Unfair credit allocations

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.

Appendix

1.1 Proof of Eq. 5

Starting with Eq. 3:

$$ \frac{\partial U_{i}}{\partial e_{i}}\,=\,p^{\prime }(e_{i})\left( U(W^{S})-U(W^{F})\right) -1 $$

By the Implicit Function theorem and due to decreasing absolute risk aversion, we simply observe that:

$$ \begin{aligned} &\left[ p^{\prime \prime }(e_{i})\left( U(W^{S})-U(W^{F})\right) \right] de+ \left[ p^{\prime }(e_{i})\left( U^{\prime }(W^{S})-U^{\prime }(W^{F})\right) \right] dw\,=\,0\\ &\left[ p^{\prime \prime }(e_{i})\left( U(W^{S})-U(W^{F})\right) \right] de\,=\,- \left[ p^{\prime }(e_{i})\left( U^{\prime }(W^{S})-U^{\prime }(W^{F})\right) \right] dw \end{aligned} $$

which implies that:

$$ \frac{de}{dw}\,\,=\,\,-\frac{p^{\prime }(e_{i})\left( U^{\prime }(W^{S})-U^{\prime }(W^{F})\right) }{p^{\prime \prime }(e_{i})\left( U(W^{S})-U(W^{F})\right) } <0 $$

1.2 Proof of Eq. 6:

Starting with Eq. 1:

$$ U_{i}\,=\,p(e_{i})U(Y-X+w_{i})+(1\,-\,p(e_{i}))U(w_{i}-c)-e_{i} $$

Let us assume that W ^S = Y − X + w _i and W ^F = w _i  − c, we can simply rewrite that:

$$ U_{i}\,=\,p(e_{i})U(W^{S})+(1\,-\,p(e_{i}))U(W^{F})-e_{i} $$

By envelope theorem and differentiating with respect to X and c, it follows that:

$$ \begin{aligned} &\left[ -p(e_{i})U^{\prime }(W^{S})\right] dX-\left[ (1\,-\,p(e_{i}))U^{\prime }(W^{F})\right] dc\,=\,0\\ &\left[ -p(e_{i})U^{\prime }(W^{S})\right] dX\,=\,\left[ (1\,-\,p(e_{i}))U^{\prime }(W^{F})\right] dc \end{aligned} $$

which implies that:

$$ \frac{dX}{dc}\,=\,-\frac{(1\,-\,p(e_{i}))U^{\prime }(W^{F})}{p(e_{i})U^{\prime }(W^{S})}\,=\,s<0 $$

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:

$$ \begin{aligned} \frac{d^{2}X}{dc^{2}}&=\,-\begin{array}{c} \left\{ \frac{\left[ -(1\,-\,p(e_{i}))U^{\prime \prime }(W^{F})-p^{\prime }(e_{i}) \frac{\partial e}{\partial c}U^{\prime }(W^{F})\right] \left( p(e_{i})U^{\prime }(W^{S})\right) }{\left[ p(e_{i})U^{\prime }(W^{S})\right] ^{2}} -\frac{\left[ (1\,-\,p(e_{i}))U^{\prime }(W^{F})\right] \left[ -p(e_{i})U^{\prime \prime }(W^{S})\frac{\partial X}{\partial c}+p^{\prime }(e_{i})\frac{\partial e}{\partial c}U^{\prime }(W^{S})\right] }{\left[ p(e_{i})U^{\prime }(W^{S})\right] ^{2}}\right\} \end{array} \\ &=\,- \begin{array}{c} \left\{-\frac{(1\,-\,p(e_{i}))U^{\prime \prime }(W^{F})}{\left[ p(e_{i})U^{\prime }(W^{S})\right] }-\frac{p^{\prime }(e_{i})\frac{\partial e}{\partial c} U^{\prime }(W^{F})}{\left[ p(e_{i})U^{\prime }(W^{S})\right] }+\frac{ p(e)(1\,-\,p(e))U^{\prime }(W^{F})U^{\prime \prime }(W^{S})\frac{\partial X}{ \partial c}}{\left[ p(e_{i})U^{\prime }(W^{S})\right] ^{2}} -\frac{p^{\prime }(e_{i})\frac{\partial e}{\partial c}U^{\prime }(W^{S})(1\,-\,p(e_{i}))U^{\prime }(W^{F})}{\left[ p(e_{i})U^{\prime }(W^{S}) \right] ^{2}}\right\} \end{array} \\ &=\, \begin{array}{c} \left\{ \frac{(1\,-\,p(e_{i}))U^{\prime \prime }(W^{F})}{p(e_{i})U^{\prime }(W^{S})} -s\frac{(1\,-\,p(e_{i}))U^{\prime \prime }(W^{S})U^{\prime }(W^{F})}{ p(e_{i})\left( U^{\prime }(W^{S})\right) ^{2}} + \frac{p^{\prime }(e_{i})\frac{\partial e}{\partial c}U^{\prime }(W^{F})}{p(e_{i})U^{\prime }(W^{S})}+\frac{p^{\prime }(e_{i})\frac{\partial e}{\partial c} (1\,-\,p(e_{i}))U^{\prime }(W^{F})}{p(e_{i})^{2}U^{\prime }(W^{S})}\right\} \end{array} \\ &=\, \begin{array}{c} \left\{ \frac{(1\,-\,p(e_{i}))}{p(e_{i})}\frac{U^{\prime }(W^{F})}{U^{\prime }(W^{S})} \left( \frac{U^{\prime \prime }(W^{F})}{U^{\prime }(W^{F})}-s\frac{U^{\prime \prime }(W^{S})}{U^{\prime }(W^{S})}\right)+\left[ \frac{p^{\prime }(e_{i})\frac{\partial e}{\partial c} U^{\prime }(W^{F})}{p(e_{i})U^{\prime }(W^{S})}\left( 1+\frac{(1\,-\,p(e_{i}))}{ p(e_{i})}\right) \right] \right\} \end{array} \\ &=\,\left\{-s\left[ -A(W^{F})+sA(W^{S})\right] +\frac{p^{\prime }(e_{i})\frac{\partial e}{\partial c}U^{\prime }(W^{F})}{p(e_{i})U^{\prime }(W^{S})}\left( \frac{1}{p(e_{i})}\right) \right\} \\ &=\,\left\{-s \left[ sA(W^{S})-A(W^{F})\right] +\left[ \frac{ \partial e}{\partial c}\frac{p^{\prime }(e_{i})}{\left( p(e_{i})\right) ^{2}} \frac{U^{\prime }(W^{F})}{U^{\prime }(W^{S})}\right] \right\}\,\lessgtr\,0 \end{aligned} $$

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:

$$ \frac{dX}{dc}\,=\,M(w)R(w) $$

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:

$$ \frac{\partial }{\partial w}\left( \frac{dX}{dc}\right) \,=\,M(w)R^{\prime }(w)+M^{\prime }(w)R(w) $$

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 $M(w)R^{\prime }(w):$

$$ \begin{aligned} M(w)R^{\prime }(w) &=\, -\frac{(1\,-\,p(e_{i}))}{p(e_{i})}\left[ \frac{U^{\prime \prime }(W^{F})U^{\prime }(W^{S})-U^{\prime \prime }(W^{S})U^{\prime }(W^{F}) }{\left( U^{\prime }(W^{S})\right) ^{2}}\right] \\ &=\,-\frac{(1\,-\,p(e_{i}))}{p(e_{i})}\left[ \frac{U^{\prime \prime }(W^{F})}{ U^{\prime }(W^{S})}-\frac{U^{\prime \prime }(W^{S})U^{\prime }(W^{F})}{ \left( U^{\prime }(W^{S})\right) ^{2}}\right] \\ &=\,-\frac{(1\,-\,p(e_{i}))}{p(e_{i})}\frac{1}{U^{\prime }(W^{S})}\left[ U^{\prime \prime }(W^{F})-\frac{U^{\prime \prime }(W^{S})U^{\prime }(W^{F})}{ U^{\prime }(W^{S})}\right] \\ &=\,-\frac{(1\,-\,p(e_{i}))}{p(e_{i})}\frac{U^{\prime }(W^{F})}{U^{\prime }(W^{S}) }\left[ \frac{U^{\prime \prime }(W^{F})}{U^{\prime }(W^{F})}-\frac{U^{\prime \prime }(W^{S})}{U^{\prime }(W^{S})}\right] \\ \end{aligned} $$

Let us define A(W) as the coefficient of decreasing absolute risk aversion, we can then rewrite $M(w)R^{\prime }(w)$ as:

$$ \begin{aligned} M(w)R^{\prime }(w) &=\,-\frac{(1\,-\,p(e_{i}))}{p(e_{i})}\frac{U^{\prime }(W^{F}) }{U^{\prime }(W^{S})}\left( A(W^{S})-A(W^{F})\right) \\ &=\,\frac{dX}{dc}\left( A(W^{S})-A(W^{F})\right)\,>\,0 \\ \end{aligned} $$

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 $M^{\prime }(w)R(w)$:

$$ \begin{aligned} M^{\prime }(w)R(w) &=\,\left[ -\frac{-p^{\prime }(e_{i})\frac{\partial e}{ \partial w}p(e_{i})-(1\,-\,p(e_{i}))p^{\prime }(e_{i})\frac{\partial e}{\partial w}}{\left( p(e_{i})\right) ^{2}}\right] \frac{U^{\prime }(W^{F})}{U^{\prime }(W^{S})} \\ &=\,\left[ \frac{p^{\prime }(e_{i})\frac{\partial e}{\partial w}}{\left( p(e_{i})\right) }+\frac{(1\,-\,p(e_{i}))p^{\prime }(e_{i})\frac{\partial e}{ \partial w}}{\left( p(e_{i})\right) ^{2}}\right] \frac{U^{\prime }(W^{F})}{ U^{\prime }(W^{S})} \\ &=\,\frac{p^{\prime }(e_{i})}{p(e_{i})}\frac{\partial e}{\partial w}\left[ 1+ \frac{\left( 1\,-\,p(e_{i})\right) }{p(e_{i})}\right] \frac{U^{\prime }(W^{F})}{ U^{\prime }(W^{S})} \\ &=\,\frac{p^{\prime }(e_{i})}{p(e_{i})}\frac{\partial e}{\partial w}\left[ \frac{U^{\prime }(W^{F})}{U^{\prime }(W^{S})}-\frac{dX}{dc}\right] \\ &=\,-\frac{p^{\prime }(e_{i})}{p(e_{i})}\frac{\partial e}{\partial w}\left[ \frac{dX}{dc}-\frac{U^{\prime }(W^{F})}{U^{\prime }(W^{S})}\right]\,<\,0 \\ \end{aligned} $$

Therefore,

$$ \frac{\partial }{\partial w}\left( \frac{dX}{dc}\right) \,=\,\frac{dX}{dc}\left( A(W^{S})-A(W^{F})\right) -\frac{p^{\prime }(e_{i})}{p(e_{i})}\frac{\partial e }{\partial w}\left( \frac{dX}{dc}-\frac{U^{\prime }(W^{F})}{U^{\prime }(W^{S})}\right)\,\lessgtr\,0 $$

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,

$$ \begin{aligned} \frac{\partial }{\partial w}\left( \frac{dX}{dc}\right) =\,s\left( A(W^{S})-A(W^{F})\right) -\frac{\partial e}{\partial w}\frac{p^{\prime }(e_{i})}{p(e_{i})}\left( s-\frac{U^{\prime }(W^{F})}{U^{\prime }(W^{S})} \right) \\ =\,s(1\,-\,p(e_{i}))\left( A(W^{S})-A(W^{F})\right) -\frac{\partial e}{\partial w }p^{\prime }(e_{i})\frac{(1\,-\,p(e_{i}))}{p(e_{i})}\left( s-\frac{U^{\prime }(W^{F})}{U^{\prime }(W^{S})}\right) \\ =\,s(1\,-\,p(e_{i}))\left( A(W^{S})-A(W^{F})\right) -\frac{\partial e}{\partial w }p^{\prime }(e_{i})\left( \left( \frac{(1\,-\,p(e_{i}))}{p(e_{i})}\right) s+s\right) \\ =\,s(1\,-\,p(e_{i}))\left( A(W^{S})-A(W^{F})\right) -\frac{\partial e}{\partial w }p^{\prime }(e_{i})\frac{s}{p(e_{i})} \\ =\,s\left( (1\,-\,p(e_{i}))\left( A(W^{S})-A(W^{F})\right)\quad-\frac{\partial e}{ \partial w}\frac{1}{p(e_{i})\left( U(W^{S})-U(W^{F})\right) }\right) \\ =\,\frac{s}{p(e_{i})\left( U(W^{S})-U(W^{F})\right) }\left( p(e_{i})(1\,-\,p(e_{i}))\left( A(W^{S})-A(W^{F})\right) -\frac{\partial e}{ \partial w}\right) \\ \end{aligned} $$

The impact of moral hazard prevails if and only if:

$$ \frac{\partial e}{\partial w}>p(e_{i})(1\,-\,p(e_{i}))\left( A(W^{S})-A(W^{F})\right) $$