# «IZA DP No. 4351 Reputation and Credit Market Formation: PAPER How Relational Incentives and Legal Contract Enforcement Interact Ernst Fehr DISCUSSION ...»

In this appendix we show the existence of reputation equilibria in a two-period version of the game implemented in our experiment. It is not our objective to provide a complete formal analysis of our experimental game. We rather want to show that in the RC and the TPC there are perfect Bayesian equilibria in which the reputation mechanisms intuitively described in Section III.B are at work.

As outlined in the text, we assume that there are two types of borrowers: A share x ∈ (0,1) are trustworthy types, i.e. they are willing to honor the credit terms set by the lender as long as they perceive them as fair. The rest of the borrowers are purely selfish profitmaximizers. We assume that the trustworthy borrowers perceive a credit offer as fair if the lender asks for the efficient project and if the desired repayment rd ensures that the return in case of project success is shared in a fair manner, i.e. a fair contract offer must be of the form (pd = A, rd ≤ rt), where rt is the maximal repayment a trustworthy borrower considers as fair.21 For tractability reason we slightly simplify the posting and acceptance of credit offers.

In the experiment the trading mechanism is a continuous auction. However, as continuous auctions have defied a fully rigorous analysis so far, we approximate the bargaining process with a posted contract mechanism. Specifically, we assume that each lender can only make one credit offer, either a public offer or a private offer in every period. Borrowers then choose in random order from the available offers. Each borrower is free to accept one of the loans available to him or not to borrow at all.

**A. Credit Market Trading in the OC**

We first analyze the behavior of risk-neutral lenders and borrowers in the OC, in which neither legal enforcement of debt repayments nor the possibility for reputation formation are present. Proposition A1 shows that in the OC credit market trading can only take place if there is a sufficiently large share of trustworthy borrowers.

**Proposition A1:**

In the OC lenders are only willing to offer credit contracts to borrowers if the fraction of trustworthy borrowers satisfies x ≥ k / (wArt), otherwise lenders make use of the endowmentstoring technology.

It is plausible that rt depends on the presence of legal enforcement of debt repayments. As the presence of third party enforceability of repayments puts lenders into a stronger position trustworthy borrowers may be prepared to regard higher repayment requests as fair when repayments are enforceable than when they are not. To keep our exposition as simple as possible, we do not explicitly include this possibility in our notation.

**Proof:**

In the OC selfish borrowers simply maximize their period profits, i.e., they choose project A and never repay after project success. Trustworthy borrowers, in contrast, are willing to honor the contract terms if they are offered a contract of the form (pd = A, rd ≤ rt). If a lenders offer such a contract to a borrower of unknown type, his expected profit is E[πL] = xwArd. Since the lender’s expected profit is increasing in rd it is profit-maximizing to set rd = rt. As a lender can always realize a profit of k by choosing the endowment-storing technology, he is only prepared to offer a credit contract to an unknown borrower if the following offer condition is satisfied x ≥ k / (wArt). s

**B. Credit Market Trading in the RC**

We next consider the RC. For simplicity and expositional clarity we consider only a twoperiod version of the game. In line with our empirical observations in the laboratory we assume that the fraction of trustworthy borrowers is insufficient to make credit contracts profitable in the OC, i.e., x k / (wArt). Proposition A2 shows that in the RC reputation effects make it possible that there is a perfect Bayesian equilibrium in which lenders are willing to extend credit even if the parameters are such that no credit market can exists in the OC.

**Proposition A2:**

Consider the RC and assume that the fraction of trustworthy borrowers lies in the following range: [k / (wArt)]2 ≤ x k / (wArt). There is a perfect Bayesian equilibrium with the following characteristics: In period 1 all lenders make a public credit offer of the form (pd = A, rd = rt).

A random selection of 7 borrowers accepts these offers and chooses project A. In case of project success trustworthy borrowers repay r = rt with certainty, while selfish borrowers repay r = rt with probability s = x (wArt – k) / [(1 – x)k]. In period 2 each lender whose incumbent borrower has repaid the loan in period 1 privately offers a credit contract of the form (pd = A, rd = rt) with probability l = rt / (wARA – b) to his incumbent borrower, while each lender whose incumbent borrower has defaulted in period 1 always chooses the endowment-storing technology. Those borrowers who get a credit offer choose project A. In case of project success trustworthy borrowers repay r = rt with certainty while selfish borrowers never repay.

**Proof:**

**Proof is by construction and is established in three steps:**

**Step 1 (project choice and repayments of trustworthy borrowers):**

We have assumed that trustworthy borrowers who get a contract of the form (pd = A, rd ≤ rt) honor the contract terms suggested by the lender if possible. Thus, all trustworthy borrowers who succeed in accepting a contract (pd = A, rd = rt) in period 1 and 2 choose project A and repay after project success.

**Step 2 (project choice and repayments of selfish borrowers):**

Since period 2 is the final period selfish borrowers behave exactly as in the OC: whenever they succeed in getting a contract they maximize their period profit by choosing project A and not repaying in case of project success. Thus, if a selfish borrower gets a contract in period 2, his expected profit is E[πB] = wARA.

In period 1 the situation is different. Let us start with the repayment decision. Assume that a selfish borrower has accepted a contract (pd = A, rd = rt) and has successfully realized a project (it may be A or B). The borrower must now choose one of two repayments: Either he imitates the behavior of a trustworthy borrower and repays (r = rt) or he does not repay at all (r = 0).22 In period 1 repaying may make sense if lenders condition the probability of a contract renewal in period 2 on the borrower’s repayment behavior. Define l(r) as the probability with which a lender renews his contract with a borrower in period 2 after observing the repayment r in period 1. After repaying r in period 1 a selfish borrower then faces the following continuation payoff for period 2: V(r) = l(r)wARA + (1 – l(r))b. This implies that a selfish borrower is willing to make a repayment r = rt if the following repayment condition is satisfied: rt ≤ V(rt) – V(0) = (l(rt) – l(0))(wARA – b).

Let us now move on to the project choice. Using the notation from above, we can write the expected stream of utility over both periods, which is implied by the choice of a project p ∈ {A,B} in period 1 as: wp[s(RA – rt + V(rt)) + (1 – s)(RA + V(0))] + (1 – wp)V(0).

Thus, a selfish borrower prefers project A over project B as long as the following project choice condition is satisfied: s(wA – wB)[– rt + V(rt) – V(0)] + wARA – wBRB 0. Since wA wB and wARA wBRB, it is straightforward to see that the repayment condition is a sufficient (but not necessary) condition for the project choice condition.

The lenders’ contract renewal probabilities given in Proposition A2 imply that the repayment condition is satisfied with equality, i.e., a selfish borrower is indifferent between repaying and not repaying after the realization of a successful project in period 1.

A positive repayment r rt is never optimal. Such a repayment is not in line with a trustworthy borrower’s behavior and would therefore reveal the selfish borrower’s type. However, if the borrower reveals his type anyway, then he is always better off by repaying nothing.

Accordingly, any repayment probability s ∈ [0,1] is optimal. Furthermore, the fact that the repayment condition is satisfied implies that also the project choice condition is satisfied.

**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) = x / [x + (1 – x)s], while the lender’s belief after default (r = 0) is given by y(0) = (1 – wA)x / [(1 – wA) + wA(1 – x)(1 – s)].

Let us now turn to the credit offers of lenders. In period 2 lenders anticipate that borrowers face the same incentives as in the OC. Accordingly, Proposition A1 implies that a lender is only willing to make a credit offer to a specific borrower if his belief satisfies the offer condition: y k / (wArt). Since we assume that x k / (wArt) (no credit market trading in OC) a borrower who does not repay in period 1 does not get a credit offer in period 2. The reason is that the lender’s belief cannot satisfy the offer condition: y(0) ≤ x k / (wArt). This implies that the contract renewal probability after default is zero: l(0) = 0.

In order to get a credit offer after repaying in period 1 the selfish borrower's repayment probability 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) ≥ k / (wArt).

This yields the following condition for the selfish borrower’s repayment probability: s ≤ x(wArt – k) / (1 – x)k 1. Given that l(0) = 0 the repayment 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 l(rt) = rt / (wARA – b). However, this contract renewal probability can only be a best a response of the lender, if the lender is indifferent between offering a contract and choosing the endowment-storing technology.

Accordingly, the lender’s belief must be exactly at the threshold level, i.e., y(rt) = k / (wArt).

This, in turn, implies that s = x(wArt – k) / (1 – x)k. Furthermore, in period 1 lenders are only willing to offer a contract if the total probability of getting a repayment ensures that they are at least indifferent between offering a contract and their outside-option. This requires that x + (1 – x)s = k / (wArt). Given the repayment behavior of selfish borrowers in period 1 this

C. Credit Market Trading in the TPC We now turn to the TPC, in which repayments of borrowers after project success are legally enforced. In Section III.A we show that in this setup maximization of short-term borrower profits requires the choice of project B. In the absence of trustworthy borrowers, lenders anticipate that project A is never chosen and they therefore offer a contract of the form (pd = B, rd = rs), where rs = (wBRB – b) / wB is a high repayment which makes the borrower only slightly better off than his outside option.23 If there is a large fraction of trustworthy borrowers, lenders may – even in the absence of reputational incentives – prefer to offer a contract of the form (pd = A, rd = rt). However, we assume that the fraction of trustworthy borrowers is not large enough to render such a contract profitable in a one-shot interaction, i.e., x ≤ wB(rs – rt) / (wA – wB)rt.24 Proposition A3 shows that in the TPC reputation effects make it possible that there is a perfect Bayesian equilibrium in which lenders offer contracts of the form (pd = A, rd = rt) despite the fact that the parameters are such that these contracts are not profitable in one-shot interactions.

**Proposition A3:**

Consider the TPC and assume that the fraction of trustworthy borrowers lies in the following range: (wB)2rs(rs – rt) / wA(wA – wB)(rt)2 ≤ x wB(rs – rt) / (wA – wB)rt. There is a perfect Bayesian equilibrium with the following characteristics: In period 1 all lenders make a public credit offer of the form (pd = A, rd = rt). A random selection of 7 borrowers accepts these offers. Trustworthy borrowers who have accepted a contract choose project A with certainty, while selfish borrowers who have accepted a contract choose project A with probability z = [wA(wA – wB)xrt – wB(xwA – (1 – x)wB) (rs – rt)] / [wB(wA – wB)(1 – x)(rs – rt)]. In period 2 each lender, whose borrower has repaid in period 1, privately offers a credit contract of the form (pd = A, rd = rt) with probability m = [wB(RB – rt) – wA(RA – rt)] / [(wA – wB)(wB(RB – rt) – b)] to his incumbent borrower, while each lender whose incumbent borrower has defaulted in period 1 always makes a public credit offer of the form (pd = B, rd = rs) to his incumbent borrower. Trustworthy borrower choose project A if they receive a contract offer of the form (pd = A, rd = rt) and project B if they receive a contract offer (pd = B, rd = rs). Selfish borrowers choose project B irrespective of the form of their contract.

Under the parameter conditions in the experiment this equivalent to a repayment of 166 (see Section III.A).

In the absence of reputational incentives the expected profit if the lender offers the contract (pd = B, rd = rs) is w r, while the expected profit if he offers the contract (pd = A, rd = rt) is (xwA + (1 – x)wB)rt.

Bs

**Proof:**

**Proof is by construction and is established in three steps:**

**Step 1 (project choice of trustworthy borrowers):**

We have assumed that trustworthy borrowers who get a contract of the form (pd = A, rd ≤ rt) honor the contract terms suggested by the lender if possible. Thus, all trustworthy borrowers who succeed in accepting a contract (pd = A, rd = rt) in period 1 and 2 choose project A.

However, trustworthy borrowers who are offered a contract (pd = B, rd = rs) choose project B.