A. The Model

Consider a credit-constrained LIC that is unable to finance high-yielding human capital investments in education, training, and health care. In the absence of collateral or assets that can be seized, private lenders from creditor countries are not willing to finance these investments. In this situation, an IFI possessing powers to penalize defaulting countries could remedy the market failure.

There are two periods. The first-period income available to a representative member of the LIC is given by y. The LIC government taxes current income, and receives transfers and loan disbursements from the IFI. Earnings in period two are given by wh(x). The wage rate paid to a unit of human capital, w, is constant and reflects LIC total factor productivity and its state of institutions and technology. h(x) is a strictly increasing and strictly concave human capital production function. The government taxes second-period earnings to repay the loan. For now, taxes are assumed to be lump-sum.

We begin by viewing the IFI as a “social planner” with control over a full set of policy instruments. The IFI disburses loans (l) to the LIC in period one and lump-sum transfers (T₁,T₂) in periods one and two, respectively. The IFI’s altruism may cause it to charge the LIC an interest rate μr that is lower than the world interest rate r, where −1/r ≤ μ ≤ 1 is an IFI policy variable.

The LIC government chooses investment levels (and the corresponding taxes) to maximize the utility function of its representative household

In equation (2.1) c₁ = y – x + l + T₁, c₂ = wh(x) – (1 + μr)l + T₂, u is a strictly increasing and strictly concave function, and 1 > β > 0 is the LIC’s subjective time discount factor.

The IFI has been set up to provide assistance to countries in need. It is endowed with a certain amount of capital in period 1, B₁, provided to it by creditor countries. For simplicity, in this section this endowment is exogenously given. The IFI makes loans to either LICs or middle-income countries. The IFI is altruistic toward LICs, but not other countries, where it always demands full repayment of loans at the going world interest rate. The IFI also cares about maintaining its ability to provide loans in the future, which is proxied by the end-of-period capital stock, B₂. The IFI’s preferences are given by

In equation (2.2), B₂ = (B₁– l–T₁)(1 + r) +l(1 + rμ)–T₂, V is a strictly concave function, and γ > 0 is the strength of the IFI’s altruism toward the LIC.

B. First-Best Redistribution

We first characterize first-best allocations assuming that the IFI has the powers of a world social planner possessing a full set of policy tools. These tools include setting investment levels and making lump-sum transfers to (or levying lump-sum taxes on) the LIC. The remainder of the paper is devoted to describing allocations that are possible when the IFI possesses a more limited set of policy instruments.

With a complete set of instruments, the IFI’s problem is to choose a nonnegative value of x* and (T₁*,T₂*) to maximize

Assuming x*>0, the necessary and sufficient conditions for maximization can be written as

The first order conditions (2.4a–2.4c) implicitly determine first-best investment x* and transfers (T₁*, T₂*). The resulting consumption allocation combines productive and intertemporal efficiency with optimal redistribution across the IFI and the poor country. The first determines the optimal redistribution. The second condition implies that the allocation of resources in the LIC is intertemporally optimal. The marginal rate of substitution in the LIC, $u_1^{\prime }/\beta u_2^{\prime }$, is equated to the world’s opportunity cost of current funds, 1+ r. Combining the last two conditions implies that 1 + r = wh′. This condition guarantees productive efficiency in the LIC, i.e. equality of the marginal returns to human capital in the LIC with the world interest rate. The following example illustrates the first-best allocation for a simple economy.

C. Loans Under Commitment

In this section we assume that the IFI no longer has the tax or transfer powers but can extend loans to the LIC at market rates and can commit to requiring full loan repayment. Without lump-sum transfers or the ability to provide interest subsidies, the IFI has no tools to redistribute resources and will fail to realize the first-best. The question remains whether the resulting allocation of resources is Pareto Optimal.

When the IFI can commit to require full repayment, unconditional loans lead to a Pareto Optimum, as can be seen by considering the unconditional loan game. This game unfolds in a single-period in two stages. In the first stage, the IFI chooses the loan amount. In the second stage, the LIC chooses investment given the loan presented by the IFI in the first stage. The game is solved working backwards. In the second stage, the LIC selects x for an arbitrarily given loan amount (and for T₁=T₂=0 and μ = 1), to maximize (2.1). The implicit solution for the investment schedule as a function of the IFI loan is

Next, the IFI’s first-stage problem is to choose l to maximize (2.2) subject to (2.5). This leads us to the following result. The proofs of all propositions are in the Annex I.

In this setting, there is no conflict of interest between the IFI and the recipient. The common objective of the IFI and of the LIC government is to maximize the utility of the representative household. In addition, commitment rules out any strategic behavior (there is no Samaritan’s dilemma). Given that the IFI is charging the world interest rate for funds, the representative household’s utility is maximized when the IFI loans an amount that gives rise to the efficient level of investment. One can also easily show that the form of the loan policy does not matter under commitment; efficiency is obtained either with unconditional or conditional loans. But while the resulting allocations are efficient, they are not fully optimal because the IFI wants to redistribute resources to the poor country (i.e., they want to satisfy (2.4a)). The question is whether IFIs can allow themselves to be altruistic, causing less than full loan repayment, without destroying the efficient outcomes obtained under commitment.

D. Loans With out Commitment

Without commitment, the IFI is free to choose μ in the second period. For its loan policies to be time-consistent, it must properly account for this fact when choosing the loan amount in the first period. This creates a dynamic game that is solved by backward induction. The IFI, and the LIC, must first anticipate the IFI’s optimal second-period choice of μ for all possible choices of first-period loan amounts and investment levels. In the first period the IFI and the LIC choose optimal loan and investments levels, given the optimal repayment policy in the second period. The timing of choices in the first period depends on whether IFI loans are conditional or unconditional.

The IFI’s optimal second-period repayment policy can be determined independently of its first-period loan policy since μ is chosen to maximize (2.2) for given x and l. The repayment policy is implicitly defined by

Now, we turn to the first period, and examine the unconditional and conditional loan policies.

Conditional Loans

We view conditional loans as an entire loan schedule, presented to the poor country, indicating how much the IFI is willing to loan at each investment level. Thus, the loan quantity is conditioned on the investment level chosen and the poor country has full knowledge of this.

In the first stage, the IFI determines its loan schedule by choosing l, given x and (2.6), to maximize (2.2). The loan schedule is implicitly defined by

$\begin{array}{ccc}{u}_1^{\prime }(y-x+\tilde{l}(x))\equiv \beta (1+r)u_2^{\prime }\left(h(x))-(1+\tilde{\mu }\left(x,\tilde{l}(x)\right)r)\tilde{l}(x)\right)& (2.9)& \end{array}$

For the same reasons as in the unconditional case, loans are chosen so as to generate an efficient intertemporal allocation of consumption.

In the second stage, the LIC takes the entire loan policy, defined by (2.6) and (2.9), as given and chooses x to maximize (2.1). This generates the following important result.

Proposition 2.3

Without commitment, conditional loans lead to a fully efficient outcome (conditions (2.4a)–(2.4c) are satisfied).

If the IFI sets its conditional loan policy optimally, as given by (2.6) and (2.9), the net marginal benefit to the LIC of increasing x is positive whenever the return to investment exceeds 1 + r. While the LIC does not receive the entire difference between the return to a unit investment and 1+ r, since some of the gain is reduced by increases in μ, it does receive a portion of it. Thus, the LIC must set investment levels so that the rate of return to investment is equated to r, if their wealth is to be maximized.

The fact that the recipient optimally chooses the investment level eliminates the need to monitor their behavior. It is in their best interest to carry out the investment, and in this sense they fully “own” the policy. In fact, one can show that recipients would choose efficient investment levels even if they had full discretion to determine their loan and investment levels independently. However, this would lead to “too much” borrowing in order to finance inefficient levels of current period consumption. The recipients would equate the marginal value of first period consumption to the subsidized, rather than the market, interest rate. Conditional loans retain the rationing feature needed to achieve an efficient allocation of consumption, while at the same time, generating efficient investment levels.

In short, conditional loans lead to an efficient outcome, where unconditional loan do not, because the LIC accounts for the fact that higher levels of investment will induce greater (rationed) loans from the IFI. Unless the LIC clearly recognizes this advantage, the full benefits of investing are not accounted for and investment is set too low.

A second interpretation of the result can be based on Becker’s (1974) famous “Rotten Kid Theorem.” Becker argues that even if a child cares nothing about other family members, he will act so as to maximize family income. However, this is only true if the child understands that his actions will affect the level of altruistic transfers received from parents. In order for a poor country, that cares nothing about the donor, to maximize consolidated resources, they too must fully understand the connection between their behavior and the altruistic transfer received from the donor. When loans and interest payments are both made conditional on recipient’s actions, then the necessary connection between behavior and transfers is made.

^{Example 2.}

In Example 1, let h(x) = x^α, 0 < α < 1. The IFI’s second-period choice of μ is,

$\tilde{\mu }=\frac{wh-\gamma \beta B_1(1+r)-(1-\gamma \beta r)l}{rl(1+\gamma \beta )}$

The IFI will require partial interest repayment, $0<\tilde{\mu }<1$, if the NPV of the LIC’s second-period income is in the range

$\gamma \beta B_1-\frac{1-\gamma \beta r}{1+r}l<\frac{wh}{1+r}<\gamma \beta B_1+l.$

Second-period LIC consumption and IFI capital are proportional to consolidated world resources wh/(1 + r) + (B₁ – l) :

$\begin{array}{l}\frac{c_2}{1+r}=\frac{\gamma \beta }{1+\gamma \beta }\left\{\frac{wh}{1+r}+(B{}_1-l)\right\}\\ \frac{B_2}{1+r}=\frac{1}{1+\gamma \beta }\left\{\frac{wh}{1+r}+(B{}_1-l)\right\}.\end{array}$

To derive the IFI’s optimal time-consistent loan policy, maximize the IFI’s objective function while taking into account its second-period choice of $\tilde{\mu }$. This yields the following.

$\tilde{l}(x)=\frac{1}{1+\gamma \beta +\gamma }\left\{\gamma \left[\frac{wh}{1+r}+B_1\right]-(1+\gamma \beta )(y-x)\right\}.$

The first-stage LIC consumption allocation and IFI capital under the conditional loan are:

$\begin{array}{c}{c}_1=\frac{\gamma }{1+\gamma \beta +\gamma }\left\{B_1+y-x+\frac{wh}{1+r}\right\}\\ \frac{c_2}{1+r}=\frac{\gamma \beta }{1+\gamma \beta +\gamma }\left\{B_1+\frac{wh}{1+r}+y-x\right\}\\ \frac{B_2}{1+r}=\frac{1}{1+\gamma \beta +\gamma }\left\{B_1+\frac{wh}{1+r}+y-x\right\}.\end{array}$

In the second stage, the LIC chooses x taking $\tilde{\mu }\text{and}\tilde{l}(x)$ as given to maximize its utility:

${\max }_x\{\ln \left[\frac{\gamma }{1+\gamma \beta +\gamma }\left\{B_1+\frac{wh}{1+r}+y-x\right\}\right]+\beta \ln \left[\frac{\gamma \beta (1+r)}{1+\gamma \beta +\gamma }\left\{B_1+\frac{wh}{1+r}+y-x\right\}\right]\}.$

Ignoring constant terms, this is equivalent to ${\max }_x\left\{(1+\beta )\ln \left[B_1+\frac{wh}{1+r}+y-x\right]\right\}.$

Maximizing the last expression is equivalent to maximization of the NPV of consolidated resources. It follows that conditional loans lead to productive efficiency, or wh′ (x) = 1 + r. Hence, conditional loans support the first-best (see Example 1). It should be noted that quite large interest rate subsidies may be required to support the first-best.

Under an unconditional loan policy, the IFI’s second-period repayment policy is unchanged, $\tilde{\mu }=\tilde{\mu }(x,l)$. In the second stage of the game, the LIC chooses investment $x=\tilde{x}(l)$ while taking into account the impact of its choice on the loan’s repayment terms. LIC investment and consumption and IFI capital satisfy

$\begin{array}{c}w(1+\beta \alpha )\alpha x^{\alpha }-w\beta \alpha (1+y)x^{\alpha -1}+(B_1-1)(1+r)=0\\ c_1=\frac{w{x}^{\alpha }+(B_1-l)(1+r)}{\beta w\alpha x^{\alpha -1}}\\ \frac{c_2}{1+r}=\frac{\gamma \beta }{1+\gamma \beta }\left\{\frac{w{x}^{\alpha }}{1+r}+(B_1-l)\right\}\\ \frac{B_2}{1+r}=\frac{1}{1+\gamma \beta }\left\{\frac{w{x}^{\alpha }}{1+r}+(B_1-l)\right\}.\end{array}$

In the first stage, the IFI selects l to maximize its utility while taking as given the country’s investment function $x=\tilde{x}(l)$.

This equilibrium is not Pareto Optimal. There is underinvestment relative to the first-best, as demonstrated in the numerical computations of Table 1.

Table 1.

Optimal, Time-Consistent Loan Policies ^1/

Source: Authors’ calculations

^1/on the economy of Example 2, with B₁=2, r=2, α= 0.5, y = β = = 1. A period is a generation (30 years). r=2 corresponds to an annual real interest rate of 3.73 percent.

Table 1.

Optimal, Time-Consistent Loan Policies ^1/

Productivity	Autarky	Perfect Markets	Unconditional Loans			Conditional Loans
w	r1	l	μu	lu	xu	μc	lc	xc
10	8.66	3.67	1.00	…	…	0.97	3.70	2.78
9	7.79	2.88	1.00	…	…	0.88	3.00	2.25
8	6.93	2.17	0.84	1.59	1.48	0.74	2.37	1.78
7	6.06	1.54	0.57	1.43	1.25	0.55	1.81	1.36
6	5.20	1.00	0.24	1.24	0.99	0.25	1.33	1.00
5	4.33	0.54	-0.22	1.00	0.68	-0.25	0.93	0.69
4	3.46	0.17	-1.14	0.66	0.34	-1.16	0.59	0.44
3	2.60	-0.13	-3.57	0.33	0.12	-3.13	0.33	0.25
2	1.73	-0.33	-11.02	0.12	0.04	-8.75	0.15	0.11
1	0.87	-0.46	-51.50	0.03	0.01	-39.13	0.04	0.03

Source: Authors’ calculations

^1/on the economy of Example 2, with B₁=2, r=2, α= 0.5, y = β = = 1. A period is a generation (30 years). r=2 corresponds to an annual real interest rate of 3.73 percent.

View Table

Table 1.

Optimal, Time-Consistent Loan Policies ^1/

Productivity	Autarky	Perfect Markets	Unconditional Loans			Conditional Loans
w	r1	l	μu	lu	xu	μc	lc	xc
10	8.66	3.67	1.00	…	…	0.97	3.70	2.78
9	7.79	2.88	1.00	…	…	0.88	3.00	2.25
8	6.93	2.17	0.84	1.59	1.48	0.74	2.37	1.78
7	6.06	1.54	0.57	1.43	1.25	0.55	1.81	1.36
6	5.20	1.00	0.24	1.24	0.99	0.25	1.33	1.00
5	4.33	0.54	-0.22	1.00	0.68	-0.25	0.93	0.69
4	3.46	0.17	-1.14	0.66	0.34	-1.16	0.59	0.44
3	2.60	-0.13	-3.57	0.33	0.12	-3.13	0.33	0.25
2	1.73	-0.33	-11.02	0.12	0.04	-8.75	0.15	0.11
1	0.87	-0.46	-51.50	0.03	0.01	-39.13	0.04	0.03

Source: Authors’ calculations

^1/on the economy of Example 2, with B₁=2, r=2, α= 0.5, y = β = = 1. A period is a generation (30 years). r=2 corresponds to an annual real interest rate of 3.73 percent.

A. Endogenous IFI Capital

The magnitude of the IFIs’ financial assistance depends critically on their capital base, which is determined by the creditor governments’ subscriptions of convertible currencies. Thus far, we have assumed that the IFI’s initial capital B₁ is exogenously given. In this section we endogenize the IFI’s endowment by viewing the IFI as an organization that represents altruistic lenders from around the world.

We assume the lenders are identical. The representative lender cares about his lifetime consumption and about the welfare of LIC citizens. The lender makes loans to either LICs (l) or to middle- and high-income countries (b). The IFI is altruistic toward LICs, but not other countries, where it always demands full repayment of loans at the going world interest rate. Since the populations of citizens in the LIC and creditor countries are exogenous, we assume, for notational convenience only, that they are equal. The representative lender can then be thought as indirectly making loans to the representative citizen of the LIC. The lender’s preferences are given by

Where${\overline{c}}_1=y_1-b-l,{\overline{c}}_2=y_2+(1+r)b+(1+\mu r)l$, yⁱ are exogenous income levels, and γ > 0 gauges the strength of the IFI’s altruism toward the LIC.

As before, the first-best allocation is the one achieved by assuming that the IFI is a benevolent social planner that chooses x, T₁, T₂ and b to maximize

The necessary and sufficient conditions for maximization are

The interpretations of these conditions are as in equations (2.4a–2.4c).

In the absence of foreign assistance or the ability to borrow, the LIC country’s government chooses x and l to maximize u(y – x + l) + βu(wh – (1 + r)l), subject to l ≤ 0. The resulting autarkic allocation, with a binding constraint on l, is characterized by

Equation (4.3) implies that investment is inefficiently low (wh’ > 1 + r). In addition, the marginal value of current consumption, in terms of forgone future consumption, exceeds the world’s opportunity cost of funds. This inefficiency creates the possibility that an IFI loan could improve the situation, but can it achieve the first-best without a full set of policy tools?

B. Loans Without Commitment

We proceed working backwards. In period two, the IFI chooses μ optimally, taking as given first-period loan amounts and investment levels. The IFI’s repayment policy is defined by maximizing (3.1) with respect to μ, given x, l, b:

In period 1, the IFI and the LIC choose optimal loan and investments levels, given the optimal period-2 repayment policy. The timing of choices in period 1 again depends on whether the IFI is following a conditional or an unconditional loan policy.

C. Uncertainty

We have suggested that IFIs behave as if they are unwilling or unable to commit to a given interest subsidy in advance. However, under our assumptions, it is unclear why this would be the case. In practice, IFIs probably do not commit to how much they will require in loan repayment from LICs because of uncertainty. At the time the loan is made no one knows the precise conditions that will prevail in the poor country during the second period. The better the economic conditions in the second period, the smaller is the portion of the loan repayment that the IFI would be willing to forgive.

In this section we introduce uncertainty by assuming that the rental rate on human capital, w, is a random variable. To assess the performance of conditional loans in an uncertain environment, we must begin by reconsidering the first-best allocations and those obtained under autarky.

D. Government Consumption

Thus far we have assumed that the poor country’s benevolent government uses lump-sum taxes for the sole purpose of financing investment projects that increase the welfare of its citizens. Now we show the results extend to the case where the poor country’s government uses income tax revenues to finance its own consumption as well as finance investment projects. With a selfish government, taxes are set independently of investment choices. This adds another layer of complexity to the problem, but does not alter it fundamentally.

Suppose the government now sets income tax rates τ₁ and τ₂ to collect revenue in the two periods. The government uses some of these revenues for its own selfish purposes. Let the government’s objective function take the form

where 0 < ρ < 1. The government may be quite selfish (have a ρ close to 0), but not completely so.

The analysis of conditional loans with a selfish government follows in the same fashion as with benevolent governments, except now tax rates have to be determined along with investment choices. Without commitment we again have a two-period game, but the government’s choice of taxes adds a stage to the games in each period.

In the second period, the first stage has the government choosing τ₂ to maximize (3.11), for given values of all other variables. The second stage has the IFI choosing μ to maximize (3.8), taking into account how this will affect the government’s tax policy.

In the first period, there is now a three-stage game. All choices in this period take into account their effects on the second-period behavior of the government and the IFI. The first stage has the government choosing τ₁ to maximize (3.11). For conditional loans, the second stage has the IFI choosing its loan schedule for given values of the poor country’s investment and accounting for how the quantity of loans impacts tax policy in both periods. Finally, in the third stage, the poor country chooses its investment level accounting for how all other behaviors are altered. The solution to this dynamic game gives the extension of Proposition 5 to the case of a selfish government.

More selfish governments set higher taxes to increase the fraction of the country’s resources consumed in the public sector. However, whatever this fraction, there are always gains from investment to be shared by all. Expanding the country’s resources is in the best interest of the government, its citizens (since they will share at least some of the expansion), and the IFI (since they care about the citizens and about receiving repayment). Just as in the case with a perfectly benevolent government, conditional loans provide the incentive for the government to recognize this and invest at efficient levels.