A.4. Application to the experiment
5.3 Theoretical framework
5.3.1
Contracting under moral hazard
The lending game is based on Innes (1990). To bring out most clearly the e¤ect of collateral on borrower behavior and credit volume, e¤ort is simpli…ed to a …nite set with …ve elements and two potential project outcomes are possible, success or failure. More precisely, the borrower has an investment project, which yields 300 points in case of success, while it yields 0 points in case of failure. E¤ort, e=f1;2;3;4;5g; is the main determinant of success. Exerting more e¤ort, e.g. spending more time and resources, increases the probability of success linearly (p(e) = e
6). However, this e¤ort is costly,
g(e) = 4e2; and the borrower requires a loan of 100 points to start the
project and cover these expenses.
E¤ort is neither contractible nor observable by the lender. Thus, after extending a loan, the lender cannot in‡uence the borrower’s e¤ort. Neverthe- less, she anticipates that, if the borrower does not bear any costs of project failure, his e¤ort will be small. To incentivize the borrower to increase his e¤ort, the lender may request the borrower to pledge his assets as collateral. The borrower has an endowment of 100, which are assets that are very costly for him to liquidate (e.g. the piece of land on which he lives). But, a por- tion C of these assets, where 0 C 100; may be pledgeable as collateral. We assume that the value of collateral, C;is the same for the borrower and the lender, and no transaction costs or loss in collateral value ensue from default. Our interest is in the e¤ect of increases in collateral,C, on borrower behavior. To identify this e¤ect we varyC exogenously across treatments.6
5.3. THEORETICAL FRAMEWORK 153
The contracting process is structured as follows. The lender decides …rst whether or not to o¤er a loan, { o¤er, no o¤er}. If she chooses to o¤er and collateral is available, the lender can choose to request collateral or no collateral. To simplify notation, if the lender chooses no collateral, C is set to 0. If collateral is chosen, C is set to the amount of collateral available. By design, the lender does not decide on the repayment, R; which is varied exogenously across treatments. This allows the identi…cation of the impact of collateral on e¤ort, for a given level of repayment, avoiding the endogeneity in the joint e¤ect of collateral and repayment.
If the lender o¤ers a loan, the borrower decides whether toaccept orreject it. If she accepts, the borrower decides on e¤ort. The relationship between e¤ort, the probability of success and e¤ort costs is displayed in Table 5.1.
E¤ort (e) 1 2 3 4 5
Probability of success 1/6 2/6 3/6 4/6 5/6
Costs 4 16 36 64 100
Table 5.1: E¤ort, probability of success, and costs
If the lender decides not to o¤er a loan or the borrower rejects an of- fer, no loan is extended. Then, the lender and borrower keep their initial endowments, of 150 and 100 respectively. If a loan is o¤ered and accepted, two outcomes are possible. First, if the project succeeds, the lender is paid back repayment R; which includes the loan principal of 100 and an interest payment: Thus, no strategic default is allowed. Second, the project may fail, in which case the lender receives the requested collateral, C; instead of repayment. This leads to the following payo¤s for the lender:
L= 8 > < > :
150 if no loan is o¤ered or accepted
50 +R if project succeeds
50 +C if project fails
The payo¤s for the borrower are:
titles on the borrower’s endowment, or changes in regulation, which increase the type of assets that are pledgeable.
B = 8 > < > :
100 if no loan is o¤ered or accepted
500 R 4e2 if project succeeds
200 C 4e2 if project fails7
In this environment, how are borrower e¤ort and credit volume expected to vary with increases in collateral? We …rst examine the e¤ect of collateral under the standard assumption that borrowers are risk neutral and then that of risk aversion.
5.3.2
Risk neutrality
We start by assuming that borrowers are risk neutral and their utility from income is linear. A borrower who accepts a loan o¤er has expected utility,
E( B) = e 6(500 R) + (1 e 6)(200 C) 4e 2
Her optimal e¤ort, which is also the incentive compatibility constraint (IC), is
e = 1
48(300 R+C) (IC)
The IC clearly depicts the incentive e¤ect of collateral: an increase in C;
increasese ;i.e. @e
@C >0:Importantly, it also reveals that the incentive e¤ect is independent of the repayment, @2e
@C@R = 0:Additionally, the IC also reveals that an increase in repayment decreases e¤ort, @e
@R < 0: Furthermore, the borrower is willing to accept a loan o¤er provided
e
6(400 R) + (1
e
6)(100 C) 4e
2 0 (PC)
To determine the repayment level, we use the lender’s optimal contract as a benchmark. The level of collateral and repayment that maximize the lender’s pro…ts are determined by maximizing the lender’s expected pro…ts, i.e. max C;R e 6R+ (1 e 6)C 100
subject to the borrower’s incentive compatibility constraint (IC), the bor-
7Note that, even if the project fails andC= 100, the borrower never makes a net loss.
5.3. THEORETICAL FRAMEWORK 155
rower’s participation constraint (PC) and the lender’s own participation con- straint. Requesting collateral is always optimal for the lender because the …rst derivative with respect to C is always positive. The optimal interest rate, for values of collateral between 0 and 100, is
R = 150 +C
The intuition behind the optimal interest rate is as follows. When the amount of collateral is low, the incentive e¤ect of collateral is also low. The borrower must pay a low interest to have an incentive to provide a large e¤ort. As the amount of collateral comes closer to 100, a low interest becomes unnecessary. The larger amount of collateral provides the borrower with su¢ cient incentive to exert e¤ort. Thus, the lender can charge a higher interest rate and still elicit a large e¤ort (see also Besley and Ghatak, 2009a). Given the relationship between R and C, we will assume in what follows that 150 R 250:
If R = 150 +C; the lender’s participation constraint is satis…ed if C
100 25 258 = 2178: Thus, for any C 2178; it is optimal for the lender to o¤er. For the borrower, it is optimal to accept in all cases, because her PC, which can now be rewritten as 25162+100 C;has been taken into account and thus is satis…ed. These results yield Proposition 1. The proof is presented in Appendix A.
Proposition 1: If the lender and the borrower are risk neutral, an increase in collateral from 0 to 100% of the loan amount (1) reduces the problem of moral hazard: e¤ort increases, and (2) increases credit supply, but it does not a¤ect credit demand, and therefore it increases credit volume. These e¤ects do not vary across di¤erent interest rate levels.
Proposition 1 highlights two main e¤ects of collateral. Pledging more collateral increases the e¤ort provided by the borrower and thus the probab- ility of repayment. This e¤ect, together with the fact that lending becomes more secure for lenders, makes lending pro…table. Credit supply increases and since credit demand remains pro…table, collateral leads to an increase in credit volume.
5.3.3
Risk aversion
We allow for risk aversion of the borrower by assuming that the borrower’s utility takes the form u = x and 0 < < 1: The expected utility from accepting a loan can then be formulated as follows:
E(uB) = e 6 (500 R) + (1 e 6) (200 C) 4e 2
where e¤ort costs are assumed to be separable.8 The borrower’s optimal
e¤ort is then
e = (500 R) (200 C)
48 (ICRA)
Risk aversion decreases the borrower’s optimal e¤ort, compared to risk neutrality. This follows from the fact that a risk averse borrower weighs the incentive e¤ect of collateral with her desire for insurance (as in Holmstrom, 1979). In addition, risk aversion also makes the reaction to collateral in- creases weaker, i.e. @e@C is smaller under risk aversion than under risk neut- rality. However, since repayment only a¤ects utility in case of project success, even if the borrower is risk averse, the e¤ect of collateral on e¤ort is still in- dependent of the repayment, @C@R@2e = 0.9 The e¤ects of risk aversion are
summarized in Proposition 2. The proof is presented in Appendix A.
Proposition 2: If the borrower is risk averse, an increase in collateral from 0 to 100% of the loan amount reduces the problem of moral hazard: e¤ort increases. As in the case of risk neutrality, this e¤ect is independent of the interest rate. In comparison to risk neutrality, the optimal e¤ort of risk averse borrowers is lower and their reaction to collateral requests weaker. Additionally, the e¤ect of collateral on credit demand may interact with the interest rate: credit demand is more likely to fall with collateral increases at high interest rates.
Interestingly, risk aversion leads to an interaction between the e¤ect of collateral on credit demand and repayment. If the interest rate on a loan is
8Assuming separability simpli…es the analysis and is in line with previous literature. 9This result follows from the assumption that e¤ort is separable. Should e¤ort costs
not be separable, the sign of @2e
@C@R depends on the parameter values chosen. Numerical