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Academic Year 2011-2012

UNIVERSITÀ DEGLI STUDI DI PISA

Dipartimento di Economia e Management

Corso di Laurea Triennale in Economia e Commercio

Financial Fragility and Contagion

in Interbank Networks

Advisor:

Ch.mo Prof. Giorgio Fagiolo

Co-Advisor:

Ch.mo Dott. Mauro Napoletano

Candidate:

Stefano Pegoraro

Universit`a degli Studi di Pisa Master thesis

Beware of topology! An analysis of

contagion in banking networks

Supervisors Candidate

Prof. Giorgio Fagiolo Dott. Matteo Benetton Prof. Giulio Bottazzi

Academic year 2011-2012

University of Pisa

Department of Economics

Financial and Labor Market

Frictions in a New Keynesian

DSGE Model

Supervisor

:

Student

:

Prof. Giulio Bottazzi

Paolo Martellini

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Contents

1 Introduction 2

2 A selected literature review 6

2.1 The nancial accelerator . . . 6

2.2 The DMP labor market . . . 8

3 The model set-up 11 3.1 The representative household . . . 11

3.2 The demand for capital . . . 13

3.3 The labor market . . . 16

3.4 The return on capital and the labor demand . . . 17

3.5 The wage contracting line . . . 19

3.6 Capital producers . . . 20

3.7 Final good producers . . . 21

3.8 The public sector . . . 22

3.9 The resource constraint . . . 22

4 Calibration 23 5 The model simulation 26 5.1 Impulse response functions . . . 27

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1 Introduction

DSGE models are now one of the most common tools in both the study of business cycle uctuations and the design of policy prescriptions aimed at stabi-lizing the economy. For this reason, much literature has recently focused on the improvement of this kind of models, and many modications to the baseline New Keynesian framework have been adopted. In this regard, two strands of literature have particularly contributed to xing some of the major weaknesses of previous DSGE models. Interestingly enough, both these lines of research hinge on features of the business cycle that were already highlighted in the past decades, although not always embedded in a fully-edged general equilibrium model.

On the one hand, the smooth allocation of capital across rms has been strongly questioned by evidence showing how the amount of rms' equity may heavily im-pact their access to capital markets. Some seminal papers in this eld stressed the existence of an external nance premium (Townsend (1979), Carlstrom and Fuerst (1997), Bernanke et al. (1999, henceforth BGG)) in the credit market 1.

Accord-ing to this literature, as the return on capital is uncertain and banks must pay a monitoring cost in order to observe it, the amount of funds available to rms turns out to be increasing in their net worth endowment, i.e. in their creditworthiness. Therefore, during a boom, higher return on capital drives up the price of capital, investment and output, while the opposite occurs in a recession. In both cases, the friction in the capital market creates a so-called `nancial accelerator' that amplies economic uctuations.

On the other hand, the search and matching approach to the labor market, also known as the DMP framework, named after Mortensen and Pissarides (1994)

1Other papers focused on the credit constraint channel pioneered by Kiyotaki and Moore (1997).

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and Diamond (1985), has started to be used in alternative to the traditional set-ting, characterized by perfect (or monopolistic) competition among workers, and marginal productivity driven labor demand by rms. According to the DMP ap-proach, a new match is generated when an unemployed worker lls a vacancy posted by a rm. The incentive for a rm to post vacancies is given by the net surplus it gets from hiring a new worker. At the same time, the job search process is triggered by the worker's benet from being employed instead of unemployed. Finally, a Nash bargaining rule determines - through the wage setting - the share of surplus raising from the new match that accrues to each of the two parties.

Although much has been written with regard to these two stand-alone frictions in the markets for production inputs, only a very few piece of works have put them together in a single model, and studied their interaction at business cycle frequencies. Among them, Mumtaz and Zanetti (2013) analyze how adding search and matching frictions to a model with an external nance premium increases or reduces the responses of macroeconomic variables to dierent kind of shocks. According to the writer, the presence of some implicit simplifying assumptions and theoretical inconsistencies in the description of agents' behavior, cast heavy doubts on the meaningfulness of their results. In a dierent fashion, Christiano et al. (2011, henceforth CTW) built a small open economy DSGE model estimated by use of Swedish data. Their model includes both an external nance premium and a DMP-style labor market. Furthermore, the dynamics of foreign trade is taken into account. The results in CTW are partially in line with those presented here, and, in section 5, I briey highlight the extent to which they dier from each other. As a general comment, CTW have the merit of introducing a rich set of features like wage rigidity, worker-specic productivity and international trading. At the same time such richness of details implies much complexity in the interaction of a great number of variables, and less model tractability. In addition, results related to

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the dynamics of a small open economy may not apply to many sizeable advanced economies.

This paper aims at contributing to this stream of literature by building a simple model where both nancial and labor market frictions coexists. The process describing capital demand and the design of the optimal contract between rms and the nancial intermediary strictly follow the standard framework presented by BGG. However, while BGG employes a perfectly competitive labor market - and similar assumptions are made by later models equipped with BGG-style nancial markets - here search frictions aect the equilibrium in the labor market and create a wedge between the real wage and the marginal productivity of labor. In particular, as in Gertler et al. (2008, henceforth GST), rms post vacancies in every period but do not pay a ow cost for each of them. Instead, they just pay a hiring cost if a match occurs and a vacancy is lled.

The response of macroeconomic variables to aggregate shocks is compared to those that the model generates if either of the two frictions is switched o. The main results are the following. After a contractionary monetary policy shock, the baseline full model (FM) is able to create a stronger and much more persistent contraction in output and employment with respect to a model with only search and matching frictions (S&M). What makes this nding of particular interest, is that such remarkable increase in unemployment occurs without relying on any form of wage rigidity - indeed the real wage reaction is about the same between FM and S&M. Furthermore, while in the external nance premium model (EFP) consumption and employment recovers immediately after the shock, and the slight persistence of output is driven by the sluggish fall in investments, in the FM model the collapse of employment creates an additional channel that drives the heavy response of real variables. A similar argument applies to the reaction to an unexpected technology shock, although in a somewhat dierent manner, for

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reasons that will be clear below.

In addition, the FM model is confronted with the macroeconomic responses to less standard `friction-specic' kinds of shocks. On the one hand, the baseline model's reaction to a rms' net worth shock is compared to that of the EFP model (the S&M model is unaected by this shock). On the other, FM and S&M's reac-tions to a workers' bargaining power shock are computed. In both cases, net worth and bargaining power shocks, the FM model shows a result that is qualitatively similar to the one friction models. Yet, it is characterized by higher volatility and higher persistence of real - but not nominal - variables.

The major intuition underlying these results is the existence of a `risk chan-nel' that goes from the nancial to the labor market and creates a feedback loop that worsens the recessions and delays economic recovery. As a shock aects the creditworthiness of a rm - for example by reducing its return on capital - the bor-rowing contract conditions deteriorate. Therefore, the rm faces a negative shift in the distribution of its payos and a higher default probability. Since labor is a state variable and cannot be adjusted freely, it turns out that such higher risk takes the form of a higher cost of hiring (for a given desired number of employees). In equilibrium, higher hiring cost implies a lower number of new matches and, consequently, a fall in the employment rate. The reduction in the number of work-ers in turn aects the protability and then the nancial constraint rms. That triggers an additional rise in the default risk and in the cost of hiring.

The mechanism presented in this paper is in line with the intuition proposed by Hall (2014). Hall analyzes the determinants of the value to a rm of hiring an additional worker - the so-called job value. He criticizes the standard use of a risk-free interest rate in discounting future cash ows and claims that, as a new worker can be considered a risky asset to a rm, the value of future earnings attributed to that worker should take into account variations in market interest

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rate spreads. Therefore, as nancial stress drives risky interest rates up, the job value may shrink for reasons that are independent from the per-period ow value generated by the worker itself. This paper also links nancial stress to rms' hiring activity. However, while Hall does not take a stand on what drives variations in nancial risk and just take changes in interest rate spreads as given, here a simple and very well know mechanism, i.e. the external nance premium, is used in order to endogenize the capital market dynamics.

The paper proceeds as follows. The next section presents a brief review of the recent models that adopt either nancial or labor market frictions. This is helpful in order to fully grasp the improvement in the performance of the two-friction model over the previous ones. Section 3 describes the model set-up. Section 4 includes the calibration of the model's parameters. Section 5 shows the simulation results and presents a comment of the most relevant features of business cycle uctuations that emerge from the FM, S&M and EFP models. Section 6 concludes.

2 A selected literature review

2.1 The nancial accelerator

Much research has been recently focused on developing business cycle models in which frictions in the nancial markets play a role in the dynamics of output and other real variables. In this section, I only present those papers that are directly related to a BGG-style EFP. However, some relevant results may also be found in models that dier from BGG in terms of both the structure of the nancial intermediation process and the agents aected by the nancial friction2.

2Among the most famous works, Gertler and Karadi (2011) and Gerali et al. (2010) aim at explaining banks' behavior by putting more structure into the lending relationships. The former impose a friction in the lending of households to banks, while borrowing by entrepreneurs occurs

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One of the most direct tests of the importance of external nance premia (EFP) is carried out by Brzoza-Brzezina and Kolasa (2013). These authors equip a standard New Keynesian DSGE model with the main ingredients of BGG. They estimate and evaluate their model using three dierent Bayesian techniques, and then compare it with a frictionless benchmark. They nd that only in one case out of three nancial frictions lead to a higher marginal likelihood. In another recent paper, Del Negro et al. (2013) test the out-of-sample performance of an EFP model with regard to the behavior of ination during the recent nancial crisis. On the one hand, their model is able to replicate the absence of a deationary spiral despite the presence of remarkable economic slackness. On the other, their result rests on the unusually high estimate of the price stickiness Calvo coecient. Mixed results can also be found in Christensen and Dib (2008). Their maximum likelihood estimation shows the high statistical signicance of the elasticity of the EFP with respect to changes in the entrepreneurs' leverage position. However, except for a weak increase in persistence, no improvement in the t to data -attributable to the presence of nancial frictions - can be detected in terms of real variable responses to shocks (this is especially true with respect to output and working hours). Finally, a very interesting result emerges from the work by Christiano et al. (2013). A wide set of shocks is included in a model in which the variance of ln(ω), the natural logarithm of the idiosyncratic disturbance to the return on capital, varies over time and can also be aected by an unexpected shock - the so-called `risk shock'. After estimating the model, the authors show that, from 1980s on, risk shocks have accounted for about 60% of the variability of output - although this result does not hold if the nancial variables are dropped

smoothly. The latter assume that a set of impatient households and the entrepreneurial sector borrow from banks and are aected by a collateral constraint. Frictions in the granting of loans to households are also central to Cúrdia and Woodford (2010).

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out the model.

In light of the evidence reported in the existing literature, two main conclusions can be drawn. First, the framework proposed by BGG is a valuable way to include a frictional nancial system in business cycle models, as it is able to create a link between nancial variables and the real economy. Second, such interactions are not always eective enough in making the moments of the model closer to data. In particular, DSGE models have still some diculties in producing endogenous variations, without relying on excessive shock persistence. In this regard, enriching these models with some sort of rigidity in the labor market provides an additional channel through which dierent kinds of shocks have an impact on the dynamics of aggregate variables. As I will show in detail below, the search and matching framework is a promising step in this direction.

2.2 The DMP labor market

The original DMP model summarized at the beginning of the previous section has provided the successive business cycle literature (Blanchard and Galí (2010), GST, Gertler and Trigari (2009, henceforth GT)) with a tractable framework in which issues related to unemployment can be fruitfully dealt with. Nevertheless, a major challenge to the development of this approach was rst posed by the so-called `Shimer puzzle'. Indeed, Shimer (2005) observed that for reasonable parametriza-tion of the economic environment, the predicted variability of the ratio between the number of vacancies and that of unemployed workers - the so-called labor market tightness - was extremely low. In other words, a shock ten times bigger than those experienced by advanced economies would have been needed in order to account for the actual dynamics of labor market variables. Strictly related to this issue, it turned out that the search and matching framework was associated with an underestimation of the real eects of technology and monetary policy shocks.

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In order to address this shortcoming, three main answers to the Shimer puzzle have been proposed. First, Hagedorn and Manovskii (2008) suggested a widely dierent parametrization of the search and matching model. They started by mea-suring the accounting prots of rms and by deriving - using the Nash bargaining rule - the implied value of workers' nonmarket activity (for a given workers' bar-gaining power, η). Then they calibrated η in order to match the elasticity of wages with respect to productivity. The authors show that the parametrization they pro-pose is able to replicate the volatility of unemployment in the data. However, their result crucially depends on the choice of an extremely high value of nonmarket activity and of a very low η. Such parametrization has been criticized by Hall and Milgrom (2008) and Pissarides (2009) and has not been adopted frequently in later works 3.

An alternative approach has consisted in substituting vacancy posting costs with hiring costs (GST, GT). Indeed, in the original model, as the increase in the job value pushes up vacancies and more workers nd a job, the labor market tightens and the probability of lling additional vacancies shrinks. Therefore, if posting a vacancy is costly irrespectively of whether a new match is created or not, then the search eort rapidly declines and the response in terms of employment is limited. In a dierent fashion, if a rm bears a cost when it hires a new worker -given that a posted vacancy is lled only with some probability - the search eort is not dampened ex-ante and the impact of aggregate shocks on employment is remarkably higher 4.

Finally, starting from Hall (2005), wage stickiness has proved to be a promi-nent ingredient of many of the recent DMP models. Indeed, if wages are not free

3As an example of a search and matching model in which η has been estimated, see GST. They get η = 0.9, which is astonishingly dierent from 0.19 in Hagedorn and Manovskii (2008).

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to adjust after, say, a positive technology shock, the job value increases, more vacancies are posted and more matches are then created. Wage rigidity has been adopted assuming either i) a constant elasticity of wage with respect to changes in technology (Blanchard and Galí (2010), Hall (2005)), ii) a Calvo adjustment mechanism, where only a given fraction of rms is allowed to change the wage of-fered in every period (GST, GT), or iii) an alternating-oer wage bargaining (Hall and Milgrom (2008), Christiano et al. (2013)). In all of these model, wage rigidity is recognized to improve the ability to replicate the response of real variables to aggregate shocks.

The model presented in this paper addresses the Shimer puzzle from a totally dierent perspective. Although it employs hiring instead of vacancy posting cost, the mechanism at work is highly independent from this assumption. Furthermore, wage rigidity of no kind is used in order to boost the response of real variables. At the same time, the driving channel in the paper holds irrespectively of the nonmarket value to workers and of their bargaining power 5.

Financial frictions introduce a new channel that is not directly related to the volatility of the share of matching surplus that accrues to rms. Indeed, tighter nancial conditions increase the hiring cost for given wage level and employment level. The next section presents the model set-up and describes in detail the linkage between capital and labor markets.

5As a matter of fact, a calibration in line with Hagedorn and Manovskii (2008) generates a stronger reaction of employment and weaker uctuations of real wages. However, as this is true for both FM and S&M, the additional eect of nancial variables is almost unchanged, whatever the choice of those parameters.

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3 The model set-up

A fully microfounded description of the model economy would require that both capital and labor were state variables to rms. However, that assumption would imply that capital and labor were employed in dierent proportion by rms, and a time-varying distribution of marginal returns would emerge in the economy. The presence of heterogeneity would heavily hinder aggregation of rm-level variables and would prevent from the analysis of economy-wide shocks.

As a simplifying assumption, I assume that each of the innitely many rms in the economy has, in every period, a given amount of net worth that evolves according to both the rm's performance and aggregate conditions. At the same time, workers can freely move across rms but are aected by search and matching frictions in moving in and out of employment. Therefore, while free labor mobility allows for the return on each production input to be the same across rms, un-reasonably high changes in hiring and ring within two successive periods - still possible in BGG - are ruled out the model.

More in detail, the model economy includes ve dierent kinds of agents: house-holds, nancial intermediaries, an entrepreneurial sector, capital producers, the public sector. In what follows, I deal with the behavior of each of them and stress the mechanism through which the nancial market aects rms' labor demand.

3.1 The representative household

The economy is populated by a representative household comprising a con-tinuum of members of measure one. Irrespectively of whether she is employed or unemployed in a given period, each member of the household consumes the same amount of nal good6. The household maximizes its life-time utility

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E0 ( X t=0 βt " lnCt− %Lt #) (3.1) where Ctis real consumption, Ltis the fraction of household members who are

employed in period t, and % is a coecient of labor disutility.

Notice that the labor supply is allowed to vary on the extensive margin only. Shimer (2004) shows that this assumption is highly realistic at business cycle fre-quencies. Also, Christiano et al. (2011, henceforth CTW) nd out that uctuations on the extensive margin explain four fth of the variability in labor supply. There-fore, in line with Shimer (2009), I implicitly assume an innite Frisch elasticity and a constant disutility of labor - for a given consumption level 7.

The household has access to a risk-free asset Dt+1 whose nominal return is

Rt+1. It earns real wage WPtt and unemployment benet Ub provided by those who

work and do not work, respectively, the prots from the rms they own, ∆t, and

a lump sum net transfer from the public sector, Tt. Therefore, the household's

budget constraint is Ct+ Dt+1 PtRt+1 = Wt Pt Lt+ (1 − Lt)U b + Dt Pt + ∆t+ Tt. (3.2)

From the rst order condition with respect to Ct and Dt+1 it follows that

1 Ct

= Λt (3.3)

and

7Search and matching models do not typically provide a thorough specication of the value household members get from leisure. Most of them simply include it in a comprehensive index of nonmarket activity. The results presented in this paper do not depend on the choice of a particular value of the Frisch elasticity.

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β Et ( Λt+1 Λt ) = Et ( Πt+1 Rt+1 ) , (3.4)

where Λt is the marginal utility of consumption at time t.

3.2 The demand for capital

The entrepreneur-intermediary lending relationship, described in this section, strictly follows BGG.

The economy is populated by innitely many entrepreneurs. Each of them uses her net worth Nt+1 and borrows some funds Bt+1 in order to buy an amount of

capital Kt+1 at price Qt in terms of nal goods

QtKt+1j = N j t+1+ B

j

t+1. (3.5)

For reasons that will be clear below, the return on capital RK

t+1 is equal across

rms and is taken as given in the choice of capital. At the end of every period, before repayment occurs, the actual return on capital becomes equal to ωj

tRKt+1,

where ωj

t is an idiosyncratic iid distrurbance with c.d.f. F (ω) and E{ω} = 1, that

satises8

∂(ωh(ω))

∂ω > 0, h =

dF (ω)

1 − F (ω). (3.6)

Firms borrow funds from a perfectly competitive nancial intermediary sector that has access to a risk-free bond market. Let Zj

t+1 be the interest rate paid to

an intermediary by rm j. Thus, it is possible to pin down a value ¯ωtj such that

¯ ωt+1j RKt+1QtKt+1j = Z j t+1B j t+1. (3.7)

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Equation (3.7) shows that when ωj

t+1> ¯ω j

t+1 the entrepreneur is able to repay

the loan and earns some net prot equal to ωj

t+1RKt+1QtKt+1j − Z j t+1B j t+1. To the contrary, when ωj t+1 < ¯ω j

t+1, the entrepreneur defaults and loses all her wealth.

In that case, the intermediary is able to appropriate nothing but a fraction of the gross return on capital and gets (1−µ)ωj

t+1Rt+1K QtKt+1j . It is easy to see that F (¯ωt)

is the default probability at time t.

Before dealing with the entrepreneurs' optimal choice of capital, a few more words on the structure of the loan contract are needed.

The loan contract is characterized by both aggregate uncertainty, related to the state of the economy, and idiosyncratic uncertainty, attributed to the variability of ωj. Since the idiosyncratic disturbance is iid and there are innitely many

rms in the economy, the intermediaries are able to diversify that risk. As for the aggregate risk, the contract is designed so to guarantee a xed state-contingent repayment to the intermediary. Dening G(¯ωj) =´ω¯j

0 ωdF (ω)and being Γ(¯ω j) =

[1 − F (¯ωj)]¯ωj+ G(¯ωj)the share of nished capital going to the intermediary, the free entry condition in the intermediary sector implies

[1 − F (¯ωjt+1)]Zt+1j Bt+1j + (1 − µ) ˆ ω¯jt+1

0

ωdF (ω)RKt+1QtKt+1j = Rt+1Bt+1j . (3.8)

or, putting (3.5) and (3.7) into (3.8), Γ(¯ωt+1j ) − µG(¯ωt+1j ) RK

t+1QtKt+1j = Rt+1(QtKt+1− Nt+1). (3.9)

In other words, the entrepreneur bears all the aggregate risk and oers the intermediary a schedule of ¯ωj

t+1, one for each realization of RKt+1.

If the condition (3.6) holds, Γ(¯ωj

t+1) − µG(¯ω j

t+1)is increasing in ¯ω j

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neigh-borhood of the value of ¯ω that maximizes the expected entrepreneur's prot 9.

Therefore, when RK

t+1 is lower than expected, both ¯ω j

t+1, the default probability

and the cost of external nance go up 10.

Given the contract design, each entrepreneur chooses her capital Kj

t+1 and the

optimal contract ¯ωj

t+1, so to maximize her expected return

Et ( Rt+1K QtKt+1j (1 − Γ(¯ω j t+1)) ) (3.10) subject to (3.9). From the rst order condition, dening ςt = E

( RK t+1 Rt+1 ) , the following key relation can be derived

QtKt+1j

Nt+1j =

QtKt+1

Nt+1

= ψ(ζt), ψ(1) = 1, ψ0(.) > 0. (3.11)

Equation (3.11) states that the nancial leverage chosen by the entrepreneur - QtKt+1

Nt+1 - is an increasing function of the spread between the expected return on

capital and the risk-free rate11.

Notice that as ψ(ςt) is the same for each entrepreneur, the nancial leverage,

¯

ωt+1 and Zt+1 are the same as well.

Such homogeneity in entrepreneurs' behavior, together with the iid nature of the idiosyncratic disturbance, allows to determine the dynamics of aggregate net worth. In order to assure that entrepreneurs do not accumulate so much net worth to fully nance their capital expenditure using only internal funds, a fraction (1−γ) is assumed to exit the market in every period, and to rebate net worth to household

9That means that the increase in the amount of the repayment triggered by higher ¯ω more than compensates the rise in the default probability.

10A state-contingent debt contract may be justied through the assumption that the debt is, at least partially, rolled over before the debt repayment occurs but after the realized return on capital is known (BGG).

11Etωj

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12. At the same time, an equal amount of entrepreneurs enter the market with zero

net worth, and both incumbents and entrants receive lump sum trasfer - l - from households. Thus, Nt+1 = γ " RKt Qt−1Kt− Rt+ µ´ω¯ j t 0 ωdF (ω)R K t Qt−1Kt Qt−1Kt− Nt ! (Qt−1Kt−Nt) # +l (3.12) where µ´¯ω j t 0 ωdF (ω)RKt Qt−1Kt

Qt−1Kt−Nt is the external nance premium.

3.3 The labor market

At the beginning of every period, the total number of unemployed workers is equal to

Ut= 1 − Lt−113. (3.13)

New hirings occur according to the CES matching function Mt= σmUtσV

1−σ

t . (3.14)

where V is the number of vacancies posted by the aggregate entrepreneurial sector and σm stands for the matching technology. Therefore, the probability of

nding a job is St = Mt Ut (3.15) 12Entrepreneurs live 1 1−γ periods on average.

13Following Pissarides (2009), changes in the size of the workforce are assumed to be negligible along the business cycle.

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and

Xt=

Mt

Lt−1

(3.16) is the economy hiring rate. A standard assumption in search and matching models is that a fraction (1−ρ) of workers exogenously separate at the end of every period. Here, there is also a fraction of workers, F (¯ωt−1), that become unemployed

as the rms they work for default. Endogenous capital market driven layos dene a direct linkage between nancial and labor market frictions. Thus, the dynamics of employment is described by

Lt= (ρ − F (¯ωt−1) + Xt)Lt−1. (3.17)

3.4 The return on capital and the labor demand

Firms produce an homogenous intermediate good adopting the production function

Yt= AtL1−αt K α

t (3.18)

where

εa,t= ρaεa,t−1+ ςa,t, εa,t= lnAt (3.19)

and ςa,t ∼ N (0, σa) is an i.i.d technology shock.

The (gross) return on capital is given by the sum of a share of current output and the capital gain raising from the dierence between the selling and buying price of capital

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RKt = (Φ∗ t Kt + (1 − δ)Qt Qt−1 ) , (3.20)

where Φ∗ is the maximum amount of prot attainable by the entrepreneurs.

It is determined at the beginning of period t, before ω is realized, through the formulation of rms' demand for labor. Since workers are allocated smoothly from one rm to another one, and matching frictions only aect the entrepreneurial sector as a whole, the labor demand is proportionally equal across rms and can be dealt with in aggregate terms.

The entrepreneurs sell intermediate goods, Yt, at real price Ptw, pay wage bill Wt

PtLt and face hiring cost

ϕXt2

2 Lt−1. Thus, they choose labor in order to maximize

Φ = PtwYt− Wt Pt Lt− ϕX2 t 2 Lt−1 (3.21)

subject to (3.17). The third term on the right-hand side can be seen as an administrative cost or some expenses related to job training 14. The rst order

condition, i.e. the labor demand, reads (1 − α)PtwYt

Lt

= Wt Pt

+ ϕXt. (3.22)

Plugging (3.22) into (3.21) determines the maximum share of output that re-munerates capital

Φ∗t = PtwαYt+ ϕXtLt−

ϕX2 t

2 Lt−1. (3.23)

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3.5 The wage contracting line

In standard real business cycle models, since labor contracts last only one pe-riod, labor supply is fully determined by the marginal utility of leisure. Instead, when search and matching frictions exist, each household member takes into ac-count the life-time disac-counted stream of utility rising from being employed,Wft, or unemployed, Uet. The rst one is

f Wt= Wt Pt − Ξ Λt + β E t ( Λt+1 Λt ((1 − ρ + F (¯ωt) eUt+1+ (ρ − F (¯ωt))fWt+1) ) (3.24) while the second one is

e Ut= U b + β Et ( Λt+1 Λt (St+1fWt+1+ (1 − St+1) eUt+1) ) . (3.25) f

Wt is given by the sum of the current period real wage, the marginal disutility

of work and the expected discounted future value of employment and unemploy-ment, weighted by the respective transition probabilities (ρ is the share of workers that do not separate while S is the probability of nding a job). Similarly, Uet is

the unemployment benet augmented by the weighted expected discounted future values of employment, Wft+1, and unemployment, Uet+1.

Dene Ht(Wt) = fWt − eUt and let Jt be the job value to the entrepreneurial

sector. ηt is the time-varying workers' bargaining power that evolves according to

ηt = ηεb,t, εb,t= ρrεb,t−1+ ςb,t, (3.26)

where ςb,t ∼ N (0, σr)is an i.i.d bargaining power shock.

The Nash contracting rule, dividing the total surplus between workers and rms, states

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ηtJt(Wt) = (1 − ηt)Ht(Wt). (3.27)

Following GST, Jtis the change in the value of rms' from having an additional

worker, after hiring costs have been paid, i.e. considering Xt as xed. From (3.12)

Jt = ∂Nt+1 ∂RK t ∂RK t ∂Lt = ϕXt  (1 − Γ( ¯ωt)) + (1 − F (¯ωt))(Γ(¯ωt) − µG(¯ωt))2RKt KtQt−1 Rt(Qt−1Kt− Nt)(Γ0(¯ωt) − µG0(¯ωt))  . (3.28) Substituting for the denition of Ht into the Nash bargaining rule gives the

`wage contracting line'

Wt Pt = % Λt + U b + ηt 1 − ηt  Jt− β Et  Λt+1 Λt Jt+1(ρ − F ( ¯ωt) − St+1)  , (3.29)

where Jt is dened as in (3.28) 15. Interestingly, the job value is decreasing

in ¯ω, i.e. it is lower when credit conditions are tighter. Therefore, without wage rigidities, an increase in ¯ω drives down the equilibrium real wage.

3.6 Capital producers

Capital producers buy nal goods and transform them into investment goods, i.e. new physical capital, sold at price Qt. They maximize

15Firms' are assumed to take into account the eect of changes in RK

t on the value of ¯ωt- the second term in brackets. Dropping that component of the job value, the second term in square brackets, does not aect the results.

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Qt[Kt+1− (1 − δ)Kt] − It (3.30)

with respect to It, subject to

Kt+1 = (1 − δ)Kt+  1 −χ 2  It It+1 − 1 2 It, (3.31)

where χ is the investment adjustment cost coecient. The rst order condition, determining the real price of capital, is

1 = Qt  1 − χ 2  It It−1 − 1 2 − χ  It It−1 − 1  It It−1  . (3.32)

3.7 Final good producers

The economy is also populated by innitely many retailers that buy intermedi-ate goods from entrepreneurs and sell nal goods in a monopolistically competitive market. Adopting the Dixit-Stiglitz aggregator for prices and nal goods, it can be proved that each retailer k faces the following demand curve for its own product

Yt(k) = " Pt(k) Pt #−τ Yt, (3.33)

where τ is the elasticity of demand for each nal good Yt(k) with respect to

Pt(k). During each period, a fraction (1 − ξ) of retailers is allowed to adjust prices

optimally while the remaining fraction charges the previous period's price. Com-bining the behavior of adjusting and non-adjusting retailers, the standard (log-linearized) Phillips curve under Calvo pricing holds

ˆ Πt = β Et{ ˆΠt+1} + (1 − βξ)(1 − ξ) ξ ( ˆP w t ). (3.34)

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3.8 The public sector

The central bank runs the following Taylor-rule monetary policy ln Rt R ! = ρπln Πt Π ! + ρyln Yt Y ! + εr,t, (3.35) where εr,t= ρrεr,t−1+ ςr,t, (3.36)

and ςr,t ∼ N (0, σr) is an i.i.d monetary policy shock.

At the same time, government spending is Gt=  1 − 1 εg,t  Yt (3.37) where εg,t= ρgεg,t−1+ ςg,t, (3.38)

and ςg,t ∼ N (0, σr) is an i.i.d government spending shock.

Government also provides households with lump sum transfer Tt and

unem-ployment benet Ub, and maintains a balanced budget constraint

Tt+ (1 − Lt)U b + Gt= 0. (3.39)

3.9 The resource constraint

The total amount of output produced in each period is divided into consump-tion by the household, investment by capital producers, government spending, adjustment cost in the hiring activity, monitoring cost by the nancial interme-diaries (related to the share of entrepreneurs that default). Thus, the economy

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resource constraint is Yt = Ct+ It+ Gt+ ϕX2 tLt−1 2 + µQt−1KtR K t ˆ ω¯t−1 0 ωdF (ω). (3.40)

4 Calibration

The model is calibrated on a quarterly basis. The values of economic parameters are in line with those in the existing literature and aim at replicating long run averages in the data (Table 1). In this section, I briey discuss the choice of some parameters related to the nancial and labor market aspects of the model, in as much as they inuence the responses of macroeconomic variables to aggregate shocks.

In the external nance premium literature, the value of parameters governing the nancial accelerator mechanism do not vary much across dierent models. In this paper, I set χ = 20 as in Carlstrom and Fuerst (1997), γ = 0.97 as in BGG, µ = 0.2, which very close to 0.215 in Christiano et al. (2013, henceforth CMR) and somewhat midway between 0.12 in BGG and 0.25 in Carlstrom and Fuerst (1997), l = 0.005 as in CMR. Furthermore, the distribution of the idiosyncratic disturbance omega is assumed to be log-normal and such that the standard error of ln(ω) = 0.26 - equal to the estimate in CMR and slightly lower than 0.28 (BGG). It follows that in steady-state the ratio K

N is equal to 2.1, a little higher that 2

(BGG), while the default probability is 0.9%, within the range 0.74 (BGG) and 0.974 (Carlstrom and Fuerst (1997)). Furthermore, the steady state ratio I

Y = 0.25

and C

Y = 0.54 (both equal to the values in CMR).

As for labor market parameters, the workers' steady state bargaining power, η, is set equal to 0.5 - like in Shimer (2004) and Pissarides (2009) - and close to

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0.54 (Hall and Milgrom (2008)) and to 0.61 (the estimate in GST under perfectly exible wages). Furthermore, I set σ = 0.5, as in almost all the DMP models mentioned above, σm = 1 - such that the steady state ratio UV is 2.04, which is

very close to 2, the value in Hall and Milgrom (2008), and somewhat higher than 1.42 in Pissarides (2009). Ub is equal to 25% of the steady state real wage, as in Hall and Milgrom (2008), while % is equal to 0.44.

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Parameter Value Parameter name

β 0.99 household discount factor

α 0.33 power on capital in the production function

χ 20 investment adjustment cost

δ 0.025 capital depreciation

τ 10 elasticity of demand for nal goods

ξ 0.75 Calvo price parameter

ρs 0.8 Taylor rule inertia

ρπ 5.5 Taylor rule response to ination

ρy 1 Taylor rule response to output gap

γ 0.971 share of net worth rebated to the household

µ 1 monitoring cost

std, ln(ω) 0.26 uncertainty about idiosyncratic disturbance

l 0.005 lump sum transfer from household

η 0.5 workers' bargaining power

% 0.44 disutility of labor

σ 0.5 elasticity of matches to unemployment

σm 1 matching technology

ϕ 11.33 hiring cost coecient

1 − ρ 0.96 separation rate

Table 1. Model Calibration - Economic Parameters

It follows that the total steady state value of nonmarket activity is 57% of steady state real wage 16. Then, setting ϕ = 11.3352 implies a steady state share

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of hiring cost equal to 0.33% of steady state output. This value is lower than the value in Blanchard and Galí (2010), 1%, but, once again, it helps to pursue a more conservative departure from standard New Keynesian models. Finally, ρ = 0.9667 as in Shimer (2005), and consistent with a steady state job nding rate S = 0.7 (Blanchard and Galí (2010)) and steady state unemployment U = 0.057 - the long-run average in the US and the value adopted by almost all the papers previously mentioned.

5 The model simulation

In this section, I show the reaction of the model economy presented to far (the so-called full model, FM) to four aggregate shocks. In addition, I compare it with a model in which the nancial friction is switched o (search and matching model, S&M) and with one in which the labor market is perfectly competitive (external nance premium, EFP). On the one hand, the S&M model is obtained by keeping ¯

ω xed at its steady state level and by ruling out the bank parity condity (3.9). In this framework, the realized return on capital and the contract terms are no longer linked to each other, and the nancial accelerator mechanism does not hold whatsoever. On the other hand, in the EFP model, all workers separate at the end of every period and rms pay no hiring costs 17 (i.e. the labor wedge is constantly

equal to 0).

nonmarket activity varies along the business cycle. In particular, it is higher the smaller the marginal utility of consumption. Furthermore, although much literature set that value close to 70% of the real wage (GST, Hall and Milgrom (2008)), employing a similar assumption in this paper would, if any, increase the eect of aggregate shocks of real variables. Therefore, the approach I take here generates substantial variation in employment for conservative choice of parameters.

17In switching to the EFP model I try to keep the model as similar to FM as possible. The only change I make is setting % = 0.47 and an implied steady state ratio I

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5.1 Impulse response functions

A shock that hits εi rises or lowers ςi by one standard deviation of its

distribu-tion. The standard deviations and autocorrelations of shocks are included in Table 2. As a standard procedure, the former are chosen in a way that allows to replicate

ˆ a 1% maximum increase in output (technology shock)

ˆ a 25 basis point increase of the nominal interest rate on impact (monetary policy shock)

ˆ a 1% decrease in net worth (net worth shock)

ˆ a 1% increase in workers' bargaining power (bargaining power shock) As for the shock autocorrelations, ρa is set equal to 0.95, a standard value in

the literature, ρr = 0.241(SW), ρb = 0.26 (GST), ρn = 0.818(CTW).

Figure 1-4 show the impulse response functions of the full model (black line), the S&M model (green dotted line) and the EFP model (red broken line). The nominal interest rate, the return on capital, and ination are expressed in terms of annual basis points. All the other graphs show percentage deviations from steady state values.

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Parameter Value Parameter name

ςa 0.012 std, technology shock

ςr 0.00065 std, monetary policy shock

ςb 0.02 std, bargaining power shock

ςn 0.0015 std, net worth shock

ρa 0.95 autocorrelation, technology shock

ρr 0.241 autocorrelation, monetary policy shock

ρb 0.26 autocorrelation, bargaining power shock

ρn 0.818 autocorrelation, net worth shock

Table 2. Model Calibration - Shocks

When a monetary policy shock hits the economy (Figure 1), output, ination, employment and consumption fall on impact by almost the same amount in FM and S&M, and almost twice as much in EFP. The absence of a fully-edged hump-shaped response of output is due to the lack of any sources of real rigidities in consumption, like the most common `internal habit'. As expected, the impact on investment and on net worth is much bigger and persistent in those models, FM and EFP, in which the nancial accelerator holds - the eect on EFP being a little stronger than on FM. However, the most remarkable dierence across models consists in the path of recovery of output, consumption and employment. Indeed, while in EFP and S&M the fall in those variables is shortlived (10 quarters) -except for a small medium run decline in output in EFP, driven by the lagged fall in investment - in the full model employment recovers much more slowly (after almost 25 quarters) and that drives down consumption and output. The source of the stronger impact of a monetary policy shock on real variables lies in the interaction between nancial and labor market frictions.

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Figure 1:Response to a 25 basis point contractionary monetary policy shock

In the full model, the rise in the interest rate triggers a spike in the cost of external nance - proxied by ¯ω - which in turn aects the labor market in two ways. First, since the job value shrinks, the wage contracting line pins down a lower equilibrium real wage level. Yet, since the reaction of wage is very similar to that in the S&M model, in which ¯ω is xed, this eect proves to be negligible 18.

Second, since the default probability rises, the total hiring cost, for a given number of employees, goes up as well. This is clear by plugging (3.17) into (3.22). Therefore, in equilibrium, the number of employees demanded by rms reduces in rst place. In a search and matching model, that implies a persistently lower employment level, notwithstanding a higher hiring rate in successive periods. Since this transmission channel is not active is the S&M model, the latter is not able to replicate the size and persistence that characterize the uctuation of unemployment in the full

18Notice that the wage level in FM is always slightly lower or equal to that in the S&M model. That means that the stronger response of employment is not driven by the emergence of any form of wage rigidity.

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model 19.

Figure 2:Response to a 1.2% technology shock

Figure 2 shows the response to a stationary technology shock. The qualitative behavior of variables is in line with most of the literature and is equal across models. Once again, the full model generates stronger and more persistent dynamics in terms of output, consumption and employment. Notice that although nancial conditions improve, employment slightly reduces on impact. This is due to the standard wealth eect that allows households to benet from a higher consumption level while oering less work in the labor market. After this transitory stage, employment goes up dramatically more in FM than in EFP and S&M. This is so because of the increase in labor demand boosted by the fall in ¯ω, i.e. in the default risk. Hiring becomes less costly and rms react by increasing the number of employees. Interestingly enough, the dierence between the green and the black lines in the IRF of employment is similar to that in GST between their exible

19At the same time, the perfectly competitive labor market in the EFP model quickly adjusts as the shock dies out.

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wage model and their formulation with staggered Nash bargaining (Figure 4 in GST). That is, the interaction with nancial frictions is able to trigger real eects after aggregate shocks, as if wages were not free to adjust in every period. At the same time, the model presented here is not able to generate the rather mute response of ination that can be found in the data and in the sticky wage GST model.

Furthermore, taking a dierent perspective, while in standard BGG-style mod-els the nancial accelerator aects the economy mostly through its impact on investment and net worth, here it enacts an additional channel that inuences the labor market as well.

Figure 3:Response to a 0.15% net worth shock

In order to go deeper in the comparison to the EFP and the S&M model, I introduce two additional shocks that are closely related to the external nance premium and search and matching descriptions of the business cycle. The rst one is a contractionary shock to net worth. As it creates no endogenous variation in the S&M model, I just compare EFP to FM. The second one is a positive

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workers' bargaining power shock. Of course, this shock is meaningless in a perfectly competitive labor market.

Figure 4:Response to a 2% bargaining power shock

A contractionary net worth shock (Figure 3) reduces the demand for and price of capital, and that in turn drives down investment. If the labor market is friction-less, the reaction of macroeconomic variables in driven by the capital market and the contraction of output is weak and delayed - essentially it traces the collapse in investment. In a dierent fashion, with search and matching frictions, the rise in credit risk and in the size of loan repayments increase the cost of hiring and puts negative pressure on employment and output. Therefore, the nancial accelerator is even more eective.

The previous arguments also hold with respect to a positive workers' bargain-ing power shock (Figure 4). The latter drives up wages and reduces the share of prot that accrues to entrepreneurs. While a lower return on capital has no im-plications for the borrowing conditions of rms in the S&M model, it gives rise to a nancial accelerator mechanism under variable default probability. Therefore,

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a small change in the labor market alters the equilibrium in the capital market which in turn aects employment according to the hiring cost channel discussed so far. Thus, not only the fall in investment and in the price of capital is stronger in FM than in the S&M model, but also it has greater impact on the economy for a given percentage deviation.

6 Concluding Remarks

This paper shows how the interaction of nancial and labor market frictions generates business cycle dynamics that dier from those of stand-alone one-friction models. More specically, the eects of aggregate shocks on employment and out-put are remarkably higher and more persistent in a model equipped with both frictions. The main force driving this result is the higher equilibrium hiring cost associated with periods of tighter credit market conditions and higher probabil-ity of default. With respect to the existing search and matching literature, the model presented here has proved to be able to amplify the real eect of aggregate shocks, even without resting on any sort of wage rigidity. At the same time, the contribution of the full model to the literature on nancial frictions is twofold. First, introducing the DMP framework allows to deal with issues related to hiring, unemployment and job searching activity. Second, the interaction with the labor market provides an additional channel - beyond investment - through which the nancial accelerator aects the dynamics of output.

The model described in this paper puts together, in a very simple way, two of the best-known streams of literature about the business cycle. Future research may attempt to enrich this model along many directions. In particular, as the coexis-tence of two state variables - capital and labor - creates endogenous persiscoexis-tence in the dynamics of real variables, a detailed microfoundation of the model

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econ-omy, including heterogeneity and state-dependency, is likely to be benecial for grasping the features of aggregate uctuations. In this regard, the present model can be easily extended to incorporate multiple sets of rms, such that workers are free to move within a given set but face some search frictions in moving to rms belonging to dierent sets20. In that framework, the (dierence among the)

distri-butions of capital, labor and net worth would aect aggregate input demands and aggregate production. Finally, and still related to this point, the degree of uncer-tainty surrounding the agents' choices crucially inuence the eect of unexpected shocks. In this paper, I assume that, while capital demand is a function of the expected future state of the economy, hiring is chosen after aggregate conditions are realized, and is allowed to absorb part of the eect of the shocks. Therefore, binding employment to be a predetermined variable would probably enhance the macroeconomic impact of even very small aggregate shocks.

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Brzoza-Brzezina, M. and M. Kolasa (2013). Bayesian evaluation of dsge models

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Carlstrom, C. T. and T. S. Fuerst (1997). Agency costs, net worth, and busi-ness uctuations: A computable general equilibrium analysis. The American Economic Review, 893910.

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