Secured Loans and Risky Assets in a Monetary Economy

Yuchi Chu∗ Yiting Li†‡

July 16, 2020

Abstract

We study the liquidity of an asset used as collateral, focusing on the risk of asset payoffs andforeclosure costs. The intermediary structure has the feature of costly state verification, withthe verification cost interpreted as the cost of foreclosing assets once a default occurs. Withoutforeclosure costs, uncertainty in asset payoffs does not matter for allocation and the asset price;the risk premium is zero though loans are exposed to default risk. There exists an equilibriumwhere borrowers always default and banks issuing credit function like an asset market. Toeconomize on the contingent loss associated with foreclosure costs, a risk premium arises, banksreduce lending, and the default probability falls. By explicitly capturing the trade-offs betweenthe loan amount and default probability, we show that if the risk premium decreases withthe asset risk, higher default rates are accompanied by larger aggregate liquidity and output;otherwise, liquidity is scarcer as assets becomes riskier.

JEL Classification: E41; E50Keywords: liquidity; money; collateral; asset risk; foreclosure cost

∗Department of Economics, University of Wisconsin Madison, 7222 Social Science Building, Madison, WI 53706,USA. E-mail: ychu3789@gmail.com

†Corresponding author. Department of Economics, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd.,Taipei 10617, Taiwan. E-mail: yitingli@ntu.edu.tw

‡We are very grateful to Randy Wright and Zhigang Ge for helpful comments.

1 Introduction

Collateralized borrowing is a widespread phenomenon in economies with limitations on debt enforce-

ment. The amount of liquidity generated by an asset used as collateral in securing debt, captured

by loan-to-value ratios, has an impact on both aggregate liquidity and output. From mortgages to

repurchase agreements (repos), what concerns lenders most in determining the loan-to-value ratios

are two things: the probability of a borrower’s defaulting on debt and the recovery value of liquidat-

ing collateral if default occurs. However, many previous studies either assume fixed resalability of

an asset or they are silent about issues related to default; therefore, the mechanism through which

an asset affects aggregate liquidity is not fully addressed.1 Default is an especially relevant issue,

as the 2008 financial crisis revealed an unprecedented surge of foreclosure rates in mortgages loans;

in the repo market the default spreads increased, margin requirements on borrowing were raised,

and lending decreased.2 To study these phenomena, one needs to have a model with endogenously

derived loan-to-value ratios, defaults, and credit allocation.

This paper presents a unified framework that jointly determines loan-to-value ratios, default

rates, and the price of an asset used as collateral, and links them to aggregate liquidity and output

in a monetary economy with limited commitment and enforcement. We focus on two factors:

uncertainty in the recovery value, captured by the risk of payoffs of the pledged asset, and foreclosure

costs, caused by limitations in information and enforcement, which work to reduce the recovery

value of collateral.3 We investigate how loan-to-value ratios, defaults, asset prices, and the trade-

offs among them are influenced by asset risks and foreclosure costs. We address concerns such as

whether higher default rates are necessarily accompanied by lower aggregate liquidity. Moreover,

1For instance, Kiyotaki and Moore (2001, 2005) consider fixed resalability of assets; Ferraris and Watanabe (2008)and Li and Li (2013) derive endogenously the amount of credit generated by an asset as collateral, but defaults donot occur in equilibrium.

2Mayer, Pence, and Sherlund (2009) reported that roughly 1.7 million foreclosures were started in the first threequarters of 2008, an increase of 62 percent from the first three quarters of 2007. Gorton and Metrick (2012) constructan index as an average haircut for collateral used in repo transactions, not including U.S. treasury securities, andthey found this index rises from zero percent in early 2007 to nearly 50 percent at the peak of the crisis in late 2008.

3The foreclosure cost includes the cost of legal, administration, or auctioneer fees for claiming the ownership andresale of the assets pledged as collateral. The foreclosure cost of mortgage loans, captured by the percentage of eitherthe loan balance or the proceeds of the sale, ranges from 1.5% in Finland to 18% in Belgium, with the average closeto 9% of the loan value (see Chart 9 in ECB 2009). In this paper, we broadly interpret the foreclosure cost as thepledgeability of assets and relate it to the properties of assets such as portability and tangibility.

1

though it is commonly assumed that the asset price and lending will fall when the asset pledged

as collateral becomes riskier, we derive explicitly the conditions under which a riskier asset may be

better suited to serve as collateral in generating liquidity.

The model is based on Berentsen, Camera, and Waller (2007), where agents can borrow from

banks to finance consumption by pledging assets as collateral as in Ferrais and Watanabe (2008)

and Li and Li (2013), but defaults can occur in our model. To study issues related to default,

we consider an asset with uncertain dividends and an intermediary structure that has the feature

of costly state verification (see, e.g., Townsend 1979, Williamson 1987), with the verification cost

interpreted as the cost of foreclosing the pledged asset once a default occurs. As shown in these

previous studies, a debt contract is the optimal arrangement between lenders and borrowers. The

debt contract indicates a loan amount in exchange for a constant repayment, and it maximizes the

borrower’s expected utility while satisfying the bank’s participation. Because the repayment is not

contingent on asset payoff realizations, the model permits default as an equilibrium outcome: When

the value of collateral falls short of the debt obligation, the borrower defaults.

In the benchmark model with no foreclosure costs, uncertainty in asset payoffs does not matter

for consumption, allocation, and the asset price. This is so because a bank can diversify the risk of

payoffs of the collateral that it seizes, and default incurs no extra costs to the lender or the borrower.

An equilibrium can be one of two types: an equilibrium with credit rationing wherein borrowers

always default and an equilibrium with no credit rationing. In the equilibrium with credit rationing,

a situation in which the borrower’s marginal benefit of borrowing is greater than the marginal cost,

the asset price exhibits a liquidity premium. A new insight is that in this equilibrium, borrowers

always default and banks seize collateral, and thus, the credit arrangement is as if the asset were

sold to the bank at the price equal to the loan amount when the debt contract was issued. This

implies that banks extending credit secured by assets function like an asset market in which people

who need liquidity can liquidate assets. In the equilibrium with no credit rationing, the asset is

priced at the discounted sum of expected dividends.4

4Li and Li (2013) show that there are two types of equilibria, characterized by whether or not there is creditrationing; moreover, if seizing collateral is the only punishment on defaulters, borrowing money from banks andselling assets to banks result in the identical asset price and allocation. Though we consider a stochastic environmentand implement different lending strategies from Li and Li (2013), we share similar results because banks can freelydiversify the asset risks when there are no foreclosure costs.

2

Once we introduce foreclosure costs, only the no-credit-rationing equilibrium exists, and the

default probability is strictly less than one. In this equilibrium, the marginal benefit of borrowing

is equal to the marginal cost, whereas the asset is priced above the expected fundamental value,

unlike in the benchmark case without foreclosure costs. The risk premium is zero when there are no

foreclosure costs, because banks can freely diversify the risk of collateral. However, diversification

cannot reduce the loss associated with the foreclosure cost because that cost is per contract and is

identical across borrowers. Thus, a risk premium arises here as a compensation for banks to make

risky loans. Moreover, the larger the debt obligation, the higher the incentive to default and the

higher the probability that the foreclosure cost occurs, which makes it suboptimal to set the loan

amount at such a level that the borrower always defaults.

Using the same reasoning, we observe that as the foreclosure cost increases to drive up the

risk premium, borrowers find it even more expensive to trade off the loan amount with the default

probability. It turns out that the loan amount, default probability, and loan-to-value ratios all fall

with the foreclosure cost, but the asset price increases. The reason is that the fall in loan-to-value

ratios makes borrowers need more assets to meet their liquidity needs, which drives up the asset

price. The resulting implication is that financial development or a policy that lowers the foreclosure

cost can have positive impacts on aggregate liquidity and total output, though the default rate will

rise.

We then proceed to illustrate the effects of increased asset risk on aggregate liquidity and output.

The exercise we consider is the mean-preserving spread that amounts to a second-order stochastic

dominance shift in the distribution of asset dividends. We show that though defaults increase as

the asset becomes riskier, the risk premium and lending may not. Increased asset risks exert two

opposite effects on the risk premium in our model: the loan contract circulating in the economy

becomes riskier, which reduces the expected return on the loan for a given repayment, and the

mean-preserving spread directly decreases the hazard rate, which implies the tendency to default is

mitigated. If the latter effect dominates, the risk premium falls with increased risk, and the value

of money and the real loan amount rise; moreover, the asset price is more likely to fall, a situation

which, together with higher loan amounts, results in higher loan-to-value ratios. In contrast, if the

former effect dominates, the risk premium rises, while the asset price and lending decrease, leading

3

to the fall of aggregate liquidity and total output. Our results suggest that increased asset risk has

an ambiguous effect on aggregate liquidity and output, and one needs to be careful about the asset’s

risk properties when specifying the relationship between the risk of an asset and its liquidity and

price.

While the debt contract in our model with foreclosure costs is similar to that in Williamson (1986,

1987, 2012) wherein lenders incur the monitoring cost when defaults occur, there are important

distinctions as follows. In Williamson’s models, the loan amount is exogenously given, and borrowers

would lower interest rates as much as possible provided that the loans remained accessible, and credit

rationing may occur. In the present paper, however, borrowers can trade off among the loan amount,

repayment, and default probability, so that no credit rationing occurs; moreover, this trade-off yields

the result that the increased risk may increase or decrease the risk premium and lending. In our

model, loans are backed by market-traded assets, while in Williamson’s models loans are backed

by entrepreneurs’ endowed investment projects. Thus, another value-added of our paper is that

we endogenize the price of an asset, which constitutes an additional mechanism that can affect

loan-to-value ratios and aggregate liquidity.

The current paper is related to the literature that resorts to credit market imperfections to

motivate credit constraints or resalability constraints, such as in Kiyotaki and Moore (1997, 2005).5

There are studies that use search monetary models with limited enforcement and commitment

and focus on how government regulations, recognizability, and credit market frictions affect the

acceptability of an asset as a means of exchange; for example, Lagos (2010), Rocheteau (2011),

Lester, Postlewaite, and Wright (2012), and Li, Rocheteau, and Weill (2012). Studies that focus

on the role of assets as collateral include the following. Ferrais and Watanabe (2008) consider a

productive input as collateral with the aim of determining the effect on capital accumulation. Li

and Li (2013) examine how the endogenously derived loan-to-value ratios vary with the efficiency

of debt enforcement. Geromichalos et al. (2016) study how the degree of liquidity in the secondary

market affects the price of the asset used as collateral, and in Geromichalos et al. (2020) an agent’s

5In Kiyotaki and Moore (1997), financial contracts are imperfectly enforceable, and creditors protect themselvesfrom the threat of repudiation by collateralizing borrowers’ debt. Kiyotaki and Moore (2005) assume constraints ondebt issuance and resalability of private claims due to limited commitment so that borrowers can sell an exogenousfraction of their capital to finance investment.

4

decision about entering the secondary market would affect the liquidity of a risky asset. Phelan

(2017) considers a general equilibrium model to study how increasing risk in endowments and future

payoffs affects the loan amount, leverage, and asset prices. The distinction of the current paper

from the literature is that we study how the liquidity and price of an asset used as collateral and

default rates are influenced by asset risk and foreclosure costs, and how these effects transmit to

impact aggregate liquidity and total production.

The rest of the paper is organized as follows. Section 2 describes the model. In section 3, we

derive the equilibrium conditions without foreclosure costs. In Section 4, we derive the equilibrium

with foreclosure costs and we discuss some comparative statics regarding changes in the foreclosure

cost, asset risk, inflation, and macroeconomic conditions. Section 5 concludes. All proofs and

omitted derivations of equations are contained in the Appendix.

2 The Model

The model is based on Berentsen, Camera, and Waller (2007), in which we incorporate an asset

used as collateral as in Ferrais and Watanabe (2008) and Li and Li (2013). Time is discrete and

continues forever. Each period is divided into two subperiods, and in each subperiod trades occur

in competitive markets. There are two consumption goods, one produced in the first subperiod, and

the other (called the general good) in the second subperiod. Consumption goods are perishable and

perfectly divisible. There is a [0, 1] continuum of infinitely lived agents. The discount factor across

periods is β ∈ (0, 1), but there is no discounting between subperiods.

In the beginning of the first subperiod, an agent receives a preference shock that determines

whether he consumes or produces. With probability n an agent can produce but cannot consume,

while with probability 1 − n the agent can consume but cannot produce. This is a simple way to

capture the uncertainty of the opportunity to trade. Consumers get utility u(q) from q consumption.

Producers incur disutility c(q) from producing q units of output. Assume u(0) = c(0) = 0, u′(q) >

0, c′(q) > 0, u′(0) =∞, u′′(q) < 0 and c′′(q) ≥ 0. We refer to producers as sellers and consumers as

buyers. Assume that all goods trades are anonymous, and there is no public record of individuals’

trading histories, a situation that yields a role for fiat money.

In the second subperiod, agents get utility U(x) from x consumption, with U ′(x) > 0, U ′(0) =∞,

5

U ′(∞) = 0 and U ′′(x) ≤ 0. Agents can produce one unit of the general good with one unit of labor,

which generates one unit of disutility. This setup allows us to introduce an idiosyncratic preference

shock while keeping the distribution of asset holdings analytically tractable.

There are two types of infinitely lived assets in the economy: fiat money and a real asset like

the claims to Lucas (1978) trees. A government controls the supply of fiat money. The money stock

evolves deterministically at a gross rate γ by means of lump-sum transfers, Mt = γMt−1, where

γ > 0, andMt denotes the per capita money stock in period t. There is a fixed supply, A, of divisible

Lucas trees per capita, and these can be interpreted as private equity. The asset held by an agent is

subject to an idiosyncratic shock—it generates a dividend of ρt units of the general good, and ρt is

an independently and identically distributed variable with the density function f(·) on [ρ, ρ], with

0 < ρ < ρ <∞. The density function is positive and differentiable, with the mean E(ρt) = ρµ. Let

F (ρ) be the cumulative density function. The dividend is realized in the second subperiod before

the asset is traded, and the realization of dividends is public information. For simplicity, we do not

consider aggregate uncertainty in this economy.

Competitive banks accept deposits from people with idle cash and extend credit to those with

liquidity needs. They accept nominal deposits and make nominal loans. In the first subperiod,

sellers can deposit their money holdings in banks, and buyers may borrow money from banks. The

financial contracts are settled in the second subperiod wherein depositors are entitled to withdraw

funds and borrowers need to repay their loans. In this environment, without loss of generality, we

consider that deposits and loans are not rolled over, and so all financial contracts are one-period

contracts. Banks have zero net worth, and there are no operating costs or reserve requirements.

We assume that banks can commit to repay deposits; however, they have limited ability to

force repayment of debts. Because of this financial friction, borrowers need to pledge some assets

as collateral to secure loans. Once borrowers renege on debts, banks are entitled to the collateral;

that is, banks seize the pledged asset and its accrued dividend. Because the dividend of collateral is

uncertain and because repayment is not contingent on dividend realizations, if the value of collateral

(the asset price plus realized dividends) falls below the debt obligation, a borrower can choose to

default. Except for seizing collateral, there is no other punishment on defaulters such as exclusion

from the credit market (but see, for example, Berentsen, Camera, and Waller 2007, and Li and Li

6

2013). Moreover, banks cannot exclude defaulters from the future asset market, and they are not

able to seize a defaulter’s assets or income in the future. Therefore, instances of default are treated

as discrete, and borrowers’ behavior has no effect on their future transactions. In the benchmark

model, we assume that banks incur no costs of seizing collateral. We then incorporate another credit

market friction—the foreclosure cost—and assume that a bank incurs a cost (in terms of general

good), ξ, to seize collateral when default occurs.6

3 Equilibrium

Let φt and ψt denote the values of money and the real asset in terms of the general good produced

in the second subperiod, respectively. We study symmetric stationary equilibria in which the value

of money holdings is constant: φtMt = φt−1Mt−1, which implies φt−1

φt= γ; the inflation rate is equal

to the money growth rate. The dividend is realized before the asset is traded, and so the real asset

price does not depend on the realized dividend, denoted as ρt. Moreover, because we focus on the

stationary equilibrium, and the asset supply and aggregate realized dividends do not change over

time, it is reasonable to consider a constant price of the real asset; hence, ψt−1 = ψt = ψ for all t.

Let Vt(m, a) denote the expected value from entering in the first subperiod of the period t with

m units of money and a units of real assets. Let W kt (m, a, `r, d), k = b, s, denote the expected value

(before the dividend is realized) from entering the second subperiod of the period t to a buyer and

seller, respectively, with m units of money, a units of real assets, `r debt, d deposits, where loans

and deposits are in the units of fiat money. We study a representative period t and work backwards

from the second to the first subperiod. We restrict our attention in stationary equilibria, and thus,

in what follows we drop the time subscript t when there is no confusion.

3.1 The second subperiod

In the second subperiod an agent produces h goods and consumes x, repays or defaults on the loan,

redeems deposits, and adjusts his holdings of fiat money and real assets. If an agent has deposited

6We focus on the role of risky assets as collateral, and thus, we do not consider the possibility that the borrowermay sell assets to banks or in the secondary market as in, e.g., Geromichalos et al. (2016, 2020). Suppose that thereis a transaction cost when the buyer sells assets to the bank. If the transaction cost is big enough, the buyer willprefer using the asset as collateral to selling it to the bank.

7

d in the first subperiod, he receives (1 + id)d units of money in the second subperiod. If he has

borrowed `b, he either repays `r units of money or defaults on the loan. A unit of the real asset

brought to the second subperiod is worth ψ + ρ units of the general good, because it generates a

dividend, ρ, and can be traded at the price ψ.

The default strategy. After the dividend is realized, a borrower decides whether to repay or

default. Let wj(m, d, `r, d, ρ), j = R,D, be the value to an agent who chooses to repay the debt,

and default, respectively. If a borrower chooses to repay the debt, his continuation value is given by

wR(m, a, `r, d, ρ) = maxx,h,m,a

U(x)− h+ βV (mR, aR)

s.t. x+ φmR + ψaR = h+ φ(m+ T ) + φ(1 + id)d+ (ψ + ρ)a− φ`r,

where T denotes transfers, and V (mR, aR) is the continuation value with a new portfolio, (mR, aR),

in the first subperiod of next period. A defaulter’s continuation value is given by

wD(m, d, `r, d, ρ) = maxx,h,m,a

U(x)− h+ βV (mD, aD)

s.t. x+ φmD + ψaD = h+ φ(m+ T ) + φ(1 + id)d.

Because the only punishment on defaulters is seizing collateral, the benefit to a defaulter is to enjoy

more leisure, while the cost is the loss of collateral.

Let (xj∗, mj , aj), j = R,D, be the optimal choice for an agent who chooses to repay the debt

and default, respectively. Substituting h from the budget constraint to the objective function in

the above problems, we obtain the following first-order conditions for nondefaulters and defaulters:

U ′(xj∗) = 1, and

φ ≥ βVm(mj , aj), “ = ” if mj > 0, (1)

ψ ≥ βVa(mj , aj), “ = ” if aj > 0, (2)

for j = R,D, where Vm(mj , aj) and Va(mj , aj) are the marginal values of an additional unit of

money and the real asset to an agent j taken into the next period. Because of the linearity of w(·),

the portfolio (mj , aj) is independent of type j’s holdings of m and a in the second subperiod. Thus,

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the optimal portfolio choice does not depend on an agent’s status of repaying the debt, and all

agents choose the same optimal portfolio, denoted as (m, a).

With the optimal choices, (x∗, m, a), we rewrite wD and wR as

wD(m, a, `r, d, ρ) = U(x∗)− x∗ − φ(m+ T ) + φ(1 + id)d− φm− ψa+ βV (m, a), (3)

and

wR(m, a, `r, d, ρ) = U(x∗)− x∗ + φ(m+ T ) + φ(1 + id)d− φm− ψa

+(ψ + ρ)a− φ`r + βV (m, a). (4)

A borrower chooses to repay the debt if wR ≥ wD; which is

(ψ + ρ)a ≥ φ`r. (5)

He repays when the collateral value (the asset price plus the realized dividend) exceeds the real debt

amount; otherwise, he defaults and abandons collateral. Note that we obtain the simple decision

rule of default, (5), because of the linearity of wj(·) and no exclusion of defaulters.

Because in equilibrium sellers do not take loans and buyers do not make deposits, in what follows

we let `r denote the debt of buyers and d the deposits of sellers. For notational simplicity, we also

drop these arguments in W k(m, a, `r, d) when no confusion is caused. Let ρ∗ denote the value of

the dividend at which wD = wR; that is, a borrower is indifferent between repaying and defaulting

on the debt. From (5), ρ∗ = φ`ra −ψ. The expected utility of a buyer and a seller from entering the

second subperiod are, respectively,

W b(m, a, `r) ≡ Emax(wD, wR) =

∫ ρ∗

ρwD(m, a, `r, ρ)f(ρ)dρ+

∫ ρ

ρ∗wR(m, a, `r, ρ)f(ρ)dρ,

W s(m, a, d) = U(x∗)− x∗ + φ(m+ T ) + φ(1 + id)d− φm− ψa+ (ψ + ρµ)a+ βV (m, a).

Again, because of the linearity of W k(m, a, `r, d), agents choose the same optimal portfolio, (m, a),

regardless of whether they were a seller or a buyer in the first subperiod of period t.

Let Π denote the default probability. Then, given ρ∗, a borrower’s default probability is Π =

Pr[(ψ + ρ)a < φ`r] = F (ρ∗). That is,

F (ρ∗) =

∫ ρ∗

ρf(ρ)dρ. (6)

9

The envelope conditions are

W bm = W s

m = φ, (7)

W sd = φ(1 + id), (8)

W b`r = −φ [1− F (ρ∗)] , (9)

W ba =

∫ ρ

ρ∗(ψ + ρ)f(ρ)dρ, (10)

W sa = ψ + ρµ. (11)

(See Appendix A for the derivation.) Condition (9) says that bringing an additional unit of debt to

the second subperiod results in the expected repayment cost, φ [1− F (ρ∗)]. Condition (10) says that

a borrower’s expected payoff to holding an additional unit of the asset as collateral is the expected

return of claiming it back if not defaulting on the debt.

3.2 The first subperiod

We now consider trade in the goods market in the first subperiod. Let qb and qs denote the quantities

consumed by a buyer and produced by a seller, respectively, and let p denote the nominal price of

the good. An agent may be a buyer with probability 1 − n, spending pqb units of money to get

qb consumption, or he may be a seller with probability n, receiving pqs units of money from qs

production. An agent holding a portfolio of (m, a) entering the first subperiod has the expected

lifetime utility

V (m, a) = (1− n)[u(qb) +W b(m+ `b − pqb, a, `r)] + n[−c(qs) +W s(m− d+ pqs, a, d)]. (12)

As agents trade in the centralized market, they take the price, p, as given. A seller solves

maxqs,d

−c(qs) +W s(m− d+ pqs, a, d)

s.t. d ≤ m.

Let λd denote the multiplier on the deposit constraint. The first-order conditions are

−c′(qs) + pW sm = 0,

−W sm +W s

d − λd = 0.

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Using (7) and (8), the first-order conditions become

p =c′(qs)

φ, (13)

λd = φid.

Equation (13) implies that a seller’s production is such that the marginal cost of producing, c′(qs),

equals the marginal real revenue, pφ. The production qs is independent of the seller’s initial portfolio

brought to the first subperiod. Moreover, for any id > 0, the deposit constraint binds, and sellers

deposit all money balances; that is, d = m.

There are credit market frictions that interfere with lending activity: if a default occurs, the

bank incurs cost γ to seize collateral. Thus the debt contract has the feature of the costly state-

verification problem (see, e.g., Townsend 1979 and Williamson 1987). Banks are assumed to be risk

neutral and face an opportunity cost of funds, which is the deposit rate, id. The debt contracts

offering zero expected profit are sufficient to ensure the participation of competitive banks. We

consider a one-period contract that specifies the nominal loan amount, `b, and the repayment of

debt, `r, where (`b, `r) ∈ R2+ ∪ {0, 0}. The contract maximizes the borrowers’ expected utility while

satisfying the bank’s participation constraint.

Given a loan contract, (`b, `r), and the borrower’s default rule, (5), a bank’s expected profit (per

borrower) is

πB =

∫ ρ∗

ρ(ψ + ρ)af(ρ)dρ+

∫ ρ

ρ∗φ`rf(ρ)dρ− φ(1 + id)`b

=

∫ ρ∗

ρ[(ψ + ρ)a− φ`r]f(ρ)dρ+ φ`r − φ(1 + id)`b, (14)

where φ(1 + id)`b is the real cost of funds from lending `b. Given that a bank serves a large number

of borrowers, it holds a perfectly diversified portfolio of assets and earns zero profits. The first term

in (14),∫ ρ∗ρ [(ψ+ρ)a−φ`r]f(ρ)dρ, is negative, and is a loss to the bank once default occurs because

the collateral value cannot compensate for the real repayment, φ`r. Alternatively, define

S ≡∫ ρ∗

ρ[φ`r − (ψ + ρ)a]f(ρ)dρ, (15)

which represents the option value that could be retrieved by a borrower when he decides to exercise

the put option by defaulting on the loan and leaving the collateral to the bank. Thus, (14) can be

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written as πB = −S + φ`r − φ(1 + id)`b; that is, the borrower’s default option value is subtracted

from the bank’s profit. Note that S ≥ 0, and equality holds when ρ∗ ≤ ρ, under which no default

occurs.

We now characterize the debt contract. A buyer uses money holding, m, plus the loan, `b,

to finance his consumption, qb. A borrower’s problem is to choose `b, `r, and qb, to maximize his

expected utility, satisfying his budget constraint and the bank’s participation constraint:

maxqb,`b,`r

u(qb) +W b(m+ `b − pqb, a, `r)

s.t. pqb ≤ m+ `b,

πB ≥ 0.

The participation constraint, πB ≥ 0, must hold with equality; otherwise, one can raise `b to increase

qb, and hence, the value of the objective function, while the constraint still holds. Thus, πB = 0 in

equilibrium, from which we obtain

`b =

∫ ρ∗ρ [(ψ + ρ)a− φ`r]f(ρ)dρ+ φ`r

φ(1 + id)≡ `b(`r), (16)

where ρ∗ = φ`ra − ψ.

We emphasize that in the economy without foreclosure costs, the ex ante loan rate of the risky

loan is the risk-free rate, which implies that the risk premium, a compensation for banks to make

risky loans, is zero. To see this, let ie denote the ex ante loan rate of borrowing `b when a debt

contract is issued. Then,

1 + ie =Emin[φ`r, (ψ + ρ)a]

φ`b=

∫ ρ∗ρ (ψ + ρ)af(ρ)dρ+

∫ ρρ∗ φ`rf(ρ)dρ

φ`b= 1 + id,

where the last equality is obtained by using πB = 0 from (14). The ex ante marginal borrowing

cost, ∂Emin[φ`r,(ψ+ρ)a]∂(φ`b)

, is also equal to 1 + id. Given that one unit of money deposited in the bank

receives 1 + id with certainty, we take id as the risk-free rate in this economy. Moreover, banks’ zero

profit condition implies that the deposit rate, id, would be the loan rate if there were no uncertainty

in asset dividends. The reason for the ex ante loan rate’s being equal to the risk-free rate is that, by

serving a large number of borrowers, a bank can diversify the risk of dividends of the collateral that

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it seizes if default incurs no extra costs such as a foreclosure cost or monitoring cost. Alternatively,

let ir ≡ `r`b− 1 denote the ex post loan rate if the borrower repays the debt. Using (14) and πB = 0,

we have

1 + ir =−∫ ρ∗ρ [(ψ + ρ)a− φ`r]f(ρ)d(ρ) + φ`b(1 + id)

φ`b= 1 + id +

S

φ`b. (17)

Then, ir ≥ id because option value S ≥ 0. That is, the bank charges a loan rate higher than the

risk-free rate (also the funding cost), id, to compensate for the loss caused by default.

The trade-off between the loan amount, `b, and repayment, `r, depends on the default probability

and the ex ante borrowing cost. From (16),

∂`b∂`r

=1− F (ρ∗)

1 + id. (18)

Equation (18) shows that the extra loan amount that a borrower can get by increasing repayment

is negatively related to the probability of default and the expected borrowing cost.

Using `b = `b(`r) from (16), we now solve for the debt contract. The buyer’s problem becomes

maxqb,`r

u(qb) +W b (m+ `b(`r)− pqb, a, `r)

s.t. pqb ≤ m+ `b(`r).

Let λb denote the multiplier on the budget constraint. The first-order conditions are as follows:

u′(qb)

c′(qs)= 1 +

λbφ, (19)

φ∂`b∂`r− φ [1− F (ρ∗)] + λb

∂`b∂`r

= 0 (20)

Substituting λbφ from (19) into (20), we obtain

∂`b∂`r

[u′(qb)

c′(qs)− (1 + id)

]= 0, (21)

Note that u′(qb)c′(qs)

is the marginal benefit of an additional unit of money borrowed, while the ex ante

marginal loan rate equals 1 + id. The difference, u′(qb)c′(qs)

− (1 + id), thus measures the degree of credit

rationing.7 From (21), when there is credit rationing, that is, u′(qb)c′(qs)

> 1 + id, it must be ∂`b∂`r

= 0,

7This is different from papers with limited enforcement and commitment but without uncertainty (for example,Berentsen, Camera, and Waller 2007, and Li and Li 2013), wherein borrowers are credit constrained because lendersimpose the credit limit such that borrowers will not default.

13

meaning that the buyer borrows up to the point where an increase in the repayment can no longer

raise the loan amount.

Using (18) and (21), we obtain the relationship between the default probability and the degree

of credit rationing as follows:

[1− F (ρ∗)]

[u′(qb)

c′(qs)(1 + id)− 1

]= 0. (22)

We summarize results in the following proposition.

Proposition 1 In an economy with no foreclosure costs, the default probability, F (ρ∗), and the

severity of credit rationing, u′(qb)c′(qs)

− (1 + id), are related in equilibrium: (i) if F (ρ∗) < 1, thenu′(qb)c′(qs)

= 1 + id and there is no credit rationing; (ii) if there is credit rationing, u′(qb)c′(qs)

> 1 + id, then

F (ρ∗) = 1; and (iii) u′(qb)c′(qs)

= 1 + id and F (ρ∗) = 1. Moreover, the risk premium is zero.

Finally, from (12) the marginal values of money and the real asset are given by

Vm(m, a) = φ

[(1− n)

u′(qb)

c′(qs)+ n(1 + id)

](23)

Va(m, a) = (ψ + ρµ) + (1− n)

[u′(qb)

c′(qs)(1 + id)− 1

][∫ ρ∗

ρ(ψ + ρ)f(ρ)dρ

]. (24)

In a stationary monetary equilibrium, we use (1) and (2) lagged one period to eliminate Vm(m, a)

and Va(m, a) from (23) and (24), respectively. Then, using the stationarity condition, we obtain

γ − ββ

= (1− n)

[u′(qb)

c′(qs)− 1

]+ nid. (25)

In a symmetric equilibrium, the goods market clearing condition in the first subperiod is nqs =

(1− n)qb. The market clearing conditions (in per capita terms) for loans, money, and the real asset

are (1 − n)`b = nd,m = M−1 where M−1 is the money supply of the last period, and a = A,

respectively.89

8Banks do not demand for the real asset in the asset market if the asset bears a positive liquidity premium abovethe expected fundamental value.

9Substituting market clearing conditions of money and loans into the binding budget constraint, we obtainpqb = m + ` = M−1 + n

1−nM−1. Hence, (1 − n)pqb = M−1. Substituting M−1 = 1−nn` into the previous expression,

we have ` = npqb. Using p = c′(qs)φ

, we obtain φ` = nc′(qs)qb, shown in Definition 1.

14

Definition 1 A monetary equilibrium is a list, (φ, ψ, id, qb, `b, `r, ρ∗), satisfying the market clearing

conditions, πB = 0, (22), (25),

ψ = β

{(ψ + ρu) + (1− n)

[u′(qb)

c′(1−nn qb)(1 + id)− 1

][∫ ρ∗

ρ(ψ + ρ)f(ρ)dρ

]}, (26)

ρ∗ = φ`ra − ψ, nc

′(1−nn qb)qb = φ`b, and (1− n)`b = nM−1.

3.3 Two types of equilibria

From Proposition 1, there are two types of equilibria, characterized by whether or not there are

credit rationing and the incidence of default.

Case 1. u′(qb)c′(qs)

= 1 + id and F (ρ∗)≤1. Substituting u′(qb)c′(qs)

= 1 + id into (25) and (26), we obtain

id =γ − ββ

, (27)

u′(qb)

c′(1−nn qb)=

γ

β, (28)

ψ =βρu

1− β. (29)

Although loans are risky, the uncertainty of dividends does not matter for the allocation and prices.

When there is no credit rationing, consumption is determined by γβ , and the real asset is priced at

the present value of expected dividends, the asset’s fundamental value.

Case 2. u′(qb)c′(qs)

> 1 + id and F (ρ∗)= 1. When the buyer is constrained by ∂`b∂`r

= 1−F (ρ∗)1+id

= 0, he

cannot borrow more by increasing repayment. From (16) the loan amount is

`b =(ψ + ρµ)a

φ(1 + id). (30)

In this equilibrium borrowers always default, and banks seize collateral. This case functions as if

an asset were sold to a bank at the price of `b shown in (30) when the debt contract is issued, and

the bank receives the asset return, (ψ + ρ)a, in the second subperiod. Banks’ issuing credit thus

functions like an asset market wherein people with a liquidity need can liquidate assets. The loan

amount in (30) depends on the average return of the asset, and borrowers transfer all the risk to

banks.

15

Full diversification of a bank’s portfolio implies that the asset price does not hinge on the risk

of asset returns.10 To see this, substituting F (ρ∗) = 1 into (26), we have

ψ =βBρµ

1− βB, (31)

where

B = 1 + (1− n)

[u′(qb)

c′(qs)(1 + id)− 1

].

The asset price is the expected present value of dividends, with βB as the effective discount factor.

Given that borrowers face credit rationing, u′(qb)c′(qs)

> 1 + id, and B > 1, the asset price in this

equilibrium is higher than the fundamental price, βρu1−β . And this liquidity premium is increased by

the severity of credit rationing, u′(qb)c′(qs)

− (1 + id).11

As a final remark, the distribution of dividends does not matter for the asset price and allocation

in this economy; what matters is the average dividend, ρµ. Although agents do not write contracts

contingent on asset dividend realizations, allowing agents to default and banks to seize collateral and

diversify risks make the model works like one wherein agents could have written complete contracts

so that consumption, qb, is not affected by the uncertainty of asset returns. This result no longer

holds once we introduce additional credit market frictions, captured by the foreclosure cost.

4 An economy with foreclosure costs

Banks bear a cost, ξ > 0, per contract of seizing and selling collateral once default occurs. In this

economy, there is only one asset used as collateral and borrowers are ex ante identical and, thus, we

assume that the foreclosure cost, ξ, is identical across borrowers.12 We will show how the liquidity

of an asset is intertwined with its risk and foreclosure costs, and we will also examine the effects

10In this economy financial intermediaries can diversify risks, whereas diversification is not available to individuallenders; see, for example, Williamson (1986). In Williamson’s paper, because a loan contract needs more than onelender’s savings, banks are more efficient in gathering all the funds necessary to lend to a borrower, and incurringthe monitoring cost at most once per borrower.

11We compare asset prices, allocation, and liquidity in both types of equilibria from numerical examples. In theequilibrium with no credit rationing, the loan-to-value ratio (defined as φ`b

ψa), real loan amount, φ`b, and consumption

qb are higher, while the real repayment, φ`r, is lower than in the equilibrium with credit rationing.12One can include many assets or heterogenous types of borrowers, and foreclosure costs vary across assets or

across borrowers. This situation implies that banks may reduce the loss associated with foreclosure costs to someextent by diversification.

16

of asset risks, foreclosure costs, and macroeconomic conditions on default rates, aggregate liquidity,

and output.

4.1 The debt contract under foreclosure costs

The borrower’s default strategy remains the same as in the benchmark model: he repays φ`r if the

realized dividend ρ ≥ ρ∗, while he defaults and forgoes the collateral value, (ψ + ρ)a, if ρ < ρ∗,

where ρ∗ = φ`ra −ψ. A bank obtains (ψ+ ρ)a− ξ if default occurs, and it receives φ`r if the debt is

repaid. The bank’s expected profit becomes

πξB =

∫ ρ∗

ρ[(ψ + ρ)a− ξ]f(ρ)dρ+

∫ ρ

ρ∗φ`rf(ρ)dρ− φ`b(1 + id)

=

∫ ρ∗

ρ[(ψ + ρ)a− φ`r]f(ρ)dρ− ξF (ρ∗) + φ`r − φ`b(1 + id). (32)

Compared with (14), the expected foreclosure cost, ξF (ρ∗), which captures additional losses in the

incidence of default, is subtracted from the bank’s profit in (32).

The contract problem is similar to that described in Section 3 except that here the loan amount

is derived from πξB = 0:

`b =

∫ ρ∗ρ [(ψ + ρ)a− φ`r]f(ρ)dρ+ φ`r − ξF (ρ∗)

φ(1 + id)≡ `b(`r). (33)

From (33), we derive the relationship between the loan amount and repayment

∂`b∂`r

=1− F (ρ∗)− ξ

af(ρ∗)

1 + id. (34)

Compared to (18), (34) incorporates the foreclosure cost, which, by reducing the loan amount, `b,

or increasing repayment, `r, acts as a hindrance to receiving more credit.

The first-order condition to the contract problem is

∂`b∂`r

[u′(qb)

c′(qs)− (1 + id)−

ξ

φ

∂F (ρ∗)

∂`b

]= 0, (35)

from which we derive the following lemma.

Lemma 1 In equilibrium, ∂`b∂`r

> 0, and therefore, u′(qb)c′(qs)

− (1 + id)− ξφ∂F (ρ∗)∂`b

= 0.

17

Note that when there are foreclosure costs, the ex ante loan rate, ie, is no longer the risk-free

rate. Now ie satisfies

1 + ie =Emin[φ`r, (ψ + ρ)a]

φ`b=

∫ ρ∗ρ (ψ + ρ)af(ρ)dρ+

∫ ρρ∗ φ`rf(ρ)dρ

φ`b= 1 + id +

ξF (ρ∗)

φ`b, (36)

where the last equality is obtained by using πξB = 0 from (32). In contrast to the benchmark

model, the ex ante loan rate here incorporates the expected foreclosure cost per unit of loan, ξF (ρ∗)φ`b

,

beyond the risk-free rate. Although banks can diversify the asset risk, they cannot reduce the loss

associated with foreclosure costs by diversification because that cost is per contract and is identical

across borrowers.13

The ex ante marginal loan rate is

∂Emin[φ`r, (ψ + ρ)a]

∂(φ`b)= 1 + id +

ξ

φ

∂F (ρ∗)

∂`b. (37)

In the last term of (37), ∂F (ρ∗)∂`b

captures the increase in the default rate driven by an additional unit

of money borrowed, which results in extra default costs, ξφ∂F (ρ∗)∂`b

. From Lemma 1, because ∂`b∂`r

> 0

we can use ∂F (ρ∗)∂`b

= ∂F (ρ∗)∂`r

/ ∂`b∂`r, ∂F (ρ∗)

∂`r= φ

af(ρ∗) > 0, and (34) to derive

∂F (ρ∗)

∂`b=

φaf(ρ∗)(1 + id)

1− F (ρ∗)− ξaf(ρ∗)

> 0, (38)

where the inequality is obtained by using (34) and the fact that ∂`b∂`r

> 0.14

We measure the risk premium, a compensation for banks to make risky loans, as the difference

between the ex ante marginal loan rate and the risk-free rate, which would be the ex ante marginal

loan rate if there was no uncertainty in the recovery value of collateral.15 Using (37) and (38), we

13The ex post loan rate if the borrower repays, ir, in this economy satisfies

1 + ir ≡`r`b

=S + ξF (ρ∗) + φ(1 + id)`b

φ`b= 1 + id +

S + ξF (ρ∗)

φ`b,

where S defined in (15) is the default option value.14If the condition f(ρ) + ξ

af ′(ρ) > 0 holds, then (34) means that when 1 − F (ρ∗) − ξ

af(ρ∗) = 0, the amount of

liquidity generated by the asset is maximized. Lemma 1 shows that ∂`b∂`r

> 0; that is, 1−F (ρ∗)− ξaf(ρ∗) > 0, so that

the maximum amount of liquidity is not achievable in this economy.15The risk premium can be interpreted as the wedge—the difference between the marginal benefits of borrowing,

u′(qb)c′(qs)

, with and without foreclosure costs. Without foreclosure costs, the marginal benefit of borrowing equals the

risk-free rate, 1 + id. Hence, the wedge is the difference between u′(qb)c′(qs)

with foreclosure costs, and 1 + id.

18

have

risk premium =ξ

φ

∂F (ρ∗)

∂`b=

ξaf(ρ∗)(1 + id)

1− F (ρ∗)− ξaf(ρ∗)

, (39)

which is strictly positive when ξ > 0. Without foreclosure costs (ξ = 0), banks can freely diversify

the risk of dividends of the collateral so that the risk premium is zero.

We summarize the results regarding the equilibrium allocation, credit rationing, and default

rates in the following proposition.

Proposition 2 In an economy with foreclosure costs, buyers borrow up to the point where the

marginal benefit of borrowing equals the ex ante marginal loan rate; that is,

u′(qb)

c′(qs)= 1 + id +

ξ

φ

∂F (ρ∗)

∂`b, (40)

and no credit rationing occurs in equilibrium. Moreover, ∂F (ρ∗)∂`b

> 0, the risk premium is strictly

positive, and F (ρ∗) < 1.

Results in Proposition 2 are in contrast to the benchmark model with ξ = 0, where the ex ante

loan rate is the risk-free rate, and credit rationing occurs if u′(qb)c′(qs)

> (1 + id), and buyers cannot

borrow more because of the upper bound on the default probability. Proposition 2 shows that

when ξ > 0, the marginal benefit always equals the marginal borrowing cost and no credit rationing

occurs in equilibrium. The ex ante marginal loan rate in (37) shows that as the borrower wishes

to increase the loan amount, `b, he needs to raise repayment, and that in turn, causes ρ∗ and the

default probability, F (ρ∗), to rise, and so does the expected foreclosure cost, ξF (ρ∗). The existence

of the foreclosure cost thus prevents agents from borrowing to the point at which defaulting on the

debt with probability one is optimal.

For the following discussion, it is convenient to use the concept of hazard functions. Define the

hazard function as

H(ρ) =f(ρ)

1− F (ρ).

Then, H(ρ∗) can be interpreted as a measure of the tendency to default.16 We assume

H ′(ρ) > 0. (41)16Loosely speaking, given ρ∗, H(ρ∗) can be interpreted as the probability of failing to repay debt in a very small

time interval, t to t+ ∆t, given that the borrower does not default until t.

19

Under this assumption, the ex ante marginal loan rate (that is, the marginal cost) of raising `b is

increasing if H ′(ρ∗) > 0.17

From (40) the risk premium equals u′(qb)c′(qs)

− (1 + id), where the marginal benefit of borrowing,u′(qb)c′(qs)

, measures the liquidity needs, and id is the deposit rate, which positively affects the supply

of funds. Therefore, the larger the risk premium, the higher the degree of liquidity shortage. Let

η = u′(qb)c′(qs)(1+id)

represent a (normalized) measure of the risk premium. From (39) and (40), we derive

η =u′(qb)

c′(qs)(1 + id)=

1

1− ξaH(ρ∗)

. (42)

The measure of the risk premium, η, is positively affected by the hazard rate.

Finally, the marginal value of the real asset is

Va(m, a) = ψ+ρµ+(1−n)

[u′(qb)

c′(qs)(1 + id)− 1

]{[1− F (ρ∗)] (ψ + ρ∗) +

∫ ρ∗

ρ(ψ + ρ)f(ρ)dρ

}. (43)

The market clearing conditions for goods, loans, money, and the real asset are as in Section 3. We

have the following definition of equilibrium.

Definition 2 A monetary equilibrium with foreclosure costs is a list, (φ, ψ, id, qb, `b, `r, ρ∗), satisfy-

ing πξB = 0, (25), (40),

ψ = β

{(ψ + ρµ) + (1− n)(

u′(qb)

c′(qs)(1 + id)− 1)

[ψ + (1− F (ρ∗))ρ∗ +

∫ ρ∗

ρρf(ρ)dρ

]},

ρ∗ = φ`ra − ψ, nc

′(1−nn qb)qb = φ`b, and (1− n)`b = nM−1.

We show in Appendix B that under some conditions, the equilibrium value of ρ∗ exists and is

unique. Given the parametric assumptions about u(q) and c(q), the equilibrium endogenous vari-

ables, (φ, ψ, id, qb, `b, `r), can be expressed as functions of ρ∗;18 therefore, the monetary equilibrium

exists uniquely. The key to this result is the assumption, H ′(ρ) > 0. In the following discussion, we

will show the comparative statics regarding changes in the foreclosure cost, risk of assets, inflation,

and macroeconomic conditions.

17From (38), ∂F (ρ∗)∂`b

=φa(1+id)H(ρ∗)

1− ξaH(ρ∗)

. Then, ∂2F (ρ∗)∂`2b

= H′(ρ∗)aφ(1+id)[a−ξH(ρ∗)]2

∂ρ∗

∂`r

∂`r∂`b

> 0 if H ′(ρ∗) > 0. Thus, the ex ante

marginal loan rate in (37) is increasing in `b.18We thank Zhigang Ge for his suggestion.

20

4.2 Effects of changes in the foreclosure cost and asset risk

We first use numerical examples to illustrate the effects of changes in the foreclosure cost (see Figure

1).19 Increases in foreclosure costs reduce the recovery value of collateral, and banks have incentives

to reduce lending and raise the loan rate. Thus, ρ∗ becomes lower and the default probability,

Π = F (ρ∗), falls, implying that loans are less risky. Moreover, lower bank loans cause aggregate

liquidity and output to fall. The asset price rises, and this, together with a lower real loan amount,

implies lower loan-to-value ratios (LTV), defined as φ`bψa . Our numerical examples show that the risk

premium, η, rises with the foreclosure cost. We thus observe that financial development or a policy

that lowers the foreclosure cost can have positive impacts on loan-to-value ratios and aggregate

liquidity and can reduce the risk premium, though default rates will rise.

The foreclosure cost works to reduce the recovery value of collateral and thus can be interpreted

as a measure of an asset’s pledgeability. If one considers multiple assets, our results can be applied

to predict that liquidity differentials across assets increase with the pledgeability of assets.

To show the effects of increased risk in the dividend of assets, we consider a mean preserv-

ing spread in the distribution of dividend about ρe, the equilibrium level of ρ∗ (see, for example,

Williamson 1986). The probability density function is changed to

f∗(ρ) = f(ρ) + δg(ρ),

where δ ∈ (0, 1], and g(ρ) is continuous on [ρ, ρ]. Let G(ρ) =∫ ρρ g(w)dw. We assume, for ρ ∈ [ρ, ρ],

(i) f(ρ) + g(ρ) > 0; (ii) G(ρe) = 0 (which implies that the quantities of probability mass to the

right and to the left of ρe do not depend on δ); (iii)∫ ρρ wg(w)dw = 0 (which states that the change

in the distribution is mean-preserving), and (iv)∫ ρρ G(w)dw ≥ 0 (which states that the distribution

F (·) second-order stochastically dominates F ∗(·)).

The new hazard function under the mean-preserving spread is

H∗(ρ) =f(ρ) + δg(ρ)

1− F (ρ)− δG(ρ), (44)

19In Figures 1-6, the functional forms are u(qb) = qb1−σ

1−σ , c(qs) = qs. We let the dividend be uniformly distributed(ρ ∼ U(ρµ − bε, ρµ + bε)), except in Figure 2, wherein the dividend follows a truncated normal distribution. Thebenchmark parameter values are σ = .2, n = .4, β = .95, ρµ = .006, b = 2.5, γ = 1.01, ε = 0.001, ξ = .0001, M = 100and A = 2.

21

The effects of increased risk depend on how risk affects the hazard function; that is, the sign ofdH∗

dδ |δ=0 (a differential change in δ, evaluated at δ = 0) is important in determining the effects of

the mean-preserving spread on liquidity and output. We summarize the results in the following

proposition (see Appendix C for the proof).

Proposition 3 Increasing risk about asset dividends raises the equilibrium level of ρ∗, (∂ρ∗

∂δ |δ=0 > 0)

and the equilibrium default rate, F (ρ∗). There are two cases: (i) if higher risk reduces the hazard

rate (dH∗

dδ |δ=0 < 0), then increased asset risk lowers the risk premium (dηdδ |δ=0 < 0), and dqbdδ |δ=0 >

0, diddδ |δ=0 > 0, dφdδ |δ=0 > 0, dφ`bdδ |δ=0 > 0; (ii) otherwise (dH∗

dδ |δ=0 > 0), increased asset risk raises the

risk premium (dηdδ |δ=0 > 0), and dqbdδ |δ=0 < 0, diddδ |δ=0 < 0, dφdδ |δ=0 < 0, dφ`bdδ |δ=0 < 0.

Proposition 3 shows that the increased risk of the dividends of an asset serving as collateral

results in higher default probability and, therefore, loans are more risky; however, higher default

rates are not necessarily accompanied by lower liquidity, which depends on how the hazard rate

responds to higher risk. Totally differentiating the hazard function, H∗(ρ∗), from (44), we have

dH∗

dδ|δ=0 =

∂H∗

∂ρ∗︸ ︷︷ ︸(+)

∂ρ∗

dδ|δ=0︸ ︷︷ ︸

(+)

+∂H∗

∂δ|δ=0︸ ︷︷ ︸

(−)

.

where the sign of dH∗

dδ |δ=0 depends on two opposing forces: higher asset risk increases the hazard

rate indirectly by raising ρ∗, which makes loans riskier, and the mean-preserving spread directly

decreases the hazard rate. If the second effect dominates, dH∗

dδ |δ=0 < 0, then the risk premium falls,

and the value of money, aggregate liquidity, and output rise with the risk, as shown in case (i) of

Proposition 3. Otherwise, liquidity and output are reduced by higher risk, as shown in case (ii) of

Proposition 3.

As a result of the general equilibrium effects caused by an asset used as collateral, increased

risk of the asset dividends does not necessarily lower the asset price. Specifically, there are three

channels through which a mean-preserving spread affects the asset price; that is,

dψ

dδ|δ=0 =

∂ψ

∂ρ∗∂ρ∗

dδ|δ=0︸ ︷︷ ︸

(+)

+∂ψ

∂η︸︷︷︸(+)

dη

dδ|δ=0︸ ︷︷ ︸(?)

+∂ψ

∂δ|δ=0︸ ︷︷ ︸

(−)

(45)

22

The first term on the right side of (45) means that the increased risk results in higher ρ∗ and

higher default probability, so buyers need to pledge more assets in order to borrow more; this raises

the demand for assets and thus drives up the asset price. In the second term, ∂ψ∂η > 0 captures

the effect that a higher risk premium implies a larger liquidity shortage and thus the demand for

collateral; however, the effect of the increased risk on the risk premium, dηdδ |δ=0, depends on the

sign of dH∗

dδ |δ=0, as indicated in Proposition 3. The third term captures the usual notion that

the price of an asset would be lower when it becomes riskier. From case (i) of Proposition 3 ifdH∗

dδ |δ=0 < 0, then dηdδ |δ=0 < 0, and it is more likely that the combined effect of the last two terms

dominates, and the asset price falls with the increased risk. Moreover, in this case, higher risk leads

to higher real loan amount, φ`b, which, together with lower asset prices, implies higher loan-to-value

ratios. Consequently, as the asset risk increases, higher default rates are accompanied by higher

loan-to-value ratios, asset prices, and aggregate liquidity. On the other hand, if dH∗

dδ |δ=0 > 0, thendηdδ |δ=0 > 0, and it is more likely that the effects of the first two terms dominates, and increased risk

results in higher asset prices.

To illustrate further the effects of increased risk, we consider two specific distributions of div-

idends for numerical exercises, and change the variance of asset dividends while holding constant

the mean return. The results shown in Figure 2 and Figure 3 are consistent with case (i) and case

(ii), respectively, of Proposition 3. In Figure 2, the dividend ρ is drawn from a uniform distribution,

U(ρµ− bε, ρµ + bε), where ρµ is the mean, and ε features the dispersion of asset returns and is used

as a measure of the asset risk. Figure 2 shows that the increased risk raises the loan-to-value ratio

and the real loan amount but lowers the asset price; the economy enjoys higher liquidity and output,

while facing higher default rates.

We then consider the case wherein the dividend ρ is drawn from a truncated normal distribution,

and the distribution before the truncation is N(ρµ, ε2).20 In Figure 3 we depict a situation in which

the foreclosure cost is above a certain level. (The numerical results with the foreclosure cost below

that level are similar to those in the case with uniform distributions.) Figure 3 shows that increased

risks lower real loan amounts and loan-to-value ratios; moreover, default rates rise, implying the

20To avoid negative realized dividends, we restrict the support of ρ to lie within b standard deviations from themean; that is, ρ = ρµ − bε, and ρ = ρµ + bε. In the numerical exercises, we set b = 2.5. Because a truncated normaldistribution is symmetric, one can increase the variance by simply increasing ε.

23

quality of loans becomes worse. As the asset risk increases, the risk premium rises, so does the asset

price.

4.3 Effects of changes in inflation and macroeconomic conditions

We first discuss the effects of changes in the inflation rate.

Proposition 4 Higher inflation reduces equilibrium ρ∗ and the equilibrium default rate, F (ρ∗).

Moreover, ∂id∂γ > 0 and the risk premium falls.

Higher inflation reduces the risk premium, ξφ∂F (ρ∗)∂`b

, because a lower default rate implies banks

face lower uncertainty in terms of incurring foreclosure costs; on the other hand, inflation raises

the bank’s cost of funds, id. Therefore, the effect of inflation on the marginal borrowing cost,

1+id+ξφ∂F (ρ∗)∂`b

, depends on these two opposing forces. Observing from (40), consumption and output

thus may rise or fall by increased inflation. If the marginal borrowing cost rises with higher inflation,

borrowers will choose the contract that generates lower loan amount, and aggregate liquidity falls,

so do consumption and output. This situation is depicted in our numerical examples (see Figure

4), in which we also observe that asset prices and loan-to-value ratios fall.21

We illustrate how aggregate liquidity and output respond to changes in macroeconomic condi-

tions. In the first scenario, we assume that the utility function takes the form, λu(q), and λ > 0,

and changes in λ capture fluctuations in aggregate demand. In the second scenario, we consider

changes in the mean of dividend, ρu, holding the variance constant. A positive shock in the mean

of asset returns can be interpreted as an increase in the productivity of Lucas trees. Our numerical

examples show that, in the first scenario, an increase in λ results in higher ρ∗, asset prices, and

loan-to-value ratios (see Figure 5). Hence, during a boom induced by higher aggregate demand,

aggregate liquidity and loan-to-value ratios increase, and this increase is accompanied by higher de-

fault rates. In the second scenario, as assets display higher productivity, we observe a boom wherein

asset prices and aggregate liquidity rise, while loan-to-value ratios and default rates fall (see Figure

21There are two channels through which inflation affects asset prices—inflation increases the borrowing cost, asituation which dampens the demand for the asset as collateral; and inflation reduces real loan amount, so buyersdemand more assets in order to borrow more. With our parametrization, the former effect dominates, so asset pricesare lowered by inflation.

24

6).22 The results imply that, though asset prices and liquidity rise in a boom, depending on the

source of changes in macroeconomic conditions, there may be opposite movements in loan-to-value

ratios and default rates. The lesson is that it is beneficial to understand the factor causing the boom

and bust because different underlying factors may result in different macroeconomic consequences

that may call for different policy responses.

5 Conclusion

This paper considers a monetary economy in which agents can borrow from banks to finance their

consumption and risky assets are pledged as collateral to secure debt. In contrast to studies that

feature limited commitment and complete contracts wherein lenders offer to lend only as much as

borrowers are willing to repay, borrowers in this economy retain the option to default. We have

shown how aggregate liquidity, default rates, and output are affected by asset risk and foreclosure

costs. We also identify the conditions under which more risky assets are better suited to serve as

collateral in the sense that they generate higher liquidity and loan-to-value ratios, though using

more risky assets as collateral may result in higher borrower default rates.

In our model, we assume that dividends are realized in the second subperiod and that the

realization of dividends is public information. Alternative timing for dividend realization to occur

is the first subperiod, and results depend on whether the dividend realization is public information.

If it is public information, then the recovery value of collateral is known with certainty, and banks

offer to lend only as much as borrowers are willing to repay. If the realization of dividends is private

information to the borrower, then the loan contract problem must take into account a truth-telling

condition.

If the loan considered here is a mortgage, our results imply that those with negative home

equity will default. From the evidence, however, only 10% of households with negative home equity

default on mortgage loans (Foote, Gerardi, and Willen 2008). In reality, people have utility by

holding collateral (for example, enjoying living in a house), and they will not default on a mortgage

simply because the loan has a higher principal than the home’s free-market value, particularly

22The numerical results are robust to different distributions of dividends and values of other parameters, such asthe foreclosure cost.

25

because other factors, including emotional attachment, are often at play. One can consider the

scenario wherein people obtain some utility from holding the asset besides enjoying dividends in

order to obtain results more in line with the evidence in mortgage loans.

26

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[10] Kiyotaki, Nobuhiro, and John Moore (2001). “Liquidity, Business Cycles and Monetary Policy.”

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nomic Review 46, 317-349.

27

[12] Lagos, Ricardo (2010) “Asset Prices and Liquidity in an Exchange Economy,” Journal of Mon-

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Asset Prices, and Monetary Policy,” Review of Economic Studies 79, 1209-38.

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28

2 4 6

10-5

0.985

0.99

0.995LTV

2 4 6

10-5

0.1435

0.144

0.1445

2 4 6

10-5

0.2844

0.2845

0.2846 l

b

2 4 6

10-5

0.6

0.8

1

2 4 6

10-5

0.711

0.7112

0.7114 q

b

2 4 6

10-5

6

8

1010-3

2 4 6

10-5

1.0174

1.0176

1.0178

2 4 6

10-5

4.266

4.268

4.2710-3

2 4 6

10-5

0.052

0.0521

0.0522id

Figure 1: Effects of changes in foreclosure costs

29

Figure 2: Effects of changes in asset risk (uniform distribution)

1 1.1 1.2

10-3

0.9845

0.985

0.9855LTV

1 1.1 1.2

10-3

0.1442

0.1443

0.1444

1 1.1 1.2

10-3

0.2843

0.28435

0.2844 l

b

1 1.1 1.2

10-3

0.4

0.5

0.6

1 1.1 1.2

10-3

0.7107

0.7108

0.7109 q

b

1 1.1 1.2

10-3

5.5

6

6.510-3

1 1.1 1.2

10-3

1.01775

1.0178

1.01785

1 1.1 1.2

10-3

4.2645

4.265

4.265510-3

1 1.1 1.2

10-3

0.0519

0.05195id

30

Figure 3: Effects of changes in asset risk (truncated normal distribution)

1 1.1 1.2

10-3

0.9834

0.9836

0.9838LTV

1 1.1 1.2

10-3

0.1445

0.14455

0.1446

1 1.1 1.2

10-3

0.28436

0.284365

0.28437 l

b

1 1.1 1.2

10-3

0.15

0.2

0.25

1 1.1 1.2

10-3

0.7109

0.71092

0.71094 q

b

1 1.1 1.2

10-3

5.14

5.16

5.1810-3

1 1.1 1.2

10-3

1.01773

1.01774

1.01775

1 1.1 1.2

10-3

4.2654

4.2655

4.265610-3

1 1.1 1.2

10-3

0.051955

0.05196

0.051965id

31

1.02 1.04 1.06 1.08 1.10.8

0.9

1LTV

1.02 1.04 1.06 1.08 1.10.1

0.15

1.02 1.04 1.06 1.08 1.10.1

0.2

0.3 l

b

1.02 1.04 1.06 1.08 1.10

0.5

1.02 1.04 1.06 1.08 1.10.4

0.6

0.8 q

b

1.02 1.04 1.06 1.08 1.12

4

610-3

1.02 1.04 1.06 1.08 1.11.01

1.015

1.02

1.02 1.04 1.06 1.08 1.12

4

610-3

1.02 1.04 1.06 1.08 1.10

0.1

0.2id

Figure 4: Effects of changes in inflation rates

32

1 1.05 1.1 1.150.98

0.985

0.99LTV

1 1.05 1.1 1.150.1

0.2

0.3

1 1.05 1.1 1.150.2

0.4

0.6 l

b

1 1.05 1.1 1.150.4

0.6

0.8

1 1.05 1.1 1.150.5

1

1.5 q

b

1 1.05 1.1 1.154

6

810-3

1 1.05 1.1 1.151

1.05

1 1.05 1.1 1.150

0.005

0.01

1 1.05 1.1 1.150.02

0.04

0.06id

Figure 5: Effects of changes in aggregate demand

33

4 5 6

10-3

0.98

0.985

0.99LTV

4 5 6

10-3

0.135

0.14

0.145

4 5 6

10-3

0.27

0.28

0.29 l

b

4 5 6

10-3

0.4

0.6

0.8

4 5 6

10-3

0.68

0.7

0.72 q

b

4 5 6

10-3

5

5.5

610-3

4 5 6

10-3

1

1.02

1.04

4 5 6

10-3

4

4.510-3

4 5 6

10-3

0.02

0.04

0.06id

Figure 6: Effects of changes in the mean of asset returns

34

Appendix A.

Deriving the envelope conditions. The expected utility of a buyer and a seller from entering

the second subperiod are, respectively,

W b(m, a, `r) ≡ Emax(wD, wR) =

∫ ρ∗

ρwD(m, a, `r, ρ)f(ρ)dρ+

∫ ρ

ρ∗wR(m, a, `r, ρ)f(ρ)dρ,

W s(m, a, d) = U(x∗)− x∗ + φ(m+ T ) + φ(1 + id)d− φm− ψa+ (ψ + ρµ)a+ βV (m, a).

Using wD(m, a, `r, d, ρ) and wR(m, a, `r, d, ρ) from (3) and (4), the default rule from (5), and the

fact that wD(m, a, `r, d, ρ) = wR(m, a, `r, d, ρ) when ρ = ρ∗, we derive the envelope conditions

(7)-(10) as follows.

W bm =

∫ ρ∗

ρφf(ρ)d(ρ) +

∫ ρ

ρ∗φf(ρ)d(ρ) = φ,

W ba = wDf(ρ∗)

∂ρ∗

∂a+

∫ ρ

ρ∗(ψ + ρ)f(ρ)dρ− wR(ρ∗)f(ρ∗)

∂ρ∗

∂a

=[wD − wR(ρ∗)

]f(ρ∗)

∂ρ∗

∂a+

∫ ρ

ρ∗(ψ + ρ)f(ρ)dρ

=

∫ ρ

ρ∗(ψ + ρ)f(ρ)dρ,

W b`r = wDf(ρ∗)

∂ρ∗

∂`r+

∫ ρ

ρ∗(−φ)f(ρ)dρ− wR(ρ∗)f(ρ∗)

∂ρ∗

∂`r

=[wD − wR(ρ∗)

]f(ρ∗)

∂ρ∗

∂`r− φ [1− F (ρ∗)]

= −φ [1− F (ρ∗)] ,

W sd = φ(1 + id).

Deriving the marginal values of holding money and the real asset in the first subperiod.

We consider the case where the buyer’s budget constraint is binding, and the seller deposits all his

money holding, d = m. Then, rewrite (12) as

V (m, a) = (1−n) max`r

{u

(m+ `b(`r, a)

p

)+W b(0, a, `r)

}+nmax

qs{−c(qs) +W s(pqs, a, d)} . (46)

Taking the derivative of V (m, a) and using (13), we have

35

Vm(m, a) = (1− n)u′(qb)

p+ nW s

d

= φ

[(1− n)

u′(qb)

c′(qs)+ n(1 + id)

].

From (16) we have

∂`b∂a

=

∫ ρ∗ρ (ψ + ρ)f(ρ)dρ

φ(1 + id),

and therefore, using (10), (11) and (13) we have

Va(m, a) = (1− n)u′(qb)

p

∂`b∂a

+ (1− n)W ba + nW s

a

= (1− n)

u′(qb)p

∫ ρ∗ρ (ψ + ρ)f(ρ)dρ

φ(1 + id)+

∫ ρ

ρ∗(ψ + ρ)f(ρ)dρ

+ n(ψ + ρµ)

= (1− n)

[u′(qb)

c′(qs)(1 + id)

][∫ ρ∗

ρ(ψ + ρ)f(ρ)dρ

]− (1− n)

[∫ ρ∗

ρ(ψ + ρ)f(ρ)dρ

]+ ψ + ρµ

= (1− n)

[u′(qb)

c′(qs)(1 + id)− 1

][∫ ρ∗

ρ(ψ + ρ)f(ρ)dρ

]+ ψ + ρµ.

Deriving the first order condition in the economy with foreclosure costs. To simplify

notations we substitute S ≡∫ ρ∗ρ [φ`r − (ψ + ρ)a]f(ρ)dρ from (15) into πξB = 0 from (32) to obtain

`b =φ`r − S − ξF (ρ∗)

φ(1 + id)≡ `b(`r) (47)

Using `b = `b(`r), the buyer’s problem becomes

maxqb,`r

u(qb) +W b (m+ `b(`r)− pqb, a, `r)

s.t. pqb ≤ m+ `b(`r).

Using W b`r

= −φ [1− F (ρ∗)] from (9), we have the first order conditions:

u′(qb)

c′(qs)= 1 +

λbφ,

φ∂`b∂`r− φ[1− F (ρ∗)] + λb

∂`b∂`r

= 0. (48)

36

Substituting λb = φ[u′(qb)c′(qs)

− 1]into (48), we have

∂`b∂`r

u′(qb)

c′(qs)= 1− F (ρ∗), (49)

which implies that the marginal benefit of acquiring an additional unit of loan indued by higher

repayment equals the probability of repayment.

Differentiating (47) with respect to `r, and using ∂S∂`r

= φF (ρ∗) and ∂F (ρ∗)∂`r

= φaf(ρ∗), we have

∂`b∂`r

=1− F (ρ∗)− ξ

af(ρ∗)

1 + id, (50)

which is (34) in the text. From (50), 1 − F (ρ∗) = (1 + id)∂`b∂`r

+ ξaf(ρ∗), which we substitute into

(49) to obtain∂`b∂`r

[u′(qb)

c′(qs)− (1 + id)

]− ξ

af(ρ∗) = 0. (51)

To further simplify the above expression, we use ∂`b∂`r

= ∂F (ρ∗)∂`r

/∂F (ρ∗)∂`b

and ∂F (ρ∗)∂`r

= φaf(ρ∗) to obtain

the following first order condition:

∂`b∂`r

[u′(qb)

c′(qs)− (1 + id)−

ξ

φ

∂F (ρ∗)

∂`b

]= 0, (52)

which is (35) in the text. Because ∂`b∂`r

> 0, we use ∂F (ρ∗)∂`b

= ∂F (ρ∗)∂`r

/ ∂`b∂`rand (50) to obtain

∂F (ρ∗)

∂`b=

φaf(ρ∗)(1 + id)

1− F (ρ∗)− ξaf(ρ∗)

.

which is (38).

Proof of Lemma 1. Substituting ∂`b∂`r

=1−F (ρ∗)− ξ

af(ρ∗)

1+idfrom (34) into (49), the first order condi-

tion of the buyer’s problem becomes[1− F (ρ∗)− ξ

af(ρ∗)

]u′(qb)

c′(qs)(1 + id)− [1− F (ρ∗)] = 0 (53)

We want to prove ∂`b∂`r

> 0. Suppose not, and consider first ∂`b∂`r

= 0, which means from (50),

1−F (ρ∗)− ξaf(ρ∗) = 0; i.e., 1−F (ρ∗) = ξ

af(ρ∗) > 0. However, substituting 1−F (ρ∗)− ξaf(ρ∗) = 0

into (53), the first order condition becomes 1− F (ρ∗) = 0, a contradiction. Then consider the case∂`b∂`r

< 0, which means 1− F (ρ∗)− ξaf(ρ∗) < 0. Substituting 1− F (ρ∗)− ξ

af(ρ∗) < 0 into (53), the

first order condition is satisfied unless 1− F (ρ∗) < 0, a contradiction, because F (ρ∗) ≤ 1.

37

Proof of Proposition 2. From Lemma 1, we obtain (40). Because ∂`b∂`r

> 0, ∂F (ρ∗)∂`b

= ∂F (ρ∗)∂`r

/ ∂`b∂`r,

and ∂F (ρ∗)∂`r

= φaf(ρ∗) > 0, we have ξ

φ∂F (ρ∗)∂`b

> 0. By (53) we rewrite the first order condition as

[1− F (ρ∗)]

[u′(qb)

c′(qs)(1 + id)− 1

]=

u′(qb)

c′(qs)(1 + id)

ξ

af(ρ∗). (54)

The right-side of (54) is strictly positive, and in the left-side, because ξφ∂F (ρ∗)∂`b

> 0, (40) impliesu′(qb)c′(qs)

> (1 + id); hence, we verify F (ρ∗) < 1.

Deriving the marginal value of the real asset in the economy with foreclosure costs.

From (33), we derive

∂`b∂a

=

∫ ρ∗ρ (ψ + ρ)f(ρ)dρ+ ξ

af(ρ∗)(ψ + ρ∗)

φ(1 + id),

and then taking derivative of (46) we have

Va(m, a) = (1− n)u′(qb)

p

∂`b∂a

+ (1− n)W ba + nW s

a

= (1− n)

u′(qb)p

∫ ρ∗ρ (ψ + ρ)f(ρ)dρ+ ξ

af(ρ∗)(ψ + ρ∗)

φ(1 + id)+

∫ ρ

ρ∗(ψ + ρ)f(ρ)d(ρ)

+ n(ψ + ρµ)

= (1− n)

[u′(qb)

c′(qs)(1 + id)− 1

][∫ ρ∗

ρ(ψ + ρ)f(ρ)dρ

]+ ψ + ρµ

+ (1− n)

[u′(qb)

c′(qs)(1 + id)− 1

] [ξ

af(ρ∗)(ψ + ρ∗)

]= ψ + ρµ + (1− n)

[u′(qb)

c′(qs)(1 + id)− 1

]{ψ + [1− F (ρ∗)] ρ∗ +

∫ ρ∗

ρρf(ρ)dρ

}.

where the last equality is obtained by replacing ξaf(ρ∗) from using (34) and (51).

38

Appendix B. Solving the model

We assume buyer’s utility is given by u(qb) =q1−σb −11−σ , and the seller’s cost is given by c(qs) = qs.

The market clearing conditions for loans and money are (1 − n)`b = nd,m = M−1 where M−1 is

money supply of last period. With the market clearing condition and d = m, we have `b = n1−nM−1

and nc′(qs)qb = φ`b in Definition 2. Thus,

qb =φ

n`b =

φM−11− n

. (55)

Substituting qb = φM−1

1−n and `b = nM−1

1−n into the equilibrium conditions in Definition 2, a monetary

equilibrium is a list of (φ, ψ, id, `r, ρ∗) satisfying

γ − ββ

= (1− n)[(φM−11− n

)−σ − 1] + nid (56)

ψ = β

{(ψ + ρµ) + (1− n)

[(φM−1

1−n )−σ

1 + id− 1

][ψ + (1− F (ρ∗))ρ∗ +

∫ ρ∗

ρρf(ρ)dρ

]}(57)

ρ∗ =φ`ra− ψ (58)

n

1− nM−1 =

ψaF (ρ∗) + φ`r [1− F (ρ∗)] + a∫ ρ∗ρ ρf(ρ)dρ− ξF (ρ∗)

φ(1 + id)(59)

ξ

af(ρ∗) = [1− F (ρ∗)− ξ

af(ρ∗)]

[(φM−1

1−n )−σ

1 + id− 1

]. (60)

From (55), u′(qb) = q−σb = (φM−1

1−n )−σ. To simply the exposition, we define the marginal utility

x = (φM−11− n

)−σ. (61)

Notice that x (and thus, qb) and φ are one-to-one. Combining (56) and (60) to solve for x and id,

in terms of ρ∗, we obtain

x =

γβ

1− nξf(ρ∗)a(1−F (ρ∗))

, (62)

id =

[1− ξ

a

f(ρ∗)

1− F (ρ∗)

]x− 1. (63)

Substituting φ`r from (58) into (59), we have

n

1− nM−1 =

ψaF (ρ∗) + (ψ + ρ∗)a [1− F (ρ∗)] + a∫ ρ∗ρ ρf(ρ)d(ρ)− ξF (ρ∗)

φ(1 + id),

39

and rearrange to get

ψ =n

a(1− n)M−1φ(1 + id) +

ξ

aF (ρ∗)− ρ∗[1− F (ρ∗)]−

∫ ρ∗

ρρf(ρ)d(ρ). (64)

Rewriting (57) we have[1− βn− β(1− n)

x

1 + id

]ψ − β(1− n)

(x

1 + id− 1

){ρ∗[1− F (ρ∗)] +

∫ ρ∗

ρρf(ρ)dρ]

}= βρµ.

(65)

Combining (64) and (65), we obtain[1− βn− β(1− n)

x

1 + id

]{n

a(1− n)M−1φ(1 + id) +

ξ

aF (ρ∗)− ρ∗[1− F (ρ∗)]−

∫ ρ∗

ρρf(ρ)dρ

}

−β(1− n)

(x

1 + id− 1

){ρ∗[1− F (ρ∗)] +

∫ ρ∗

ρρf(ρ)dρ]

}= βρµ,

which can be rearranged to become

[1− βn− β(1− n)

x

1 + id

] [n

a(1− n)M−1φ(1 + id) +

ξ

aF (ρ∗)

]− (1− β)

{ρ∗[1− F (ρ∗)] +

∫ ρ∗

ρρf(ρ)d(ρ)

}= βρµ. (66)

The right-side of the equality in (66) is βρµ > 0; in the left-side, x, φ and id are functions of ρ∗ from

equation (62) and (63) and the definition of x, so the left-side of (66) is a function of ρ∗. Because

the term in the second bracket and the last term in the left-side of (66), and βρµ are strictly positive,

the term in the first bracket must be strictly positive; i.e.,

1− βn− β(1− n)x

1 + id> 0. (67)

We now proceed to prove that equilibrium ρ∗ exists and is unique. Once we solve for equilibrium

ρ∗, we obtain x, and id from (62) and (63), by plugging equilibrium ρ∗. Then, using (61), (55), (58),

and (65), we can solve for the equilibrium values of φ, qb, `r and ψ.

From (66) we define J(ρ∗) as

40

J(ρ∗) =

[1− βn− β(1− n)

x

1 + id

] [n

a(1− n)M−1φ(1 + id) +

ξ

aF (ρ∗)

]− (1− β)

{ρ∗[1− F (ρ∗)] +

∫ ρ∗

ρρf(ρ)dρ

}− βρµ.

Given the hazard function,

H(ρ) =f(ρ)

1− F (ρ),

and using φM−1 = (1− n)x−1σ and (1 + id) = (1− ξ

aH(ρ∗))x from (63), we have

J(ρ∗) =

[1− βn− β(1− n)

1

1− ξaH(ρ∗)

] [n

a(1− ξ

aH(ρ∗))x1−

1σ +

ξ

aF (ρ∗)

](68)

− (1− β)

{ρ∗[1− F (ρ∗)] +

∫ ρ∗

ρρf(ρ)dρ

}− βρµ.

Lemma 2 If H ′(ρ) > 0, ∀ρ ∈ [ρ, ρ], then J ′(ρ∗) < 0.

Proof. Rearrange J(ρ∗) from (68) to get

J(ρ∗) =n

ax1−

1σ

[(1− βn)(1− ξ

aH(ρ∗))− β(1− n)

]︸ ︷︷ ︸

(A1)

− β(1− n)1

1− ξaH(ρ∗)

ξ

aF (ρ∗)︸ ︷︷ ︸

(A2)

+ξ

aF (ρ∗)(1− βn)− (1− β)

{ρ∗[1− F (ρ∗)] +

∫ ρ∗

ρρf(ρ)dρ

}− βρµ︸ ︷︷ ︸

(A3)

. (69)

Let

A1 ≡ n

ax1−

1σ

[(1− βn)(1− ξ

aH(ρ∗))− β(1− n)

],

A2 ≡ −β(1− n)1

1− ξaH(ρ∗)

ξ

aF (ρ∗),

A3 ≡ ξ

aF (ρ∗)(1− βn)− (1− β)

{ρ∗[1− F (ρ∗)] +

∫ ρ∗

ρρf(ρ)dρ

}− βρµ.

41

To prove J ′(ρ∗) < 0, we will show that A1, A2, and A3 are decreasing in ρ∗. First,

∂x1−1σ

∂ρ∗= (1− 1

σ)x−1σγ

β[1− nξ

aH(ρ∗)]−2

nξ

aH ′(ρ∗) < 0, (70)

where we have used the assumption of H ′(ρ) > 0, and given that the empirical estimate of elasticity

of intertemporal substitution is greater than 1 (σ < 1).

A1′(ρ∗) =n

a

∂x1−1σ

∂ρ∗

[(1− βn)(1− ξ

aH(ρ∗))− β(1− n)

]+n

ax1−

1σ (1− βn)(−1)

ξ

aH ′(ρ∗).

If we can show[(1− βn)(1− ξ

aH(ρ∗))− β(1− n)]> 0, then A1′(ρ∗) < 0. Recall from (67) that

1− βn− β(1−n) x1+id

> 0, and then using (1 + id) = (1− ξaH(ρ∗))x from (63) and 1− ξ

aH(ρ∗) > 0

(otherwise, we will have negative id), we have

(1− βn)

[1− ξ

aH(ρ∗)

]− β(1− n) > 0. (71)

Therefore, A1′(ρ∗) < 0. Now,

A2′(ρ∗) = −β(1− n)ξ

a

[1− ξ

aH(ρ∗)]f(ρ∗) + ξ

aF (ρ∗)H ′(ρ∗)[1− ξ

aH(ρ∗)]2 < 0,

and

A3′(ρ∗) =ξ

af(ρ∗)(1− βn)− (1− β) [1− F (ρ∗)]

= [1− F (ρ∗)]

[ξ

aH(ρ∗)(1− βn)− (1− β)

]< (1− F (ρ∗)) [1− βn− β(1− n)− (1− β)] = 0,

where the inequality in the last line comes from (71).

Proposition 5 Given a distribution, f(ρ), with support [ρ, ρ], and H ′(ρ) > 0, if J(ρ) > 0, then the

equilibrium value of the default rule, denoted as ρe, which satisfies J(ρe) = 0, exists and is unique.

Proof. First, we show J(ρ) < 0. Notice that limρ→ρH(ρ) =∞, and x(ρ) = 0. Then,

J(ρ) =n

a(0)1−

1σ

[(1− βn)

[1− lim

ρ→ρH(ρ)

]− β(1− n)

]− β(1− n)

ξa

1− limρ→ρH(ρ)

+ξ

a(1− βn)− (1− β)ρµ − βρµ < 0.

42

Moreover,

J(ρ) =n

a(

γβ

1− nξa f(ρ)

)1−1σ

[(1− βn)(1− ξ

af(ρ))− β(1− n)

]− (1− β)ρ− βρµ.

If J(ρ) ≥ 0; i.e., if

βρµ ≤n

a(

γβ

1− nξa f(ρ)

)1−1σ

[(1− βn)(1− ξ

af(ρ))− β(1− n)

]− (1− β)ρ,

which holds when ρµ is sufficiently small. Then from Lemma 2, J ′(ρe) < 0, and using Intermediate

Value Theorem we obtain the result.

43

Appendix C. Comparative statics

In this Appendix, we first show the mean preserving exercise, and then the comparative statics

results for the inflation rate, γ.

Mean preserving exercise We consider a mean preserving spread in the distribution of dividend

about ρe, the equilibrium level of ρ∗ (see, e.g., Williamson 1986). The probability density function

is changed to

f∗(ρ) = f(ρ) + δg(ρ),

where δ ∈ (0, 1], and g(ρ) is continuous on [ρ, ρ]. Let F (ρ), and F ∗(ρ) be the CDF before and after

mean preserving spread. Let G(ρ) =∫ ρρ g(w)dw. We assume, for ρ ∈ [ρ, ρ], (i) f(ρ) + g(ρ) > 0;

(ii) G(ρe) = 0 (which implies that the quantities of probability mass to the right and to the left

of the equilibrium, ρe, do not depend on δ); (iii)∫ ρρ wg(w)dw = 0 (which states that the change

in the distribution is mean-preserving), and (iv)∫ ρρ G(w)dw ≥ 0 (which states the distribution

F (·) second-order stochastically dominates F ∗(·) ). Figure 7 shows the type of mean preserving we

consider.

We first derive a property under the mean preserving spread we consider here that will be useful

for the following discussion. Using the integration by part,∫ ρe

ρG(ρ)dρ = G(ρe)ρe −G(ρ)ρ−

∫ ρe

ρρg(ρ)dρ

= 0−∫ ρe

ρρg(ρ)dρ ≥ 0.

where we have used the assumptions of∫ ρeρ G(ρ)dρ ≥ 0 and G(ρe) = 0. Therefore,

∫ ρe

ρρg(ρ)dρ ≤ 0.

In Figure 7, the slope of F ∗(ρ∗) is smaller than F (ρ∗) at ρ∗ = ρe, which also verifies that g(ρe) < 0.

Proof of Proposition 3. In this proof we first show that ∂ρ∗

∂δ |δ=0 > 0 and dF ∗(ρ∗)dδ |δ=0 > 0. Then

we show that the effects of mean preserving on other variables depend on the sign of dH∗

dδ |δ=0.

44

Figure 7: Graph of mean preserving

By replacing the mean-preserving spread density function to (69), we get

J∗(ρ∗) = A∗1(ρ∗) +A∗2(ρ

∗) +A∗3(ρ∗), (72)

where

A∗1 ≡ n

ax1−

1σ

[(1− βn)(1− ξ

aH∗(ρ∗))− β(1− n)

],

A∗2 ≡ −β(1− n)1

1− ξaH∗(ρ∗)

ξ

a[F (ρ∗) + δG(ρ∗)] ,

A∗3 ≡ ξ

a(F (ρ∗) + δG(ρ∗))(1− βn)− (1− β)

{ρ∗[1− F (ρ∗)− δG(ρ∗)] +

∫ ρ∗

ρρ [f(ρ) + δg(ρ)] dρ

}− βρµ,

and

H∗(ρ) =f(ρ) + δg(ρ)

1− F (ρ)− δG(ρ)

is the new hazard function after mean preserving. Using (72) and according to the implicit function

theorem, we obtain∂ρ∗

∂δ= −

∂J∗(ρ∗)∂δ

∂J∗(ρ∗)∂ρ∗

= −∂A∗1∂δ +

∂A∗2∂δ +

∂A∗3∂δ

∂A∗1∂ρ∗ +

∂A∗2∂ρ∗ +

∂A∗3∂ρ∗

. (73)

As in Williamson (1986), we study local comparative statics by setting δ = 0.

We begin by determining the signs of ∂A∗1∂ρ∗ ,

∂A∗2∂ρ∗ , and

∂A∗3∂ρ∗ at δ = 0. As δ → 0, we replace ρ∗

with ρe, which is the equilibrium value of default rule before mean preserving. To proceed, we first

45

derive the following two properties:

∂H∗(ρ)

∂ρ|δ=0 =

[1− F (ρ)− δG(ρ)] [f ′(ρ) + δg′(ρ)] + [f(ρ) + δg(ρ)]2

(1− F (ρ)− δG(ρ))2|δ=0

=[1− F (ρe)] [f ′(ρe)] + [f(ρe)]2

(1− F (ρe))2

≡ H ′(ρe) > 0. (74)

The above property says that the new hazard function is increasing in ρ, ∀ρ ∈ [ρ, ρ]. This the

condition ensures the existence and uniqueness of the equilibrium after mean preserving. The

second property is

∂H∗(ρ∗)

∂δ|δ=0 =

[1− F (ρ∗)− δG(ρ∗)] g(ρ∗) + [f(ρ∗) + δg(ρ∗)]G(ρ∗)

(1− F (ρ∗)− δG(ρ∗))2|δ=0

=[1− F (ρe)] g(ρe) + f(ρe)G(ρe)

[1− F (ρe)]2

=[1− F (ρe)] g(ρe)

[1− F (ρe)]2< 0, (75)

where the second equality comes from the condition (ii) G(ρe) = 0, and the last inequality comes

from g(ρe) < 0.

Now we have

∂A∗1∂ρ∗|δ=0 =

n

a

∂x1−1σ

∂ρ∗

[(1− βn)(1− ξ

aH∗(ρe))− β(1− n)

]− n

ax1−

1σ (1− βn)

ξ

aH∗′(ρe) < 0,

∂A∗2∂ρ∗|δ=0 = −β(1− n)

ξ

a

[1− ξ

aH∗(ρ∗)

][f(ρ∗) + δg(ρ∗)] + ξ

a [F (ρ∗) + δG(ρ∗)]H∗′(ρ∗)[1− ξ

aH∗(ρ∗)

]2 |δ=0

= −β(1− n)ξ

a

[1− ξ

aH∗(ρe)

]f(ρe) + ξ

aF (ρe)H∗′(ρe)[1− ξ

aH∗(ρe)

]2 < 0,

∂A∗3∂ρ∗|δ=0 =

ξ

a[f(ρ∗) + δg(ρ∗)] (1− βn)− (1− β)(1− F (ρ∗)− δG(ρ∗))|δ=0

=ξ

af(ρe)(1− βn)− (1− β) [1− F (ρe)]

= [1− F (ρe)]

[ξ

aH∗(ρe)(1− βn)− (1− β)

]< 0,

46

where we have used (70) and ∂H∗(ρ)∂ρ |δ=0 > 0 from (74) to obtain the inequality.

Next we proceed to derive the sign of ∂A∗1

∂δ , ∂A∗2

∂δ , and ∂A∗3∂δ at δ = 0.

∂x1−1σ

∂δ|δ=0 = (1− 1

σ)x−1σγ

β[1− nξ

aH∗(ρe)]−2

nξ

a

∂H∗(ρe)

∂δ> 0.

∂A∗1∂δ|δ=0 =

n

a

∂x1−1σ

∂δ

[(1− βn)(1− ξ

aH∗(ρe))− β(1− n)

]+n

ax1−

1σ (1− βn)(−1)

ξ

a

∂H∗(ρe)

∂δ> 0,

∂A∗2∂δ|δ=0 = −β(1− n)

ξ

a

(1− ξaH∗(ρ∗))G(ρ∗) + ξ

a(F (ρ∗) + δG(ρ∗))∂H∗(ρ∗)∂δ

(1− ξaH∗(ρ∗))2

|δ=0

= −β(1− n)ξ

a

ξaF (ρe)∂H

∗(ρe)∂δ

(1− ξaH∗(ρe))2

> 0,

where we have used ∂H∗

∂δ |δ=0 < 0 from (75). Also,

∂A∗3∂δ|δ=0 =

ξ

aG(ρ∗)(1− βn)− (1− β)

[−ρ∗G(ρ∗) +

∫ ρ∗

ρρg(ρ)dρ

]|δ=0

= −(1− β)

∫ ρe

ρρg(ρ)dρ ≥ 0,

where we have used∫ ρeρ ρg(ρ)dρ ≤ 0.

From (73), because ∂A∗1∂δ |δ=0 +

∂A∗2∂δ |δ=0 +

∂A∗3∂δ |δ=0 > 0 and ∂A∗1

∂ρ∗ |δ=0 +∂A∗2∂ρ∗ |δ=0 +

∂A∗3∂ρ∗ |δ=0 < 0, we

have∂ρ∗

∂δ|δ=0 > 0.

We will use the total derivative to discuss the following comparative statics because of changes

of the two variables (ρ∗ and δ). One should be aware that other parameters are held constant except

for the change in δ.

dF ∗(ρ∗)

dδ|δ=0 =

[G(ρ∗) +

(f(ρ∗) + δg(ρ∗)

)∂ρ∗∂δ

]|δ=0

= G(ρe) + f(ρe)∂ρ∗

∂δ|δ=0 > 0.

47

The sign ofdH∗

dδ|δ=0 =

∂H∗

∂ρ∗︸ ︷︷ ︸(+)

∂ρ∗

∂δ|δ=0︸ ︷︷ ︸

(+)

+∂H∗

∂δ|δ=0︸ ︷︷ ︸

(−)

is important to determine the sign of the effects of the mean-preserving spread on φ, qb and id. We

consider the following two cases: dH∗

dδ |δ=0 < 0 and dH∗

dδ |δ=0 > 0.

Case 1: The risk premium decreases after mean preserving ( dH∗

dδ |δ=0 < 0) With

c(q) = q, the risk premium becomes

η ≡ u′(qb)

c′(qs)(1 + id)=

1

1− ξaH∗(ρ∗)

,

dη

dδ|δ=0 =

ξ

a[1− ξ

aH∗(ρ∗)]−2

dH∗

dδ|δ=0 < 0,

so the risk premium decreases in this case. Next, we solve for dxdδ |δ=0. From that we can determine

the sign of dφdδ |δ=0 and dqbdδ |δ=0, because φ and qb are negatively correlated with x according to (55)

and (61).

dx

dδ|δ=0 =

γ

β

nξ

a[1− nξ

aH∗(ρ∗)]−2

dH∗

dδ|δ=0 < 0,

so

dφ

dδ|δ=0 > 0,

dqbdδ|δ=0 > 0.

From (63),

diddδ|δ=0 =

γ

β

ξa(n− 1)dH

∗

dδ |δ=0[1− nξ

a H∗(ρ∗)

]2 > 0.

By replacing the mean preserving spread density function to (65), we get

ψ =βρµ + β(1− n)(η − 1)

{ρ∗[1− F (ρ∗)− δG(ρ∗)] +

∫ ρ∗ρ ρ[f(ρ) + δg(ρ)]dρ

}1− βn− β(1− n)η

,

48

anddψ

dδ|δ=0 =

∂ψ

∂ρ∗∂ρ∗

∂δ|δ=0 +

∂ψ

∂η

dη

dδ|δ=0 +

∂ψ

∂δ|δ=0.

The derivations of ∂ψ∂η ,∂ψ∂ρ , and

∂ψ∂δ are as follows.

∂ψ

∂η|δ=0 =

[1− βn− β(1− n)η]β(1− n)[ρ∗[1− F (ρ∗)− δG(ρ∗)] +

∫ ρ∗ρ ρ(f(ρ) + δg(ρ))dρ

][1− βn− β(1− n)η]2

|δ=0

+β(1− n)

{βρµ + β(1− n)(η − 1)

[ρ[1− F (ρ)− δG(ρ)] +

∫ ρρ ρ(f(ρ) + δg(ρ))dρ

]}[1− βn− β(1− n)η]2

|δ=0

=[1− βn− β(1− n)η]β(1− n)

[ρe [1− F (ρe)] +

∫ ρeρ ρf(ρ)dρ

][1− βn− β(1− n)η]2

+β(1− n)

{βρµ + β(1− n)(η − 1)

[ρe [1− F (ρe)] +

∫ ρeρ ρf(ρ)dρ

]}[1− βn− β(1− n)η]2

> 0.

∂ψ

∂ρ∗|δ=0 =

β(1− n) {[1− F (ρ∗)− δG(ρ∗)]− ρ∗f(ρ∗)− ρ∗δg(ρ∗) + ρ∗f(ρ∗) + δρ∗g(ρ∗)}1− βn− β(1− n)η

|δ=0

=β(1− n)(η − 1) [1− F (ρe)]

1− βn− β(1− n)η> 0,

∂ψ

∂δ|δ=0 =

β(1− n)(η − 1)[−ρ∗G(ρ∗) +

∫ ρ∗ρ ρg(ρ)dρ

]1− βn− β(1− n)η

|δ=0

=β(1− n)(η − 1)

(∫ ρeρ ρg(ρ)dρ

)1− βn− β(1− n)η

< 0,

where we have used∫ ρeρ ρg(ρ)dρ < 0, and 1− βn− β(1− n)η > 0 by (67).

There are three opposing forces affecting how the mean preserving affects ψ.

dψ

dδ|δ=0 =

∂ψ

∂ρ∗∂ρ∗

∂δ︸ ︷︷ ︸(+)

|δ=0 +∂ψ

∂η

dη

dδ︸ ︷︷ ︸(−)

|δ=0 +∂ψ

∂δ︸︷︷︸(−)

|δ=0.

If the last two effects dominate, the asset price decreases with risks, and loan to value ratio, φ`bψa ,

increases with risk.

49

Case 2: The risk premium increases after mean preserving ( dH∗

dδ |δ=0 > 0) HeredH∗

dδ |δ=0 > 0, the risk premium increases after mean preserving. The effects of mean preserving

on η, φ and qb take the opposite signs from those in Case 1. That is, dηdδ |δ=0 > 0, dφdδ |δ=0 < 0 and

dqbdδ |δ=0 < 0. Moreover,

diddδ|δ=0 =

γ

β

ξa(n− 1)dH

∗

dδ |δ=0[1− nξ

aH∗(ρe)

]2 < 0.

Also,dψ

dδ|δ=0 =

∂ψ

∂ρ∗∂ρ∗

∂δ︸ ︷︷ ︸(+)

|δ=0 +∂ψ

∂η

dη

dδ︸ ︷︷ ︸(+)

|δ=0 +∂ψ

∂δ︸︷︷︸(−)

|δ=0

If the first two effects dominate, mean preserving spread has positive effects on asset price and

negative effects on loan to value ratio.

Proof of Proposition 4. Using J(ρ∗) = 0 from (68) we have

∂ρ∗

∂γ= −

∂J∂γ

∂J∂ρ∗

∂J

∂γ=

n

a(1− 1

σ)x−1σ

{(1− βn)

[1− ξ

aH(ρ∗)

]− β(1− n)

}1

β[1− nξa H(ρ∗)]

< 0.

From Proposition 2, J ′(ρ∗) < 0, we have

∂ρ∗

∂γ< 0,

∂F (ρ∗)

∂γ< 0,

∂H(ρ∗)

∂γ< 0,

and∂η

∂γ=

1[1− ξ

aH(ρ∗)]2 ξaH ′(ρ∗)∂ρ∗∂γ < 0.

50

For the following variables, both γ and ρ∗ change. We use total derivatives to derive the following

comparative statics:

diddγ

=∂id∂γ

+∂id∂ρ∗

∂ρ∗

∂γ

=

[1− ξ

aH(ρ∗)]

β[1− nξ

a H(ρ∗)]︸ ︷︷ ︸

(+)

+

γβξa(n− 1)H ′(ρ∗)[1− nξ

a H(ρ∗)]2︸ ︷︷ ︸

(−)

∂ρ∗

∂γ︸︷︷︸(−)

> 0;

dx

dγ=

∂x

∂γ+∂x

∂ρ∗∂ρ∗

∂γ

=1

β[1− nξ

a H(ρ∗)]︸ ︷︷ ︸

(+)

+γ

β

nξa H

′(ρ∗)

[1− nξa H(ρ∗)

]2︸ ︷︷ ︸(+)

∂ρ∗

∂γ︸︷︷︸(−)

There are two opposing forces through which an increase in γ affects x: a direct effect from the

change in γ, and an indirect effect from the change in ρ∗. If the direct effect dominates, x increases.

Moreover, because q−σb = x = (φM−1

1−n )−σ, qb and φ are negatively related to the change in x. Then

we would expect that qb and φ decreases with γ.

51

Appendix D. Numerical exercises

We use two specific distributions of asset dividends in numerical examples to investigate the effects

of changes in risk by changing the variance of asset dividends while preserving the mean return.

Note that the mean preserving exercise in the previous section is restricted to a small change in risk

(δ → 0). To make numerical exercises comparable to Proposition 3, we focus on a narrow range of

asset risks.

Example 1: Uniform distribution U [ρµ − bε, ρµ + bε] The hazard function of the uniform

distribution is

H(ρ) =1

ρµ + bε− ρ,

so H ′(ρ) > 0, and

dH

dε=

−(b− ∂ρ∗

∂ε )

(ρµ + bε− ρ∗)2< 0

if ∂ρ∗

∂ε is not too big.

Example 2: Truncated Normal Distribution Assume a normal distribution N(ρµ, ε2). We

restrict the random dividend, ρ, to be drawn within the b standard deviation from the mean, so

ρ = ρµ − bε, and ρ = ρµ + bε. Assume F (·) be the cumulative density function, and f(·) be the

probability density function before the distribution is truncated. Then, the cumulative density

function after the distribution is truncated is

F T (ρ) =F (ρ)− F (ρ)

F (ρ)− F (ρ),

the pdf is

fT (ρ) =f(ρ)

F (ρ)− F (ρ).

and the hazard function is

HT (ρ) =f(ρ)

F (ρ)− F (ρ).

52

It is hard to know the sign of the hazard function of truncated normal distribution after mean

preserving. However, in the following numerical exercise with truncated normal distribution, the

wedge is increasing in asset risks, ε.

We show how to calculate∫ ρρ wf(w)dw .

∫ ρ

ρwfT (w)dw =

∫ ρ

ρw

f(w)

F (ρ)− F (ρ)dw

=1

F (ρ)− F (ρ)

∫ ρ

ρwf(w)dw

=1

F (ρ)− F (ρ)

[ ∫ ρ

−∞wf(w)dw −

∫ ρ

−∞wf(w)dw

]=

1

F (ρ)− F (ρ)

[E(ρ|ρ < ρ)F (ρ)− E(ρ|ρ < ρ]F (ρ)

]Then we can use the inverse mills ratio to get the conditional expectation of normally distributed

random variable.

E(ρ|ρ < ρ) = ρµ + ε−f(

ρ−ρµε )

F (ρ−ρµε )

Numerical examples We choose the following parameters for numerical exercises, γ = 1.01,

β = 0.95, σ = 0.2, n = 0.4, M = 100, A = 2, ρµ = 0.006,, ξ = 0.00010, and ε = 0.001. We consider

a case where dividend is uniformly distributed, and the other case where the asset dividend is

drawn from a truncated normal distribution, and we assume the same support of dividends in both

cases. Specifically, we consider ρ = ρµ − 2.5ε, and ρ = ρµ + 2.5ε. Because uniform distributions

and truncated normal distributions are symmetric distributions, we can change ε while preserve the

mean.

53