«IZA DP No. 4351 Reputation and Credit Market Formation: PAPER How Relational Incentives and Legal Contract Enforcement Interact Ernst Fehr DISCUSSION ...»
Step 2 (project choice and repayments of selfish borrowers):
Since period 2 is the final period selfish borrowers behave exactly as in a one-shot interaction:
whenever they succeed in getting a contract they maximize their period profit by choosing project B. Thus, if a selfish borrower gets a contract (pd = A, rd = rt) in period 2, his expected profit is E[πB] = wB(RB – rt). If he gets a contract (pd = B, rd = rs) his expected profit is identical to his outside option E[πB] = wB(RB – rs) = b.
In period 1 the situation is different. Assume that a selfish borrower has accepted a contract (pd = A, rd = rt). Since project A has a higher success probability, choosing project A may make sense if the lender conditions the probability with which he offers another contract of the form (pd = A, rd = rt) to his borrower in period 2 on the borrower’s repayment behavior.
Define m(r) as the probability with which a lender renews his contract with his borrower in period 2 after observing the repayment r in period 1. After a repayment r in period 1 a selfish borrower’s continuation payoff for period 2 is: V(r) = m(r)wB(RB – rt) + (1 – m(r))b. This implies that a selfish borrower is willing to choose project A if the following project choice condition is satisfied: wB(RB – rt) – wA(RA – rt) ≤ (wA – wB)(V(rt) – V(0)), where the difference in continuation payoffs can be rewritten as V(rt) – V(0) = (m(rt) – m(0))( wB(RB – rt)– b).
The lenders’ contract renewal probabilities given in Proposition A3 imply that the project choice condition is satisfied with equality, i.e., a selfish borrower is indifferent between project A and project B. Accordingly, any probability z ∈ [0,1] of choosing project A is optimal.
Step 3: Sequential Rationality and Credit Contract Offers of Lenders
Sequential rationality requires that a lender’s belief y about the trustworthiness of a borrower is defined at every information structure in the game. The initial prior, that is the probability that a lender assigns to the event that an unknown borrower is trustworthy, is given by the population fraction of trustworthy borrowers: y(∅) = x. If a lender interacts with a borrower in period 1 he updates his belief about the trustworthiness of this borrower based on the observed repayment using Bayes’ Rule. Accordingly, the lender’s belief after a repayment of r = rt is given by y(rt) = wAx / [wA(x + (1 – x)z) + wB(1 – x)(1 – z)], while the lender’s belief after default (r = 0) is y(0) = (1 – wA)x / [(1 – wA)(x + (1 – x)z) + (1 – wB)(1 – x)(1 – z)].
Let us now turn to the credit offers of lenders. In period 2 lenders anticipate that borrowers face the same incentives as in a one-shot interaction. Accordingly, a lender is only willing to make a credit offer to a specific borrower if his belief about this borrower satisfies the condition: y wB(rs – rt) / (wA – wB)rt (see above). Since we assume that the population fraction of trustworthy borrowers satisfies x wB(rs – rt) / (wA – wB)rt, a borrower who does not repay in period 1 does not get a renewed contract from his lender in period 2. The reason is that the lender’s belief cannot satisfy the required condition: y(0) ≤ x wB(rs – rt) / (wA – wB)rt. This implies that the contract renewal probability after default is zero: m(0) = 0.
In order to get a credit offer after repaying in period 1 the selfish borrower's probability of choosing project A has to be low enough such that the lender's updated belief at the beginning of period 2 has at least increased to the necessary threshold value: y(rt) ≥ wB(rs – rt) / (wA – wB)rt. This yields the following condition for the selfish borrower’s probability of choosing the project B: z ≤ [wA(wA – wB)xrt – wB(xwA + (1 – x)wB) (rs – rt)] / [wB(wA – wB)(1 – x)(rs – rt)] 1. Given that m(0) = 0 the project choice condition from Step 2 implies that this repayment probability can only be best response of a selfish borrower if the lender’s contract renewal probability after repayment is given by m(rt) = [wB(RB – rt) – wA(RA – rt)] / [(wA – wB)(wB(RB – rt) – b)]. However, this contract renewal probability can only be a best response of the lender, if the lender is indifferent between offering the contract (pd = A, rd = rt) to his incumbent borrower and making a public credit offer of the form (pd = B, rd = rs).
Accordingly, the lender’s belief must be exactly at the threshold level, i.e., y(rt) = wB(rs – rt) / (wA – wB)rt. This, in turn, implies that z = [wA(wA – wB)xrt – wB(xwA + (1 – x)wB) (rs – rt)] / [wB(wA – wB)(1 – x)(rs – rt)]. Furthermore, in period 1 lenders are only willing to offer a contract of the form (pd = A, rd = rt) if the total fraction of borrowers who choose project A ensures that they are at least indifferent between offering this contract and offering the contract (pd = B, rd = rs). This requires that the following condition holds: [wA(x + (1 – x)z) + wB(1 – x)(1 – z)]rt ≥ wBrs. Given the repayment behavior of selfish borrowers in period 1 this condition can only be satisfied if the initial fraction of trustworthy borrowers is not too low: x ≥ (wB)2rs(rs – rt) / wA(wA – wB)(rt)2. s
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The table displays the characteristics of the two projects available to borrowers.
Project A has a high probability of success and maximizes the expected returns on investment. Project B is an inefficient high-risk project. Due to limited liability and wealth constraints Project B can be attractive to borrowers when they plan or have to make a positive repayment in case of project success.