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Structuring Subsidies in a Long-Term Credit Relationship

Eric Van Tassel

Abstract

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.

JEL Classification: D86; G21; O16

Acknowledgements

The author would like to thank two anonymous referees for their very helpful comments and suggestions. Any remaining errors are author’s alone.

Appendix

Proof of Lemma 3. First note that the two partial derivatives of VS1,Sg,0,0 with respect to S1 and Sg 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 (S1,Sg), the present value of the total expected subsidy obligation is S1+λ1iSg. To minimize the size of this obligation, the lender chooses a pair (S1,Sg) where the first period strategic default constraint in eq. (4) binds. This is because the agent’s income is increasing in both S1 and Sg. If we solve the default constraint for S1 we have S1=1λ1(1+i)iϕR(1+i)(1Sg), and so, S1>0 whenever λ<λ, where λ(1+i)iϕR(1+i). (Note that, if λλ, the lender does not require any subsidies for the credit relationship, so S1=Sg=0.) Plugging S1 into into S1+λ1iSg, we get 1λ1(1+i)iϕR(1+i). Obviously this sum is independent of the actual pair (S1,Sg). This means that it does not matter which subsidy pair the lender picks. All (S1,Sg) 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 VS1,Sg,Sn;1, one can easily confirm that all partial derivatives with respect to Sj 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 S1=Sg=0 and Sn=S_n. This is the lowest collection of subsidies that the lender can use when θ=1. If we plug these subsidies into the strategic default constraint in eq. (4), whether the inequality holds or not depends on what λ is. If we solve the inequality for λ we get

(9)λ1+iR1+i1i(ϕR(1+i))R1+i,and we defineλ1+iR1+i1i(ϕR(1+i))R1+i.

First, consider the case where λλ. In this event, the subsidies S1=Sg=0 and Sn=S_n induce loan repayment in all periods of the credit relationship.

Second, consider λ<λ. 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 11+iR. To find the optimal collection of subsidies, the lender aims to keep the total subsidy obligation as low as possible. If we plug S1 from the first period strategic default constraint into the total subsidy obligation S1+1iλSg+(1λ)Sn, the result is that obligation is independent of the actual combination (S1,Sg,Sn) selected by the lender. That is, just like the other lending plan, it doesn’t matter which subsidy the lender raises. All (S1,Sg,Sn), such that the strategic default constraint binds, cost the lender the same amounts of funds. Hence, when λ<λ, an optimal combination of subsidies is Sg=0, Sn=S_n and S1 such that eq. (4) holds at θ=1. QED

Proof of Proposition 1. In 4 we confirmed that if θ=0, then the resulting aggregate income is FS1R1+i, where S1 is defined in Lemma 3. Alternatively, under the θ=1 plan, the aggregate income depends on what λ is. In particular, the lender must use S1>0 if λ<λ, and if λλ, then S1=0. \

(i.) First, consider the case where λ<λ. Then the θ=1 plan generates aggregate income FS1L4+1i(1λ)S_nR1+i, where S1L4 is defined in Lemma 4. Thus, the θ=1 plan yields higher income than the θ=0 plan if

(10)FS1L4+1i(1λ)S_nR1+i>FS1R1+iS1>S1L4+1i(1λ)S_n0>(1λ)R(1+i)2+1i(1λ)1R(1+i)2R>1+i.

(ii.) Second, consider λλ. λ<λ.

In this case, for the θ=1 plan, S1=0. This means that using the subsidy S_n in the θ=1 plan, the strategic default constraint in eq. (4) is binding at λ=λ, and not binding at λ>λ. Thus, when λ[λ,λ), the discounted expected income for an agent under a θ=1 lending relationship is not less than R1+i . Consequently, the θ=1 plan generates an aggregate income not less than F1i(1λ)S_nR1+i. Hence, it is sufficient to show that\

(11)F1i(1λ)S_nR1+i>FS1R(1+i)S1>1i(1λ)S_n1λ1(1+i)iϕR(1+i)>1i(1λ)1R(1+i)2R(1i)(1+i)2λR+ϕR(1+i)2(1+i)2.

Before we proceed further, we must confirm that both sides of this inequality are positive.

(a.) For the left-hand side, A3 ensures that R>(1i)(1+i)2.

(b.) The right-hand side is positive if

(12)R+ϕR(1+i)2(1+i)2>0,orR>2(1+i)21+ϕ(1+i).

Since we know R exceeds (1i)(1+i)2 (due to A3), it is sufficient to show that

(13)(1i)(1+i)2>2(1+i)21+ϕ(1+i)ϕ(1i2)>1+iϕ(1+i)(1i)>1+i1i>1ϕ,

which holds due to A3.

Consequently, we can now claim that the inequality in eq. (11) holds if λλˆ where λˆR(1i)(1+i)2R+ϕR(1+i)2(1+i)2. This proves that the θ=1 plan generates a higher income than the θ=0 plan when λλˆ. QED

Proof of Corollary 1.Under the θ=1 lending plan, the value of the aggregate income depends on how λ compares to λ.

First, consider the case where λλ. We must show that VS1L4,Sg=0,S_n,θ=1>F, or

(14)FS1L4+1i(1λ)S_nR1+i>FR>(1+i)[1λ1i(1+i)(ϕR(1+i))(1λ)R(1+i)2+1i(1λ)[1R(1+i)2]]R>1+iλϕRi+1+ii(1λ)R1+i1+iiλϕR+(1λ)R>(1+i)2iR

It is sufficient to show that R>(1+i)2iR, or R>1+i, which holds due to A1.

Second, consider the case where λ>λ. We must show that V(S1=0,Sg=0,S_n,θ=1)>F, or

(15)F1i(1λ)S_nR1+i>FR>(1λ)1i(1+i)2R1+i

Since R>1+i, it is sufficient for us to demonstrate that 1+i1i(1+i)2R1+i, or R(1+i)21i, which holds due to A3. QED

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Published Online: 2017-8-11

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