Rationing despite screening: A motif based on price-discrimination

Florian Heider∗

Department of Finance, Stern School of Business, New York University

November 2002Preliminary and Incomplete

∗44 West 4th. St., Suite 9-190, New York, NY 10012, USA; Tel.: +1 212 998 0311, Fax: +1 212 995 4233, email:fheider@stern.nyu.edu

1

Rationing despite screening: A motif based onprice-discrimination

Abstract

The paper revisits Stiglitz and Weiss (1981)’ seminal Credit Rationing result in more detail. Ithas often been argued that rationing is irrelevant since a) it only pertains to specific assumptionsabout the quality of investment projects and b) it is eliminated by offering screening contracts toborrowers. The paper argues the opposite. It shows that Credit Rationing holds for a wide rangeof assumptions about the quality of investment projects and it explains that if a bank has marketpower then screening contracts may not only fail to eliminate rationing they may even create it.

2

In their seminal contribution Stiglitz and Weiss (1981) show how asymmetric information between

banks and borrowing firms can lead to a failure of market clearing in credit markets. When a bank

does not know the quality of firms’ investment projects then it may not want to raise loan interest

rates even if there is currently an excess demand for bank loans. Doing so may drive firms with good

(safe) investment projects out of the loan market and hence may leave the bank with a costly Adverse

Selection of bad (risky) borrowers.1

Since Adverse Selection only arises when the uninformed party (the bank) cannot distinguish

between different quality types of the informed party (the firm) the question arises whether a bank

could not use more sophisticated debt contracts to screen borrowing firms into different quality classes.

Indeed, Bester (1985) shows that banks can use collateral requirements as a screening device. Safe

firms prefer debt contracts with high collateral requirements and low interest repayments while risky

firms prefer debt contracts with low collateral requirements and high interest repayments. By offering

both types of debt contracts and observing which firm chooses which, a bank can overcome asymmetric

information. Consequently, there is no issue of Adverse Selection and hence, there cannot be rationing.

Stiglitz and Weiss’ credit rationing result begs further questions. If debt contracts can lead to

Adverse Selection and hence rationing, why not use other financing contracts? Indeed, DeMeza and

Webb (1987) show that equity eliminates rationing in the Stiglitz and Weiss model. Moreover, they also

show that their credit rationing result is not robust to an alternative specification of the parameters

that describe the quality-difference between good/safe and bad/risky firms.

The state of affairs therefore seems to be that credit rationing is impossible. It is eliminated by

screening contracts or simply by equity finance. And in any case, it pertains only to a specific choice

of parameters. The impossibility of rationing is troublesome in so far as rationing has become a useful

ingredient of the micro-foundations of modern macroeconomics. As one textbook states: ”If credit is

rationed, then it is possible that the interest rate is not a reliable indicator of the impact of financial

variables on aggregate demand. It is quite likely in that case that quantity variables, such as the

amount of credit, have to be looked at in appraising monetary and financial policy.” (Blanchard and

Fischer (1989), p.479).2

The aim of this paper is to re-establish the possibility of rationing. Rationing may not be eliminated1Adverse Selection was first discussed by Akerlof (1970).2See Greenwald et al. (1984) for one of the first applications of rationing a la Stiglitz and Weiss to obtain macroeco-

nomic dynamics.

3

by screening contracts (they may even create it!) because screening allows monopolistic banks to

make profits trough price-discrimination. Any clearing away of excess demand, i.e. an elimination of

rationing, would interfere with price discrimination and consequently lead to lower profits. Moreover,

I show that pure equity finance will not necessarily elimination rationing and that rationing pertains

to a wide range of parameters that describe the quality-difference between safe and risky firms.

Stiglitz and Weiss’ formulated their credit/debt rationing result for the case when safe and risky

firms’ cash-flows are Mean-Preserving Spreads. The paper makes more general assumptions about

the difference in cash flows between safe and risky firms. It shows that credit rationing also occurs

under Second-Order Stochastic Dominance or even when the cash-flows are not ordered by either

First-Order or Second-Order Stochastic Dominance. The only case when credit rationing never occurs

is First-Order Stochastic Dominance which is the case that DeMeza and Webb originally considered.

DeMeza and Webb’s other result, the absence of Adverse Selection with equity finance, is at odds

with Myers and Majluf (1984)’s seminal Pecking-Order result. The Pecking-Order relies on the very

fact that equity finance leads to a stronger Adverse Selection effect than debt finance. The contra-

diction can also be resolved by making more general assumptions about the difference in cash flows

between safe and risky firm. Equity finance never leads to Adverse Selection under Mean-Preserving

Spreads (i.e. the Stiglitz and Weiss assumption). But under First-Order Stochastic dominance it may

well entail Adverse Selection and thus lead to rationing too. Moreover, the paper shows that equity

finance can also lead to Propitious Selection, i.e. risky firms leaving the market, under Second-Order

Stochastic Dominance.

Having established a) that debt rationing is pervasive and b) that equity cannot eliminate rationing,

the paper then shows that screening contracts, here combinations of debt and equity finance, cannot

do not always eliminate rationing when a bank has market power. They may even create rationing.

The assumption of market power in banking is crucial. Bester obtains opposite results because he

assumes perfect competition.3

In the present paper the bank uses debt and equity finance to screen and discriminate borrowers.

There are many other screening mechanisms that the bank could use and that have been pointed out

by the literature. For example collateral requirements (Bester (1985), Bester (1987), Deshons and

Freixas (1987), Mattesini (1990)), loan size (Milde and Riley (1988)) or the probability of receiving3I will argue that Stiglitz and Weiss (1981) original model, although termed competitive, really is one where banks

have market power. Yanelle (1997) shows that is it not obvious how to model competition among banks.

4

funds (Gillet and Lobez (1992)).4 The logic of my persistence result however does not depend on what

screening instruments are actually used. It is the desire to carry out price discrimination when you

have market power that is important.5

Chiesa (2001) obtains rationing and monopolistic banking endogenously in a different context where

outside deposit holders cannot fully control the actions of inside bank managers. Bank managers can

improve projects they fund and makes sure they have a positive Net Present Value if they monitor

them. Since they find monitoring costly and since limited liability protects them from downside risks,

they take on many projects but do not monitor them. Instead they gamble on the upside potential, i.e.

they hope that projects turn out to be successful after all. A natural way to control bank managers

then is to impose upper limits on the volume of lending relative to their own inside capital that is at

stake. Capital requirements, i.e. rationing, can thus be seen as a way of disciplining bank managers

and providing them with incentives to monitor projects.

Conditional on bank managers monitoring projects, it is efficient to fund as many as possible. Thus

capital requirements are socially costly. The key question then is how relax the monitoring constraint

of bank managers in order to reduce costly capital requirements and to maximize lending. The answer

is to guarantee bank managers large profit margins, i.e to grant them market power. The more bank

managers profit from funding projects the more own incentives they have to monitor them so that one

can do with less capital requirements. Chiesa thus provides an alternative explanation for associating

market power in banking with rationing.

Hellmann and Stiglitz (2000) also re-establish rationing outside a screening or discrimination con-

text, but their focus is on competition. Borrowers can choose between raising money from a competi-

tive debt market with Adverse Selection or a competitive equity market with Adverse Selection. Due4Furthermore, Besanko and Thakor (1987a) show that screening borrowers using collateral requirement may not

work if borrowers do not have enough initial wealth to pledge the required amount of collateral. Schmidt-Mohr (1997)

circumvents the wealth problem by showing that collateral will not be used as a screening instrument when the loan

size can be used a screening instrument instead. Since the loan size will vary across borrowers who all want the same

amount of funds, some borrowers will necessarily feel rationed. Loan size rationing (or type I rationing) is a different

phenomenon from all-or-nothing rationing (or type II rationing) which is the theme of this paper. Moreover, loan size

rationing assumes that investment projects are perfectly divisible. See also the very useful survey by Jaffee and Stiglitz

(1990).5The use of debt and equity has the advantage to allow costless screening. I do not have to assume an exogenous cost

of signalling. The crucial Single-Crossing Property arises endogenously from firms’ preferences towards debt and equity

finance. See the second chapter of the thesis for further discussion.

5

to simultaneous rationing in both equity and debt markets, rationing not only survives intra-market

but also inter-market competition.

Rationing therefore continues to be plausible microeconomic phenomenon. The plausibility is

important because rationing it is part of the micro-foundations of modern macroeconomics.

The organization of the paper is as follows. Section 1 introduces the model and section 2 discusses

the assumptions about the quality difference between investment projects. Section 3 examines the

case of pure debt finance, which may lead to classical rationing a la Stiglitz and Weiss, while section 4

examines the case of pure equity finance. Sections 5 and 6 consider screening contracts that use both

debt and equity finance. Section 7 finally concludes.

1 Model

In this section we present the elements of the formal model that we use to make our point about

the relevance of rationing. There is one monopolistic bank with a limited amount of funds. The

bank knows that borrowers have attractive positive NPV projects that need funding but it does not

know the exact characteristics or quality of these projects. The model is a screening model where

the uninformed party (the bank) moves first and proposes one or several financing contracts to the

informed party (the firms) who can then choose among the contracts or reject them. Thereafter,

the demand and supply of funds determines whether all or just some firms receive financing for their

investment projects.

1.1 Financing projects with debt or equity

Each firm has access to one investment opportunity of type t if it spends an amount I right now. The

investment is risky: it succeeds with probability pt returning an amount xt and it fails with probability

1− pt returning nothing. The investment has a positive Net Present Value, ptxt > I. Since each firm

has exactly one investment opportunity of type t we speak of both ”a type t firm” or ”a type t project”.

There are only two sorts of investment projects: safe ones, t = s, and risky ones, t = r. A safe

project succeeds more often than a risky project but in the case of success a safe project returns less

than a risky project: pr ≤ ps and xr ≥ xs with not both as an equality. The ex-ante probability that

a firm has a safe investment project or, equivalently, the percentage of firms with safe projects is q.

6

Each firm has identical assets-in-place whose value, denoted by W > I.6 We denote the total value

of a type t firm with Vt = W + ptxt.

Assets-in-place however cannot be used to pay for the investment opportunity since they are not

marketable right now when the investment opportunity must be seized. Also, the opportunity cannot

be postponed nor can the investment project be partially funded. A firm therefore needs to raise the

entire amount I from the bank. The bank is willing to finance a firm’s investment project using either

debt or equity. More precisely, debt is a zero-coupon bond with a repayment (face value) R and equity

gives the bank an α%-stake in the firm. The expected value of debt that finances a project of type t

is

Dt(R) = ptR

and the expected value of equity that finances a project of type t is

Et(α) = α(W + ptxt)

Note that debt is worthless if the project fails and that equity gives right to a fraction of what the

firm is worth including assets-in-place.

1.2 A pure asymmetric information model of monopolistic rationing

We want to concentrate solely on the effect of asymmetric information between the bank and the firms

on rationing. A number of assumptions remove issues that are not central to our analysis. First, there

are no taxes. Second, we normalize the interest rate to zero. Third, everybody is risk-neutral. Fourth,

there are no direct costs of bankruptcy such as legal or liquidation costs. And fifth, there is no conflict

of interests between the management of a firm or the bank and their respective owners. We make no

distinction between them and talk instead of ”a firm” and ”the bank”.

The crucial assumption is that the bank does not know which firm has a safe investment project

and which firm has a risky project. But the firms themselves do know the type t of their project.

All other parameters in the set-up are known to everybody. We model the information asymmetry

by having the bank move first. It starts the game by offering debt and/or equity contracts to firms.

Then firms either accept a particular financing contract or reject all of them in which case they forego6The assumption is for simplicity only. It guarantees the existence of screening equilibria even if the expected value

of safe firms is higher than the expected value of risky firms. As we shall see below, we do not actually need screening

in those cases to rule out rationing. Pure equity finance is sufficient.

7

their investment opportunity. If a firm has accepted either a debt or an equity contract then there is

still the possibility that so many firms want to invest that the bank cannot finance them all. In that

case some firms will have to forego their investment opportunity. Those firms that accept a financing

contract and actually receive financing go ahead and undertake the project. Some time later, the

game ends when the project delivers its returns and the bank is paid off according to the debt and/or

equity contracts it holds. The following figure illustrates the sequence of events.

-time

t=0Bank offers debt

and/or equity

finance to firms.

t=1Firms choose

preferred financing

contract or do not

invest.

t=2Depending on

demand, bank may

not be able to finance

all firms that wish to

invest at current

conditions.

t=3Investment returns

are realized, firms

are liquidated and

bank is paid-off.

Let us describe in more detail the stages of the game. We begin at the last stage t=4 when the

game has ended. If the bank finances a type t firm with debt then it receives Dt(R) and the firm is

worth Vt−Dt(R). If the bank finances a type t firm with equity then it receives Et(α) and the firm is

worth Vt −Et(α). A firm can only invest in the project if it was granted finance at the previous stage

t=3. If a firm is denied finance then it is worth W, the value of its assets-in-place.

Whether a firm is granted finance at t=3 depends on the supply and demand of funds. The bank

is the only source of funds. We assume that the bank is endowed with a limited amount of funds F.

Also, we assume that there are more safe firms than risky firms and that there are enough funds either

to finance just all safe firms or to finance all risky firms with some funds left over, F = qI > (1− q)I.

There not enough funds to finance all positive NPV projects.7 It is therefore possible that a firm does

not receive financing although it is strictly worse off by having to forgo the investment opportunity.

Let λ denote the probability with which a firm receives funds from the bank. Then we have three7Clearly, a necessary condition for credit rationing to occur is the presence of scarce funds at the equilibrium loan

rate (see Stiglitz and Weiss (1981), Theorem 5, p. 397). For analysis where this is not the case see Besanko and Thakor

(1987a).

8

possibilities. If only safe firms or only risky firms demand funds then λ = 1, if all firms demand funds

then λ = q.

While the supply of funds is fixed, the demand depends on whether at t=2 firms accept or reject

the bank’s offer of debt and/or equity contracts. Normally, each firm wants funds to undertake its

positive NPV investment opportunity. But it is possible that the bank offers unattractive debt and

equity contracts so that firms prefer to forego the opportunity. For example a firm of type t only

wants to undertake its project using equity finance if

λ(Vt − Et(α)) + (1− λ)W ≥ W (1)

i.e. if financing the project, taking into account the possibility of not receiving funds in the next stage,

gives a higher firm value than foregoing the project. Equation (1) is a firm’s Participation Constraint

for equity finance. For debt finance, the Participation Constraint is

λ(Vt −Dt(R)) + (1− λ)W ≥ W (2)

Hence, whether a firm accepts or rejects the bank’s offer depends on the debt repayment R and

the equity stake α the bank asks for. At t=1, the bank first chooses what sort of financing contract

it offers and then it decides on the terms of the contract, i.e. R and α. The bank has four possible

financing policies. It can either offer only debt or only equity finance. Or, it can offer both (offer debt

to safe firms and equity to risky firms and vice versa). We will go through the various possibilities in

the following sections of the paper. Here, we only emphasize that the bank aims at maximizing the

payment it receives from the financing contracts.

2 On the quality difference between investment projects and the

related literature

In this section we discuss in more detail the way in which safe and risky firms differ. It will be useful to

describe the quality difference between safe and risky investments not in absolute terms (e.g. ps− pr)

but in normalized relative terms. For this we define,

Definition 1 The relative differences in success probabilities γ and project returns ε are given by

ε =xr − xs

xr(3)

9

γ =ps − pr

ps(4)

Figure ?? illustrates how changes in xt and pt translate into changes in ε and γ. In the figure we

fix the characteristics of the safe project, xs and ps, and vary the characteristics of the risky project,

xr and pr. For example, if pr = ps and xr > xs(risky project A) then γ = 0 and ε > 0. Alternatively,

if pr < ps and xr = xs (risky project B) then ε = 0 and γ > 0. The following figure demonstrates the

usefulness of these measures.

-

ε6

γ

1

1¡¡

¡¡¡¡

¡¡¡¡

¡¡¡¡

¡¡

Safe dominates

Risky by SOSD

Neither FOSD

nor SOSD

»»»»»»9

Safe and Risky are

MPS

³³³³³³)

Safe dominates

Risky by FOSD

XXzRisky dominates

Safe by FOSD

When there is no difference in return, ε = 0, then the safe project dominates the risky project by

First-Order Stochastic Dominance.8 The reverse is true when there is no difference in success proba-

bilities, γ = 0. The safe project dominates the risky project by Second-Order Stochastic Dominance

when ε ≤ γ. When ε = γ we have a special case of Second-Order Stochastic Dominance: projects then

are Mean-Preserving Spreads. Finally, neither First- nor Second-Order Stochastic Dominance apply

when ε > γ.

Contributions to the credit rationing literature make different assumptions about the difference

in quality between investment projects.9 Stiglitz and Weiss (1981) original analysis of rationing was

cast in a model that assumed that investment projects are Mean-Preserving Spreads. In our simpler

two-type set-up this corresponds to values of γ and ε that lie on the diagonal. Bester (1985)’s screening

critique made the same assumption (he subsequently relaxed it in Bester (1987)). In addition to the

case of Mean-Preserving Spreads, DeMeza and Webb (1987) also considered a case of First-Order

Stochastic Dominance which corresponds to the ε = 0 in our model.8For a definition and discussion of these concepts see Laffont (1989).9We do not consider here cases of loan-size rationing such as Besanko and Thakor (1987b).

10

3 Classical debt rationing

Suppose that the bank offers only debt contracts. The bank then has to decide what repayment R it

asks for bearing in mind that firms can reject the bank’s offer and ”leave the market” if the repayment

is excessive. The key insight is that safe firms leave the market earlier, i.e. at a lower repayment, than

risky firms. A safe firm’s Participation Constraint (2) holds as long as

λ(Vs −Ds(R)) + (1− λ)L ≥ W

R ≤ xs

while a risky firm’s Participation Constraint holds as long as

λ(Vr −Dr(R)) + (9− λ)W ≥ W

R ≤ xr

We call the repayments Rt = xt for which a firm is indifferent between accepting debt finance or not,

a firm’s ”debt capacity”. The simple insight that a risky firm has a higher debt capacity than a safe

firm drives the classical credit rationing result. If the bank raises the interest repayment then it is

the safe firms who leave the market before a risky firm does since their debt capacity is smaller. The

bank may therefore face an Adverse Selection of borrowers, i.e. risky firms, if it raises the interest

repayment.10

More precisely the bank has to decide between the following two scenarios. If it asks for a low

repayment Rs then both firms accept and the demand for funds qI + (1 − q)I is larger than the

maximum supply qI. There is excess demand and the bank is able to lend all its funds to finance

positive NPV projects. Alternatively, the bank can ask for a high repayment Rr but then only risky

firms accept. Now, the demand for funds (1− q)I is smaller than the maximum supply qI. There is

no excess demand and the bank is not able to lend all its funds to finance positive NPV projects.11

The bank prefers the low repayment if

q[qDs(Rs) + (1− q)Dr(Rs)− I] ≥ (1− q)(Dr(Rr)− I) (5)

The term in square brackets is the expected profit from lending to both types of firms. The bank has

to form the expectation across firms since it does not know which firms have a safe and which firms10In the special case ε = 0 (xs = xr) safe and risky firms leave the market at the same time so that there cannot be

credit rationing.11The excess funds are invested in a safe asset with zero net interest.

11

have risky investment projects. The term in square brackets is multiplied by the probability of being

granted financing (λ = q) when all firms apply in order to obtain the average profit per firm. The

term on the right hand side is the average profit per firm from lending to risky firms only but asking

for a high repayment.

Equation (5) captures the main point of Stiglitz and Weiss (1981)’s explanation of why loan

interest rates may not rise in order to clear away excess demand. If the equation holds then it is

not in the bank’s interest to raise the interest repayment despite there being an excess demand for

funds. Although the bank foregoes higher interest repayments it lends to more firms and those firms

on average pay back the loan more often.

If the bank decides to ask for a low interest repayment Rs so that there is excess demand then,

although the chance of not receiving financing is the same for all firms, 1− λ = 1 − q, it is the risky

firms that would be strictly better off if they did receive financing since they have not exhausted their

debt capacity,

λ(Vr −Dr(Rs)) + (1− λ)W > W

Rs < xr = Rr

In other words, risky firms are rationed.

Definition 2 A firm is rationed if it does not receive financing with some probability but at the same

time, the firm would be strictly better off if it did receive financing.

If the bank decides to ask for a high interest repayment Rs then no type of firm is rationed. Risky

firms exhaust their debt capacity and safe firms do not apply for financing in the first place.

The bank’s decision in equation (5) not to raise the interest repayment depends on the relationship

between the amount of interest foregone, xr−xs, and the increase in the probability of repayment, ps−pr. Using the definitions for ε and γ we can rewrite equation (5) to yield the following characterization

of credit rationing.

Proposition 1 With only debt finance, the bank does not raise the interest repayment to clear away

excess demand, thus rationing risky firms, if the relative difference in return ε is sufficiently low:

0 < ε ≤ φ(γ) =X + (Y + Z)γX + Y + Zγ

(6)

where X = (2q − 1)(psxs − I), Y = (1− q)psxs and Z = (1− q)(−qpsxs).

12

-

ε6

γ

1

1¡¡

¡¡¡¡

¡¡¡¡

¡¡¡¡

¡¡

¾ φ

Rationing.

Safe and risky

firms invest.

No Rationing.

Only risky

firms invest.

XX+Y

−XZ

Proposition 1 extends Stiglitz and Weiss’ original credit rationing logic in the context of our model

where project returns are not necessarily Mean-Preserving Spreads. To discuss the extension it is

useful to consider the following figure. It shows the threshold function φ in the ε, γ-plane. The plane

represents all possible differences in success probabilities and project returns between a safe and a

risky project. The following corollary to proposition (1) confirms the shape of the threshold function

φ and how it intersects with 45 degree line.

Corollary 1 The threshold function φ(γ) i) is strictly convex, ii) φ(0) = XX+Y , iii) φ(1) = 1, iv)

φ(γ) = γ ⇐⇒ γ = −XZ

Proof: In the appendix.

The area under φ indicates for which differences in ”quality”, i.e. relative differences in success

probabilities γ and relative differences in project returns ε, there is rationing. We make two observa-

tions. First, debt rationing is a pervasive phenomenon that applies well outside the original Stiglitz and

Weiss case (the diagonal). Second, we cannot link rationing to a particular type of stochastic ordering.

We see that rationing may occur under Mean-Preserving spreads (the diagonal), under Second-Order

Stochastic dominance (below the diagonal), when there is no ordering (above the diagonal) or when

the risky project dominates the safe project by First-Order Stochastic dominance (the vertical axis).

Alternatively there may be no rationing in each of these cases. The only case where there is never

debt rationing is when there is no difference in successful project returns, ε = 0 (see also DeMeza and

Webb (1987)).

13

We when restrict ourselves to the original Stiglitz and Weiss case of Mean-Preserving Spreads (the

diagonal), then we can easily derive some intuitive comparative statics. In that special case rationing

occurs when

ε = γ ≤ φ(γ)

ε = γ ≤ −X

Z

ε = γ ≤ (1

1− q− 1

q)

r

r + 1

where r = (psxs − I)/I is the safe project’s expected rate of return. Debt rationing occurs when the

degree of information asymmetry (measured by either γ or ε) is smaller than the return of the safe

project weighted by a measure how many safe firms where are in relation to risky firms. Debt rationing

is more pervasive if either the safe project becomes more profitable or if there are more safe firms.

Note that rationing is more pervasive when the inefficiency, i.e. the presence of indistinguishable risky

firms is less pervasive.

4 Equity rationing

Now let us suppose that the bank only offers equity contracts and repeat the analysis of the previous

section. Can there be rationing with equity too? To answer this question we derive the ”equity

capacity” of safe and risky firms by asking what is the maximal equity stake α they are willing to give

to the bank. A safe firm’s Participation Constraint (1) holds as long as

λ(Vs −Es(α)) + (1− λ)W ≥ W

α ≤ psxs

W + psxs

while a risky firm’s Participation Constraint holds as long as

λ(Vr −Er(α)) + (1− λ)W ≥ W

α ≤ prxr

W + prxr

14

Unlike in the pure debt case, it is no longer true that risky firms always have a larger ”capacity”. In

fact a risky firm has a larger equity capacity than a safe firm if and only if

αr > αr

prxr

W + prxr>

psxs

W + psxs

ε > γ

We then have three cases. First, when ε = γ then safe and risky firms have the same debt capacity,

αs = αr. In that case both types of firms leave the market at the same time so that there cannot

be an Adverse Selection of firms that stay in the market. Another way of saying this is that equity

is never mispriced. Both firms have the same expected gross value W + psxs = W + prxr so that

an α%-stake in a firm always has the same value irrespective of the firm being a risky or a safe one,

Es(α) = Er(α). Suppose now that the bank can offer both debt and equity contracts in this special

case then it is immediate that the bank cannot do better than to offer equity contracts that extract

all the surplus from firms, α∗ = αs = αr. In the case of Mean-Preserving Spreads equity therefore

eliminates rationing, a result which is originally found in DeMeza and Webb (1987).

When ε < γ then safe firms have a larger equity capacity than risky firms. If the bank increases

the required equity stake α then risky firms drop out of the market before safe firms do. Since any

equity stake in a safe firm is more valuable than in a risky firm, ε < γ ⇐⇒ Es(α) > Er(α), and since

it is the less valuable risky firms that drop out of the market, we now have a situation of Propitious

Selection.

If the bank can issue both debt and equity when ε < γ then it cannot do better than offer an

equity contract that extracts all surplus from safe firms, α∗ = αs, since it lends all its funds to finance

the more profitable firms. There is no rationing since risky firms do not receive financing while all

safe firms do. In the case in which a safe project dominates a risky project by either First-Order or

Second-Order Stochastic Dominance equity therefore eliminates rationing.

Finally when ε > γ then it is the risky firms that have a larger equity capacity than safe firms.

Since risky firms drop out of the market later than safe firms and since risky firms’ equity is more

valuable than safe firms’ equity, ε > γ ⇐⇒ Es(α) < Er(α), there can now be Adverse Selection for

equity in addition to Adverse Selection for debt. In the case when the risky project dominates the

safe project by First-Order Stochastic Dominance or in the case of no ordering both debt and equity

15

may give rise to Adverse Selection. It is therefore no longer obvious what the bank should do in order

to maximize profits, let alone whether there is rationing or not.

5 Screening contracts fail to eliminate rationing

The bank can hope to find out the quality of borrowing firms if it offers a menu of different financing

contracts and if firms of different quality-types choose different financing contracts. By observing

which firms choose which contract the bank can infer the quality of their investment projects. In our

case there are just two quality-types, safe and risky, and two financing contracts, debt and equity.

The bank could for example offer a menu of debt and equity contracts where it intends safe firms to

choose the equity contract, αs, and risky firms to choose the debt contract, Rr. In order for screening

to work, it must be the case the contracts offered are incentive compatible, i.e. that safe firms indeed

prefer equity to debt and that risky firms indeed prefer debt to equity.

Vs −Es(αs) ≥ Vs −Ds(Rr) (7)

Vr −Dr(Rr) ≥ Vr −Er(αr) (8)

Alternatively, the bank could offer a menu of debt and equity where it intends safe firms to choose

debt and risky firms to choose equity.

Vs −Ds(Rs) ≥ Vs −Es(αr) (9)

Vr − Er(αr) ≥ Vr −Dr(Rs) (10)

However, only the first possibility works.

Lemma 1 Only offering equity to safe firms and debt to risky firms is Incentive Compatible.

Proof: Writing out and combining (7) and (8) yields

W + prxr

pr≥ Rr

αs≥ W + psxs

ps

16

Such a combination as and Rr exists since we know from our initial assumption, pr ≤ ps and xr ≥ xs,

with not both as an equality, that

W (pr − ps) < pspr(xr − xs)

W + prxr

pr>

W + psxs

ps

Writing out and combining (9) and (10) on the other hand yields

pr

W + prxr≥ αr

Rs≥ ps

W + psxs

so that it must be that

W (pr − ps) ≥ pspr(xr − xs)

which is impossible given our initial assumption that pr ≤ ps and xr ≥ xs with not both as an equality.

The intuition for lemma 1 is that equity is a convex claim that has a higher value if issued by

risky firms and that debt is a concave claim that has higher value if issued by safe firms. If the menu

of screening contracts intends safe firms to issue debt then by switching to equity, safe firms would

gain since they are now issuing an overvalued claim. The same holds for risky firms switching from

equity to debt. Safe firms mimicking risky firms, and vice versa, means that Incentive Compatibility

is violated.

Next we examine the firms’ Incentive and Participation Constraints in more detail under the

assumption that both types of firms accept the Incentive Compatible contracts offered to them (this

need not necessarily be the case since the bank may prefer to target just one group of borrowers).

It turns out that the incentive compatibility constraint for safe firms, equation (7), does not matter.

When offered equity, a safe firm (strictly) never wants to mimic the behavior of a risky firm by choosing

the debt contract.

Lemma 2 The Incentive Compatibility constraint of safe firms never binds.

Proof: From the proof of lemma 1 we know that

W + prxr

prαs ≥ Rr ≥ W + psxs

psαs

17

with not both weak inequalities holding as an equality. The bank’s profits are increasing in Rr so that

it will set Rr away from its lower incentive compatible bound

Rr >W + psxs

psαs

which means that the inequality in equation (7) is never binding.

The result indicates that any distortions that may arise when the bank uses screening contracts

result from the need to induce risky firms to reveal their type by choosing the debt contract. Since

safe firms need not be induced to reveal their type, the bank can extract all their surplus.

Lemma 3 The Participation Constraint of safe firms always binds.

Proof: From lemma 2 we know that

Rr >W + psxs

psαs

so that the bank can always increase αs until the safe firm’s participation constraint binds.

αs =psxs

W + psxs

Lemma 3 tells us that when the bank offers a menu of screening contracts acceptable to both types

of firms then safe firms will never feel rationed. Even if there is a chance that a safe firm do no not

receive financing because the demand for funds exceeds the supply, it will not feel rationed since it

will not be strictly better off than if it did receive financing.

We argued that the interesting case for rationing is when ε > γ, i.e. when risky firms are more

valuable than safe firms, Vr > Vs so that either risky firms dominate safe firms by First-Order Stochas-

tic Dominance (γ = 0), or no ordering is possible. In that case we can confirm that it is the risky

firms that cause distortions. It is them who want to mimic safe firms and the best the bank can do is

to just make risky firms indifferent between their own debt contract and safe firms’ equity contract.12

The bank must leave some rent to risky firms for them to be willing to reveal their type.13

12The convention is that in this case risky firms stick with their own debt contract.13The case where ε < γ is not interesting from our viewpoint since we are interested in rationing. We know that when

ε < γ equity eliminates rationing.

18

Lemma 4 If ε > γ then the Incentive Compatibility Constraint of risky firms always binds while their

Participation Constraint never binds. If ε < γ then the opposite holds. If ε = γ then both constraints

bind.

Proof: Using lemma 3 a risky firm’s Incentive Compatibility Constraint (8) can be rewritten as

Rr ≤ W + prxr

pr

psxs

W + psxs

A risky firms Participation Constraint (2) provides another upper bound on the repayment Rr

Rr ≤ xr

The bank wants to increase the repayment Rr until one of these constraints binds. Which of them

binds earlier depends on ε and γ, in fact we have

W + prxr

pr

psxs

W + psxsS xr ⇔ ε S γ

Lemma 4 indicates that although we do not yet exactly know what financing contracts the bank

will offer when ε > γ, there is a parallel to the classical rationing case of section 3. It will be risky

firms that may feel rationed. As in section 3 it is them who manage to capture some surplus from

investing in the project and consequently will be strictly worse off if they do not receive financing. The

reason for the surplus is that in classical rationing as well as here, the bank is unwilling to raise the

interest repayment to extract that surplus. But the motif for this unwillingness is different. In classical

rationing it is the danger of loosing safe borrowers. Here it is the need to offer Incentive Compatible

contracts. Lemma 4 says that if ε > γ then risky firms’ Incentive Compatibility Constraint will be

binding, i.e. the interest repayment has to satisfy

Rr = R =W + prxr

W + psxs

ps

prxs (11)

If the bank were to increase the interest repayment beyond the incentive compatible amount then

risky firms would not longer want to reveal their type by choosing debt contracts. An excess demand

for funds can therefore persist even if the bank offers a menu of screening contracts.

If in those cases where classical debt rationing persisted (i.e. the area above the diagonal and

below the curve φ in the figure) the bank prefers offering the menu of screening contracts to offering

19

just debt contracts then we can conclude that screening fails to eliminate rationing. The reason why

the bank indeed prefers the screening contracts is apparent form lemma 2 and equation (11). From the

lemma we know that with screening contracts the bank extracts all the surplus from safe borrowers.

In classical debt rationing the bank charges the low repayment Rs = xs which also extracts all the

surplus from safe borrowers. With screening contracts however the bank can extract a higher surplus

from risky borrowers. They now have to repay Rr = R > xs (since ε > γ) given in equation (11)

which is more than what they have to repay under classical debt rationing, Rs = xs. It is intuitive

that screening contracts dominate since they force the revelation of information which improves the

situation of the previously uninformed bank.

6 Screening contracts may create rationing

Could it be that screening contracts ”import” rationing into situations where there was no classical

rationing, i.e. into the area above and diagonal and above the curve φ?

The reason why there is no classical rationing for values of ε and γ above φ was that the bank

finds it more profitable to lend to risky firms only, extract all their surplus and store the unused funds

q− (1− q)I in a riskless zero-interest asset. As a result, no type of firm felt rationed. Safe firms never

applied for financing and risky firms have all their surplus extracted.

By offering screening contracts the bank now has the advantage of investing all its funds in positive-

interest assets: the investment projects. The disadvantage is that it can no longer extract all the

surplus from risky firms (lemma 4). However, it can extract all surplus from safe firms. If the

bank finds offering screening contracts more profitable than just offering debt contracts, i.e. if the

advantage of lending previously unused funds is large enough, then there is now rationing where there

was previously none. As a result, some risky firms will feel rationed. Due to the scarcity of funds, it

is no longer 100% certain that they will receive financing and if they do not then they will be worse

off since they could have gained some surplus from investing.

If ε > γ then the bank prefers to offer screening contract, with the consequence of rationing, when

q[qEs(αs) + (1− q)Dr(R)− I] ≥ (1− q)(Dr(Rr)− I) (12)

The left hand side represents the bank’s profits from offering equity to safe firms and debt to risky

firms. We know from lemma 1 that offering debt and equity the other way round is not incentive

20

compatible. Furthermore, we know from lemma 3 that the bank can extract all surplus from safe

firms, hence αs = αs, and from lemma 4 that the bank can extract surplus from risky firms up to the

point where they just are just indifferent between revealing their type or not, hence Rr = R. The right

hand side represents the bank’s profits from extracting all surplus from risky borrowers using only

debt. We can rewrite equation (12) using the definitions of ε and γ to obtain the following proposition.

Proposition 2 Suppose that ε > γ. The bank prefers lending to everybody by offering screening

contracts (which entails rationing of risky borrowers) instead of lending to risky firms only (which

entails no rationing) if the relative difference in return ε is sufficiently low:

ε ≤ ψ(γ) =XVs + (Y Vs + Z(Vs −W ))γ

XVs + Y Vs + Z(Vs −W )(13)

where X = (2q − 1)(psxs − I), Y = (1− q)psxs and Z = (1− q)(−qpsxs).

The proposition is similar to proposition 1. Again, there is a threshold function, here ψ, that

determines whether the bank prefers to lend to both types of firms or just to risky firms: for any given

difference in success probabilities γ, the difference in success returns ε must not be too large. When

ε > γ then risky firms are more profitable than safe firms and the only reason why the bank may

prefer to lend to safe firms too is to use all its funds. As mentioned before, lending all funds comes at

the cost of no longer being able to extract all investment surplus from profitable but risky firms. For

any given γ, as ε increases risky firms become more and more profitable relative to safe firms so that

eventually the opportunity loss of not being able to capture all the surplus from risky firms outweighs

the gain from additional lending to safe firms.

In order to illustrate proposition 2 in terms of γ and ε we use the following properties of the

threshold function ψ.

Corollary 2 The threshold function ψ(γ) i) is linear, ii) ψ(0) = XVsXVs+Y Vs+Z(Vs−W ) > φ(0) and iii)

ψ(1) = 1.

The following figure shows both threshold functions φ and ψ in the ε, γ-plane which represents all

possible differences in success probabilities and successful project returns. We observe that below the

diagonal there is no rationing since the bank only targets safe firms using equity finance (section 4).

Above the threshold function ψ, which lies above the diagonal, there is no rationing since the bank

prefers to target only risky firms. The interesting area lies between the diagonal and ψ. The part of

21

-

ε6

γ

1

1¡¡

¡¡¡¡

¡¡¡¡

¡¡¡¡

¡¡

©©©©©©©©©©©©

¾ ψ

No Rationing.

Only safe firms

with equity.

No Rationing.

Screening fails

to eliminate

rationing.

-

Screening

creates

rationing.

-

the area that lies below the threshold function φ is the one where screening is optimal but fails to

eliminate rationing (section 5). The part of the area that lies above the threshold function φ is the

one where screening is optimal and creates rationing.

Two things have changed. We allowed equity contracts and we allowed screening contracts com-

posed of debt and equity. Pure equity finance eliminates rationing wherever it was present below the

diagonal (Second-Order Stochastic Dominance or First-Order Stochastic Dominance of safe firms).

Screening never eliminates rationing. Although screening dominates pure debt finance, Incentive

Compatibility prevents a hardening of screening contracts up to the point where excess demand is

cleared away. In fact screening creates rationing since it allows a better use of funds than pure debt

finance.

The final observation is that the threshold function ψ lies above the threshold function φ. The

intuition is that lending to both types of firms using screening contracts always dominates lending to

both types of firms using only debt finance. The consequence is that there is always the possibility

that screening creates rationing even if risky firms dominate safe firms by First-Order Stochastic

Dominance.

7 Conclusion

The aim of the paper was to affirm the relevance of rationing in situations where banks have market

power but are imperfectly informed about the quality of the investment projects of its borrowers.

Stiglitz and Weiss (1981) were the first to explain how rationing can occur in such a situation but

22

subsequent analyses criticised their explanation. Their explanation is based of a variation of Akerlof

(1970)’s Adverse Selection theme. Bester (1985) argues that more complicated debt contracts allow to

screen borrowers into different quality classes which overcomes Adverse Selection. DeMeza and Webb

(1987) show how equity finance does not give rise to Adverse Selection in the first place.

The argument by which we reaffirm the relevance of rationing is as follows. First, we show how

Stiglitz and Weiss original explanation can be broadened to hold in situations where both criticism

no longer hold. Our main modifications are i) to relax the assumption that banks are perfectly

competitive banks and ii) that investment returns are ordered by First- or Second-Order Stochastic

Dominance or that they are Mean-Preserving Spreads.

Second, we extend DeMeza and Webb’s result that equity does not give rise to Adverse Selection,

a result that only holds if project returns are Mean-Preserving Spreads. We show that equity can

eliminate rationing in other cases too since it may create Propitious Selection.

Third, we ask whether screening contracts can eliminate rationing. The answer is no since Incentive

Compatibility prevents a hardening of the screening contracts. Without such a hardening a possible

excess demand for funds can persist. Fourth, we argue that offering screening contracts may even

create rationing in situations where there is no rationing without screening contracts. The reason is

that screening contracts, although not being able to clear away excess demand, are more profitable

for banks that have market power since they allow a larger rent extraction from borrowers. More rent

can be extracted because screening helps to overcome the information disadvantage of banks. The

bottom line of our analysis is that rationing is relevant.

In order to relax the assumptions of perfect competition and specific stochastic orderings, we

limited our analysis to the case where there are just two types of firms (we called them ”safe” and

”risky”), one single monopolistic bank and either debt or equity finance. Although it is reasonable

to think that the same arguments establish the relevance of rationing for more complex settings, our

analysis can only be the first step. Recently, there have been significant advances in the theory of

screening that should enable us to make the next steps. In particular, Jullien (2000) shows how to

deal with cases where participation constraints are not monotonic and Chone and Rochet (1998) show

how to tackle multi-dimensional screening.

23

References

Akerlof, G. (1970). The market for lemons: Qualitative uncertainty and the market mechanism.

Quarterly Journal of Economics, 84:488–500.

Besanko, D. and Thakor, A. V. (1987a). Collateral and rationing: Sorting equilibria in monopolistic

and competitve credit markets. International Economic Review, 28:671–689.

Besanko, D. and Thakor, A. V. (1987b). Competitive equilibrium in the credit market under asymetric

information. Journal of Economic Theory, 42:167–182.

Bester, H. (1985). Screening vs. rationing in credit markets with imperfect information. American

Economic Review, 75:850–855.

Bester, H. (1987). The role of collateral in credit markets with imperfect information. European

Economic Review, 31:887–899.

Blanchard, O. J. and Fischer, S. (1989). Lectures on Macroeconomics. MIT Press.

Chiesa, G. (2001). Incentive-based lending capacity, competition, and regulation in banking. Journal

of Financial Intermediation, 10:28–53.

Chone, P. and Rochet, J.-C. (1998). Ironing, sweeping and multi-dimentional screening. Econometrica,

66:783–826.

DeMeza, D. and Webb, D. C. (1987). Too much investment: A problem of asymmetric information.

Quarterly Journal of Economics, 102:281–292.

Deshons, M. and Freixas, X. (1987). Le role de la garantie dans le contrat de pret bancaire. Finance,

8:7–32.

Gillet, R. and Lobez, F. (1992). Rationnement du credit, asymetrie de l’information et contrats

separants. Finance, 13:57–71.

Greenwald, B., Stiglitz, J. E., and Weiss, A. (1984). Informational imperfections in the capital market

and macroeconomic fluctuations. American Economic Review, 74:194–199.

Hellmann, T. and Stiglitz, J. (2000). Credit and equity rationing in markets with adverse selection.

European Economic Review, 44:281–304.

24

Jaffee, D. and Stiglitz, J. E. (1990). Credit rationing. In Friedman, B. M. and Hahn, F. H., editors,

Handbook of monetary economics, pages 837–88. Elsevier Science, New York, USA.

Jullien, B. (2000). Participation constraints in adverse selection models. Journal of Economic Theory,

93:1–47.

Laffont, J.-J. (1989). The Economics of Uncertainty and Information. MIT Press, Cambridge, USA.

Mattesini, F. (1990). Screening in the credit market: The role of the collateral. European Journal of

Political Economy, 6:1–22.

Milde, H. and Riley, J. G. (1988). Signaling in credit markets. Quarterly Journal of Economics,

103:101–129.

Myers, S. C. and Majluf, N. (1984). Corporate financing and investment decisions when firms have

information investors do not have. Journal of Financial Economics, 13:187–221.

Schmidt-Mohr, U. (1997). Rationing versus collateralization in competitive and monopolistic credit

markets with asymmetric information. European Economic Review, 41:1321–1342.

Stiglitz, J. and Weiss, A. (1981). Credit rationing in markets with imperfect information. American

Economic Review, 71:393–410.

Yanelle, M.-O. (1997). Banking competition and market efficiency. Review of Economic Studies,

64:215–239.

25

Appendix

A Proof of lemma 1

Parts ii), iii) and iv) are straight forward. To show i), i.e. that φ is concave, we note that the first

derivative

ψ′ =(X + Y + Z)Y(X + Y + Zγ)2

is positive since Y + Z = (1− q)2psxs > 0. The second derivative

ψ′′ = −2(X + Y + Z)Y Z

(X + Y + Zγ)3

is also positive since the numerator is negative (X > 0, Y > 0, Y + Z > 0 and Z < 0) and the

denominator is positive. To see the latter, note that the denominator is linear in γ and that at γ = 0

it is given by

(1− q)I + (psxs − I)q > 0

and at γ = 1 it is given by

(1− 2q)I + q2psxs

which strictly positive since psxs−II > −(1−q)2

q2 .

26