Group Formation and Endogenous Information Collection in Microcredit

Abstract

This paper attempts to address the effects of different types of loan contract on a borrower’s incentive for investment in information. We model the trade-off that a borrower faces when she collects information about the potential of her intended projects both under individual and joint liability loan contracts. Even under limited liability, the borrower faces a trade-off at information collection stage between the cost of signal collection, and the cost of her time and effort for project execution in case the project fails. We show that joint liability contract induces borrowers to invest more in information than individual liability for low rates of interest. However, for some high rates of interest, borrowers invest positive amount in information collection under individual liability, but do not take up the project under joint liability.

Appendix

Proof of Proposition 5 Notice that at $r=r_{c}^{I}$,

$$ g_{1}\left( r\right) =\frac{1}{2}\left[ \hat{\theta }\left( r\right) \left( Y-r\right) -e\right] -c\left( \hat{\theta }\left( r\right) \right) =0 $$

and

$$\begin{aligned} g_{1}^{J}\left( r\right)= & {} \frac{1}{2}\left[ \hat{\theta }_{J}\left( r\right) \left( Y-r-(1-\hat{\theta }_{J}\left( r\right) )r\right) -e\right] -c\left( \hat{\theta }_{J}\left( r\right) \right) \\= & {} \frac{1}{2}\left[ \hat{\theta }_{J}\left( r\right) \left( Y-r\right) -e \right] -c\left( \hat{\theta }_{J}\left( r\right) \right) -\hat{\theta } _{J}\left( r\right) (1-\hat{\theta }_{J}\left( r\right) )r \\< & {} \frac{1}{2}\left[ \hat{\theta }_{J}\left( r\right) \left( Y-r\right) -e \right] -c\left( \hat{\theta }_{J}\left( r\right) \right) \le \frac{1}{2} \left[ \hat{\theta }\left( r\right) \left( Y-r\right) -e\right] -c\left( \hat{ \theta }\left( r\right) \right) \\= & {} 0 \end{aligned}$$

where the last inequality follows from the fact that $\hat{\theta }\left( r\right)$ maximizes $\left[ \frac{1}{2}\left[ \theta \left( Y-r\right) -e \right] \right. \left. -c\left( \theta \right) \right]$. Since $g_{1}^{J}\left( r_{c}^{I}\right) <0$, $\left( g_{1}^{J}\left( r\right) \right) ^{\prime }<0$ and $g_{1}^{J}\left( r_{c}^{J}\right) =0$, it must be the case that $r_{c}^{I}>r_{c}^{J}$ (please see, Fig. 3).

Suppose $g\left( 0\right) <0$. Notice that

$$ g^{J}\left( r\right) =\frac{1}{2}\left[ \hat{\theta }_{J}\left( r\right) \left( Y-r-(1-\hat{\theta }_{J}\left( r\right) )r\right) -e\right] -c\left( \hat{\theta }_{J}\left( r\right) \right) -\left( \frac{Y}{2}-\frac{3r}{4} -e\right) $$

and

$$ g^{I}\left( r\right) =\frac{1}{2}\left[ \hat{\theta }\left( r\right) \left( Y-r\right) -e\right] -c\left( \hat{\theta }\left( r\right) \right) -\left( \frac{Y}{2}-\frac{r}{2}-e\right) $$

Hence, for any r, $g^{J}\left( r\right) >g\left( r\right)$ if and only if

$$\begin{aligned}&\frac{1}{2}\left[ \hat{\theta }_{J}\left( r\right) \left( Y-r-(1-\hat{\theta }_{J}\left( r\right) )r\right) -e\right] -c\left( \hat{\theta }_{J}\left( r\right) \right) -\left( \frac{Y}{2}-\frac{3r}{4}-e\right) \\> & {} \frac{1}{2}\left[ \hat{\theta }\left( r\right) \left( Y-r\right) -e\right] -c\left( \hat{\theta }\left( r\right) \right) -\left( \frac{Y}{2}-\frac{r}{2} -e\right) \end{aligned}$$

$$ \Leftrightarrow \frac{r}{4}-\frac{1}{2}\hat{\theta }_{J}\left( r\right) (1- \hat{\theta }_{J}\left( r\right) )r>c\left( \hat{\theta }_{J}\left( r\right) \right) -c\left( \hat{\theta }\left( r\right) \right) -\left( \hat{\theta } _{J}\left( r\right) -\hat{\theta }\left( r\right) \right) \frac{Y-r}{2} $$

Now, the RHS of the above expression

$$\begin{aligned}&c\left( \hat{\theta }_{J}\left( r\right) \right) -c\left( \hat{\theta } \left( r\right) \right) -\left( \hat{\theta }_{J}\left( r\right) -\hat{\theta } \left( r\right) \right) \frac{Y-r}{2} \\= & {} \int \limits _{\hat{\theta }}^{\hat{\theta }_{J}}\left[ c^{\prime }\left( \theta \right) -\frac{Y-r}{2}\right] d\theta \\< & {} \int \limits _{\hat{\theta }}^{\hat{\theta }_{J}}\left[ c^{\prime }\left( \hat{\theta }_{J}\right) -\frac{Y-r}{2}\right] d\theta \end{aligned}$$

since $c^{\prime \prime }\left( \theta \right) >0$. But from (11)

$$\begin{aligned} \int \limits _{\hat{\theta }}^{\hat{\theta }_{J}}\left[ c^{\prime }\left( \hat{\theta } _{J}\right) -\frac{Y-r}{2}\right] d\theta= & {} \int \limits _{\hat{\theta }}^{\hat{ \theta }_{J}}\left[ \frac{Y-2\left( 1-\hat{\theta }_{J}\right) r}{2}-\frac{Y-r }{2}\right] d\theta \\= & {} \int \limits _{\hat{\theta }}^{\hat{\theta }_{J}}\frac{\left( 2\hat{\theta } _{J}-1\right) r}{2}d\theta \\= & {} \left( \hat{\theta }_{J}-\frac{1}{2}\right) \left( \hat{\theta }_{J}-\hat{ \theta }\right) r \end{aligned}$$

We now show that for any $r>0$,

$$ \frac{r}{4}-\frac{1}{2}\hat{\theta }_{J}(1-\hat{\theta }_{J})r>\left( \hat{ \theta }_{J}-\frac{1}{2}\right) \left( \hat{\theta }_{J}-\hat{\theta }\right) r\Leftrightarrow \frac{1}{2}>\hat{\theta }_{J}^{2}-2\hat{\theta }_{J}\hat{ \theta }+\hat{\theta } $$

which holds since $\hat{\theta }_{J},\hat{\theta }\in \left( \frac{1}{2} ,1\right)$. Hence, we can conclude that for any $r>0$, $g^{J}\left( r\right) >g^{I}\left( r\right)$. Since, at $r=r_{d}^{I}$, $g^{I}\left( r\right) =0$, we can conclude that $g^{J}\left( r_{d}^{I}\right) >0$. Since $g^{J\prime }\left( r\right) >0$ and $g^{J}\left( r_{d}^{J}\right) =0$, it must be the case that $r_{d}^{J}<r_{d}^{I}$. The interest rate at which the borrower starts investing in signal quality is higher under individual liability.