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Journal of Financial Stability 1 (2005) 451–465 Imperfect financial contracting and macroeconomic stability Thomas Steinberger 1 Center for Studies in Economics and Finance (CSEF) at Dipartimento di Scienze Economiche e Statistiche, Universit` a di Salerno, Via Ponte Don Melillo, I-84084 Fisciano (SA), Italy Received 22 October 2004; received in revised form 27 July 2005; accepted 3 September 2005 Abstract This paper studies the implications of imperfect financial contracting for macroeconomic stability in the context of a stochastic dynamic general equilibrium model. We find that the equilibrium growth path might be indeterminate in an economy with financing frictions even if the aggregate production function exhibits constant returns to scale. Self-fulfilling expectations about the future price of capital lead to macroeconomic fluctuations in this economy. © 2005 Elsevier B.V. All rights reserved. JEL classification: E32; E44; D58 Keywords: Local indeterminacy; Endogenous fluctuations; Imperfect financial markets 1. Introduction In recent years, various authors have proposed dynamic general equilibrium models of the business cycle incorporating some kind of financial market imperfection (see Bernanke et al. (1999), Kiyotaki and Moore (1997) or Carlstrom and Fuerst (1997)). This paper explores the con- sequences of relaxing the assumption of perfect financial markets for the local stability properties of equilibrium paths. This is an important theoretical issue because if the equilibrium of the econ- omy is characterized by local indeterminacy expectational errors provide a source of shocks which might lead to persistent fluctuations in many aggregate time series such as output, employment E-mail address: [email protected] (T. Steinberger). 1 We have benefited from discussions with Roger Farmer, Thomas Hintermaier, Omar Licandro, Marco Pagano and Mirko Wiederholt. Any remaining errors are our own. 1572-3089/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jfs.2005.09.001

Imperfect financial contracting and macroeconomic stability

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Page 1: Imperfect financial contracting and macroeconomic stability

Journal of Financial Stability 1 (2005) 451–465

Imperfect financial contracting andmacroeconomic stability

Thomas Steinberger1

Center for Studies in Economics and Finance (CSEF) at Dipartimento di Scienze Economiche e Statistiche,Universita di Salerno, Via Ponte Don Melillo, I-84084 Fisciano (SA), Italy

Received 22 October 2004; received in revised form 27 July 2005; accepted 3 September 2005

Abstract

This paper studies the implications of imperfect financial contracting for macroeconomic stability in thecontext of a stochastic dynamic general equilibrium model. We find that the equilibrium growth path mightbe indeterminate in an economy with financing frictions even if the aggregate production function exhibitsconstant returns to scale. Self-fulfilling expectations about the future price of capital lead to macroeconomicfluctuations in this economy.© 2005 Elsevier B.V. All rights reserved.

JEL classification: E32; E44; D58

Keywords: Local indeterminacy; Endogenous fluctuations; Imperfect financial markets

1. Introduction

In recent years, various authors have proposed dynamic general equilibrium models of thebusiness cycle incorporating some kind of financial market imperfection (seeBernanke et al.(1999), Kiyotaki and Moore (1997)or Carlstrom and Fuerst (1997)). This paper explores the con-sequences of relaxing the assumption of perfect financial markets for the local stability propertiesof equilibrium paths. This is an important theoretical issue because if the equilibrium of the econ-omy is characterized by local indeterminacy expectational errors provide a source of shocks whichmight lead to persistent fluctuations in many aggregate time series such as output, employment

E-mail address: [email protected] (T. Steinberger).1 We have benefited from discussions with Roger Farmer, Thomas Hintermaier, Omar Licandro, Marco Pagano and

Mirko Wiederholt. Any remaining errors are our own.

1572-3089/$ – see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.jfs.2005.09.001

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452 T. Steinberger / Journal of Financial Stability 1 (2005) 451–465

and stock market wealth (seeBenhabib and Farmer (1996)for an excellent survey of the litera-ture on locally indeterminate equilibrium). Here, the existence of financial market imperfectionsimplies that the first fundamental theorem of welfare cannot be applied to these models and it istherefore no longer ensured that a unique, pareto-efficient equilibrium exists. Fluctuations due toexpectational errors are likely to be independent of fundamental technological shocks and wouldtherefore lead to excessive volatility of the macroeconomy and a welfare loss for indivduals whoprefer to intertemporally smooth consumption. We find that the equilibrium paths of economieswith imperfect financial markets might indeed be characterized by local indeterminacy even ifthe aggregate production function displays constant returns to scale.

Contrary to other models in the indeterminacy literature which rely on self-fulfilling expecta-tions about future consumption, fluctuations here originate from self-fulfilling expectations aboutthe future price of capital. The mechanism can briefly be described as follows. Agents expectinga high price of capital in the next period increase their investment expenditures today. With con-stant returns to scale and imperfect financial markets this investment increase triggers two effects:agents consume less and monitoring costs associated with investment increase. Indeterminacyarises if together with consumption also employment decreases. As a result of the employmentdecrease, the marginal product of capital decreases and the price of capital in the future rises asexpected, but then returns to its steady-state level relatively quickly. Despite the short-lived devi-ation of the price of capital from its steady-styate value, the model generates persistent effects onoutput and employment due to the persistent effects on entrepreneur’s net worth which mitigatesthe financing frictions in the future. Investment and the capital stock remain above their steady-state values for many periods and also the changes in consumption and employment are quitepersistent. The model therefore provides a financial propagation mechanism that is completelyabsent from business cycle models which rely only on technology shocks.

To understand the source of indeterminacy of equilibrium it is important to notice that thispaper takes the view that private investment is restricted by capital market imperfections seri-ously. Investing at a higher rate than warranted by the optimal debt contract given the currentstate variables, leads to an increase in monitoring costs per unit of investment making invest-ment less productive and dampening the increase in the capital stock. The increase in per unitmonitoring costs is clearly inefficient, but necessary to maintain the correct incentives for thetruthful revelation of the private information possessed by entrepreneurs. The constant returns toscale production function and optimal choice of labour supply imply a contemporaneous drop inconsumption and employment for the higher investment rate to be consistent with the aggregateresource constraint. In the period after the shock was realized, the capital stock and the net worth ofentrepreneurs are slightly above their steady-state levels. Entrepreneurs continue to invest abovethe steady-state rate, incentivized by the higher price of capital and improved financing conditionsdue to their higher net worth. At this point, the initial expectation has been corroborated and boththe price of capital and the net worth of entrepreneurs have risen above their steady-state levels.

As the shock subsides, the price of capital reverts relatively quickly to its steady-state value,but the aggregate capital stock and net worth of entrepreneurs continue to drift away from thebalanced growth path fueled by above steady-state investment rates due to improved financingconditions. Only after a considerable number of periods, the persistent rise of the capital stockultimately leads to increased consumption and labour supply which bring down the rate of capitalaccumulation and pull the economy back to the steady-state. Clearly, fluctuations in economicactivity arising through this channel are welfare reducing from the point of view of the overalleconomy because they induce fluctuations in worker’s consumption and inefficient investmentrates.

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T. Steinberger / Journal of Financial Stability 1 (2005) 451–465 453

The main point of the paper is to document rigorously a theoretical mechanism that might be aplausible source of persistent and recurring fluctuations in economic activity and which in our viewhas been unwarrantedly neglected in the business cycle literature so far. We argue that self-fulfillingexpectations about the future price of capital contribute to the volatility of macroeconomic timesseries. Aggregate data suggest that asset prices are excessively volatile and at least a part of thisexcess volatility might derive from forecast errors made by agents in the economy. Monitoring offirms is an important function of financial intermediaries and there is empirical evidence that boththe intensity of monitoring and firm default rates are varying countercyclically as suggested by ourmodel. While we admit that it is not likely that these expectational shocks are responsible for themain part of the observed fluctuations in the data, we think that the high persistence of deviationsfrom the steady-state arising from the inefficient intertemporal allocation of ressources followingthese shocks potentially adds important information at medium and long run frequencies of morethan 2 years. Giving a definite answer to the question whether our mechanism in conjunctionwith fundamental shocks allows us to match better the features of business cycle data than othermodels is outside the scope of this paper, and must be left for future research.

In the next section, we lay out the model and derive the stationary equilibrium. In Section3we parametrize the utility function of the representative agent and characterize numerically andgraphically the parameter space in which the equilibrium of the economy is indeterminate. Wealso compute impulse response functions for an economy with indeterminate equilibrium andcompare them to the standard empirical results. Section4 concludes and suggests directions forfuture research.

2. The model set-up

There is a continuum of two types of infinitely-lived agents, referred to as workers and en-trepreneurs, with mass normalized to 1. Only entrepreneurs have access to the capital-productiontechnology and can obtain financing through a competitive sector of financial intermediaries thatoperates without cost. The risks associated with any individual business are diversified away bythe financial intermediaries and workers therefore obtain a certain rate of return on their savings.Let me begin with describing the economic problem of workers.2

2.1. Workers

The fraction of workers in the population is (1− η). They are assumed to be risk-averse andmaximize

E0

[ ∞∑t=0

βtU(ct, 1 − ht)

]

with respect toct andht and subject to their budget constraint. HereEt denotes the expectationconditional on information available at timet β ∈ (0, 1) is the discount rate of workers,ct isconsumption,ht is the amount of hours worked and the time endowment is normalized to 1. Thebudget constraint is

ct = Wtht + Rtkt − Qt(kt+1 − (1 − δ)kt)

2 The model we use has been studied also byCarlstrom and Fuerst (1997). Lowercase letters denote individual quantitiesand uppercase letters denote aggregate quantities, while greek letters denote parameters.

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454 T. Steinberger / Journal of Financial Stability 1 (2005) 451–465

whereWt, Rt andkt are the wage rate, the return to capital and the capital stock of each worker,respectively.δ is the depreciation rate andQt denotes the price of capital goods in terms ofconsumption goods. The optimal choices of workers are described by the labour supply curve

−Uh(t)

Uc(t)= Wt (1)

and the consumption Euler equation

QtUc(t) = βEtUc(t + 1)[Qt+1(1 − δ) + Rt+1] (2)

In each period, aggregate output of the consumption good in this economy, denoted byYt , isproduced competitively with a constant returns to scale technology using aggregate capitalKt

(which in slight abuse of notational conventions includes both the capital of workers and that ofentrepreneurs), aggregate labour and aggregate entrepreneurial skills, which are denoted byHE

t .

Yt = F (Kt, Ht, HEt )

The price of the consumption good is normalized to 1 and competitive factor markets ensurethat the three factors of production receive their respective marginal products:Rt = F1(t), Wt =F2(t), Xt = F3(t). Xt represents the wage rate for entrepreneurial skills. The assumption of en-trepreneurial labour income is necessary in order to ensure positive net wealth of entrepreneursin each period, which is required in order for the financial contracting problem described belowto be well defined.

2.2. The debt contract

Before describing the behavior of entrepreneurs, we will now turn to the derivation of theoptimal debt contract. Debt is welfare-improving in this economy, because only entrepreneurscan produce the capital good, but may not have enough current wealth to purchase the optimalquantity of factor inputs. The presence of asymmetric information is the reason why access tothe financial market is restricted for entrepreneurs. As mentioned before, we assume the exis-tence of financial intermediaries, which collect the savings of workers and then lend to manyentrepreneurs simultaneously. In this way the financial intermediaries are able to diversify awaythe risks associated with any individual debt contract and guarantee a certain rate of return tothe workers3. The most important restriction that is used in order to simplify the implementationof the contracting problem into the dynamic general equilibrium model is that only intraperiodcontracts are feasible and that there is no stochastic monitoring. Also there is no possibility oftracking the behavior of entrepreneurs over time.

Optimal debt contracts are written between the financial intermediaries and the entrepreneurs.Both parties are risk-neutral and entrepreneurs have positive net wealthn (in this part we dropthe time indices, since there are no dynamic considerations). The capital production technologyoperated by each entrepreneur is linear and allows them to convert 1 consumption good intoω unitsof capital.ω is an idiosyncratic random variable with non-negative support, a mean of 1, somedensity functionφ(ω) and a cumulative distribution functionΦ(ω). It is assumed to be i.i.d. acrosstime and across entrepreneurs. The contracting problem arises, because only the entrepreneur

3 In fact, because we assume that there is free-entry into lending and the loans are intraperiod loans the rate of returnwill be exactly 0.

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T. Steinberger / Journal of Financial Stability 1 (2005) 451–465 455

operating the technology can observe the realization ofω costlessly. Other agents must pay somemonitoring costµi, which is proportional to the size of the projecti, in order to learn this value.This asymmetry of information creates a moral hazard problem, because entrepreneurs have anincentive to report a low value ofω. By the revelation principle, however, the optimal contractwill be structured such that the entrepreneurs always truthfully reportω.

In the absence of the possibility of stochastic monitoring and with all economic rents flowing tothe entrepreneur (that means free entry into lending), the optimal contract maximizes the expectedreturn to investing of entrepreneurs subject to the participation constraints of entrepreneurs and theno-profit-condition for financial intermediaries. Being the residual claimant, the expected returnof the entrepreneur is given by the difference between the expected output of the project and theexpected cost of finance:

Q

[∫ ∞

ω

ωiφ(ω)dω − (1 − Φ(ω))(1 + rk)(i − n)

]

where the expected return is measured in terms of consumption goods,rk is the interest ratespecified by the optimal contract and (i − n) is the amount borrowed by the individual entrepreneur.Because of truthful reporting, monitoring and default will occur exactly when the entrepreneur isunable to repay the debt, i.e. if

ω <(1 + rk)(i − n)

i≡ ω

Truthful reporting and the definition of the optimal cutoff level imply the following proposition.

Proposition 1. With truthful reporting, the definition of the cutoff-level for monitoring implies thatfor fixed values of entrepreneurial net worth n and price of capital Q, the optimal debt contractspecifies the size of investment projects i and the cutoff-level for productivity ω in such a way thatthe probability of default Φ(ω) and monitoring costs per unit of investment Φ(ω)µ are increasingin the size of the investment project i.

Proof. For any investment project of positive size, any positive level of entrepreneurial net worthn and any positive gross interest rate on lending contracts (1+ rk), the derivative of the optimalcutoff levelω with respect toi given by

∂ω

∂i= (1 + rk)n

i2> 0

is positive. SinceΦ(ω) is a cumulative distribution function monotonically increasing in its argu-ment andµ is a positive fraction of investment lost due to monitoring, ¯ω increasing ini directlyimplies thatΦ(ω)µ is increasing ini as well.

Effectively, the optimal contract specifies a pair of decision variables (i, ω), whereω is thecutoff level of the reported idiosyncratic shock below which monitoring (and default) will occur.To derive analytical expressions for the determination of the optimal values of (i, ω) defining thedebt contract, we take the price of capitalQ and the net wealth of entrepreneursn as parametersof the problem.

Substituting the definition of ¯ω into the objective function above yields

Qi

[∫ ∞

ω

ωφ(ω)dω − (1 − Φ(ω))ω

]≡ Qif (ω)

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456 T. Steinberger / Journal of Financial Stability 1 (2005) 451–465

wheref (ω) = [∫ ∞ω

ωφ(ω)dω − (1 − Φ(ω))ω]

can be interpreted as the fraction of expected netcapital output received by the entrepreneur. Since the loans are intraperiod loans, the maximizationis subject to the intermediaries and the entrepreneurs not receiving a negative return in expectation,

Q

[∫ ω

0ωiφ(ω)dω − Φ(ω)µi + (1 − Φ(ω))(1 + rk)(i − n)

]≡ Qig(ω) ≥ (i − n)

and

Qif (ω) ≥ n

Analogously tof (ω), g(ω) =[∫ ω

0 ωφ(ω)dω − Φ(ω)µ + (1 − Φ(ω))ω]

can be interpreted as

the fraction of expected net capital output received by the financial intermediaries and adding thetwo terms yields

f (ω) + g(ω) = 1 − Φ(ω)µ

which shows that a fractionΦ(ω)µ of the produced capital is destroyed by monitoring and theremaining capital is divided between the entrepreneurs and the financial intermediaries. Therelevant first-order conditions pinning down the optimal values of ¯ω and i in each period asfunctions ofQ andn are

Q

1 − Φ(ω)µ + φ(ω)µ

[f (ω)

f ′(ω)

]= 1 (3)

i = n

1 − Qg(ω)(4)

In the following, it will be important to note that the optimal cutoff level ¯ω is the same for allthe entrepreneurs, regardless of their current net worth and only depends on the price of capitalQ.This is due to the linear technology of entrepreneurs, which equalizes their marginal productivityregardless of the size of operations and the sources of finance.

2.3. Entrepreneurs

Entrepreneurs are assumed to be risk-neutral, but discount the future more heavily than workersdo. This assumption is contrary to the “received wisdom” that entrepreneurs are more patient thanother people and therefore willing to accumulate more capital. In the presence of financial marketimperfections this statement must be qualified. Since with imperfect financial markets there is apositive wedge between the expected rates of return of lenders and entrepreneurs, entrepreneursdo not save because they are more patient, but rather because they can get a higher return. In fact,what is needed in order to prevent them from saving too heavily in initial periods and becomecompletely self-financing, is that entrepreneurs are less patient than other agents. This is preciselywhat is assumed here and we chose the spread between the discount rates of entrepreneurs andworkers such that it neutralizes the expected internal rate of return in the steady-state. Specifically,entrepreneurs maximize

E0

[ ∞∑t=0

(βγ)tcEt

]

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T. Steinberger / Journal of Financial Stability 1 (2005) 451–465 457

wherecEt denotes consumption by entrepreneurs at time andγ ∈ (0, 1) is the additional rate of

discounting and defined by

γ = 1

rE

whererE denotes the expected internal rate of return in steady state to be defined below. En-trepreneurs supply inelastically one unit of entrepreneurial skills and accumulate their own capitalholdings, denoted byzt at timet. Their budget constraint is

cEt = ntr

Et − zt+1Qt

wherent is the individual entrepreneur’s net wealth andrEt the individual expected rate of return

on internal funds in the production of new capital goods at timet. From the optimal contractingproblem above, this expected internal rate of return is

rEt = Qtitf (ωt)

nt

= Qt

[f (ωt)

1 − Qtg(ωt)

]> 1

Because this expected internal rate of return is greater than 1, the risk-neutral entrepreneurs willalways pour their entire net worth into the loan contract to be negotiated. Net worth at timet interms of the consumption goods is given by the sum of entrepreneurial wages, rental income fromthe capital stock and the value of the depreciated capital stock

nt = Xt + Rtzt + Qtzt(1 − δ) (5)

If optimal entrepreneurial consumption is always positive, the optimal consumption decisions arecaptured by the following Euler equation

Qt = βγEt

[Qt+1(1 − δ) + Rt+1]Qt+1

[f (ωt+1)

1 − Qt+1g(ωt+1)

](6)

It should be noted that, because of the linearity of utility, no individual variables appear in this equa-tion (if we recall that the optimal cutoff level ¯ω only depends on the price of capitalQ) and hencethe dispersion in entrepreneurial net worth that is caused by the idiosyncratic productivity shocksdoes not matter for the entrepreneur’s consumption-saving decisions. All solvent entrepreneurswill consume according to this Euler equation and those that had to declare bankruptcy at datetloose all their wealth, consume zero consumption goods in periodt, and have to start from scratchin the next period. The linearity of utility in consumption makes entrepreneurs indifferent to theserisks. In the definition of equilibrium below, we will further use the aggregated budget constraintof entrepreneurs, which is given by

Zt+1 = η(ntrEt − cE

t )

Qt

= ηXt + Zt [Qt(1 − δ) + Rt ][

f (ωt)

1 − Qtg(ωt)

]− ηcE

t

Qt

(7)

2.4. Equilibrium

The model is closed by the market clearing conditions for the four markets in the economy,the two labour markets, the consumption good market and the capital market, respectively

Ht = (1 − η)ht

HEt = η

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458 T. Steinberger / Journal of Financial Stability 1 (2005) 451–465

(1 − η)ct + ηcEt + ηit = Yt (8)

Kt+1 = (1 − δ)Kt + It (9)

whereIt stands for aggregate realized investment and is equal to total production of new capitalgoods net of monitoring costs

It = ηit(1 − Φ(ωt)µ)

A recursive competitive equilibrium in this economy is defined by decision rulesfor Ct+1, Kt+1, C

Et , Ht, It, Nt, ωt , which are stationary functions of the state variables

(Ct, Kt, Qt, Zt) and satisfy Eqs.(1)–(9).

3. Detecting indeterminate equilibria

As mentioned before, the purpose of this paper is to examine the robustness of the property ofdeterminacy of equilibrium in SDGE-models with imperfect financial markets. From the exampleof Carlstrom and Fuerst (1997), we know that at least for one specific set of calibrated parametersthe present model possesses a determinate equilibrium. The question, we are asking in this sectionis how robust this property of determinate equilibrium is to changes in the calibration. In otherwords, we want to see whether there are other plausible calibrations of the same model, for whichthe model possesses an indeterminate equilibrium. Before going on to do that, we briefly wantto review the requirements for and the interpretation of indeterminacy and the main empiricalproblems these models face4.

Put simply, the property of indeterminacy means that the model possesses infinitely manyequilibrium paths arbitrarily close to each other. This implies that a problem of equilibriumselection arises. Technically, indeterminacy of equilibrium corresponds to “excess stability” ofthe underlying system of difference (or differential) equations. In determinate models, the numberof stable roots equals the number of predetermined variables (variables whose next period valueis completely determined by the current state variables). If this is the case, there are exactlyenough initial conditions to pin down a unique equilibrium path (the saddle path). In models withindeterminate equilibrium, the number of stable roots of the system is larger than the numberof predetermined variables. It follows that one or more initial conditions are missing and manyequilibrium paths (i.e. sequences of the state variables which satisfy the system of differenceequations derived from the equilibrium conditions) exist. In the business cycle literature thisproperty is exploited by letting expectational errors generate fluctuations of real variables in themodel. Since these fluctuations are driven by expectational errors, they do not have a relation toeconomic fundamentals and generally produce non-pareto-optimal allocations if agents are risk-averse. The possibility of indeterminate equilibrium has been known at least sinceBlanchard andKahn (1980)but only recently some economists have taken this possibility seriously and showedthat for perfectly plausible parameter values SDGE-models of the business cycle may displayindeterminacy (seeBenhabib and Farmer (1996), Schmitt-Grohe (2000)).

The methodology used in the business cycle literature is by now very well-known. One firstsolves for the steady-state of the underlying growth model and then log-linearizes the equilibriumconditions around the steady-state. In the present case, we get a 9-dimensional system of linear

4 An excellent survey on this has recently been published byBenhabib and Farmer (1999).

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T. Steinberger / Journal of Financial Stability 1 (2005) 451–465 459

equations, which can be reduced to a 4-dimensional system of difference equations5 of the form

Et

Ct+1

Kt+1

Qt+1

Zt+1

= J

Ct

Kt

Qt

Zt

(10)

whereJ is a (4× 4) matrix and ˆxt = xt−xx and denotes the percentage deviation ofx from its

steady-state valuex. As state variables we have chosen the aggregate consumption of workerhouseholdsCt , the capital stock of worker householdsKt , the price of capitalQt and the capitalstock of the entrepreneursZt . The local stability properties of the model can by determined byexamining the eigenvalues of the matrixJ characterizing this system and in this model, indeter-minacy occurs if there are more than two eigenvalues inside the unit circle, since there are twopredetermined variables,Kt andZt . The expectations operator on the LHS is very important,because only in the case of determinate equilibrium these expectations are uniquely determinedby the initial conditions on the predetermined variables. With an indeterminate equilibrium, wemust write the resulting system as

Ct+1

Kt+1

Qt+1

Zt+1

= J

Ct

Kt

Qt

Zt

Et(Ct+1) − Ct+1

0

Et(Qt+1) − Qt+1

0

(11)

so that expectational errors can affect the actual values of the free variables. To simulate the model,the matrixJ is diagonalized and the effect of the unstable roots ofJ on the system is excluded. Asa consequence of excluding these variables the expectational shocks to the free variables becomefunctions of fundamental innovations or are required to be zero (if there are no fundamentalinnovations in the system, as is the case in(11). If there are not enough unstable roots, we cannotexclude enough variables and the expectational errors cannot be restricted to zero or functions offundamental innovations. Expectational errors then lead to fluctuations in real variables.

In this model, the dimension ofJ is relatively large and the entries ofJ are each complex func-tions of the underlying parameters and functions. Therefore, we cannot analytically determinethe conditions under which indeterminacy arises. Rather we have to analyze numerically the setof plausible calibrations to determine whether and under which conditions the model’s equilib-rium is indeterminate. The calibration procedure in total requires the choice of three functionalforms and six parameters. The functional forms need to be chosen for the production function ofconsumption goodsF (K, H, HE), the distribution of the idiosyncratic productivity shock to thecapital production technologyΦ(ω) and the utility function of the representative worker householdU(ct, 1 − ht). The parameters to be calibrated are the share of entrepreneurs in the populationη, the shares of capital and entrepreneurial skills in the production of the consumption goodα

andΓ , the fraction of capital goods destroyed by declaring bankruptcyµ , the discount rates ofworker and entrepreneur householdsβ andβγ, and finally the depreciation rate of capitalδ.

A plausible choice of the technology parameters requires them to be in line with empiricalestimates of corresponding measures. Therefore, we do not assume strong externalities in the

5 The log-linearized version of the equations can be found in the appendix.

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460 T. Steinberger / Journal of Financial Stability 1 (2005) 451–465

production function, but instead assume a Cobb-Douglas production function with constant returnsto scale, setting the share of capital toα = 0.36. This choice is standard in the macro literature,but most models with perfect financial markets, require some degree of increasing returns toobtain an indeterminate equilibrium. Since the assumption that entrepreneurial skills are usedin the production of the consumption good is made only to make the asymmetric informationproblem well-defined, we set the elasticity parameter to be very smallΓ = 0.0001 , which yieldsa production function for consumption goods of the form

Yt = (Kt)0.36(Ht)

0.6399(HEt )0.0001

The model is calibrated with a horizon of one quarter for each period and so we choose thedepreciation rate to equalδ = 0.02. The fraction of entrepreneurs in the population is set toη = 0.1 (close to the share reported byGentry and Hubbard (2004)) and the fraction of capitalgoods destroyed during monitoring is set toµ = 0.25. This seems to be a reasonable value, giventhat liquidation is costly. The distribution of the idiosyncratic productivity shock is assumed to be

log-normal, with cumulative distribution functionΦ(ω) = 12

(1 + Erf

[σ2+2 ln[ω]

2√

]), which only

depends on the parameterσ since the mean of the distribution must be 1. FollowingCarlstromand Fuerst (1997), we choose to set the standard-deviationσ of the distribution toσ = 0.2,in order to arrive at an empirically plausible bankruptcy rate of 1%, which is given byΦ(ω)in the model. In the sensitivity analysis, it turned out that altering the choice of any of theseparameters exceptΓ , does not significantly change the results of the simulation. The structureof the resulting system of difference equations is stable with respect to small variations in theseparameters. Increasing the weight of entrepreneurial skills in the production of the consumptiongoodΓ leads to strong interactions between worker’s consumption and labour supply choices andthe net worth of entrepreneurs, potentially yielding unstable or indeterminate equilibria. With thesmall value ofΓ chosen here, no such problems arise however.

What remains to be chosen at this point is the utility function of the representative household andthe discount rates of the agents. Since the time horizon of the period is one quarter, we setβ = 0.99,to arrive at a real interest rate of approximately 4% per year. The discount rate of entrepreneursis then given by the requirement that the spread exactly offsets the higher internal rate of returnon entrepreneurial capital, which implies a value of 0.947 forγ. We allow for a lot of flexibilityin the calibration of the utility function of the representative worker household. Without givinga justification,Carlstrom and Fuerst (1997)choose a utility function that is additively separablein consumption and labour, with the coefficient of relative risk aversions set to 1 and the utilityfunction linear in labour with a steady state labour supply of 1/3 of the time endowment. Weretain their choice ofh = 0.33 as the steady-state labour supply, but allow for the full class ofpreferences concave inct and 1− ht .

Rather than restricting the utility function to some specific functional form, we use a set ofparameters to identify properties of utility. After log-linearization, the only parameters of theutility function, which appear in the equilibrium conditions are the “second-order” elasticities ofthe function, evaluated at the steady-state. Since local determinacy of equilibrium can be verifiedalso by checking the linearized version of the equilibrium conditions, we can conveniently workwith these four parameters, which are defined as

εcc = c

Uc(c, 1 − h)Ucc(c

, 1 − h) εch = 1 − h

Uc(c, 1 − h)Uch(c, 1 − h)

εhc = c

Uh(c, 1 − h)Uhc(c

, 1 − h) εhh = 1 − h

Uh(c, 1 − h)Uhh(c, 1 − h)

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T. Steinberger / Journal of Financial Stability 1 (2005) 451–465 461

whereUc denotes the partial derivative ofU(.) with respect toc and Uh denotes the partialderivative ofU(.) with respect to 1− h. There is an important restriction on the “second-order”cross-elasticities and the “first-order” elasticities, which is derived from the symmetry of theHessian ofU(.). It is given byεch = εh

εcεhc and can be expressed asεch = w(1−h)

c εhc by usingthe optimal labour supply condition. In order to make sure that the resulting equilibrium is autility maximum, the instantaneous utility function must also be concave. The concavity condi-tions requireεcc < 0, εhh < 0 andεhhεcc ≥ εchεhc (seeHintermaier (2003)). Obeying all of theserestrictions, three parameter of the utility function remain to be chosen. These are the elasticityof marginal utility from consumption with respect to changes in consumptionεcc, the elasticityof marginal utility from leisure with respect to changes in leisureεhh and one of the the crosselasticitiesεch. The concavity condition links these three parameters and the resulting inequalitymust be fulfilled.

We investigate the stability of the saddle-path equilibrium by calculating the eigenvalues ofJ

for 100.000 random draws from the parameter spaceΩ = εcc, εch, εhh with the restrictionsεcc ∈] − 100, 0] andεch] − 100, 100[ andεhh ≤ εchεhc

εcc. The upper bound of the first interval is given by

the restriction thatεcc < 0 from the concavity condition and the lower bound represents a relativelyarbitrarily chosen extreme value (for separable utility functions,−εcc equals the coefficient ofrisk aversion). The bounds of the second interval are chosen to be 100 for reasons of symmetryand computational efficiency. The restriction onεhh is sufficient to ensure concavity, since theright hand side of the relevant inequality is always negative. The result of our computations areconveniently summarized inFig. 1.

The graph shows the location of parameter combinations yielding an indeterminate equilibriumof the economy in the 3-dimensional parameter space described above. The concavity restrictionis not depicted in the graph, but the points all lie close to its surface. The fact that parametercombinations are close to the surface implied by the concavity restriction implies that the utilityfunction cannot be too concave in order for the equilibrium of the economy to be indeterminate.Another structural feature of the indeterminate equilibria is that the first eigenvalue ofJ is always

Fig. 1. Indeterminacy region in parameter space.

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462 T. Steinberger / Journal of Financial Stability 1 (2005) 451–465

unstable. This implies that it is not the consumption shock that drives the fluctuations of theeconomy, but rather the shock to the expected price of capitalEt(Qt+1) − Qt+1. The mechanicalexplanation for this is that an unstable root implies that the shocks to the system of differenceequations associated with this root drive away the system from the steady state and inplies anunstable trajectory for the economy. This trajectory would at some point violate the transversalitycondition imposed on the decision problem of the representative agent and would therefore notbe consistent with rational expectations. In our model, this is the case for the expectational errorin the consumption equation. Schematically, the calibrated system has the following form:

J =

j11 ∼ 0 j13 ∼ 0

∼ 0 j22 ∼ 0 ∼ 0

∼ 0 ∼ 0 j33 ∼ 0

j41 j42 j43 j44

wherejxy denotes a non-zero element and∼ 0 denotes matrix elements which are very small inabsolute value. Given this particular structure the eigenvalues of the matrix are largely determinedby the entries on the main diagonal. It turns out thatj44 is the eigenvalue related to the consumptionequation and that|j44| > 1 is a very stable result. For indeterminacy to arise from expectationalerrors about future consumption, one would need the first root to be smaller than one in absolutevalue. The analytical expression forj44 is given by

j44 = 1 + N − X

(CE + I)QZ

whereX is a very small number (remember thatΓ = 0.0001) denoting the marginal product ofentrepreneurial skills in the aggregate production function.

That the slopes of labour supply curves play an important role for the stability properties ofthe equilibrium in dynamic general equilibrium models is well known (seeBenhabib and Farmer(1999)). We now illustrate the properties of the labour market needed to obtain an indeterminateequilibrium in order to understand better what is required from the utility function to generateindeterminacy. We have computed the implied coefficients for the Frisch labour supply curve

w = const+ αhh + αλλ

whereαh = εhh − εchεhc

εccandαλ = εhc

εcc− 1.Fig. 2shows the combinations of Frisch labour supply

curve coefficients computed from the parameter sets generating indeterminacy. The Frisch laboursupply curve is derived by holding constant the marginal utility of consumption and as one can

Fig. 2. Indeterminacy region in space of Frisch labour supply curve coefficients.

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T. Steinberger / Journal of Financial Stability 1 (2005) 451–465 463

see both coefficients must be negative and smaller than 1 in absolute size. This means that thelabour supply curve must be downward-sloping, but very flat and stable with respect to changesin the marginal utility of wealth. Small changes in the real wage have a strong effect on hoursworked and hence output will be volatile in response to changes in the capital stock.

Another way to study the mechanism that gives rise to indeterminacy is to look at the impulseresponse functions. Below we plot the impulse response function for output (Y), employment (h),aggregate capital (K), price of capital (Q), total investment spending (I) and entrepreneur’s networth (N) in response to a positive 1% shock to the expected price of capital. All six graphs havethe same format with the vertical axis measuring the percentage deviation from the steady stateand the horizontal axis measuring the number of quarters passed after the initial shock (seeFig. 3).

If the agents expect a higher price of capital in the future they invest more and consume less.While the rise in the capital stock is contained by an increase in monitoring costs, employmentfalls with consumption, and as a result of this, also output falls, but by not as much as consumption.The price of capital increases because of the higher demand, which raises the future net worth of

Fig. 3. Impulse response functions.

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464 T. Steinberger / Journal of Financial Stability 1 (2005) 451–465

entrepreneurs. In the following periods investment remains high because of the persistent net worthincrease but the price of capital quickly falls back to its initial level. This reduces entrepreneurialnet worth, and despite the adjustment of the threshold level of monitoring, as investment slowlydecreases also the rise in the capital stock is slowing. The capital stock peaks after about 16quarters and for some time consumption, hours, and output actually rise above their steady-statevalues pulling the total capital stock towards its steady-state value over the long term.

4. Conclusion

We have shown that in a 1-sector SDGE-model of the business cycle with imperfect financialmarkets, the equilibrium paths of real variables might be locally indeterminate even if the aggregateproduction function for output has constant returns to scale. This result is interesting in itselfbecause in the standard perfect financial markets case, some degree of increasing returns toscale is necessary in order to get locally indeterminate equilibria (seeHintermaier (2003)). Themechanics of the model with financial market imperfections are therefore closer to those of a 2-sector model for which it is well known that even with aggregate constant returns to scale modelswith locally indeterminate equilibria exist.

Another interesting innovation of the financial market imperfections model is that the source ofbusiness cycle fluctuations are not shocks to consumer’s expectations about future consumption,but shocks to entrepreneur’s expectations of the future price of capital which are then transmittedvia labor demand to consumption and output. This type of shock corresponds more closely tothe “animal spirits” that Keynes and his followers focused on in their explanation of businesscycles. These authors have argued that self-fulfilling expectations about the future price of capitalcontribute to the volatility of macroeconomic times series. Aggregate data also suggest that assetprices are excessively volatile and at least a part of this excess volatility might derive from forecasterrors made by agents in the economy. While we admit that it is not likely that these expectationalshocks are responsible for the main part of the observed fluctuations in the data, we think thatthe high persistence of deviations from the steady-state arising from the inefficient intertemporalallocation of ressources following these shocks potentially adds important information at mediumand long run frequencies of more than 2 years.

Future research has to be directed towards studying more closely the type of financial marketimperfection that gives rise to local indeterminacy and comovement between stock prices and realoutput growth. The main empirical issue that needs to be resolved is the lack of positive correlationbetween investment and consumption which is a consequence of the elimination of technologyshocks as a source of output variation. Adding fundamental shocks to the model or allowing formultiple sectors in the economy might be well suited to overcome this type of problem.

Appendix A

The log-linearized system:

(−εcc + εhc)Ct = (−α − Γ − εch + εhh)Ht + αKt (1*)

εccCt − εchHt + Qt − εccCt+1 =(

R(1 − α − Γ )

R + Q(1 − δ)− εch

)Ht+1 + R(−1 + α)

R + Q(1−δ)Kt+1

+ Q(1 − δ)

R + Q(1 − δ)Qt+1 (2*)

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T. Steinberger / Journal of Financial Stability 1 (2005) 451–465 465

Qt =µω

(f (ω)φ(ω)2

(1−Φ(ω))2− f (ω)φ′(ω)

−1+Φ(ω)

)Q ωt (3*)

It = Nt + Qg(ω)

1 − Qg(ω)Qt + NQω(1 − µφ(ω) − Φ(ω))

I ωt (4*)

Q(1 − δ)ZQt + (N − ηX)Zt = (−1 + α + Γ )(ηX + RZ)Ht

+(−ηXα − R(−1 + α)Z)Kt + NNt (5*)

Qt −(

Q(1 − δ)

R + Q(1 − δ)+ 1

1 − Qg(ω)

)Qt+1 = R(1 − α − Γ )

R + Q(1 − δ)Ht+1

− R(1 − α)

R + Q(1 − δ)Kt+1 +

(ωf ′(ω)

f (ω)+ ωQg′(ω)

1 − Qg(ω)

)ωt+1 (6*)

CE

QZ + CE CEt + QZ

QZ + CE Zt+1 = Nt +(

CE

QZ + CE + Qg(ω)

1 − Qg(ω)

)Qt

+(

ωf ′(ω)

f (ω)+ ωQg′(ω)

1 − Qg(ω)

)ωt (7*)

(1 − α − Γ )Ht + αKt = C

Y Ct + CE

Y CEt + I

Y It (8*)

− (1 − δ)

δKt + 1

δKt+1 = It + µωφ(ω)

1 − µΦ(ω)ωt (9*)

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