Copyright

by

Gregory William Weitzner

2020

The Dissertation Committee for Gregory William Weitzner

certifies that this is the approved version of the following dissertation:

Essays on Information and Contracts in Financial Markets

Committee:

Andres Almazan, Supervisor

Sheridan Titman, Supervisor

Aydogan Altı

William Fuchs

Thomas Wiseman

Essays on Information and Contracts in Financial Markets

by

Gregory William Weitzner

Dissertation

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

The University of Texas at Austin

December 2020

Acknowledgments

I would like to thank my dissertation committee: Andres Almazan (co-chair), Sheridan Titman

(co-chair), Aydogan Altı, William Fuchs and Thomas Wiseman. The hours I spent in Andres’s

office, not just talking about research, but economics more generally were incredibly intellectually

satisfying. I will always appreciate how he kept pushing me to further clarify my ideas. Working

with Sheridan was such a joy. He shared my ideal to connect finance research to reality and provided

endless fuel for my creative drive. Aydogan provided me incredible support and encouragement

throughout the entire process. I cringe thinking back to some of the ideas I came to his office with,

but he always seemed to push me in the right direction and challenge me intellectually. Although

he joined the department later in the process, Willie provided invaluable critiques and advice which

drove me to improve my papers. I also want to thank my coauthors Cesare Fracassi and Travis

Johnson. Cesare taught me how to turn ideas into real empirical papers. Travis provided me

support and mentorship from a very early stage of the PhD. I always enjoyed talking through

both theoretical and empirical issues with him. I also would like to thank Jonathan Cohn, Daniel

Neuhann, Jangwoo Lee, Iman Dolatabadi, Garrett Schaller and Lee Seltzer. Thank you to my

family and friends for supporting me throughout the entire process.

Gregory Weitzner

The University of Texas at Austin

October 2020

iv

Essays on Information and Contracts in Financial Markets

Publication No.

Gregory Weitzner, Ph.D.

The University of Texas at Austin, 2020

Supervisors: Andres Almazan and Sheridan Titman

My dissertation contains two chapters. In chapter one I explore the relationship between debt

maturity and information production in a theoretical model. In my model, long-term financing

creates an excessive tendency for financiers to acquire information and screen out lower quality

borrowers. In contrast, short-term financing deters information production at origination but in-

duces it when firms are forced to liquidate, depressing the market value of assets due to adverse

selection. Through the feedback effect between firms’ maturity structures and asset prices, in-

creases in uncertainty can impair the aggregate volume of short-term financing and investment.

The analysis can jointly rationalize i) the widespread use of short-term debt by financial firms,

ii) periodic disruptions in short-term funding markets and iii) regulations that curb short-term

funding markets in normal times and support them in periods of market stress. In the second

chapter, I analyze information externalities in the interbank market. In the model, banks use their

information to adjust the size of loans rather than the prices they offer to counterparties because

of adverse selection. Each banks’ individual rationing decision creates an information externality

that increases the efficiency of trade. This information externality occurs even though information

is not shared and banks compete with each other. However, banks do not internalize the cost their

contracts impose on other banks through the counterparty’s likelihood of default, which creates

v

a counteracting negative externality that exacerbates as the number banks increases. The model

provides a microfoundation for interbank discipline and has implications for the optimal structure

of interbank markets.

vi

Contents

Acknowledgments iv

Abstract v

Chapter 1 Debt Maturity and Information Production 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Agents and Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.2 Financial Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.3 Baseline Model Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Model Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.1 Benchmark: Information Acquisition is Contractible . . . . . . . . . . . . . . 13

1.3.2 Information Acquisition Non-Contractible . . . . . . . . . . . . . . . . . . . . 15

1.3.3 Second-Best Financial Contract . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.4 Comparative Statics and Discussion . . . . . . . . . . . . . . . . . . . . . . . 20

1.4 Market Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.4.1 Market Equilibrium Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.4.2 Market Equilibrium Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.4.3 The Effect of Uncertainty on Aggregate Short-Term Debt and Investment . . 29

1.4.4 Welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.6.1 Data Description for Figure 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 34

vii

1.6.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.6.3 Firm Can Raise Additional Funds at t = 0 and Store Funds Across Dates . . 45

1.6.4 Firm Knows Project Type and Lender Cannot Acquire Information . . . . . 48

1.6.5 Firm Knows Project Type and Lender Can Acquire Information . . . . . . . 49

1.6.6 Both Firm and Lender Can Acquire Information . . . . . . . . . . . . . . . . 50

1.6.7 Firm Receives Exogenous Information . . . . . . . . . . . . . . . . . . . . . . 50

1.6.8 NPV Positive Bad Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

1.6.9 Non-Zero Project Payoff in the Case of Failure . . . . . . . . . . . . . . . . . 57

1.6.10 Auction for Market Equilibrium Mechanism . . . . . . . . . . . . . . . . . . . 59

Chapter 2 Counterparty Information Externalities 61

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.2 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.2.1 Agents and Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.2.2 Contracting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.3 Model Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.3.1 Team-Efficient Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.3.2 Market Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.4 Endogenous Information Production . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.6.1 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Bibliography 74

viii

Chapter 1

Debt Maturity and Information

Production

The maturity of a firm’s liabilities affects the information financiers produce about the firm’s assets.

In my model, long-term financing creates an excessive tendency for financiers to acquire information

and screen out lower quality borrowers. In contrast, short-term financing deters information pro-

duction at origination but induces it when firms are forced to liquidate, depressing the market value

of assets due to adverse selection. Through the feedback effect between firms’ maturity structures

and asset prices, increases in uncertainty can impair the aggregate volume of short-term financing

and investment. The analysis can jointly rationalize i) the widespread use of short-term debt by

financial firms, ii) periodic disruptions in short-term funding markets and iii) regulations that curb

short-term funding markets in normal times and support them in periods of market stress.

1.1 Introduction

Short-term debt is used pervasively by many types of financial firms. While commercial banks’ ac-

cess to deposits can rationalize their use of short-term financing, it is less clear why other financial

institutions, such as hedge funds, non-bank broker-dealers, mortgage REITs and mortgage origina-

tors also rely so heavily on it. This reliance on short-term debt seems puzzling given that it exposes

financial firms to potentially costly asset liquidations. Indeed, many observers have argued that

the extensive use of short-term debt by financial firms may adversely affect asset prices, investment

1

and the functioning of short-term funding markets.1

In this paper, I present a model in which the maturity of firms’ liabilities affects the incen-

tives of financiers to produce information about firms’ assets. When financiers provide long-term

financing, they tend to produce an excessive amount of information. Since firms ultimately bear

the cost of inefficient information production, they have an impetus to deter it by borrowing short-

term. Specifically, short-term financing limits financiers’ exposure to the underlying quality of

firms’ assets, reducing their incentives to produce information. However, the downside of short-

term financing is that firms may face costly asset liquidations when they refinance.2

I first characterize the choice between long and short-term financing in a security design

problem in which liquidation costs are exogenous. I then show how these costs arise endogenously

when buyers of assets can produce information when firms liquidate. Specifically, while short-term

financing deters information production at origination, it triggers information production in the

asset market, creating adverse selection. However, since firms’ maturity choices ultimately depend

on liquidation costs, a severe enough adverse selection problem impairs the aggregate volume of

short-term financing. Furthermore, I show that firms’ maturity structures can be either inefficiently

short or long depending on the slope of the demand curve in the asset market.

Formally, I consider a three period model in which a firm seeks financing for an asset from

a lender. The asset’s return depends on its type (good or bad) which is initially unknown to both

the firm and lender, and an aggregate state that occurs at the interim date. There are two key

ingredients in the model: 1) at the interim date the firm can liquidate a portion of the asset in

a competitive outside financial market and 2) before committing funds to finance the asset, the

lender can privately incur a cost to learn the asset’s type. Information production is inefficient,

i.e. the cost of learning the asset’s type exceeds the social value of avoiding financing the bad

asset; however, the lender cannot commit to not doing so. I show that when providing long-term

financing, the lender’s private value of information always exceeds the social value of information.

Intuitively, the lender bears the full downside of financing the asset but only receives a portion its

cash flows, which makes screening out the bad asset more attractive for the lender. As a result,

1E.g. see Geithner (2006). For a detailed discussion of the malfunctioning of short-term debt markets in the2008/2009 financial crisis see Brunnermeier (2009) and Krishnamurthy (2010).

2Henceforth, I will refer to liquidation and refinancing interchangeably. Liquidation costs may encompass actualcosts of transferring assets or any other cost related to the issuance of financial securities.

2

the lender may acquire information despite it being inefficient.

In some cases the optimal security, which can be interpreted as short-term financing, forces

the firm to liquidate a portion of the asset at the interim date following a negative shock. Asset

liquidations blunt the lender’s incentives to acquire information by reducing the lender’s expected

losses from the bad asset. Hence, the firm uses short-term financing when the benefit of deterring

inefficient information production exceeds the expected cost of liquidation. Otherwise, the firm

uses long-term financing which inefficiently reduces its investment scale or triggers information

production.

Next, I examine a market equilibrium in which there are many firms whose financing de-

cisions affect the size of liquidation costs. When firms liquidate at the interim date, an outside

investor can incur a cost to acquire information about individual assets and make anonymous offers

to buy them. If a firm does not sell its asset to the informed investor, it sells it in a competitive pool

of uninformed investors. Information is privately valuable to the investor because it allows to buy

good assets at a reduced price. However, this “cream skimming” worsens the quality of assets that

flows to the uninformed, leading to adverse selection and higher liquidation costs. Consequently,

firms’ initial financing choices both affect and are affected by the adverse selection at liquidation.

Shorter maturities incentivize more information production in the asset market which results in

higher liquidation costs. When the adverse selection problem in the asset market is mild, all firms

use short-term financing. However, in some cases, if all firms were to use short-term financing, liq-

uidation costs would be too high to sustain any short-term financing. Hence, some firms resort to

long-term financing, resulting in a reduced level of aggregate short-term financing and investment.

Finally, I perform a normative analysis by asking whether a planner that changes the number

of firms using short-term financing can increase welfare. While firms internalize the social cost of

inefficient information production at origination, they do not at liquidation because asset prices are

market-determined. However, firms also do not internalize how higher levels of short-term financing

raises the profits of the informed investor by allowing it to buy more assets at a cheap price. The

relative magnitude of these two forces, which can be summarized by the slope of the demand curve,

determines the efficiency of the equilibrium. Intuitively, the more downward sloping the demand

curve is, the more wasteful information production is induced by a marginal increases in aggregate

short-term debt. However, if the demand curve is flat, increasing aggregate short-term debt simply

3

results in a transfer between firms and the informed investor, which has no effect on total welfare.

Therefore, when the demand curve is sufficiently downward sloping the planner prefers lower levels

of aggregate short-term debt. In contrast, when not all firms are using short-term financing and

the demand curve is sufficiently flat, the planner prefers higher levels of short-term debt.

Several of the model’s implications are consistent with stylized facts. First, the model can

rationalize the widespread use of short-term debt by many types of financial firms. For example,

Figure 1.1 shows that over 80% of hedge funds, mortgage originators and mortgage REITs’ debt

is short-term, compared to just over 20% for industrial firms. In practice, financial firms borrow

extensively from other financial institutions.3 These types of lending relationships may be par-

ticularly prone to inefficient information generation because the lender has the ability to evaluate

the same types of assets as the borrower.4 Furthermore, financial assets are generally less costly

to liquidate than real assets.5 Hence, the trade-off between inefficient information production and

liquidation costs may make short-term debt particularly attractive for financial firms.

Second, the endogenous relationship between asset prices and the volume of short-term

financing can help explain the disruptions in short-term funding markets observed in periods of

heightened uncertainty. For example, during the 2008/2009 financial crisis, both the total volume

and maturity of short-term debt decreased across a variety of markets.6 In my model, higher uncer-

tainty increases individual lenders’ incentives to acquire information. Firms respond by shortening

their debt maturities to deter information production. However, shorter debt maturities lead to

higher liquidation costs through information production in the asset market. If liquidation costs

become large enough, the aggregate volume of short-term debt becomes impaired.

3For example hedge funds predominantly borrow from prime brokers within large banks (Ang, Gorovyy, andVan Inwegen (2011)), and mortgage REITs and mortgage originators borrow extensively from banks (Pellerin, Sabol,and Walter (2013b) and Kim et al. (2018)). This raises the question as to why financial firms borrow from banksin the first place. Given their expertise in evaluating financial assets, banks are likely best suited to perform duediligence on other financial firms. However, this expertise can be a double-edged sword in that lenders may betempted to collect too much information while performing due diligence. In practice prime brokers appear to engagein due diligence on hedge funds (Aikman (2010)).

4The following statement by the former head of the Financial Services Authority is consistent with this idea, “Ifind it difficult, if not impossible, to identify an activity carried out by a hedge fund manager which is also not carriedout by the proprietary trading desk within a large bank, insurance company or broker dealer” McCarthy (2006).

5Although I consider an information-based liquidation cost, real assets may also be subject to second-best usercosts (e.g. Shleifer and Vishny (1992)).

6See Hordahl and King (2008), Krishnamurthy (2010), Covitz, Liang, and Suarez (2013), Gorton, Metrick, andXie (2014), Gabrieli and Georg (2014) and Perignon, Thesmar, and Vuillemey (2018) for evidence of shorteningmaturities and Hordahl and King (2008), Gorton and Metrick (2012a), Gorton and Metrick (2012b) Kim et al. (2018)for evidence on reductions in the volume of short-term debt. Gallagher et al. (2020) find evidence of informationproduction by informed investors as debt markets dry up in the context of money market funds.

4

Third, the normative analysis implies that firms’ debt maturities may be inefficiently short,

resulting in too little information production by financiers at origination and too much information

production in the asset market. This implication is consistent with policymakers’ concerns that

there is insufficient credit analysis by institutions that lend to financial firms.7 However, I also show

that firms may use too little short-term debt when short-term funding markets become impaired.

This result is consistent with policymakers efforts to support short-term funding markets during

the LTCM crisis the 2008/2009 financial crisis.8 If adverse selection is already severe and increased

asset sales do not induce substantially more information production, the model would suggest that

policymakers should support short-term funding markets.

There are two main strands of literature rationalizing financial institutions borrowing short-

term. In one class of models, financial firms produce short-term liabilities to meet investors’ liquidity

needs (e.g. Diamond and Dybvig (1983) and Goldstein and Pauzner (2005)), or demand for safe

assets (e.g. Stein (2012), Krishnamurthy and Vissing-Jorgensen (2012) and Diamond (2016)).

Short-term debt can also provide a disciplining role for banks (e.g. Calomiris and Kahn (1991),

Flannery (1994), Diamond and Rajan (2001) and Eisenbach (2017)).9 While the aforementioned

models tend to be used to understand banks borrowing short-term from depositors, I argue that my

model helps to understand lending between financial institutions in which lenders are predisposed

to produce information about borrowers’ assets.

In a more general context, debt maturity can be used to avoid or induce debt overhang (e.g.

Myers (1977), Shleifer and Vishny (1992), Hart and Moore (1995) and Diamond and He (2014)).

Firms may also borrow short-term to signal their quality when information arrives over time (e.g.

Flannery (1986), Diamond (1991), Titman (1992) and Stein (2005)). While in signaling models it

is generally irrelevant which party refinances the initial loan, in my setting the firm must refinance

from an outside party to induce the initial lender to not acquire information. This contractual

7For example in referring to banks’ lending practices prior to the LTCM episode: “There was a lack of balancebetween the key elements of the credit risk management process... banks compromise[d] other critical elements ofeffective credit risk management, including upfront due diligence,... ongoing monitoring of counterparty exposure”and “In managing relationships with [hedge funds], banks clearly relied on significantly less information on thefinancial strength... of these counterparties than is common for other types of counterparties” Basel Committee onBanking Supervision (1999).

8For instance, the Federal Reserve imposed the Term Auction Facility, Term Securities Lending Facility and thePrimary Dealer Credit Facility offered short-term financing during the 2008/2009 financial crisis (see Gorton, Laarits,and Metrick (2018) for more details).

9Short-term debt can also be used as a disciplining device (Leland (1998), Benmelech (2006), Diamond (2004) andCheng and Milbradt (2012)).

5

feature resembles the ubiquitous variation margins used by financial firms in practice. Finally,

Brunnermeier and Oehmke (2013) show that if firms lack commitment in their debt maturity

decisions debt maturities may be inefficiently short.10

This paper builds on the literature analyzing security design to prevent endogenous adverse

selection. Gorton and Pennacchi (1990) show that risk-free debt can prevent informed traders from

taking advantage of uninformed investors with liquidity needs. When assets are risky, Dang, Gorton,

and Holmstrom (2012) find that debt is the optimal security to induce counterparties to not acquire

information. Under general assumptions about the information acquisition technology, standard

debt is the uniquely optimal security to avoid endogenous adverse selection (Yang (2019)).11 To my

knowledge, this is the first paper to consider the role of maturity to deter information production.

I also show how inefficient information production not only leads firms to use short-term debt, but

can generate fire sales when firms raise new funds to repay their lenders.12

Building on the microfoundations of Dang, Gorton, and Holmstrom (2012), Gorton and

Ordonez (2014) analyze the dynamics of information production in debt markets and its macroeco-

nomic implications. Information about collateral decays over time producing a credit and output

boom; however, after an aggregate shock, lenders may suddenly have an incentive to produce in-

formation, which can lead to a large drop in output. My model differs in several respects. First,

in Gorton and Ordonez (2014) there can only be short-term debt because generations live for one

period, while in my model firms choose between short-term and long-term financing. Second, I in-

corporate endogenous information production in the asset market, to study the interaction between

debt maturity, asset prices and aggregate investment. Third, I show that debt maturities can be

both inefficiently short or long, depending on the slope of the demand curve in the asset market.

The link between debt maturity and asset prices relates to the literature examining financial

contracts and market liquidity (e.g. Myers and Rajan (1998), Gromb and Vayanos (2002), Brunner-

meier and Pedersen (2008), Acharya and Viswanathan (2011), He and Milbradt (2014) and Biais,

10Other models of dynamic debt maturity include He and Milbradt (2016), Huang, Oehmke, and Zhong (2019) andGeelen (2019).

11Other models relating investors’ endogenous or exogenous information to security design include Boot and Thakor(1993), Fulghieri and Lukin (2001), Inderst and Mueller (2006), Axelson (2007), Farhi and Tirole (2012b) and Yangand Zeng (2018). In Dang et al. (2017) banks can create liquid, information-insensitive liabilities by keeping assetson the balance sheet.

12In the context of securitization, Pagano and Volpin (2012) show how an issuer’s decision to withhold informationhelps liquidity in the primary market but hurts it in the secondary market.

6

Heider, and Hoerova (2018)).13 In my setting, the feedback effect between information production

at origination and liquidation jointly determines debt maturity and asset market liquidity.

This paper also relates to the body of literature in which information production generates

adverse selection in financial markets. In Fishman and Parker (2015) buyers information production

can generate credit crunches and multiple equilibria in the primary market and in Bolton, Santos,

and Scheinkman (2016) the equilibrium acquisition of information is generically inefficient in OTC

markets. In the context of primary market securitization, Hanson and Sunderam (2013) show that

originators’ production of informationally-insensitive assets ex-ante leads to too little informed

capital ex-post.

Finally, a growing literature explores the macroeconomic implications of adverse selection

(e.g. Eisfeldt (2004), Kurlat (2013), Malherbe (2014), Bigio (2015), Moreira and Savov (2017),

Neuhann (2017) and Asriyan, Fuchs, and Green (2018)). A distinction between my model and

the existing literature is that I consider how adverse selection endogenously arising from firms’

financing choices affects output.

The paper is organized as follows. Section 1.2 describes the baseline model setup. Section

1.3 analyzes the model. Section 1.4 introduces the market equilibrium and Section 1.5 concludes.

All proofs are in the Appendix unless otherwise stated.

1.2 Model Setup

There are three dates, (t = 0, 1, 2) and three agents: a firm that raises funds from a lender at

t = 0 and can raise funds from an outside financial market at t = 1. All agents are risk-neutral

and there is no discounting.

13The market equilibrium also relates to models of fire sales and pecuniary externalities (e.g. Shleifer and Vishny(1992), Kiyotaki and Moore (1997), Lorenzoni (2008), Shleifer and Vishny (2011), Stein (2012), Malherbe (2014),Kurlat (2016), Davila and Korinek (2017), Biais, Heider, and Hoerova (2018), Dow and Han (2018) and Kurlat(2018)).

7

Figure 1.1: Debt Maturity and Leverage. See the Appendix for a detailed description of thedata sources and variable definitions.

8

1.2.1 Agents and Technology

Firm

The firm has no wealth and limited liability. In addition, the firm has access to a technology that

requires an investment k ∈ [0, 1] at t = 0 to produce a random output kR at t = 2. The per-unit

output R can take values R (success) or 0 (failure). Although I refer to the investment technology

as a single project, in the case of a financial firm it can be thought of as an investment strategy or

portfolio of assets. Two factors affect the project’s probability of success: i) the project’s type and

ii) a publicly observable state at t = 1. Specifically, there are two project types υ ∈ {g, b} (good

and bad) which occur with the following probabilities

υ =

g with prob. θ ∈ (0, 1)

b with prob. 1− θ.

The project’s type is initially unknown to all agents. There are two states z ∈ {h, l} (high or low),

which occur with the following probabilities

z =

h with prob. πh ∈ (0, 1)

l with prob. πl ≡ 1− πh.

The good project succeeds with certainty regardless of the state. In contrast, the bad project

succeeds with certainty if the state is high and with probability µ ∈ (0, 1) if the state is low.14

Figure 1.2 provides a visual description of the project’s probability of success. I define φz the

probability that the ex-ante average project succeeds following state z

φh ≡ 1, φl ≡ θ + (1− θ)µ.

I make the following assumptions

Assumption 1.1.

14The probability of success need not be 1 for the good project in both states and the bad project in the high state.The key is that there is a difference in the probability of success across project types in one of the states.

9

Pr(R = R)

1

1z = l

z = h

1

µz = l

z = hυ = b

υ = g

t = 1t = 0 t = 2

Figure 1.2: Payoff Summary.

1. (πh + πlµ)R < 1,

2. (πh + πlφl)R > 1.

Assumption 1.1.1 says that the bad project is NPV negative, while Assumption 1.1.2 says the

ex-ante, average project is NPV positive.15

Lender

The firm borrows from a single, deep-pocketed lender that belongs to a competitive pool of lenders.

The lender has access to an information acquisition technology at t = 0, whereby incurring a cost

c ≥ 0 it learns the project’s type υ. I denote the lender’s information acquisition decision by

a ∈ {0, 1} where a = 0 refers to the lender not acquiring information and a = 1 refers to the lender

acquiring information. For convenience, I assume if a = 1, υ becomes public knowledge at the end

of t = 0.16 In addition, I focus the analysis on the case where c is within the following bounds:

15The assumption that the bad project is NPV negative (Assumption 1.1.1) is not strictly necessary; however,it simplifies the exposition by ensuring it is without loss of generality to focus on a single contract rather than amenu. It also ensures that the lender’s participation constraint always binds at the optimum, which further limitsthe number of forms the optimal contract can take. In the Appendix, I characterize the optimal contract in the casewhere this assumption is relaxed and show the model’s main qualitative predictions regarding the use of short-termcontracts remain.

16This assumption eliminates the possibility of signaling problems in the outside financial market and is also madein Gorton and Ordonez (2014). All of the results go through if I relax this assumption and impose reasonableoff-equilibrium beliefs for the outside financial market.

10

Assumption 1.2. The cost of information acquisition c is between c and c, c ∈ (c, c), where

c ≡ (1− θ)(1− πhR− πlµR), c ≡ θ(1− φl)(1− πhR)

φl.

As shown below, this assumption implies that information acquisition is inefficient, but the cost of

information acquisition is sufficiently low to materially affect the financial contracting problem.

Outside Financial Market

At t = 1 the firm can raise additional funds from a competitive outside financial market. The firm

can raise funds by: i) new claims on the existing project’s output, i.e. securities, or ii) the sale of

a portion of its existing project, i.e. asset sales. For ease of exposition, I refer to asset sales as the

method of raising funds.17

Formally, at t = 1 in state z the firm sells a portion of its project qz ∈ [0, k] for a price

paz = (1 − γaz )E[R|F ] where γaz ∈ [0, 1] is a liquidation cost that can depend on the state and the

lender’s information acquisition decision and E[R|F ] is the expected value of the project given all

public information after z has been realized at t = 1. Although in principle there can be numerous

sources of the liquidation cost, a relevant friction for financial assets is adverse selection, which is

what I consider in Section 1.4.

1.2.2 Financial Contracts

At t = 0 the firm offers the lender a financial contract C, which consists of an investment level and

state-contingent liquidations and payments from the firm to the lender at each date. For simplicity,

I restrict focus on contracts in which the firm only raises funds for investment at t = 0 and does not

store funds across periods, which I show is without loss of generality in the Appendix. In addition,

Assumption 1.1.1 ensures it is without loss of generality for the firm to offer the lender a single

contract rather than a menu.18 Formally,

Definition 1.1. A financial contract C ≡ {k, qh, ql, d1h, d1l, d2h, d2l} consists of an investment level

17In the context of the model there is no difference between asset sales and issuing securities because of the project’sbinary payoff.

18If the lender acquires information and discovers the project is bad, the initial cost of financing cannot be recoupedin expectation regardless of the terms of the contract.

11

t = 0

1. Firm offers contract C

2. Lender acquires info about υ?

3. Lender accepts C?

t = 1

1. State z is realized

2. Liquidation qz

3. Payments d1z made

t = 2

1. Output R realized

2. Payments d2z made

Figure 1.3: Model Timing.

k, state-contingent liquidations qz and state-contingent payments d1z and d2z from the firm to the

lender at t = 1, 2, respectively for each state z.

After C has been offered, and before the lender accepts or rejects it, the lender decides whether to

acquire information; however, C cannot be made contingent on the lender’s information acquisition

decision a. As shown below, this friction implies that the lender will have an excessive tendency to

acquire information.

I assume that the firm cannot contract with the outside financial market at t = 0.19 Fur-

thermore, the firm and lender can commit to not renegotiate C.20 If the firm fails to make the

contractual payments at t = 1 the remainder of the project is liquidated in the outside financial

market and all proceeds are given to the lender. Since contracts are state-contingent, it is without

loss of generality to focus on cases where default does not occur in equilibrium. Figure 1.3 displays

the timing of the model.

1.2.3 Baseline Model Discussion

In this section I discuss some of the main features of the baseline model. An important assumption

is the firm and lender begin symmetrically uninformed. One rationale for this occurring in financial

markets is that lenders and firms both invest in the same types of assets. For instance, banks employ

traders and research analysts that have expertise in the same securities as the hedge funds they

19This assumption can be rationalized if the lender must incur an initial cost of due diligence which is too costlyfor the outside financial market to incur. A cost of due diligence can also rationalize the firm borrowing from a singlelender at t = 0 to avoid duplicative monitoring costs (e.g. Diamond (1984)).

20As shown in Section 1.4, the specific form of liquidation cost I focus on is information-based; hence, the lender’svalue of the asset at t = 1 coincides with that of the outside financial market, eliminating the incentive to refinancewith the lender at t = 1.

12

lend to. Nonetheless, in the Appendix I analyze the case in which the firm is endowed with private

information about its project quality and show the main qualitative results remain so long as the

initial information is not too precise.

What type of information could be socially inefficient to produce? While financial firms

certainly generate information about their own assets, certain pieces of information may not be

worth producing. For instance, a hedge fund would probably not pay their analysts to investigate

individual houses before purchasing a mortgage-backed security. As I show below, however, a bank

lending long-term to that hedge fund may want to investigate the houses. Consistent with this

example, I show in the Appendix that if the firm is also given the option to acquire information

about the project, the firm does not.

1.3 Model Analysis

1.3.1 Benchmark: Information Acquisition is Contractible

To gain intuition, I first analyze the case in which the lender’s information acquisition decision is

contractible. For information acquisition to be efficient, the value from acquiring information (i.e.

the avoiding financing the bad project) must offset its cost. Formally this occurs when

k(1− θ)(1− πhR− πlµR) ≥ c. (1.1)

Assumption 1.2 and k ≤ 1 imply that (1.1) does not hold. Therefore, information acquisition is

inefficient. Because information acquisition is contractible, the firm can induce the lender to not

acquire information by offering sufficiently low payments if the lender acquires information.21 The

21More specifically, the firm could offer a contract in which the lender breaks even from financing the good projectif a = 1. If a = 1 the lender’s profits would be −c, so the lender would never acquire information.

13

firm’s problem is

maxC

∑z

πz

[qzp

0z − d1z + φz ((k − qz)R− d2z)

](1.2)

s.t.

k ≤ 1, (1.3)

k ≤∑z

πz(d1h + φzd2h), (1.4)

qz ∈ [0, k], d1z ≤ qzp0z, d2z ≤ (k − qz)R z = h, l, (1.5)

p0z = (1− γ0

z )φzR z = h, l. (1.6)

The firm chooses a contract to maximize its expected profits subject to investment not exceeding

the maximum scale (1.3), the lender’s participation constraint (1.4), which states that the expected

payments the lender receives when it does not acquire information must be at least as large as the

firm’s initial investment, and the promised payments at t = 1 and t = 2 not exceeding the available

funds from liquidation and the project’s output (1.5). Finally, (1.6) reflects the asset prices in each

state when the lender does not acquire information.

Proposition 1.1. When information acquisition is contractible, the lender does not acquire infor-

mation and the optimal contract CFB is

CFB ≡ {kFB, qFBh , qFBl , dFB1h , dFB1l , d

FB2h , d

FB2l } = {1, 0, 0, 0, 0, dFB2h , d

FB2l },

where ∑z

πzφzdFB2z = 1,

dFB2z ∈ [0, R] z = h, l,

and the firm’s expected profits are

V FB = (πh + πlφl)R− 1.

Because the asset price in both states is always less than the expected output of the project

14

p0z = (1 − γ0

z )φzR ≤ φzR, liquidations reduce the firm’s expected profits. Hence there are no

liquidations or payments at t = 1. The firm also invests at full scale because investment is NPV

positive. Since the firm has all of the bargaining power, any feasible combination of d2h and d2l

that leads (1.4) to bind constitutes an optimal benchmark contract. The firm’s profits V FB equal

the expected value of the ex-ante, average project at the full investment scale, which I henceforth

refer to as the first-best level of surplus.

1.3.2 Information Acquisition Non-Contractible

In this section, I show that when information acquisition is non-contractible, if the firm offers the

lender CFB, the lender acquires information. In order for the lender to not acquire information,

the lender’s payoff from acquiring information must be less than its cost. Formally,

(1− θ) [k − πh(d1h + d2h)− πl(d1l + µd2l)] ≤ c. (1.7)

If we insert the terms of CFB into (1.7) we have

θ(1− φl)(1− πhdFB2h )

φl≤ c. (1.8)

The LHS of of (1.8) is at least as large as c because dFB2h ≤ R. However, this implies that (1.8)

is violated because Assumption 1.2 states that c is less than c. Thus, the lender would acquire

information if the firm offered the lender CFB.

Lemma 1.1. When information acquisition is non-contractible, if the firm offers the lender the

optimal benchmark contract CFB, the lender acquires information and accepts CFB if the project is

good (υ = g) and rejects it otherwise. The firm earns profits strictly less than V FB.

To gain more intuition for why the lender acquires information if offered CFB, compare the lender’s

private benefits to the social benefits of information acquisition for any contract in which there are

no t = 1 payments d1h = d1l = 0

(1− θ)(k − πhd2h − πlµd2l)︸ ︷︷ ︸Lender’s benefits

≥ k(1− θ)(1− πhR− πlµR)︸ ︷︷ ︸Social benefits

.

15

Because of limited liability, the contract must pay the lender less than the full output of the project

when it succeeds d2z ≤ kR. Intuitively, when deciding whether to produce information, the lender

does not internalize the loss in the firm’s expected profits from the bad project which can be seen

from subtracting the social benefits of information from the lender’s benefits

(1− θ)(k − πhd2h − πlµd2l)︸ ︷︷ ︸Lender’s benefits

− k(1− θ)(1− πhR− πlµR)︸ ︷︷ ︸Social benefits

= (1− θ) [πh(kR− d2h) + πlµ(kR− d2l)]︸ ︷︷ ︸Firm’s profits from bad project

.

Given that the project yields R when it succeeds regardless of its type, the lender’s private benefits

of information acquisition only coincide with the social benefits when the firm earns zero profits.

Assumption 1.2 ensures that c is small enough so that this misalignment of incentives materially

affects the financial contracting problem. As shown below, contracts with payments at t = 1 can

be a way to improve the incentives of the lender because they reduce the lender’s expected loss

from financing the bad project.

1.3.3 Second-Best Financial Contract

In this section, I characterize the optimal contract when information acquisition is non-contractible.

I separate the search for the optimal contract into two cases i) the optimal contract in which the

lender finds it optimal to not acquire information C0∗ and ii) the optimal contract in which the

lender finds it optimal to acquire information C1∗. I then compare the profits between C0∗ and

C1∗ to find the optimal contract C∗. Contracts which include payments at t = 1, I refer to as

“short-term”, while contracts that do not I refer to as “long-term”.22

Optimal Contract that Deters Information Acquisition

To find the optimal contract that deters information acquisition C0∗, the firm faces the benchmark

problem (1.2) with the addition of the lender’s incentive compatibility constraint (1.7). It is useful

to first establish the following lemma.

22This definition is consistent with regulatory standards where the maturity is tied to when the lender can demandrepayment versus the unconditional repayment date (see SEC (2019b)).

16

Lemma 1.2. In the optimal contract that deters information acquisition,

i) q0∗h = d0∗

1h = 0

ii) d0∗2h = k0∗R

iii) d0∗1l = q0∗

l p0l

iv) The incentive compatibility constraint (1.7) binds.

Liquidations and payments at t = 1 when z = h lead to liquidation costs but do not affect the

lender’s incentives (given that the project succeeds regardless of its type). The reason that all

output is paid to the lender in the high state, d0∗2h = k0∗R can be seen from the LHS of (1.8).

Fixing the expected payments to the lender at t = 2, the private benefits from the lender acquiring

information are decreasing in d2h. Hence, the optimal contract includes as large a value of d2h as

possible to minimize risky payments in the low state.23 Finally, the firm only liquidates enough of

the project to meet t = 1 payments in the low state because the incentives of the lender do not

improve if the firm keeps the proceeds from liquidation. The following proposition characterizes

C0∗.

Proposition 1.2. In the optimal contract that deters information acquisition the participation

constraint (1.4) binds and the contract and respective profits take two forms depending on parameter

23Since the bad project is NPV negative it must be the case that πhR < 1. Therefore, the contract must includepayments in the low state for the lender to break-even.

17

values:

C0∗ = C0L ≡ {k0L, q0Lh , q0L

l , d0L1h , d

0L1l , d

0L2h , d

0L2l }

=

{cφl

θ(1− φl)(1− πhR), 0, 0, 0, 0, k0LR,

c

θπl(1− φl)

},

V 0∗ = V 0L = k0L (πhR+ πlφlR− 1) ,

or

C0∗ = C0S ≡ {k0S , q0Sh , q0S

l , d0S1h , d

0S1l , d

0S2h , d

0S2l }

=

{1, 0,

d0S1l

p0l

, 0,1− πhR− cφl

θ(1−φl)

πl, R,

c

θπl(1− φl)

},

V 0∗ = V 0S = πl

(φl(R− d0S

2l

)−

d0S1l

1− γ0l

).

There are two potential channels to deter the lender from acquiring information. First, reducing

investment k lowers the expected payments required for the lender to break-even. This makes it

more difficult for the lender to recoup c through avoiding financing the bad project. Hence, when

C0∗ = C0L investment is reduced just enough so that the incentive compatibility constraint (1.7)

binds.

More centrally to the paper, holding the expected payments to the lender constant, shifting

payments from t = 2 to t = 1 in the low state, deters the lender from acquiring information.

Because payments at t = 1 are independent of the project’s quality, shifting payments to t = 1

decreases the lender’s expected loss from financing the bad project, which in turn lowers the lender’s

private value of information. Hence, for parameters in which C0∗ = C0S , the firm fully invests and

the contract includes the minimum payment d1l so that (1.7) binds.

In summary, the optimal contract that deters information acquisition either includes reduced

investment or interim liquidations and payments.

18

Optimal Contract that Induces Information Acquisition

To find the optimal contract that induces information acquisition C1∗, the firm faces the following

problem

maxC

θ∑z

πz(qzp

1z − d1z + (k − qz)R− d2z

)(1.9)

s.t.

k ≤ 1,

c ≤ θ

(∑z

πz(d1z + d2z)− k

), (1.10)

c ≤ (1− θ) [k − πh(d1h + d2h)− πl(d1l + µd2l)] , (1.11)

qz ∈ [0, k], d1z ≤ qzp1z, d2z ≤ (k − qz)R z = h, l,

p1z = (1− γ1

z )R z = h, l. (1.12)

The main differences between (1.9) and the problem that deters information acquisition are i)

the participation constraint (1.10) which reflects the lender’s payoff if it acquires information, ii)

the incentive compatibility constraint (1.11) which states that the lender must prefer to acquire

information and iii) asset prices (1.12) because the lender only finances the good project when

a = 1. As in the benchmark case, there are no benefits of liquidations, demanding payments at

t = 1 or reducing investment. Hence, (1.11) is slack (i.e. the lender will always acquire information)

and (1.10) binds and the firm captures the full surplus.

Proposition 1.3. The optimal contract that induces information acquisition takes the following

form:

C1∗ = C1L ≡ {k1L, q1Lh , q1L

l , d1L1h , d

1L1l , d

1L2h , d

1L2l } = {1, 0, 0, 0, 0, d1L

2h , d1L2l },

where

θ

(∑z

πzd1L2z − 1

)= c,

d1L2z ∈ [0, R] z = h, l.

19

The firm’s expected profits V 1∗ = V 1L where:

V 1L = θ(R− 1)− c.

The firm ultimately bears the cost of the lender’s inefficient information production and its profits

are strictly less than the first-best V 1L < V FB.

Optimal Contract

The optimal contract can be found by comparing the expected profits between the three classes of

contracts (C0S , C0L and C1L).

Proposition 1.4. Let V ∗ ≡ max{V 0S , V 0L, V 1L}. The optimal contract C∗ is:

C∗ =

C0S if V ∗ = V 0S

C0L if V ∗ = V 0L

C1L if V ∗ = V 1L

Figure 1.4 depicts example regions of the parameter space in which each of the candidate contracts

is optimal, varying the probability of the low state πl and the liquidation cost in the low state when

the lender does not acquire information γ0l . Henceforth, I refer to C0S as the short-term contract

and C0L and C1L as long-term contracts.

1.3.4 Comparative Statics and Discussion

In this section, I discuss comparative statics and particular features of the short-term contract.

Proposition 1.5. The lower the liquidation cost in the low state when the lender does not acquire

information γ0l , the more likely the short-term contract is optimal C∗ = C0S.

The profits from the short-term contract V 0S are decreasing in γ0l , while the profits of both of the

long-term contracts are unaffected by γ0l . Hence, the lower γ0

l the more likely the firm chooses the

short-term contract. In reality, financial assets likely have lower liquidation costs than real assets

which makes short-term debt a cost-effective method to deter inefficient information production for

20

COL

(Long-TermRed. Inv.)

C1L

(Long-TermAcquire)

COS

(Short-Term)

0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.490.00

0.05

0.10

0.15

0.20

0.25

πl

γl0

Figure 1.4: Optimal Contract. This figure depicts regions of the parameter space in whicheach of the candidate contracts is optimal (C0S , C0L and C1L) for the example parameters: θ = µ =0.5, R = 1.2, c = 0.05, varying πl and γ0

l .

financial firms. This result will also play an important role in market equilibrium where certain

shocks can cause an endogenous increase in the liquidation cost, which then affects firms’ decisions

to use short-term financing.

I now analyze the maturity of the short-term contract C0S where I refer to its maturity

being “shorter” if d0S1l is larger. A higher probability of the low state πl leads to a shorter maturity

of the short-term contract.24

Proposition 1.6. The maturity of the short-term contract is decreasing in the probability of the

low state, i.e. d0S1l is increasing in πl.

There is only uncertainty in payoffs across project types in the low state; thus, as πl increases, the

lender’s value of information increases. In order to induce the lender not to acquire information,

the interim payment in the low state must increase.25 Because firms must liquidate more assets

24This is also true if maturity is defined as the relative ratio of payments in the low stated0S1l

d0S2l

+d0S1l

.25To isolate the effect from higher uncertainty, one can easily show that this result holds if the NPV of the project

(πh + πlφl)R− 1 is held constant while πl varies.

21

as the interim payment increases, a higher probability of the low state also makes the short-term

contract less likely to be optimal as can be see in Figure 1.4. This result also plays a role in the

market equilibrium as the aggregate maturity structure will endogenously affect liquidation costs.

The payment from the firm to the lender at the t = 1 in the low state resembles the variation

margins used by financial firms which require counterparties to post cash collateral when the value

of their assets drops. This form of short-term debt is distinct from standard short-term debt

contracts that can be rolled over by any party (e.g. Flannery (1986) and Diamond (1991)). In my

setting it is key that the initial lender does not refinance the interim payment, otherwise the lender

would anticipate this and acquire information at t = 0. Hence, the particular form of short-term

financing that arises in this setting shares the feature of short-term loans used by financial firms in

practice.

1.4 Market Equilibrium

In this section I incorporate information acquisition in the outside financial market to endogenize

liquidation costs. Since there is no uncertainty in payoffs across project types in the high state, I

focus the analysis on information production in the low state at t = 1.

1.4.1 Market Equilibrium Setup

There is a unit mass of firms indexed by i that are distributed uniformly in the interval [0, 1].

Firms’ project qualities υ(i) are iid and have the same distribution as in the baseline model. Each

firm makes an offer to a single lender at t = 0 where a(i) refers to firm i’s lender’s information

acquisition decision at t = 0. The outside financial market is composed of a single deep-pocketed,

informed investor and a competitive fringe of uninformed investors.26 The timing is the same as

the baseline model; however the investor in the outside financial market can acquire information

at t = 1, which affects the endogenously determined liquidation costs.

After the low state has been realized at t = 1, the informed investor can randomly match

with a measure of η ≥ 0 firms by incurring κ(η) > 0, where κ(·) > 0 is a strictly increasing and

26I assume there is one informed investor rather than many to avoid a the potential for a multiplicity of equilibriumin the outside financial market (e.g. Fishman and Parker (2015) and Bolton, Santos, and Scheinkman (2016)) and tosimplify the exposition.

22

convex function of η. If the investor matches with firm i, both parties learn the firm’s project type

and the informed investor makes an offer to buy its assets for sale ql(i) for ql(i)pl(i). Matching and

offers are anonymous so the uninformed investors cannot observe which firms previously received

offers from the informed investor. Therefore, a firm that does not sell its asset to the informed

investor can sell its assets to the uninformed investors for the competitively determined market price

pal , where there can be different prices received by firms whose lenders have acquired information

and those that have not (i.e. p0l and p1

l ). Figure 1.5 displays the timing in the liquidation stage.27

I make the following assumptions regarding the cost function κ(·).

Assumption 1.3 (Cost Function).

1. κ′(1) <d0S

1l θ(1−φl)(1−θ(1−φ2l ))

φl(1−θ(1+(1−φl)φl))

2. κ′′(η) >d0S

1l θ2(1−φl)

(1−ηθ(1+(1−φl)φl))2 ∀ η

Assumption 1.3.1 says that investor would never become informed about all firms’ projects.28

Assumption 1.3.2 ensures that the informed investor’s profits are concave in η, which allows for

interior solutions.

Let me briefly discuss the role of the informed investor and uninformed investors. Informed

investors can be thought of as other financial firms with sufficient funds to purchase assets in the

low state. For instance Mitchell, Pedersen, and Pulvino (2007) show that multi-strategy hedge

funds become net buyers of convertible bonds when convertible arbitrage hedge funds become dis-

tressed. Uninformed investors can represent less sophisticated institutional investors. For example

Ben-David, Franzoni, and Moussawi (2012) find that insurance companies, pension funds and re-

tail investors stepped in to buy assets sold by hedge funds during the 2008/2009 financial crisis.

Although each firm has one project, it is best to think of firms liquidating individual assets within

their portfolio to meet interim payments and the informed investor producing information about

these individual assets rather than the portfolio as a whole.

27The discussed mechanisms are a stylized representation of how trading occurs in reality. Financial assets tradein several different ways. For example most stocks trade in organized exchanges, convertible bonds trade in dealermarkets and structured products may trade in even less transparent ways through bilateral agreements. The keyingredient is that the informed investor is able to distinguish good assets from bad when they buy them at the marketprice. In the Appendix, I show that an auction yields the same asset prices as the market mechanism in the maintext.

28This assumption is not strictly necessary but simplifies the exposition so that I can ignore corner solutions inwhich all investors become informed.

23

4. Any unsold assets are sold to uninformed for pal

3. Informed investor make offers to buy firms’ assets

2. Informed investor incurs κ(η) to the quality of η projects

1. State z = l is realized

Figure 1.5: Liquidation Stage Timing at t = 1.

1.4.2 Market Equilibrium Analysis

In this section I characterize equilibrium asset prices, investors’ decisions to become informed and

firms’ financial contracts. In the high state both good and bad projects payoff R with certainty;

hence, there is no potential for adverse selection and p0h = p1

h = R. Consider I0 the set of firms

in which their lenders do not acquire information at t = 0 and I1 the set of firms in which their

lenders do acquire information

I0 ≡ {i : a(i) = 0}, I1 ≡ {i : a(i) = 1}.

Firms i ∈ I1 are financed only if the project is good, implying that the uninformed are willing

to pay p1l = R. Therefore, if the informed investor matches with a firm i ∈ I1 in the low state,

the informed investor offers pl(i) = p1l = R and the firm accepts. If a firm i ∈ I1 does not match

with the informed investor, it sells its assets to the uninformed for p1l = R. Hence, regardless of

matching outcomes all firms i ∈ I1 receive p1l = R.

The remaining price to characterize is p0l . Since the informed investor’s matching and offers

are anonymous, uninformed investors cannot distinguish the quality of individual assets in the pool

sold by firms i ∈ I0. Therefore, p0l ∈ [pR,R] where p0

l depends on the inferred proportion of good

assets sold to the uninformed. Consider a firm i ∈ I0 that matches with the informed investor. If

its project is good its expected payoff R is always greater than p0l . Since each firm’s outside option

is p0l , the informed investor offers pl(i) = p0

l and the firm accepts. In contrast, if the project is

bad its expected payoff is pR which is always less than p0l .

29 Thus, the firm rejects any offer the

29I confirm in equilibrium that the inequality is strict.

24

Table 1.1: Liquidation Costs

a = 1 a = 0

z = h γ1h = 0 γ0

h = 0

z = l γ1l = 0 γ0

l = ηθ(1−φl)(1−ηθ)φl ≥ 0

informed investor makes and sells the asset to the uninformed investors for p0l .

30 A firm that does

not match with the informed investor also sells its assets to the uninformed for p0l . Summarizing,

Lemma 1.3.

1. If the informed investor matches with a firm i ∈ I1 at t = 1 it buys the asset at p1l = R

2. If the informed investor matches with a firm i ∈ I0 at t = 1,

(a) it buys the asset at p0l if the asset is good

(b) it does not buy the asset otherwise.

3. Any assets that go unsold to the informed investor are purchased at pal by the uninformed

investors

Lemma 1.3 implies that the market price received by firms i ∈ I0 is

p0l (η) =

(1− ηθ(1− φl)

(1− ηθ)φl︸ ︷︷ ︸γ0l (Adverse selection)

)φlR,︸︷︷︸

Expected value

(1.13)

where p0l (η) is decreasing in η as a larger portion of low quality assets flow to the uninformed due to

the informed investor “cream skimming” good assets. Table 1.1 summarizes the liquidation costs

in each state borne by firms whose lenders acquire and do not acquire information.

Next, I turn to the informed investor’s decision to produce information taking firms’ con-

30Note that even if the firm does not keep any of the proceeds itself, it would default if pRql(i) < d1l(i) whichmeans it always prefers selling the assets for a higher price.

25

tracts as given. Define the aggregate asset sales and payments made by firms i ∈ I0

Q ≡∫I0

ql(i)di, D ≡∫I0

d1l(i)di.

The expected payoff from the informed investor producing information about η firms is

Π(η) =

∫ η

0

(θQ(R− p0

l (η))︸ ︷︷ ︸

Information rents

dη)− κ(η). (1.14)

If the the informed investor matches with a firm i ∈ I1 it pays p1l = R for ql(i) units of the asset

and earns zero profits. If the informed investor matches with a firm i ∈ I0 there is a θ probability

the firm’s asset is good in which case the investor pays p0l (η) for ql(i) units of the asset that yield

R with certainty where (1.14) integrates over all firms i ∈ I0. From Lemma 1.2, d0∗1l = q0∗

l p0l , so

for convenience (1.14) can be written as

Π(η) =

∫ η

0θD

(R

p0l (η)

− 1

)dη − κ(η). (1.15)

Assumption 1.3.2 ensures that the informed investor never produces information about all

assets and 1.3.2 ensures that (1.15) is concave in η.31 If Π(0) < 0 then the informed investor

does not produce information about any projects η∗ = 0. Otherwise, η∗ solves Π′(η∗) = 0. It

immediately follows that η∗ is increasing in D. Formally,

Lemma 1.4. Shorter aggregate maturities induce the informed investor to produce information

about a larger number of projects, i.e. dη∗

dD ≥ 0, where the inequality is strict when η∗ > 0.

Shorter aggregate maturities in turn affect equilibrium asset prices which can be seen by differen-

tiating p0l (η∗) with respect to D

dp0l (η∗)

dD=∂p0

l

∂η∗dη∗

dD≤ 0. (1.16)

The first term is negative and dη∗

dD is positive, and strictly positive when η∗ > 0, from Lemma 1.4.

In summary when η∗ > 0, as D increases, the informed investor produces information about more

31This can clearly be seen by replacing D with the largest amount of t = 1 aggregate payments in the low stated0S

1l .

26

projects, which causes p0l (η∗) drops. Hence, the informed investor’s information production yields

a downward sloping demand curve in the asset market.

Now that I have established the main properties of the liquidation stage, I can define the

market equilibrium.

Definition 1.2. A market equilibrium consists of contracts C(i) for all i, information acquisition

decisions by each firm’s lender a(i) for all i, a mass of firms η∗ that the informed investor produces

information about at t = 1 when z = l, and asset prices {paz}z=h,l, a=0,1, such that:

1. Firms’ contracts are optimal C(i) = C∗ for all i

2. Lenders information acquisition decisions are optimal given contracts

3. The informed investor’s decision to produce information is optimal given firms’ contracts and

asset prices

4. Asset markets clear in each state.

The following definitions will be useful for characterizing the equilibrium.

Definition 1.3. The long-term contract that yields the highest profits and its corresponding profits

are

CL∗ =

C0L if V 0L ≥ V 1L

C1L otherwise,

and

V L∗ = max{V 0L, V 1L}.

I also define aggregate short-term debt α as the mass of firms that borrow short-term which is

equivalent to the amount of investment funded by short-term contracts since k0S = 1 in the short-

term contract. Formally,

Definition 1.4. Aggregate short-term debt α is

α ≡∫ 1

0I(C(i) = C0S)di.

27

Proposition 1.7. The equilibrium mass of firms using short-term contracts α∗, the mass of firms

the informed investor produces information about η∗ and liquidation costs in the low state γ0∗l borne

by firms i ∈ I0 are one of the following types depending on the parameter values

Type 1: α∗ = 1, η∗ = 0 and γ0∗l = 0

Type 2: α∗ = 1, η∗ > 0 and γ0∗l > 0

Type 3: α∗ ∈ (0, 1), η∗ > 0 and γ0∗l > 0,

where in all equilibrium types a mass 1− α∗ of firms choose CL∗.

In the Type 1 equilibrium all firms use short-term contracts and the informed investor does not

produce information about any firms which results in a liquidation cost of zero in the asset market.

In the Type 2 equilibrium all firms use short-term contracts and a positive mass of investors

become informed, where all firms still find it optimal to use short-term contracts because liquidation

costs are not too high. Finally, in the Type 3 equilibrium, if all firms chose short-term contracts,

liquidation costs would be so high that it would no longer be optimal for any firms to use short-term

contracts. When this is the case, a mass of firms less than 1 choose short-term contracts such that

firms are indifferent between the short-term contract and the long-term contract that yields the

highest profits.

When the long-term contract that deters information is more profitable than the long-term

contract that induces information acquisition CL∗ = C0L, there are infinitely many equilibria such

that all firms’ profits equal V 0L.32 However, D∗ and γ0∗l are the same regardless of the equilibrium.

Therefore, for convenience I focus on the case where firms simply choose between C0S and C0L.

Nonetheless, I characterize the full set of equilibria in the proof of Proposition 1.7.33

To understand the real consequences of the market equilibrium, I define aggregate investment

as the sum of realized investment across firms accounting for i) firms that choose C0L invest less

than 1 and ii) firms that choose C1L are only financed when the project is good which occurs with

probability θ. Formally,

32For example there is also a symmetric equilibrium in which all firms choose d1l > 0 and reduce their investmentk < 1.

33The notion of aggregate short-term debt can potentially vary based on the choice of equilibrium (i.e. the amountof investment funded by contracts with payments at t = 1 in the low state); however, the comparative statics aredirectionally the same across equilibria.

28

Definition 1.5. Aggregate investment K is:

K =

α+ (1− α)I0L if CL∗ = C0L

α+ (1− α)θ otherwise.

Therefore, equilibrium aggregate investment K∗ can be characterized in the follow corollary.

Corollary 1.1. If the equilibrium is Type 1 or Type 2, K∗ = 1, otherwise K∗ < 1.

Hence, if the adverse selection in the asset market is severe enough aggregate investment becomes

impaired through a portion of firms resorting to long-term financing.

1.4.3 The Effect of Uncertainty on Aggregate Short-Term Debt and Investment

In this section I show how an increase in uncertainty can lead to a reduction in aggregate short-

term debt and investment. Recall that the only uncertainty in payoffs across project types occurs

in the low state and from Proposition 1.6, the maturity of the short-term contract is decreasing

in the probability of the low state, i.e. d0S1l is increasing in πl. Hence, in the market equilibrium

whenever α∗ = 1 (i.e. Type 1 or Type 2 equilibrium) an increase in πl leads to an increase in

D∗. In the Type 2 equilibrium, higher aggregate payments at t = 1 in the low state lead to an

increase in the mass of investors that become informed causing γ0l to increase. If πl becomes large

enough the equilibrium can switch from Type 2 to Type 3 where some firms must use long-term

contracts, resulting in a drop in aggregate investment and short-term debt. This is summarized in

the following proposition.34

Proposition 1.8. Aggregate investment K∗ and short-term debt α∗ are decreasing in πl.

Although I have only considered exogenous changes in πl, other changes in parameters that increases

the value of information for lenders lead to an increase in d0S1l which in turn increases adverse

selection in the asset market.

Proposition 1.8 can help explain why short-term funding markets periodically become im-

paired. These episodes are often associated with increases in uncertainty and reduced maturities

of short-term debt (e.g. the 2008/2009 financial crisis).

34This is true even if the NPV of the project (πh + πlφl)R− 1 is held constant while πl varies.

29

1.4.4 Welfare

In this section I ask whether firms’ financing decisions in the market equilibrium are efficient. I do

so by considering a concept of constrained efficiency in which a planner can manipulate the mass of

firms using short-term contracts α to maximize welfare, but cannot directly affect the information

production decisions of lenders and the informed investor. For simplicity, I follow Gromb and

Vayanos (2002) by considering an infinitesimal change in α. Total welfare W (α) is the sum of firm

and the informed investor’s profits

W (α) ≡ αV 0S + (1− α)V L∗ + πlΠ(η∗).

Differentiating W (α) with respect to α

dW (α)

dα= V 0S − V L∗ + α

dV 0S

dα+dΠ(η∗)

dα. (1.17)

The first two terms reflect the difference in profits between firms using short-term contracts and

long-term contacts. In the Type 2 equilibrium this is strictly positive, while in the Type 3 equi-

librium it equals zero because firms are indifferent between short and long-term contracts. The

last two terms reflect the two externalities that individual firms do not take into account when

determining their own maturity structures. The third term is the price impact of firms’ short-

term contracts on asset prices which is negative because the demand curve is downward sloping

from (1.16). Finally, the last term is the impact of firms’ short-term contracts on the informed

investor’s profits, which is positive because shorter aggregate maturities allow the informed investor

to buy more assets at a cheap price. The following proposition characterizes the efficiency of the

equilibrium.

Proposition 1.9 (Inefficient Short-Term Debt). The efficiency of each equilibrium type is as follows

1. The Type 1 equilibrium is always constrained efficient

2. The planner can raise welfare in the Type 2 equilibrium by reducing α below α∗ = 1 when

30

demand curve is sufficiently downward sloping

dp0l (η∗)

dα∗< −

p0l (η∗)(d0S

1l Rη∗θ + (V 0S − V L∗ − d0S

1l η∗θ)p0

l (η∗))

d0S1l Rα

∗ (φl − η∗θ).

3. The planner can raise welfare in the Type 3 equilibrium by reducing α below α∗ < 1 when the

demand curve is sufficiently downward sloping

dp0l (η∗)

dα∗< −

η∗θ(R− p0l (η∗))p0

l (η∗)

α∗R (φl − η∗θ),

and raise welfare by increasing α above α∗ < 1 otherwise.

In the Type 1 equilibrium there is no information production at any point and the first-best is

achieved W (α∗) = V FB. However, the Type 2 and Type 3 equilibrium are inefficient if the demand

curve is sufficiently downward sloping. Intuitively, the demand curve slopes downward because

shorter aggregate maturities induce information production in the asset market, which is purely

wasteful from a social perspective. Hence, the planner would like to avoid this if possible. However,

fixing the amount of information the informed investor produces, higher levels of short-term debt

also lead to higher profits from more firms facing an adverse selection discount. While individual

firms would like to avoid the adverse selection discount, it is irrelevant to the planner because it has

no effect on total welfare. The relative magnitude of these two effects determines the efficiency of

the Type 2 and Type 3 equilibrium. In the Type 2 equilibrium there can only be too much short-

term debt since all firms are using short-term contracts (α∗ = 1), while in the Type 3 equilibrium

there can be too little short-term debt when the demand curve sufficiently flat.

Proposition 1.9 can rationalize taxes of short-term debt (i.e. repo contracts or margin loans)

in good times (Type 2 equilibrium) and subsidizing short-term debt when short-term debt markets

become impaired (Type 3 equilibrium). In practice, the latter situation may occur in periods when

adverse selection is already severe and the cost of producing information about marginal projects is

high or informed investors have already devoted substantial capital to investing in distressed assets.

A caveat to my normative analysis is that I have only considered the inefficiency that may

arise from short-term debt through pecuniary externalities in asset markets. There may also be

coordination problems that arise between creditors (e.g. He and Xiong (2012) and Brunnermeier

31

and Oehmke (2013)). In addition, when regulating debt maturity and choosing monetary policy,

policymakers must distinguish between short-term debt demanded by investors with liquidity needs

(e.g. Diamond and Dybvig (1983), Stein (2012) and Diamond (2016)) and short-term debt being

used to prevent inefficient information production by lenders. In Stein (2012), the government can

reduce the incentives of financial institutions to issue safe securities by issuing them itself. This

policy would not have an effect in the context of my model because firms borrow short-term for

reasons unrelated to the demand for safe assets.

In practice, regulators can also directly intervene in asset markets. For example if the a

regulator committed to buying all assets at t = 1 in the low state at R, there would be no incentive

for investors to become informed and the first-best would be achieved. However, asset market

interventions may have other costs such as moral hazard (e.g. Farhi and Tirole (2012a) and Lee

and Neuhann (2017)) or direct costs of intervention. Therefore, I leave the consideration of these

types of interventions for future work.

1.5 Conclusion

In this paper, I propose a new rationale for the widespread use of short-term debt by non-bank

financial firms. In particular, I argue that short-term financing allows financial firms to avoid

excessive information production by their financiers. This problem may be particularly severe for

financial firms that borrow heavily from financial institutions, such as banks, that have expertise

in the same types of assets.

Although short-term financing deters information production at origination, it leads to ex-

cessive information production when firms liquidate to repay their initial lenders following a negative

shock. This delay in information production endogenously raises the cost of short-term financing

through information-based fire sales. Moreover, when market conditions deteriorate, short-term

funding markets can become impaired, i.e. short-term debt maturities shorten, while the volume of

total credit decreases. These implications are consistent with the behavior of numerous short-term

debt markets in the 2008/2009 financial crisis.

From a welfare perspective, my analysis rationalizes 1) policymakers’ concerns that financial

firms’ are overly reliant on short-term debt as opposed to detailed credit analysis (Basel Commit-

32

tee on Banking Supervision (1999)) in normal times and 2) short-term debt markets should be

supported when they malfunction in financial crises.

Future empirical work could directly test whether shorter debt maturities reduce information

production by lenders, while inducing information production in the asset market. In addition,

testing how sensitive asset prices are to short-term debt through deferring information production

may be useful for policymakers in determining whether or not short-term funding markets should

be curbed or supported.

33

1.6 Appendix

1.6.1 Data Description for Figure 1.1

The hedge fund data is from SEC (2019a) where short-term debt is defined as debt with a maturity

under 365 days (Table 48) and Debt/Assets is derived from Borrowing/NAV (Figure 4a). The

mortgage REIT data is from Pellerin, Sabol, and Walter (2013a) where short-term debt is defined

as repo agreements. The mortgage originator debt maturity data comes from Kim et al. (2018)

where they define short-term debt as warehouse loans (typically under a year maturity) and “other

forms of short-term”. Mortgage originator leverage is defined as secured debt to tangible assets and

is taken from a poll of 10 firms by Moody’s (Moody’s Investor Service, 2016). Industrial firms are

from Compustat fiscal year-end 2018 and short-term debt is defined as debt with maturity under

one year. I remove any financial firms within the SIC range of 6000-6999 and exclude firms with

leverage greater than 1.

34

1.6.2 Proofs

Proof of Proposition 1.1. Denote CFB ≡ {kFB, qFBh , qFBl , dFB1h , dFB1l , d

FB2h , d

FB2l } the optimal bench-

mark contract. To show qFBh = 0, suppose to the contrary that qFBh > 0. First suppose that

dFB1h = 0. If the firm reduces qFBh by any ε > 0 where qFBh − ε ≥ 0, (1.4) and (1.5) would not be

violated while the firm’s profits (1.2) would increase because γ0h ≥ 0, which contradicts that CFB

is optimal. Next suppose that dFB1h > 0. If the firm reduces qFBh by any ε > 0 where qFBh − ε ≥ 0,

decreases dFB1h by εp0h

and increase dFB2h by ε, (1.4) and (1.5) would not be violated and (1.2) would

increase. The same steps can be used to show that qFBl = 0. Since qFBh = qFBl = 0, (1.5) implies

that dFB1h = dFB1l = 0.

To show that kFB = 1, suppose to the contrary kFB < 1. If the firm increases kFB by any

ε > 0 where kFB + ε ≤ 1, increases dFB2h by εR and increases dFB2l by 1−πhRπlφl

, (1.4) and (1.5) would

not be violated and the firm’s profits (1.2) would increase. Therefore kFB = 1. We can rewrite the

problem as follows

maxC

∑z

πzφz(R− d2h),

s.t.

1 ≤∑z

πzφzd2z, (1.18)

d2z ∈ [0, R] z = h, l. (1.19)

Since the firm has all of the bargaining power (1.18) binds with equality. Thus any contract that

leads (1.18) to bind and satisfies (1.19) constitutes an optimal benchmark contract and the firm

earns profits

V FB = (πh + πlφl)R− 1.

�

Proof of Lemma 1.1. From (1.8) the lender would produce information if offered CFB. The

firm’s profits would be

θ

(∑z

πz(R− d2z)

). (1.20)

Note that the benchmark optimal contract that makes (1.20) largest while satisfying (1.18) is

d2h = R and d2l = 1−πhRπlφl

which yields

θ (πhR+ πlφlR− 1)

φl. (1.21)

However, (1.21) is strictly less than V FB because θφl< 1 . �

35

Proof of Lemma 1.2. Denote C0∗ ≡ {k0∗, q0∗h , q

0∗l , d

0∗1h, d

0∗1l , d

0∗2h, d

0∗2l } the optimal contract that

deters information acquisition. The same steps from the proof of Proposition 1.1 can be used to

show that q0∗h = d0∗

1h = 0. To show that d0∗1l = q0∗

l p0l , suppose to the contrary q0∗

l p0l > d0∗

1l . If the

firm reduces q0∗l by an ε > 0 where (q0∗

l − ε)p0l ≥ d0∗

1l , (1.5) would not be violated while the firm’s

profits (1.2) would increase because γ0l ≥ 0. Therefore d0∗

1l = q0∗l p

0l . The problem can then be

rewritten as

maxC

πh(kR− d2h) + πlφl

((k − d1l

p0l

)R− d2l

),

s.t.

k ≤ 1,

k ≤ πhd2h + πl(d1l + φld2l), (1.22)

(1− θ)(k − πhd2h − πl(d1l + µd2l)) ≤ c, (1.23)

d2h ≤ kR, (1.24)

d2l ≤(k − d1l

p0l

)R. (1.25)

Next we can prove the incentive compatibility constraint (1.23) binds. By Lemma 1.1, (1.23) is

violated if the following three conditions are true: i) d1l = 0, ii) k = 1 and iii) the participation

constraint (1.22) binds; hence at least one of the above conditions do not hold. I then prove by

contradiction that (1.23) binds in each of these three cases.

Case 1. d0∗1l > 0

Suppose to the contrary that d0∗1l > 0 and (1.23) is slack. There exists a small enough ε > 0 such

that if the firm decreases d0∗1l by ε and increases d0∗

2l by εφl

, then (1.22), (1.23) and (1.25) would not

be violated. This would reduce q0∗l which causes the firm’s profits to increase because γ0

l ≥ 0.

Case 2. k0∗ < 1

Suppose to the contrary that k0∗ < 1 and (1.23) is slack. There exists a small enough ε > 0 such

that if the firm increases k0∗ by ε and increases d0∗2h by εR and increases d0∗

2l by 1−πhRπlφl

, then (1.22),

(1.23), (1.24) and (1.25) would not be violated, while the firm’s profits would increase.

Case 3. The participation constraint (1.22) is slack

Suppose to the contrary (1.22) and (1.23) are slack. There exists a small enough ε > 0 such that if

the firm decreased d0∗2h, d0∗

1l , or d0∗2l by ε, then (1.22), (1.23), (1.24) and (1.25) would not be violated,

while the firm’s profits would increase. These three cases imply that the incentive compatibility

constraint must bind.

Finally to prove d0∗2h = k0∗R, suppose to the contrary d0∗

2h < k0∗R. If the firm increases d0∗2h

by ε > 0 where d0∗2h + ε ≤ k0∗R and decreases d0∗

2l by εφl

, then (1.22), (1.24) and (1.25) would not

36

be violated and (1.23) would slacken. However, as just shown (1.23) must bind at the optimum,

which contradicts C0∗ being optimal. �

Proof of Proposition 1.2. Following Lemma 1.2 and ignoring (1.25) for now, the problem can

be written as

maxC

πlφl

[(k − d1l

p0l

)R− d2l

],

s.t.

k ≤ 1, (1.26)

k(1− πhR) ≤ πl(d1l + φld2l), (1.27)

(1− θ) [k(1− πhR)− πl(d1l + µd2l)] = c. (1.28)

The Lagrangian is

L = πlφl

[(k − d1l

p0l

)R− d2l

]− λ1

[k(1− πhR)− πl(d1l + φld2l)

]−

λ2

[(1− θ) (k(1− πhR)− πl(d1l + µd2l))− c

]− λ3(k − 1).

The Kuhn-Tucker necessary conditions are

Ld1l=πl(p0l (λ1 + (1− θ)λ2)− φlR

)p0l

≤ 0,

Ld2l= πl

[φl (1− λ1)− µ(1− θ)λ2

]≤ 0,

Lk = πlφlR− (1− πhR)(λ1 + (1− θ)λ2)− λ3 ≤ 0,

Ld1ld1l = 0, Ld2l

d2l = 0, Lkk = 0,

λ1

[k(1− πhR)− πl(d1l + φld2l)

]= 0,

(1− θ)[k(1− πhR)− πl(d1l + µd2l)

]− c = 0,

λ3(k − 1) = 0,

λ1 ≥ 0, λ2 > 0, λ3 ≥ 0,

(1.27), (1.28), (1.26).

There are many potential solutions to the system of equations; however, several can be eliminated

immediately upon inspection. After doing so we are left with three potential solutions

k = 1, d1l =1− πhR− cφl

θ(1−φl)

πl, d2l =

c

(1− φl)θπl, (1.29)

λ1 =φl(p

0l − µR)

(1− µ)θp0l

, λ2 =φl(φlR− p0

l )

(1− φl)θp0l

, λ3 =φlR

[πlp

0l − (1− πhR)

]p0l

.

37

k =φlc

(1− φl)θ(1− πhR), d1l = 0, d2l =

c

(1− φl)θπl, (1.30)

λ1 =φl(1− πhR− πlµR)

(1− µ)θ(1− πhR), λ2 =

φl(πhR+ πlφlR− 1)

(1− φl)θ(1− πhR), λ3 = 0.

k = 1, d1l = 0, d2l =1− c− θ − πhR(1− θ)

µ(1− θ)πl, (1.31)

λ1 = 0, λ2 =φl

(1− θ)µ, λ3 =

φl(πhR+ πlµR− 1)

µ.

First, (1.31) can be ruled out because λ3 < 0 from Assumption 1.1.1. In order for (1.29) to be a

solution, it must be the case that γ0l ≤ 1 − 1−πhR

πlφlRso that λ3 ≥ 0. (1.30) is a potential solution

because λ1 > 0 by Assumption 1.1.1. However, Ld1l> 0, only if γ0

l > 1 − 1−πhRπlφlR

. Therefore, if

γ0l ≤ 1− 1−πhR

πlφlR(1.29) is the solution while (1.30) is the solution otherwise. Also, note that (1.25)

is not violated in any of the candidate solutions. �

Proof of Proposition 1.3. Denote C1∗ ≡ {k1∗, q1∗h , q

1∗l , d

1∗1h, d

1∗1l , d

1∗2h, d

1∗2l } the optimal contract

that induces information acquisition. The same steps from the proof of Proposition 1.1 can be

used to show there are no liquidations or payments at t = 1 and k1∗ = 1; however, in all cases

(1.11) must remain slack which is always true because of Assumption 1.2. We can then rewrite the

problem as follows

maxC

θ∑z

πz(R− d2z), (1.32)

s.t.

θ

(∑z

πzd2z − 1

)≥ c, (1.33)

d2z ∈ [0, R] z = h, l. (1.34)

Since the firm has all of the bargaining power (1.33) binds. Therefore any contract that leads (1.33)

to bind and satisfies (1.34) constitutes an optimal contract. The expected profits can be found by

plugging the terms of C1∗ into (1.32). �

Proof of Proposition 1.4. The proof comes immediately from comparing the profits from C0∗

and C1∗ from Propositions 1.2 and 1.3.

�

Proof of Proposition 1.5. Differentiating V 0S with respect to γ0l we have

cφlθ(1−φl) + πhR− 1

(1− γ0l )2

< 0.

38

V 0L and V 1L do not vary with γ0l , hence the lower γ0

l the more likely C∗ = C0S . �

Proof of Proposition 1.6. Differentiating d0S1l with respect to πl we have

R− 1 + cφlθ(1−φl)

π2l

> 0.

Notice this also true if we define maturity asd0S

1l

d0S1l +d0S

2l

∂(

d0S1l

d0S1l +d0S

2l

)∂πl

=cθR

(1− φl) (c+ θ (1− πhR))2 > 0.

�

Proof of Lemma 1.3. See text. �

Proof of Lemma 1.4. To see the effect of shorter maturities on the equilibrium number of in-

formed investors we can apply the implicit function theorem

dη∗

dD=

0 if Π(0) < 0

−∂Π(η∗)∂η∗

Π(η∗)∂D

=θ(1−φl)(1−ηθ−ηθ(1−φl)φl)(1−ηθ+ηθφ2

l )φl((1−ηθ−ηθ(1−φl)φl)2κ′′(η)−Dθ2(1−φl))

if Π(0) ≥ 0,(1.35)

Focusing on −∂Π(η∗)∂η∗

Π(η∗)∂D

, the numerator is always positive and the denominator is positive from As-

sumption 1.3.2. Hence dη∗

dD ≥ 0 and the inequality is strict when Π(0) > 0. �

Proof of Proposition 1.7. For convenience, define the set of contracts that deter the lender from

producing information.

C0 ≡ {C : a = 0}.

Let V 0S(D) denote the profits from the short-term contract and Π(η,D) the informed investor’s

profits each as a function of the aggregate payments at t = 1 in the low state D. I break the proof

into three cases.

Case 1. Π(0, d0S

1l

)< 0

If α = 1 no investors would find it optimal to become informed η = 0. Hence, η∗ = 0, γ0∗l = 0 and

α∗ = 1 (from Proposition 1.4).

Case 2. Π(0, d0S

1l

)≥ 0 and V 0S(d0S

1l ) ≥ V L∗

Since Π(0, d0S

1l

), if α = 1 then a positive mass of investors η∗ > 0 would find it optimal to become

informed resulting in a positive liquidation cost γ0∗l > 0. However, the resulting profits from the

39

short-term contract V 0S(d0S1l ) are sufficiently high enough such that all firms still find it optimal to

choose C0S by Proposition 1.4. Hence α∗ = 1, η∗ > 0, and γ0∗l > 0.

Case 3. V 0S(d0S1l ) < V L∗

If α = 1, the liqudation cost γ0l would be so high that no firms would find it optimal to choose

C0S . However, if α = 0, then γ0l = 0 and all firms would find it optimal to choose C0S . Thus, all

firms must choose contracts C(i) such that they are indifferent between C(i) and the most profitable

long-term contract V (i) = V L∗ for all i.

Lemma 1.5. In Case 3 V (i) = V L∗ for all i.

Proof. Suppose to the contrary there is some contract C′ used in equilibrium in which V ′ 6= V L∗.

First it can never be the case that V ′ < V L∗ by Proposition 1.4. Now suppose that V ′ > V L∗. Since,

C1L = C1∗, if V ′ > V L∗ then C′ ∈ C0. Since V ′ > V 0L it must be that C′ = C0S by Proposition 1.2.

However, if V 0S > V L∗, then all firms would choose C(i) = C0S , which is a contradiction because

V 0S(d0S1l ) < V L∗. Therefore, V (i) = V L∗ for all i in Case 3. �

There are two sub-cases within Case 3.

Subcase 1. CL∗ = C1L

First note that it cannot be that C(i) = C1L for all i because if this were true γ0l = 0 which would

induce all firms to choose C0S . Therefore there must be some contract C′ ∈ C0 used in equilibrium

with profits V ′ where from Lemma 1.5, V ′ = V 1L. From Proposition 1.2, it must be the case that

C′ = C0S . Thus, the only contracts that are used in equilibrium are C0S and C1L where α is the

fraction of firms that choose C0S . Hence, the following equation characterizes α∗

G(α∗d0S1l ) = V 0S

(α∗d0S

1l

)− V 1L = 0.

First note that α∗ 6= 1 because V 0S(d0S1l ) < V 1L. Define α as the largest α such that no investors

become informed η∗ = 0

α = arg maxα

α s.t. η∗ = 0.

Since V 0S(αd0S

1l

)= V FB > V 1L, it must be that α∗ ∈ (α, 1). In this region γ0

l is strictly increasing

in α from (1.35). Hence, V 0S(αd0S

1l

)is strictly decreasing in α and there exists a unique α∗ ∈ (0, 1)

such that G(α∗d0S1l ) = 0.

Subcase 2. CL∗ = C0L

From Lemma 1.5 V (i) = V 0L for all i. It will be useful to establish the following lemma

Lemma 1.6. In Case 3 when CL∗ = C0L, V 0S(D∗) = V 0L.

40

Proof. Suppose V 0S(D∗) < V 0L then from Proposition 1.2 all firms to choose C∗ = C0L. However,

if this is the case γ0l = 0 which is a contradiction. Suppose V 0S(D∗) > V 0L then from Proposition

1.2 all firms would choose C∗ = C0L. However, if this is the case V 0S(d0S1l ) < V 0L which is also a

contradiction. �

Lemma 1.6 implies that the Lagrange multiplier (λ3) on the constraint k ≤ 1 for the candidate

solution (1.29) equals zero. Hence, there can potentially be other contracts C ∈ C0 other than C0S

and C0L used in equilibrium. The following conditions characterize the equilibrium

k(i)(1− πhR) = πl (d1l(i) + φld2l(i)) ∀i,

(1− θ)[k(i)(1− πhR)− πl (d1l(i) + µd2l(i))

]= c ∀i,

d1l(i) ≥ 0 ∀i,

V (i;D∗) = V 0L ∀i.

Each firm’s participation constraint and incentive compatibility constraint bind and all firms must

earn profits equal to the long-term contract with reduced investment given D∗. Further simplifying,

d2l(i) =c

θπl(1− φl)∀i,

k(i) =cφl + d1l(i)(1− φl)θπl

(1− φl)θ(1− πhR)∀i, (1.36)

d1l(i) ≥ 0 ∀i,

V (i;D∗) = V 0L ∀i.

Integrating (1.36) over i,

K∗ =

∫ 1

0k(i)di =

cφl +D∗(1− φl)θπl(1− φl)θ(1− πhR)

.

Since K∗ and D∗ do not depend on the exact distribution of firms’ contracts, I focus on the

case where firms choose between C0S and C0L. Therefore, the following equation characterizes the

equilibrium

G(α∗d0S1l ) = V 0S

(α∗d0S

1l

)− V 0L = 0.

As in the previous sub-case, V 0S(αd0S

1l

)is strictly decreasing in α when α ∈ [α, 1]. Therefore, there

exists a unique α∗ ∈ (0, 1) such that G(α∗d0S1l ) = 0. �

Proof of Corollary 1.1. From Proposition 1.7 in the Type 1 and 2 equilibria α∗ = 1. Therefore

K∗ = 1 because k0S = 1. In the Type 3 equilibrium α∗ < 1 which implies K∗ < 1 because θ < 1

and k0L < 1. �

41

Proof of Proposition 1.8. Recall in the Type 3 equilibrium, α∗ solves

G(α∗d0S1l ) = V 0S

(α∗d0S

1l

)− V L∗ = 0.

Applying the implicit function theorem

dα∗

dπl= −

∂G∂πl

∂G∂η∗

dη∗

dD∂D∂α∗

= − (−)

(−)(+)(+)< 0. (1.37)

Differentiating K∗ with respect to πl when CL∗ = C0L

dK∗

dπl

∣∣∣CL∗=C0L

= (1− α∗) ∂k0L

∂πl︸ ︷︷ ︸(−)

+(1− k0L)dα∗

dπl︸︷︷︸(−)

< 0.

and when CL∗ = C1L

dK∗

dπl

∣∣∣CL∗=C1L

= (1− θ) dα∗

dπl︸︷︷︸(−)

< 0.

Next we need to show that K∗ is decreasing in πl at the point where V 0L = V 1L. First note that

V 0L − V 1L is decreasing in πl

∂(V 0L − V 1L)

∂πl= −

c(R− 1)Rφ2l

(1− φl)θ(1− πhR)2< 0.

This implies that as πl increases it becomes more likely that CL∗ = C1L. Finally we need to show

that the realized investment level for C1L is lower than C0L when the profits from those contracts

are equal i.e. θ < k0L when V 1L = V 0L

k0L − θ =cφl

θ(1− φl)(1− πhR)− θ,

which is negative when V 1L = V 0L. Hence, when the equilibrium is Type 3, both K∗ and α∗ are

decreasing in πl. The last step is to show that as πl increases the equilibrium moves from Type 1

to Type 2 to Type 3. When the equilibrium is Type 1 the following condition holds

Π(0, d0S

1l

)< 0. (1.38)

Since d0S1l is increasing in πl, an increase in πl tightens (1.38). When the equilibrium is Type 2, the

42

following conditions are true

Π(0, d0S

1l

)> 0, (1.39)

V 0S(d0S

1l

)≥ V L∗ (1.40)

An increase in πl relaxes (1.39) and from (1.37) tightens (1.40). Therefore, as πl increases the

equilibrium moves from Type 1 to Type 2. When the equilibrium is Type 3, the following condition

is true:

V 0S(d0S

1l

)< V L∗. (1.41)

An increase in πl relaxes (1.41). Therefore, as πl increases the equilibrium moves from Type

2 to Type 3. Note that the equilibrium cannot jump from Type 1 to Type 3 from increasing πl

continuously because (1.38) implies (1.40). Summarizing, if an increase in πl causes the equilibrium

to switch types, the new equilibrium type results in a lower K∗ and α∗. Together all of these pieces

imply that α∗ and K∗ are decreasing in πl. �

Proof of Proposition 1.9 . In the Type 1 equilibrium, there is no information production by

lenders or the informed investor and all firms fully invest K∗ = 1. Hence the Type 1 equilibrium

achieves the first-best: V 0S = V FB. In the Type 2 equilibrium η∗ > 0 and α∗ = 1. Since all firms

choose the short-term contract there can never be too little short-term financing.

From the terms in (1.17) first notice that

dV 0S

dα∗=d0S

1l Rα∗φl

dp0l (η∗)

dα∗

p0l (η∗)2

. (1.42)

Also notice that

dΠ(η)

dα∗=

d

dα∗

∫ η∗

0θD

(R

p0l (η∗)− 1

)dη − κ(η∗)

=

[θD

(R

p0l (η∗)− 1

)]dη∗

dα∗η∗ +

∂

∂η∗dη∗

dα∗

∫ η∗

0

[θD

(R

p0l (η∗)− 1

)]dη − κ(η∗)

=

[θD

(R

p0l (η∗)− 1

)]dη∗

dα∗η∗

=d0S

1l η∗θ(R(p0l (η∗)− α∗ dp

0l (η∗)

dα∗

)− p0

l (η∗)2)

p0l (η∗)2

(1.43)

where the third line comes from the fact that∫ η∗

0ddα

[θD(

Rp0l (η∗)− 1)]dη − d

dακ(η∗) = 0 at the

informed investor’s optimal information production choice. Substituting (1.42) and (1.43) into

43

(1.17) we have

dW (α)

dα= V 0S − V L∗ + d0S

1l

(Rη∗θp0

l (η∗) +Rα (φl − η∗θ)

dp0l (η∗)

dα∗

p0l (η∗)2

− η∗θ

)

Simplifying in terms ofdp0l (η∗)

dα∗ we have the following condition for dV 0S

dα∗ < 0 in the Type 2 equilibrium

dp0l (η∗)

dα∗<p0l (η∗)(d0S

1l Rη∗θ + (V 0S − V L∗ − d0S

1l η∗θ)p0

l (η∗))

d0S1l Rα

∗ (φl − η∗θ). (1.44)

Hence, whenever the demand curve is sufficiently downward sloping there is too much short-term

financing in the Type 2 equilibrium. In the Type 3 equilibrium α∗ < 1, hence there can potentially

be too much or too little short-term financing. At the Type 3 equilibrium point V 0S − V L∗ = 0 so

(1.44) simplifies to

dp0l (η∗)

dα∗< −

η∗θ(R− p0l (η∗))p0

l (η∗)

α∗R (φl − η∗θ),

Hence whenever the demand curve is sufficiently downward sloping, the planner can raise welfare by

reducing the amount of short-term financing. Otherwise, the planner can raise welfare by increasing

the amount of short-term financing. �

44

1.6.3 Firm Can Raise Additional Funds at t = 0 and Store Funds Across Dates

In the main text I do not allow the firm to raise additional funds beyond k at t = 0 or store funds

across dates. In this section I show that this is without loss of generality. Consider the following

revised definition of a financial contract.

C ≡ {k, qz, d0, d1z, d2z(R), e0, e1z, e2z(R)}z=h,l, R=R,0 (1.45)

The differences between Definition 1.1 and (1.45) are i) d0 which is the funds the firm raises at

t = 0 which can potentially exceed k and ii) the payment at t = 2 d2z(R) can be conditioned on the

project’s success or failure, and iii) e is the firm’s consumption that may depend on the state and

project output. The revised definition of the contract will only be relevant for the optimal contract

without information acquisition C0∗. For the optimal contract with information production C1∗ as

long as the firm captures the full surplus and the lender acquires information its profits will always

be V 1∗ = θ(R − 1) − c. The timing of the firm’s consumption is irrelevant because the firm is

risk-neutral and there is no discounting. Hence, it is without loss of generality to assume the firm

stores any excess funds raised at t = 0 to t = 1. The firm’s problem can be written as

maxC

d0 − k +∑z

πz

[qzp

0z − d1z + φz [(k − qz)R− d2z(R)]− (1− φz)d2z(0)

],

s.t.

k ≤ d0, k ≤ 1,

d0 ≤ πh(d1h + d2h) + πl [d1l + φld2l(R) + (1− φl)d2l(0)] ,

(1− θ) (d0 − πh(d1h + d2h)− πl [d1l + pd2l(R) + (1− p)d2l(0)]) ≤ c, (1.46)

qz ∈ [0, k], d1z ≤ qzp0z + d0 − k z = h, l,

d2h(R) ≤ d0 − k + qhp0h − d1h + (k − qh)R,

d2l(R) ≤ d0 − k + qlp0l − d1l + (k − ql) R R = R, 0,

p0z = (1− γ0

z )φzR, z = h, l.

Using the same steps from Proposition 1.1 we can show q0∗h = 0. Since there is no difference in the

lender’s expected payments or the firm’s profits if the lender receives payments in the high state

at t = 1 or t = 2 it is w.l.o.g to set d0∗h = 0. In addition, using the same steps from Lemma 1.2 we

can show the incentive compatibility constraint (1.46) binds and d0∗2h = k0∗(R− 1) + d0∗

0 . Define δ1

as the firm’s cash at the beginning of t = 1

δ1 ≡ d0 − k,

and δ2z as the firm’s cash at the beginning of t = 2:

δ2z ≡ d0 − k + qzp0z − d1z z = h, l.

45

We can then rewrite the problem as follows

maxC

πl

[φl [δ2l + (k − ql)R− d2l(R)] + (1− φl) [δ2l − d2l(0)]

], (1.47)

s.t.

k ≤ d0, k ≤ 1,

d0 ≤ πh(δ1 + kR) + πl [d1l + φld2l(R) + (1− φl)d2l(0)] , (1.48)

(1− θ) (d0 − πh(δ1 + kR)− πl [d1l + µd2l(R) + (1− µ)d2l(0)]) = c, (1.49)

ql ∈ [0, k], d1l ≤ qlp0l + δ1

d2l(R) ≤ δ2l + (k − ql) R R = R, 0,

p0z = (1− γ0

z )φzR, z = h, l.

It will be useful to establish the following lemma.

Lemma 1.7. d2l(0)0∗ = δ0∗2l

Proof. Suppose to the contrary d2l(0)0∗ < δ0∗2l . First consider the case in which d0∗

2l (R) > 0. If the

firm increases d2l(0)0∗ by ε > 0 where d2l(0)0∗ + ε ≤ δ0∗2l and reduces d0∗

2l (R) by(

1−φlφl

)ε, (1.48)

would not be violated while (1.49) would slacken which contradicts C0∗ being optimal. Next consider

the case in which d0∗2l (R) = 0. Since πhR < 1, it must be that q0∗

l > 0 in order for (1.48) to not be

violated. To see this, suppose that q0∗l = 0, and insert the largest value of d2l(0)0∗ = d0 − k − d1l

into (1.48) and we have

k(1− πhR)− πlφl(d0 − k − d1l) ≤ 0 (1.50)

Inserting the largest possible value of d0∗1l = d0 − k into (1.50) we have

k(1− πhR) ≤ 0

Which is violated so long as k > 0. Hence q0∗l > 0. The firm can then reduce q0∗

l by ε and no

constraints are violated while the firm’s profits (1.47) increase since γ0l ≥ 0. �

46

Once again rewriting the problem following Lemma 1.7

maxC

πlφl [δ2l + (k − ql)R− d2l(R)] ,

s.t.

k ≤ d0, k ≤ 1,

d0 ≤ πh(δ1 + kR) + πl [d1l + φld2l(R) + (1− φl)δ2l] , (1.51)

(1− θ) (d0 − πh(δ1 + kR)− πl [d1l + µd2l(R) + (1− µ)δ2l]) = c, (1.52)

ql ∈ [0, k], d1l ≤ qlp0l + δ1

d2l(R) ≤ δ2l + (k − ql)R,

p0l = (1− γ0

l )φlR.

Expanding (1.51) and (1.52),

d0 ≤ πh(δ1 + kR) + πl[φl(d1l + d2l(R)) + (1− φl)qlp0

l

],

(1− θ)(d0 − πh(δ1 + kR)− πl

[µ(d1l + d2l(R)) + (1− µ)qlp

0l

])= c.

Upon inspection, we can see the lender’s incentives to acquire information and the firm’s profits

are invariant between d1l and d2l(R), i.e. storing funds from t = 1 to t = 2. Hence, it is without

loss of generality to set δ0∗2l = 0 and rewrite the problem as

maxC

πlφl [δ1 + (k − ql)R− d2l(R)] ,

s.t.

k ≤ d0, k ≤ 1,

d0 ≤ πh(δ1 + kR) + πl [d1l + φld2l(R)] ,

(1− θ) (d0 − πh(δ1 + kR)− πl [d1l + µd2l(R)]) = c,

ql ∈ [0, k], d1l ≤ qlp0l + δ1, d2l(R) ≤ (k − ql)R,

p0l = (1− γ0

l )φlR.

We can use the same steps from Lemma 1.2 to show that d0∗1l = q0∗

l p0l + δ0∗

1 . The problem can be

47

written as

maxC

πlφl [(k − ql)R− d2l(R)] ,

s.t.

k ≤ 1,

k ≤ πhkR+ πl(qlp

0l + φld2l(R)

),

(1− θ)(k − πhkR− πl(qlp0

l + µd2l(R)))

= c,

ql ∈ [0, k], d2l(R) ≤ (k − ql)R,

p0l = (1− γ0

l )φlR,

where d0 simply drops out. Hence, it is without loss of generality that the firm only raises k initially

and does not store funds across periods.

1.6.4 Firm Knows Project Type and Lender Cannot Acquire Information

In this section, I analyze the case in which the firm knows υ and the lender cannot acquire infor-

mation. This is useful to show that exogenous asymmetric information alone will not lead to firms’

using short-term financing.

Because the firm knows its project type, its contract offer may be a signal about its type.

First note that there can never be a separating equilibrium because the bad project is NPV negative.

Depending on off-equilibrium beliefs, there can be a multiplicity of equilibria. To narrow down the

potential equilibria, I apply undefeated equilibrium refinement from Mailath, Okuno-Fujiwara, and

Postlewaite (1993).

This refinement allows the firm with the good project to choose a contract that yields the

highest profits subject to being mimicked by the firm with the bad project. The problem can be

written as

maxC

∑z

πz(qzp

0z − d1z + (k − qz)R− d2z

)s.t.

k ≤ 1,

k ≤∑z

πz(d1z + φzd2z), (1.53)

qz ∈ [0, k], d1z ≤ qzp0z, d2z ≤ (k − qz)R z = h, l,

p0z = (1− γz)φzR, z = h, l.

The following lemma immediately follows.

Lemma 1.8. In the optimal contract in which the firm knows its project type and the lender cannot

acquire information,

48

i) q∗h = q∗l = 0

ii) d∗1h = d∗1l = 0

iii) k∗ = 1

iv) d∗2h = R.

Proof. i) and ii) follow from the same steps as Proposition 1.1. Suppose k∗ < 1. If we increase k∗

by ε and increase d∗2h by εR and increase d∗2l by ε(1−πhR)πlφl

(1.53) is not violated and the objective

increase which is a contradiction. Suppose that d∗2h < R. If we increase d∗2h by ε and decrease d∗2lby πhε

πlφl(1.53) is not violated and the objective increase which is a contradiction. �

After Lemma 1.8, the problem reduces to

maxC

πl(R− d2l)

s.t.

(1− πhR) ≤ πlφld2l. (1.54)

Since the firm has all of the bargaining power (1.54) binds and the optimal contract is

C∗ ≡ {k∗, q∗h, q∗l , d∗1h, d∗1l, d∗2h, d∗2l} =

{1, 0, 0, 0, 0, R,

1− πhRπlφl

}.

1.6.5 Firm Knows Project Type and Lender Can Acquire Information

In this section, I analyze the case in which the firm knows υ and the lender has the same information

acquisition technology as in the baseline model. I apply the same undefeated equilibrium refinement

from Mailath, Okuno-Fujiwara, and Postlewaite (1993). Therefore, the firm with the good project

chooses the contract that maximizes its profits subject to pooling with the firm with the bad project.

Note that because the lender can acquire information, the lender may be able to distinguish the

types after the contract offer. Similarly, when the firm knows its type the optimal contract without

information production coincides with the optimal contract from the main text C1∗ = C1L. Since

from Lemma 1.1, the lender will acquire information if the firm offers a contract in which k = 1

and ql = d1l = 0, the firm with the good project’s profits from C1L is

R− 1− c. (1.55)

For simplicity, assume information acquisition is contractible. As in the benchmark case there is

no value to liquidations; therefore, the lender’s break-even condition is

k ≤∑z

πzφzd2z. (1.56)

49

To minimize pooling costs, the firm with the good project chooses d2h = kR and (1.56) binds.

Therefore, the firm with the good project’s profits are

πlk

(R− 1− πhR

πlφl

),

which is strictly less than (1.55). Hence, the firm with the the good project wants to induce the

lender to acquire information to avoid the pooling cost with the firm with the bad project. This

result is similar to Fulghieri and Lukin (2001). Therefore, the optimal contract is C∗ = C1L.

1.6.6 Both Firm and Lender Can Acquire Information

In this section I analyze the case in which the firm can also incur c to learn υ. Before the firm offers

the lender a contract, the firm decides whether to acquire information or not. If the firm produces

information the choice of producing information becomes immediately public, while υ is revealed

at the end of t = 0. I assume the firm’s information production choice becomes public to avoid the

problem of the firm discovering it has a bad project then attempting to pool with a firm that does

not produce information. The rest of the timing is exactly the same as in the baseline model in

Section 1.2.

From Section 1.6.5, the optimal contract conditional on the firm knowing its project type

is C1L. If the firm acquires information then offers C1L, the firm with the good project’s expected

profits will be V 1L−c which is strictly less than V 1L, hence the firm would not acquire information.

Summarizing,

Proposition 1.10. If the firm has access to the same information technology as the lender at the

beginning of t = 0, it does not acquire information and the optimal contract is the same as in the

main text.

1.6.7 Firm Receives Exogenous Information

In the baseline model, the firm and lender begin symmetrically uninformed; however in practice, it

seems plausible firms have some form of information advantage over their lenders. In this section

I show that if the firm receives an exogenous noisy signal regarding its project type the optimal

contract may still deter information production. Suppose that prior to offering the lender a contract

the firm receives a signal s ∈ {G,B} regarding υ. Specifically, s = G with probability 12 and

Pr(υ = g|s = G) = θ + ε and Pr(υ = b|s = B) = θ − ε otherwise where ε ≤ max{θ, 1 − θ}.The change in probability need not be symmetric, but this simplification makes the unconditional

probability of the project succeeding the same as in the baseline model (i.e. Pr(υ = g|s = G)Pr(s =

G) + Pr(υ = g|s = B)Pr(s = B) = θ). The firm’s signal is private information and the lender has

access to the same information acquisition technology in the baseline model. Once again, I use the

undefeated equilibrium refinement from Mailath, Okuno-Fujiwara, and Postlewaite (1993) so the

50

problem amounts to the firm that receives the good signal s = G maximizing its profits subject to

pooling with the firm that receives the bad signal s = B. Define

φh(G) ≡ 1, φl(G) ≡ θ + ε+ (1− θ − ε)µ.

Then the firm that receives signal s = G faces the following problem to deter information production

maxC

∑z

πz

[qzp

0z − d1z + φz(G) ((k − qz)R− d2z)

]s.t.

k ≤ 1,

k ≤∑z

πz(d1z + φzd2z),

(1− θ) [k − πh(d1h + d2h)− πl(d1l + µd2l)] ≤ c,

qz ∈ [0, k], d1z ≤ qzp0z, d2z ≤ (k − qz)R z = h, l,

p0z = (1− γz)φzR, z = h, l.

Proposition 1.11. When the firm receives an exogenous private signal regarding the project type,

the optimal contract that deters information production takes the form C0L or C0S depending on

parameters.

The steps are the same as in the main text so I omit them. For the problem that induces information

acquisition the firm that receives the good signal solves the following problem

maxC

(θ + ε)∑z

πz(qzp

1z − d1z + (k − qz)R− d2z

)s.t.

k ≤ 1,

c ≤ θ

(∑z

πz(d1z + d2z)− k

),

c ≤ (1− θ) [k − πh(d1h + d2h)− πl(d1l + µd2l)] ,

qz ∈ [0, k], d1z ≤ qzp1z, d2z ≤ (k − qz)R z = h, l,

p1z = (1− γ1

z )R z = h, l.

The following proposition immediately follows

Proposition 1.12. When the firm receives an exogenous private signal regarding the project type,

the optimal contract that induces information production takes the form C1L.

51

The firm that receives the good signal’s expected profits from the three classes of contracts are

V 0L(G) = k0L(πhR+ πlφl(G)R− 1),

V 0S(G) = πlφl(G)

(R− d0S

2l −d0S

1l

(1− γ0l )φl

),

V 1L(G) = (θ + ε)(R− 1)− c.

Both V 1L(G) − V 0S(G) and V 1L(G) − V 0S(G) are increasing in ε. Hence, when the firm begins

with private information, it is more likely the optimal contract induces information production.

However, there are still parameter ranges in which the optimal contract is short-term (e.g. when ε

and γ0l are small).

1.6.8 NPV Positive Bad Project

In this section, I characterize the optimal contract when the bad project is NPV positive. Specifi-

cally,

(πh + πlµ)R > 1. (1.57)

Condition (1.57) implies that c < 0, therefore I impose the additional assumption that c > 0, i.e.

c ∈ (0, c). To keep the problem interesting, I also impose the following condition

πhR < 1 (1.58)

Condition (1.58) ensures the lender cannot breakeven by only receiving payments in the high state

where there is no uncertainty across project types. The firm’s problem for the optimal contract

without information acquisition C0∗ remains the same. However, if we inspect the potential solutions

from Proposition (1.2), note that now (1.30) can be ruled out because of (1.57) and (1.31) is the

solution when γ0l ≥

1−πhRπlφlR

and (1.29), C0S , is the solution otherwise.

The optimal contract that induces information acquisition C1∗ can differ from the case in

which the bad project was NPV negative because the firm can offer a menu to induce the lender to

accept different terms depending on υ. I use superscripts to refer to the contract terms intended for

the specific project type. To keep notation manageable, I suppress the references to the optimality

of the contract throughout the proofs. For instance if I state qbl = 0 in a Lemma, this refers to the

value of qbl at the optimum unless otherwise stated. The firm’s problem that induces information

acquisition is

52

maxkυ ,dυ2z

θ[πh((kg − qgh)R− dg2h) + πl((k

g − qgl )R− dg2l)]

+

(1− θ)[πh((kb − qbh)R− db2h) + πlµ((kb − qbl )R− db2l)

],

s.t.

kυ ∈ [0, 1] υ = g, b,

c ≤ θ(∑

z

πz(dg1z + dg2z)− k

g)

+

(1− θ)(πh(db1h + db2h) + πl(d

b1l + µdb2l)− kb

), (1.59)

kg ≤∑z

πz(dg1z + dg2z), (1.60)

kb ≤ πh(db1h + db2h) + πl(db1l + µdb2l), (1.61)∑

z

πz(db1z + db2z)− kb ≤

∑z

πz(dg1z + dg2z)− k

g, (1.62)

πh(dg1h + dg2h) + πl(dg1l + µdg2l)− k

g ≤ πh(db1h + db2h) + πl(db1l + µdb2l)− kb, (1.63)

qυz ∈ [0, kυ], dυ1z ≤ qυz (1− γ1z )E[R|υ, z], dυ2z ≤ kυR z = h, l, υ = g, b,

where (1.59) is the lender’s ex-ante participation constraint, (1.60) and (1.61) are the lender’s

participation constraints conditional on discovering the project is good (bad) and (1.62) says if the

firm discovers the project is good it must be incentive compatible to accept the terms for the good

project and vice versa for (1.63).

Lemma 1.9. qυh = dυ1h = 0, υ = g, b.

Proof. The proof is the same regardless of υ. Suppose that qυh > 0 for any υ ∈ {g, b}. First suppose

that dυ1h = 0, then the first can reduce qυh by ε and no constraints are violated and the firm’s profits

increase because γ1h ≥ 0. Next suppose dυ1h > 0, the firm can decrease qυh by ε, decrease dυ1h by

ε(1−γ1

h)E[R|υ,z=h]and increase dυ2h by ε

(1−γ1h)E[R|υ,z=h]

and none of the constraints are affected and the

firm’s profits increase because γ1h ≥ 0. Hence, qυh = 0 and thereby dυ1h = 0 for all υ. �

To simplify the proof I make the following claim which I confirm is true at the conclusion

of the proof

Claim 1.1. (1.63) is slack.

I then establish the following lemma.

Lemma 1.10. qυl = dυ1l = 0 υ = g, b.

53

Proof. The same steps from Lemma 1.9 can be taken to show that qgl = dg1l = 0. Next, suppose

that qbl > 0. There are two cases to consider. First suppose that db1l = 0, then the first can

reduce qbl by ε and no constraints are violated and the firm’s profits increase because γ1l ≥ 0. Next

suppose db1l > 0, the firm can decrease qbl by ε, decrease db1l by ε(1−γ1

l )E[R|υ=b,z=l]and increase db2l by

εµ(1−γ1

l )E[R|υ,z=l] and (1.62) is relaxed while none of the other constraints are affected and the firm’s

profits increase because γ1l ≥ 0. �

Hence, the contract is long-term and we can rewrite the problem as

maxkυ ,dυ2z

θ[πh(kgR− dg2h) + πl(k

gR− dg2l)]

+

(1− θ)[πh(kbR− db2h) + πlµ(kbR− db2l)

],

s.t.

kυ ∈ [0, 1] υ = g, b,

c ≤ θ(πhdg2h + πldg2l − k

g) + (1− θ)(πhdb2h + πlµdb2l − kb),

kg ≤ πhdg2h + πldg2l,

kb ≤ πhdb2h + πlµdb2l,

πhdb2h + πld

b2l − kb ≤ πhd

g2h + πld

g2l − k

g,

dυ2z ≤ kυR z = h, l, υ = g, b.

Lemma 1.11. (1.60) is slack and kg = 1

Proof. Suppose (1.60) binds, then we can replace kg = πhdg2h + πld

g2l into (1.62) which simplifies

to kb ≥ πhdb2h + πld

b2l; however, this violates (1.61). Suppose kg < 1, then we can increase kg by ε

such that kg + ε ≤ 1 and increase dg2h by πhε and dg2l by πlε and (1.62) and (1.59) are not violated

and the firm’s profits increase.

�

Lemma 1.12. (1.59) is slack if kb = 1

Proof. Suppose to the contrary that (1.59) binds if kb = 1, then we can rewrite (1.62) as

πhdb2h + πlφld

b2l − 1− c

θ≤ 0. (1.64)

For (1.64) to hold, we need πhdb2h + πlφld

b2l to be sufficiently small while not violating (1.60) and

db2z ≤ R for all z ∈ {h, l}. It easily shown that the LHS of (1.64) is smallest when (1.60) binds and

dg2h = R; however, even when this is the case (1.64) is violated. �

54

Lemma 1.13. (1.62) binds.

Proof. Suppose to the contrary (1.62) is slack. First consider the case in which kb < 1, we can

increase kb by ε and increase db2h by ε and db2l by εµ such that remains (1.62) is not violated while

(1.59) remains unchanged and the firm’s profits increase. Now consider the case in which kb = 1.

From Lemma 1.12, (1.59) is slack. Hence we can decrease dg2h by a small enough ε such that (1.59),

(1.60) and (1.62) remain slack while increasing the firm’s profits. �

Lemma 1.14. (1.61) binds

Proof. Suppose to the contrary that (1.61) is slack. First consider the case in which kb = 1. When

this is the case (1.59) is slack from Lemma 1.12. Hence we can decrease db2l (note db2l > 0 because

otherwise (1.61) would be violated) by a small enough ε such that (1.59) is not violated, (1.62)

would not be violated and the firm’s profits would increase. Now consider the case in which kb < 1

if we increase kb by ε and increase dg2l or dg2h by ε θπh(1−θ)µ such that (1.61) remains slack, then (1.59)

remains unchanged and (1.62) slackens while the firm’s profits increase.

�

Lemma 1.15. db2h = kbR

Proof. Suppose to the contrary that db2h < kbR, then we can increase db2h by ε and decrease db2l byπhεπlµ

(db2l must be positive because otherwise (1.59) is violated) such that (1.59) and (1.61) remain

unchanged; however, (1.62) slackens which leads to a contradiction. �

Hence from we can rewrite the problem as

maxkb,dg2h,d

g2l,d

b2l,θ[πh(R− dg2h) + πl(R− dg2l)

]+ (1− θ)πlµ

(kbR− db2l

), (1.65)

s.t.

kb ≤ 1 (1.66)

c ≤ θ(πhdg2h + πldg2l − 1) + (1− θ)(πhkbR+ πlµd

b2l − kb), (1.67)

kb = πhkbR+ πlµd

b2l,

πhkbR+ πld

b2l,−kb = πhd

g2h + πld

g2l − 1, (1.68)

dg2z ≤ kgR z = h, l.

Notice that (1.65), (1.67), (1.68) depend on the expected payments from the good project πhdg2h +

πldg2l, but not the relative values of dg2h and dg2l. Hence, we can set dg2h = R (which will ensure

55

Claim 1.1 holds). The Lagrangian is

L = θ[πh(R− dg2h) + πl(R− dg2l)

]+ (1− θ)πlµ

(kbR− db2l

)− λ1(kb − 1)−

λ2

[c− θ(πhdg2h + πld

g2l − 1)− (1− θ)(πlµdb2l − kb(1− πhR))

]−

λ3(kb(1− πhR)− πlµdb2l)− λ4

(kb(πhR+

1− πhRµ

− 1

)− πhdg2h − πlR+ 1

)−

λ5(dg2l −R)− λ6(db2l − kbR).

The Kuhn-Tucker necessary conditions are

Lkb = πl(1− θ)µR− λ1 − (1− πhR) ((1− θ)λ2 + λ3 − λ4) + λ6R ≤ 0,

Ldb2l = πl [(1− θ)µ (λ2 − 1) + µλ3 − λ4]− λ6 ≤ 0,

Ldg2l = πl (θ (λ2 − 1) + λ4)− λ5 ≤ 0,

Lkbkb = 0, Ldb2ldb2l = 0, Ldg2ld

g2l = 0,

λ1(kb − 1) = 0,

λ2

[c− θ(πhdg2h + πld

g2l − 1)− (1− θ)(πlµdb2l − (1− πhR)kb)

]= 0,

kb(1− πhR)− πlµdb2l = 0,

πhkb(R− dg2h) + kb

(1− πhR

µ− 1

)− πlR+ 1 = 0,

λ1 ≥ 0, λ2 ≥ 0, λ3 > 0, λ4 > 0, λ5 ≥ 0, λ6 ≥ 0,

(1.66), (1.67).

There are two potential solutions

kb =cµ

θ(1− µ)(1− πhR), dg2l =

c+ θ(1− πhR)

θπl, db2l =

c

θπl(1− µ), (1.69)

λ1 = 0, λ2 =(µ+ θ(1− 2µ)) (1− πhR)−R(1− θ)µ2πl

θ(1− µ) (1− πhR),

λ3 =(1− θ)φl (πhR+ πlµR− 1)

θ(1− µ) (1− πhR), λ4 =

(1− θ)µ (πhR+ πlµR− 1)

(1− µ) (1− πhR),

λ5 = 0, λ6 = 0,

56

and,

kb = 1, dg2l =c+ θ(1− πhR)

θπl, db2l =

c

θπl(1− µ), (1.70)

λ1 = πl(1− θ)µR−(µ+ θ(1− 2µ)) (1− πhR)

µ, λ2 = 0,

λ3 = 1 + θ

(1

µ− 1

), λ4 = θ, λ5 = 0, λ6 = 0,

where, (1.69) is the solution when (µ+ θ(1− 2µ)) (1− πhR)− R(1− θ)µ2πl ≥ 0 and (1.70) is the

solution otherwise. We can also now confirm Claim 1.1. When (1.69) is the solution (1.63) reduces

to

µ(

1 +c

θ

)+ πh(1− µ)R− 1 ≤ 0,

which holds. When (1.70) is the solution, the LHS of (1.63) reduces to 0, hence (1.63) holds. The

respective profits for (1.69) and (1.70) are

θ(R− 1)− c+c(1− θ)µ (πhR+ πlµR− 1)

θ(1− µ) (1− πhR), (1.71)

φl (πhR+ µπlR− 1)

µ.

Notice that (1.71) is strictly greater than V 1L. Hence, the optimal contract with information

acquisition is either (1.69) or (1.70). To find C∗ we can simply compare the profits from the

optimal contract with information acquisition and that without information acquisition.

Intuitively, the firm is able to finance the project regardless of its type; however, there is

still a welfare loss relative to the first-best because the lender produces information. Hence, for γ0l

sufficiently close to 0 the optimal contract is short-term C∗ = C0S .

1.6.9 Non-Zero Project Payoff in the Case of Failure

Because the project yields 0 in the case of failure, it is not possible to distinguish between equity

and debt for payments at t = 2. In this section I show that state-contingent debt is the optimal

contract when the project yields a positive payoff in the case of failure. Suppose that R = r < 1

when the project fails. To analyze the interesting case I revise Assumption 1.1 as follows

Assumption 1.4. The bad project is NPV negative:

πhR+ πl(µR+ (1− µ)r) < 1,

while the ex-ante, average project is NPV positive:

πhR+ πl(φlR+ (1− φl)r) > 1,

57

and Assumption 1.2 as follows,

Assumption 1.5. c ∈ (c, c), where

c ≡ (1− θ)(1− πl ((1− µ)r + µR)− πhR),

and

c ≡ θ(1− φl)(1− πhR− πlr)φl

.

Let d2z(r) denote the promised payment from the firm to the lender if the project fails in state z.

Then consider the revised definition of a financial contract

C = {k, qh, ql, d1h, d1l, d2h, d2h(r), d2l, d2l(r)}

To find the optimal contract to induce the lender to not acquire information the firm solves

maxC

∑z

πz

(qzp

0z − d1z + φz [(k − qz)R− d2z] + (1− φz) [(k − qz) r − d2z(r)]

)s.t.

k ≤ 1,

k ≤∑z

πz (d1z + φzd2z + (1− φz)d2z(r)) ,

(1− θ) [k − πh(d1h + d2h)− πl(d1l + µd2l + (1− µ)d2l(r))] ≤ c,

qz ∈ [0, k], d1z ≤ qzp0z, d2z ≤ (k − qz)R, d2z(r) ≤ (k − qz)r z = h, l,

p0z = (1− γ0

z )(φzR+ (1− φz)r) z = h, l.

The project never fails in the high state so we can ignore d2h(r). Using the same arguments from

Proposition 1.1 and Lemma 1.2, q0∗h = d0∗

h = 0, d0∗2h = k0∗R, the incentive compatibility constraint

58

binds and d0∗1l = q0∗

l p0l . We can rewrite the problem as follows

maxC

πl

(qlp

0l − d1l + φl [(k − ql)R− d2l] + (1− φl) [(k − ql) r − d2l(r)]

),

s.t.

k ≤ 1,

k(1− πhR) ≤ πl(d1l + φld2l + (1− φl)d2l(r)), (1.72)

(1− θ) [k(1− πhR)− πl(d1l + µd2l + (1− µ)d2l(r))] = c, (1.73)

d2l ≤(k − d1l

p0l

)R, (1.74)

d2l(r) ≤(k − d1l

p0l

)r, (1.75)

p0l = (1− γ0

l )(φlR+ (1− φl)r).

We can then show that (1.75) binds implying the optimal contract is debt.

Lemma 1.16. In the optimal contract (1.75) binds

Proof. Suppose to the contrary (1.75) is slack. If the firm increases d0∗2l (r) by any ε > 0 where

d0∗2l (r)+ε ≤

(k0∗ − d0∗

1l

p0l

)r and decreases d0∗

2l by (1−µ)εφl

, then (1.72) and (1.74) would not be violated

while (1.73) would slacken. However, because (1.73) binds at the optimum this is a contradiction.

�

Lemma 1.16 implies that payments at t = 2 are equivalent to debt because all of the cash

flows from the project in the case of failure are paid to the lender. I omit the remaining steps to find

the optimal contract that induces the lender to not acquire, however it follows from Proposition

1.2.

1.6.10 Auction for Market Equilibrium Mechanism

In the main text I assume that the informed investor can make unobservable offers to firms to buy

their assets before the firms sell to a pool of uninformed investors. In this section I show how a

second-price auction yields the same price as the mechanism in the main text.

The matching process works exactly as in Section 1.4; however, the informed investor simply

learn project types and do not enter a bilateral bargaining game with the firm whose project they

learn. Instead, each firm sells ql(i) units of good in an auction after the low state has been realized.

Uninformed investors do not observe if they are bidding against the informed investor which leads

to the winner’s curse. Uninformed investors are symmetric and have no private information so I

can restrict focus to symmetric bidding strategies among the uninformed. Let bU (i) denote each

uninformed investors per unit bid for firm i’s project. Let bI(i) denote the informed investor’s bid.

If bI(i) ≥ bU (i) the informed investor wins the auction and pays bU (i)ql(i) for the ql(i) units of

59

the project and vice versa if bI(i) < bU (i). For simplicity I assume if an uninformed investor wins

the auction there is a random tiebreaker so that one uninformed investor pays bU (i)ql(i) for ql(i)

units of the project. If the informed investor does not match with firm i, I assume they bid zero

bI(i) = 0. Hence the uninformed problem for firm i’s asset is

maxbU (i)

E[(R− bU (i))ql(i)|bI(i) < bU (i)].

When i ∈ I1 all investors know firm i is good quality. Therefore, investors bid bU (i) = bI(i) = R.

Henceforth I restrict focus to firm bidding strategies for firms i ∈ I0. When i ∈ I0 firm

i’s project type is not publicly known. The uninformed never bid bU (i) < µR because even if the

project is bad with certainty its expected payoff is µR. In addition, bU (i) > µR for all i. Suppose

to the contrary that bU (i) = µR for some i ∈ I0. Then, the informed investor’s best response

would be to bid bI(i) = µR and win the good. However, with probability 1−η there is no informed

investor bidding. Therefore, an uninformed investor could always bid ε more than µR and earn

positive profits. Hence, bU (i) > µR for all i.

The uninformed investor also always bids bU (i) < R. Suppose to the contrary bU (i) = R,

then the informed investor would bid bI(i) = R when υ(i) = g and win the asset and bid bI(i) = µR

when the asset is bad and lose the bid. Therefore, the uninformed would earn negative profits by

bidding bU (i) = R. Since bU (i) ∈ (µR,R) the informed bidder bids its valuation and earns profits

R − bU (i) when υ(i) = g and loses the bid otherwise. Hence, the uninformed optimal bid b∗ must

satisfy:

(1− θ)(µR− b∗) + θ(1− η)(φlR− b∗) = 0.

Solving for b∗

b∗ =

(1− ηθ(1− φl)

(1− ηθ)φl

)φlR. (1.76)

Notice that the winning bid in each auction (1.76) is the same as the price in the main text (1.13).

Therefore, the solution is identical to the bargaining process in Section 1.4 of the main text.

60

Chapter 2

Counterparty Information

Externalities

I analyze interbank markets with 1) asymmetric information about counterparty risk and 2) non-

exclusive contracts. Due to adverse selection, banks use their information to adjust the size of loans

rather than the prices they offer to counterparties. Each banks’ individual rationing decision creates

an information externality that increases the efficiency of trade. This information externality occurs

even though information is not shared and banks compete with each other. However, banks do not

internalize the cost their contracts impose on other banks through the counterparty’s likelihood of

default, which creates a counteracting negative externality that exacerbates as the number banks

increases. The model provides a microfoundation for interbank discipline and has implications for

the optimal structure of interbank markets.

2.1 Introduction

In the interbank market, banks contract with many counterparties at once. Banks are intercon-

nected: their contracts affect other banks who contract with the same counterparties. Intercon-

nectedness can lead to negative externalities, such as excessive risk or contagion, when banks only

consider their own welfare when choosing contracts (e.g. Zawadowski, 2013, Acharya and Bisin,

2014, Acemoglu, Ozdaglar, and Tahbaz-Salehi, 2015, Farboodi, 2014). However, banks may also

monitor or produce valuable information about their counterparties, which can in turn benefit other

banks. Academics have speculated that this form of discipline has important aggregate effects. For

example:

“Interbank discipline is not the product of one bank’s assessment of another. Rather, itspower lies in the judgments that many banks constantly make about one another. Thesedecisions may have only small effects on the disciplined bank, if viewed in isolation, butcan have significant effects in the aggregate.”— Judge (2012)

61

Furthermore, policymakers have argued that counterparty evaluation is essential for achieving fi-

nancial stability:

“We need to adopt policies that promote private counterparty supervision as the firstline of defense for a safe and sound banking system.” — Greenspan (2001)

Despite the practical importance of counterparty externalities, it is unclear how they arise. Moreo-

ever, understanding the underlying mechanism of counterparty externalities is critical for the design

and regulation of interbank and OTC markets.

In this paper, I analyze a model in which each bank’s evaluation of a shared counterparty

creates a positive externality that increases the efficiency of trade. Adverse selection induces banks

to use their private information to ration the amount they lend to the counterparty, rather than

adjust prices. In equilibrium, each bank’s individual rationing decision creates an information ex-

ternality which arises even though information is not shared and banks’ compete with each other.

However, because contracts are non-exclusive, banks do not fully internalize the effect of their

loans on the probability of the counterparty defaulting, creating a counteracting negative exter-

nality due to excessive risk-taking. Hence, the model features two main concerns of policymakers:

positive externalities from counterparty evaluation and negative externalities from excessive risk-

taking. Furthermore, while regulations that make interbank markets exclusive, e.g. central clearing

mandates, eliminate excessive risk-taking, they may still lower welfare by eliminating information

externalities in interbank markets.

Formally, I consider a two date model in which n banks lend simultaneously to a shared

counterparty. The counterparty prefers consuming at the first date and is endowed with an illiquid

asset which it can use to borrow against. The value of the counterparty’s asset affects its ability

to repay its loans. Specifically, if the counterparty has insufficient funds to repay its loans at the

second date, it defaults and all banks receive a pro rata share of the bank’s asset’s payoff minus

a bankruptcy cost. Importantly, the counterparty’s asset value is private information, creating

asymmetric information between banks and counterparties. Prior to offering the counterparty a

contract, each bank receives a signal about the counterparty’s asset value. Because the asset value

is not perfectly observable, the market exhibits adverse selection: banks cannot adjust the prices

of their loan offers to account for the expected quality of the counterparty because it would drive

out solvent counterparties from the market. Hence, banks compete on the size of the loans they

offer the counterparty.1

In the case of a single bank contracting with the counterparty, i.e. exclusive contracting, the

bank chooses a loan size that trades off the potential for higher interest payments at the second date

with the increased probability of the counterparty defaulting. However, when there are two or more

banks, i.e. non-exclusive contracting, the asymmetric information problem becomes attenuated.

Although information is not shared, banks rely on the unobserved information embedded in other

banks’ loans when determining the size of their own loans. Specifically, when a bank receives a

1This rationing effect is similar to Stiglitz and Weiss (1981).

62

positive signal about the counterparty, having more banks contracting with the counterparty acts

as a hedge for when that individual bank’s information is wrong. This allows banks with positive

information to offer larger loans to the counterparty. The opposite effect causes banks receiving

negative information to offer smaller loans to the counterparty as the number of banks grows. The

more banks there are, the less an individual bank’s mistaken information matters and the more

strongly banks act on their signals. I show that in isolation, this information externality always

raises efficiency.

While the information externality increases efficiency, banks do not internalize the cost their

contracts impose on other banks through the counterparty’s probability of default. This creates

a counteracting negative externality that exacerbates as the number of banks increases.2 Because

the information externality and strategic externality have opposite effects on welfare, the number

of banks contracting with a shared counterparty has an ambiguous effect on efficiency. For some

cases, increasing the number of banks raises welfare, while in other cases it lowers it.

Finally, I analyze an extension, in which rather than being endowed with information, banks

pay a fixed cost to produce information. I show that there is a unique equilibrium in which all banks

produce information if the cost of information is below a certain threshold. Perhaps surprisingly,

this threshold is increasing in the number of banks contracting with the shared counterparty. Hence,

the larger the number of banks, the more likely all banks produce information. This result suggests

that information externalities are stronger the more banks that participate in the interbank market.

Two key features of the model drive the counterparty information externality 1) asymmetric

information about counterparty risk and 2) unobservability of other banks’ contracts. The former

feature induces banks to ration the amount of credit they supply to the counterparty because of

adverse selection. Indeed, several papers find evidence consistent with adverse selection inducing

rationing in interbank markets (e.g. Furfine (2001), King (2008), Angelini, Nobili, and Picillo

(2011), Afonso, Kovner, and Schoar (2011), Acharya and Merrouche (2013)). The latter feature is

consistent the fact that interbank loans and other OTC contracts are generally spot contracts in

practice, which may be because of their opacity (e.g. Acharya and Bisin (2014)) or because of the

speed at which they often have to executed.

To my knowledge this is the only paper formally analyzing positive externalities that arise

from banks’ private information about each other. Rochet and Tirole (1996) analyze a reduced-

form version of peer monitoring when banks can engage in moral hazard; however, they admit that

adverse selection is likely the more realistic channel.3 My model shows how both counterparty

information externalities and excess risk taking have an ambiguous net effect on welfare as the

2This externality is similar to the counterparty risk externality in Acharya and Bisin (2014) and the strategicexternalities that occur in Cournot models.

3“Anecdotal evidence suggests that the adverse selection version is a better description of the current state ofinterbank monitoring” Rochet and Tirole (1996).

63

structure of the interbank market changes.4

My model relates to the literature analyzing imperfections in the interbank market arising

from asymmetric information (e.g. Bhattacharya et al. (1985), Broecker (1990), Flannery (1996),

Rochet and Tirole (1996), Freixas and Holthausen (2005), Freixas and Jorge (2008) and Heider,

Hoerova, and Holthausen (2015)). Freixas and Jorge (2008) show that asymmetric information can

cause rationing in the interbank market, which in turn can cause banks to reduce their lending

when depositors withdraw funds. Broecker (1990) shows how increased screening by banks can

lead to a winner’s curse problem. The main difference in his setting is higher prices do not drive

out higher quality borrowers, which is critical to the information externality in my model. Heider,

Hoerova, and Holthausen (2015) also analyze a setting in which there is asymmetric information

about counterparty risk. When counterparty risk becomes more dispersed, adverse selection can

cause the interbank market to freeze. In contrast to my setting, the interbank market is competitive

and banks do not receive any private information regarding other banks.

In another class of models analyzing the interbank market, there is no asymmetric infor-

mation and banks trade in interbank markets to smooth liquidity shocks; however, government

interventions can raise efficiency when there are aggregate liquidity shortages (e.g. Allen and Gale

(2004), Diamond and Rajan (2005), Allen, Carletti, and Gale (2009), Freixas, Martin, and Skeie

(2011)). In my model, banks have sufficient capital to lend; however, capital is endogenously

rationed due to adverse selection.5

My analysis also relates to the role of networks in financial markets with counterparty risk

(e.g. Allen and Gale (2000), Zawadowski (2013), Acemoglu, Ozdaglar, and Tahbaz-Salehi (2015)

and Farboodi (2014)). To my knowledge this is the first paper analyzing information externalities

between counterparties.

This paper also relates to the literature studying non-exclusive contracts in credit markets

(e.g. Bizer and DeMarzo (1992), Brunnermeier and Oehmke (2013), Acharya and Bisin (2014)

DeMarzo and He (2016), Admati et al. (2018) Donaldson, Gromb, and Piacentino (2019), Demarzo

(2019) and DeMarzo, He, and Tourre (2019)). Axelson and Makarov (2016) also analyze an infor-

mation externality in non-exclusive credit markets; however, they study a sequential credit market

rather than a simultaneous one. While the existing literature generally focus on the costs of non-

exclusive contracts, I uncover a benefit when lenders receive private information about a borrower’s

risk. Furthermore, while the main application of my model is interbank loans, it is also well suited

to understand the role of non-exclusivity in opaque markets, e.g. derivatives and repo, in which

contracts are generally non-exclusive.6

4There is also a literature studying the monitoring behavior of financial institutions outside the interbank market.In Diamond (1984) intermediaries can act as delegated monitors by reducing monitoring costs and Calomiris andKahn (1991) show how dispersed holders of demand deposits can prevent banks from absconding with funds. Higherlevels of bank capital can also increase monitoring incentives (e.g. Holmstrom and Tirole (1997), Allen, Carletti, andMarquez (2011), and Mehran and Thakor (2011)).

5Imperfections in the interbank market may also arise from market power Acharya, Gromb, and Yorulmazer(2012).

6The paper also relates to the literature on multiprincipal externalities (e.g Bernheim and Whinston (1990)Martimort and Stole (2002), Bizer and DeMarzo (1992)).

64

Finally, from a modeling perspective there are some similarities between my setup and

Cournot models with uncertain aggregate demand (Palfrey, 1985, Vives, 1988, Vives, 2001 and

Vives, 2010). Banks choose quantities and there is uncertainty about a state that jointly affects

banks’ payoffs. In the aforementioned models the demand curve is linear, while in my model

default is non-linear which induces the information externality. Furthermore, in Cournot models

increasing the number of firms generally raises welfare by reducing the deadweight loss from the

underproduction of goods. In my model, increased competition can either increase or decrease

welfare depending on the relative magnitude of information and strategic externalities.

2.2 Model Setup

There are two dates t = 0, 1 and n identical banks indexed by i ∈ I and one counterparty. At t = 0

agents enter contracts and at t = 1 payoffs are realized and payments are made.

2.2.1 Agents and Technology

Banks have preferences represented by the utility function E[c0 + c1] and are endowed with e of

consumption good at t = 0 and receive no endowment at t = 1. The counterparty has limited

liability and its preferences are given by E[αc0 + c1], where it is risk-neutral, but derives α > 1 per

unit of consumption at t = 0.7 The preference for consumption at t = 0 can be thought of as an

NPV positive loan opportunity for the bank.

The counterparty is endowed with an illiquid project that yields nA at t = 2, where I

scale the payoff of the project A by the number of banks n to maintain comparison across market

structures.8 The prior distribution of A is

A ∼ U [0, A] w.p1

2

A = A otherwise.

Effectively, the distribution is uniform between 0 and A with a mass point at A.9 For convenience

I normalize A = 1, which is without loss of generality. This distributional assumption is highly

tractable and allows me to solve the equilibrium analytically. Importantly, I assume that A is

privately known by the counterparty, while banks only know the prior distribution of A.

2.2.2 Contracting

At t = 0 each bank offers a loan contract to the counterparty. The contract between bank i and

the counterparty specifies a loan quantity qi ∈ [0, e] and a gross interest rate ri, in which the bank

7This reduced-form for early consumption is standard in the security design literature (e.g. DeMarzo and Duffie(1999)).

8This practice is common in Cournot models with n firms (e.g. Vives (2001) and Vives (2010)).9See Caramp (2017) for a similar distribution.

65

pays the counterparty qi at t = 0 and the counterparty promises to repay qiri at t = 1. Throughout

the analysis I assume that e is sufficiently large so that the constraint qi ≤ e is always slack. I refer

to set of all banks’ contracts as C.Before offering a loan contract, each bank receives a conditionally independent and identi-

cally distributed signal si ∈ {G,B}. The distribution of the signal depends on the payoff of the

counterparty’s project. Specifically, if A = 1 there is a λ ∈ [1/2, 1) probability that s = G, while

if A < 1 there is a 1 − λ probability that s = G. Hence, by Bayes’ rule Pr(A = 1|G) = λ and

Pr(A < 1|G) = 1 − λ. The signal reveals information as to whether A = 1 or A < 1, but does

not provide additional information about A conditional on A < 1. After bank receive signals,

they simultaneously makes loan offers to the the counterparty. Importantly, each bank’s signal

and loan offer are unobservable to the other banks. I focus on simple spot contracts in which A

and other banks contracts cannot be directly contracted upon. In practice, interbank loans, repos

and derivatives do not usually contain covenants, which may be due to a lack of transparency or

enforceability (Acharya and Bisin (2014)). I define bank i’s bankruptcy share χi as follows

χi ≡qi

qi +∑−i∈I

q−i.

If the counterparty’s wealth exceeds the promised t = 1 payments to banks∑

i∈I qiri ≤ A, all banks

receive their promised payments from the counterparty. Otherwise, the counterparty defaults and

all banks receive χizA, where z ∈ [0, 1) is a linear bankruptcy cost.

To simplify the analysis, I make the following assumption regarding λ

Assumption 2.1. λ ≤ λ ≡ min{

2α−z+n(3−n−z+(n−1)α)n(2−z+2n(α−1))+2α−z ,

n2(n−1)

}.

Assumption 2.1 ensures that the total payments each counterparty makes at t = 1 is always less

than n (the largest possible payoff of the counterparty’s project). This assumption dramatically

simplifies the problem by allowing me to use a first-order approach to characterize banks’ optimal

strategies.

Finally, to eliminate the possibility of unrealistic equilibria I make two assumptions. First

I assume that there is an arbitrarily large limit d on the promised payment to all banks at t = 1,

i.e. qiri < d ∀i where d ∈ (n,∞). Furthermore, I assume that there is an ε probability that A = nd

and analyze the limiting case in which ε tends to 0. This assumption eliminates the possibility

of an equilibrium in which all banks offer loans with extremely high promised t = 1 repayments

knowing that the counterparty will default with certainty. Given that other banks t = 1 payments

are so high, no individual bank has an incentive to reduce theirs since the counterparty will default

regardless. Introducing an arbitrarily small probability that the counterparty cannot default breaks

this equilibrium.

66

2.3 Model Analysis

Define the set of banks who offer the counterparty loans in which the interest rate is smaller than

the counterparty’s time preference

I ′ ≡ {i ∈ I : ri ≤ α},

and the following function

D(C) =

1 if∑

i∈I αqi −A ≥∑

i∈I′(α− ri)qi,

0 otherwise.

If D(C) = 1, the counterparty accepts bank i’s contract offer whenever ri ≤ α. In contrast, if

D(C) = 0, the counterparty always accepts bank i’s contract offer regardless of the interest rate.

Intuitively, if the counterparty knows it will not default, it accepts any loan in which the interest

rate is less than its time preference α. In contrast, if the counterparty knows it will ultimately

default, it accepts any contract offer since it will default regardless. I refer to cases in which ri > α

as counterparty risk being priced. When counterparty risk is priced, the counterparty only accepts

the loan offer when it will default, which creates adverse selection. This phenomenon is similar

to Stiglitz and Weiss (1981) in which high interest rates cause safe borrowers to drop out of the

market. In the following Lemma I show that counterparty risk cannot be priced in equilibrium.10

Formally,

Lemma 2.1 (Adverse Selection). ri ≤ α ∀i ∈ I.

Lemma 2.1 implies that all contract offers are ultimately accepted in equilibrium. Because each

bank’s loan offer may depend on the signal it receives I include an s superscript on all contract

terms. Hence, bank i’s ex-ante expected utility conditional on receiving signal s is

W si ≡ Pr (D(C) = 0|s) qsi (1 + rsi ) + Pr (D(C) = 1|s)E[χsi z|D(C) = 1]− qsi .

and bank i’s expected utility is

Wi ≡1

2

(WGi +WB

i

).

I also define aggregate welfare as the sum of all banks expected utility

W ≡∑i∈I

Wi.

10The fact that counterparty risk is not priced across counterparties is empirically consistent with Arora, Gandhi,and Longstaff (2012).

67

2.3.1 Team-Efficient Benchmark

Before analyzing the equilibrium in which banks act strategically, I establish a “team-efficient”

benchmark in which banks play decentralized strategies to maximize welfare, but are unable to

share information (Radner, 1962, Vives, 1988 and Angeletos and Pavan (2007)).11 This can be

thought of as a planner telling each bank the action to take if they get a particular signal. I assume

that the planner does not know the counterparty’s project payoff and cannot force it to accept or

reject loan offers. As shown below, analyzing the team-efficient benchmark allows us to isolate the

efficiency gains from information externalities from the efficiency losses from strategic externalities.

The team-efficient problem is thus

maxqsi ,r

si ∀i,s

W. (2.1)

Because banks maximize their aggregate expected utility, the interest rates on loans will be as low

as possible so long as the counterparty always accepts all offers. Formally,

Lemma 2.2. rsi = α ∀i, s.

Finally, I restrict focus to symmetric strategies, which is without loss of generality given that banks

are risk-neutral.

Assumption 2.1 ensures that in equilibrium the sum of the counterparty’s repayments to

banks at t = 1 never exceeds the maximum level of counterparty wealth∑

i∈I qiri ≤ n. Further-

more, given the distributional assumptions regarding A, (2.1) is concave in each the bank’s loan

choice following each signal. Hence, there is a unique solution that can be found from the first

order conditions. Formally,

Proposition 2.1. The solution to the team-efficient problem is

qG∗∗

=(α− 1)(2 + 2λ(n− 1)− n)

2(2α− z)(1− λ), qB

∗∗=

(α− 1)(n− 2λ(n− 1))

2(2α− z)λ,

where the following are increasing in n

i) The difference in loan size between banks with good and bad signals qG∗∗ − qB∗∗

ii) Expected loan size 12

(qG∗∗

+ qB∗∗)

iii) The difference in expected utility between banks receiving good and bad signals WG∗∗ −WB∗∗

iv) Team-efficient welfare W ∗∗

The key result is Part iv) of Proposition 2.1, which arises through an information externality. Intu-

itively, as n increases, the average signal becomes more informative about A. The planner increases

11I ignore the counterparty’s profits in the benchmark to isolate the two main effects of interest, but I confirm themain results in the Appendix when my benchmark includes the counterparty’s profits.

68

welfare by increasing the unconditional average loans as well as the difference in conditional average

loans across counterparty types.

To gain intuition, define Q as the realized average loan size

Q ≡ 1

n

(kqG + (n− k)qB

),

where k is the the number of good signals banks receive. The conditional variance of Q is

V ar(Q|A = 1) = V ar(Q|A < 1) =(qG − qB)2(1− λ)λ

n.

Because of the law of large numbers, as n becomes large, the conditional aggregate loan sizes are

known with certainty limn→∞ V ar(M |A = 1) = 0. Thus, as n increases, the planner tells banks

to act more aggressively on their signals (Part i) of Proposition 2.1, which enables higher average

loan sizes (Part ii), which in turn increases welfare (Part iv). As shown below, this information

externality persists in the market equilibrium in which banks choose loan sizes that maximize their

own expected utility; however, there is also a negative externality that arises from non-exclusive

contracting.

2.3.2 Market Equilibrium

I now analyze the market equilibrium where banks act strategically in their choice of contracts.

I first establish that the interest rate on all loans are equal to the counterparty’s time

preference. Formally,

Lemma 2.3. ri = α ∀i.

From Lemma 2.1 ri ≤ α. If ri < α, then bank i can always increase ri by a small enough amount

and the counterparty would still accept the contract regardless of the counterparty’s wealth.

I consider symmetric equilibrium in which each bank chooses payments that maximize its ex-

pected utility conditional on the signal it receives, taking as given other banks’ strategies. Formally,

Definition 2.1. A symmetric market equilibrium consists of a loan size qs∗

for s ∈ {G,B} in which

each bank’s loan size conditional on each signal realization maximizes its expected utility given all

other banks’ strategies.

qs∗

= arg maxq≥0

αqPr (D(C) = 0|s) + Pr (D(C) = 1|s)E[χzA|D(C) = 1, s]− q ∀s. (2.2)

Given my assumptions regarding the distribution of A, (2.2) is concave in q. Hence, I can use

a first-order approach to solve for banks’ equilibrium loan sizes. Furthermore, Assumption (2.1)

ensures that the sum of banks’ loan sizes never exceeds 1; hence the counterparty never defaults

69

when A = 1.12

Proposition 2.2. The market equilibrium loan sizes are

qG∗

=(α− 1)(2λ(n− 1)− n+ 3)

2α(2α− z)(1− λ)(n+ 1), qB

∗=

(α− 1)(n+ 1− 2λ(n− 1))

2α(2α− z)λ(n+ 1),

and the following are increasing in n

i) The difference in loan sizes between banks receiving good and bad signals qG∗ − qB∗

ii) Average loan size 12(qG

∗+ qB

∗)

iii) The difference in expected utility between banks receiving good and bad signals WG∗ −WB∗

Depending on parameters W ∗ is increasing or decreasing in n.

As n increases, banks put more weight on the distribution of other banks’ signals rather than their

own. When a bank receives a good signal but A < 1, the counterparty may end up defaulting.

However, the higher n, the lower the average quality of signals is when a bank receives a good

signal. Hence, other banks’ signals act as a hedge for when a bank with a good signal’s information

is wrong. This causes the bank to increase the size of its loan as n increases. In contrast, when a

bank receives a bad signal and A < 1, some of the other banks may receive good signals. In fact,

the higher n, the higher the average number of mistaken good signals, which forces banks with bad

signals to be more conservative. In equilibrium, this effect causes banks to act more strongly on

their signals as n increases, raising efficiency. However, as n increases banks’ internalize less the

effect of their contracts on the counterparty’s default probability, which results in excessive loan

sizes relative to the team-efficient benchmark.

For a large or small n, receiving a good or bad signal has the same meaning for a bank

in terms of its new expectation about the wealth of the counterparty. However, as n increases,

the difference in expected utility between banks receiving a good and bad signal increases. This

is in contrast to the winner’s curse in which bidders adjust their bids downwards as the number

of bidders increases. In this setting, being more optimistic is more valuable when there are more

banks because this leads to more banks receiving bad signals when the counterparty may default,

thereby mitigating the risk of default.

Compared to the team-efficient benchmark, welfare in the market equilibrium need not be

increasing in n. This result is due to the strategic externality from non-exclusive contracting. In

particular, banks do not internalize the cost larger loans impose on other banks by increasing the

counterparty’s likelihood of default.

Proposition 2.3. Market equilibrium loan sizes are excessive compared to the team efficient bench-

mark, i.e. qG∗ ≥ qG∗∗ and qB

∗ ≥ qB∗∗, where the inequalities are strict when n > 1.

12Relaxing Assumption 2.1 would lead to cases in which banks may choose payments off the support of thecounterparty’s project payoff distribution. This complicates the problem by making banks’ payoffs discontinuousin qs.

70

2.4 Endogenous Information Production

In this section rather than being endowed with information, I assume banks can incur a cost to

receive a signal. Formally, prior to offering contracts, each bank simultaneously decides whether

to incur a cost c to receive a signal with precision λ. Let ai ∈ {0, 1} denote bank i’s decision to

produce information where ai = 0 if the bank produces information and ai = 1, otherwise. After

receiving the signal, the timing is exactly as in the baseline model. I restrict focus to symmetric

strategies after banks have decided whether to produce information but banks need not all make

the same information production decisions.

Definition 2.2. A market equilibrium in the endogenous information production game consists of

information acquisition decisions a∗i ∀i, a loan size qni∗

for banks that have not produced information

and loan sizes for those that have qs∗

for s ∈ {G,B} in which each bank’s loan size maximizes its

own expected utility given its information and other banks’ loan sizes. Each bank’s information

acquisition decision maximizes their expected utility given {qni∗ , qG∗ , qB∗}.

The following proposition shows that there is a unique equilibrium in which all banks either produce

or do not produce information depending on a threshold value of c.

Proposition 2.4. If c ≤ c ≡ n(α−1)2(1−2λ)2

4(2α−z)α(1−λ)λ all banks acquire information. Otherwise, no banks

produce information. The threshold to produce information c is increasing in λ, α and n.

Because of the information externality, information is more valuable the more banks that can

potentially produce it. This causes c to increase in n.

2.5 Conclusion

Increased interconnectedness in interbank markets can reduce welfare by inducing excessive risk

taking by banks. However, in this paper I show that increased interconnectedness can also raise

efficiency in markets plagued by asymmetric information through information externalities that

arise from banks’ private information. This mechanism provides a microfoundation for interbank

market discipline which policymakers should take into account when implementing regulations in

interbank and OTC markets. The current model only allows for information to be aggregated

through a decentralized market equilibrium; however, in future work I plan on taking a mechanism

design approach to analyze the optimal design of a central clearing mechanism when there is

asymmetric information about counterparty risk.

71

2.6 Appendix

2.6.1 Proofs

Proof of Lemma 2.1. I can compare bank i’s payoff from lending to the counterparty when ri > α

to ri ≤ α. The counterparty only accepts a contract in which ri > α when A <∑

i∈I qi. Hence,

the interest rate ri does not affect bank i’s payoff when it offers a contract in which ri > α. Bank

i’s expected payoff from offering a contract in which ri > α is

Pr(D(C) = 1|s) (E[χizA|D(C) = 1, s]− qi) . (2.3)

However, if bank i deviates and offers a contract in which ri = α then its expected payoff is

Pr(D(C) = 0|s)qi(α− 1) + Pr(D(C) = 1|s) (E[χizA|D(C) = 1, s]− qi) , (2.4)

where (2.4) is larger than (2.3) because Pr(D(C) = 1|s)qi(α−1) is always positive from the fact that

α > 1 and Pr(D(C) = 1|s) is strictly positive. The latter is true because there is an ε probability

A = nd and∑

i∈I qiri < nd. �

Proof of Lemma 2.2. Suppose there was a loan contract such that rsi < α. The counterparty

would always accept this contract regardless of whether it defaults or not. However, the planner

could change the contract offer to rsi = α and the counterparty would still accept the offer and W

would increase. �

Proof of Proposition 2.1. Differentiating (2.1) with respect to qG and qB I have the following

first order conditions

[qG]1

2

(α− 1− (2α− z)

[(1− λ)λqB − (1− λ)λ(qG − qB)

n− (1− λ)2qG

])= 0,

[qB]1

2

(α− 1− (2α− z)

[λ2qB +

(1− λ)λ(qG − qB)

n− λ(1− λ)qG

])= 0.

Solving for two equations and two unknowns I have

qG∗∗

=(α− 1)(2 + 2λ(n− 1)− n)

2n(2α− z)(1− λ), qB

∗∗=

(α− 1)(n− 2λ(n− 1))

2n(2α− z)λ.

I now show that the sum of equilibrium loan offers never exceeds 1 (the maximum wealth of the

counterparty). First note that qB∗∗< qG

∗∗. Hence, the largest possible sum of loan offers is nqG

∗∗.

Also, notice that nqG∗∗

is increasing in λ. Hence if I substitute the largest possible value of λ into

qG∗∗

I have

qG∗∗∣∣∣λ=λ

= max{4(n− 1)(α− 1)

(n− 2)(2− z)α,n(2− z)− z + 4α− 2

(1 + n)(2− z)α

}≤ 1.

72

In addition, the second order conditions confirm that the solution is a maximum.

[qG] − 1

2(1− λ)(λ+ (1− λ)n)(2α− z) < 0,

[qB] − 1

2(2α− z)λ(1 + (n− 1)λ) < 0.

To show Part i) I have

∂

∂n

(qG∗∗ − qB∗∗

)=

(α− 1)(2λ− 1)

(2α− z)(1− λ)λ≥ 0,

where the inequality comes from λ ≥ 12 and α > 1. To show Part ii) I have

∂

∂n

(qG∗∗

+ qB∗∗)

=(α− 1)(2λ− 1)2

(2α− 1)(1− λ)λ> 0.

To show Part iii) I have

∂

∂n

(WG∗∗ −WB∗∗

)=

(α− 1)2(2λ− 1)

(2α− 1)(1− λ)λ> 0.

Part iv) can immediately be seen from

∂W ∗∗

∂n=

(α− 1)2(2λ− 1)2

4(2α− z)(1− λ)λ> 0.

�

Proof of Lemma 2.3. See text. �

Proof of Proposition 2.2. Differentiating with respect to qi I have the following first order con-

ditions

[qG] α− 1− (2α− z)(1− λ)((n− 1)(qG(1− λ) + qBλ) + 2qi)

2n= 0, (2.5)

[qB] α− 1− (2α− z)λ((n− 1)(qG(1− λ) + qBλ) + 2qi)

2n= 0. (2.6)

To find a symmetric equilibrium, I replace qi with qG in (2.5) and with qB in (2.6) and solve for

two equations and two unknowns. I then have

qG∗

=(α− 1)(2λ(n− 1)− n+ 3)

2(2α− z)(1− λ)(n+ 1), qB

∗=

(α− 1)(n+ 1− 2λ(n− 1))

2(2α− z)λ(n+ 1).

I need to now ensure the sum of total loan offers never exceeds 1. Notice that qG∗> qB

∗; hence,

the largest possible loan size is qG∗. Also, notice that qG

∗is increasing in λ. Hence, the largest

73

possible value of qG∗

is when λ = λ, but this equals 1

qG∗∣∣∣λ=λ

= max{ 6(n− 1)n(α− 1)

(n− 2)(1 + n)(2α− z),n(2− z)− z + 4α− 2

(1 + n)(2− z)α

}≤ 1.

Hence, the first order approach is valid. The second order conditions confirm the solution is indeed

a maximum

[qG] − (2α− z)(1− λ)

n< 0,

[qB] − (2α− z)λn

< 0.

To show Part i) I have

∂

∂n(qG

∗ − qB∗) =(α− 1)(2λ− 1)

(2α− z)(1− λ)λ≥ 0,

where the inequality holds because λ ≥ 12 and α > 1. To show Part ii) I have

∂

∂n

(qG∗

+ qB∗)

=(α− 1)

((1 + n)2 − 4λ(1− λ)(n(2 + n)− 1)

)(1 + n)2(2α− z)(1− λ)λ

. (2.7)

Notice that the term in the numerator 4λ(1− λ)(n(2 + n)− 1) < n(2 + n)− 1 because λ > 12 . Also

notice that (n + 1)2 − (n(2 + n) − 1) = 2. Hence the numerator of (2.7) must always be positive,

implying the entire expression is positive. To show Part iii) I have

∂

∂n(WG∗ −WB∗) = (2.8)

(α− 1)2(2λ− 1)((1 + n)3 − 4λ(1− λ)(n− 1)(n(4 + n)− 1)

)2(1 + n)3(2α− z)(1− λ)λ

.

Notice that the term in the numerator 4λ(1 − λ)(n − 1)(n(4 + n) − 1) < (n − 1)(n(4 + n) − 1)

because λ > 12 . Also notice that (n + 1)3 − (n − 1)(n(4 + n) − 1) = 8n. Hence the numerator of

(2.8) must always be positive, implying the entire expression is positive. �

Proof of Proposition 2.3. The proof can immediately be seen by comparing the market equilib-

rium loan sizes to the team-efficient benchmark

qG∗ − qG∗∗ =

2(n− 1)(α− 1)

(n+ 1)(2α− z)≥ 0,

and

qB∗ − qB∗∗ =

2(n− 1)(α− 1)

(n+ 1)(2α− z)≥ 0.

Notice that the inequalities are strict when n > 1. �

74

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