Diamond (1989) - Reputation Acquisition In Debt Markets

• This page is referenced in BPP Field Exam Papers

Reference(s)

Diamond, D. (1989), "Reputation Acquisition in Debt Markets," Journal of Political Economy, 97(4): 828 862 pdf

Abstract

This paper studies reputation formation and the evolution over time of the incentive effects of reputation to mitigate conflicts of interest between borrowers and lenders. Borrowers use the proceeds of their loans to fund projects. In the absence of reputation effects, borrowers have incentives to select excessively risky projects. If there is sufficient adverse selection, reputation will not initially provide improved incentives to borrowers with short credit histories. Over time, if a good reputation is acquired, reputation will provide improved incentives. General characteristics of markets in which reputation takes time to work are identified.

The Model

There are three types of borrowers:

[math]G\,[/math] - good types, in proportion [math]f_G\,[/math], who make investments that return [math]G \gt r\,[/math] [math]B\,[/math] - bad types, in proportion [math]f_B\,[/math], who make investments that return [math]B \gt G\,[/math] with probability [math]\pi\,[/math] and [math]0\,[/math] with probability [math]1-\pi\,[/math], where [math]\pi B \lt r\,[/math] [math]GB\,[/math] - Bad or Good types, in proportion [math]f_GB\,[/math], that have access to both investments

Lenders have measure [math]m \gt 1\,[/math], and an outside option of [math]r\,[/math]. They are endowed with 1 unit to lend.

The model has the following assumptions:

• There is a competitive market for loans and lenders last only one period (so no long term contracts)
• Realized output is private
• Payments are public
• If a borrow defaults, all parties get zero and this credit history is recorded.
• Borrowers have limited liability (i.e. [math]c_t \ge 0\,[/math])

Utilities of the borrowers are based on their consumption, [math]c\,[/math], and a discount factor, [math]\beta\,[/math].

[math]U(\tilde{c})=\sum_{t=1}^{T}\beta ^{t}\mathbb{E}[\tilde{c}_{t}]\,[/math]

1 Period Economy

There are four facts about returns:

1. If the contract requires payment of [math]r_T\,[/math], no-one would repay more (it signals nothing)
2. If the contract requires payment of [math]r_T\,[/math], no-one would repay less (as it would imply default and payment of 0)
3. [math]r_T \ngtr G\,[/math] as investments would be unprofitable
4. [math]r_T \ge r\,[/math] as the probability of repayment is less than or equal to 1 and r is the outside option

The utilities for [math]G\,[/math] and [math]B\,[/math] project choices are:

[math]U_G = G-r_{T} \mbox{ and } U_B = \pi (B-r_{T})\,[/math]

Therefore the [math]GB\,[/math] type will choose the [math]G\,[/math] project if:

[math]\pi (B-r_{T})\gt G-r_{T} \;\therefore r_{T}^{b}\equiv \frac{r}{\pi f_{B}+\pi f_{BG}+f_{G}}\,[/math]

Given the competitive debt market and the outside option it must be that beliefs for [math]G\,[/math] and [math]B\,[/math] choices by [math]GB\,[/math] types that:

[math]r_{T}^{b}\equiv \frac{r}{\pi f_{B}+\pi f_{BG}+f_{G}}\,[/math]

[math]r_{T}^{g}\equiv \frac{r}{\pi f_{B}+f_{BG}+f_{G}}\,[/math]

It is clear that:

[math]r\lt r_{T}^{g}\lt r_{T}^{b}\,[/math] which gives four cases:

1. If [math]r_{T}^{g}\gt G\,[/math] or if ( [math]r_{T}^{g}\gt \frac{G-\pi B}{1-\pi }\,[/math] and [math]r_{T}^{b}\gt G\,[/math] ) then no loans are made
2. If [math]r_{T}^{g}\gt \frac{G-\pi B}{1-\pi }\,[/math] and [math]r_{T}^{b}\leq G\,[/math] then the unique equilibrium has [math]r_{T}=r_{T}^{b}\,[/math], and [math]GB\,[/math] types choose [math]B\,[/math] projects
3. If [math]r_{T}^{b}\lt \frac{G-\pi B}{1-\pi }\,[/math] then the unique equilibrium has [math]r_{T}=r_{T}^{g}\,[/math] and [math]GB\,[/math] types choose [math]G\,[/math] projects
4. If [math]r_{T}^{g}\lt \frac{G-\pi B}{1-\pi }\,[/math] and [math]\frac{G-\pi B}{1-\pi }\lt r_{T}^{b}\leq G\,[/math] then there are three equilibria:
   1. [math]r_{T}=r_{T}^{b}\,[/math] and [math]GB\,[/math] types choose [math]B\,[/math] projects
   2. [math]r_{T}=r_{T}^{g}\,[/math] and [math]GB\,[/math] types choose [math]G\,[/math] projects
   3. [math]r_{T}=\frac{G-\pi B}{1-\pi }\,[/math] and [math]GB\,[/math] types choose [math]G\,[/math] projects with probability [math]p=\frac{(1-\pi )r-(G-\pi B)[\pi(f_{B}+f_{BG})+f_{G}]}{(G-\pi B)(1-\pi )f_{BG}}\,[/math]

In (1) lending doesn't take place. Either the rate is too high for good types, or lenders believe that [math]GB\,[/math] types will choose good projects, but they would actually choose bad projects and ten the rate is too high. In (2) [math]GB\,[/math] types would choose bad projects, lenders believe this and the rate supports it. In (3) [math]GB\,[/math] types would choose good projects, lenders believe this and the rate supports it. And in (4) the rates and beliefs are consistent with both choices and so a mixed strategy is possible.

The model then assumes that all [math]GB\,[/math] types choose the risky project (to make things interesting) and then it must be that:

[math]G\gt r\gt \frac{G-\pi B}{1-\pi}\,[/math]

This gives us the first Lemma:

Lemma 1: At the final period [math]T\,[/math], (a) all borrowers offer a debt contract with the lowest interest rate that provides an expected return of [math]r\,[/math]; (b) all borrowers who can repay do so; and (c) Only borrowers for whom beliefs suggest a high enough probability of being a [math]G\,[/math] type will get a loan in period [math]T\,[/math].