Diamond (1989) - Reputation Acquisition In Debt Markets

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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.

Summary

The model is one of endogenous reputations. One possible outcome of the model is that types that can choose project choose bad projects to start with, gain a reputation through luckily not defaulting, have sufficient value from their reputation to behave (i.e. choose good projects) for some periods, and then when insufficient time remains they choose bad projects again.

The solution concept is Sequential Equilibrium

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]

One 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].

More than One Periods

There are a series of lemmas that follow directly and need essentially no explanation:

• If at time [math]t\le T\,[/math] a borrower is revealed to be a type [math]B\,[/math] or [math]GB\,[/math], then he will get no loans afterward
• If a loan is made at date [math]t\,[/math] then [math]r_{t}\in [r,G]\,[/math]
• If returns are above [math]r_{t}\,[/math] then borrowers repay exactly [math]r_{t}\,[/math], while if returns are below [math]r_{t}\,[/math] then the project is liquidated.
• Any payment less than [math]r_{t}\,[/math] at time <matht\,</math> leads to no loans for all [math]t^{\prime }\ge t\,[/math].
• All surviving borrowers at time [math]t\,[/math] offer the same rate [math]r_{t}\,[/math] that gives lenders an expected return of [math]r\,[/math].

To characterize the equilibrium we need to specific the actions of [math]GB\,[/math] types and show that the rate sequences are best responses to these actions. This is done through continuation values for the [math]GB\,[/math] types in periods [math]T\,[/math] (where [math]GB\,[/math] types will choose the bad project) and [math]t\lt T\,[/math] as follows (where the subscripts indicate choosing that action for period [math]t\,[/math] but then continuing with optimal choices forward):

[math]V_{T}=\beta \pi (B-r_{T})\,[/math]

[math]V_{t}^{b}=\beta \pi (B-r_{t}+V_{t+1})\,[/math]

[math]V_{t}^{g}=\beta (G-r_{t}+V_{t+1})\,[/math]

In equilibrium it must be that:

[math]V_{t}=\max \{V_{t}^{b},V_{t}^{g}\}\,[/math]

So the good project is choosen at time [math]t\,[/math] iff:

[math]\beta(G-r_{t}+V_{t+1})\geq \beta \pi (B-r_{t}+V_{t+1}) \; \therefore r_{t}-V_{t+1}\leq \frac{G-\pi B}{1-\pi}\,[/math]

Which leads to the lemma:

Lemma: Good projects are optimal at time [math]t\,[/math] if and only if [math](1-\pi )V_{t}\geq \beta \pi (B-G)\,[/math]

Notice that as [math]r_t\,[/math] goes down good projects become more attractive (holding [math]V_{t+1}\,[/math] constant). In addition [math]V_t\,[/math] goes up, as does [math]V_{t-1}\,[/math], meaning that if a good project is choosen at time [math]t\,[/math], it will be choose \forall [math]t' \lt t\,[/math]. However, for a good to be choosen at all reputation effects must kick in. This is in the next lemma and its proof.

Lemma: Type [math]BG\,[/math] borrowers will select the safe project on some date [math]t\,[/math] only if (i.e. a necessary condition):

[math]\frac{\beta (G-r)}{1-\beta }\geq \frac{\beta \pi (B-r)}{1-\beta \pi }\,[/math]

The proof is as follows:

Choosing a good project at time [math]t'\,[/math] means that this condition is satisfied:

[math]r_{t^{\prime }}-V_{t^{\prime }+1}\leq \frac{G-\pi B}{1-\pi }\,[/math]

A bad project is choosen in the last period, so choosing bad projects in all periods after [math]t'\,[/math] implies:

[math]V_{t^{\prime }+1}=\sum_{t=1}^{T-t^{\prime }}(\beta \pi )^{t}(B-r_{t})\leq \sum_{t=1}^{T-t^{\prime }}(\beta \pi )^{t}(B-r)\lt \sum_{t=1}^{\infty }(\beta \pi )^{t}(B-r)=\frac{\beta \pi (B-r)}{1-\beta\pi }\,[/math]

And as :[math]r_{t}\geq r\,[/math]:

[math]r-\frac{\beta \pi (B-r)}{1-\beta \pi }\lt \frac{G-\pi B}{1-\pi }\,[/math]

which gives the result.

A sufficient condition on the interest rates follows (it's proof and feasibility is ignored here):

Fix [math]t^{\prime }\,[/math]. If for all [math]t\in \{t^{\prime },t^{\prime}+1,...,T\}\,[/math]:

[math]r_{t}\lt \beta G+\frac{(1-\beta )(G-\pi B)}{(1-\pi )}\,[/math]

then there exists [math]T\lt \infty\,[/math] such that:

[math]r_{t^{\prime }}-V_{t^{\prime }+1}\leq \frac{G-\pi B}{1-\pi }\,[/math]

and a safe project is chosen at [math]t^{\prime }\,[/math]

We then need the Bayesian rational interest rates. These depend on the number of survivers of [math]B\,[/math] types and the actions (and survivors) of [math]GB\,[/math] types. For the [math]G\,[/math] types:

[math]f_{Gt}=f_{G}\,[/math]

For the [math]B\,[/math] types:

[math]f_{Bt}=\pi ^{t-1}f_{B}\,[/math]

For the [math]GB\,[/math] types, if in period [math]t\,[/math] they choose [math]G\,[/math] with probability [math]\sigma \in [0,1]\,[/math]:

[math]f_{BGt}=\sigma f_{BGt-1}+(1-\sigma )\pi f_{BGt-1}\,[/math]

Which has bounds:

[math]r_{t}^{b}=r\cdot \frac{f_{Bt}+f_{BGt}+f_{G}}{\pi f_{Bt}+\pi f_{BGt}+f_{G}}\,[/math]

and

[math]r_{t}^{g}=r\cdot \frac{f_{Bt}+f_{BGt}+f_{G}}{\pi f_{Bt}+f_{BGt}+f_{G}}\,[/math]

using the notation from before.

We now (ab)use measure theory to get three results. The (ab)uses are by assuming that some population is of measure zero but that by continuity we can allow some positive measure into the population and the result will hold.

First we solve the end game with [math]f_{BG}=0\,[/math] - the end game being the part of the game from which the [math]GB\,[/math] types choose bad projects forward to infinity. If the end game is bounded then the part of the game in which reputation works is unbounded. With [math]f_{BG}=0\,[/math] the interest rates are deterministic and tend towards r:

[math]r_{t}=r\cdot \frac{\pi ^{t-1}f_{B}+1-f_{B}}{\pi ^{t}f_{B}+1-f_{B}}\,[/math]

Building on this there is the case of moral hazard without adverse selection - that is there are no bad types. In this case we get the following proposition:

Proposition: With no adverse selection interest rates are constant, and if reputation ever works it works immediately at [math]t=1\,[/math] and stops working at some [math]t^{\prime }\lt T\,[/math].

With adverse selection as well we get the alternative proposition:

Proposition: If [math]r_{t}\,[/math] falls over time and a type [math]GB\,[/math] borrower optimally selects good projects at time [math]t^{\prime \prime }\,[/math] and bad projects at some [math]t^{\prime }\lt t^{\prime \prime }\,[/math], then bad projects are optimal for all [math]t\lt t^{\prime }\,[/math]. This implies that if good projects are optimal at two dates [math]t_{1}\lt t_{2}\,[/math], then good projects are optimal for all [math]t\in \{t_{1},t_{1}+1,...,t_{2}\}\,[/math].

For further detail see either the paper or Tadelis' write-up of the paper:

• Tadelis, Steve (2007), "Topics in Contracts and Organizations: Lecture Notes", UC Berkeley, September pdf tex