Real Business Cycles, Investment Finance, and Multiple Equilibria

Real Business Cycles, Investment Finance, and Multiple Equilibria

Journal of Economic Theory 86, 100122 (1999) Article ID jeth.1999.2511, available online at http:www.idealibrary.com on Real Business Cycles, Inve...

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Journal of Economic Theory 86, 100122 (1999) Article ID jeth.1999.2511, available online at http:www.idealibrary.com on

Real Business Cycles, Investment Finance, and Multiple Equilibria Brian Hillier* Department of Economics and Accounting, University of Liverpool, Eleanor Rathbone Building, Bedford Street South, Liverpool L69 7ZA, United Kingdom Brian.Hillierliverpool.ac.uk

and Jonathan Rougier Department of Mathematical Sciences, University of Durham, United Kingdom J.C.Rougierdurham.ac.uk Received January 16, 1998; revised January 12, 1999

This paper develops a simple neo-classical model of the business cycle characterised by multiple equilibria produced by the interaction of the investment and production processes. For the usual symmetric information model the source of the multiple equilibria may be found in non-linearities in the production function. Under asymmetric information another possible source of multiple equilibria is non-linearities in the investment-wealth relationship. Persistence is generated by changes in the supply of funds and, hence, in the capital stock. Journal of Economic Literature Classification Numbers: D50, E32.  1999 Academic Press

1. INTRODUCTION Empirical evidence indicates that the economy may display a tendency to oscillate around either a high output locally stable equilibrium or a low output locally stable equilibrium with an unstable equilibrium in between [13]. This paper examines a simple neo-classical real business cycle model capable of producing multiple equilibria as a result of the interaction between the investment and production processes. In a symmetric information context the multiple equilibria may arise as a result of non-linearities * Corresponding author. We thank an anonymous referee and seminar participants at Durham, Hull, Keele, Kent, and Loughborough Universities for their helpful comments. Much of this research took place while the authors were members of the Department of Economics at the University of Durham.

100 0022-053199 30.00 Copyright  1999 by Academic Press All rights of reproduction in any form reserved.

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arising from the production function under standard concavity assumptions. Alternatively, informational asymmetry may produce another source of multiple equilibria by producing appropriate non-linearities in the relationship between profits and investment even if the model would not exhibit multiple equilibria in the symmetric information case. The plan of the paper is as follows: Section 2 describes the model. Section 3 analyses the model and demonstrates the possibility of multiple equilibria in a standard full-information setting. Section 4 introduces the problem of asymmetric information and shows how it produces another possible source of multiple equilibria. Section 5 concludes.

2. THE MODEL Consider a version of the overlapping generations model of Samuelson [4] akin to that presented in Bernanke and Gertler [5]. Each generation, distinguished by its date of birth, is of equal size and consists of a continuum of risk-neutral agents who live for two periods, their youth and oldage. There are two goods in the model, an output or consumption good, and a capital good. The latter is produced using the output good as an input via an investment technology to be described below. The output good is produced using the capital good and labour as inputs to a production process. We initially treat the production process as deterministic but later introduce stochastic shocks. Output goods may also be produced by placing output goods in a safe storage process to which all old agents have access. The storage technology yields an exogenously given gross return of s1 units of the output good at the end of the period for each unit placed in storage at the start of the period. 1 Agents can only invest in the storage technology; they cannot borrow from it. In their youth, agents are endowed with a single unit of homogenous labour. Old agents hire out capital goods, which they own, to the young agents in return for a price, q, in units of the output good. The capital market is perfectly competitive hence q is equated to the marginal product of capital measured in output goods. Young agents use their labour and hired capital to produce output goods, keeping the returns, w, in excess of capital costs for themselves; where w is measured in output goods. Agents do not consume the wealth earned when young, which is equal for all young agents; instead they use it to provide for consumption in old age. The old use their wealth in one or more of three different ways: they may place all or part of it into the safe, constant returns to scale storage 1 There is not reason, in principle, to set s1. This inequality, however, guarantees that the gross interest rate on bank loans and deposits exceeds one.

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technology; they may deposit all or part of it in a bank; or they may use all or part of it to finance an investment project. The banking sector is perfectly competitive; each bank makes zero profits in equilibrium and holds a well-diversified portfolio of loans used to support investment projects. At the end of their period of old age, agents receive their returns from storage, deposits or investment projects, consume and die. The aim of agents is to maximise consumption. Each old agent is endowed with access to an investment project which only he may undertake. A project i undertaken by the i th old agent at the end of period t (beginning of period t+1) produces an amount, z i , of capital goods in period t+1. The value z i is stochastic with a common and independent distribution function G which has support on the interval [ g, g ]. Denote the expected value of z by z e. Capital depreciates totally  within one period. Heterogeneity is introduced by assuming that the cost, x i , measured in output goods, of investment projects differs across old agents. This cost lies in the interval [x, x ], and is described across old agents by the distribution function L, with  mean x e. Investment projects are indivisible such that the ith agent's project requires x i funds if it is to be undertaken; it is impossible to invest more or less than x i funds in the i th project. Agents with lower x i values may be thought of as being more efficient than other agents in undertaking investment projects and, as we shall see, are more likely to undertake their investment projects. This heterogeneity provides a simple way of generating an interest elastic demand for investment funds (or, equivalently, supply of capital goods). The return to a project undertaken by an old agent is used first to repay any debt acquired by the project owner when financing his project and any remainder is supplied to the capital market in exchange for output goods to consume. Let k be the aggregate quantity of capital supplied per member of the old age generation. The labour supply per member of the young generation is unity and aggregate production of the output good (expressed per member of the young generation) is, in the deterministic case, given by f (k), which is nonnegative increasing and concave in k. Assume that an old agent who chooses to undertake his project engages in maximum equity participation, that is his own investment is equal to min[x i , w]. Outside funds, max[0, x i &w], are provided by borrowing under a standard debt contract with a bank. Banks acquire funds for loans by taking deposits from old agents who either choose not to undertake their investment projects or who have funds left over after doing so and who choose to place those funds on deposit with a bank rather than in the storage technology. The assumptions of credit finance and maximum equity participation are easily defended. For the full information case the assumptions are innocuous

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since the ModiglianiMiller theorem [6] holds, so the method of finance does not affect real investment decisions and the value of the firm. For the asymmetric-information case, to be dealt with in section 4, the assumptions may be justified from first principles. The asymmetry of information is introduced by assuming that when an agent i undertakes an investment project only he can observe costlessly the actual number of units, z i , of the capital good produced. 2 Any other agent must pay a monitoring cost of m units of the capital good to observe the value z i , where m is independent of z i or x i . We thus have a problem of costly state verification in which standard credit contracts intermediated through banks arise naturally to economise on monitoring costs [8, 9, 5]. 3 Lenders need to monitor defaulting projects or else borrowers, knowing that the lender cannot observe whether announcement of a low return is true or not, will claim default to minimise payments to lenders. We further assume that the credit contract can specify the equity participation of the owner and that in the event of default on the loan the bank can costlessly access the returns to the owner's savings placed on deposit with another bank or in storage. It then follows from [11] that a contract with maximum equity participation weakly dominates any other contract. In the following section we examine the behaviour of the model under the assumption of symmetric information where lenders face no costs in observing the returns to projects.

3. EQUILIBRIUM WITH FULL INFORMATION To find the within-period equilibrium we proceed in stages. We begin by examining the market for investment funds. The equilibrium in the investment funds market may then be used in determining the equilibrium in the market for capital goods which, in turn, may be used in determining the overall solution of the process of investment and production of output. 2 We do not consider other informational problems such as selection problems, where lenders cannot observe x i , or moral hazard with hidden actions. For a survey of the effect of such problems on investment behaviour and business cycles see [7]. 3 For simplicity we consider only pure strategy contracts and rule out stochastic monitoring. See, however, [10] for consideration of stochastic monitoring in the optimal contract. Ruling out stochastic monitoring simplifies the analysis (and might be argued to be more realistic) without changing the macroeconomic implications of the model. Furthermore, our comparison of the full-information case and the asymmetric-information case is facilitated by being able to hold the form of contract constant across the two cases.

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The Market for Investment Funds Let the standard debt contract specify a gross repayment, R i , in terms of output goods for the i th borrower. The actual repayment to the bank is min[qz i , R i ], that is when the project yields less than R i q capital goods, which we term ``default'' or ``failure,'' the entire yield in capital goods, z i , is taken by the lender. The expected loan repayment, denoted p(R i ), is then given by def

p(R i ) =

|

z

min[qz, R i ] dG(z).

(1)

z 

Let the gross interest rate on deposits in the banking sector be r output goods per unit of deposit. Under perfect competition, full information and risk-neutral agents the banking sector makes normal (defined as zero) profits when the gross interest rate on deposits is equal to the expected gross interest rate earned on loans, both in the aggregate and for an individual loan. 4 Hence, for a given equilibrium interest rate in the loan market, banks determine R i by solving p(R i )=r(x i &w),

x i >w.

(2)

Given the above we may now examine the decision of an old agent choosing whether to proceed with funding his project. In making this decision each old agent treats the equilibrium values of r and q as parametric. An old agent will choose to fund his project whenever the expected returns from doing so exceed the expected returns from the next best alternative action. When the loan market operates it must be the case that rs, so that the next best alternative use of his funds is to place them on deposit at a bank. 5 In this case, the old agent for whom x i >w will choose to fund his project if the following holds,

|

z

max[0, qz&R i ] dG(z)wr,

(3)

z 

which, using (1) and (2), simplifies to qz e x i r.

(4)

Similar logic also yields (4) as the condition for an old agent with wealth greater than the the cost of his investment. The interpretation is obvious: 4

This will be shown in Section 4 as a special case when monitoring costs are zero. When the loan market does not operate either wealth is sufficiently large to allow all agents with worthwhile projects to fully self-finance, or there are no projects offering a better return than the storage technology. 5

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an old agent undertakes his project whenever its gross expected rate of return exceeds the opportunity cost of funds. Thus an old agent will have a gross demand for funds (that is from his wealth plus credit if required) of x i if x i qz er and zero otherwise, and the aggregate gross demand for investment funds 6 will be the function c d(r, q), where def

c d(r, q) =

|

qzer

x dL(x)

(5)

x 

providing that x
c s(w, r) =

0

{w

r
(6)

Figure 1 shows the aggregate supply and demand functions for investment funds. The shape of the demand function is implied by a continuous unimodal distribution for x. In order to describe the equilibrium, define c^(q) as def

c^(q) = c d(s, q),

(7)

and define the function r(w, q) such that c d(r(w, q), q)#w.

(8)

In these terms the market-clearing amount of investment funds, c*, and interest rate, r*, are given by 0, s,

{

c*, r*= c^(q), s, w, r(w, q),

c^(q)=0 0
(9)

c^(q)>w.

6 All aggregate values are expressed per capita, as the number of members in each generation is the same.

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FIG. 1.

The market for investment funds.

For q=0, c^ =0, and c*=0, there is no investment because the result of that investment in capital goods could not be sold for any output goods (remember that q is output goods per capital good). At the point where q=sxz e, c^ rises above zero, the cheapest project is funded and the interest  rate jumps to s. What happens next depends upon whether w is less than or greater than x e. In the former case, when q reaches a critical value q m(w), such that c d(s, q m(w))#w,

(10)

at which point c^ =w, the market is supply constrained, and further increases in q, which increase c^ above w, can only bid up the rate of interest, leaving the cost of the marginal project unchanged. In the latter case c^ will never rise as far as w, the interest rate will remain at s, and the amount of investment will approach its maximum value x e as q approaches sxz e. In this case, all surplus wealth is placed in the storage technology. The Market for Capital The number of projects funded by the amount of investment c* is n(w, q), where def n(w, q) =

|

qzer* x 

dL(x)=L(qz er*);

(11)

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note that, as all aggregate expressions are per capita, the maximum number of projects is one. For a large number of agents, the Law of Large Numbers implies that the supply of capital, k s(w, q), is given by k s(w, q)rz en(w, q).

(12)

The demand for capital is determined by the wealth-maximising behaviour of young agents, who face the problem (which we initially consider for a non-stochastic production function) max w=f (k)&qk

subject to k0.

(13)

k

For simplicity, we will require that either (i) f (0)>0, or (ii) f (0)=0 and lim k Ä 0 f $(k)=, which ensures that the inequality constraint will always be satisfied. It then follows that the demand for capital by the young agents is k d(q)0, such that f $(k d(q))#q.

(14)

The market for capital is shown in Fig. 2, drawn for three different w, w 1
(15)

and following on from these two definitions, define w^ such that k s(w^, q^ )=k.

(16)

k d(q(w))#k s(w, q m(w)),

(17)

Finally, define q(w) such that

where q m(w) was previously defined in (10). The equilibrium amount of capital, k*, and price of capital, q*, is now given by k*, q*=

{

k s(w, q m(w)), q(w), k, q^,

0
(18)

When w=0 then k*=0, since there can be no capital without wealth. As w rises above zero to a value such as w 1 so k* rises and q* falls, until w reaches the value w^. At this point the supply of capital reaches its maximum value k*=k. A larger value of w than w^, such as w 2 , has no impact upon the amount of capital, k*, because a supply of capital in excess of k

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FIG. 2.

The market for capital.

would only be forthcoming under a value of q greater than the marketclearing value q^. We are assuming here that the old agents have perfect foresight (or, in the stochastic case to be considered below, rational expectations). Our first proposition concerns the relationship between k* and w. Proposition 1. The equilibrium amount of capital, k*, is concave in w for w in the interval 00. At the point where the supply curve k s(w, q) becomes vertical, we have q=q m(w) and r*=s, from which it follows that dk* d s d = k (w, q m )=z e L(q mz es) dw dw dw =z e dL(q mz es)(z es)

dq m . dw

(19)

Differentiating the identity (10) and rearranging, we find that dq m d d m = c (q , s) dw dq

\

+

&1

.

(20)

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d d c (q, r)=(z er)(qz er) dL(qz er), dq

(21)

d d m c (q , s)=(z es)(q mz es) dL(q mz es)>0, dq

(22)

Now from (5),

hence

which implies that dq mdw>0. Substituting (22) into (20), and this in turn into (19) gives the results, after some cancellation, dk* s = >0 dw q m d 2k* &s dq m = <0, dw 2 (q m ) 2 dw

(23) (24)

which are sufficient to complete the proof in the continuous case. In the general case, a discrete distribution for x will imply that c d(q, r) will have one or more horizontal sections, in which case dq m(w)dw=0 for some w, and in general we can only assert that dq m(w)dw0. It follows from (24) that d 2k*dw 2 0, hence k*(w) is concave but not necessarily strictly concave. K It is straightforward to provide an economic rational for the fact that k* is concave over 0
w(k) = f (k)&f $(k) k.

(25)

The star superscript indicates that the reward to the young agents is an equilibrium outcome dependent upon the capital stock of the old agents. Moving forward one generation, the young agents become old agents, and a path for wealth and capital evolves, of the form w0 Ä k* 0 Ä w* 1 Ä k* 1 Ä w* 2 Ä }}} .

(26)

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The path is determined entirely by the starting point, w 0 # (0, w^ ]. 7 We would like to know the equilibrium point or points for which w e =w(k*(w e ))

(27)

e for starting and to identify whether these are stable, i.e., lim t Ä  w *=w t e points in the region around w . The following sufficient conditions for a unique stable equilibrium are straightforward.

Proposition 2. For a unique stable equilibrium, it is sufficient that w(0)>0 and w"(k)<0 for all k>0. If w(0)=0, it is sufficient that w$(0)>xz e and w"(k)<0 for all k>0.  Proof. If w"(k)<0 for all k>0, then the inverse function of w(k), denoted k &1(w), is convex on w. Providing that this inverse function starts below k*(w), a single stable intersection is guaranteed by the concavity of k*(w). Since lim w Ä 0 k*(w)=0, this could be achieved by w(0)>0, proving the first part of the proposition. If w(0)=0, then k &1(w) will start below k*(w) provided that dk &1dw
:>0, 0<;<1.

(28)

It follows that w(k)=(1&;) f (k) hence w(0)=0, but lim w$(k)= lim :;(1&;) k ;&1 =,

kÄ0

kÄ0

(29)

which is greater than xz e. This case is shown in the top panel of Fig. 3. However, there are also conditions under which multiple equilibria are possible. A necessary but not sufficient condition for multiple equilibria is w"(k)>0 for some k>0. Following on from the previous proof, if w"(k)>0 for some k>0 then w(k) is not concave, and k &1(w) is not convex in w. This makes possible multiple crossings of k*(w) and w(k). To take a simple example, if w(0)>0, then k &1(w) may cross k*(w) from below, then from above, then from below again. In this configuration, shown in the lower panel of Fig. 3, the outer equilibria are stable, while the interior equilibrium is unstable. 7 Of course, technically w 0 # R ++ #(0, w^ ], but it has been shown above that all w 0 >w^ will give a path identical to that of w 0 =w^.

INVESTMENT AND MULTIPLE EQUILIBRIA

FIG. 3. equilibria.

111

The relation between capital and wealth. (a) Unique equilibrium. (b) Multiple

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It is important to stress that the condition w"(k)>0 is achievable with a standard concave production technology, and so does not require increasing returns to scale. In this respect the multiple equilibria arising in our model are qualitatively different from those suggested by [12, 13], as will be discussed in more detail below. In terms of the production function the condition w"(k)>0 is equivalent to &f "(k)< kf $$$(k). As a simple example of a production function satisfying this condition, consider f (k)=(:+k) ;,

:>0, 0<;<1,

(30)

for which w(k) is convex on 00 and w$(0)=lim k Ä  w$(k)=0, as depicted in Fig. 3b. Insofar as (30) is simply a linear reparameterisation of the CobbDouglas function, the local convexity in w(k) should not be seen as an unusual special case, but rather as a reasonable generalisation. We summarize this discussion in the following proposition. Proposition 3. The set of admissible production functions contains a subset for which it is possible, for certain values of s, G( } ) and L( } ), to have multiple equilibria under full information; in particular there is a subset which gives rise to ``low'' and ``high'' stable equilibria, with an unstable equilibrium in between. Stochastic Shocks Finally, we introduce stochastic shocks in the production technology. As in [5] we suppose that there is a non-negative random variable % with an expected value of unity which acts as a multiplicative shock to the output technology. Decisions about investment and finance are made before % is known, but capital is hired after % is known. Thus the problem for the young agents is now max w=%f (k)&qk

subject to k0,

(31)

k

and the old agents must forecast % and the demand curve for capital before making their decisions about where to place their wealth. Since the old are risk neutral, however, they act as if % were certain to take its mean expected value of unity and the supply of capital function is unaffected by the introduction of uncertainty over % (and, hence, q), that is it continues to be given by z en(w, q). 8 Therefore k*(w) remains unchanged. On the other 8

Neither the aggregate gross demand nor the aggregate gross supply functions for investment funds will be affected by the introduction of uncertainty under our assumptions. Hence the market-clearing values r* and k* for a given w will not be affected by the introduction of uncertainty.

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hand, wealth as a function of k becomes stochastic and the effect of a positive shock is to shift w(k) upwards for all values of k, i.e., to the right in Fig. 3. Consider an economy at position A in Fig. 3b. 9 There are asymmetries in the impact of shocks. A one-off positive shock increases w* above w A but since w A >w^ there is no change in k*=k and w* returns to w A next period. Likewise a small negative shock for which w* does not fall below w^ also produces no change in k* and w* returns to w A in the next period. But a medium-sized negative shock for which w* lies in the range (w B, w^ ) implies a fall in k* to below k and persistence in the return of k* and w* to their equilibrium values at A; the time path in this case shows a jump back to A once w* reaches w^. If we had drawn the figure so that w A
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storage technology (although there is serial correlation in consumption since a positive shock, for example, will raise q and the consumption of the old, as well as raising w* and the amount placed on storage by the young for consumption when the young become old). The effect of the assumption is to remove persistence in output from the symmetric information version of the model. The economy is placed in a kind of funding trap where an increase in funds is simply soaked up by the storage technology in a manner akin to money being soaked up by speculative demand in the famous Keynesian special case of the liquidity trap. The purpose of Bernanke and Gertler's assumption was to highlight the role of asymmetric information in creating a role for net worth in explaining business cycle dynamics with persistence in investment, capital and output. Their model was explicitly set up to exclude such persistence in the absence of asymmetric information. 10 An examination of Fig. 3b shows that the assumption also rules out multiple equilibria which rely upon intersections at values of w less than w^. The present paper, therefore, relaxes their assumption, allowing a role for net worth in producing persistence to appear and, more importantly, showing the possibility of multiple equilibria. This role for net worth, w, in explaining persistence might proxy the well documented link between profits, retained earnings and investment [17, 18, 19], whilst the non-linearity in the production function necessary to produce multiple equilibria might also be empirically plausible. The model may, therefore, have possible empirical relevance. The following section introduces monitoring costs and asymmetric information as in Bernanke and Gertler [5]. The purpose is not to repeat their analysis for the special case where w* always exceeds w^, but to show that for the more general case it is possible that asymmetric information can lead to multiple equilibria even when the model would not exhibit multiple equilibria in the symmetric information case.

4. COSTLY STATE VERIFICATION PROBLEMS It is not entirely straightforward to introduce costly state verification into the analysis; therefore we outline the general problem before considering a simple example capturing the main features. 10

As we show in the next section, the effect of the costly state verification problem is to cause the level of wealth required to produce k to rise from w^ under full information to w-. The effect of this is to introduce persistence over the range of wealth w^ to w- where previously, under full information, there was no persistence.

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The Market for Investment Funds We begin by reconsidering the credit market. The standard debt contract specifies a repayment to the bank of min[qz i , R i ] and the expected loan repayment, p(R i ), is given by equation (1) as for the symmetric information case. The difference is that if a borrower defaults (i.e., claims to be unable to meet the gross repayment R i ) under asymmetric information, the bank monitors the borrower at a cost m in capital goods. In a risk-neutral competitive banking environment the expected loan repayment net of expected monitoring costs will be equated to the promised payment to depositors. Thus Eq. (2) is replaced by p(R i )=r(x i &w)+mqG(R i ).

(32)

The interpretation of this is straightforward: the gross repayment R i must be determined so that the expected repayment on a break-even loan contract for x i &w must cover not only the opportunity cost of the loan, r(x i &w), but also the expected monitoring costs associated with the loan, mqG(R i ). In other words any expected monitoring costs are passed on to the borrower by the lender adjusting the terms of the debt contract. The higher the level of wealth of a borrower and the smaller the loan required, the lower the likelihood of default and expected monitoring costs so that higher levels of wealth are associated with more favourable loan terms to borrowers. 11 It is, on the other hand, possible that the problem of asymmetric information and monitoring costs may be so severe that the credit market collapses, that is no break-even credit contract can be found. The market collapse will be illustrated with an example below but first we continue to consider the case where the credit market operates. Since monitoring costs impact upon the terms of the debt contract they also impact upon the decision of an old agent choosing whether to fund his project. Using Eqs. (1), (3), and (32) it is easy to show that he will now choose to fund his project whenever qz e rx i +mqG(R i ).

(33)

With full information m=0, and wealth does not play a role in defining the marginal project; with asymmetric information this is not the case. In general, the higher the level of wealth for any interest rate, the lower the amount of borrowing required to support a given project, the lower the probability of default and the lower the expected monitoring costs. It is possible that, in the real world, retained earnings out of profits perform the 11 It is this mechanism which produces persistence in the model of [5] which, given their assumptions, does not produce persistence without this role for wealth.

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role taken by wealth in the model by increasing the amount of self-funding, lowering expected monitoring costs and encouraging investment. The observed relationship between profits and investment might, therefore, be due to problems of information asymmetry. The effect of asymmetric information is to kink the demand for investment funds. In the full information case, the marginal project is that for which x i =qz er. If this project is riskless, i.e. qz (qz er&w) r, then it will remain the marginal project under asymmetric  information, and aggregate demand for investment funds will be the same in the two cases. But if this project is risky, then for it to be the marginal project with asymmetric information a lower interest rate will be required to compensate for the expected monitoring costs. Consequently, the aggregate demand for investment funds under asymmetric information, c ad(w, q, r), will fall below the full information function c d(q, r) at the point where r=q(z e &z )w.  To keep the notation simple it helps if z =0, and this is assumed from  now on. It follows that the only safe projects are those which are fully selffunded and that c d(q, r) and c ad(w, q, r) diverge for values of rqz ew the equilibrium amount of investment for the asymmetric information case is the same as that for the full information case, since in this region all projects which are undertaken are fully self-funded so that monitoring costs are absent. Otherwise the equilibrium amount of investment is lower. The Market for Capital The supply of capital following the outcome of the investment process will be unchanged in the case where all projects are fully self-funded. This occurs over the range q # (0, swz e ]. For large values of q supply is reduced relative to the full information case for two reasons. First, as discussed in the previous subsection, the demand for investment funds was lower because some projects could not cover the additional cost of monitoring default. Second, some of the capital generated by the projects is dissipated in the activity of monitoring. If we redefine q m(w) to be that function for which, in the asymmetric information case, c ad(w, q m(w), s)#w

(34)

then for qq m(w) the capital supply function becomes vertical since the investment market is supply constrained. Along this part of the function, r rises in proportion with q in the investment funds market and the cost of

INVESTMENT AND MULTIPLE EQUILIBRIA

FIG. 4.

117

The market for investment funds, asymmetric information.

the marginal project remains the same. Furthermore, for any project R i rises in proportion with q and r and the repayment in capital goods corresponding to R i remains constant; hence the expected monitoring costs, in terms of capital goods, remains constant. In other words, for a given capital goods value corresponding to successful repayment of the loan, as q rises the expected repayment to the lender less monitoring costs remains constant in terms of capital goods but rises proportionately with q in terms of output goods when r rises proportionately with q. Although the maximum possible value of k remains the same, at k, when comparing a full-information with an asymmetric-information world, the level of wealth required to produce k under asymmetric information, w -, exceeds that required under full information, w^. Under asymmetric information k will only be forthcoming if there are no monitoring costs, which requires the level of wealth to equal the cost of the marginal project at k, i.e., w - =q^z es>w^. The supply of capital functions are shown in Fig. 5 for the values of w=w^, w - and wx, along with the demand function which is unchanged from the full-information case. 12 The resulting supply of capital function 12

The supply functions are shown for a continuous distribution of z; for a discrete distribution, such as given in the market collapse example below, the discontinuity will require a discrete jump in q to cover the discrete jump in expected monitoring costs as the marginal project becomes risky.

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FIG. 5.

The market for capital, asymmetric information.

k a(w) is shown in Fig. 6, as well as the function k(w) for comparison and w(k) (for which the derivation does not change in moving from the fullinformation case to the asymmetric-information case). The effect of costly state verification is to lower the amount of capital produced for any level of wealth and, as noted above, increase the value of wealth at which the function becomes horizontal from w^ to w -. Although the function k(w) must be concave up to w^, it is not necessary that the function k a(w) be concave up to w -, as the following proposition shows. Proposition 4. Under certain circumstances, notably the collapse of the market for investment funds, k a(w)=L(w). Proof. This proof is by example. Consider the case where (i) z i # [0, 2] with equal probability; (ii) x i # (0, 2); (iii) s=1; (iv) m2; (v) f (k) is such that q^ 2. From (i) the probability of bankruptcy is 0.5, in which case (33) becomes qrx i +mq_

0,

{0.5,

self financed otherwise

from which it follows by (iv) that only self-financed projects are viable. Therefore there is no demand for investment funds, the investment market fails to operate and the opportunity cost of capital is given by s. By

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FIG. 6.

119

The relation between capital and wealth, asymmetric information.

assumption (iii) self-funded agents will go ahead with their projects whenever qx i . Finally, assumption (v) ensures that q*q^ 2>x i for all x i , from which it follows that all self-financed projects will go ahead. Therefore the supply of capital in these circumstances is k a(w)=L(w). K If the distribution function L( } ) is unimodal, Proposition 4 implies that k a(w) will be S-shaped under k(w), as drawn in Fig. 6. Hence asymmetric information can introduce multiple equilibria (at A, B, or C) even if there would be a unique equilibrium under full information (at D). Shocks may be introduced into the model in exactly the same way as in the previous section, and with similar results. Comparison of Figs. 3a, 3b, and 6 shows that the multiple intersections in Fig. 6 occur because of the S-shaped k a(w) function, whereas they occur in Fig. 3b because of the reverse S-shaped w(k) function. In Fig. 6 this requires a region where k increases at an increasing rate with respect to w, while in Fig. 3b it requires a region where w increases at an increasing rate with respect to k. Thus multiple equilibria can arise as a result of non-linearities in the relationship between net worth and the capital stock in the presence of asymmetric information, or as a result of non-linearity in the production function with or without asymmetric information. Finally, note that for m<2 it is possible for the credit market to operate (the expected returns from cheap projects can feasibly exceed the cost of

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the project plus the opportunity cost of any loan and the expected monitoring costs). In this case the k a(w) function must lie in the envelope between k a(w; m2) and the full-information k(w). Figure 5 shows (for the case where G( } ) is continuous) that for low values of w the capital demand and supply functions intersect on the upper vertical segment of the k as(w) function so that the credit market operates and the savings technology is unused. As w increases beyond a critical level the credit market and the savings technology are both used as the intersection moves to the central upward-sloping part of the k as(w) function. As w increases yet further, to w - and beyond, the credit market ceases to operate as the intersection moves to the lower upward-sloping part of the k as(w) function, and k reaches the value of k.

5. CONCLUSIONS This paper has developed a simple neo-classical model of the business cycle characterised by multiple equilibria produced by the interaction of the investment and production processes. Persistence is generated by changes in the supply of funds and, hence, in the capital stock. The problem of costly state verification was shown to make it easier to produce multiple equilibria; in so far as such costs occur in the real world they may explain why economies with undeveloped capital markets and high monitoring costs locate at low outputlow investment equilibria. Although we have not carried out a formal welfare analysis, a policy implication of the model may be that economies should do all they can to avoid large negative shocks which may tend to tip them into a low level equilibrium. Once in a low level equilibrium a large shock may be needed to return the economy to the higher stable equilibrium. Policies to increase the level of investment (such as a reduction in corporate profits taxes to increase retained earnings) will shift the k a(w) function upwards which, as inspection of Fig. 6 shows, will raise the low equilibrium and lower the intermediate one, thus making it both harder to fall into the low equilibrium and easier to get out of it. Finally, although we introduce non-linearities through the production function, or through costly verification and heterogeneity over project costs, our analysis also indicates the potential importance of other sources of non-linearity or heterogeneity (see, e.g., [20] on non-linearities, or [21] for a detailed treatment of heterogeneity and stochastic dynamics using the tools of statistical physics). Similarly, although the present model is a real business cycle model, we would not preclude incorporating a role for nominal or relative price shocks in explaining cyclical behaviour and, in the

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case of multiple equilibria, the time the economy tends to spend in or around each of the stable equilibria (see, e.g., [22] for a model with relative price shocks). It seems reasonable to conjecture, however, that, as in the present analysis, the multiplicity itself is likely to stem from the real, rather than the nominal, side of the economy. REFERENCES 1. S. N. Neftci, Are economic time series asymmetric over the business cycle? J. Polit. Econ. 92 (1984), 307328. 2. J. D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica 57 (1989), 357384. 3. F. X. Diebold and G. D. Rudebusch, A nonparametric investigation of duration dependence in the American business cycle, J. Polit. Econ. 98 (1990), 596616. 4. P. A. Samuelson, An exact consumption-loan model of interest with or without the social contrivance of money, J. Polit. Econ. 66 (1958), 467482. 5. B. S. Bernanke and M. Gertler, Agency costs, net worth, and business fluctuations, Amer. Econ. Rev. 79 (1989), 1431. 6. F. Modigliani and M. H. Miller, The cost of capital, corporation finance and the theory of investment, Amer. Econ. Rev. 48 (1958), 261297. 7. B. Hillier and T. Worrall, Asymmetric information, investment finance and business cycles, in ``The New Macroeconomics: Imperfect Markets and Policy Effectiveness'' (H. Dixon and N. Rankin, Eds.), pp. 245272, Cambridge Univ. Press, Cambridge, UK, 1995. 8. R. M. Townsend, Optimal contracts and competitive markets with costly state verification, J. Econ. Theory 21 (1979), 265293. 9. S. D. Williamson, Costly monitoring, financial intermediation, and equilibrium credit rationing, J. Monet. Econ. 18 (1986), 159179. 10. D. Mookherjee and I. Png, Optimal auditing, insurance, and redistribution, Quart. J. Econ. 104 (1989), 399415. 11. D. Gale and M. Hellwig, Incentive-compatible debt contracts. I. The one-period problem, Rev. Econ. Stud. 52 (1985), 647664. 12. R. Cooper and A. John, Coordinating coordination failures in Keynesian models, Quart. J. Econ. 83 (1988), 441463. 13. J. Benhabib and R. Farmer, Indeterminancy and increasing returns, J. Econ. Theory 63 (1994), 1941. 14. H. Bohn and G. Gorton, Coordination failure, multiple equilibria and economic institutions, Econometrica 60 (1993), 257280. 15. R. E. A. Farmer, ``The Macroeconomics of Self-Fulfilling Prophecies,'' MIT Press, Cambridge, MA, 1993. 16. J. Frank, The stock market and equilibrium recession, in ``The New Macroeconomics: Imperfect Markets and Policy Effectiveness'' (H. Dixon and N. Rankin, Eds.), pp. 237244, Cambridge Univ. Press, Cambridge, UK, 1995. 17. S. M. Fazzari, R. G. Hubbard, and B. C. Petersen, Financing constraints and corporate investment, Brookings Pap. Econ. Act. 1 (1988), 141206. 18. R. Blundell, S. Bond, M. Devereux, and F. Schiantarelli, Investment and Tobin's Q: Evidence from company panel data, J. Econometrics 51 (1992), 233257. 19. A. Carruth, A. Dickerson, and A. Henley, ``Econometric Modelling of UK Aggregate Investment; the Role of Profits and Uncertainty,'' Economic Research Paper 97-06, Aberystwyth, 1997.

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20. O. Galor and H. E. Ryder, Existence, uniqueness, and stability of equilibria in an overlapping-generations model with productive capital, J. Econ. Theory 49 (1989), 360375. 21. M. Aoki, ``New Approaches to Macroeconomic modelling: Evolutionary Stochastic Dynamics, Multiple Equilibria, and Externalities as Field Effects,'' Cambridge Univ. Press, Cambridge, UK, 1996. 22. L. Ball and N. G. Mankiw, Relative-price changes as aggregate supply shocks, Quart. J. Econ. 110 (1995), 161193.