I. Introduction

Since 1982, a number of major debtor countries have faced weakened prospects for repayment of their debts and, consequently, very restricted access to foreign borrowing. The adjustment to that situation has brought about major changes in their economies, in particular, sharp drops in investment and large real depreciations of the exchange rates. This paper provides a formal framework to analyze one particular aspect of the optimal debtor response to this situation, namely that of determining the optimal level of productive investment and its composition in the face of external financing difficulties.

When a debtor country faces a situation of credit constraints, it needs to lower investment and/or raise domestic savings (or generate a smaller excess of investment over savings) to obtain the required improvement in its trade account. In the absence of policy intervention, the savings/investment adjustment would take place without consideration of the feedback of actions into the credit constraint itself because, for individual agents, these are nonexistent. Results available in the debt literature are that this market response to the credit constraint would involve an excessive reduction in investment.^1/ However, there is also another reason why investment may have a strong effect on the credit ceiling: the allocation of investment between export-producing and import-competing industries will affect the creditworthiness of the debtor country. The policy dilemma that the authorities face is the following: is it advisable to promote investment in the production of importables that would be valuable in the event of a default or is it preferable to invest in the production of exportables, which would help increase the availability of foreign financing? This paper shows that investment in export-producing activities contributes towards relaxing the foreign borrowing constraint, while investment in import-competing activities tightens the foreign borrowing constraint. Therefore, at least when terms of trade are exogenous to the debtor country, policy should be directed towards promoting investment in the export producing sector.^2/

The framework of international credit markets that we adopt is one in which debtor countries may choose to repudiate their debts even in situations in which they have enough resources to repay them.^3/ There are, of course, costs associated with a debt repudiation. At every point in time, a social planner computes the costs and benefits from the repayment and the repudiation alternatives, and decides which route to take. This option to repudiate foreign debt alters substantially the consumption/savings and resource allocation decisions even if the country never actually repudiates its debts. This framework is similar to the one studied by Eaton and Gersovitz (1981), and Cohen and Sachs (1986).

The nature of the sanctions that a debtor country may expect to be subjected to in case of repudiation is a critical factor in this framework. When a country repudiates its debt, it is obtaining debt relief at the cost of financial and economic sanctions. The exact penalties that a repudiating country would face are not explicit, and very difficult to determine. Although there has been some historical experience, it cannot be easily extrapolated. Given the critical role of sanctions in any debt model, we discuss the issue in detail below. We conclude that it is reasonable to assume that sanctions comprise a permanent exclusion from foreign borrowing and some trade-related measures that reduce the advantages of international trade for the debtor country.

In this paper, we consider the case of an economy producing two goods--an import and an export good. Then, in addition to the savings/investment balance, the adjustment to the more stringent foreign borrowing conditions implies a current account adjustment that usually requires large real exchange rate changes. This paper suggests that the optimal response, because it implies the need to promote export production relative to import-competing production, cannot be achieved through exchange rate policy alone. An increase in the production of exports helps to increase the amount of foreign borrowing available to the country. Consequently, the country needs to shift the pattern of investment towards the export sector.^4/ Although the country invests less in the export sector than in the case of no credit ceiling, it will invest more than a decentralized economy. It is noteworthy that this policy response implies an increase in the returns to the production of exports relative to that of imports, and that an exchange rate policy is not a good instrument for that purpose. Even the existence of uncertainty about the future regime (default or repayment) does not alter the basic result of the convenience of export promotion, because investment in the export-producing sector shifts out the credit supply function for the country, improving its borrowing conditions.

The plan of the paper is as follows. Section II discusses the issue of costs and benefits of default, and justifies the assumptions adopted in this paper. Section III develops a two-period framework, in which precise results can be obtained. Section IV considers the infinite horizon case, which enables us to consider issues such as the cost resulting from exclusion from future borrowing. Section V extends the model by incorporating uncertainty. Section VI concludes with a summary of results and their possible policy implications. Two appendices detail the derivation of results in section III and the numerical solution method applied in simulating the model of section IV.

II. Debt Repudiation: Possible Sanctions

A key determinant of the implications of any debt model is the assumption about the resolution of a default situation. In this paper we assume that a country defaults whenever the expected benefits from default exceed the expected costs. This seems a more relevant criterion than determining the default decision by the country’s ability to pay. Few countries are physically unable to meet their obligations (indeed sovereign lending was considered safe since true insolvency of a country is virtually impossible) but the costs of doing so may far exceed the benefits.^5/

Although the benefit to the debtor from debt repudiation is simply the avoidance of debt service, the costs of that action are even difficult to identify, let alone estimate. Although there exists a growing literature in this area,^6/ a number of issues are unresolved, and some legal issues remain to be tested in the courts. Unlike an ordinary commercial borrower, a debtor country is protected from legal sanctions under the broad concept of “sovereign immunity.” Since a foreign borrower cannot be brought to bankruptcy court, private creditors have limited recourse to legal sanctions.

The actual power of creditors to legally obtain compensation for unpaid sovereign debt is doubtful. One frequently cited possible line of defense is based on the Foreign Sovereign Immunities Act (1976) in the U.S. (and the corresponding legislation in the U.K.). However, the actual effectiveness of the FSIA is limited, because sovereign borrowers explicitly waive their immunities in the loan contracts, and because of the commercial activity exception that prevents the application of the FSIA when the act is connected with a commercial activity. Another, potentially more effective line of defense might be based on the “act of state doctrine,” which originally applied to expropriation, but that recently has also been applied to foreign exchange controls. Although the act of state doctrine would not apply in the case of default on a specific contract, it could apply if, for example, a debtor country were to impose exchange controls that make impossible to service foreign debts in foreign currency.^7/ The experience of private creditors (until the 1970s these were mainly bond holders) attempting to attach assets belong to the defaulting country has been varied, and, especially in the last century, none too successful. Eichengreen and Portes (1986), for example, have discussed the sanctions for default in the 1930s.

Therefore, it appears likely that the main penalties for default will consist of non-judicial sanctions. Of these, the most often-cited penalty would be the country’s exclusion from further participation in the capital markets.^8/ Of course, a borrower contemplating default will be unlikely to obtain much long-term credit from the capital markets in any case, which means that a future exclusion from borrowing will not represent a severe cost. Kindleberger (1982), for example, argues that several defaults during the 1920s were prompted by the perception that financial markets were collapsing so that the reputation for being a “good” borrower became less valuable and default became more attractive. More conclusively, in a recent paper Bulow and Rogoff (1988) show that, if the country can hold foreign assets after default, it will always choose default on its debt at some point in time, which renders reputation an empty concept in international borrowing.^9/

A more important direct penalty is that the defaulting country is liable to lose its short-term trade finance, and the trade intermediation services provided by international banks that go along with it. Export credit flows themselves have grown rapidly in recent years, and constitute a major proportion of developing countries’ external finance. In addition to the financial component, bank intermediation provides a number of ancillary services that may greatly facilitate international transactions for debtor countries. Should a country lose access to these lines of credit and the availability of attached services, the cost of international transactions may rise considerably. Enders and Matione (1984), for example, estimate that the rise in the costs of imports, per unit of exports, may be in the order of 5 to 10 per cent. The loss of such credits and international bank services constitute perhaps the most severe penalty creditor banks may impose on a defaulting country.

Ultimately, of course, the penalties for default will depend upon a complex interaction of political and bargaining issues. Commercial banks will also have to consider the effects of their response to a default by one borrower on their reputations vis à vis all their other borrowers.^10/ Obviously, a formal model cannot capture all, or indeed most, of these complex considerations. At a minimum, the model should incorporate the exclusion from future borrowing (which we term the “indirect” penalty) and some form of trade-related measures, or “direct” penalty. The latter is intended to capture the increased costs of trade without trade financing, and to a lesser degree, the possible seizure of the debtor’s goods in transit.^11/

Typical trade sanctions associated with default would be the lack of access to commercial credit and to bank trade intermediation, trade embargoes from some creditor countries--which could be avoided by trading through a third country--and the possibility of seizure of shipments of merchandise on international transport. It is not unreasonable to assume that all these actions would involve a cost that is proportionate to the dollar value of trade, and can therefore be represented as an increase in the unit cost of imports and a reduction in the unit return to exports, that is, simply a terms of trade deterioration. Equivalently, they could be thought of as increasing “transportation costs” or “transaction costs” which reduce export FOB prices and increase import CIF prices. Indeed, in their study, Enders and Mattione (1984), assume that “trade in both exports and imports is disrupted, and the costs of disruptions can be modeled as an X percent decrease in unit export earnings and an equivalent increase in unit import costs; these costs are due to foreign suppliers’ attempts to insure themselves against new defaults, to creditors’ attempts to attach goods and payments, and to the inefficiencies of administration such a scheme.” Accordingly, we model the direct penalty for default as a permanent change in the country’s terms of trade. In addition, we assume the existence of an indirect penalty of perennial exclusion from the capital markets.

III. Debt and Investment in a Two-period Model

We consider an economy with a two-period time horizon, producing one exportable and one importable good. The production of each good requires sector-specific capital. In this framework, three important results obtain:

1. The ceiling on foreign borrowing faced by this economy is an increasing function of the stock of capital in the export sector and a decreasing function of the stock of capital in the import sector.

2. The optimal response to the imposition of the credit ceiling is to reduce investment in both productive sectors (relative to the case of no risk of debt repudiation.)

3. The optimal amount of investment in the export sector is higher than the amount that would obtain in a decentralized economy. This means that the credit-rationed economy should follow an “export promotion” policy.

In the two-period case, the penalties for lack of repayment cannot include exclusion from future borrowing. Thus, penalties will consist entirely of trade sanctions. Consistently with the discussion of the previous section, we assume that the effect of trade sanctions is equivalent to a permanent terms of trade deterioration for the debtor country. Specifically, in the event of external debt repudiation, a fraction ρ of exports are lost and the cost of imports raises by an equivalent proportion. Therefore, if the price of exports was unity, the net return to exports will become 1-ρ after default, and if the price of imports was P, the total cost of imports will become P/(1-ρ) after default.^12/ This means that the terms of trade--relative price of imports in terms of exports--will shift from P to θP, where θ = (1-ρ)^-1

As is known from several contributions to this literature in this type of framework creditor banks will set a ceiling on lending to sovereign countries in order to avoid the repudiation of their obligations.^13/ This is because as foreign debt increases, the rewards to a repudiation of foreign debt also increase.^14/ Consequently, creditor banks will never increase lending past the point in which repudiation becomes a more attractive program than repayment.

We are interested in finding out how the credit ceiling function is affected by investment in each productive sector. We start by obtaining the credit ceiling function as of the beginning of the second period. For this purpose, it is convenient to use the indirect utility functions that correspond to the repayment and default regimes. An indirect utility function gives the maximum utility obtained by the representative consumer, as a function of relative prices and income. In the case of repayment of foreign debt, the indirect utility function, V^R (the superscript R stands for repayment) will be obtained from:

where P is the exogenous relative price of imports, y^* and y represent the the supply of the exportable and importable good, respectively, c^* and c represent consumption of each of the two goods, and D represents the level of foreign debt carried over from the previous period, which carries a (world) interest rate r. For the exportable and importable goods, production functions are given by:

y^* = f(k^*), and

y = f(k)

where k^* and k represent the capital stocks specific to the respective sector.

Let V^D represent the indirect utility function in case of repudiation (the superscript D standing for default). V^D is obtained from:

where θP represent the (net) international terms of trade received by the debtor country as a consequence of trade sanctions. The credit ceiling is the maximum amount of debt that debtors can owe and still prefer the repayment option. Since V^* is a decreasing function of D, the credit ceiling is a quantity $\overline{\mathrm{D}}$ such that:

Note that this implies that the credit ceiling $\overline{\mathrm{D}}$ is the equivalent variation in income that compensates for a terms of trade deterioration in proportion θ. In terms of expenditure functions, $\overline{\mathrm{D}}$ can be obtained as:

where e(·) is the expenditure function, and V^D is the indirect utility function under default defined in (2). The determination of the credit ceiling can be illustrated with the aid of Figure 1. In Figure 1, the point E represents the pair (y,y^*) of output of the two goods. At the default terms of trade (θP) consumption would be at point D. At the undisturbed (free trade) terms of trade P, the same utility level could be achieved consuming at point R. It can be seen that, if the resources of the country were reduced by an amount $\overline{\mathrm{D}}\left(1+\mathrm{r}\right)$ (in export good units), the country would be indifferent between repaying or defaulting on its foreign debt. This implies that the credit ceiling is equal to $\overline{\mathrm{D}}$.

View Full Size

The 2-Period Model

Citation: IMF Working Papers 1989, 016; 10.5089/9781451923216.001.A001

D: Default consumptionR: Repay consumption (when Debt = Debt ceiling)E: Endowment point (y,y”)$\overline{\mathrm{D}}$: Debt ceiling

• Download Figure
• Download figure as PowerPoint slide

Differentiating (3) we can investigate the dependency of the credit ceiling on the variables of interest. We first compute:

where V_I represents the derivative of the indirect utility function with respect to income (its second argument). The second equality follows from a well-known envelope theorem that equates marginal utility of income to marginal utility of consumption at the optimal point.^15/ Similarly, we have:

where the superscripts R and D in the utility function indicate that the derivatives are computed at the consumption bundles corresponding to the repayment and default situations, respectively, and where subscripts indicate partial derivatives with respect to the variable in the subscript. In obtaining the second equality we have also used the conditions that, at an optimal point, ${\mathrm{U}}_{\mathrm{c}}^{\mathrm{R}}\mathrm{=}{\mathrm{PU}}_{\mathrm{c}}^{\mathrm{R}}\mathrm{*}$, and ${\mathrm{U}}_{\mathrm{c}}^{\mathrm{D}}\mathrm{=}\theta {\mathrm{P}\mathrm{U}}_{\mathrm{c}}^{\mathrm{D}}\mathrm{*}$.

For fixed terms of trade, we can express the credit ceiling as a function of only the capital stock in each productive sector: $\overline{\mathrm{D}}\mathrm{=}\mathrm{h}\mathrm{\left(}{\mathrm{k}}^{\mathrm{*}}\mathrm{,}\mathrm{k}\mathrm{\right)}$.^16/ Noting that the marginal product of capital is the same in either the repayment or the default case because capital is predetermined, and because there is no further use for capital after period 1, the derivatives of the h(·) can be expressed simply as:

The sign of this derivative follows from ${\mathrm{U}}_{\mathrm{c}}^{\mathrm{R}}\mathrm{*}\mathrm{>}{\mathrm{U}}_{\mathrm{c}}^{\mathrm{D}}\mathrm{*}$, which is proved in Appendix 1. The inequality means that the export good has a higher marginal utility under repayment than under default, when evaluated at the point in which the country is indifferent between repaying or defaulting. This is because when the country defaults, the export good becomes relatively cheaper (terms of trade deteriorate by a factor of θ), and the marginal utility of the export good decreases as its relative price falls and the utility level is kept constant.

The derivative of the credit ceiling function with respect to capital installed in the import good industry is:

Similarly, the sign of (7) is determined by the fact that ${\mathrm{U}}_{\mathrm{c}}^{\mathrm{D}}\mathrm{>}{\mathrm{U}}_{\mathrm{c}}^{\mathrm{R}}$, also proved in Appendix 1. In other words, the marginal utility of the import good increases as its relative price goes up and the utility level is kept constant.

Consider now the problem of the optimal composition of investment in the debtor country given its foreign borrowing constraints. The “social planner” of the debtor economy, must decide how much to allocate to consumption and investment on each industry, given the credit ceiling function h(·). That decision is reached by solving:

where we have assumed a 100 percent depreciation rate, i.e. k₁ = I in both sectors, and that there is no initial debt. The first order conditions for this problem are:

where λ and μ are the Lagrange multipliers associated with the constraints in (8). Investment in the export sector is implicitly given by (9f). Using (6) one obtains:

This means that investment in the export sector will be less than the level that would obtain if the country did not have the option to repudiate its debt. With no risk of repudiation, the domestic interest rate would be equal to the world interest rate r and the marginal product of capital would be equal to both. The existence of risk of debt repudiation and the consequent credit ceiling increases the actual cost of borrowing for the debtor country; the optimal domestic interest rate (the one that generates the optimal amount of investment) exceeds the world interest rate by a factor--in the right-hand side of (10)--that expresses how the choice between default or repayment is affected by additional units of capital in the export sector. Investment in the import sector can be obtained using (7) and (9f):

Thus, investment in the import sector also declines relative to the level that would prevail if there were no option to repudiate foreign debt. The decline in investment in the import sector is larger than the decline in the export sector, in the sense that the optimal domestic cost of capital for the import sector is larger than for the export sector. This is clear from (6) and (7).

The reason for the differential effect on investment in the two sectors is that investment in the export sector has a beneficial effect on the credit ceiling. Because the relative price of exports falls in the event of repudiation, investment in the export sector makes repudiation more costly to the debtor country and permits a higher safe limit of indebtedness, easing the credit constraint. Then, the response to the existence of a credit constraint by an optimal planner should be a generalized reduction in investment, but with a change in its composition into export production (relative to the levels prevailing in the absence of the repudiation option).

For policy purposes, however, the relevant comparison is between the social planner decisions and those which would obtain in a (partially) decentralized economy, because this indicates the precise nature of intervention in the form of taxes or subsidies that is needed to achieve the social optimum. In this context, an economy can only be partially decentralized because the decision to repudiate foreign debt must be a coordinated one. Then, we consider an economy in which there is a central authority whose main function is to decide, at each point in time, if the country should repudiate its foreign debt. In addition, the central authority is a financial intermediary between foreign lenders and domestic borrowers. We assume the central authority conducts this function in such a way as to allow domestic prices and interest rates to be determined in competitive markets, and that it refunds to the private sector, in a lump-sum way, all profits resulting from this intermediation activity.

The first thing to note about such a decentralized economy is that it will face exactly the same credit ceiling function as does the socially planned economy. This is because the aggregate costs and benefits of repudiation are the same. In case of repudiation, the decentralized economy will face the same implicit terms of trade deterioration, and benefit from full debt relief, just as the planned economy. Thus, from the point of view of a representative consumer, the point at which default becomes optimal is the same for the two economies. This is certainly a special feature of the two-period model; in a multiperiod setup, the change in the domestic interest rate following default would be different in the two economies, and this factor alone would make the value of the repudiation option different.

The mechanics of the decentralized economy are the following: a central authority does all foreign borrowing and repayments. The foreign resources thus obtained are passed on to the public at a market-clearing interest rate. If the country is credit-constrained, the domestic interest rate will exceed the international interest rate, and the central authority will make a profit, which we assume is returned to the private sector as a lump-sum payment at the beginning of the second period.

The private sector behavior in the decentralized economy will be determined by the solution to the following optimization problem (where the superscript c identifies the decentralized economy problem):

where r^D is the domestic interest rate. The term $\frac{{\mathrm{r}}^{\mathrm{D}}\mathrm{-}\mathrm{r}}{\mathrm{1}\mathrm{+}{\mathrm{r}}^{\mathrm{D}}}\mathrm{h}\mathrm{\left(}{\mathrm{I}}_{\mathrm{0}}^{\mathrm{*}}\mathrm{,}{\mathrm{I}}_{\mathrm{0}}\mathrm{\right)}$ represents profits made by the central authority by borrowing abroad at the rate r, and competitively allocating those funds among domestic borrowers at rate r^D. If the country is credit constrained, profits will always be positive. Although profits are immediately reimbursed to the private sector, agents know that their individual actions have an insignificant impact on the central authority’s profit distribution and correspondingly ignore any such repercussion. In consequence, the first-order conditions for the private sector’s problem are:

These first-order conditions are quite standard. Rates of growth of marginal utility and marginal products of capital in each sector are equalized to the domestic interest rate r^D. There are also two market clearing conditions that, assuming that the economy is credit constrained, are given by:

These two conditions represent the current account balance in periods 1 and 2. Of course, one of these conditions is redundant by application of the budget constraints.

The comparison of the planned economy and the decentralized economy can be done in the following way. We are going to obtain the effect on utility of a representative consumer resulting from a marginal increase in investment in each sector: dV^c/dI^* and dV^c/dI. Since we know that the planned economy achieves maximum utility, it follows that if dV^c/dI^* > 0, the planned economy invests more in the export sector than the decentralized economy, and if dV^c/dI < 0, the planned economy invests less in the import sector than the decentralized economy.

Applying first-order and equilibrium conditions, the expression reduces to:

because ${\mathrm{h}}_{{\mathrm{k}}^{\mathrm{*}}}^{\mathrm{\prime }}>0$. The effect on utility of the change in the domestic interest rate vanishes because the private sector is in a balanced position with respect to r^D with any increase in the cost of borrowing refunded by the central authority. Symmetrically:

In summary, in the two-period horizon case, the optimal policy response involves a subsidy for investment in the export sector and a tax on investment in the import-competing sector. We will now study the extension of this result to the infinite horizon model.

IV. Debt and Investment in an Infinite-Horizon Model

The addition of the indirect penalty in the infinite-horizon model does not alter the results of the previous section in a significant way. Although we have been unable to show that the optimal policy involves a subsidy to investment in the export-producing sector and a tax on investment in the import-competing sector at every point in time, there is a strong pressumption that this is in fact the case. If the derivatives of the credit ceiling function have the same signs--as we conjecture--investment in the export sector will be subsidized in the steady state position. With adjustment costs to investment, it then seems natural that investment in the export sector will always be subsidized. Furthermore, we provide a numerical example that is in accordance to this argument.

As before, the credit ceiling function faced by this economy is obtained from a comparison of the value functions under the default and repayment alternatives. The determination of the credit ceiling in this model is extremely complicated. Recall that one of the benefits of repayment is the continued access to capital markets; the more the country will use the capital markets in the future, the greater its incentives to repay today. Therefore, the higher the credit ceiling the country expects tomorrow the larger the debts it can safely owe today. But the credit ceiling tomorrow will be an increasing function of the credit ceiling the period after as well. Clearly, the credit ceiling is infinitely recursive. This suggests that the credit ceiling function can be solved for by using dynamic programming techniques to compute the utility value for the debtor country of the repayment and default alternatives.

Let V^D(k,k^*) denote the present discounted utility (i.e. the value function) if the country decides to default. This value function is computed from:

where δ is the depreciation rate of capital.

Let V^R(k,k^*,D) denote the value function under repayment. This value function is computed from:

where $\overline{\mathrm{D}}$ represents the ceiling on foreing debt faced by the country. As before, the credit ceiling is obtained from:

From (18)-(20) we can compute the effects that capital accumulation in each sector has on the country’s credit ceiling. These effects are given by:

Note that V_k* and V_k are the marginal utility values of a unit of capital in the corresponding productive sector.^17/ This means that, in each sector, the credit ceiling will be an increasing function of the capital stock if and only if the marginal value of capital is larger under the repayment option than under the default option. (Note that ${\mathrm{V}}_{\mathrm{D}}^{\mathrm{R}}$ is negative). In other words, creditors will be more willing to extend credit if the country invests in projects that make the repayment alternative more valuable than the default alternative.

An analytical derivation of the signs of the derivatives of the credit ceiling functions is, in fact, quite complex. It involves comparing the relative sizes of investment q’s in two different optimization problems (the repayment and the default problems) which display different interest rates (marginal rates of substitution in consumption). However, there is a strong presumption that--as in the two-period case--the credit ceiling is increasing in capital in the export sector and decreasing in capital in the import-competing sector. Consider the effect of an additional unit of investment in the import-competing sector. In the event of default the country is assumed to suffer a terms of trade deterioration, therefore, the value (measured in terms of the true CPI) of its capital stock in the import competing sector rises by a discrete jump θ. If the country repays, however, the value of its import-competing capital remains constant. This suggests that the gain from an additional unit of capital in the import competing sector is larger in the case of default than it is under repayment. Exactly the opposite holds for capital in the export sector. The larger the export sector, the larger the loss resulting from the terms of trade deterioration should the country default. Hence the utility value of export capital is greater under repayment than it is under default. Thus we conjecture:

and the credit ceiling function displays:

h_k*(k,k^*) > 0 and h_k(k,k^*) < 0

that is, the credit ceiling is an increasing function of capital in the export sector, but a decreasing function of capital in the import-competing sector.

Once again, debt repudiation will never take place in this framework. This is because rational creditors foresee the debtor country’s incentives to repudiate its foreign debt and restrict credit in such a way that the debtor country always finds repayment preferable to default. Consider now the optimal plan for the debtor economy given the existence of the credit ceiling function h(·) described above. The first-order conditions for that problem are given by:

where V_Dt+1 is a shorthand for ${\mathrm{V}}_{\mathrm{D}}^{\mathrm{R}}\mathrm{\left(}{\mathrm{k}}_{\mathrm{t}\mathrm{+}\mathrm{1}}^{\mathrm{*}}\mathrm{,}{\mathrm{k}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{,}{\mathrm{D}}_{\mathrm{t}\mathrm{+}\mathrm{1}}\mathrm{\right)}$ etc. Also, the following envelope theorems can be proved:

where μ is the Lagrange multiplier on the external credit ceiling for the economy. While the credit ceiling is not binding, μ is equal to zero, and the above conditions imply the standard optimal rules for consumption and investment:

That is, the marginal utility of consumption grows at a rate inverse to the world interest rate, and the marginal product of capital equals the interest plus depreciation rates in each sector.

Once the credit constraint is binding, however, μ is positive, and the optimal policies must be modified:

where S_t is equal to ${\mathrm{U}}_{{\mathrm{c}}_{\mathrm{t}}^{\mathrm{*}}}/\beta {\mathrm{U}}_{{\mathrm{c}}_{\mathrm{t}\mathrm{+}\mathrm{1}}^{\mathrm{*}}}$, which we will refer to as the implicit domestic interest rate. Therefore, when the credit constraint is binding, the marginal product of capital in each sector is set equal to a weighted average of the implicit domestic interest rate and the international interest rate, with weights h_k* and h_k, respectively. It can be shown that this policy implies a subsidy to investment in the export sector relative to the import-substitution sector in the following way:

Since the interest rate in the credit-constrained economy must be greater than the international interest rate, the first term in square brackets is positive and since h_k* > 0 and h_k < 0, the second term in square brackets is negative. This implies that there is a bias towards greater investment in the export sector relative to the import sector. The reason is, once again, that altering the investment mix of the economy can relax the credit ceiling, which increases welfare in a credit constrained country.