Abstract
This article investigates the impact of credit allocation on heterogeneous wealth entrepreneurs. We show that with decreasing risk aversion and unobservable wealth, poorer borrowers exert more effort. As a consequence of endogenous adverse selection, they are either excluded from the market or necessarily subsidize richer borrowers in a pooling equilibrium resulting in a paradoxical and inequitable redistribution. Alternatively, a less likely separating equilibrium may occur, in which poor types bear the entire weight of separation in the form of excess risk taking.
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Notes
Kan and Tsai (2006) test this conjecture and discover a positive effect of wealth on transition into self-employment controlling for the impact of risk aversion. Further, a recent experiment conducted by Elston and Audretsch (2011) finds that capital constraints negatively affect the probability of start-up, although in their case the distribution of risk attitudes is similar between borrowers and non-borrowers.
Most contributions are in the field of development economics; for a survey, see Benabou (1996)
Notwithstanding the fact that borrowers are risk averse and the bank risk neutral, posting collateral (and a lower interest rate) increases the surplus from the project. The additional surplus, in equilibrium, accrues entirely to borrowers and may compensate, especially for low values of collateral where the incentive effect is supposed to be larger and risk aversion lower, the additional risk for borrowers. This explains why the optimal contract, C POOL, is not necessarily a zero-collateral one.
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Acknowledgments
We are indebted to David de Meza, Alberto Bennardo, Emma Carter, Daniel Kraus, Alireza Naghavi and Alberto Zazzaro for useful comments. The suggestions of the associate editor and an anonymous referee helped to improve considerably the paper. We are also grateful to the participants at the SIE conference (October, 2010), the Meeting of ECINEQ in Catania (July, 2011), the conference of the European Economics and Finance Society in London (June, 2011) and the European Economic Association Conference in Oslo (August, 2011). Responsibility for mistakes remains entirely ours. Giuseppe Coco gratefully acknowledges the financial support of the MIUR Prin Grant No. 2007RR7HCN.
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Appendix
Appendix
1.1 Proof of Eq. 5
Starting with Eq. 3:
By the Implicit Function theorem and due to decreasing absolute risk aversion, we simply observe that:
which implies that:
1.2 Proof of Eq. 6:
Starting with Eq. 1:
Let us assume that W S = Y − X + w i and W F = w i − c, we can simply rewrite that:
By envelope theorem and differentiating with respect to X and c, it follows that:
which implies that:
1.3 Second order condition \(\left( \frac{d^{2}X}{dc^{2}}\right)\)
Let us define as s the slope of the indifference curve. The curvature of the indifference curve can be studied after differentiating Eq. 6 with respect to c:
The first expression in curly brackets gives the negative risk aversion effect which makes the indifference curve more concave while the second positive term is the effort disincentive effect which renders the indifference curve more convex. The former is larger, thus the higher the degree of risk aversion, the more sensitive is the probability of success to the amount of collateral provided. However, to simplify in Figs. 1 and 2, we design the indifference curves with a negative second derivative because convexity due to the dominance of the moral hazard impacts does not have significant implications for the existence and nature of equilibrium.
1.4 Proof of Eq. 7:
We can again rewrite the slope of the indifference curve as:
where \(M(w)\,=\,-\frac{(1\,-\,p(e_{i}))}{p(e_{i})}\) while \(R(w)\,=\,\frac{U^{\prime }(W^{F})}{U^{\prime }(W^{S})}. \) The curvature of the indifference curve with respect to change in wealth is then:
where \(M(w)R^{\prime }(w)\) captures the effect of risk preference while \(M^{\prime }(w)R(w)\) explains the moral hazard effect. First, let us solve \(M(w)R^{\prime }(w):\)
Let us define A(W) as the coefficient of decreasing absolute risk aversion, we can then rewrite \(M(w)R^{\prime }(w)\) as:
Since W 1 > W 2 and considering decreasing absolute risk aversion, i.e. risk aversion decreases with wealth, \(A(W^{F})\,>\,A\left( W^{S}\right) \) and considering that by construction \(\frac{dX}{dc}\) is negative, we can surely say that the effect of risk preferences \(M(w)R^{\prime }(w)\) is positive.
Then we can solve \(M^{\prime }(w)R(w)\):
Therefore,
As shown, the sign of Eq. 7 is uncertain due to the combination of the positive effect of risk aversion \(\left( \frac{dX}{dc}\left( A(W^{S})-A(W^{F})\right) \right) \) and the negative moral hazard impact \(-\frac{p^{\prime }(e_{i})}{p(e_{i})}\frac{\partial e}{\partial w}\left( \frac{dX}{dc}-\frac{U^{\prime }(W^{F})}{U^{\prime }(W^{S})}\right)\).
1.5 Proof of Eq. 8:
After some algebraic manipulations,
The impact of moral hazard prevails if and only if:
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Coco, G., Pignataro, G. Unfair credit allocations. Small Bus Econ 41, 241–251 (2013). https://doi.org/10.1007/s11187-012-9422-3
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DOI: https://doi.org/10.1007/s11187-012-9422-3