Supplemental Appendix to Enforcement Frictions and Optimal Lending Contracts , Economic Theory

Similarly, we can rewrite the incentive constraint as follows: u(ct(s )) + βvt+1(s ) ≥ O(F (kt(s), st)− et(s)). In this section, we conjecture that ct(s ), kt(s t−1), et(s t−1) and vt+1(s ) can be expressed as a function of vt(s t−1), st−1 and st. I derive a recursive version of the problem and then I show that the conjecture is correct: the recursively generated contract is the optimal one. If an agent has some promised expected utility v, and the past realization of the productivity shock was s−1, the contract consists of capital k, enforcement capacity e, consumption c(s) as a function of the current productivity shock s, and continuation utility v′(s) as a function of the shock. The recursive contract must satisfy the following constraints: