger the increase in π T . The parameter η T affects T F F T the derivative in (73) in two different ways. On a hand, given an increase in g0 , the larger the country T ’s shock volatility, the higher the increase in dS,T must be to push π T up to the level at which a new 0 F equilibrium is reached. On the other hand, it follows from (70) that η T has a direct and ambiguous effect F on dS,T and then on the increase in the interest expenses caused by a higher g0 . 0 T 3.4 Foreign credit supply for the nontradable sector As we saw in subsection (2.3), the strong assumption that the foreign creditors realize that the sector NT has no incentive to repay loans was introduced into the model by imposing φN T = 0 in (11). Consequently, it follows from (46) and (49) that it ends up having no access to foreign funds. Formally, this means that 17 the equilibrium amount of foreign credit supplied for the nontradable producers is always zero, that is, dS,NT (w0 , h0 ; ΦF ) = 0 , 0 (75) for any w0 , h0 and ΦF . 3.5 Tradable sector ’s foreign credit demand Now, we derive the equilibrium sector T ’s labor supply and credit demand, denoted by l0 and dD,T S,T 0 T T respectively, as functions of g0 , h0 and ΦH . Note that competitive individuals in home country take g0 as given. By deÞnition, l0 and dD,T give, respectively, the effective labor supply and outstanding foreign S,T 0 debt such that • (C3) both sectors are maximizing the lifetime utility function; • (C4) all good markets are cleared in both periods; • (C5) home country sectors ’s expectations are formed rationally, that is, period 0-expectations about future prices are consistent to the actual allocative decisions. For sake of simplicity, we assume that the home country’s parameters vector ΦH is such that default S,T never occurs in home country’s belief. As a consequence, it follows from the result (44)-(46) that l0 and dD,T must satisfy the condition 0 ¡ ¢¡ ¢ 1 + λT 1 + g0 dD,T T 0 h0 µT H kT ≥ ³ ´1+λT − . (76) ¯H T S,T (1 − h0 ) (1 − h0 ) [1 − exp (−²H )] l0 The inequality (76) and the equilibrium condition (C3) under competitive markets imply that the tradable sector’s optimal choices of dT and l0 must satisfy the marginal conditions 0 T ¡ ¢ " ¡ ¢# 1 ∂u0 cT 0 ¡ T ¢ 1 ∂u1 0, cT 1 − 1 + g0 βE0 = 0; (77) p0 ∂cT 0 p1 ∂cT1 " ¡ ¢# 1 £ ¤ ¡ T ¢λi ∂u1 0, cT1 βE0 (1 − h0 ) kT + h0 µT l0 1 H T − l0 = 0 . (78) p1 ∂cT 1 18 Given the equilibrium conditions (C4)-(C5), we can substitute the equations for prices and consumption in (33)-(40) into the system (77)-(78) to Þnd the equations system that l0 and dD,T must satisfy for the S,T 0 home country to be in equilibrium, which is given by   1 ¡ T ¢  1  − β 1 + g0 E0  ³ ´1+λT =0; (79) dD,T (1−h0 )kT +h0 µT S,T 0 1 H 1+λT l0 − (1 + g0 ) dD,T T 0  £ ¤ ³ S,T ´λT   (1 − h0 ) kT + h0 µT 1 H l0  S,T βE0  ³ ´1+λT  = l0 . (80) T +h µT (1−h0 )k 1 S,T 1+λT 0 H l0 − (1 + g0 ) dD,T T 0 As the solution for this system must satisfy the inequality (76), this condition imposes some constraints on the parameters in ΦH . In this sense, the proposition (1) below sets a sufficient condition for the existence and uniqueness of a equilibrium solution with the property that the tradable sector never defaults in home country’s belief: Proposition 1 Consider the function γ T = γ T (h0 ) , (81) where h0 ∈ I ≡ (−υ , υ), with 1 > υ > 0, deÞned implicitly by the equation £ ¤ A γ T (h0 ) , h0 = 0 (82) and by the condition γ T (h0 ) ∈ (0, ξ) , (83) where A(x, h0 ) 1 ≡ (84) [1 − exp (−²T )] [(1 H − h0 )xkT + h0 µT ] H · ¯ ¸ 1 −βE0 [(1 − h0 )k1 + h0 µH ] − [1 − exp (−²T )] [(1 − h0 )xkT + h0 µT ] T T H ¯H H 19 and 1 ξ≡ . (85) 1 − exp (−²T ) H Suppose that the parameters in ΦH are such that γ T (0) < 1 (86) and J ⊂ I is a interval such that for all h0 ∈ J, γ T (h0 ) < 1 . (87) T Then, given g0 ≥ r0 e h0 ∈ J, there is an unique equilibrium solution for the tradable sector ’s labor supply and net foreign debt such that default never occurs in home country’s belief. In addition, this solution is given by S,T S,T ¡ T ¢ l0 = l0 g0 , h0 ; ΦH v " # u ¡ ¢ u 1 + λT [(1 − h0 ) kT + h0 µT ] = tβE0 1 H (88) [(1 − h0 ) kT + h0 µT ] − [1 − exp (−²T )] [(1 − h0 ) γ T (h0 ) kT + h0 µT ] 1 H H ¯H H ¡ T ¢ dD,T = dD,T g0 , h0 ; ΦH 0 0 £ T T T ¤£ ¡ T ¢¤ ³ S,T ´1+λT (1 − h0 ) γ (h0 ) kH + h0 µH 1 − exp −²H l0 = ¯ ¡ ¢ (89) 1 + λT (1 + g0 )T The proof of this proposition is in the appendix. Comparative statistics for l0 and dD,T S,T 0 Next, we get comparative statistics results for l0 and dD,T . S,T 0 First, derivating (88) with respect to hT , we have that 0 q ¡ ¢ β 1 + λT ½ · N T (h0 ) ¸¾− 2 · µ T ¶¸ 1 S,T ∂l0 ∂ N (h0 ) = E0 E0 (90) ∂h0 2 DT (h0 ) ∂h0 DT (h0 ) 20 where N T (h0 ) ≡ kT (1 − h0 ) + h0 µT 1 H (91) £ T ¤ £ ¡ ¢¤ £ T ¤ DT (h0 ) ≡ k1 (1 − h0 ) + h0 µT − 1 − exp −²T H H γ (h0 ) kT (1 − h0 ) + h0 µT . H (92) ¯ S,T A change in h0 has two different effects on l0 , which can be distinguished when the left-hand side of the equation (80) is written as  £ ¤ ³ S,T ´λT   (1 − h0 ) kT + h0 µT 1 H l0  E0  ³ ´1+λT  T +h µT (1−h0 )k 1 0 H S,T T D,T 1+λT l0 − (1 + g0 ) d0   ³ ´λT 1 S,T   = µT l0 H E0  ³ ´1+λT  (93) T +h µT (1−h0 )k1 0 H S,T T D,T 1+λT l0 − (1 + g0 ) d0   ³ ´λT 1 S,T   + l0 COV0 (1 − h0 ) kT , 1 ³ ´1+λT  T +h µT (1−h0 )k 1 0 H S,T T D,T 1+λT l0 − (1 + g0 ) d0 The Þrst term of the right-hand side in (93) sets up that a higher h0 reduces the period 1-wealth volatility and then the labor’s marginal utility, so that producers have incentive to supply less labor. On the other hand, a higher h0 makes the labor’s return and the period-1 wealth less positively covaried, increasing the labor’s marginal utility and then the labor supply. The last effect can be better understood when we note that · ¸ (1 − h0 ) kT + h0 µT ³ S,T ´1+λT ¡ 1 H ¢ D,T CORR0 (1 − h0 ) kT 1 , l0 T − 1 + g0 d0 =1 (94) 1 + λT so that · ³ ´1+λT ¡ ¸ T T (1 − h0 ) k1 + h0 µH S,T T ¢ D,T COV0 (1 − h0 ) k1 , l0 − 1 + g0 d0 1 + λT · ¸ £ T ¤ (1 − h0 ) kT + h0 µT ³ S,T ´1+λT ¡ 1 H T ¢ D,T = DP0 (1 − h0 ) k1 DP0 l0 − 1 + g0 d0 (95) 1 + λT ³ ´1+λT 2 S,T (1 − h0 ) l0 £ ¤ = T V AR0 kT 1 1+λ 21 Since the labor return and the period-1 wealth are linearly correlated, the positive covariance between them decreases when a higher h0 makes both less volatile. As to the effect of h0 on dD,T , derivating (89) with respect to h0 , we have that 0 £ ¡ ¢¤ h i T ∂dD,T 0 1 − exp −²T H S,T λ = T ¡ T ¢ l0 (96) ∂h0 (1 + g0 ) 1 + λ (· ¸ ) S,T ∂γ T (h0 ) T S,T ¡ ¢£ ¤ ∂l0 µT − γ T (h0 ) kT + (1 − h0 ) H k l + 1 + λT (1 − h0 ) γ T (h0 ) kT + h0 µT ¯H ∂h0 ¯H 0 ¯H H ∂h0 We can better understand this effect by observing the expression (79). On a hand, we have to consider the effect on l0 : dD,T increases with l0 in order to smooth consumption overtime. On the other hand, S,T 0 S,T a higher h0 reduces the period 1-wealth’s volatility and then the consumption’s marginal utility in this period, encouraging consumers to transfer more wealth to present. Therefore, we can conclude that dD,T 0 increases unambiguously with a higher l0 . When l0 decreases, the net effect on dD,T depends on the S,T S,T 0 relative strength of the two effects explained above. The effect of a change in the parameters µT and η T on dD,T and l0 can also be inferred from (79) H H 0 S,T and (93)-(95) respectively. On a hand, a higher µT or ηT increases the mean and reduces the volatility H H of the wealth’s marginal utility at period-1, so that sector T has incentive to work less and borrow more loans. On the other hand, a higher µT increases the expected labor return and a higher ηT reduces the H H covariance between the labor return and the period-1 wealth, so that sector T has incentive to work more, which in turn has a positive effect on the credit demand. Finally, as the labor return does not depend on the credit cost, a higher g0 reduces dD,T while l0 gets T 0 S,T unaltered, that is, S,T ∂l0 T = 0; (97) ∂g0 £ ¤£ ¡ ¢¤ £ T ¡ T ¢¤1+λT ∂dD,T 0 (1 − h0 ) γ T (h0 ) kT + h0 µT 1 − exp −²T H H H l0 g0 ; h0 ; ΦH = − ¯ ¡ ¢ <0. (98) T T 2 ∂g0 1 + λT (1 + g0 ) 22 3.6 General equilibrium solution Finally, we derive the general equilibrium solution for the world economy as a function of h0 and Φ = ³ ´ ΦH UΦF , which is deÞned as a vector z0 ≡ di , ˆ0 , g i such that ˆ ˆ li ˆ 0 ¡ T ¢ ˆ dT = dD,T g0 , h0 ; ΦH = dS,T (w0 , h0 ; ΦF ) ; ˆ ˆ (99) 0 0 0 ³ ´ dNT = dD,NT g0 , g0 , dT , ˆ0 , h0 ; ΦH = dS,NT (w0 , h0 ; ΦF ) ; ˆ 0 0 ˆNT ˆT ˆ0 lT 0 ˆ (100) ¡ ¢ ˆT = lS,T g T , h0 ; ΦH ; l0 ˆ0 (101) 0 ³ ´ ˆNT = lS,NT g NT , g T , dT , ˆT , h0 ; ΦH . l0 ˆ0 ˆ0 ˆ0 l0 (102) 0 As the general equilibrium solution for the other endogenous variables of the model, namely, exports, prices, consumption and production, can be directly derived as functions of z0 through the equations (31)- ˆ (40), it is enough to limit the deÞnition of general equilibrium on the endogenous variables in the vector z0 . Note that all conditions (C1)-(C5) are met when z0 = z0 : both the home and foreign economies are in ˆ equilibrium. The proposition (2) below delivers sufficient conditions for the existence and the uniqueness of a general equilibrium solution for the world economy with the property that default never occurs in home country’s beliefs: Proposition 2 Suppose that the parameters vector ΦH meets the same conditions set in the preposition (1). Then, there is an unique general equilibrium solution for the world economy such that the condition (76) is satisÞed if and only ˜ ¯ kT (h0 ) < kT , (103) F where µ ¶ ˜ γ T (h0 ) kT 1 h0 µT kT (h0 ) ≡ ¯H + −1 H . (104) φT φT (1 − h0 ) 23 Moreover, the solution for the labor supply and the net foreign credit are given by ˆT = ˆT (h0 ; Φ) l0 l0 (105) v ( ) u ¡ ¢ u 1 + λT [(1 − h0 ) kT + h0 µT ] = tβE0 1 H ; [(1 − h0 ) kT + h0 µT ] − [1 − exp (−²T )] [(1 − h0 )γ T (h0 ) kT + h0 µT ] 1 H H H H ¯ ˆT = dT (h0 ; Φ) d0 ˆ (106) £0 ¡ ¢¤ £ ¤ 1 − exp −²T H (1 − h0 )γ T (h0 ) kT + h0 µT ³ˆT ´1+λT H H = ¡ ¢ ¯ l0 ; 1 + λT (1 + g0 )ˆT ˆNT = ˆNT (h0 ; Φ) = l0 l0 (107) q ¡ ¢ β 1 + λNT v " # u u 1−θ [kT − [1 − exp (−²T )] [(1 − h0 )γ T (h0 ) kT + h0 µT ]] tE0 θ 1 H ¯ H H 1−θ [kT − [1 − exp (−²T )] [(1 − h0 )γ T (h0 ) kT + h0 µT ]] + (kT − µT ) h0 1 H 1 θ ¯H H H ˆ ˆ dNT = dNT (h0 ; Φ) = 0 , (108) 0 0 T whereas the solution for g0 is given by ˆT ˆT ˜ g0 = g0 (h0 ; Φ) = r0 , if kT (h0 ) ≤ kT , (109) ¯F ¡ ¢ ¯ (1 + r0 ) kT − kT T T 1 + g0 = 1 + g0 (h0 ; Φ) = ˆ ˆ F ¯ F ˜ ¯ , if kT < kT (h0 ) < kT , (110) ¯T − kT (h0 ) kF ˜ ¯F F NT and the solution for g0 is given by ˆNT = 1 + g0 (h0 ; Φ) 1 + g0 ˆNT (111) ¡ ¢ ˆT 1 + g0 1 = · ¸ βγ T (h0 ) [1 − exp (−²T )] kT H H 1 ¯ E0 [kT (1−h0 )+h0 µT ] 1 H −γ T (h0 )[1−exp(−²T )]kT H ¯H ˆT where g0 is given in (109)-(104). The proof of this proposition is in the appendix. We can distinguish two different cases. In the case in (110), foreign creditors are so pessimistic about the sector T ’s ability/willingness to repay their loans that is strictly positive, pushing up. We can interpret the higher credit cost as a kind of credit constraint faced by the home country. In the case in (109), foreign creditor are not enough pessimistic to cause any effect on the equilibrium credit demand and supply. 24 Stability of the solution Now, we set up a sufficient condition for the stability of the equilibrium solution above. Substituting (105) and (110) into (73) and (98), we have that ¡ ¢ £ ¡ ¢¤ ³ T ´1+λT h i φT 1 − exp −²T ˆ l0 ∂dS,T 0 (w0 , h0 ; ΦF ) ˆ ∂dD,T 0 ˆT g0 , h0 ; ΦH ¯T − kT (h0 ) ˜ H T − T = (1 − h0 ) kF ¡ ¢ 2 (112) ∂g0 ∂g0 1 + λT (1 + g0 ) ˆT It follows from (103)-(104) in the proposition (2) that, if there is an equilibrium solution such that the condition (76) is satisÞed, the equation above is positive in an interval H enough small around h0 = 0. Therefore, stability requires that the parameters vector ΦF is such that the equation ³ ´1+λT ½ · ¸ · ¸¾ £1 − exp ¡−²T ¢¤ ˆT l0 ∂dS,T (w0 , h0 ; ΦF ) 0 ˆ ¯ γ T (h0 ) kT 2 F T = (1 − h0 ) kT − 2 F ¯H − h0 µT H −1 ¡ ¢ 2 (113) ∂g0 φT φT 1 + λT (1 + g0 ) ˆT is positive for all h0 in H. Comparative Statistics for z0 ˆ Before examining the welfare effects of a change in h0 , we must know how this change affects the general equilibrium solution, given by the vector z0 . For reasons that will be ˆ clear in the next section, we are now particularly interested in the effects of a higher h0 on g0 and ˆ0 . The ˆT lT effect on ˆ0 was already explained in the subsection (3.5). The equilibrium solution for this variable is lT determined only by the country H’s demand and supply of labor and does not depends on the parameters that measure the country F ’s beliefs. Therefore, it follows from (101) that S,T ¡ T ¢ ∂ ˆ0 (h0 ; Φ) lT ∂l0 g0 , h0 ; ΦH ˆ = , (114) ∂h0 ∂h0 ˆT where the right-hand side of the equation above is given by (90)-(92). As to the effect on g0 , note in (99) that S,T ∂dS,T (w0 ,h0 ;ΦF ) ˆ ∂dS,T (w0 ,h0 ;ΦF ) ∂l0 (g0 ,h0 ;ΦH ) ˆ ˆT ∂dD,T (g0 ,h0 ;ΦH ) 0 ˆT gT ∂ˆ0 (h0 ; Φ) 0 ∂h0 + 0 T ∂l0 ∂h0 − ∂h0 =− . (115) ∂h0 ∂dS,T (w0 ,h0 ;ΦF ) ˆ ∂dD,T (g0 ,h0 ;ΦH ) 0 ˆT 0 T ∂g0 − T ∂g0 25 Then, substituting the derivatives in (71)-(73), (90)-(92) and (96)-(98) into (115), we have that gT ∂ˆ0 (h0 ; Φ) ˜ = 0 ; kT (h0 ) ≤ kT ; (116) ∂h0 ¯F ¡ T ¢h T ³ ´ T i ¯ − kT kH ∂γ T (h0 ) + 1 − 1 (1 + r0 ) kF ¯T µH gT ∂ˆ0 (h0 ; Φ) ¯ F φ ∂h0 φT (1−h0 )2 ˜ ¯ = h ³ ´ T i2 ; kT < kT (h0 ) < kT (117) ∂h0 ¯ T γ T (h0 )kT 1 h0 µH ¯F F kF − φT ¯H − φT − 1 1−h0 ˆT It is clear from (116) that a change in h0 has no effect on g0 when information asymmetry does not make D,T foreign credit more expensive. The positive effect of a higher h0 on l0 is accommodated by an increase in dS,T at the same interest rate. The effect in (117) is better understood when we examine the two ways 0 S,T D,T through which h0 affects l0 and l0 . First, we can see from (72) and (96) that, holding all other variables S,T D,T constant, a change in h0 affects both l0 and l0 directly. As it was explained in the subsections (3.4)-(3.5), D,T S,T the effect on l0 is unambiguously positive, whereas the effect on l0 depends, among other things, on the S,T D,T sector F ’s beliefs. Second, it follows from (71) and (90) that h0 also affects both l0 and l0 indirectly S,T S,T through its direct and ambiguous effect on l0 . Since only the direct effect on l0 is unambiguous, the net ˆT effect on g0 depends on the parameters vector Φ, which determines the relative strength of the effects of a S,T D,T S,T D,T higher h0 on l0 and l0 : the stronger the effect on l0 , relative to the effect on l0 , the lower the new S,T ˆT ˆT equilibrium level for g0 . Note also that the net effect of h0 on g0 decreases with the elasticity of l0 and D,T ˆT l0 with respect to g0 , which can be derived from (73) and (98). This is another way that the parameters of the country F ’s beliefs may affect the effect of a higher h0 on the equilibrium solution. It is important to observe that an increase in ˆ0 does not necessarily leads to a decrease in g0 . It is lT ˆT possible that a change in hT push both ˆ0 and g0 up or down. A reason for this is that the effect of a 0 lT ˆT S,T D,T higher hT on l0 and l0 goes in the same way. Other reason is that, although the direct effect on is 0 unambiguously positive, the direct effect on depends on the parameters. 4 Welfare effect of a change in h0 This section derives and interpret the welfare effects of a change in h0 . More precisely, we derive sufficient conditions for this change to result in a Pareto-improvement for the home country. We assume that the 26 world economy rests initially on a stable general equilibrium solution as the one deÞned above. Analytical tractability restricts us to examine changes around h0 = 0. 4.1 Pareto-improvement deÞnition Consider Þrstly the sector i’s lifetime utility, denoted by U i , as a function of the vector z0 = (di , l0 , g0 ), 0 i i when default does not occurs, which is given by ¡ ¢ £ ¡ ¢¤ 1 ¡ i ¢2 U i = U i (z0 ) ≡ ln ci + βE0 ln ci − 0 1 l , (118) 2 0 where ci and l0 are deÞned in (33)-(40). This function follows directly from (12)-(16) by doing δ i = 1. 0 i Next, we deÞne V i as the sector i ’s lifetime utility as a function of h0 and Φ, so that V i = V i (h0 ; Φ) ≡ U i (ˆ0 ) , i = T, NT , z (119) whereas ci , pi and pt , deÞned as ˆt ˆt ˆ ci = ci (h0 , Φ) ≡ ci (ˆ0 ) ; ˆt ˆt t z (120) pi = pi (h0 , Φ) ≡ pi (ˆ0 ) ; ˆt ˆt t z (121) pt = pt (h0 , Φ) ≡ pt (ˆ0 ) , ˆ ˆ z (122) give the general equilibrium solution for consumption and prices as a function of h0 and Φ. The vector z0 is the general equilibrium solution as deÞned in (105)-(111) and is also written as a function of the ˆ parameters, such that ³ ´ z0 = z0 (h0 ; Φ) ≡ dT , dNT , ˆ0 , ˆ0 , g0 , g0 , h0 ˆ ˆ ˆ0 ˆ0 lT lNT ˆT ˆNT , (123) and ˆ ˆ di ≡ di (h0 ; Φ) ; (124) 0 0 ˆi ≡ ˆi (h0 ; Φ) ; l0 l0 (125) ˆi ˆi g0 ≡ g0 (h0 ; Φ) . (126) 27 The effect of a change in h0 on the sector i’s lifetime utility is given by ∆V i ≡ V i (h0 ; Φ) − V i (0; Φ) , (127) Starting from h0 = 0, a change in h0 leads to a Pareto-improvement for the home country if and only if ∆V i ≥ 0 for i = T, N T , with strict inequality for at least one sector. We just analyze changes in h0 enough small to be well approximated by a Þrst-order Taylor expansion, so that the change in the lifetime utility is given by i ∼ ∂V (0; Φ) h0 , ∆V =i (128) ∂h0 such that ∂V T (0; Φ) = − (1 − θ) K (Φ) + L (Φ) ; (129) ∂h0 ∂V NT (0; Φ) = θK (Φ) + L (Φ) , (130) ∂h0 where ( ) 1 ∂U T [ˆ0 (0; Φ)] ∂ ˆ0 (0; Φ) ∂U T [ˆ0 (0; Φ)] ∂ ˆ0 (0; Φ) ∂U T [ˆ0 (0; Φ)] z lT z lNT z K (Φ) ≡ − T + + (131) 1−θ ∂l0 ∂h0 ∂l0 T N ∂h0 ∂h0 ( ) 1 ∂U NT [ˆ0 (0; Φ)] ∂ ˆ0 (0; Φ) ∂U NT [ˆ0 (0; Φ)] ∂ ˆ0 (0; Φ) ∂U NT [ˆ0 (0; Φ)] z lT z lNT z = T + NT + ; (132) θ ∂l0 ∂h0 ∂l0 ∂h0 ∂h0 ∂U T [ˆ0 (0; Φ)] ∂g0 (0; Φ) z T ∂U NT [ˆ0 (0; Φ)] ∂g0 (0; Φ) z T L (Φ) ≡ T = T (133) ∂g0 ∂h0 ∂g0 ∂h0 and the derivatives of U i with respect to z0 , when evaluated at z0 (0; Φ), are given by ˆ ∂U i [ˆ0 (0; Φ)] z = 0; (134) ∂dT0 ∂U i [ˆ0 (0; Φ)] z = 0; (135) ∂dNT 0 28 µ ¶ ∂U T [ˆ0 (0; Φ)] z θ ∂U NT [ˆ0 (0; Φ)] z T = − T (136) ∂l0 1−θ ∂l0 v " # u ¡ ¢ u 1 + λT kT = − (1 − θ) tβE0 T 1 ; (137) k1 − [1 − exp (−²H )] γ T (0) kT T H ¯ µ ¶ ∂U T [ˆ0 (0; Φ)] z θ ∂U NT [ˆ0 (0; Φ)] z NT = − NT (138) ∂l0 1−θ ∂l0 q ¡ ¢ = (1 − θ) β 1 + λNT ; (139) £ T ¤ £ T ¤ ∂U T z0 (0; Φ) ˆ ∂U N T z0 (0; Φ) ˆ T = T (140) ∂g0 ∂g0       ˆ dT (0; Φ) 0 = −θβE0 (141)  kT hˆT  1 i1+λT  1+λT l0 (0; Φ) − [1 + g0 (0; Φ)] dT (0; Φ)  ˆT ˆ 0 θ = − ; (142) 1+ ˆT g0 (0; Φ) ∂U i [ˆ0 (0; Φ)] z =0; (143) ∂g0 T N £ T ¤ µ ¶ ∂U T z0 (0; Φ) ˆ θ ∂U NT [ˆ0 (0; Φ)] z = − (144) ∂h0 1−θ ∂h0 · ¸ kT − µT 1 H = −βE0 T . (145) k1 − [1 − exp (−²T )] γ T (0) kT H H ¯ i Before proceeding with the derivation of V , it is helpful to understand the intuition behind the sign of the derivatives above. The null derivatives in (134)-(135) lacks generality and follows directly from (75). As to the derivatives in (136 )-(138), competitive markets assumption explains why the own labor’s marginal utility is negative for both sectors. Moreover, the sector i’s welfare increases with the other sector’s labor supply because the relative price of its output is pushed up. The derivative in (141) shows that the sector ˆT T ’s wealth and welfare increases with a fall in g0 as its foreign liabilities are reduced. This implies that the home country must export less to Þnance the capital account’s deÞcit, increasing in this way the supply 29 ˆT of tradable goods for the home market. Therefore, the sector NT is also beneÞted by a lower g0 due to an increase in the relative price of its output. This reasoning also explains why the derivative in (143) is ˆ zero. As we always have dNT = 0 in equilibrium, a change in g0 has no effect on the country H’s wealth ˆNT 0 and welfare. The derivatives in (144)-(145) show the direct effect of a higher h0 on the sector i’s welfare, holding everything else constant: it is the utility gain for an individual producer in sector i when the rest of its sector is not provided with the smoothing security. By the envelope theorem, the second-order effects of a higher h0 on the welfare by changing dT and ˆ0 are zero. ˆ 0 lT ˆT Substituting (134)-(144) and (105 )-(111), when evaluated at z0 (0; Φ), into (131)-(133), we have that ¡ ¢£ ¡ ¢¤ " T £ T ¤ T # β 1 + λT 1 − exp −²T H µH k1 − γ T (0) kT + ∂γ (0) kT kT H ∂h0 ¯H 1 K (Φ) = E0 ¯ 2 2 {kT − [1 − exp (−²T )] γ T (0) kT } 1 H ¡ ¢ · ¸ ¯H θ β 1 + λNT kT − µT 1 H + E0 T (146) 1−θ 2 k1 − [1 − exp (−²T )] γ T (0) kT H H · ¸ ¯ 1 kT − µT 1 H +β E0 T >0, 1−θ k1 − [1 − exp (−²T )] γ T (0) kT H H ¯ whereas ˜ L (Φ) = 0 , if kT (0) ≤ kT , (147) ¯F and h ³ ´ i ∂γ T (0) k T 1 θ ∂h0 φT ¯H + φT − 1 µT H L (Φ) = − ˜ ¯ , if kT < kT (0) < kT . (148) ¯ kT − γ T (0)kT ¯H ¯F F F φT Next, the comparative statistics results are derived for both cases above. We just consider changes around ˜ in h0 around 0 such that kT (0) remains in the interior of the interval in (147) or (148). 4.2 ˆT Pareto-improvement when g0 > r0 ˜ ¯ Consider the case in (148), where kT φ ≡ ¯ ¯ ; H (150) kTF µ ¶ T T 1 T T γ T (0) kT H α > α ≡ T ρ ηH + ˆ ¯ −1 ; (151) µH φT µ ¶ T T 1 T T γ T (0) kT H ρ < ρ ≡ T α µH − ˆ ¯ +1 . (152) ηH φT Then, deÞning Λ as µ ¶ kT ∂γ T (0) 1 Λ ≡ ¯H + − 1 µT , H (153) φT ∂h0 φ T it follows from (148) and from the fact that the function K (Φ) does not depend on φT , αT and ρT that ∂V i (0; Φ) lim = lim T L (Φ) = ∞− (154) ˆ φT −→φ+ T ∂h0 ˆ φT −→φ+ ∂V i (0; Φ) lim = lim L (Φ) = ∞− (155) αT −→ˆ T α+ ∂h0 αT −→ˆ T α+ ∂V i (0; Φ) lim = lim L (Φ) = ∞− (156) ρT −→ˆT ρ− ∂h0 ρT −→ˆT ρ− when Λ > 0 and ∂V i (0; Φ) lim = lim L (Φ) = ∞+ (157) ˆT φT −→φ+ ∂h0 ˆT φT −→φ+ ∂V i (0; Φ) lim = lim L (Φ) = ∞+ (158) αT −→ˆ T α+ ∂h0 αT −→ˆ T α+ ∂V i (0; Φ) lim = lim L (Φ) = ∞+ (159) ρT −→ˆT ρ− ∂h0 ρT −→ˆT ρ− 31 when Λ < 0. When Λ = 0, L (Φ) = 0. The results (154)-(156) and (157)-(159) show that there is a range for the vector Φ such that Pareto- improvement is possible by increasing h0 . For this, we need further that the sector T (N T ) be provided with the smoothing security when Λ < 0 (> 0). These results are better understood by noting how the effects in (73) are affected by the country F ’s beliefs. A higher η T and a lower µT decreases the elasticity F F T T of the foreign credit’s supply with respect to g0 , so that the size of the effect of a higher h0 on g0 becomes stronger. A lower φT affects not only this elasticity but also the size of the effect of a higher h0 on dS,T 0 and dN,T for a given g0 . 0 T As we saw, the public provision of the smoothing security amounts to a compulsory redistribution of the exposure to the productivity shocks across the home country’s sectors. But why competitive markets do not provide incentive to this risk reallocation? The answer is that individual producers can not prevent its sector as a whole from sharing the beneÞts provided by its position in the security. Therefore, individual producers have incentive to behave as a free rider. Although we don’t introduce a home private market for the smoothing security into the model, we can show that the allocative market inefficiency is not caused only by market incompleteness. When θ = 0.5, it follows from (144)-(145) that the existence of this market is irrelevant, because no amount of the smoothing security would be traded in equilibrium. Moreover, given that the asset in (23) can be seen as a future contract on the tradable good, the future price of this good would be exactly equal to µT in equilibrium. In this case, we can assure that the allocative inefficiency F does not result from market incompleteness. Even so, the results (154)-(159) still shows that there is a scope for Pareto improvement by smoothing the exchange rate volatility. Supposing, for sake of simplicity, that θ = 0.5, we can better understand why competitive markets fail to signal the correct incentives for a fully efficient risk reallocation. For this, assume that there is a scope for Pareto improvement when Λ < 0, so that both sectors would proÞt if the sector N T sold the smoothing security to the sector T . Note then that the derivatives in (144)-(145) give the welfare gain for an individual Þrm in each sector when it buys one unit of the smoothing security and the rest of its sector does not. In addition, these derivatives have always opposite signs. Assume Þrst that the sign of the sector T ’s derivative is negative. Using a game theory approach, we can see that for this sector the strategy of 32 buying the security is strictly dominated by the strategy of not buying. As foreign creditors can observe only the aggregate levels of the labor supply and of the security traded in the market, tradable sector’s Þrms have an incentive to behave as a free rider. Assume now that the sign of the sector NT ’s derivative is negative. In this case, selling the smoothing security is a strictly dominated strategy for this sector. Even if foreign creditors are fully informed, nontradable sector’s Þrms can not prevent the rest of its sector from sharing a higher relative price for its output. The same reasoning can be used when Λ > 0. An important result is that Pareto-improvement does not require that the smoothing security be always provided to the sector T , even if ˆ0 increases with h0 . As we know from the section (3), neither ˆ0 increases lT l necessarily with h0 nor a higher ˆ0 must result in a lower g0 . As a change in h0 leads to a Pareto improvement l ˆ if only if the new general equilibrium is reached with a lower g0 , the sign of Λ determines which sector ˆ should be provided with the security: if it is negative (positive), we need a positive (negative) change in h0 , that is, the security should be transferred to the sector T (N T ). However, suppose that the derivative in (145) is positive when Λ < 0. In this case, one can argue that the sector T ’s individual Þrms would refuse to add the security to their portfolios. For the same argument above, not accepting the security is a strictly dominant strategy because it could behave as a free rider. To go around this problem, the government could provide the sector NT with another asset whose pay-off is just the opposite of the one described in (23). The sector N T would accept the offer because this is now a strictly dominant strategy, whereas the balanced government budget restriction would force the sector T to face an exposure to the productivity shocks equivalent to that stemming from a higher h0 . Another related question is why the tradable producers does not have incentive to supply more labor so as to make foreign credit cheaper? Why public intervention is necessary to provide the socially correct incentive? The answer starts with noting that, by assumption, the individual labor supply can not be directly monitored by foreign creditors. This implies that each individual producer is not able to exclude the rest of the sector from taking advantage of a lower interest rate caused by the increase in this output. Therefore, as we have a large number of Þrms in the sector, the lower cost of the loans borrowed by the individual producer is not enough large to pay the marginal desutility of the labor. This market failure to signal the right incentives to exports production provides another theoretical justiÞcation for ERVS 33 policies. 4.3 ˆT Impossibility for Pareto-improvement when g0 = r0 ˜ Consider now the case in (147), where kT (0) ≤kT . Thus, the equilibrium solution for g0 , given by the ˆT ¯F T equation (109), is r0 : the country F ’s beliefs are not enough pessimistic to make loans for the home country more expensive. Therefore, it follows from (129)-(130) that ∂V T (0; Φ) 1 − θ ∂V NT (0; Φ) =− (160) ∂h0 θ ∂h0 T In this case, there is no scope for a Pareto-improvement because g0 is already at its lower level. The effect of an higher h0 on the foreign credit’s supply and demand curves can change only the equilibrium level for T the foreign debt. Neither can the sector T proÞts from a lower g0 , nor can the sector NT proÞts from a higher relative price for its output. Note that different beliefs across countries with respect to economic performance and reputational costs, is not a necessary condition for a Pareto improvement. What comes ˆT to be necessary is g0 higher than r0 , that is, the interest rate on the sector T ’s debt must be above the default risk-free interest rate. This condition arises even without this kind of information asymmetry: it is enough that both home and foreign countries be enough pessimistic about the sector T ’s ability or willingness to repay their loans. 5 Conclusion The model shows that ERVS policies may bring a Pareto improvement for a small open economy if there is some positive externality underlying the exchange rate risk realignment not efficiently allocated by the market. Pareto improvement requires that the welfare gain of the sector having its wealth volatility increased be enough large to compensate it for its broader exposure to the exchange rate risk. More precisely, when competitive markets fail to provide the correct signs for an efficient redistribution of the exchange rate risk exposure across the tradable and the nontradable sectors, ERVS policies are theoretically justiÞable. 34 This may occur when the home economy is paying a large spread on the default risk-free world interest rate and market imperfections, such as nontradable goods and imperfect information, prevent home econ- omy’s Þrms from internalizing all beneÞts and costs of the risk realignment into their allocative decisions. The reason is that the wealth volatility of an individual Þrm impacts on its foreign credit’s supply and demand curves and then on the interest rate it pays on its foreign liabilities. The effects of the ERVS policies on the debt cost are ambiguous and go in two different ways: directly, by changing the borrowers’s default probability, and indirectly, by changing the incentives for production. The relative strength of these effects depends, to a large extent, on the foreign creditors’s beliefs about the home economy’s ability and willingness to repay. References [1] Backus, David K. and Kehoe, Patrick J. (1989). On the denomination of government debt: a critique of the portfolio balance approach. Journal of Monetary Economics, 23, 359-376. [2] Bohn, Henning (1990). A Positive Theory of Foreign Currency Debt. Journal of International Eco- nomics, 29, 273-292. [3] Bohn, Henning (1990). Tax Smoothing with Financial Instruments. American Economic Review, 1217- 1230. [4] Burnside, C., Eichenbaum, M. and Rebelo, S.(1999). Hedging and Financial Fragility in Fixed Ex- change Rate Regimes. NBER WP no.7143. [5] Eichengreen, Barry and Hausmann R. (1999). Exchange Rates and Financial Fragility. NBER, WP no.7418. [6] Gale, D. (1990). The efficient design of public debt, in Public Debt Management: Theory and History, ed. by R. Dornbush and M. Draghi, Cambridge University Press. 35 [7] Goldfajn, Ilan (1998). Public Debt Indexation and Denomination: The Case of Brazil. IMF Working Paper. WP/98/18. [8] Goldfajn, Ilan (1995). On Public Debt Indexation and Denomination. Brandeis University Working Paper No.345. [9] Kroner, Kennetth F. and Claessens, Stijn (1991). Optimal dynamic hedging portfolios and the currency composition of external debt. Journal of International Money and Finance, 10, 131-148. [10] Miller, Victoria (1997). Why a goverment might want consider foreign currency denominated debt. Economic Letters, 55, 247-250. [11] Missale, Alessandro (1997). Managing the Public Debt: The Optimal Taxation Approach. Journal of Economic Surveys, vol.11, No.3. 6 Appendix Proof of the proposition 1 Firstly, we have to prove that there is an interval I = (−υ, υ) , with 0 < υ < 1, such that the function γ T (h0 ) is deÞned according to (82)-(85). As the function A is differentiable in x and h0 , it is enough to prove that lim A(x, h0 ) = ∞+ ; (x,h0 )−→(0+ ,0) lim A(x, h0 ) = ∞− (x,h0 )−→(ξ− ,0) and, by using the Leibnitz’s rule, ∂A(x, h0 ) <0, ∂x for all x and h0 < 1. Secondly, we have to prove that there is a range for the vector of parameters Φ, denoted by R, such that, given any Φ ∈ R, we can Þnd an interval J (Φ) ⊂ I such that γ T (h0 ) ≤ 1 for 36 any h0 ∈ J (Φ). For this, it is enough to prove that there is a set R such that, for any Φ ∈ R, we have A(1, 0) ≤ 0. It follows from (2) and (84) that · ¸ 1 1 A(1, 0) = − βE0 T [1 − exp (−²T )] kT H H k1 − [1 − exp (−²T )] kT H ¯ £ ¡¯H ¢¤ 1 β ¯T − 1 − exp −²T kT k = T T − ¯T ln H H ¯H . [1 − exp (−²H )] kH kH − kHT kT − [1 − exp (−²T )] kT ¯ ¯ ¯H H ¯H Therefore, by noting that (3)-(5), we have lim A(1, 0) = ∞− . (ηT ,²T )−→(0,∞+ ) H H Now, we prove that for any Φ ∈ R and for any h0 ∈ J (Φ) , the equations (88)-(89) are a solution for the system (79)-(80), such that the restriction (76) is satisÞed. Note that, as γ T (h0 ) ≤ 1 for hT ∈ J (Φ), we 0 have that ¡ ¢¡ ¢ T ˆ 1 + λT 1 + g0 dT 0 h0 µT H kT ≥ γ T (h0 ) kT = ³ ´1+λT − (161) ¯H ¯H (1 − h0 ) (1 − h0 ) [1 − exp (−²T )] ˆ0 H lT Therefore, we can substitute (88)-(89) directly into (79)-(80) in order to get  £ ¤ ³ T ´λT  T T (1 − h0 ) k1 + h0 µH ˆ l0   ˆT βE0  ³ ´1+λT  − l0 = 0 (162) (1−h0 )kT +h0 µT ˆT T ˆ T 1 1+λT H l0 − (1 + g0 ) d0 and   1 ¡ T ¢  1  − β 1 + g0 E0  ³ ´1+λT  ˆT d0 (1−h0 )kT +h0 µT 1 H ˆT l0 T ˆ T − (1 + g0 ) d0 1+λT 1 = (163) [1 − exp (−²T )] [(1 H − h0 )γ T (h0 ) kT + h0 µT ] H · ¯ ¸ 1 −βE0 [(1 − h0 )kT + h0 µT ] − [1 − exp (−²T )] [(1 − h0 )γ T (h0 ) kT + h0 µT ] 1 H H ¯H H = 0 Rearranging (162), we get (88). The second equality in (163) follows from the deÞnition of the function γ T (h0 ) in (82)-(85). Finally, we prove the uniqueness of the solution. For this, suppose that (dT , ¯0 ) is a ¯ lT 0 37 solution for the system in (79)-(80) such that the restriction (76) is satisÞed. Then, there is some τ ≤ 1, such that (dT , ¯0 ) satisÞes the conditions ¯ lT 0 ¡ ¢¡ ¢ T ¯ 1 + λT 1 + g0 dT 0 h0 µT H kT H ≥ τ kT = H ¡ T ¢1+λT − ; ¯ ¯ T (1 − h0 ) [1 − exp (−²H )] l0¯ (1 − h0 )  £ ¤ ¡ T ¢λT  T T (1 − h0 ) k1 + h0 µH l0 ¯ βE0   − ¯0 = 0 lT (164) (1−h0 )kT +h0 µT ¡¯ ¢1+λT T T ¯ T 1 1+λT H l0 − (1 + g0 ) d0 and   1 ¡ ¢ 1 ¯T − β 1 + g0 E0  ˆT T +h µT ¡ T ¢1+λT  d0 (1−h0 )k1 0 H ¯ l0 T ¯ − (1 + g0 ) dT 1+λT 0 1 = (165) [1 − exp (−²H )] [(1 − h0 )τ kT + h0 µT ] T H · ¯ ¸ 1 −βE0 . (166) [(1 − h0 )kT + h0 µT ] − [1 − exp (−²T )] [(1 − h0 )τ kT + h0 µT ] 1 H H H H ¯ = A(τ , h0 ) = 0 Since Φ ∈ R and h0 ∈ J (Φ), it follows from (82) and from the last equality in (165) that τ = γ T (h0 ). Therefore, dT = dT and ¯0 = ˆ0 . ¯ 0 ˆ 0 l T lT Proof of the proposition 2 It follows from (101) that ˆ0 is given by (88). Then, it follows from (66), (89) and (99) that lT h i ˆ (1 − h0 ) kT + h0 µT φT = (1 − h0 ) γ T (h0 ) kT + h0 µT , (167) H ¯H H ˆT where we use the deÞnition in (11). Substituting (63)-(65) into (167), we get g0 in (109)-(104). Next, ˆ substituting g0 into (89), we get dT . It follows from (75) and (100) that ˆT 0 ˆ dN T = 0 . (168) 0 Finally, we have that g0 and ˆ0 solve the equation system ˆNT lNT ¡ ¢ " ¡ ¢# pNT ∂u0 cNT ˆ0 ˆ0 ¡ ¢ pNT ∂u1 0, cNT ˆ1 ˆ1 ˆNT − 1 + g0 βE0 = 0; (169) p0 ˆ ∂cN T 0 p1 ˆ ∂cN T 1 " ¡ ¢# pNT NT ³ˆNT ´λN T ∂u1 0, cNT ˆ1 ˆ1 βE0 k l0 − ˆ0 lNT = 0 , (170) p1 1 ˆ ∂cNT 1 38 where prices and consumption in equilibrium are given by (120)-(122). 39