Global dynamics in a model with search and matching in labor and capital markets

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Journal of Economic Dynamics & Control

Journal of Economic Dynamics & Control 34 (2010) 1651–1679

0165-18

doi:10.1

$ The

Society

research

empiric� Cor

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journal homepage: www.elsevier.com/locate/jedc

Global dynamics in a model with search and matching in labor andcapital markets$

Ekkehard Ernst a,�, Willi Semmler b

a ILO, International Institute for Labour Studies, Geneva, CH, Switzerlandb New School for Social Research, Department of Economics, New York, USA

a r t i c l e i n f o

Available online 21 August 2010

Keywords:

Search and matching frictions

Global macroeconomic dynamics

Credit constraints

Equilibrium unemployment

JEL classification:

C61

C63

E24

E32

G14

J63

89/$ - see front matter & 2010 Elsevier B.V. A

016/j.jedc.2010.06.016

authors would like to thank participants at t

for Non-linear Dynamics and Econometrics

assistance as well as to three anonymous ref

al version of this paper was presented at the

responding author.

ail address: [email protected] (E. Ernst).

a b s t r a c t

In this paper global macroeconomic dynamics are studied when search frictions are

present in both labor and capital markets. On the basis of the Merz (1995) macro-

economic model with labor market frictions and capital accumulation, our paper offers

an extension to frictions in capital markets, analogously modeled as a search and

matching process. Using the Merz model as limit case, we consider exogenous as well as

endogenous borrowing constraints. We also allow the cost of issuing bonds to change

endogenously. As we show, capital market frictions exacerbate and accentuate the

interaction between the two markets and magnify the effects of shocks on output,

consumption, employment, and welfare. This interaction of the frictions in labor and

capital markets are also shown to give rise to multiple equilibria. On the basis of

numerical solution techniques, instead of relying on first or second order approxima-

tions around a (unique) steady state, our paper uses dynamic programming techniques

to compute decision variables and the value function directly and assess the local and

global dynamics of the model. The steady state solutions are studied by using the

Hamiltonian and the dynamics are assessed for various model variants by using

dynamic programming techniques.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Labor market search and matching has been increasingly used in the literature on dynamic stochastic generalequilibrium models (DSGE), starting with the seminal paper by Merz (1995) that models labor market frictions in aReal Business Cycle (RBC) model. This approach has proved to be very fruitful to understand the dynamics of outputand employment in an intertemporal decision framework. Here consumption, the search effort for jobs by workers, andvacancies posted by firms, are decision variables and both the capital stock and employment are state variables. In thisway employment adjustments are costly, and sudden jumps in the employment level of firms are excluded. Nevertheless,the first generation of such search and matching models for the labor market has not been particularly successful inmatching empirical facts on volatility of unemployment and vacancies. By modifying the basic model, recent models have

ll rights reserved.

he Conference on Computing in Economics and Finance, Paris, 2008, and the 2007 Conference of the

for helpful comments. Many thanks also to Uwe Koller who provided very valuable and timely

erees whose comments helped substantially in improving upon the original version of the paper. An

North American Winter Meeting of the Econometric Society, Atlanta, January 3-5, 2010.

www.elsevier.com/locate/jedc

dx.doi.org/10.1016/j.jedc.2010.06.016

mailto:[email protected]

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E. Ernst, W. Semmler / Journal of Economic Dynamics & Control 34 (2010) 1651–16791652

performed better in terms of explaining the volatility of unemployment and vacancies (e.g. see Hagedorn and Manovskii,2008).1

DSGE models have also been extended to include more realistic features of the capital market, such as the financialaccelerator. The research on capital markets since the 1980s has pointed out that imperfections and frictions in the capitalmarket prevent supply and demand for financial funds to match easily. Research conducted, for instance, by Bernanke andGertler (1989) or Stigltiz and Greenwald (2003) demonstrates that there are informational asymmetries and search effortsinvolved in obtaining credit that makes capital markets behave markedly different from what has been perceived instandard theory. In particular, capital market borrowing may be subject to constraints and financing costs (such as costs toissue new corporate bonds). Those constraints and costs may be given exogenously (from the point of view of the firm) orendogenously, depending on the financial conditions of the firm, such as its net worth.

One step toward including more realistic features of capital markets has been to model capital markets in analogyto the labor market as a search and matching process, introducing a market for loanable funds in the spirit ofthe Wicksellian tradition. Due to such search frictions the supply and demand for bonds are not matched completelyand savings may be left unused or there may be an excess demand of savings. Models along this line have recentlybeen developed by Becsi et al. (2000), Den Haan et al. (2003), and Wasmer and Weil (2004). Credit constraints forfirms are exogenous in these models since firms credit constraints are given through an aggregate search and matchingprocess.

Following this strand of the literature, our model presented here, combines both search and matching in the labormarket as well as in the capital market in a model with capital accumulation.2 Yet, going a step further to reflect the workon imperfect capital markets and the financial accelerator, we endogenize financial market frictions and link them tofinancial conditions of firms. In this context, frictions arising in search and matching are called exogenous financialfrictions (exogenous to the firm). On the other hand the frictions arising from collateralized borrowing will be calledendogenous financial frictions, since they arise from endogenous credit constraints of firms. We then show that there areexacerbating and accentuating interactions of the two markets that exert—similar to the financial acceleratortheory—strong effects on output, consumption, employment, and welfare. In particular, we demonstrate that multipleequilibria and low level steady states can easily arise in such a framework. In contrast to the labor market search literature,in our approach, financial market frictions can significantly alter the global dynamics and global outcomes, possiblyleading to severe financial and real contractions.3

In contrast to some recent papers that have used default premia to construct financial accelerator models, we prefer themechanism of endogenous credit constraints to demonstrate such magnifying mechanism of finance for the real economy.We want to take this as a starting point since the use of default premia in consumption-based macro-economic models isin an early stage and their success is still debated.4 We can justify the use of endogenous credit constraints instead ofdefault premia in such models, as both yield similar results but models based on credit constraints have a more solid basein the academic literature (see, for instance, Grune et al., 2008a).

A further innovation of our paper relates to the numerical solution methodology, which attempts to meet the challengesof the particular characteristics and the out-of-steady-state behavior of such models. Typically, when intertemporaldecision models are numerically solved, current approaches have used first or second order approximation methods tosolve the model around a (unique) steady state.5 Since in our context the global dynamics needs to be explored we usedynamic programming techniques that allow to better understand the global dynamics of the economy and to account forthe possibility of destabilizing mechanisms, bifurcations and multiple steady states.6 Yet, in order to usefully employdynamic programming it is helpful to explore steady states by the use of the Hamiltonian and the computed steady statesthereby.

The remainder of the paper is organized as follows. In Section 2 some further literature on search and matching inlabor and capital markets are discussed. Section 3 presents our numerical solution technique. Section 4 introducesand solves a basic labor market search and matching model, the Merz (1995) model. Section 5 adds capital marketsearch and matching in addition to the one on the labor market. Hereby the Merz model is considered the limit caseof our extended model variants. Section 6 explores the global dynamics of our model variants by using dynamicprogramming techniques, assuming that credit constraints are exogenous. Section 7 studies two model variants withendogenous credit frictions, the first with endogenous credit constraints and the second with endogenous bond issuingcosts. In both cases financial conditions of firms matter when borrowing from capital markets. A final section concludes thepaper.

1 Yet those types of models seem to be far from explaining some drift in unemployment or vacancies. For a number of further puzzles that can arise in

this type of models see Krause and Lubik (2006, 2008).2 Note that we follow here the standard practice by using the term financial market ‘‘frictions’’. In contrast, recent research would rather suggest to

speak about the magnifying effect of financial intermediation.3 Financial market frictions not only affect the transmission speed of shocks but also the long-run behavior of the economy.4 For the use of finance or default premia in those models, see De Fiore and Tristani (2008). In most of the literature it is stated that the quantitative

effect of the default premia is small.5 For a more general evaluation of the accuracy of first and second order approximation methods see Becker et al. (2007).6 Along those lines Brunnermeier and Sannikov (2009) have recently argued that the destabilizing mechanisms, arising through nonlinearities,

cannot be captured by local linearization techniques.

E. Ernst, W. Semmler / Journal of Economic Dynamics & Control 34 (2010) 1651–1679 1653

2. Financial markets in macroeconomic models

Labor market search and matching has been a successful analytical tool for understanding the functioning of the labormarket. Since Pissarides (2000) popularized the approach by introducing wage negotiations (instead of the—slightly—

more complicated approach via wage posting), it has been used intensively, but mainly for partial equilibrium analysis.Most analysis has concentrated on comparative statics, leaving out any complications resulting from the dynamics that aredescribed by the underlying Bellman equations. Its intricate features and the unemployment rate that arises due tomatching frictions—multiplied by labor rigidities and policies—has granted this approach a prominent place in labormarket theory and policy analysis.7

What was missing until recently, however, was a rigorous introduction of savings and investment decisions in a modelwith capital accumulation. First in an RBC context (Merz, 1995; Andolfatto, 1996), then extended to introduce endogenousgrowth (Eriksson, 1997), the approach has finally also been retained in the New Keynesian literature (Walsh, 2005;Blanchard and Galı, 2006; Trigari, 2009 among others) to address the issue of lacking output persistence, which in the NewKeynesian case proved important for understanding inflation persistence. The real rigidity that labor market matchingintroduces in these models has proved useful and relevant to explain output persistence and higher volatility ofemployment in comparison to wages (Andolfatto, 1996). Some issues—such as the higher volatility of vacancies comparedto productivity—have subsequently been addressed by making wages rigid as well (Shimer, 2007), instead of assumingconstant renegotiation of wages in the event of a shock. Others have modified the model to explain the empiricallyobserved volatility of vacancies and unemployment (see Hagedorn and Manovskii, 2008).

Financial market considerations, however, have not been at the fore in these models, an issue that seems to warrantsome considerations as the recent events have demonstrated. Indeed, as observers such as Stiglitz and others havedemonstrated in numerous contributions, credit exerts a strong influence on real activity, albeit less through the interestrate channel than through the availability of credit. In their view, it is the quantity of loans, not just the interest ratecharged, that is critical for real economic activity. Credit does not need to be completely rationed by borrowers, but thereare information and search costs as well default probabilities that drive a wedge between the lenders and borrowers.As Stigltiz and Greenwald (2003) point out, the informational and search efforts involved in obtaining credit let marketbehave markedly different from how this is conventionally viewed. In particular, rising interest rates do not give incentivesfor the lender to increase loans as they may be a signal of rising default risk and loan losses. By implication, the loan supplymay not be increasing and the demand for loans may not be satisfied. Frictions in credit markets are thus essentialto generate non-clearing credit markets. In addition to such credit constraints, the actual credit cost also needs to bede-linked from the interest rate, and the imbalances between credit supply and demand may even express themselves assevere financial disintermediation.8

The financial accelerator literature as being put forward by Bernanke and Gertler (1989) or Bernanke and Gilchrist(1999), has also applied information economics to justify significant and pro-cyclically moving finance premia. In thiscontext, such premia arise due to the financial conditions of firms. If the firm’s collateral is low—due to low networth—screening and auditing cost are high which will give rise to substantial finance premia if a firm seeks externalfunds. This way credit constraints of firms become endogenous, depending on its balance sheet conditions such as networth.

In this regard, an extension of the labor market matching framework to financial markets promises an interestingavenue as the transmission channel runs through the availability of credit rather than variations in the interest rate.9 Someauthors have already started to apply the search and matching methodology to include more realistic features of capitalmarkets in macro models (Becsi et al., 2000; Den Haan et al., 2003; Wasmer and Weil, 2004; Pierrard, 2005; Nicoletti andPierrard, 2006). While the literature here is still in its infancy, it has already managed to show interesting insights based onsuch an approach, in particular in relation to possible interactions between financial and labor market and theconsequences for output and employment growth. In particular, these models have shown how such financial marketmatching frictions can deepen existing imbalances on the labor market by impacting negatively on job creation of firms.However, part of this literature lacks the inclusion of capital accumulation, which has somehow hampered its morewidespread use and empirical testing. Moreover, the search framework typically considers financial market frictions asexogenous, leaving out issues related to asymmetric information, moral hazard and adverse selection.10

Our paper starts with a search and matching model of the Merz type that includes capital accumulation, and employsthe Merz model as a limit case. We start from this limit case and then introduce more realistic features of the capitalmarket by introducing first (exogenous) credit market matching frictions. We extend this framework further to account forthe endogeneity problem mentioned, and include endogenous frictions arising from the net worth of a firm. We also

7 See the papers by Krause and Lubik (2006, 2008).8 The default premia may move in a different direction as compared to interest rate movements, see Curdia and Woodford (2009a,b) and Grune et al.

(2008a).9 Recently developed alternative approaches attempt to explain risk premia on the credit market based on consumption-based asset pricing models

(see De Fiore and Tristani, 2008; De Graeve et al., 2008; Curdia and Woodford, 2009a,b). Thus far, these papers have not yet completely been able to

sufficiently match actual credit market risk premia that is found in the data.10 See Amable and Ernst (Forthcoming) for an example of how to introduce endogenous financial market matching frictions.


consider the possibility for endogenous issuing costs of corporate bonds to render financial market frictions endogenous.To evaluate the dynamics that arises from these models, we make use of both local steady state analysis and numericaltechniques to assess the global dynamics of the economic systems and the possibility of instability, bifurcations andmultiple steady states.

3. The numerical solution of the model

We use the dynamic programming method by Grune and Semmler (2004) to study the dynamics of the model withsearch and matching for labor and capital markets. The dynamic programming method can explore the local and globaldynamics by using a coarse grid for a larger region and then employing grid refinement for smaller region. Since it does notuse first or second order Taylor approximations to solve for the local dynamics, dynamic programming can provide onewith the truly global dynamics in a larger region of the state space without losing much accuracy (see Becker et al., 2007).In contrast, local linearization, as has been argued there, does not give sufficient information on the global dynamics.

Hence, before going into the model discussion, we start by briefly describing this dynamic programming algorithm andthe way by which it enables us to numerically solve our dynamic model variants. The feature of the dynamic programmingalgorithm is an adaptive discretization of the state space which leads to high numerical accuracy with moderate use ofmemory. In particular, the algorithm is applied to discounted infinite horizon dynamic decision problems of the typeintroduced for the study of our search and matching models. In our model variants we have to numerically compute V(x)

VðxÞ ¼maxu

Z 10

e�rtf ðx,uÞdt

s:t: _x ¼ gðx,uÞ, xð0Þ ¼ x0 2 R1

where u represents the decision variable and x a vector of state variables.In the first step, the continuous time optimal decision problem has to be replaced by a first order discrete time

approximation given by

VhðxÞ ¼maxu2U

Jhðx,uÞ

where Jhðx,uÞ ¼ hP1

i ¼ 0ð1�rhÞf ðxhðiÞ,uiÞ and xh is defined by the discrete dynamics

xhð0Þ ¼ x, xhðiþ1Þ ¼ xhðiÞþhgðxhðiÞ,uiÞ

and h40 is the discretization time step. Note that U denotes the set of discrete control sequences u=(u1,u2,y) for ui 2 U.The value function is the unique solution of a discrete Hamilton–Jacobi–Bellman equation such as

VhðxÞ ¼maxu2Ufhf ðx,uÞþð1�rhÞVhðxhð1ÞÞg ð1Þ

where xh(1)=x+hg(x,u) denotes the discrete solution corresponding to the control and initial value x after one time step h.Using the operator

ThðVhÞðxÞ ¼maxu2Ufhf ðx,uoÞþð1�rhÞVhðxhð1ÞÞg

the second step of the algorithm now approximates the solution on a grid G covering a compact subset of the state space,i.e. a compact interval [0,K] in our setup. Denoting the nodes of G by xi with i=1,y,P, we are now looking for anapproximation VG

h satisfying

VGh ðx

iÞ ¼ ThðVGh Þðx

iÞ ð2Þ

for each node xi of the grid, where the value of VGh for points x which are not grid points (these are needed for the evaluation

of Th) is determined by linear interpolation. We refer to Grune and Semmler (2004) for the description of iterative methodsfor the solution of (2). This procedure allows then the numerical computation of approximately optimal trajectories.

In order to distribute the nodes of the grid efficiently, we make use of an a posteriori error estimation. For each cell Cl ofthe grid G we compute

Zl :¼maxk2cl

jThðVGh ÞðkÞ�VG

h ðkÞj

More precisely we approximate this value by evaluating the right hand side in a number of test points. It can be shownthat the error estimators Zl give upper and lower bounds for the real error (i.e., the difference between Vj and VG

h ) andhence serve as an indicator for a possible local refinement of the grid G. It should be noted that this adaptive refinement ofthe grid is particularly effective for computing steep value functions, non-differential value functions and models withmultiple equilibria, see Grune et al. (2005) and Grune and Semmler (2004). These are all cases where local linearizationsare not sufficiently informative.


4. Matching frictions on labor markets

We start with the well-known study of search and matching frictions for the labor market in the context of a dynamicgeneral equilibrium framework, introduced by (Merz, 1995), and that allows for capital accumulation.

4.1. Search and matching on labor markets

In this model two types of agents are considered: entrepreneurs and workers. Entrepreneurs rent capital on africtionless financial market and workers produce output but have no capital. A productive firm is thus a relationshipbetween an entrepreneur and workers.

Labor market frictions are present under the form of a matching process �a la Pissarides (2000), with a constant returnsmatching function11 mLðs � U ,VÞ. Matches between workers and firms depend on the number of job vacancies V,unemployed workers U and the intensity with which the unemployed search for a job s. The instantaneous probability offinding a worker is thus mLðs � U ,VÞ=V while for the worker the instantaneous probability to find a firm writes asmLðs � U ,VÞ=U . Moreover, we have mL

s 40. Finally, the aggregate matching process yielding N filled jobs, and normalizing thelabor force to unity, i.e. L¼ 1, we have U ¼ L�N) ul � ðL�NÞ=L¼ 1�n, with n�N=L.12

4.2. The base model with perfect capital markets

Contemporaneous utility is composed of utility from today’s consumption taking into account the labor effort forproducing today’s output:

uðCt ,NtÞ ð3Þ

where we specify uðCt ,NtÞ ¼ C1�Zt =ð1�ZÞ�eðNtÞ, Za1. The functional form of the disutility of effort has been chosen to be

eðNtÞ ¼ eNwt ð4Þ

where w40 measures the inverse of the Frisch elasticity of labor supply.The dynamic decision model allows to maximizes utility against a resource constraint given by

YtðKt ,ANtÞ ¼ CtþFðstÞð1�NtÞþz � Vtþ It ð5Þ

with Ct: aggregate consumption, Yt: aggregate production, A: (exogenous) labor productivity, Nt: employment, Vt:vacancies, st: search effort on the labor market and It: investment. Search for open vacancies is costly, with search costsincreasing in a convex way, i.e. Fu40, F0040. Jobs are destroyed with the exogenous probability s. Holding vacancies openis costly as well, with per-period flow costs measured by z.

The dynamic decision problem is subject to two dynamic constraints, one for the change in employment, the other forcapital accumulation:

_Nt ¼mLðst � Ut ,VtÞ�sNt ð6Þ

_K t ¼ It�dKt ð7Þ

Hence, the decision problem writes as

maxst ,Ct ,Vt

Z 10

UðCt ,NtÞe�rt ¼

Z 10

C1�Zt

1�Z�eNwt

( )e�rt

subject to (5), (6), (7) and the initial conditions K040,N040. This problem can be solved by setting up the current-valueHamiltonian:

maxst ,Ct ,Vt

H¼ C1�Zt

1�Z�eNw

t

þn1½mLðst � Ut ,VtÞ�sNt�

þn2½YtðKt ,ANtÞ�Ct�FðstÞð1�NtÞ�z � Vt�dKt�:

Solving the Hamiltonian yields the following first-order conditions:

@H@Ct¼ 0 3 C�Zt ¼ n2 ð8Þ

11 mL has positive and decreasing marginal returns on each input.12 Note that U refers to all those not in employed, i.e. both job seekers and inactive people. The rate ul is therefore the inactivity rate rather than the

unemployment rate.


@H@Vt¼ 0 3 n2z¼ n1

@mL

@V ð9Þ

@H@st¼ 0 3 n2FuðstÞ ¼ n1

@mL

@sð10Þ

as well as the two transversality conditions:

limt-1

n1Nte�rt ¼ lim

t-1n2Kte

�rt ¼ 0:

Using Pontryagin’s maximum principle, the co-state variables for K and N evolve according to

_n1 ¼ rn1�@H@Nt¼ ðrþsÞn1�

@Y

@NtþFðstÞ

� �� n2þewNw�1

t ð11Þ

_n2 ¼ rn2�@H@Kt¼ ðrþdÞn2�

@Y

@Kt� n2 ð12Þ

From (8) and (12) the optimal path for consumption can be derived as

_C t

Ct¼

1

Z@Y

@Kt�d�r

� �

Moreover, at steady state, employment is determined by

@Y

@NtþFðstÞ

� �� C�Zt �ðrþsÞ

zC�Zt@mL

@Vt

¼ ewNw�1t

and the optimal search effort is determined implicitly by

Fuðs�Þ ¼ z@mL

@s

@mL

@V

� ��ð13Þ

4.3. Steady state and numerical solution

To determine the steady state of the model, we start by focusing on the equilibrium of the two dynamic constraints (6)and (7). At the steady state, employment and capital dynamics yield:

_Nt ¼ 0 3 N� ¼mLðs� � U�,V�Þ

s

_K t ¼ 0 3 I� ¼ dK�:

hence the steady state employment falls with the (exogenous) separation rate and increases with the efficiency of thematching process. In the absence of frictions in the capital market, capital accumulation simply equals the depreciationrate. Moreover, output will be determined by Y*=Y(K*,AN*). Letting _n2 ¼ 0, we can determine the steady state capital stockfrom (12) after rearranging terms such that

@YðK�,AN�Þ

@K¼ rþd

Finally, substituting (8) and (9) into (11) and rearranging terms leads to

YNþFðs�Þ�ðrþsÞz@mLðs��U� ,V�Þ

@V

!½YðK�,AN�Þ�I��Fðs�Þð1�N�Þ�z � V��Z ¼ ewðN�Þw�1

In order to determine numerically the steady state and to assess the dynamics of the model, functional forms for searchcosts, the matching function and the production function need to be specified. Key to the determination of the steady stateis the selection of an appropriate search cost function FðsÞ. As introduced by Merz (1995), a convenient functional form is:FðsÞ ¼ c0ðc1 � sÞ

y,y41. In addition, the labor market matching function is specified as

mLðs,V,UÞ ¼ q0V1�q1 ðs � UÞq1

with q0 ¼ q1 ¼ 0:5, which correspond to typically values found in the empirical search and matching literature (seePissarides and Petrongolo, 2001).13 We will take the search effort here and in the remainder of the paper as a constant ats*=0.2. This is in essence the Merz model (1995: 281).

13 Using c0-0 and y-1 in the search cost function FðsÞ, this allows for a constant optimal search activity. Employing (13) allows to derive the

optimal search effort as

Table 1Steady state values in the Merz case.

Steady state

GDP (level) Y 1.887

Capital stock K 11.324

Employment rate N 0.689

Vacancies V 0.049

Consumption C 1.544

Box 1–Summary of parameter specifications—Merz model.

The Merz model has been solved using the following functional specifications and parameter values:

� Production function:

Y ¼ FðKANÞ ¼ KaðANÞ1�a

where A=1 and a¼ 0:36.

� Utility: The constant relative risk aversion, Z, has been set to unity and the logarithmic form chosen. Moreover, wehave e¼ 1;w¼ 5. Finally, the intertemporal time preference rate has been set at r¼ 0:03. Hence:

uðCt NtÞ ¼ logðCtÞ�eNwt ¼ logðCtÞ�N5

t

� Labor market matching function:

mLðsVUÞ ¼ q0V1�q1 ðs �UÞq1

with q0 ¼ q1 ¼ 0:5.

� Structural parameters:

s 0:04 job separation rate

z 0:07 cost of opening a vacancy

d 0:03 capital depreciation rate


Together with the functional specifications presented in Box 1, the following numerical solution for the unique steadystate obtains14 (see Table 1).

This implies a savings ratio of around 18.2%. Note that N is defined as the employment rate and has to lie in the interval(0,1). The Jacobian of the 4-dimensional dynamic system set up by the vector ðn1,n2,K ,NÞ has the following eigenvalues atthe steady state:

l1 ¼�0:7841þ1:0336i

l2 ¼�0:7841�1:0336i

l3 ¼ 1:4151

l4 ¼�0:0034

Note that the steady state is stable with three negative and one positive real part of the eigenvalues. This predicts,in terms of a solution using dynamic programming, a stable path toward the steady state.15

(footnote continued)

s� ¼1

c1zð1�q1ÞV

c0yq1

� �1=y

¼1

c1

zV

c0y

� �1=y

Thus the search effort will be parametrically fixed by: s*=0.2. Note that this decision variable becomes independent of the state variable and thus a

constant.14 Parameters have been chosen such as to roughly match observed employment-to-population ratios which fall between 65% and 72% for advanced

OECD countries. Note that given the lack of labor market participation decisions in our model, these ratios do not allow to deduct unemployment rates

but rather inactivity rates.15 This is so because dynamic programming operates only in the state space and there are no co-state variables that create saddle-path dynamics as

would arise when using the Hamiltonian to solve the optimal decision problem.

Table 2Comparative static exercises in the Merz case.

Increase vacancy costs Increase in labor disutility

z0 ¼ 0:07-z1 ¼ 0:1 D e0 ¼ 1-e1 ¼ 2 D

GDP (level) Y1 1.886 �0.1% Y1 1.644 �12.9 %

Capital stock K1 11.318 �0.1% K1 9.862 �12.9%

Employment rate N1 0.688 o�0:1pp N1 0.600 �8.9pp

Vacancies V1 0.049 �0.2% V1 0.029 �41.0%

Consumption C1 1.542 �0.1% C1 1.346 �12.8%

Increase in job separation rate Increase in capital depreciation rate

s0 ¼ 0:04-s1 ¼ 0:06 D d0 ¼ 0:03-d1 ¼ 0:05 D

GDP (level) Y1 1.887 0.0% Y1 1.622 �14.1%

Capital stock K1 11.320 0.0% K1 7.299 �35.5%

Employment rate N1 0.689 0.0pp N1 0.696 0.7pp

Vacancies V1 0.110 124.7% V1 0.051 4.5%

Consumption C1 1.539 �0.3% C1 1.253 �18.8%

Note: Percentage changes are report in the column noted and are computed with respect to steady state values. For the employment rate, percentage

point changes (pp) are reported.

15.000

12.667

10.3330.533 0.543 0.731 0.919

N

K

Fig. 1. Labor market search and matching. Note: The figure displays stable dynamics in the N�K space when only labor market frictions are present,

with K*=11.35 and N*=0.73.


In Table 2, some comparative static results are reported regarding changes in the main structural parameters of themodel, such as the capital depreciation rate, the job destruction rate, costs of issuing vacancies and the preferenceparameter indicating the disutility of labor. For all four cases, the changes in the steady state values arise in the expecteddirection. Notice the interesting case of an increase in the job destruction rate that exclusively affects steady stateconsumption through an increase in the number of vacancies issued (and hence the corresponding aggregate cost, whichdiminishes consumption possibilities).

4.4. The dynamics

On the basis of the above parameter sets we compute the global dynamics and the value function, applying the dynamicprogramming algorithm as described in the previous section. Fig. 1 shows in the quadrant set up by the capital stock(vertical line) and the employment rate (horizontal line) the out-of-steady-state dynamics. As one would expect, the

9.59

8.58

7.57

6.5

5.56[V

]

0.450.4 0.550.5 0.650.6 0.750.7 0.850.8 0.958.599.51010.51111.51212.513

0.9[N]

[K]

Fig. 2. Labor market search and matching (value function).


trajectories of all initial conditions lead, in a globally stable manner, to the vicinity of the steady state for N* and K* ascomputed earlier.16 Moreover, the dynamic programming algorithm allows us to draw the value function for each point inthe state space N�K on a grid as defined in Section 3.

Regardless of the initial condition, all trajectories move toward the steady state, driven by the optimal controls V and C

at each point in the N�K space. Hence, if the actual capital stock and employment are below their steady state values ourdynamic programming solution shows exactly the optimal size of the decision variables consumption, C, and vacancy, V,that will push the capital stock and employment levels towards the steady state: Consumption will be low, and thussavings high, such that the capital stock growth is greater than depreciation. The employment dynamics generated byfilling open vacancies will be such that employment moves towards the steady state employment since the separation ratewill be lower than the employment rate obtained through the search effort. From our dynamic programming solution wecan exactly determine the dynamic decisions needed to be taken when the economy is out of equilibrium (even far awayfrom the steady state). This is the advantage of our globally operating solution method.

In Fig. 2 the value function is the surface above the space spanned by N (front line) and K (line going to the back).Each projection from the surface to the N�K coordinates represent the value (welfare) obtained at those coordinates. Thevalue is the integral obtained by the optimal decisions V and C starting from initial conditions defined by the coordinates ofN�K and following the trajectories, as represented in Fig. 1, to the steady state given by N* and K*. All vertical projectionsonto the surface N � K represent thus the value (welfare) obtained when the trajectories, for example represented by theones as depicted in Fig. 1, move toward the steady state of N�K. Note that the steady state N* and K* will be a certain pointon the surface toward which the trajectories will move dynamically. This holds if the steady state of N* and K* are stable,which is the case for the Merz model.

5. Introducing capital market frictions

In this section, we are extending the basic Merz-framework by introducing frictions in the capital market and derive thedynamics of the optimal decision program. In this approach, financial intermediation is not perfect as in the Merz case anda wedge exists between aggregate savings and investment. In this section, we concentrate on the case where capitalmarket frictions are exogenous, in the sense that there is an aggregate search and matching process on capital markets thatis exogenous to the firm and independent of its own financial conditions. In Section 7, we will broaden the scope of theinteractions by introducing a different form of financial intermediation, where the financial condition of the firm, forexample the net worth of firms, matters. The latter we call endogenous credit frictions.

16 Note that the slight deviations from the exact steady state result from some numerical rounding errors. Note also that with the above parameters

we have a unique equilibrium, but it is known that with slight modification of the Merz model one could also obtain multiple equilibria, see Krause and

Lubik (2004).


5.1. Search and matching on capital markets

When financial intermediation is introduced and capital market frictions have to be made explicit, a third sector needs to bediscussed: the financial sector. If firms want to increase their capital stock, they are now obliged to search for funds in the capitalmarket, a resource-intensive activity. This assumes that all finance is external and no internal funding is available. We representthe financial intermediation by a bond market. When firms generate returns on their capital stock, we assume that all profits flowback to the shareholders. Yet, the search for funds may be costly, and it is likely to be more costly the higher the leverage is.17

In extending the search and matching methodology to capital market models, we make use of the framework firstintroduced by Wasmer and Weil (2004) and increasingly used in the literature on financial market frictions in macro-economic models (e.g. Becsi et al., 2000; Koskela and Stenbacka, 2002, 2004; Pierrard, 2005). In this approach—similar tolabor market matching—a firm needs to find external funding on financial markets that are subject to search frictions(no internal funding is supposed to be available). This creates a wedge between the supply of funds and bonds, withsome funds left uninvested in each period. This creates an endogenous holding of money, and the aggregate demandimplications of unmatched funds can worsen any initially existing unemployment problem as discussed by Wasmer andWeil (2004).

Matching on capital markets is characterized by a matching function mBðD,BÞ where D is the deposit amount byhouseholds to be intermediated into corporate bonds B to finance an increase of the firm’s capital stock. From the point ofview of firms, credit market tightness is measured by f� B=D and 1=f is an index of credit market liquidity, i.e. the easewith which entrepreneurs can find financing. The instantaneous probability that an entrepreneur will find financing ismBðD,BÞ=B� 1=f, i.e. it is inversely proportional to credit market tightness.

In the following, we will make use of this first extension to the basic Merz framework to understand the role of capitalmarket frictions for the global dynamics of our model.18 Staying as close as possible to the original framework, we firstconsider a constant-returns-to-scale framework for the credit market matching function. Here each firm faces anexogenous (aggregate) matching process. The alternative is to endogenize the credit supply in a capital market matchingframework: In such a case, the firm faces endogenous borrowing constraints in the sense that available funds may belimited by tighter lending or capital market standards.19 In Section 7, we will also introduce variable costs of issuing bonds(the capital market equivalent to labor market vacancies) and analyze the implications of those basic modifications for thedynamics of our model.20

It should be stressed that we concentrate in this paper on bond markets as a source of enterprise finance and neglectdynamics in a firm’s equity holdings. This may be justified on the ground of recent research that has shown that equity assource of financing of firms’ investment appears to be limited and the equity market seems to be too volatile to be tightlylinked to firms’ financing of investment.21 We, therefore, consider the credit market as more crucial. As the financialmarket events of the years 2007–9 and related research have shown, the perception of risk in the bond market and banklending standards were essential factors in determining the cost and availability of financial funds for economic activity.22

5.2. Equilibrium conditions with capital market frictions

We start again with the decisions on consumption and labor effort. The contemporaneous utility of consumption andleisure remains defined by (3) and (4). In the presence of search frictions on the financial market, the stream of deposits todetermine the capital accumulation path is given by the optimal choice of consumption. Hence, the resource constraint isnow given by

Dt ¼ YtðKt ,ANtÞ�Ct�FðstÞð1�NtÞ�z � Vt�k � Bt ð14Þ

with Ct: aggregate consumption, Yt: aggregate production, A: (exogenous) labor productivity, Dt: total deposits, Bt: totalbonds, Nt: employment and Vt: vacancies. In addition to costly search on the labor market, issuing bonds adds another costfactor to the macroeconomic resource constraint, with per-period flow costs for unmatched bonds measured by k.

17 For the variation of lending cost over the business cycle, as resulting from default premia, see Grune et al. (2008a).18 Adrian and Shin note that speaking of capital market frictions may in fact understate the role of the financial sector in driving the financial

expansions and contractions, see Adrian and Shin (2009).19 In US the Fed undertakes regularly a survey on senior loan officers lending practices as an estimate of the change of lending standards of credit

market institutions in US. Since 2003, the ECB has undertaken a similar effort with its quarterly bank lending survey (available at: http://www.ecb.int/

stats/money/lend/html/index.en.html). Depending on their net worth, individual firms may face individual lending standards. But note that the net worth

depending borrowing could also be expressed through a risk premium, see Gilchrist et al. (2009).20 The latter will allow us to consider increasing returns to scale in the credit matching process for the dynamic analysis as both the size of installed

capital and of corporate bonds issues impact the matching rate. It should be noted that credit market matching is a relatively recent concept that is still in

its infancy in the theoretical literature. No empirical studies exist to our knowledge about the precise micro-foundations to determine the functional

specification of such a matching process.21 Indeed, there is now empirical evidence that investment is less determined by Tobin’s equity q than by Tobin’s bond q (see Philippon, 2009).22 See Gorton and He (2008), Grune et al. (2008a) and recent speeches by Bernanke (2007) and Mishkin (2008). In particular, the latter two stress the

credit market and the financial accelerator as transmission mechanisms of financial disturbances to the real economy.

http://www.ecb.int/stats/money/lend/html/index.en.html

http://www.ecb.int/stats/money/lend/html/index.en.html


The dynamic decision problem is subject to our two dynamic constraints, one for the change in employment, the otherfor capital accumulation, which-in the presence of search frictions-writes as

_K t ¼mBðD,BÞ�dKt ð15Þ

Hence, the decision problem writes as

maxst ,Ct ,Bt ,Vt

Z 10


Z 10

C1�Zt

1�Z�eNw

t

( )e�rt

subject to ð6Þ,ð14Þ,ð15Þ and K040, N040

which can again by solved by setting up the current-value Hamiltonian:

maxst ,Ct ,Bt ,Vt

H¼ C1�Zt

1�Z�eNwt

þn1½mLðst � Ut ,VtÞ�sNt �

þn2½mBððYtðKt ,ANtÞ�Ct�FðstÞð1�NtÞ�z � Vt�k � BtÞ,BÞ�dKt�:

Solving the Hamiltonian yields a new set of first-order conditions:

@H@Ct¼ 0 3 C�Zt ¼ n2

@mB

@D ð16Þ

@H@Bt¼ 0 3 k @mB

@D ¼@mB

@B ð17Þ

@H@Vt¼ 0 3 n2z

@mB

@D ¼ n1@mL

@V ð18Þ

@H@st¼ 0 3 n2FuðstÞ

@mB

@D ¼ n1@mL

@sð19Þ


limt-1

n1Nte�rt ¼ lim

t-1n2Kte

�rt ¼ 0:



@Y

@NtþFðstÞ

� �� n2

@mB

@DtþewNw�1

t ð20Þ

_n2 ¼ rn2�@H@KT¼ ðrþdÞn2�

@Y

@Kt� n2

@mB

@Dtð21Þ

The equilibrium conditions can be further simplified. From (17), we conclude that

@mB

@Dt¼

1

k@mB

@Bt

Moreover, letting _n2 ¼ 0, we conclude from (21) that

@YðK�,AN�Þ

@K

@mB

@D ¼ rþd 3@YðK�,AN�Þ

@K

@mB

@B ¼ kðrþdÞ ð22Þ

Substituting (16) and (18) into (20) and rearranging terms leads to


@V

!½YðK�,AN�Þ�D��Fðs�Þð1�N�Þ�z � V��k � B��Z ¼ ewðN�Þw�1

ð23Þ

The steady state of employment and capital dynamics yield

_Nt ¼ 0 3 N� ¼mLðs� � U�,V�Þ

sð24Þ

_K t ¼ 0 3 K� ¼mBðD�,B�Þ

d: ð25Þ

Box 2–Summary of parameter specifications—Exogenous capital frictions.

The model with exogenous capital market frictions has been solved using the following functional specifications andparameter values:






t



with q0 ¼ q1 ¼ 0:5.

� The matching function on the financial market is specified as:

mBðDBÞ ¼ p0 �Dp1B1�p1

and hence

@mB

@B¼ p0 � ð1�p1Þ

B

D

� ��p1

@mB

@D¼ p0 � p1

B

D

� �1�p1

with p0=1, p1=0.85. Moreover, from the steady state condition we have:

@mB

@D¼ 1

k@mB

@B() D¼ p1

1�p1kB

� The remaining structural parameters are specified as:



k 0:10 cost of issuing bonds



Hence, the equilibrium for the model with exogenous capital market frictions is determined by (22)–(25). The aboverepresents the derivations of the basic model with labor market and capital market search and matching which will beused subsequently. Note that for simplicity we have assumed above that consumption can be smoothed intertemporallydue to frictionless markets for deposits23 but for firms’ decisions on investment, frictions in capital markets are relevant.For the household there are only frictions in the labor market.24

6. Exogenous credit frictions

To explore numerically the model variant with exogenous frictions in the capital market we study the effect ofparameter variations in the capital market matching function, mB, on the steady state and analyse the out-of-steady-statedynamics. In particular, we concentrate on the elasticity of capital market matching with respect to deposits—representedin the following by the parameter p1. The Merz model will be contained as a special case—with p1=1—whereas exogenouscapital market frictions arise as soon as p1o1. We undertake first a comparative static analysis, computing again only

23 For a model where frictions for consumption smoothing exist and where a high correlation between employment and consumption can be

observed, see Ernst et al. (2006).24 Note that we abstract here from the fact that there could be capital market frictions, or credit constraints, for households as well. Such aspects

could be developed in a subsequent paper.

Table 4Comparative static exercises.

Increase of bond issuing costs Increase vacancy costs Increase in labor disutility

k0 ¼ 0:1-k1 ¼ 0:2 D z0 ¼ 0:07-z1 ¼ 0:1 D e0 ¼ 1-e1 ¼ 2 D

GDP (level) Y1 1.704 �5.7% Y1 1.806 �0.1% Y1 1.574 �12.9 %

Capital stock K1 8.529 �15.0% K1 10.029 �0.1% K1 8.739 �12.9%

Employment rate N1 0.689 0.0pp N1 0.688 �0.1pp N1 0.600 �8.9pp

Vacancies V1 0.049 �0.0% V1 0.049 �0.0% V1 0.029 �41.0%

Bonds B1 0.230 �52.8% B1 0.488 �0.1% B1 0.425 �12.9%

Deposits D1 0.261 �5.7% D1 0.276 �0.1% D1 0.241 �12.9%

Consumption C1 1.394 �5.7% C1 1.476 �0.2% C1 1.288 �12.8%

Decrease of financial market efficiency Increase in job separation rate Increase in capital depreciation rate

p0 ¼ 1:0-p0 ¼ 0:8 D s0 ¼ 0:04-s1 ¼ 0:06 D d0 ¼ 0:03-d1 ¼ 0:05 D

GDP (level) Y1 1.593 �11.8% Y1 1.806 0.0% Y1 1.553 �14.1%

Capital stock K1 7.079 �29.4% K1 10.031 0.0% K1 6.468 �35.5%

Employment rate N1 0.689 0.0pp N1 0.689 0.0pp N1 0.696 0.7pp

Vacancies V1 0.049 �0.1% V1 0.110 124.7% V1 0.051 4.5%

Bonds B1 0.430 �11.8% B1 0.488 0.0% B1 0.524 7.4%

Deposits D1 0.244 �11.8% D1 0.276 0.0% D1 0.297 7.4%

Consumption C1 1.303 �11.8% C1 1.473 �0.3% C1 1.200 �18.8%



Table 3Steady state values with exogenous credit frictions.

Steady state

GDP (level) Y 1.807



Vacancies V 0.049

Bonds B 0.488

Deposits D 0.276

Consumption C 1.478


steady state solutions. Thereafter, we also study the dynamics by using our dynamic programming approach. In addition,we explore whether for some parameter values of p1o1 there is a globally stable dynamics converging toward a singlesteady state and compare its characteristics with the situation of the Merz model. We will also report the eigenvalues ofthe steady state and show how they change with the parameter p1.

6.1. Functional specification and comparative statics

In order to study the comparative statics of the model we continue with the model variant of Section 5, assumingconstant search effort, constant bond issuing cost and exogenous credit constraints. This model variant has four decisionvariables, V, B, s and D, and two state variables, K and N. The remaining data are parameters, r,d,k,Z and s. The searcheffort, s, is again treated as constant. At the steady state, output will be determined by Y*=Y(K*, AN*). In order to be able toderive a numerical solution, several functional specifications have again to be made that are summarized in Box 2.

With these parametric and functional specifications, the following steady state solution obtains25 (see Table 3).This implies a savings ratio of around 18.2%. Here again, N is defined as the employment rate and has to lie in the interval

(0,1). Note that in this case the capital stock, income, and consumption is lower than in the model variant of Section 4, i.e.compared to the situation without capital market frictions. The Jacobian of the 4-dimensional dynamic system that is set upby the vector ðn1,n2,K ,NÞ has the following eigenvalues at the steady state, indicating a stable equilibrium:

l1 ¼ 0:100þ0:162i

l2 ¼ 0:100�0:162i

25 Note that there might be a possibility of more than one steady state, which is not explored in the current context, using the Hamiltonian. This will

be done in Section 6.2, using dynamic programming.


l3 ¼�0:683

l4 ¼�0:332

Before turning to the global dynamics of this system, we first carry out some comparative statics. The results of thisexercise are reported in Table 4, showing the change of the main economic variables of our model in reaction to (small)variations of the structural parameters of the model. As summarised in the table, an increase in bond issuing costs and anincrease in the degree of capital depreciation decrease the capital stock, output, consumption and employment, whereas anincrease in financial market efficiency increases the capital stock, output and consumption.

6.2. Global dynamics with exogenous credit frictions

The model with capital market frictions where credit constraints arise solely due to the search and matching process isequivalent to our case of p0=1 and p1o1. The capital market frictions are exogenous here since they do not enter the firm’s

11.000

10.000

9.000

8.000

7.000

6.0000.010 0.198 0.386 0.574 0.762 0.950

[N]

[K]

Fig. 3. Global dynamics with exogenous credit constraints (upper steady state, p1=0.85). Note: The figure displays the dynamics in the N � K space

around the upper steady state, with K*=10.45 and N*=0.69.

9876543210

1 67

89

[K]

1011

[N]

-1

[V]

00.1

0.20.30.40.5

0.50.70.80.9

Fig. 4. Value function with exogenous credit constraints (upper steady state, p1=0.85). Note: At the upper steady state, the value function takes V*=7.87.

5.000

4.002

3.004

2.006

1.008

0.0100.010 0.198 0.386 0.574 0.762 0.950

[N]

[K]

Fig. 5. Global dynamics with exogenous credit constraints (lower steady state, p1=0.85). Note: The figure displays the dynamics in the N � K space at the

lower steady state with K*=0.35 and N*=0.54.


decision problem but arise from the aggregate matching process. We start the analysis of the global dynamics of this modelby exploring the possibility of multiple steady states. We indeed confirm the possibility of the existence of a second, lower-level steady state and show through dynamic programming how the two steady states are approached. Then we show howthe lower steady state eventually becomes a global attractor when increasing the degree of capital market frictionsas exemplified by a decrease of the parameter p1. Here again, we explore the global dynamics by using our dynamicprogramming algorithm as sketched in Section 3.

6.2.1. Dynamics with multiple equilibria

Studying the steady states and dynamics of a model that exhibits multiple equilibria becomes intricate. In general,multiple equilibria can arise in dynamic decision models with non-concave pay-off functions, externalities and increasingreturns, as well as with feedback effects of state variables on the pay-off function. Multiple equilibria can also arise inconcave economies. A survey of recent work on such models with multiple equilibria is given in Deissenberg et al. (2003).In terms of the dynamics, multiple equilibria are difficult to analyze, since the domains of attraction can usually not becomputed analytically and a proper global solution technique needs to be employed to numerically finding the domains ofattraction.

As mentioned before, in the context of our decision model with search and matching on labor and capital marketsand exogenous credit constraints—the latter being activated with p1o1, which adds further nonlinearities in thefeedback mechanisms (now from both labor and capital markets)—multiple equilibria can arise. We want to give asimple example by just varying one parameter for the simplest model of the labor and capital market interaction.As aforementioned a natural parameter to vary is p1. When employing the parameter settings in Box 2—and in particularp1=0.85—we can show by way of dynamic programming that there are at least two equilibria with distinct levels of capitalstocks.

The upper equilibrium is located in the region of the capital stock 6oKo11. As Fig. 3 shows, for a large set of initialconditions of the capital stock (vertical axis) and employment (horizontal axis) all trajectories move to an upperequilibrium at K*=10.45 and N*=0.69 . It coincides with the one that is reported in Table 3. On the other hand exploring thelower region we detect another, lower-level equilibrium towards which the trajectories converge (see Fig. 5). The corre-sponding value functions are depicted in Fig. 4 for the upper region, and in Fig. 6, for the lower region.

This means that the optimal decisions on consumption, vacancies and bond issuing will make capital stock andemployment move far up or far down, depending on the initial condition. Since the two visible equilibria, the upperone depicted in Fig. 3 and the lower one in Fig. 5, are attractors there must exist a separation line that separatesthe dynamics toward the two equilibria.26 Towards which steady state the capital stock and employment will move ishistory-dependent in the sense that the initial conditions matter for the final outcome of the dynamics of capital stock andemployment.

26 In technical terms such a separation line has been called a Skiba line, honoring the work of Skiba who has first conjectured such points or lines,

see Deissenberg et al. (2003). Such a separation line is computed in Grune and Semmler (2004).

25

20

15

10

5

0

[V]

00.1

0.2

0.40.3

0.50.60.7

0.80.9

1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5[N]

[K]

Fig. 6. Value function with exogenous credit constraints (lower steady state, p1=0.85). Note: At the lower steady state, the value function takes V*=10.85.

Table 5Eigenvalues with exogenous credit constraints.

p1 0.9 0.8 0.7 0.6

l1 �0.692 �0.666 �0.641 �0.529

l2 �0.331 �0.334 �0.337 �0.361

l3 0.100+0.162i 0.100+0.162i 0.100+0.161i 0.101+0.160i

l4 0.100�0.162i 0.100�0.162i 0.100�0.161i 0.101�0.160i


Note that such a model with multiple domains of attraction may be of great relevance for policy studies. As the outcomein Figs. 3 and 5 shows the dependence of the final outcome on initial conditions means that policy shocks not only havetemporary but possibly permanent effects on economic activity. The possible choices that policy makers can undertake willtilt the dynamics of capital stock and employment either to the lower or the upper steady states.27 Also note that suchmodels with multiple domains of attraction are in principle empirically testable.28

In the context of this model, in both cases it is easily feasible to track the optimal decisions out of the steady state asdiscussed for the model in Section 4. The magnitude of the decision variables, C, B and V are numerically available evenwell off the equilibrium, in contrast to approaches using linear approximation methods. This can give guidance to actualdecisions even if the economy is not at the steady state. Note that with a lower capital stock-as observable at the lowersteady state—the value function does not necessarily take on a lower value. This comes from the fact that the magnitude ofthe value function is determined by both the optimal consumption as well as leisure time (employment). Since in ourlower steady state the employment is lower (with more available leisure time) the value function can rise instead of falling.This is what apparently has occurred in our above example, when the capital stock moves down from the higher (upper)steady state to the lower steady state. We will come back to this issue below.29

27 Gertler and Karadi (2009) explore how a preferable steady state can be approached through unconventional monetary policy, such as direct credit

intervention and quantitative easing.28 A model that predicts distinct domains of attraction, with separating dynamics in the middle, can be suitably tested by Markov transition matrices

if an appropriate data set can be constructed, see Semmler and Ofori (2007). A first attempt to test the above model using a multiple-regimes VAR analysis

has been undertaken in Ernst et al. (2010).29 As concerning the above case of two attractors, one could speak of a local determinacy. Yet, since different initial conditions lead to different

outcomes, the model exhibits global indeterminacy.

11.000

8.802

6.604

4.406

2.208

0.0100.010 0.198 0.386 0.574 0.762 0.950

[N]

[K]

Fig. 7. Global dynamics with the low steady state as attractor (p1=0.60). Note: There is convergence to K*=1.23; N*=0.54.

121086420

00.10.2

0.30.40.50.60.70.80.9

1 02

46

810

12

[K]

[N]

[V]

Fig. 8. Value function with the low steady state as attractor (p1=0.60). Note: The value function takes V*=11.2.


6.2.2. Global dynamics with increase in credit market frictions

What happens to the global attractor when credit market frictions increase and financial intermediation declines?The parameter p1 is a natural candidate for studying such a change in the degree of capital market frictions. If thisparameter value falls, the frictions in the capital market become more severe. As discussed above, with p1=1 we recoverthe Merz case, where the financial intermediation of financial flows work perfectly well and no frictions are present. Withrising capital market frictions financial intermediation becomes more and more difficult and one naturally expects thecapital stock to go down. The following table explores the eigenvalues around the upper attractor.

With increasing capital market frictions, the local dynamics become more complicated and the negative eigenvalueseven switch to a complex-conjugate couple for values of p1 below 0.6 (not presented in Table 5). This also influences theglobal dynamics as shown in Fig. 7 where we use p1=0.60.

Note that in the above Figs. 7 and 8, we have shown the global dynamics, in the entire state space, namely for the capitalstock 0oKo11 and labor 0oNo1. So, in Fig. 7 we see only one global attractor. The solution we have obtained bydynamic programming points to the fact that there is really only one optimal global attractor for very large capital marketfrictions. We also want to note that here again, with an increase of frictions, the value function does not necessary fall,since employment also falls.


7. Endogenous credit frictions

As discussed in Sections 2 and 5, models with credit frictions are increasingly receiving attention in the literature. In thefollowing, we want to explore an extension for such models in our search and matching model by considering endogenouscredit frictions. In particular, we consider two extensions: The first one considers a varying matching efficiency dependingon the net worth of the firm (Section 7.1), the second one assumes that bond-issuing costs are dependent on the size of thebond being issued (Section 7.2). As before, we put a particular emphasis on the exploration of the interactions betweencredit and labor market frictions.

7.1. Endogenous credit constraints

First we want to explore the effect of endogenous credit constraints for capital accumulation, employment,consumption and welfare. In this case, the constraints are endogenous in the sense that creditworthiness of firmsbecomes relevant for borrowing and enters its optimal decision problem. As above noted, when firms look for funds tofinance the desired investments they may not obtain the funds they desire. There may be borrowing constraints in thesense that for the borrower credit may be restricted by lending standards of banks or risk premia maybe required by thecapital market for the bond issuer.

As recent financial market events—in particular in the years 2007–9—and recent research has shown, bank lendingstandards and the perception of risk in the bond market are crucial factors in determining the availability of financial fundsfor economic activity. These affect both direct lending from credit market institutions as well as issuing of new corporatebonds over the business cycle. This is captured in the above discussed financial accelerator theory. In the US, in particularBernanke (2007) and Mishkin (2008), have stressed the importance of the financial accelerator as a transmissionmechanism of financial disturbances to the real side of the economy.30

The financial accelerator discusses how shocks to the real side of the economy are magnified either through varying riskpremia or through tightening or loosening credit constraints. Crucial for both is the net worth of the firm, NW, and how itaffects the financial market matching process.31 Following the theory of collateralized borrowing, as put forward by thefinancial accelerator literature,32 let a : NW/aðNWÞ with a(0)=0, a(K)=1 and auðNWÞ40 for 0oNWoK . When net worthis maximum (i.e. equal to K), the company does not face credit constraints. On the other hand, when net worth is zero, nofurther bonds can be issued any more. In other words the bond issuing firm can go up only to the value of the capital stock,K, if it issues bonds on the capital market. The value of collateral K acts here as defining the borrowing limits. In the contextof our model we need to specify the function further, making the coefficient a is defined by net worth: aðNWÞ ¼ ðK�BÞ=K .The coefficient a should have the above stated limits of 0 and 1. If the debt capacity of B¼ K is reached the coefficient is 0.If there is no borrowing the coefficient is 1. Note that these upper and lower limits of collateralized borrowing areconstraints in the optimization problem which can be taken into account by our numerical methods.33

Given these endogenous credit constraints, the search and matching function is modified to mB ¼mBðaðNWÞD,BÞ and thecapital stock evolves according to

_K t ¼mBðaðNWÞD,BÞ�dKt ð26Þ

with a=1 equivalent to no endogenous credit constraints and ao1 indicating the existence of state dependent creditconstraints, for example triggered by a change of bank lending standards or by the willingness of the bond market to acceptrisk. Note that this definition of the credit-constraint function a(NW) is similar to the function aðB,KÞ, as used in Grune et al.(2005, 2008b). Such a function can also be used to define a default premium to be paid by firms that want to raise capitalfrom the capital markets but have low collateral (see Bernanke and Gertler, 1989; Gilchrist et al., 2009; Grune et al., 2005,2008b).

In our set-up, we prefer making use of the net worth of firms to define (endogenous) credit constraints instead ofreferring directly to default premia. This is similar to Yaffee and Stiglitz (1990) who consider the maximization problem ofbanks when providing loans to customers. Since a rise of the interest rate offered by customers is likely to increase thedefault loss, a bank will, by maximizing its profits, ration loans at certain returns that are appropriate for it to avoid losses.Then it can be shown that the amount of loans offered by banks to customers are not market-clearing offers. Loans arerationed as compared to the demand for loans expressed by the customers. This is why there is an equivalence of theresults concerning the risk premium and credit constraints. Thus either risk premia or credit constraints can be used inmacro models. An equivalence of the results of the two approaches concerning investment and firm value is also shown in

30 Recently, the Federal Reserve Bank of the U.S. has devoted an entire conference to the importance of the financial accelerator for the current

financial market meltdown, see http://www.federalreserve.gov/events/conferences/fmmp2009.31 As recent literature has also shown, net worth of financial institutions (such as banks) may affect the available credit at the lenders side, and thus

reduce financial funds for non-financial firms, see Adrian and Shin (2009) and Gertler and Karadi (2009).32 See Bernanke and Gertler (1989).33 Dynamic decision problems with constraints in the control variables are theoretical not easily solved. Yet, numerically we can define the control

variables to be constrained by a certain control space within which the dynamic programming algorithm then operates.

http://www.federalreserve.gov/events/conferences/fmmp2009


(Grune et al., 2008a, b). In the latter paper there it is also shown that historically an increase in risk premia alwayscoincides with a tightening of credit constraints.34

7.1.1. Equilibrium conditions with endogenous credit constraints

Using (26) as the new dynamic constraint for capital accumulation, the dynamic decision problem now writes as

maxst ,Ct ,Bt ,Vt

Z 10


Z 10

C1�Zt

1�Z�eNwt

( )e�rt

subject to ð6Þ,ð14Þ,ð26Þ and K040, N040

which can again by solved by setting up the current-value Hamiltonian:

maxst ,Ct ,Bt ,Vt

H¼ C1�Zt

1�Z�eNw

t


þn2½mBðaðNWÞðYtðKt ,ANtÞ�Ct�FðstÞð1�NtÞ�z � Vt�k � BtÞ,BÞ�dKt�:

Solving the Hamiltonian yields a new set of first-order conditions:

@H@Ct¼ 0 3 C�Zt ¼ n2

@mB

@D ð27Þ

@H@Bt¼ 0 3 k @mB

@D ¼@mB

@B þ@mB

@D@aðNWÞ

@B ð28Þ

@H@Vt¼ 0 3 n2z

@mB

@D ¼ n1@mL

@V ð29Þ


@mB

@D ¼ n1@mL

@sð30Þ


limt-1

n1Nte�rt ¼ lim

t-1n2Kte

�rt ¼ 0:



@Y

@NtþFðstÞ

� �� n2

@mB

@DtþewNw�1

t ð31Þ


@Y

@Kt� aðNWtÞþ

@aðNWtÞ

@Kt� ðÞ

� �� n2

@mB

@Dtð32Þ

The equilibrium conditions can be further simplified. From (28) and noting that @aðNWÞ=@B¼�1=K , we conclude that

@mB

@D ¼1

k� @aðNW�Þ

@B

@mB

@B ¼K�

k � K�þ1

@mB

@B

Moreover, letting _n2 ¼ 0 and noting that @aðNWÞ=@K ¼ B=K2, we conclude from (32) after rearranging terms that

@YðK�,AN�Þ

@K� aðNW�

ÞþB�

ðK�Þ2ðYðK�,AN�Þ�C��Fðs�Þð1�N�Þ�z � V��k � B�Þ

!@mB

@D ¼ rþd ð33Þ

Substituting (27) and (29) into (31) and rearranging terms leads to


@V

!½YðK�,AN�Þ�D��Fðs�Þð1�N�Þ�z � V��k � B��Z ¼ ewðN�Þw�1

ð34Þ

The steady state of employment and capital dynamics yield

_Nt ¼ 0 3 N� ¼mLðs� � U�,V�Þ

s ð35Þ

34 DSGE models have attempted to construct model variants with risk premia to account for the effects of shocks, but since risk premia are not

quantitatively significant in consumption-based macro models, the success of those models is currently still explored, see De Fiore and Tristani (2008),

De Graeve et al. (2008). Here, we suggest an alternative route.

Box 3–Summary of parameter specifications—Endogenous credit constraints.

The model with endogenous credit constraints has been solved using the following functional specifications andparameter values:






t



with q0=q1=0.5.


mBðDBÞ ¼ p0 � ðaðNW ÞDÞp1B1�p1

where aðNW Þ ¼ K�BK and hence

@mB

@B¼ p0 � ð1�p1Þ

B

D

� ��p1

�p0 � p11

KDp1B1�p1

@mB

@D¼ p0 � p1aðNW Þp1

B

D

� �1�p1


@mB

@D¼ 1

k@mB

@B() B¼ Kð1�p1Þ

p1ðk � Kþ1ÞD

� The remaining structural parameters are specified as:



k 0:10 cost of issuing bonds


Table 6Steady state values with endogenous credit constraints.

Steady state

GDP (level) Y 1.645



Vacancies V 0.048

Bonds B 0.192

Deposits D 0.248

Consumption C 1.374


_K t ¼ 0 3 K� ¼mBðD�,B�Þ

d: ð36Þ

Hence, the equilibrium for the model with endogenous credit constraints is determined by (33), (34), (35) and (36). Notethat for simplicity we have assumed that borrowing for consumption does not face capital market frictions, which areassumed to be relevant only for investment decisions of firms.

12.000

11.000

10.000

9.000

8.000

7.0000.010 0.198 0.386 0.574 0.762 0.950

[N]

[K]

Fig. 9. Global dynamics with endogenous credit constraints (p1=0.85). Note: with K*=7.8 and N*=0.68.

14121086420

00.1

0.20.3

0.40.5

0.60.7

0.80.9

718

910

[K]

1112

[V]

[N]

Fig. 10. The value function with endogenous credit constraints (p1=0.85). Note: the value function takes V*=8.12.

Table 7Steady states and eigenvalues with endogenous credit constraints.

p1 0.9 0.8 0.7 0.6

l1 �0.74 �0.76 �0.75 �0.71

l2 �0.30 0.09+0.19i 0.08+0.21i 0.07+0.23i

l3 0.09+0.17i 0.09�0.19i 0.08�0.21i 0.07�0.23i

l4 0.09�0.17i �0.26 �0.22 �0.19


7.1.2. Global dynamics with endogenous net worth

Using the above net worth function together with the parameters in Box 3 and plugging it into the model of Section 6,we can again compute the steady state for different parameter values of p1. For p1=0.85 we obtain the values as shown inTable 6.


Note that for the case of endogenous credit constraints and our standard case p1=0.85, the size of bonds issued as wellas the capital stock are smaller compared to the case of exogenous credit constraints (see Table 3). We also report theeigenvalues of the steady state for a range of values of the financial market elasticity p1 (see Table 7). As the table presents,the local dynamics remain broadly the same over the full range of parameter values, albeit the strength of the negativeeigenvalues of the equilibria becomes weaker.

In order to further analyze the global dynamics, we again use our dynamic programming approach to undertake twoexercises. We first solve the model with p1=0.85, which gives us a stable steady state at K*=7.8 and N*=0.68. The globaldynamics and the value function are shown in Figs. 9 and 10.

As mentioned above, when introducing endogenous credit constraints, in comparison to exogenous credit constraints inSection 6, the capital stock and the volume of bonds (as well as output and consumption)—are reduced. As one canobserve, the steady state results have changed in comparison to the results reported in Table 3 and Figs. 3 and 4 wherecredit market frictions remained exogenous.35 One can also observe that the two value functions, representing the welfareof the households for the case of exogenous credit constraints and with endogenous credit constraints are only slightlydifferent.36

By decreasing the parameter further to p1=0.6 a low equilibrium becomes an attractor. As a consequence, we observethat all trajectories move down for the original area of starting values, making all trajectories globally converging (Fig. 11and 12).

Note that here too, the capital stock and GDP are smaller compared to the cases of no credit market frictions (seeTable 1) and exogenous credit constraints (see Table 3). As capital market frictions worsen, moving from p1=0.85 to 0.60,financial intermediation is likely to be disrupted similar to the situation described by Gertler and Karadi (2009).37

Nevertheless, the value function is higher at the latter steady state. This again has to come from the fact that employmentis lower (and leisure is higher) than in the previous steady state.

7.2. Endogenous bond issuing costs

Finally we want to study the alternative case where there is some endogenous easing of capital market frictions.Frequently the important empirical issue has been discussed that for different size classes of firms there may be differentfinancing costs. This is, for example, postulated by Gertler and Gilchrist (1994) who show that financing cost of small andlarge firms are significantly different. Furthermore, Curdia and Woodford (2009a, b) introduce models with credit marketfrictions where financial intermediaries vary the markup on loans, which drives a wedge between the interest paid to thesupplier of financial funds and the borrowers. They develop model variants of exogenous and endogenous markupdetermination.38

In this final version of our model we want to capture this basic idea by introducing endogenous bond issuing costs. Wedo this by assuming that for large bonds the issuing costs may be relatively lower as compared for bonds that are supposedto raise smaller amounts of capital. If bond issuing costs go down with the size of the firm and thus the size of the bond thisis similar to driving down the credit costs for the borrower as in the model versions of Gertler and Gilchrist (1994) andCurdia and Woodford (2009a, b).39 We will explore again the steady state by using the Hamiltonian and demonstrate thedynamics for a low steady state. This is sufficient in our context.

As before, the contemporaneous utility of consumption and leisure is defined by (3) and (4). The resource constraintnow includes variable bond issuing costs, that vary with the amount of bonds being issued:

Dt ¼ YtðKt ,ANtÞ�Ct�FðstÞð1�NtÞ�z � Vt�kðBtÞ � Bt ð37Þ

again with Ct: aggregate consumption, Yt: aggregate production, A: (exogenous) labor productivity, Dt: total deposits, Bt:total bonds, Nt: employment and Vt: vacancies. All functional forms and variables remain as before with the exception ofthe bond issuing costs k, which now are assumed to vary with the size of the bond issued. In particular, we want to makean assumption of increasing returns to scale in issuing bonds such that kuðBtÞo0.

35 Yet, as Table 6 as compared to Table 3 presents, employment does not seem to be much affected by switching from exogenous to endogenous

credit constraints. This may come from some general equilibrium effects: A reasonable conjecture might be that there may have been a substitution of

labor for capital as capital is more difficult to be obtained.36 The value function in the second case is slightly higher compared to the case without endogenous credit constraints (compare Fig. 4 with Fig. 10).

Here too it seems to come from the fact that along the path toward the steady state consumption and employment take on a different path, which may

end up in a value function above the case of exogenous capital market constraints. Because of space limitations, we do not explore the issue of multiple

equilibria here.37 A vivid description of such a scenario where an economy is moving into a situation of severe credit constraints can be found in Kindleberger

(1976). He sees such a scenario arising from the instability of credit.38 Curdia and Woodford, in particular, want to focus on the challenges that central banks face when they reduce the interest rate but the financial

intermediaries raise, due to rising default risk, the markup. Although our model may have also implications for central bank policies, we will not deal with

this issue here.39 In the bond market there is usually an underwriter for bonds who buys bonds from an issuer and sells them to the market. This drives a wedge

between the return of the issuer of the bond and the investor buying the bond. We are making such a wedge endogenous by making it simply depending

on firm size and the size of the bond.

Box 4–Summary of parameter specifications—Endogenous bond issuing costs.

The model with endogenous bond issuing costs has been simulated using the following functional specificationsand parameter values:






t



with q0 ¼ q1 ¼ 0:5.

� Bond-issuing costs with scale effects:

kðBtÞ ¼ka

Bt� kb

Bt

with ka ¼ 0:19, kb ¼ 0:03. This guarantees that kðBtÞ�0:41 for all Bt , i.e. bond issuing costs are bounded from above.


mBðDBÞ ¼ p0 �Dp1B1�p1

and hence

@mB

@B¼ p0 � ð1�p1Þ

B

D

� ��p1

@mB

@D¼ p0 � p1

B

D

� �1�p1


@mB

@D¼ 1

ðkþBkuÞ@mB

@B() D¼ p1

1�p1kbB:

� The remaining structural parameters are specified as follows:




Table 8Steady state values with endogenous bond issuing costs.

Steady state

GDP (level) Y 2.193



Vacancies V 0.057

Bonds B 3.896

Deposits D 0.336

Consumption C 1.509

Bond-issuing costs k 0.089


Table 9Comparative static exercises.

Increase in marginal bond issuing costs Increase in vacancy costs Increase in labor disutility

kb,0 ¼ 0:03-kb,1 ¼ 0:02 D z0 ¼ 0:07-z1 ¼ 0:1 D e0 ¼ 1-e1 ¼ 2 D

GDP (level) Y1 2.438 11.1% Y1 2.192 �0.1% Y1 1.884 �14.1%

Capital stock K1 20.604 27.5% K1 16.146 �0.1% K1 13.571 �16.0%

Employment rate N1 0.734 2.1pp N1 0.713 0.0pp N1 0.621 �9.2pp

Vacancies V1 0.065 14.0% V1 0.057 �0.2% V1 0.033 �42.7%

Bonds B1 10.829 178.0% B1 3.891 �0.1% B1 2.876 �26.2%

Deposits D1 0.373 11.1% D1 0.335 �0.1% D1 0.288 �14.1%

Consumption C1 1.455 �3.6% C1 1.506 �0.2% C1 1.302 �13.7%

Bond issuing costs k1 0.056 �36.9% k1 0.089 0.1% k1 0.102 14.7%

Decrease of financial market efficiency Increase in job separation rate Increase in capital depreciation rate

p0 ¼ 1:0-p0 ¼ 0:8 D s0 ¼ 0:04-s1 ¼ 0:06 D d0 ¼ 0:03-d1 ¼ 0:05 D

GDP (level) Y1 1.911 �12.9% Y1 2.193 0.0% Y1 1.925 �12.2%

Capital stock K1 11.030 �31.7% K1 16.153 0.0% K1 10.782 �33.3%

Employment rate N1 0.713 �0.1pp N1 0.713 0.0 pp N1 0.730 1.7pp

Vacancies V1 0.057 �0.1% V1 0.128 124.8% V1 0.063 11.3%

Bonds B1 2.957 �24.1% B1 3.894 0.0% B1 4.688 20.3%

Deposits D1 0.292 �12.9% D1 0.335 0.0% D1 0.368 9.7%

Consumption C1 1.318 �12.7% C1 1.503 �0.3% C1 1.171 �22.4%

Bond issuing costs k1 0.100 13.3% k1 0.089 0.0% k1 0.081 �8.1%




As before, the dynamic decision model has two dynamic constraints, (6) and (15), when determining levels ofemployment and capital. The optimal program remains essentially the same but the budget constraint is now given by(37). Hence, the problem can be solved by setting up the modified current-value Hamiltonian:

maxst ,Ct ,Bt ,Vt

H¼ C1�Zt

1�Z�eNw

t


þn2½mBððYtðKt ,ANtÞ�Ct�FðstÞð1�NtÞ�z � Vt�kðBtÞ � BtÞ,BÞ�dKt�:

Solving the Hamiltonian yields the following first-order conditions:

@H@Ct¼ 0 3 C�Zt ¼ n2

@mB

@D ð38Þ

@H@Bt¼ 0 3 ðkþBtkuÞ

@mB

@D ¼@mB

@B ð39Þ

@H@Vt¼ 0 3 n2z

@mB

@D ¼ n1@mL

@V ð40Þ


@mB

@D ¼ n1@mL

@sð41Þ


limt-1

n1Nte�rt ¼ lim

t-1n2Kte

�rt ¼ 0:

Using Pontryagin’s maximum principle, the co-state variables K and N evolve according to


@Y

@NtþFðstÞ

� �� n2

@mB

@DtþewNw�1 ð42Þ


@Y

@Kt� n2

@mB

@Dtð43Þ

12.000

9.600

7.200

4.800

2.400

0.0000.010 0.198 0.386 0.574 0.762 0.950

[N]

[K]

Fig. 11. Global dynamics with endogenous credit constraints (a low equilibrium as attractor, p1=0.60). Note: with K*=0.32 and N*=0.54.

1214

1086420

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1 02

46

810

12

[K]

[N]

[V]

Fig. 12. The value function with endogenous credit constraints (a low equilibrium as attractor, p1=0.60). Note: the value function takes V*=11.79.

Table 10Eigenvalues with endogenous bond issuing costs.

p1 0.9 0.8 0.7 0.6

l1 �0.75 �0.78 �0.75 �0.67

l2 �0.37 �0.35 �0.32 �0.30

l3 0.10+0.08i 0.09+0.02i 0.12 0.10

l4 0.10�0.08i 0.09�0.02i 0.05 0.03


Employing again the assumption of constant search effort, the model has three choice variables: consumption C,vacancies, V, and demand for bonds B, as well as two state variables: capital stock, K, and employment, N. The model hassix parameters r,d,k,Z,s and s. At the steady state, output will be determined by Y*=Y(K*,AN*). From (39), we conclude that

@mB

@D ¼1

ðkþB�kuÞ@mB

@B

18.000

14.404

10.808

7.212

3.616

0.0200.010 0.198 0.386 0.574 0.762 0.950

[N]

[K]

Fig. 13. Global dynamics with endogenous bond issuing costs (a low steady state, p1=0.6). Note: with K*=1.9 and N*=0.58.

25

20

15

10

5

0

[V]

[N]

00.1

0.20.30.40.50.6

0.70.80.9

1 02

4 68

[K]10

1214 16

18

Fig. 14. The value function with endogenous bond issuing costs (a low steady state, p1=0.6). Note: the value function takes V*=12.5.


Moreover, letting _n2 ¼ 0, we conclude from (43) that

@YðK�,AN�Þ

@K

@mB

@D ¼ rþd

Substituting (40) into (42) and rearranging terms leads to


@V

!½YðK�,AN�Þ�D��Fðs�Þð1�N�Þ�z � V��kðB�Þ � B��Z ¼ ewðN�Þw�1

The steady state of employment and capital dynamics yield:

_Nt ¼ 0 3 N� ¼mLðs� � U�,V�Þ

s

_K t ¼ 0 3 K� ¼mBðD�,B�Þ

d:


As indicated above, we assume that bond-issuing costs follow kðBtÞ with kðBtÞo0. Thus a larger requests makes thecompany’s bargaining position stronger and reduces, through a scale effect, the costs of issuing a new bond.40 Theparameters and functional specifications are summarized in Box 4.

With these specifications, the numerical solution for the stable steady state obtains as reported in Table 8. Moreover,the local stability around the steady state by means of the Jacobian of the 4-dimensional dynamic system, set up by thevector ðn1,n2,K ,NÞ, is given by:

40 A similar assumption has frequently been used for adjustment cost of capital

lower.41 In terms of the indeterminacy literature all the above cases exhibit local det

discussed below, multiple steady states.42 Here, as in the previous case of endogenous credit constraints, a scenario of

Stable steady state

l1
�0.773
l2
�0.360
l3
0.097�0.053i
l4
0.097+0.053i
Hence, the steady state in the state space should be stable. Other equilibria exist but are not economically viable and sowill not be considered here (e.g. exhibiting negative consumption levels). Before analyzing the global dynamics, we firstwill have another look at the comparative static results (reported in Table 9). As before, changes in parameters yieldexpected changes in the (stable) steady state values. Note, however, the interesting case of changes in (marginal) bondissuing costs: An increase in the marginal cost may lead to a substantial increase in the actual bond issued in order forthe firm to benefit from lower absolute costs (which has to do with the fact that the firm is issuing at the negative part ofthe marginal cost curve). As a consequence, this will increase the bond issuing, stock of capital, production andemployment (as also visible in the Table 9). On the other hand, the shift towards increased sizes of bonds comes at the costof a lower steady state value of the consumption-income ratio (more of the disposable income will be used for investmentrather than consumption), with likely adverse effects on steady state welfare (but note that the latter depends also onemployment N).

As can be noted from the above, for our standard parameters as well as for the comparative static exercises, all theresults show quite a high equilibrium capital stock, large successful bond issuing by firms, and mostly high employmentrates, all above N40:70. This results from the decreasing bond issuing cost resulting from scale effects. Firms are willing tooffer more vacancies, but also desire more bonds, and with the deposit available from financial intermediaries, totalinvestment and thus equilibrium capital stock can rise. So does employment. Overall, this results from an improvedfunctioning of capital markets. This at least holds for our case of only moderate capital market frictions, with p1=0.85. Yet,this may not be so for more severe capital market frictions.

Next, as before, we want to report the eigenvalues by varying the financial market matching parameter p1. Theeigenvalues for different values of this parameter are displayed in Table 10. As the table demonstrates, the local dynamicsremain saddle paths throughout for all parameter values p1, though it partly shows complex eigenvalues41.

Turning to the global dynamics, due to limited space, we report only the exercise for a low level equilibrium with severecapital market frictions, with p1=0.6 and use again our dynamic programming approach to solve our model. This gives usan attractor for the lower region, with K*=1.9 and N*=0.58. The solution is shown in Figs. 13 and 14.

As can be observed, similarly to the previous cases of Sections 6.2.1 and 7.1, with financial intermediation characterizedby severe capital market frictions, p1=0.6, all trajectories go to the lower region of the capital stock and employment fallsas well. A global attractor arises at a substantially lower capital stock and lower employment level.42 Yet, similar to theprevious cases, the value function does not necessarily decline, since with lower employment (and higher leisure) welfaretends to rise.

8. Conclusions

Encouraged by some recent work on a joint modeling of labor and capital market search and matching we started thispaper with the modeling of the labor market that follows the Merz (1995) approach and then added capital market searchand matching. The latter implies that in addition to the labor market there is also no instantaneous clearing of the capitalmarket. Following recent theories of imperfect capital markets, we have, in addition to exogenous frictions in the capitalmarket, introduced endogenous credit constraints. We also consider endogenous bond issuing costs. As we have shown,these extensions can exacerbate the feedback effects of both markets and negatively impact capital stock, employment,output and consumption. As demonstrated in this paper, the non-linear interactions of the labor market and capitalmarkets are likely to give rise to multiple equilibria and possibly low level steady states.

: for a large amount of capital, the adjustment costs of capital may be

erminacy. But this might not be so globally, since there might be, as

a severe decline of financial intermediation may emerge.


In terms of numerical solution techniques, most of the current approaches have used first or second order approxi-mations around a (unique) steady state. We demonstrate the advantage of using a dynamic programming approach andanalyzing the global—rather than local—dynamics in the context of possible multiple steady states. In particular, the use ofglobal dynamic analysis allowed for an understanding of the emergence and disappearance of multiple equilibriadepending on the parameter space, and especially the parameter for the severity of capital market frictions. Such ananalysis is particularly useful in times of economic and financial stress when small variations in market conditions can leadto abrubt changes in economic dynamics. In this regard, we could assess the effects of both exogenous and endogenouscredit constraints as well as endogenous costs of issuing bonds on economic activity and employment. We have solvednumerically for the steady state solution(s) by using the Hamiltonian. For selected examples we have also studied theglobal dynamics of our model variants by using dynamic programming techniques.

The use of both the Hamiltonian and dynamic programming have helped us to evaluate our model variants. Overallwe could observe, starting with the original Merz model as the limit case for our model extensions, that the introductionof exogenous or endogenous capital market frictions lead to a lower capital stock and reduced employment rates. Inparticular, this happens when our parameter capturing the strength of the capital market frictions moves down to p1=0.6,the trajectories go down to a lower region of the capital stock and also employment frequently falls. We have shown caseswhere a global attractor arises at a substantially lower capital stock and lower employment level, though usually theemployment is not as much affected as the capital stock. It is a bit of a surprising result, however, that the value functiondoes not necessarily always decline with more frictions, since with lower employment (and higher leisure) welfare canrise. This has often not been realized in other models with frictions in the financial intermediation where the valuefunction is typically not explicitly computed.43

We also want to note, that the model versions and tools presented here open up the possibility to get a better graspon monetary policy decisions when capital markets exhibit severe frictions. This is true for both the model variants ofexogenous as well as endogenous credit constraints. Easing exogenous credit market frictions by improving the search andmatching of financial flows will ameliorate financial market intermediation. Similarly in our model with endogenous creditfrictions the improvement of asset values and deleveraging will strengthen firms’ creditworthiness and borrowing capacityand increase capital stock and employment. Also, moving bond issuing costs down in our model version of Section 7.2. Thisis similar to driving down the credit cost for the borrower as in the model versions by Curdia and Woodford (2009a,b).

Finally as regards possible further extensions of our approach, these include the introduction of credit constraints forthe household sector, which are likely to interact in a mutually reinforcing way with the firm’s credit constraints indetermining macroeconomic performance. A recent empirical application of our analysis shows promising first results inhelping understand how financial frictions induce multiple equilibria and can be tested via a multiple regimes VAR (seeErnst et al., 2010).

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43 As we also have discussed, the transition to low level steady states, arising from the decline of the efficiency in financial intermediation (our

parameter p1) or from endogenous credit constraints, are likely to be accompanied by financial disruptions, which are however, not modeled here further.

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Global dynamics in a model with search and matching in labor and capital markets