We study a credit market using an infinite horizon model where an altruistic lender offers loans to agents for production projects that may grow over time. The lender funds the loans using a combination of external debt and subsidies. The optimal way for the lender to subsidize the credit relationships depends on the probability of project growth. When growth is less likely, it is best to commit to ongoing subsidies. However, for a range of growth probabilities, ongoing subsidization may not be credible and this can have negative efficiency implications.
The author would like to thank two anonymous referees for their very helpful comments and suggestions. Any remaining errors are author’s alone.
Proof of Lemma 3. First note that the two partial derivatives of with respect to and are both negative. This means that the lender prefers to allocate as little subsidy to each credit relationship as possible. When the lender selects the pair , the present value of the total expected subsidy obligation is . To minimize the size of this obligation, the lender chooses a pair where the first period strategic default constraint in eq. (4) binds. This is because the agent’s income is increasing in both and . If we solve the default constraint for we have , and so, whenever , where . (Note that, if , the lender does not require any subsidies for the credit relationship, so .) Plugging into into , we get . Obviously this sum is independent of the actual pair . This means that it does not matter which subsidy pair the lender picks. All that satisfy (4) cost exactly the same amount of funds. Thus while the pair of subsidies described in Lemma 3 is not unique, it is optimal. QED
Proof of Lemma 4. With income , one can easily confirm that all partial derivatives with respect to are negative. As in Lemma 3, this means that the lender prefers to minimize the amount the subsidy dedicated to each lending relationship. \
Consider the subsidies and . This is the lowest collection of subsidies that the lender can use when . If we plug these subsidies into the strategic default constraint in eq. (4), whether the inequality holds or not depends on what is. If we solve the inequality for we get
First, consider the case where . In this event, the subsidies and induce loan repayment in all periods of the credit relationship.
Second, consider . In this case, the lender requires additional subsidies to persuade the agent to repay. The task of finding exactly which subsidy to increase is made easier because the first period strategic default constraint binds, implying the agent’s income is simply . To find the optimal collection of subsidies, the lender aims to keep the total subsidy obligation as low as possible. If we plug from the first period strategic default constraint into the total subsidy obligation , the result is that obligation is independent of the actual combination selected by the lender. That is, just like the other lending plan, it doesn’t matter which subsidy the lender raises. All , such that the strategic default constraint binds, cost the lender the same amounts of funds. Hence, when , an optimal combination of subsidies is , and such that eq. (4) holds at . QED
Proof of Proposition 1. In 4 we confirmed that if , then the resulting aggregate income is , where is defined in Lemma 3. Alternatively, under the plan, the aggregate income depends on what is. In particular, the lender must use if , and if , then . \
(i.) First, consider the case where . Then the plan generates aggregate income , where is defined in Lemma 4. Thus, the plan yields higher income than the plan if
(ii.) Second, consider . .
In this case, for the plan, . This means that using the subsidy in the plan, the strategic default constraint in eq. (4) is binding at , and not binding at . Thus, when , the discounted expected income for an agent under a lending relationship is not less than . Consequently, the plan generates an aggregate income not less than . Hence, it is sufficient to show that\
Before we proceed further, we must confirm that both sides of this inequality are positive.
(a.) For the left-hand side, A3 ensures that .
(b.) The right-hand side is positive if
Since we know exceeds (due to A3), it is sufficient to show that
which holds due to A3.
Consequently, we can now claim that the inequality in eq. (11) holds if where . This proves that the plan generates a higher income than the plan when . QED
Proof of Corollary 1.Under the lending plan, the value of the aggregate income depends on how compares to .
First, consider the case where . We must show that , or
It is sufficient to show that , or , which holds due to A1.
Second, consider the case where . We must show that , or
Since , it is sufficient for us to demonstrate that , or , which holds due to A3. QED
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