---
title: "Multiple Equilibria in Markets with Screening"
authors:
  - name: "Alexis Direr"
    affiliation: "Laboratoire d'Économie d'Orléans (LEO), Université d'Orléans"
date: "2005-08"
doi: "10.1080/00036846.2011.602010"
keywords: [banks, asymmetric information, externalities, screening, credit market, multiple equilibria]
jel_codes: [G21, D82, D62]
language: en
type: research-article
---

# Multiple Equilibria in Markets with Screening

**Authors**
- Alexis Direr — Laboratoire d'Économie d'Orléans (LEO), Université d'Orléans

**DOI**: [10.1080/00036846.2011.602010](https://doi.org/10.1080/00036846.2011.602010)

**Keywords**: banks; asymmetric information; externalities; endogenous screening; multiple equilibria.
**JEL codes**: G21, D82, D62.
**Research framework**: Laboratoire d'Économie d'Orléans (LEO), Université d'Orléans. Earlier version (titled *A Dynamic Model of Credit Allocation*) presented at the 1999 Annual Meetings of the Society for Economic Dynamics and the 1999 ESEM Congress in Santiago de Compostela.

---

## Abstract

This paper adds endogenous screening to Broecker (1990) and shows the possibility of multiple screening equilibria. A high intensity of screening by a bank decreases average quality of firms applying to other banks, which then in turn have further incentives to screen.

---

## 1. Motivation and contributions

The paper builds on Broecker (1990), in which banks use imperfect and independent tests to assess potential creditors. Because screening selects higher-quality borrowers, it lowers the average quality of applicants reaching banks that have not yet been visited — a *negative screening externality* between banks. Shaffer (1997) provides supportive evidence using U.S. commercial banks over 1986–1995.

The contribution is twofold.

1. **Multiple screening equilibria with endogenous screening intensity.** Whereas in Broecker the externality operates through interest rates, here it operates through the binary screening choice. When banks expect a high-quality applicant pool they screen little, so few rejects circulate and the pool stays clean — confirming the optimistic prior. When banks expect a low-quality pool they screen, generating many rejects and degrading the pool — again confirming the prior. The paper proves (Proposition 1) that for a non-empty set of parameter values both a *permanent screening* and a *permanent pooling* equilibrium coexist. This connects the screening externality to the strategic-complementarity framework of Cooper & John (1988). The paper situates itself alongside other models of endogenous screening (Gale 1993; Thakor 1996; Manove, Padilla & Pagano 2001; Gehrig & Stenbacka 2003).

2. **Banking concentration and screening intensity.** Proposition 2 shows that the value of screening is increasing in the number of banks $N$. With more banks, an applicant rejected by one bank has more outside options, so the proportion of bad firms in circulation rises, raising each individual bank's incentive to screen. Banking competition (in the headcount sense) therefore expands the region of multiple equilibria, so a more competitive banking industry can be associated with more frequent shifts in lending standards.

A subsidiary discussion connects the multiplicity result to macroeconomic stylized facts — procyclical lending volumes with countercyclical lending standards (Berger & Udell 2003), credit crunches (Bernanke & Lown 1991), and the pump-priming logic of Lang & Nakamura (1993).

---

## 2. Model setup

### 2.1 Players and technology

- **Banks.** $N \geq 2$ risk-neutral banks. Each obtains funds at the safe gross interest rate $r$.
- **Firms.** Two risk-neutral types $j \in \{g, b\}$ (good, bad). Each project requires unit investment, yields $X > 0$ with probability $p_j$ and zero otherwise. Good projects are socially valuable, bad ones are not:

$$p_g X > r > p_b X.$$

- **Wealth and contracts.** Entrepreneurs have no initial wealth. Funding is a standard debt contract with fixed repayment $R_t$ in non-bankruptcy states. With limited liability the firm's expected return is $p_j(X - R_t)$, so firms participate iff $R_t \leq X$.

### 2.2 Search and timing

Time is discrete; firms can visit at most one bank per period. Each period a continuum of mass $1$ of $g$ projects and mass $l$ of $b$ projects enters the market. Together with carry-overs from previous periods, applicants randomly approach a bank not previously visited and privately negotiate the rate. The bank's offer can be refused, in which case the firm searches further. The setup follows the sequential-search tradition of Burdett & Judd (1983) and Bizer & DeMarzo (1992); the alternative case of publicly posted rates is treated by Broecker (1990).

The end-of-search condition for a $g$ applicant requires that today's offer dominates the discounted value of further proposals:

$$p_g(X - R_t) \geq \beta^j \, p_g(X - R_{t+j}), \quad \forall j = 0,1,\dots$$

with $\beta < 1$.

### 2.3 Screening technology

Each bank $i$ chooses $\alpha_{it} \in \{0,1\}$ at cost $c > 0$ per applicant. The screening test produces a signal $G$ or $B$. A bad firm produces $B$ with probability $q_b$, a good firm with probability $q_g < q_b$. The signal informativeness reduces to the assumption $q_b > q_g$. Binary types and binary signals are without loss because the lending decision itself is binary.

**Information sharing assumption.** Banks do *not* share screening evaluations. The paper motivates this through the literature on evaluation sharing (Avery, Resnick & Zeckhauser 1999; Jappelli & Pagano 1993, 2002; Shaffer 1997): soft, complex, non-standardized information about young or small businesses is hard to credibly transmit (Berger & Udell 1998). The model therefore applies to medium and large young firms and to small businesses with informationally opaque credit profiles, but not to markets for personal loans or trade credit where evaluation sharing is widespread.

### 2.4 Bank profits

Let $Q^g_{it}$ (resp. $l Q^b_{it}$) be the masses of $g$ ($b$) firms visiting bank $i$ at $t$. Period profit is

$$\pi(\alpha_{it}, Q^g_{it}, l Q^b_{it}) = Q^g_{it}\big[(1 - \alpha_{it} q_g)(p_g R_t - r) - \alpha_{it} c\big] + l Q^b_{it}\big[(1 - \alpha_{it} q_b)(p_b R_t - r) - \alpha_{it} c\big].$$

The objective is by nature static — a rejected firm cannot reapply to the same bank — so the screening choice does not affect the bank's *own* future profits. It does affect *other* banks' future profits through the composition of the residual applicant pool. This is the **intertemporal screening externality**.

### 2.5 Equilibrium interest rate

The Nash equilibrium repayment is $R_t = X$ regardless of the actual screening intensity:
- $R_t = X$ satisfies firm participation and the end-of-search condition;
- raising it violates participation;
- a candidate symmetric equilibrium with $R_t < X$ unravels because any bank can deviate to a slightly higher rate while still satisfying the end-of-search condition.

Attention is restricted to symmetric equilibria with a common screening level $\alpha_t$, so each bank faces the same applicant flow $Q^g_t / N$ and $l Q^b_t / N$.

### 2.6 Steady-state populations

Under permanent screening, type-$j$ firms are rejected with probability $q_j$ at each visit. After visiting all $N$ banks, surviving applicants leave the market. The stationary stock of type-$j$ firms is

$$Q^j = 1 + q_j + q_j^2 + \dots + q_j^{N-1}, \quad j \in \{g, b\}.$$

Under permanent pooling, all firms are funded on first contact, so $Q^j = 1$ for both types.

---

## 3. Equilibrium screening choices

A *permanent screening* equilibrium requires:

$$\pi(1, Q^g, l Q^b) \geq \pi(0, Q^g, l Q^b) \quad \text{and} \quad \pi(1, Q^g, l Q^b) \geq 0.$$

A *permanent pooling* equilibrium requires:

$$\pi(0, 1, l) \geq \pi(1, 1, l) \quad \text{and} \quad \pi(0, 1, l) \geq 0.$$

The relative weight of bad applicants $l Q^b / Q^g$ is the single sufficient statistic capturing how past screening decisions by competitors feed into a given bank's current screening incentive.

---

## 4. Results

### 4.1 Proposition 1 — Multiplicity

> **Proposition 1.** There exists a set of values for $(l, c, p_b, p_g, q_b, q_g)$ for which either a screening or a pooling equilibrium may occur.

**Proof sketch.** Define the per-good-applicant value of investing in screening when others do as well:

$$V(l Q^b / Q^g) \;\equiv\; \frac{1}{Q^g}\big[\pi(1, Q^g, l Q^b) - \pi(0, Q^g, l Q^b)\big] \;=\; \frac{l Q^b}{Q^g}\big[q_b(r - p_b X) - c\big] - q_g(p_g X - r) - c.$$

For the signal to be informative, $q_b > q_g$, hence $Q^b / Q^g > 1$, so $V(l Q^b / Q^g) > V(l)$: when other banks have already screened, the effective bad-to-good ratio is higher, raising each remaining bank's value of screening. $V$ is also monotone increasing in $l$ — it is negative for $l$ small enough and positive for $l$ large enough. There is therefore a range of $l$ for which

$$V(l Q^b / Q^g) > 0 \geq V(l),$$

i.e. screening is profitable at the screening steady state and pooling is profitable at the pooling steady state simultaneously. $\square$

**Interpretation.** The mechanism is self-confirming expectations. Optimism about the applicant pool ⇒ no screening ⇒ few rejects circulating ⇒ pool stays clean ⇒ optimism vindicated. Pessimism ⇒ screening ⇒ many rejects ⇒ pool degraded ⇒ pessimism vindicated.

### 4.2 Proposition 2 — Banking concentration

> **Proposition 2.** A screening equilibrium is more likely when the number of banks $N$ increases (the parameter set compatible with screening expands in $N$).

**Proof sketch.** $V$ depends on $N$ via the ratio $Q^b / Q^g = (1 + q_b + \dots + q_b^{N-1}) / (1 + q_g + \dots + q_g^{N-1})$. The required inequality

$$\frac{q_b^N}{q_g^N} > \frac{1 + q_b + \dots + q_b^{N-1}}{1 + q_g + \dots + q_g^{N-1}}$$

is true for $N=1$ (since $q_b > q_g$) and is shown by induction using $q_b > q_g$. Hence $V$ is increasing in $N$. $\square$

**Implication.** The pooling equilibrium is unaffected by $N$ (all firms are funded immediately). The screening equilibrium becomes attainable for a wider parameter set as $N$ grows. The region of multiplicity therefore *expands* with $N$. Two opposing effects on aggregate credit follow:
- More banks ⇒ more chances for any given applicant to be accepted under a fixed screening level (extensive margin).
- More banks ⇒ degraded pool ⇒ stronger incentive to screen (intensive margin).

The net effect of concentration on aggregate credit is ambiguous; the standard intuition that fewer banks imply less credit need not hold once screening is endogenous.

### 4.3 Welfare and policy

The screening equilibrium is inefficient whenever the pooling equilibrium is also implementable: each bank's screening choice exerts a negative externality on others' applicant pools, which is not internalized. This fits the strategic-complementarity framework of Cooper & John (1988).

A pump-priming policy in the spirit of Lang & Nakamura (1993) can be welfare-improving: if one bank lowers its screening, the residual pool improves for the others, who may then also lower their standards, moving the economy toward the high-credit / low-screening equilibrium.

> **Author's critical reading.** The model identifies an inefficiency conditional on the pooling equilibrium also being feasible. Whether pooling is in fact feasible at the same fundamentals — i.e. whether the average expected surplus on the unscreened pool is non-negative — is a parameter-dependent condition, not a generic property.

---

## 5. Conclusion

A small extension of Broecker (1990) — letting banks choose screening intensity rather than only interest rates — is enough to deliver multiple screening equilibria in the credit market. The result confers a key role to banks' optimism or pessimism about the type composition of loan applicants.

The paper points to three macroeconomic readings.

1. **Procyclicality of bank lending and countercyclicality of lending standards** (Berger & Udell 2003). Optimism in expansions selects the no-screening equilibrium; pessimism in downturns selects the screening equilibrium.

2. **Credit crunch episodes** (Bernanke & Lown 1991). A worsening of the perceived applicant pool — typical of recession onset — pushes the economy toward the screening equilibrium and a sharp drop in loan supply.

3. **Banking concentration debate.** Variability in screening intensity can break the often-asserted link between concentration and low aggregate credit.

The basic structure (multiple, independent testers evaluating heterogeneous applicants) is portable to other markets: skilled-labor screening, journal refereeing (cf. Nakamura & Shaffer 1991).

### Limitations and research extensions

- **Cost of funds taken as exogenous.** Embedding the screening decision in a full macroeconomic model with $r$ endogenous to aggregate credit is a natural next step.
- **Steady-state focus.** Out-of-steady-state dynamics of screening — transitions between equilibria, self-fulfilling switches — are not characterized.
- **No information sharing.** The model is silent on regimes where credit registers or bureaus partially overcome the negative externality.
- **No collateral or contract design margins** beyond the binary screen / no-screen choice; richer signal structures or contingent contracts are absent.

---

## Acknowledgments

The author thanks Patrick Fève, Pierre Picard, and the referee for useful comments. Any errors or omissions are the author's own. An earlier version was presented at the 1999 SED Annual Meeting and the 1999 ESEM Congress in Santiago de Compostela.

---

## Main references

Berger, A. N., & Udell, G. F. (1998). The economics of small business finance: The roles of private equity and debt markets in the financial growth cycle. *Journal of Banking and Finance* 22(6–8), 613–673.

Berger, A. N., & Udell, G. F. (2003). The institutional memory hypothesis and the procyclicality of bank lending behavior. BIS Working Paper No. 125.

Bernanke, B., & Gertler, M. (1990). Financial fragility and economic performance. *Quarterly Journal of Economics* 105(1), 87–114.

Bernanke, B., & Lown, C. (1991). The credit crunch. *Brookings Papers on Economic Activity* 2, 205–247.

Bizer, D., & DeMarzo, P. (1992). Sequential banking. *Journal of Political Economy* 100(1), 41–61.

Broecker, T. (1990). Credit-worthiness tests and interbank competition. *Econometrica* 58(2), 429–452.

Burdett, K., & Judd, K. (1983). Equilibrium price dispersion. *Econometrica* 51(4), 955–970.

Cooper, R., & John, A. (1988). Coordinating coordination failures in Keynesian models. *Quarterly Journal of Economics* 103(3), 441–463.

De Meza, D., & Webb, D. (1988). Credit market efficiency and tax policy in the presence of screening costs. *Journal of Public Economics* 36(1), 1–22.

Gehrig, T., & Stenbacka, R. (2003). Screening cycles. Working Paper No. 2, Swedish School of Economics.

Jappelli, T., & Pagano, M. (2002). Information sharing, lending and defaults: Cross-country evidence. *Journal of Banking and Finance* 26(10), 2017–2045.

Lang, W., & Nakamura, L. (1993). A model of redlining. *Journal of Urban Economics* 33(2), 223–234.

Manove, M., Padilla, A. J., & Pagano, M. (2001). Collateral versus project screening: A model of lazy banks. *RAND Journal of Economics* 32(4), 726–744.

Shaffer, S. (1997). The winner's curse in banking. *Journal of Financial Intermediation* 7(4), 359–392.

Thakor, A. (1996). Capital requirements, monetary policy, and aggregate bank lending: Theory and empirical evidence. *Journal of Finance* 51(1), 279–324.

*The full reference list appears in the PDF.*