Financial constraints,asset liquidity and investment volatility

..

.... GETULIO VARGAS

ECO~O\lICi\

...

EPGE

Escola de Pós-Graduação em Economia

"Financiai Conatralnta,Asaet Liquidity and

Inveatrnent, Volatility"

Prof. Heitor Almeida (University of Chicago)

.LOCAI.

Fundação Getulio Vargas

Praia de Botafogo, 190 - 100 andar - Auditório

DATA 17/12/98 (58 feira)

HORÁRIO

16:00h

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and Investment Volatility .

Heitor Almeida

University ofChicago, November 1998

We build a general equilibrium model, where we introduce credit market imperfections. The source of imperfection is the illiquidity of real assets, due to the inalienability of human capital. Besides allowing for varying degrees of collateralliquidity, we allow for a case in which the availability of credit is determined by the ability of creditors to extract debt repayment from debtors through potential debt renegotiation processes.

The most important conclusion in this paper is that higher availability of credit does not necessari1y translate into lower investment volatility, in an economy where financial constraints are binding. The implication for previous empiricalliterature is that most tests on the importance of financial constraints may be misspecified.

We also find thftt the positive relationship between liquidity and investment volatility is

more likely to happen in a highly collateralized economy, or in a sub-sample ofhighly collateralized firms. The latter should also be more sensitive to shocks to net worth, than an economy (or a sub-sample offirms) with relationship-based lending.

I am grateful to Raghuram Rajan for numerous discussions and extremely valuable advice. I also wish to thank Douglas W Diamond, José A. Scheinkman and Luigi

Zingales for their comments and advice. Special thanks to my friends Guilherme Marone, João M Rato, Guy E. Saidenberg and Olivier Vigneron for their daily help and

..

..

1

1

Introduction

Recent empirical literature has found evidence that the net worth of fums seems to affect investment, output and other real variables, in a way that is inconsistent with standard neoclassical models. More specifically, there have been studies showing that investment is correlated with measures of

cash flow, after controlling for changes in investment opportunities1 _(Fazzari

et.al,1988, Gilchrist and Himmelberg, 1995, Hoshi et.al, 1991, Hubbard and Kashyap, 1992, Hubbard et.al, 1995, Kashyap et.al, 1994, Lamont, 1997, Oliner and Rudesbusch, 1992, Schaller, 1993, and Whited, 1992 are only some of the references)2.

The explanation for the excess volatility of investment provided by the empirical studies above is based on financiaI constraints. The empirical stud-ies usually divide a sample of fums according to some a priori characteristic

that is supposed to proxy for the existence of financiaI constraints3_{, and}

compare the correlation between net worth and investment across the sub-samples. A higher correlation in the sub-sample identified as more finan-cially constrained is interpreted as evidence for the importance of financial constraints as a source of excess volatility ..

However, is it true that investment in a more constrained fum should always be more sensitive to changes in net worth?

Existing theoreticalliterature does not have much to say about this cru-cial questiono Theoretical models such as Kiyotaki and Moore (1997) and Bernanke and Gertler (1989) predict that credit-constrained firms should

re-spond more to shocks that affect net worth, as compared to fums that do

not face borrowing constraints. This excess sensitivity of real variables to net worth, due to imperfections in credit markets, has been called elsewhere the "credit multiplier", or "financial accelerator".

Nevertheless, this does not mean that investment volatility should be

monotonic in the degree to which fums are financially constrained. In other

words, there is no result showing that the credit multiplier increases

monoton-1 Furthermore, there is also evidence that the response of sales, inventories, investment

and employment deroand following a rooneta,ry contraction is significant1y stronger for furos that are likely to be liquidity-constrained (Bernanke et.al, 1996, Gertler and Gilchrist, 1993 and 1994, Kashyap et.al, 1994 are some references).

2See Almeida (1998), and Bernanke, Gertler and Gilchrlst (1996), for a survey of this literature.

3Such as dividend policy, tangibility of assets, furo size and age .

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-ically, as fums become more constrained3.

Our aim in this paper is to look more closely into the mechanisms of the credit multiplier. More specifically, v/e will attempt to characterize how the

multiplier changes, as the availability of credit to constrained firms changes.

In our setup, borrowing constraints arise from an asset illiquidity problem. Entrepreneurs cannot commit to transfer the future proceeds from their

phys-ical assets to potential creditors (inalienability of human capital) 4

• Assets are

illiquid, because they are worth more in the hands of entrepreneurs (generat-ing ex-post rents), and because ex-ante contract(generat-ing possibilities are limited.

Therefore, as assets become more liquid, the availability of credit to

con-strained entrepreneurs increases. Should this necessarily reduce the volatility

of entrepreneurial investment ? N aturally, if the increase in asset liquidity

is high enough, so that entrepreneurs are no longer constrained, the credit multiplier disappears altogether, and volatility decreases. However, as long

as entrepreneurial investment remains constrained by credit market

imper-fections, the relationship between asset liquidity and investment volatility is not so immediate.

The most important conclusion in this paper is that higher availability of credit does not necessarily translate into lower investment volatility, in an economy where borrowing constraints are binding. In other words, the credit

multiplier is non-monotomc in asset liquidity. If liquidity is high enough, the

multiplier is necessarily reduced (possibly to zero). However, for lower leveIs of liquidity, it is possible that further increases in liquidity actually increase the multiplier.

These implications are relevant both for comparisons of liquidity across fums, and across countries.

In terms of a cross-section of fums, our results imply that it is inappropri-ate to assume that investment volatility should be higher in a sub-sample of fums identified as more constrained. Empirical tests that are implicitly based on this assumption may be misspecified. For an example, it does not suflice

to compare the coefEicients of cash flow in investment equations. Our model

implies that, even if the cri teria used to classify the sample is appropriate

(in the sense that the sub-sample identified as credit-constrained is indeed

more constrained), it does not follow that the more constrained sub-sample

should respond more to changes in net worth. The only way to guarantee the

3This point was originally made by Kaplan and Zingales (1997).

4 As in Hart and Moore, 1994 .

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appropriateness of such a test is to identify precisely a sub-sample of fums that are not subject to credit constraints.

The cross-country implications are also interesting. Institutions are likely to differ across countries, in ways that are correlated with the liquidity of as-sets and the availability of credito Legal issues related to corporate finance, corporate governance systems and the leveI of financiaI deepening are ex-amples of such differences. Qur model implies that, even though institutions that facilitate credit reduce underinvestment problems associated with finan-ciaI constraints, this may happen at the cost of higher investment voIatility.

We build a three- period general equilibrium model, where we intro-duce credit market imperfections, and where asset liquidity is suitably para-metrized. Qur microeconomic setup is based on Hart and Moore (1994). Borrowing constraints arise from incomplete contracting and human capital specificity. As in Kiyotaki and Moore (1997), we embed these borrowing constraints in a general equilibrium model, in order to characterize the be-havior of the credit multiplier. In Kiyotaki and Moore, asset liquidity and the availability of credit are always determined by the amount of collateral

available in the economy (an economy with collateralized credit contracts).

Collateral liquidity is also :fixed in their setup.

Here, besides al10wing for varying degrees of collateral liquidity5, we also

allow for a case in which the availability of credit is determined by the abil-ity of creditors to extract debt repayment from debtors through potential

debt renegotiation processes6. We name such contracts relationship lending

contracts, to stress the fact that the availability of credit in such cases is

determined by an ability of creditors to bargain directly with debtors. It is

also possible to parametrize asset liquidity in this case, with the assumption that creditors have different degrees of bargaining power.

In the case of a collateralized economy7, we show that an increase in

5The assumption that allows for this parametrization is that a certain fraction of the value of collateral is lost, if creditors are forced to liquidate debtors' physical assets. As this fraction increases, there is more illiquidity, and the bOITowing constraint gets tighter. A similar parametrization of liquidity is used in Myers and Rajan (1997). As shown by Diamond and Rajan (1998), we can also interpret an economy with smaller (proportional) liquidation costs as an economy with more developed bank intermediation of financial contracts

6This is also based on Hart and Moore (1994).

7We characterize only the extreme cases in which credit is either determined entirely by collateralliquidity, or entirely by creditor bargaining power. This is mostly for illustrative

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the liquidity of colIateral decreases the underinvestment problem caused by borrowing constraints. Entrepreneurs (debtors) invest more as the

liquid-ity of colIateral increases. Since our economy is characterized by the fact

that entrepreneurial investment is below its first best leveI, this decreases the distortions in our economy. We also characterize the response of the colIateralized economy to an unexpected productivity shock, starting from a cred.it constrained equilibrium. In our economy, the productivity shock has an impact on the equilibrium alIocation of capital, due to the presence of borrowing constraints (and only if they are binding). However, the percent-age change in entrepreneurial capital (fram the initial equilibrium), folIowing

the shock, is non-monotonic in the liquidity of collateral.

The specific shape of the function, relating the percentage change in cap-ital and the liquidity of colIateral, depends on specific parameters such as

the marginal productivity of capital. Nevertheless, there is a pattern that

is robust across ali parametrizations. For high enough liquidation costs (our

measure of illiquidity), the percentage change in capital becomes decreasing in these costs, even though higher liquidation costs always make the economy more constraineds.

The intuition behind the non-monotonicity result is related to a debt

overbang eifect. Increases in the liquidity of colIateral increase the amount of debt that entrepreneurs take, and also the amount of debt repayment they must make. However, the amount of debt repayment does not change with the (unexpected) productivity shock. In other words, the impact of the overhang effect on equilibrium investment before the shock is proportionately

higher (if the shock is positive) than on investment following the shock,

causing the economy to respond more to shocks as the liquidity of colIateral

increases. If the shock is negative, the impact on investment after the shock is

proportionately higher, amplifying the effects of the shock. The result is that increases in colIateral liquidity can generate higher investment volatility ..

Similar exercises for the relationsmp lending economy show that monotonic-ity will also not hold in such an economy. This is also due to an overbang

ef-fect. As creditor bargaining power increases, the amount of debt repayments

reasons. Our main conclusions would not change if we allowed for more complicated, hybrid contracts.

8For some parametrizations, the response of the economy to the shock is initially

in-creasing in liquidation costs. For others, it is monotonically decreasing, for the whole

range of liquidation costs. It is never monotonically increasing .

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increase. Since the repayment does not vary with the shock, the impact is

proportionately higher on the equilibrium before the shock (if the shock is

positive), amplifying the effects of the shock.

The comparison between the collateralized and the relationship lending economy yields further insights into the behavior of the credit multiplier. We show that non-monotonicities are more likely to occur in the collateralized

economy. In other words, the positive effect of liquidity on volatility is higher

for the collateralized economy. Another implication is that investment in the collateralized economy is more volatile, for a given leveI of liquidity (measured bythe initial equilibrium. that precedes the shock).

These two results are caused by a different operation of the overhang effect in the two economies. In the relationship lending economy, higher creditor bargaining power increases entrepreneurial debt and investment. However, since the amount of credible debt is a always a given fraction of output, and output is a concave function of investment, the absolute magnitude of the overhang effect tends to decrease as liquidity increases. This contrasts with the linearrelationship of debt and investment in the collateralized economy,

where the amount of debt is always equal to the market value of the

ÍD-vestment good held by entrepreneurs. In other words, the fact that debt repayments are a concave function of investment in the relationship lending economy mitigates the importance of the overhang effect.

Therefore, our consideration of the relationship lending economy gener-ates further implications. The positive relation between liquidity and volatil-ity is more likely to happen in a highly collateralized economy, or in a sub-sample of highly collateralized firms. Another implication is that a highly collateralized economy (or a sub-sample of collateralized firms) should be more sensitive to shocks to net worth, than an economy with relationship-based lending.

This conclusion can have interesting implications for the theoretical analy-sis of financiaI crises. A financial crianaly-sis is often preceded by an increase in

liquidity in credit markets (as in the recent Asian crisis). Furthermore, the

increase in liquidity has been identified as an important factor leading to

the crisis9_{. According to the literature, the links between liquidity and}

vul-nerability to a crisis arise from moral hazard problems on the creditor side (excessive, and excessively risky lending), or from the possibility of financial

9See Krugman (1998), Corsetti, Pesenti and Rubini (1998), and Sachs and Redelet

..

panics. Explanations based on agency problems on the debtor side have been generally overlooked.

In fact, if we had monotonicity in the sense described above, there would be no hope of using a model based on borrowing constraints and underin-vestment to explain a positive relationship between liquidity and volatility. Higher Iiquidity would Iead both to Iess underinvestment and Iower

voIatiI-ity. However, as long as monotonicity is shown not to hold, models based on

agency probIems on the debtor side can add insight to the theoretical debate on the likelihood of financiaI crises.

The next section presents the model, characterizes the credit constrained equilibrium and the first best allocation in our economy. Section 3 examines the impact of unexpected productivity shocks, as a function of asset liquidity, in the two benchmark economies. Section 4 concludes .

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2

The Model

2.1 Technologies and Endowments

The economy described in this section is a three period (t = 0,1,2)

produc-tion economy. There are two goods. 0nly one of them is used for

consump-tion .. There are technologies that transform the producconsump-tion (investment) good

kt; into consumption (output) in the next period, Yt+1. At each period, there

is a market where agents can trade the investment good for the consumption

good, at a price qt.

There are two types of agents, that differ both in endowments and in the production technologies. The first type (that we call lenders), are

en-dowed with the entire stock of the investment good, K, and also have a

large endowment of the consumption good (the total market value of their

initial endowment is w5) . Finally, they are endowed with a technology that

produces consumption good next period (Yf+l)' from investment this period

(kf):

yf+1 = G(kf), t = 0,1 (1)

The second type of agents (that we call entrepreneurs) are endowed with

an initial amount of the consumption good (wff) , and with a production

technology :

yi+l = f(kf), t = O, 1

Assumption A.1

Both technologies are increasing and concave. Furtherrnore :

1'(0) - G'(O) = 00

flll(X) > O; GIII(x) > O \/x

These technical assumptions will be necessary ahead.

(2)

The crucial difference between the technologies

f

and G is that the

entre-preneurial technology

f

requires the input of human capital that is specific to

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each individual entrepreneurlO

• More specifically, once entrepreneur i starts

production at date t with

kL

only he has the skill necessary to produce the

t

+

1 output

f

(k;). If his human capital is withdrawn during period t

+

1, the

entire output is lost. The physical input

k;

can only be sold at the market for

the price qt+l, yielding the payoff qt+lk~ in terms of the consumption good.

2.2 Endogenous Borrowing Constraints

There is a one period credit market in this economy, opening every period,

at which agents can trade one unit of the consumption good today for R

units of consumption tomorrow. As in Kiyotaki and Moore (1997), we rule

out long term contracts by assumption 11. The short term debt contracts are

characterized by the amounts (b~, bf) that are transferred to entrepreneurs

in each period, and that must be repaid in the following period (with a gross

interest rate equal to Rt ).

U nder contractual incompleteness, the functioning of this credit market

will be affected by the human capital specificity of entrepreneurs12_{• The}

assumption is that entrepreneurs cannot write contracts at date t forcing

them to input their human capital and repay their debt at date t

+

113_.

De-pending on the amount of debt they have outstanding (Rtbn, entrepreneurs

may decide to renege on their debt and renegotiate with their creditors (the lenders). Lenders have the option to bargain with entrepreneurs, or simply

to seize the physical assets under the control of entrepreneurs (k!) 14. The

timing of events, and the specifics of the renegotiation game are as in figure

1.

We assume that the use of the liquidation option by lenders entails trans-action costs that are proportional to the value of the collateral held by

entre-preneurs15_{• More specifically, if the physical assets}

_kf

_{are seized by lenders,}

lOThis is exactly as in Hart and Moore, 1994.

11 We also experimented with long term contracts. In this three period model, the main conclusions of the paper are unchanged. This might not be true in a fully dynamic model, though.

12This is true even if there is no uncertainty, as we assume here.

13Given our assumptions about their endowments, entrepreneurs will always become

borrowers, if the credit market is active at alI.

141n the case of a long term contract, we assume that creditors can seize the total amount

of physical assets in the hands of entrepreneurs (kf) at period t = 2.

15We rule out partial liquidation of assets. All the proceeds from liquidation go to

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-a fr-action T E (0,1) of the proceeds qt+lkf is lost. An increase in T will then

decrease the liquidity of col1ateral.

The other important parameter to determine the outcome of the renego-tiation game is creditor bargaining power. Following Hart and Moore (1994),

we assume that, if the bargaining stage is reached, each party gets to make

a take-it-or-leave-it offer of a new debt repayment with a certain

probabil-ity. The probability that creditors have the right to make the offer is then

a simple measure of creditor bargaining power in this stylized scenario. We

call this probability À. As À increases, creditors can extract a higher

frac-tion of the current output through the bargaining game. The specifics of the

bargaining game are as in figure 216_.

With this set up, it is straightforward to characterize the outcome of the

renegotiation game as a function of the parameters (À, T) 1 i. The maximum

amount that entrepreneurs can credibly commit to pay in the last period is :

(3)

If total outstanding debt is higher than

P2

ax_{, entrepreneurs can}

success-fully renegotiate it down to p

₂

ax. Therefore :

R bE

_<

_p,max

1 1 - 2

In period t = 1, a similar constraint applies:

(4)

(5)

Equations 4 and 5 are the (endogenously derived) borrowing constraints

on the entrepreneur. If (total outstanding) entrepreneurial debt exceeds the

maximum between the value of their col1ateralizable asset, and the value that lenders can obtain through bargaining, entrepreneurs will renege. Anticipat-ing this, creditors are not willAnticipat-ing to lend in excess of this value.

lenders.

16We assume that creditors forego the liquidation option if they choose to bargain, and

that creditors cannot seize entrepreneurial past savings in any case. If the parties fail to

reach an agreement with bargaining, current output is lost, and the proceeds from the sale of the physical assets go to entrepreneurs .

17 See Hart and Moore (1994) for a more complete analysis of this game .

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2.3 Agents' Optimization Problems

2.3.1

Lenders

Lenders are not sub ject to borrowing constraints, since their technology is not specific. Furthermore, the assumption that they have a large endowment of the consumption good means that we can safely ignore the constraint that consumption has to be non-negative. Assuming for simplicity that the

discount rate is one, and that lenders are risk-neutral, we can write the

lenders' optimization problem asIS _:

max(~

+

cf

+

c{)s.t.

c~ - w~ - qok~

+

b~

cf - G(k~)

+

ql(k~ - kf)

+

bf - Rob~

c~ - G(kf)

+

q2k

f -

RIbf

The first order conditions yield :

Ro - RI = 1

G'(kt) - qo - ql

=

Uo

G'(kf)

_-

ql - q2

=

UI

(6)

(7)

The equilibrium interest rate must be equal to one. The marginal pro-ductivity of the investment in the lenders' sector must be equal to the user

cost of land in each period, that we define as Ut.

2.3.2 Entrepreneurs

The problem facing each entrepreneur is more interesting. We have to impose

non-negativity constraints on consumption in periods t

=

O and t

=

1, and

add the restrictions on borrowing derived above from our assumptions on contracts and technologies (equations 4 and 5) .

18We are assuming that the market price of land in period 2 (q2) is not necessarily zero,

despite the fact that this is the last period.

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max Co

+

c1

+

C2 S.t. (8)

cE _{o -} w~ - qok~

+

bff ~ O

cE _{1 -} f(k~)

+

ql(k~ - kf)

+

bf - bff ~ O

c₂ E - f(kf)

+

q2 kf - bf b_O E

<

max {(I - 7) k~qll ).f(k~)}

b₁ E

<

max {(I -7) kfq2' ).f(kf)}

Before characterizing the solution to this program, we characterize the first best allocation in this economy.

2.4 First Best Allocation

It is useful to characterize the first best allocation in this economy, as a

bench-mark. It is simply the allocation that obtains in the absence of borrowing

constraints on entrepreneurs19_•

Proposition 1 In the first best, the marginal productivity of the investment

good is equalized across sectors, and the user cost of land is constant across time:

G'(kf.Jb) = f'(kf.Jb) = u fb , t = O, 1

where kf.Jb

+

kf.Jb = K

Proof: Since marginal productivity is monotonic in both sectors by

as-sumption, the functions G' ( .) and

f' (.)

cross at most once. By equation 7

, the point at which they cross is equal to the user cost of land in the first best allocation.

Remark 1 If the functions G(.) and f(.) are identical (a case we will use

ahead to illustrate our results), the first best is simply : kE - kG - kfb _ K

t - t - t - 2

u[b

=

G' ( ~), for

t

=

O, 1

In the absence of borrowing constraints, entrepreneurs can borrow to finance any efficient level of investment. Therefore, the constant endowment of the investment good is shared efficiently across the two sectors.

19We use "fb" to refer to the first best allocations .

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2.5 Equilibrium with Borrowing Constraints

The nature of the credit constrained equilibrium will depend on the type of credit contract that is optimal to use. For an example, if the liquidity of

collateral is high (low T), and creditors do not have a lot of bargaining power

(low >'), it will be optimal to back credit contracts with entrepreneurial assets

(collateralized debt). We refer to such a contract as a collateralized contract.

If creditor bargaining power is high (high >'), it may be optimal to increase

the amount of credit to entrepreneurs over and above the amount of collateral

available in the economy. We refer to such a contract as a relationsbip lending

contract. The terminology stresses the fact that such a contract is based on

the ability of lenders to negotiate with entrepreneurs. It is also possible that the amount of collateral determines the amount of credit in the economy in a period, but the ability to negotiate increases credit in the other period. Here, we will not focus on such hybrid contracts.

Since there is no heterogeneity across agents in our economy, a single

type of contract is optimal for the whole economy. There is no room for the co-existence of different types of contracts.

Despite the fact that the relative optimality of specific contracts depends on the parameter specification, it is possible to establish some general prop-erties of the credit-constrained equilibrium. We begin with the following lemma:

Proposition 2 In any period t, if some of the entrepreneur's borrowing con-straints binds, than his optimal consumption in that period is zero.

Proof: Let

>'t

be the multiplier associated with the non-negativity

con-straint on consumption for period t, and /-Lt and be the multipliers associated

with the borrowing constraints, in period

t.

The first order conditions for

the choice of debt leveIs in program 8 are :

(9)

The first order conditions 9 imply that >'1

=

/-LI. Furthermore, since >'0

=

/-Lo

+

/-LI' and /-Lo, /-L1 ~ 0, if /-Lo is greater than zero, >'0 is also greater than

zero .

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We will focus here on two types of (non-hybrid) contracts. We start with a collateralized contract (bg = (1 - T) k~ qI, bf = (1 - T) kf q2) , which is optimal if the liquidity of collateral is high and creditors do not have a lot

of bargaining power20. We will then move on to analyze our credit

con-strained economy when relationsmp lending (bg

=

Àf(k~), bf

=

Àf(kf) is

optimal (this occurs if creditor bargaining power is high, and/or the liquidity

of collateral is low).

2.5.1

The Credit-Constrained Equilibrium with

Col-lateralized Contracts

In this case, entrepreneurs borrow up to the value of their collateralizable assets, in both periods, and repay their debt on the following period. The relevant entrepreneurial program is :

(E E E)

max Co

+

_cI

+

_c2 s.t. (10)

cff

- wff - qokff

+

bff ~ O

cE ₁ - f(kff)

+

qI(kff - kf)

+

bf - bff ~ O

cE _{2 -} f(kf)

+

q2k f - bf

b_o E

<

(1 - T) kffqI b₁ E

<

(1 - T)kfq2

The first order conditions for the optimal choices of kff and kf are2I:

f'(kff)

f'(kf)

_ uo(1

+

/-Lo )

+

/-Lo TqI

1

+

/-LI 1

+

/-LI

- UI (1

+

/-LI)

+

/-LI Tq2

(11)

Proposition 3 In a credit constrained equilibrium with collateralized con-tracts, entrepreneurs underinvest in the investment good, as compared to the first best allocation .

20This is the only type of contract in Kiyotaki and Moore (1997). They do not allow

for long term contracts, and (implicitly) assume that creditor bargaining power is low. 21/Lo and /LI are the multipliers associated with the borrowing constraints .

'," \..

"'-'-.

. , .

.

. . ... .

..-.'.: .... .;

, -" '~.

•

Proof: The first order conditions 11, together with the first order condi-tions for the unconstrained program 6, imply that (we use stars to refer to equilibrium values):

The multipliers /-Lo and /-LI are positive. Therefore, in a credit-constrained

equilibrium22

, we must have:

u;

<

ur,

t

= O, 1

kE,*

<

kE,/b

t

=

°

1

t t "

(12)

The presence of the borrowing constraints has the usual effect on the equilibrium allocation. The constrained sector invests less than it would be optimal. This lowers the equilibrium price of the investment good, as compared to the first best case.

Proposition 2 allows us to compute the credit constrained equilibrium directly. Since the borrowing constraints bind, we know that entrepreneurial

consumption is necessarily equal to zero: (c~'*, c~,*)

=

(O, O). Ali

consump-tion occurs in the last period. Given this, we can deduce entrepreneurial

holdings of the investment good (k~'*, k~'*) directly from the budget

con-straints in 8. This gives23 _:

kE_,*

o - k* _ _{o -}

_Uõ

₊

_r(ui Wo

₊

_q2) (13)

kE_,*

1

-k* _ !(kô)

+

r(ui

+

q2)k

_ô

1 -

ui

+

rq2

kG_,*

t - K - k;,

t

= O, 1

u* _o - G'(K - kô)

u* ₁

_-

G'(K - k{)

22 Again, this follows from the properties of the functions f and G, more specifically

from the monotonicity of their first derivatives.

23 Apart from the result in lemma 1, the equilibrium consumption allocations are

unessential for our discussion. Therefore, we will limit ourselves to characterizing the equilibrium leveIs of investment, and the user cost of the investment good .

_ ",:-.c.- •• ,-' : .. ...

. - ... .. ... "

•

Remark 2 The presence of borrowing constraints will not necessarily cause the distortions above (equations 12), since it is possible that they do not bind. This depends on the parameter values, such as entrepreneurial wealth (wff) , Iiquidation costs (7), and the production functions f and G. A mechanic way

to check whether the constraints bind or not is as follows : - Assume that the constraints bind;

- Compute the allocation above (13) ;

- lf u;

<

ur , this is the equiIibrium allocation in period t;

- lf u; ~ u{b, then the allocation is the first best one, as in proposition 1.

N otice that it is also possible for the constraint to bind only in period t =

o.

As the entrepreneur accumulates wealth from the first to the second period,

he may have enough to purchase the first best amount of the investment good

in period

t

=

·0, despite the presence of bOITowing constraints24.

We know ask ourselves what happens to the equilibrium as we vary our

measure of liquidation costs, the parameter 7. By equations 4 and 5, an

increase in 7 decreases the amount that can be borrowed, for a given aroount

of collateraL Therefore, we would expect entrepreneurs to become more constrained, and to reduce the amount of investment made in equilibrium

(k;). Naturally, this exercise only makes sense starting from an equilibrium

at which the borrowing constraints are binding. If this is not the case, a

small increase in liquidation costs would have no effect on the equilibrium

allocation, which is equal to the first best one anyway.

Proposition 4 Starting from an equiIibrium in which the borrowing con-straint is binding, an increase in 7 decreases entrepreneurial investment in

period t = O .

Proof: In the appendix.

The saroe result holds in period

t

= 1, under certain (mild) conditions:

Proposition 5 Starting from an equiIibrium in which the borrowing con-straint is binding, there exists a leveI of entrepreneurial investment at time zero, kõax , satisfying O

<

kõax

<

ktb ,such that for all kô

<

_kõax , an increase

in 7 decreases entrepreneurial investment in period t = 1. Furthermore, Iet:

kmax _ f(k

õ

ax )

+

7(G'(k - krax)k

_õ

ax

1 - G'(k - kiax)

24The increase in his net worth also increases the amount he can borrow, by equation 4.

: . ' • O', ,_, • ; . ~; " ,

" -"~ .. '" ~.

. .

• • ~f ... ' .... ". " ' , ' '.

-. ' . '. . . •... "

';." . '. ~:' ...•.

. :: .. :-.... ,:

..

...

•

•

. . ~",' .

]f J kmax l '

>

kfb then even zf k* O

>

kmax Ôk*jÔT <O O , 1 _ .

Proof: In the appendix .

In other words, if the initial distortion

kt

b -

kô

is high enough, it is always

true that ôk; j Ô7

<o.

Furthermore, if the parameters of the economy are such

that an investment of kõax at time zero aliows the first best to be reached at

time

t

= 1, then it is always true that ÔkUÔT ~O

It is very important to notice that the conditions above are only suflicient

conditions for ôkU ÔT

<O,

as we argue in the appendix. More generaliy, we

argue in the appendix that a case in which ôk; j ÔT

>

O, although logically

possible, does not occur for reasonable parametrizations of our model

econ-omy. For ali of the simulations below (section 3), it is true that ÔkUÔT

<o.

The intuition for these two propositions is straightforward. An increase in the liquidity of collateral increases the availability of credit, and therefore decreases the underinvestment problem caused by borrowing constraints. En-trepreneurs (debtors) invest more as the liquidity of coliateral increases.

2.5.2 The Credit-Constrained Equilibrium with

Rela-tionship Lending

In this case, entrepreneurs are able to bOITOW b~

=

)..f(k[), and bf

=

)..f(kf)

, due to the ability of lenders to negotiate with the entrepreneurs. The

entrepreneurial program in this case is :

max(cg

+

cf

+

cf)s.t.

cg -

wff - qokff

+

bff ~ O

cf -

f(k~)

+

ql(k~ - kf) - b~

+

bf ~ O

c~ - f(kf)

+

q2k f - bf

b~

<

)..f(kff) bf

<

)..f(kf)

The respective first order conditions are :

(14)

. ,\ ,," . , , .

.. t , .

", .,~

..

".. ~. ..

',' •• "c" •.

i)

k

'~;{,';.

,t;

,"<

';<';:";2"':::~1:.~;.~,:~~,?;~1%~~

~ ' l '

•

. -',- \ .

f'(k~)

f'(kf)

(15)

Proposition 6 In a credit constrained equilibrium with relationship lending, entrepreneurs underinvest in the investment good, as compared to the first best allocation.

Proof: The first order conditions 15, together with the first order condi-tions for the unconstrained program 6, imply that:

u~

_ G'(K - kE_,*)

_<

_f'(kE_{,*) = u*( 1}

+

_/-lo

+

_/-lI)

+

_{/-lo *}

o o o 1

+

_À/-lo 1

+

_{À/-lo ql}

U*I - G'(K - kE,*)

<

!'(kE,*)

=

u*( 1

+

/-lI )

+

/-lI

I I I 1

+

À/-l

I) (1

+

À/-l1) q2

Given this result, we can compute the equilibrium allocation using a similar algorithm as the one used above for the collateral case:

kE_,*

o - k* _ Wo _{0 - U}

+

Àf(k~)

ô

+ui

+

q2 (16)

kE_,*

ki = (1 - À)f(k

ô)

+

(ui

+

q2)k

ô

+

Àf(ki)

I

-ui

+

q2 kG_,*

t - K - k;,

t

=

0, 1 u* _{o -} G'(K - k~)

u* _I - G'(K - ki)

Again, if k;

<

kfb, this is indeed the equilibrium allocation in period

t. Otherwise, k; = kfb, and the economy is unconstrained. Starting from

an equilibrium at which the borrowing constraints are binding, it will be true that an increase in creditor bargaining power makes the economy less constrained, as we would expect :

: ~'-;"-' '-. ,--.-.

-" '\.'

..' , . c:;' \\'-:;'.'

;~./.i,:::{~~~,i!

':. I ' - . r ::~ ' . \ ~ ~ '.-.. ' ' . . "'.' ... ;.;.

•

..

Proposition 7 Starting from an equilibrium in which the borrowing

con-straint is binding, an increase in ,\ increases entrepreneurial investment in both periods .

Proof: In the appendix.

N otice that in the case of a relationship lending economy, it is always true

that ôk;/ô'\ ::;0

In this economy, the welfare of entrepreneurs and creditors is

unambigu-ously increasing in the ability of creditors to extract debt repayments from entrepreneurs. This increases the debt capacity of entrepreneurs, and re-duces the underinvestment problem caused by the presence of human capital specificity and incomplete contracting .

. ~ ' ... '

. ... ";.: " - ' ,

J'::;' :.:;~. :! ~ .. ': ... :

.;.- 'i," "

'./'

3

The Credit Multiplier - A Comparison of

Collater-alized Debt and Relationship Lending.

We now consider the impact of an unexpected shock to productivity25, that

hits the economy in period

t

= 126. Period 1 production for an entrepreneur

is27:

yf

= (1

+

B)f(k%)

For a given input k%, production is B% higher than expected. Since

the shock is entirely unexpected, it does not affect period

t

= O investment

(k% = kô).

We also assume that the shock hits the economy after the entrepreneur

has input his human capital. Therefore, it does not affect the renegotiation

game, and the actual amount repaid28• In a collateralized contract, b

õ

=

(1 - 7)(ui

+

_q2)kô, and in an insider contract, b

_õ

= _)..f(kô).

In an economy without credit constraints, given the assumptions of risk-neutrality, and the temporary nature of the shock, there would be no effect on the amount of investment good held by entrepreneurs. The extra production is simply consumed this period, and the amount of entrepreneurial investment remains at its first best leveI.

In an economy where the borrowing constraints are binding, however, the increase in net worth will allow entrepreneurs to invest more. This creates

an impact on current investI;nent (kf), and on the user cost of the

invest-ment good this period (u_{l )29.} This amplification of the impact of shocks

25The impact of anticipated uncertainty on a credit-constrained economy (where the borrowing constraints are derived with incomplete contracting assumptions) is analyzed in Krishnamurthy (1997), and Kiyotaki and Moore (1997-b). However, they do not allow for the presence of insider contracts as we do here, and do not evaluate the monotonicity issue. See also Almeida (1998), for a discussion of these two papers.

26This is the same exercise performed by Kiyotaki and Moore (1997).

27It is inessential whether lenders are also hit by the shock. If they are, they will simply

consume the extra prod uction in period 1.

281n the simulations below, we consider the case in which the contract is renegotiated after the shock (see figure 8).

29 As shown in Kiyotaki and Moore (1997), the dynamic impacts ofthe shock can amplify considerably its effects on the economy. The increase in production next period allows investment to increase further. This increases the future user cost of the investment good,

• '-•••. o', • ' . _ • • • " " _ _ '. =-:..~.

~ .. '

I . ' . . ,:'- .. : .. ":. '

' ; :

", ... ,~ .:' :. ~ .. '- -: ....

;::( 't:;.' • -', ": ~," '. .:'

•

. , \ .

.. ~ .

•

to net worth, due to the interaction of borrowing constraints and market equilibrium, is what is called "credit multiplier"29.

We will start by characterizing the behavior of the multiplier in a collat-eralized economy, and then move on to compare the results to the economy with insider contracts.

3.1 The Credit Multiplier in a Collateralized Economy

The equations characterizing the new equilibrium k~ (assuming k~

<

k{b), in

an economy with collateralized contracts aréo.

k~ - (1

+

O)f(k

ô)

+

(u~

+

q2)k

ô -

b

ô

UI

+

rq2

b* _{o -} (1- r)(u~

+

q2)kô

b~ - (1 - r)q2k~

UI - G'(K - k~)

Substituting the second equation on the first one :

k~ - (1

+

O)f(k

ô)

+

(u~ - ui)k

ô

+

r(ui

+

q2)k

ô

UI

+

rq2

(18)

If k~

>

k{b, then the shock causes the economy to reach the first best

allocation of the capital stock, in period 1. The new equilibrium is then

k_{1 - 1 - 1 ·} E - kG - k fb

We are specially interested in the percentage change (over the initial

equilibrium) of holdings of the investment good by entrepreneurs3l :

creating an important feedback effect via asset prices. here, we assume that the future

price of land (q2) is fixed. This corresponds to the static multiplier of Kiyotaki and Moore

(1997).

290r "FinanciaI Accelerator" .

30We use hats to refer to the new equilibrium.

31 This is ameasure of the unexpected change in investment, caused by the productivity

shock.

, .~ . ...

_-

..

-

.... - . --.~"

" , .,! ,

. ~ -' ' ... ~ ..

. " . . . ~,': ,~~.. ;,': ~. . .

,::

;~:':,.~

'''.:~.

~

.... , "

...

"/~;;'

..

~ <">j~:fi}

' I , " J ,

. . ~ .

.;. ... ". ,-'

-.. . ' , \ .

•

"

k% (k~ - k;) • fb ₍₁₉₎

1

-k* ₁ ' if k1

<

k1

k% _- (k[b - k;) _{, if k}. fb

1

>

k1

1 _k*

1

We evaluate this measure of the response of the economy to productiv-ity shocks, by performing numerical simulations of the model, for different

parameter values32

•

3.1.1 Simulation Results

In order to perform the numerical simulations, we need to parametrize the production function, and then choose values for ali the parameters. We will use a Cobb-Douglas specification for the production function :

Yt+1 = Ak~ , 0< a < 1, t = 0,1

We assume that the production functions are identical for ali agents in the economy. In particular, entrepreneurs and lenders have the same production

function (same parameters A and a). Given this, the first best allocation of

capital in thiseconomy is (see remark 1) :

k_t E - kt - t -G _ kfb _ K

2 (20)

fb _Aa

(~

_yr-\

_for

_t

= 0,1

ut

-In ali the simulations, we set K = 1 (this is just a normalization), and A =

1. The remaining parameters are () (the proportional productivity shock),

q2 (the price of the investment good in period 2), Wo (entrepreneurial initial

32Evaluating the derivative 8k~ /8T by hand did not prove very practical. We also

attempted to linearize the system 18 around the original equilibrium, but the results were not very enlightening, due to the dependence of the original equilibrium on the parameter

T. On the other hand, the picture that emerges from the simulations is very clear, so we

chose to go this way

BIBLIOTECA MARIO HENRIQUE SIMONSE~

•

"

...

wealth), and T (our measure of liquidation costs). Our main objective is to

characterize the behavior of the economy following a productivity shock, for

different values of T. We will also analyze the sensitivity of the results with

respect to the remaining parameters.

Figure 2 shows the results of the simulation of equations 18 and 19, across

the whole feasible range for T. We set a = 0.6, Wo = 0.15, () = 5%, and

q2 = u1b(the user cost of capital in the first best allocation).

For values of T that are elose to zero, the initial equilibrium is very elose

to the first best (that is, ki is elose to k1b ). Therefore, the economy does not

respond a lot to the shock. As T increases a little further, the response of the

economy to the shock increases very sharply, reaching elose to a 4% increase

in the capital stock, for T around 0.05. However, as T increases further,

the response of the economy tends to decrease, while remaining positive and

significant.

Figure 3 helps us understand the key result in figure 2. It displays the

initial equilibrium (ki ), and the equilibrium after the shock (k~ or kfb), for

different values of the liquidation cost and the same parametrization as in

figure 2. Some specific values for the initial equilibrium are singled out in

the graph as triangles. The values for the new equilibrium are singled out as

squares33.

For very low values of T, the initial distortion (kfb - ki) is not very large.

Therefore, the economy jumps to the first best capital stock, as a consequence

of the shock. Since ki is decreasing in T, the response of the economy to the

shock initially increases with the liquidation shocks. For this range, since entrepreneurs with less liquid collateral are more constrained, they respond more to the productivity shock ().

However, for large values of the liquidation cost, the distortion remains

after the productivity shock (k~

<

k{b). More importantly, k~ is also

decreas-ing in the liquidation costs. As figure 2 shows, this effect is strong enough

to cause the percentage response in the capital stock (as a percentage of the

initial capital stock) to decrease with liquidation costs.

In other words, for high enough T, a further decrease in the liquidity of

entrepreneurial collateral decreases the response of the economy to a

produc-tivity shock. Notice that this obtains even though an increase in liquidation

costs elearly makes the economy more constrained ( the difference (k{b - ki)

33Notice that the "actual" new equilibrium function k_1(T) is the lower envelope ofthe

functions kr (T) and k~ (T).

... ~ ,. ",.- ...

) ' " .

..

•

23

is always increasing in T, as shown in figure 3).

In order to understand this result, consider the following experimento

Assuming that there are no price effects following the shock (that is, u~ =

ui),

we have:

k',NPE (1

+

B)f(k

ô)

+

T(ui

+

q2)k

ô

(21)

-1

ui

+

Tq2

k%' NPE

_-

(k; - k;) Bf(k

ô)

-1 _k*

f(k

_ô)

+

T(ui

+

q2)k

_ô

1

In figure 4, we depictk~' NPE and k~ (in the full model), for the same

parameter values as in figure 234• It is clear that the forces leading to a

positive relationship between liquidity and volatility (and therefore to non-monotonicities) do not come from price effects. The price effects scale down the response of the economy to the shock, and also tend to reduce the slope

of the function k~(T), specially for low values of T.

Let us then try to understand the shape of the function k~' NPE(T).

Given the result that price effects are not the driving force behind non-monotonicities, this should give us the intuition for the positive relationship between liquidity and volatility.

The direct effect of T on the formula above (21), is to decrease the impact

of the shock, since increases in T (decreases in collateral liquidity) increase

the denominator of k~ , NPE. This is a force that pushes towards a negative

slope for the function k~ , NPE ( T). We must also consider the indirect impacts

of T, that operate via changes in

kô

and

ui

(we showed in section 2 that both

are decreasing functions of T). Since kô and

ui

decrease with T, it is possible

that the term T(ui

+

q2)k

_ô

actually decreases with T, what would tend to

revert the direct impact of changes in T.

However, as shown in table 1, the direct effect oí changes in T tends to

dominate. As T increases, the denominator of k~ , NPE ( T) increases, and the

impact of the shock tends to decrease.

34We eliminate the range of T for which the response of the economy is increasing in T.

This range is unchanged if we eliminate price effects, since the capital stock following the

shock is always equal to the first best .

,'o', . ... '.

•

•

. , .:~ ' ...

As long as í is greater than zero, there is another effect leading to a

negative slope for the function k~' NPE(í). If the term í(ui

+

q2)k

_ô

is

pos-'t' t h ' . k*' th f t' O!(kô) S' k*'

1 lve, an mcreases In o mcrease e unc lon !(k

ô)+r(ui+Q2)kô' mce o IS

d . . O!(kô) 1s t ds t d . h

ecreasmg m í , !(k

ô)+r(ui+Q2)kô a o en o ecrease wIt í.

The crucial term behind the non-monotonicity result is therefore í( ui

+

q2)k

_ô'

If this were equal to zero, than k~ ,NPE would always be equal to (j

(the size of the shock). Price effects would then cause k~ to decrease, as the

initiaI equilibrium (kô, k;) increased35.

This term is the extra cash-flow that remains with entrepreneurs after they have repaid their debts, not accounting for the proceeds from production

(f(kô)) :

Therefore, the e:ffect behind the non-monotonicity result is a debt

over-hang efEect. Increases in the liquidity of colIateral increase the amount of debt that entrepreneurs take, and also the amount of debt repayment they must make. Since the amount of debt repayment does not change with the

productivity shock, the e:ffect of this in the numerator of k~ , NPE cancels out,

and all that remains is the effect on the denominator, which tends to increase the impact of the shock. In other words, the impact of the overhang effect on

equilibrium investment before the shock (ki) is proportionately higher (if the

shock is positive) than on investment following the shock (k~), causing the

economy to respond more to shocks as the liquidity of collateral increases36•

Now, let us go back to figure 4 and try to understand the impact of

the price e:ffects on the shape of the function k~ (7). In credit..:constrained

economies, changes in the price of the investment good have complex e:ffects on the equilibrium outcome. Consider for example the case of a positive shock, as in figure 4. The user cost of the investment good tends to increase

with the increase in entrepreneurial investment (u~

-ui>

O), since

entrepre-neurs are more efficient users of the investment good at the initial distorted

35The elasticity of the user cost of land with respect to k~ is :

k"

=-(a-l)~

which is increasing in k~. Therefore, the "choking-off" effect of the increase in user cost

. is stronger for higher liquidity.

36If the shock is negative, the impact on k~ is proportionately higher, what also tends

to amplify the effects of the shock .

. .411.

~~~!i~

...• \..

:-.-';' '.# .•.. ~ .... :':7 ',' ,:",_':.' ... ~ ~-~ ~ •• "-~' ~- .•.. ~' ~ •...• ~ ... -:-.r.~ ... --:...-:~~. _o, ,7"-:". :-:.

~ ... ;

.•.. '.., .

•

•

•

' , '

-.

.' '."

equilibrium. Since investment goods become more expensive, this tends to "choke-oW' the impact of the shock. However, increases in the user cost of the investment good also increase entrepreneurial net worth following the shock. This efFect tends to amplify the impacts of the shock.

The result in figure 4 indicates that in oUI essentially static economy

the "choking-off" effect dominates the net worth efFect37

• This is why the

response of the shock is scaled down when we account for price effects.

Fur-thermore, the elasticity of the user cost of land with respect to entrepreneurial investment is increasing in the amount of investment :

(22)

The price effects increase as

ki

increases, and are therefore stronger for

low values of 7. This explains the shape of the function k~ (7) in figure 4.

Figures 5 and 6 show the effect of changing the productivity parameter

a, on the shape of the functions k~' NPE (7) and k~ (7). Figure 5 (no price effects) shows that the overhang effect above is robust to changes in the

productivity parameter. Absent price effects, k~ tends to decrease with

a38_{• As shown in equation 22 above, the elasticity of the user cost of land}

with respect to entrepreneurial investment is decreasing in 0::. Therefore, the

choking-off effect is stronger for low values of 0::. Figure 6 shows this effect

very clearly, specially for low values of 7 (due to the effect of higher

k;

on the

elasticity). If O:: and 7 are low enough, the impact of the shock can actually

become decreasing in 7. This is made more clear in figure 7. Notice that the

negative relationship between liquidity and volatility for low values of 7 in

37In Kiyotaki and Moore (1997) dynamic model, the net worth effect dominates the

choking-off effect, and tends to amplify the effects of the shock. It is interesting to check

whether this can change our basic result of non-monotonicity. There is reason to believe that non-monotonicities should be even stronger in a fully dynamic model (see below).

38In order to understand this, consider again table 1. Increase in a have opposite effects

on

kô

and

ui.

For a given T, the effect on the crucial term Tkô(ui + q2) is nearly zero.

However, since

kô

is decreasing in a, the response of the economy to the shock also tends

to decrease with a :

ô ( 8f(kõ) ) f(kô)+c > O

ôk

_ô

if c> O .

..' r

~ .;,' ,', . ' '.'.

~::, .~,,/ ..

I:",

,., ....

',-.1···.· ... _{-.: -;} ~ ."

•

. '""".

26

this figure is not due to the fact that the economy reaches the first best after

the shock, as in figure 2.

These results suggest that one of the reasons for the non-monotonicity

result in the collateralized economy is the rigidity of the debt repayment

following the shock. The fact that the amount repaid does not change with the shock is behind the crucial overhang effect. Figure 8 shows the result of assuming that the amount of debt repayment refiects the new expected price

of capital following the shoc129 :

b~

=

(1 - 7")(u~

+

q2)k~

It is clear from figure 8 that such a scheme will both reduce the size of the

shock, and the slope of the function k~ ( 7" ), specially for low values of 7". This

helps demonstrate the importance of the overhang effect in this economy.

Consider also the following experimento Assume that entrepreneurs can

borrow in period t = O, backed by their expected holdings of capital in period

t

= 2. Formally :

B_o LT - (1 - 7")q2k;

pE

1 - O

p'E

2 - (1 - 7")q2k;

p;

_-

(1 - 7")q2k~

U nder this contract40, there is no debt repayment in period t = 1.

There-fore, there is no debt overhang effect. We would expect this set up to totally

eliminate the non-monotonicity in the economy.

39If the shock is positive, this implies b~ > b

ô.

Therefore, if entrepreneurs have the option

to stick to the contracted payment b

_ô

(as in the Hart and Moore model), a repayment of

b~ cannot be the outcome of the renegotiation game, even if renegotiation occurs after

the shock hits the economy. If the shock is negative (so that b~ < b

_ô),

and renegotiation

occurs after the shock hits the economy, then it is possible that entrepreneurs manage to reduce the amount of debt that they must repay.

In any case, our main goal is to use figure 8 for illustrative purposes, without worrying

too much about the underlying contracting possibilities.

40 Again, we do this onIy for illustrative reasons. We do not claim that this contract

can be the outcome of a fully specified renegotiation garoe, or that it is preferable to the collateralized debt contract analyzed in the paper.

. . ... . . ~ ...

. ....

~.::.*Ij