Complete Contracts with Limited Enforcement

Limited commitment problems are often invoked for investments in education, because human capital is a notably poor collateral (Becker, 1975; Friedman and Kuznets, 1945). While human capital cannot be repossessed, the cost of defaulting on a loan might depend directly on the

education of the individual as it determines his earnings. Then, the amount of credit a person could obtain would be endogenously linked to his investments in education, as these investments determine the amount of credit that the borrower can credibly commit to repay (Lochner and Monge-Naranjo 2011, 2012).56

To formalize this argument, assume that once a borrower leaves school, he can always opt to default on a repayment D (z) contracted earlier. But, default is not without its costs. For simplicity, assume that a defaulting borrower loses a fraction κ ∈ (0, 1) of his labor earnings, so his post-school consumption is cD₁ (z) = (1 − κ) zaf (h). These losses could reflect punishments imposed by lenders themselves (e.g. wage garnishments) or by others (e.g. landlords refusing to rent or employers refusing to hire). Alternatively, the borrower could repay D (z) yielding post- school consumption cR

1 (z) = zaf (h) − D (z). The borrower’s decision is straightforward: repay if

the cost of defaulting exceeds the cost of repaying:

D (z) ≤ κzaf (h) . (7)

Obviously, if reneging on the debt were costless (κ = 0), then no student loan market could be sustained, since no borrower would ever repay. Similarly, if κ is high enough, the temptation to default could be eliminated, and we would be back to the first best.

The restrictions (7) can be seen as participation constraints on the borrower. As long as they are satisfied, the credit contract ensures that the borrower remains in the contractual arrangement. Any contract in which default occurs can be replicated by a contract without default by setting D (z) = κzaf (h). Since default is costly for the borrower and the lender does not necessarily recover all of those losses, optimal contracts in this setting would always prevent default. The optimal lending contract is similar to the first best problem only restricted so that condition (7) holds for all z ∈ Z.

Let λ (z) be the Lagrange multipliers associated with the inequality (7) for any realized z.57 The optimal program maximizes the value of the borrower’s lifetime utility (1) subject to the break- even or participation condition for the lender (2), the expressions (3) and (4) for consumption during and after school, and inequality (7) for all z ∈ Z.

56_{We only consider one-sided limited commitment problems where the lender can fully commit. This is natural}

when considering the optimal design of government credit arrangements.

57_{The multipliers are discounted and weighted by probabilities, i.e. the term qφ (z) λ (z) multiplies the condition}

The first order optimality conditions for this problem are straightforward. The optimal repay- ment value D (z) conditional on the realization z implies the following relationship between c1(z)

and c0:

u0(c0) = [1 + λ (z)] u0[c1(z)] .

For states of the world in which the participation constraint is not binding (i.e. D (z) < κzaf (h)), λ (z) = 0 and there is full consumption smoothing: c1(z) = c0. However, when the participa-

tion constraint is binding, λ (z) > 0 and c1(z) > c0. The participation constraint restricts the

repayment that can be asked of the borrower for high labor market realizations. In turn, those restrictions limit the capacity of the student to borrow resources while in school, resulting in low school-age consumption relative to post-school consumption in high-earnings states.

From the first order conditions for d and h, one can show that optimal human capital investment satisfies af0[h] E  z 1 + κλ (z) 1 + λ (z)  = q−1. (8)

Notice that Ehz1+κλ(z)_1+λ(z)i < E [z] as long as κ < 1 and participation constraints bind (i.e. λ (z) > 0) for some realizations of z. Comparing (8) to (6), it is clear that, given concavity in f (·), the inability to fully commit to repayment reduces human capital investment below the first best level. The presence of limited commitment effectively reduces the expected return on human capital due to the inability to effectively borrow against returns in the highest earnings states or to spread the resources from those states to other states with fewer resources.

In contrast to the unrestricted environment above, family resources W are a determinant of investment levels under limited commitment. Individuals with low wealth levels will want to borrow more while in school. This raises desired repayment amounts D(z) in all future states, causing participation constraints to bind more often and more severely. Thus, poorer students face greater distortions in their consumption and investment allocations than wealthier students. It is important to understand the nature of credit constraints that arise endogenously from the participation constraints associated with commitment problems. As with any other model of credit constraints, this environment predicts inefficiently low early consumption levels for those that are constrained (i.e. a first order gain could be attained by increasing early consumption and reducing post-school consumption for some labor market realizations). A more unusual aspect of constraints in this environment is that they arise due to an inability to extend insurance to

fully cover high earnings realizations. The participation constraints do not restrict the ability to smooth consumption across adverse labor market outcomes, since the contract allows for negative repayments for low enough realizations of z. Rather, the limits arise due to the incentives of borrowers to default on high payments associated with strong positive earnings outcomes. The lender must reduce requested repayments in those states to drive the borrower to indifference between repaying and defaulting. This reduction in repayments must be met with less credit up front.58 Finally, it is important to note that default never formally happens in equilibrium, because repayments D (z) are designed to provide as much insurance as possible while avoiding default.

The ability to write fully contingent contracts is important for many of these results. As we show next, contracts and borrower behavior differ substantially if the repayment function D (z) cannot be made contingent on labor market realizations.