Credit Frictions and Labor Market Dynamics ∗

Atanas Hristov ¶

DIW Berlin / Humboldt Universität zu Berlin

October 15, 2008PRELIMINARY WORK

Abstract

The paper develops a model in which two real frictions are embedded into an otherwiseconventional New Keynesian model: a labor search problem in the labor market anda costly state verification problem in the credit market. The first friction allows forendogenous equilibrium unemployment while the latter introduces riskiness of capitalmanagement. The claim of this paper is that credit markets may play an important rolefor the dynamics of the labor market. To illustrate the above proposition, we presenta simple framework to analyze the joint behavior of access to credit and labor marketprices. We consider the responses of unemployment in the economies to a monetaryshock, a technology shock and a government consumptions shock. Differences in creditmarkets can play role for propagating shocks that originate outside the financial sectorthat and in turn increase the persistence and impact of labor market variables.

JEL: J64, G24, E51Keywords: Credit and search frictions, unemployment, monetary policy

1 Introduction

In the last decade the New Keynesian paradigm gained a huge attention and popularity in themodern macroeconomic research. Leading central banks institutions like European CentralBank or Federal Reserve Board adopt DSGE modelling strategies for their policy analysisand forecasting purposes. In particular, the modelling approach proposed by Christiano et al.∗We are grateful to Sven Blank, Olaf Fuchs, Martin Hillebrand, Sean Holly and Vladimir Kuzin for their

comments. This document has been produced with the financial assistance of the European Union withinthe context of the FINESS program. The contents of this document are the sole responsibility of the authorand should not be regarded as reflecting the position of the European Union.¶DIW Berlin, Mohrenstraße 58, 10117 Berlin, Germany, E-Mail: ahristov@diw.de.

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(2005) and characterized by the inclusion of a relatively large range of shocks and extensiveuse of Bayesian methods became an important reference point for a large series of recentacademic papers.

However, the poor fit of estimated dynamic general equilibrium models to empiricaldata remains a notorious problem in the academic research since the introducing of RBCframework in the seminal work of Kydland and Prescott (Nov., 1982). This important short-coming leads to extending the basic structure of standard medium scale New Keynesianmodels typically including i.a. different sticky prices and wages features, adjustment costsin investment, habit persistence in preferences, see for example Christiano et al. (2005) fora reference. Bernanke et al. (1999) introduce credit market frictions in the standard NewKeynesian setup. Their results stress an important influence of endogenous development incredit markets on business cycle dynamics. The features proposed by Bernanke et al. (1999)are also integrated in the model of Christiano et al. (2007) and Queijo von Heideken (2008).They conclude that shocks originating in the financial sector accounts for a significant portionof business cycle fluctuations.

Another series of papers concentrates on integration of labour market frictions similar toMortensen and Pissarides (1994) into a standard New Keynesian model and it is argued thatthis art of friction is crucial to modelling business cycle fluctuations as well as propagationof monetary policy shocks. This branch of the literature includes Trigari (2004), Christoffeland Linzert (2006), Trigari (2006) and others. Their findings are that the presence of searchand matching frictions in a New Keyenesian model may account to better understanding ofthe empirically observed inflation inertia.

Last not but least, the impact of frictions in financial markets on the employment dynam-ics is of independent interest for many macroeconomic researches as well as policymakers.Acemoglu (2001) argues that credit market frictions may be important contributor to highunemployment in Europe. In particular, Acemoglu (2001) stresses the differences betweenEurope and the U.S. In contrast to the U.S., job creation in Europe is constrained by creditmarket imperfections, so unemployment possesses persistent dynamics. Wasmer and Weil(Sep., 2004) develop a theoretical model involving credit and labor market restrictions, anddemonstrate that both types of market imperfections can interact in a complementary wayexplaining pronounced differences in the unemployment dynamics between Europe and theU.S.

The aim of our paper is to investigate the link between credit market frictions andemployment dynamics in the New Keynesian DGSE framework. We answer the followingquestion: what role do financial markets play in the propagation of non-financial marketshocks and how these shocks impact the labor market. Our starting point is the model ofBernanke et al. (1999) involving credit market frictions. We extend the model by integratinglabor market search and matching frictions similar to Trigari (2006).

We find that financial frictions play an important role in amplifying the transmission of

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shocks that move output and price in the same direction. Monetary policy shocks fall intothis category. This means that restrictive monetary policy slows down employment creationfor a longer period and to a higher extent in an environment where monitoring costs onentrepreneurs’ activity and riskiness of entrepreneurial projects is higher. They mitigate theeffects of other shocks.

The outline of the paper is as follows. In the next section we present our theoreticalframework. Section 3 lists some of our considerations about how our theoretical frameworkrelates to the real economy and how this can be useful for policy purposes. Section 4 discussesthe importance of the financial sector for the real economy. Section 5 concludes. Varioustechnical details are relegated to appendices.

2 The Model Economy

Our model is an extension of a closed economy DSGE model from Bernanke et al. (1999)and adopts the credit sector structure in Christiano et al. (2004), without including the"debt-deflation" effect, an explicit banking sector, and other real and nominal frictions ofthe latter set-up. The key difference between their models and ours is the inclusion of alabor market subject to search frictions. Our model features search and matching frictionsalong the lines of Pissarides (1988) and Mortensen and Pissarides (1994) with exogenousjob destruction. As a result, we assume that the economy is characterized by the followingrigidities: price stickiness, capital adjustment costs, habit formation, credit market frictions,and search frictions. We also assume the economy is disturbed by three transitory shocks:technology, monetary policy, and government demand.

The model is composed of households, production agents/firms, and a government au-thority. There are six types of production agents/firms: retailers, wholesale producers, capitalproducers, entrepreneurs, financial intermediaries and labor agencies. At the beginning ofeach period, households supply labor and entrepreneurs supply capital to homogeneous fac-tor markets. Wholesale firms produce differentiated goods after purchasing labor servicesand renting capital from the factor markets. Retailers buy the differentiated wholesale goodsand then bundle the goods into final goods. The output produced by retail firms is usedfor consumption and investment. Capital producers combine investment goods with usedcapital, purchased from entrepreneurs, to produce new capital. The new capital is then soldto entrepreneurs 1. Entrepreneurs make new capital purchases using their own resources, aswell as bank loans from financial intermediaries that convert household deposits into busi-ness financing. The presence of asymmetric information between entrepreneurs and lenderscreates a credit friction which makes entrepreneurial demand for capital depend on their fi-nancial position. Changes in the supply of or demand for capital induces the price of capital

1Having the entrepreneur repurchase her entire physical capital stock is a device to insure that financialconstraints apply to the entrepreneur as a whole, and not just to her last marginal investments.

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to fluctuate and further propagate shocks. Labor survices are supplied to homogeneous labormarket by labor agencies. Each labor producer retains a large number of workers. In thiseconomy, the presence of search frictions in the labor market prevents some jobseekers fromfinding jobs and some vacant positions from being filled in each period. Wholesale firms setnominal prices in a staggered fashion a la Calvo (1983). This nominal rigidity gives monetarypolicy a role in this model.

Our model differs from Bernanke et al. (1999) in its characterization of monetary policyby a modified Taylor-type rule. We assume that the Federal Reserve manages short-terminterest rates in response to inflation and output.

2.1 Search and Matching Process

The labor market is subject to search frictions. To form new employment relationships firmsmust post vacancies. Workers do not face job-finding costs. The matching technology whichconverts unemployed workers and free vacancies into matches is given by:

lt = luψMt v1−ψMt , 0 < l, 0 ≤ ψM ≤ 1, (1)

where lt denotes the number of job matches created in period t, ut represents the size ofthe unemployment pool in period t, vt is the total number of vacancies created by the firmsin period t, l governs the efficiency of the matching process, and ψM is the elasticity ofthe matching function with respect to unemployment. After normilizing the labor force toone, ut also gives the unemployment rate, ut = 1 − nt, where nt denotes employed workersin period t. Vacancies and unemployed agents are randomly matched with each other. Theassumption of constant returns to scale matching techology is supported by Petrongolo andPissarides (2001). It implies that on average with endogenous probability ft = lθ1−ψM

t workerschange their status from unemployment to employment in period t, while with endogenousprobability st = lθ−ψMt firms turn their unfilled vacancies into filled ones in period t. Theratio of vacancies to unemployment, θt = vt

ut, is the labor market tightness in period t. In a

stationary environment, the above probabilities define the mean duration of unemploymentand vacancies, respectively. The labor law of motion is given by:

nt+1 = (1− χ) (nt + lt) , 0 ≤ χ ≤ 1, (2)

where χ is the constant exogenous separation rate. A flow (1− χ) of nt employed workerscontinues working in the next period, while a flow (1− χ) of lt matches survives into thenext period.

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2.2 Representative Household

There is a continuum of infinitely-lived households distributed uniformly on the unit interval.The representative household itself consists of continuum of family members indexed byi ∈ [0, 1]. There is a measure ni,t of employed individuals in the household and a measureui,t of unemployed individuals. A typical household’s lifetime utility specification is given bythe utility function:

E0

∞∑t=0

βt {Ut (ct, ct−1)−K (hi,t)ni,t} , 0 < β < 1, (3)

where Ut (ct, ct−1) denotes household’s utility from consumption. Household consumptionin period t is given by ct. Hours worked in period t by the i employed family member aregiven by hi,t. The E0 symbol denotes the expectation operator conditional on informationavailable at date 0, and β is the subjective discount factor of the household. As in Andolfatto(Mar., 1996) and Merz (1995), we assume that workers can insure themselves against earninguncertainty and unemployment. In order to do that workers pool all their income sourcesso as to ensure equal consumption across members. Temporary preferences defined overconsumption and work effort are given by:

Ut (ct, ct−1) = log (ct − %ct−1) , K (hi,t) = aLh1+σLi,t

1 + σL, 0 ≤ %, 0 < aL, 0 ≤ σL. (4)

The parameter % governs habit formation in consumption. The larger is %, the stronger isthe degree of habit formation. The introduction of habit formation in consumption mainlyhelps account for the hump-shaped response of consumption observed in the data after amonetary policy shock. aL is a positive scaling parameter of disutility of work. The inverseof the parameter σL reflects the elasticity of the labor supply of workers with respect towages, holding consumption constant (i.e. the Frisch elasticity). We assume symmetry in thedisutility of work effort amongst the employed, so that hi,t = ht.

2.3 Household’s Budget Constraint and Optimal Decisions

Each period, the household allocates its wealth to purchases of financial assets, purchases ofconsumption goods, expenditures on transfer payments to entrepreneurs, and expenditureson lump-sum taxes. The household owns representative shares of all firms in the economy. Ithas the following sources of income: interest income on financial asset holdings, wages, unem-ployment benefits, lump-sum transfers from entrepreneurs exiting the economy, and profitsfrom all firms. The household faces the period-by-period intertemporal budget constraint in

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real terms:

NBt+1

Pt+ dt+1 + ct + wE + tt

≤ RNt−1

NBt

Pt+ rt−1dt + wi,tni,t + bU (1− ni,t) + (1− ς) (1− ι) vet + ψt. (5)

At the end of period t, the household decides on the amount of nominal government bonds,NBt+1, and real riskless deposits with the financial intermediary, dt+1, to acquire. At thebeginning of t + 1, these pay nominal gross rate of return, RN

t , and real rate of return, rt,respectively. The price of a unit of the consumption basket is Pt. The household pays a lump-sum tax, wE, in real terms to finance the transfer payments made to the ι entrepreneurs thatsurvive and to the 1− ι entrepreneurs that come newly to the market. tt denotes real lumpsum taxes/transfers from the fiscal authority. An employed household member, indexed withi, earns a nominal wage, Wi,t, in period t, where wi,t =

Wi,t

Ptdenotes real wage. Unemployed

household members receive real unemployment benefits of size bU . In addition, the householdreceives lump-sum transfers, 1− ς, corresponding to the net worth in real terms of the 1− ιentrepreneurs, vet = V et

Pt, who exit the economy the current period. ς ∈ (0, 1) governs the

amount of net worth that the entrepreneurs consume before exiting the economy. ψt denotescumulative real profits of the firms owned by the household.

The representative household optimizes its life-time utility (3) by choosing consumption,and bonds and deposits to hold subject to the household budget constraint (5). Denoteλt the time-t Lagrange multiplier on the flow budget constraint. The following optimalityconditions must hold:

for ct : λt = Utc,t = (ct − %ct−1)−1 − βbEt {ct+1 − %ct}−1 , (6)

forNBt+1

Pt+1

: λt = βEt

{λt+1

RNt

πt+1

}, (7)

for dt+1 : λt = βEt {λt+1rt} , (8)

limj→∞

βjEt

{Λt,t+j

NBt+j

Pt+j

}= 0, ∀t, (9)

with the addition of (5) holding with equality. Equation (6) defines the marginal utilityof consumption at period t, Utc,t. Denote πt = Pt

Pt−1period-t inflation. Equation (7) and

equation (8) are the Euler conditions with respect to nominal bonds and riskless deposits,respectively. They state that the household prefers expected marginal utility to be constantacross time periods, unless an expected gross real return on deposits, rt = Et

{RNtπt+1

}, ex-

ceeding household’s time preference induces it to lower its consumption today relative to thefuture. Denote βjΛt,t+j = βj

λt+jλt

the household’s pricing kernel between periods t and t+ j.Equation (9) is a borrowing constraint that prevents the possibility of Ponzi schemes.

We can rewrite the household’s problem, given its recursive structure, through an optimal

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value function as:

V h (ni,t) = Ut (ct, ct−1)−K (hi,t)ni,t + βEt {V h (ni,t+1)} . (10)

We need to find the value enjoyed by the household from the marginal worker i. This isdefined as the change in the household’s optimal utility from having an additional memberemployed. Taking the derivative of V h (ni,t) in (10) with respect to ni,t, subject to (2) and(5) we find the worker’s surplus, V hn (ni,t):

forni,t : V hn (ni,t) = λtwi,t − λtbU −K (hi,t)

+ (1− χ) βEt {V hn (ni,t+1)} − (1− χ) ftβEt {V hn (nt+1)} . (11)

There is a continuum of labor agency firms indexed by k ∈ [0, 1]. These open new vacancies,vk,t, each period 2. Respectively, any two jobs i and k at the labor agency firm are identical.Let Et {V hn (nt+1)} = Et

{vk,tvtV hn (ni,t+1)

}be the expected average worker’s marginal value

in period t + 1 and vk,tvt

be the probability of being matched to firm k. Therefore, the valuethat the household enjoys from holding a job at firm k consists of real wage net of laborunemployment benefit (in terms of marginal utility), minus the disutility of work effortthat employed members experience, plus the future value of the job conditional on survival,(1− χ) Et {βV hn (ni,t+1)}, minus the value the worker would contribute to the household ifshe searched for a job, (1− χ) ftEt {βV hn (nt+1)}.

2.4 Production

This section provides an overview of the production agents/firms sector.

2.4.1 Retailers

The retail sector sells in perfectly competitive markets. It buys wholesale goods of typez ∈ [0, 1], yz,t, at price Pz,t, aggregates all varieties og goods into a homogenous final good,yt, and sells it at final price, Pt. Retailers produce according to a constant elasticity ofsubstitution function:

yt =

(∫ 1

0

y1−εε

z,t dz

) ε1−ε

, ε < 1, (12)

where ε is the elasticity of substitution between different types of goods. Any retailer min-imizes her costs with respect to her inputs. The minimization problem defines the currentdemand function for the product of wholesale producer z, yz,t:

yz,t =

(Pz,tPt

)−εyt. (13)

2Labor agencies are large in a sense that they do not consist of worker-job pair.

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2.4.2 Wholesale Producers

Total number of firms in the wholesale sector has unit mass. Any wholesale firm produces adifferentiated good, yz,t, according to:

yz,t =

{kαz,tx

1−αz,t − φW if kαz,tx

1−αz,t > φW

0 otherwise, 0 ≤ α ≤ 1, (14)

where φW is a fixed cost of production. Wholesale goods firms are competitive in factormarkets, where they confront a nominal rental rate on capital, PtrKt , and a nominal cost onlabor services, PtpWt . Each wholesale goods firm chooses its capital and labor services, kz,tand xz,t respectively, to minimize its total costs, taking factor prices as given:

minkz,t,xz,t

[rKt kz,t + pWt xz,t

],

subject to the production function (14). The result of the minimization problems yields:

mct =rKt

α(xz,tkz,t

)1−α =pWt

(1− α)(xz,tkz,t

)−α =

(rKtα

)α(pWt

1− α

)1−α

, (15)

wheremct = MCtPt

are wholesale goods firms’ real marginal costs. Wholesale firms set nominalprices on a staggered basis. Following Calvo (1983), we assume that each period any wholesalefirm adjusts its price with probability 1−ζ. The adjustment probability is independent acrosstime and across firms (i.e., it does not depend on how long a firm’s price has been fixed).Wholesale goods producers use the household’s pricing kernel, βjΛt,t+j, to discount theirprofits between periods t and t+ j. If the firm z is permitted to optimize its price at time t,it chooses Pz,t = P o

z,t to optimize discounted profits:

maxP oz,t

Et

∞∑j=0

(βζ)j{

Λt,t+j

[P oz,t

Pt+jyz,t+j −mct+jyz,t+j

]}, (16)

subject to (13). The current value of firm’s profit is expressed as the total real revenue of itssales, P oz,t

Pt+jyz,t+j, reduced by the total real costs, mct+jyz,t+j. The first order condition of the

retailer’s optimizing behavior gives:

Et

∞∑j=0

(βζ)j{

Λt,t+j

[(P oz,t

Pt+j

)1−ε

− (1 + µP )mct+j

(P oz,t

Pt+j

)−ε]yt+j

}= 0, (17)

where µP is a price mark-up. The mark-up is inversely related to the elasticity of demand,ε, as 1 + µP = 1

1−1/ε. In the case of perfect competition, when ε =∞, the net mark-up over

the marginal cost, mct, is zero, since 1 + µP converges to 1.

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From the Calvo pricing assumption we get law of motion of the price level:

Pt =(ζ (Pt−1)1−ε + (1− ζ) (P o

t )1−ε) 11−ε . (18)

2.4.3 Capital Producers

A continuum of competitive capital producers, each indexed by a ∈ [0, 1], produces capitalgoods by combining investment and existing (depreciated) capital stock. Capital producersbuy the undepreciated capital stock at the end of each period from entrepreneurs and afterproducing the new capital, sell it back to the entrepreneurs at nominal price Qt. The realprice of capital is qt = Qt

Pt. Capital production activity entails physical adjustment costs,

that is, the producer needs investment, ia,t, to deliver new capital goods, ξ(ia,tka,t

)ka,t. The

adjustment cost function is assumed to be strictly increasing, concave, and to satisfy ξ (δ) = δ

and ξ′ (δ) = 1, where δ denotes capital depreciation rate. These last two assumptions ensurethe absence of adjustment costs in the steady state. The functional form of the capitaladjustment is given by:

ξ

(ia,tka,t

)=ia,tka,t− ϑ

2

(ia,tka,t− δ)2

, 0 < ϑ, 0 < δ < 1.

Capital producers maximize profits:

maxia,t

[qt

(ia,tka,t− ϑ

2

(ia,tka,t− δ)2)ka,t − ia,t

], (19)

with respect to investment. This implies the following first order condition:

qt =

(1− ϑ

(ia,tka,t− δ))−1

. (20)

Aggregate capital accumulation obeys:

kt+1 = (1− δ) kt + ξ

(itkt

)kt. (21)

2.4.4 Entrepreneurs

There is a continuum of entrepreneurs indexed by b ∈ [0, 1]. Entrepreneur b purchases phys-ical stock of capital, kb,t+1, at the end of period t at real price qt = Qt

Pt. The entrepreneur

finances her capital purchases, qtkb,t+1, both with her own real net worth, nwb,t+1 =Nwb,t+1

Pt+1,

and real debt, dbb,t+1 =Dbb,t+1

Pt+1, borrowed from a financial intermediary. We suppose that

nwb,t+1 < qtkb,t+1. Entrepreneurs are risk neutral and have finite expected horizon for pro-viding their services: ι < 1 is their probability of survival to the next period. The expected

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lifetime horizon of an entrepreneur is 1/(1− ι). This assumption ensures that entrepreneurs’net worth, both now and in the future, will not be enough to fully finance their new capitalacquisitions.

Return on capital depends on both aggregate and idiosyncratic shocks. Denote rFt+1 theex-post average rate of return on capital across all entrepreneurs. Similar to Bernanke et al.(1999) and Christiano et al. (2004), we assume that after entrepreneurs have purchasedcapital, they draw an idiosyncratic shock, ωb,t+1, which changes kb,t+1 to ωb,t+1kb,t+1. ωb,t+1

is an independent and identically distributed lognormal random variable with cumulativedistribution function F (ωt+1), over non-negative support, and mean one, Et {ωb,t+1} = 1.The riskiness of entrepreneurs is determined by the variance of the idiosyncratic shock, σω.The ex-post return on capital for entrepreneur b, rFb,t+1, is:

rFb,t+1 =

(rKt+1 + (1− δ) qt+1

qt

)ωb,t+1 = rFt+1ωb,t+1. (22)

The relationship borrower-lender is subject to an agency cost problem. We follow Bernankeet al. (1999) in supposing a "costly state verification" where lenders must pay a moni-toring cost to observe any single borrower’s realized return, in case she declares default.The monitoring cost is a proportion 0 ≤ µ < 1 of the realized gross payoff to the en-trepreneur’s capital, i.e., the monitoring cost equals dt = µG (ωb,t+1) rFt+1qtkb,t+1, whereG (ωb,t+1) =

∫ ωb,t+1

0ωt+1dF (ωt+1) is the probability of monitoring. When µ = 0, we are

in the special case of frictionless financial markets.The entrepreneur signs a standard debt contract with the intermediary. This specifies the

loan amount and a gross rate of interest, rLb,t+1, to be paid if ωb,t+1 is above a cut-off level,ωb,t+1, which determines default states. Entrepreneurs who draw ωb,t+1 below the cut-off levelgo bankrupt and must turn everything they have to the bank. The cut-off satisfies:

ωb,t+1 =rLb,t+1dbb,t+1

rFt+1qtkb,t+1

. (23)

Receipts from entrepreneurs are the only source of cash to the financail firms. In equi-librium, the intermediary holds a pooled, and perfectly safe, portfolio and the entrepreneursabsorb any aggregate risk. Therefore, the financial intermediary obtains its funds from house-holds at the riskless gross rate of return, rt. Perfect competition and free entry on the financialmarket imply that intermediaries’ net cash flow must be zero in each period t + 1 state ofnature 3. Respectively, loan contract must satisfy:

(1− F (ωb,t+1)) rLb,t+1dbb,t+1 + (1− µ)G (ωb,t+1) rFt+1qtkb,t+1 = rtdbt+1. (24)

3We assume assuming that markets are incomplete. The zero profit condition and equation (23) holdingwith equality in each state implies that both rL

b,t+1 and ωb,t+1 must vary across t + 1 state of nature.

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Denote Γ (ωb,t+1) = ωb,t+1 (1− F (ωb,t+1))+G (ωb,t+1) the share of entrepreneurial earningskept by the financial intermediary. The optimal contract is chosen to maximize the expectedlinear entrepreneurial utility:

maxqtkb,t+1nwb,t+1

,ωb,t+1

Et

{[1− Γ (ωb,t+1)]

rFt+1

rt

qtkb,t+1

nwb,t+1

}, (25)

with respect to the cut-off level and the asset to net worth ratio, qtkb,t+1

nwb,t+1, conditional on

the expected return of the financial intermediary, the aggregated equation (24). The opti-mal contract between entrepreneur and financial intermediary requires the two first-orderconditions below to hold:

fornwb,t+1

qtkb,t+1

: Et

{[1− Γ (ωb,t+1)]

rFt+1

rt+ λωb,t+1

[(Γ (ωb,t+1)− µG (ωb,t+1))

rFt+1

rt− 1

]}= 0,

(26)

for ωb,t+1 : (Γ (ωb,t+1)− µG (ωb,t+1))rFt+1

rt=

(1− nwb,t+1

qtkb,t+1

), (27)

where λωb,t =Γω(ωb,t+1)

Γω(ωb,t+1)−Gω(ωb,t+1). It is clear from the first first-order condition that when

financial markets are frictionless, µ = 0, then λωb,t+1 = 1 and Et

{rFt+1

}= rt: the ex-ante

return on capital equals the risk free rate. The second first-order condition is related to thefact that the financial intermediary receives an expected return equal to the opportunity costof its funds. In this case, the lender’s expected return can simply be expressed as a functionof the average cutoff value of the firm’s idiosyncratic shock, ωt+1.

It is clear from the first order conditions that each entrepreneur’s standard debt contractis characterized by the same cut-off level and asset to equity ratio. This guarantees thataggregation is straightforward. This implies that capital expenditures by each entrepreneurb are proportional to her net worth. The law of motion of entrepreneurial net worth (inconsumption units) follows: nwt+1 = ιvet + wE. Respectively, we can express aggregate networth at the end of period t by:

nwt+1 = ι(rFt − rt−1 − µG (ωt+1) rFt qt−1Kt

)qt−1kt + wE + ιrt−1nwt. (28)

After the entrepreneurs have settled their debt to the banks in period t + 1, and en-trepreneurs’ capital have been sold to capital producers, the entrepreneurs’ period t + 1

net worth is determined. The entrepreneurs who exit the economy transfer 1 − ς of theirwealth to the household and consume the rest. Each period new entrepreneurs enter themarket in sufficient numbers so that the population of entrepreneurs remains constant. Newentrepreneurs born in period t+ 1 receive a transfer of net worth, wE, from the household.

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2.4.5 Labor Producers

The labor agency firm employs nk,t workers. Each worker provides hk,t hours of work. Anysingle, employed worker is able to produce an intermediate good xk,t, with k = i denoting thegood developed by worker i. In section 2.5 we suppress the indexes k and i when implyingany two identical jobs, and denote by ht hours invested in any labor good, and so on. Thelabor good is produced according to:

xk,t = τAt hνk,t, 0 ≤ ν ≤ 1,

where ν measures the decreasing returns of hours worked. The variable τAt is a stationaryshock to technology:

log τAt+1 = ρa log τAt + εat+1, 0 ≤ ρa < 1,

where εatiid∼ N (0, σ2

a). Total production of firm k, xk,t, is given by:

xk,t = nk,txk,t. (29)

We assume that the new matches at firm k at the beginning of period t+ 1 going to thefirm are proportionate to the ratio of its vacancies to total vacancies in the economy, vk,t

vt.

Thus, vk,tltvt

= vk,tst is hiring initiated by firm k.We assume standard linear specification for the real cost per vacancy, φΥ, following Pis-

sarides (1988). In each period, the labor firm maximizes the current profit and its discountedfuture values. Similar to the wholesale firm, the labor agency firm uses the household’s pricingkernel to discount the future profit values. The value function of the agency is characterizedby the following dynamic programming equation:

V f (nk,t) = pWt xk,t − φΥvk,t − wk,tnk,t + βEt {Λt,t+1V f (nk,t+1)} , (30)

where V f (nk,t) is the firms’ value function in period t, dependent on the employment stock,nk,t. Let V fn (nk,t) be the value to the firm of employing another worker at time t:

fornk,t : V fn (nk,t) = pWt xk,t − wk,t + (1− χ) βEt {Λt,t+1V fn (nk,t+1)} . (31)

The condition states that the value of an additional worker equals the worker’s marginalrevenue product, minus the real wage plus the job’s continuation value.

The labor agency optimizes the dynamic equation (30) above subject to (2). The firstorder condition with respect to vacancies gives:

for vk,t :φΥ

qt= (1− χ) βEt {Λt,t+1V fn (nk,t+1)} . (32)

The condition for vacancy posting equates the marginal cost of posting a vacancy with the

12

discounted marginal benefit of the job. Combining (31) and (32) we can write the job creationcondition as:

φΥ

qt= (1− χ) Et

{βΛt,t+1

[pWt xk,t+1 − wk,t+1 +

φΥ

qt+1

]}, (33)

which states that at the optimal choice, the vacancy-creation cost incurred by the firm isequated to the discounted expected value of profits from the match. Profits from a matchtake into account the wage cost of that match. This condition is a free-entry condition inthe creation of vacancies and is one of the crucial equilibrium conditions of the labor sector.

2.5 Wage and Hour Bargaining

The dynamics of wages is decisive for the determination of real marginal costs of the laboragencies since it affects their hiring incentives. We assume, as in most of the labor searchliterature, that worker and firm bargain at the individual level over the joint surplus oftheir match, according to the Nash bargaining solution. Given that in equilibrium all laboragencies behave the same and use the same technology we can drop the i and k subscripts.Bargaining takes place both over hours per worker and the wage, to maximize:

max

(V fn (nt)

)1−η(1

λtV hn (nt)

)η, (34)

where the first term in brackets is the firm’s surplus, the second term presents the worker’ssurplus, and η ∈ (0, 1) is the bargaining worker’s power in the wage negotiation process. Thefirst-order condition of the Nash product with respect to wt is:

forwt : − (1− η)

(1

λtV hn (nt)

)(∂V fn (nt)

∂wt

)︸ ︷︷ ︸

δf

= η

(V fn (nt)

)(∂ 1λtV hn (nt)

∂wt

)︸ ︷︷ ︸

δw

, (35)

where δw = 1 and δf = −1. Using the above equation and employing optimal vacancycondition (32) yields:

(1− η) Et

{1

λtV hn (Nt+1)

}= ηEt

{βt,t+1V fn (Nt+1)

}= η

1

(1− χ)

φΥ

qt. (36)

By combining conditions (11), (31), (35) and (36) we derive the following equilibriumwage rule:

wt = η(pWt xt

)+ (1− η)

(bU +

aLλt

h1+σLt

1 + σL

)+ η

(φΥftst

). (37)

The first term in brackets on the right-hand-side is the production firm’s contemporaneous

13

surplus (excluding wage payments) from consummating the match and is equal to output perworker. The second term in brackets is the worker’s threat point in bargaining. If the workerwalks away from the match, he would suffer the disutility value aLh1+σL

t /λt (1 + σL) of notworking at the production firm (expressed in consumption units) and enjoy an unemploymenttransfer bU . The third term in brackets represents the saving on hiring costs that the firmenjoys when a job is created.

Hours per worker are chosen in a competitive labor market, so as to maximize the jointsurplus of the employment relationship between a worker and a firm. However, the choiceof hours is independent of the wage 4. The joint surplus St = V fn (nt) + 1

λtV hn (nt) is the

sum of the firm’s and worker’s surpluses, both expressed in terms of consumption units.Under the specified household utility function and the assumption of the large family, themaximization of the total surplus yields:

forht : pWt νhν−1t =

aLhσLt

λt. (38)

The condition equalizes the marginal product of labor to the worker’s marginal rate ofsubstitution between leisure and consumption. It also elucidates the driving forces of hoursvariation in the search model. A higher marginal utility of wealth, Utc,t, and a higher marginalproduct of labor all increase labor supplied, whereas it falls whenever the disutility of laboror the intertemporal preferences increase.

2.6 Government

The central bank adjusts the nominal interest rate, RNt , in response to deviations of inflation

and output from their steady-state values. The monetary policy rule evolves according to:

log(RNt

)= (1− ρm) log

(π

β

)+ ρm log

(RNt−1

)+ (1− ρm)

(γπ log

(Et {πt+1}

π

)+γy4

log

(yty

))+ εmt . (39)

Let ρm ∈ [0, 1), 1 < γπ and 0 ≤ γy are response coefficients to lagged interest rate, inflationand output, respectively. Variable without an index represents the steady state value of thecorresponding variable. Let εmt

iid∼ N (0, σ2m) is an iid log-normal shock to the monetary policy

stance.Let g be the government’s long-run target level for government expenditures. Government

4It is important to note that the optimization condition can change depending on how firm and workerbargain over wages. We have assumed that bargaining is done over period wages per worker similar to Krauseet al. (2008). Respectively, Sveen and Weinke (2008) discuss the case for hourly wages.

14

consumption expenditures, gt, follow a first-order autoregressive process:

log gt+1 = (1− ρg) log g + ρg log gt + εgt+1, 0 ≤ ρg < 1, (40)

εgtiid∼ N

(0, σ2

g

). The government budget constraint is:

NBt

Pt+ tt = RN

t−1

NBt−1

Pt+ utbU + gt. (41)

The government finances its expenditures by lump-sum taxes, tt, and issue of nominal bonds,NBt+1. We assume the government elastically supplies bonds until the bond market clears.The government makes expenditures on debt repayment and coupon, unemployment benefits(the term involving bU), as well as government spending, gt.

2.7 Market Clearing

In a competitive equilibrium, all agents’ optimality conditions are satisfied and all marketsclear. We assume a symmetric equilibrium throughout, which entails identical choices forall variables. Defining aggregates as the averages of firm specific variables, we have thatkt = ka,t =

∫ 1

0ka,tda, kt = kb,t =

∫ 1

0kb,tdb, 1−ut = nt = nk,t =

∫ 1

0nk,tdk, vt = vk,t =

∫ 1

0vk,tdk,

and xt = xk,t =∫ 1

0xk,tdk. Furthermore, as Pz,t = Pt, yz,t = yt, for all t and z. Thus

wholesale firms produce the same amounts of output and face the same marginal costsmct. Capital production firms produce the same amounts of capital. Entrepreneurs rent thesame amounts of capital services. Labor agencies rent equal amounts of labor. Finally, usingthe household budget contraint, firms profits, and the government constraint, the resultingaggregate income identity is:

yt = ct + ς1− ιι

(nwt+1 + wE) + it + gt + dt + φΥvt. (42)

Equilibrium in the retail goods market requires that the production of the retail goods beallocated to private consumption by households, entrepreneurs, investment, public spending,to resource costs that originate from the lender’s monitoring of the investment activity andfrom labor firms’s job creation activity.

3 Policy Considerations

A good indicator, about the degree of inefficiency present in the credit market in our setupis the spread between the riskless interest rate and the interest rate on bank loans.

The spread depends on the following conditions: The equity ratio, the amount of asym-metric information present in the credit market, the riskiness of the projects undertaken byfirms and the monitoring cost which banks have to incur after a firm declares bankruptcy.

15

Policy interventions that aim at fostering economic activity can be distinguished accord-ingly as measures,

• that improve the equity ratio of the existing firms,

• that increase the transparency of financial markets and

• that are lowering monitoring costs.

Within the theoretical model used, this translates into:

• a "positive" shocks to the net worth of the entrepreneurs,

• a negative shock to the interest rate spread and

• a negative shock to the monitoring cost.

A change of the corporate tax system is the prime candidate for a policy intervention,intended at improving the equity position of existing firms. Lowering the corporate tax rate,changing the structure how dividends and capital gains are taxed, or changing the accountingrules that govern the determination of accounting profits are examples of such a measure.

Accounting rules, publishing requirements, the penalty system which applies in case ofa violation of the regulatory rules are all instruments under the control of the governmentthat affect the degree of asymmetric information present in financial markets.

It is less clear how policy makers can change the monitoring cost, because these are realcost of production.

One of the attractive features of a dynamic macroeconomic model is the fact that thegrowth of GDP and the business cycles are studied within the same theoretical framework,the neoclassical growth model. Nevertheless, in practice, due to the techniques used to solveinfinite time horizon rational expectations models, some dichotomy applies here too. Thedynamics of macroeconomic aggregates are analyzed by first determining the steady stategrowth path of the model at hand. Then in a second separate step the short run deviationsfrom the steady state path are studied with the help of a first- or second-order approximationaround the steady state path. Naturally this two step approach also characterizes the analysisof policy interventions.

The topic of following section is the quantification of the impact of the financial marketfrictions on the performance of the labor market. Within our setup, a labor market withsearch frictions and wage negotiations, the key variables which characterize the state of thelabor market are: the unemployment rate, the average duration of an unemployment spell,hours worked, the number of vacancies and the level of the wages.

An extension of our paper will consider including a wide range of shocks that bothoriginate inside the financial sector, the ones listed above, and others that originate outsidethe financial sector. Our future extension of the paper will quantify the importance of each

16

the shocks buffeting the economy. We will answer how, if at all, should policy react tofinancial market shocks and as well as how to react to other than financial sector shocks.

4 Results

Our goal is to analyze how greatly do changes in the structure of the financial market(we mean calibrated parameters about the financail sector) impact the observed business-cycle fluctuations in unemployment and job vacancies in response to shocks of a plausiblemagnitude. Before turning to the results we briefly discuss how the parameter values arechosen.

4.1 Calibration

We have briefly sketched the parameters that we have chosen. This section is to be extended.We consider the following structural parameter values:

β = 1.01−0.25, b = 0.6, aL = 340.8, σL = 1, F (ω) = 0.013,

µ = 0.1, ι = 1− 0.0238, wE = 0.004, ϕ = 0.25 ε = 6,

α = 0.36, δ = 0.25, ζ = 0.63, ρm = 0.75, γπ = 1.5, γy = 0.25,

χ = 0.05, ψM = 0.4, η = 0.4, ν = 0.98

We target steady-state share of government consumption of 0.2 (set in such a way thatGY

= 0.2), steady-state vacancy-filling rate s of 0.7, steady-state job-finding rate f of 0.6,ratio of recruiting expenditures to output φssυ = φΥV

Y= 0.01.

We consider the following structural parameter values that characterize the exogenousshocks:

ρa = 0.95, σa = 0.008, ρg = 0.9, σg = 0.008, σm = 0.043.

4.2 Simulation and Main Findings

We assume that our model economy is driven by monetary policy shocks, technology shocksand shocks in government consumption.

Figure 1 displays the dynamic response of the endogenous variables to a monetary policyshock, εmt , in (39). In addition to displaying the responses implied by our benchmark model,we also display the responses implied by our simulated versions of the benchmark where oncethe monitoring costs µ are increased to 0.16 and respectively the percent of bankruptciesin steady state F (ω) equals 0.02. The models are referred to as: ’benchmark’, ’benchmark+ mu=0.16’ and ’benchmark + F(omegabar)=0.02’ in the figure. The size of the monetarypolicy shock is the same in each model.

17

In the benchmark model, the monetary policy shock drives up the short term interest rateby approximately 14 basis points, and the interest rate returns monotonically to its meanafterward. The internal propagation of the model is strong in that the effects on capital,employment and other variables continue well after the roughly 2 years it takes for the effectson the interest rate to die out. Output, investment, and employee hours worked display aninverted ’U’ shape. The maximal response of investment is roughly as big, in percent terms,as the response of output. This is a weakness of the model. It relates both to the functionalform for investment as well as to the calibrated parameters. Future estimation of the modelwill strive to correct for the shortfall. The fall in investment drives down the price of capital(not shown), and the implied capital losses contribute to a fall in entrepreneurial net worth.The drop in net worth is roughly twice as big as the drop in the price of capital, presumablybecause net worth is also reduced by the fall in income earned by entrepreneurs. These effectscontribute to a rise in the external finance premium paid by entrepreneurs and reinforce thedrop in investment. The increase of nominal interest rates decreses the profitability of thelabor production firms and job creation respectively. The effects in the capital and financialmarkets further reinforce the slow down of job creation activity by decreasing wholesaleproduction and consecutively the demand for the labor product in the homogeneous labormarket.

The role of the Bernanke et al. (1999) financial frictions in the propagation of the mone-tary policy shock is substantial. The persistence in net worth, investment and unemploymentis approximately 2 times smaller in the benchmark model than it is in the versions of thebechmark model. In the benchmark model, the impact on unemployment is one half smallerthan it is in its versions. The Bernanke et al. (1999) financial frictions and their size are theprimary reason that the response of investment is so long-lasting in the benchmark model.The presence and scope of Bernanke et al. (1999) financial frictions adds substantially to thedrop in the real wage after a monetary policy shock. Presumably this reflects the substantialacceleration effects, which reduce the demand for investment and lead to a substantial fallin the demand for labor.

Figure 2 displays the dynamic response of the endogenous variables to a shock in the tech-nology neutral process, εat . In addition to displaying the responses implied by our benchmarkmodel, we also display the responses implied by our simulated versions of the benchmarkwhere once the monitoring costs µ are increased to 0.16 and respectively the percent ofbankruptcies in steady state F (ω) equals 0.02. The size of the technology shock is the samein each model.

The figure is illustrative of the acceleration effect induced by the presence of creditfrictions. Hence we see that a current rise in productivity triggers a rise in the production andinvestment. Crucially, net worth responds after a quarter upwards. The default threshold aswell as the bankruptcy rate stay roughly constant. The response of the net worth depends onthe ’credit cycle’ effect induced by the rise in production. Given that capital is roughly fixed

18

in the short-run (due to capital adjustment costs), the rise in the return for the entrepreneursinduces a slightly higher rise in net worth. While not completely evident from the figure,we can guess that the countercyclical response of the finance premium is stronger the largerthe size of the monitoring imperfections or riskiness of the projects the entrepreneurs haveundertaken. All this translates into decrease of unemployment proportional to monitoringimperfections or riskiness.

We can observe also that credit frictions work in the direction of dampening the responseof the nominal interest rate to the rise in technology. This is due to the fact that increasedcapital accumulation with accelerated investment triggers, for any given level of employment,a rise in the real marginal cost, which tends to dampen the countercyclical response ofinflation to technology shocks that is typical of sticky price models with frictionless creditmarkets. This fact together with the pronounced different responses of unemployment underthe various parameterization of the benchmark model call for a more extensive study of’optimal’ monetary policy under financial and labor market imperfections.

Figure 3 displays the dynamic response of the endogenous variables to a shock in the gov-ernment consumption, εgt . In addition to displaying the responses implied by our benchmarkmodel, we also display the responses implied by our simulated versions of the benchmarkwhere once the monitoring costs µ are increased to 0.16 and respectively the percent ofbankruptcies in steady state F (ω) equals 0.02. The size of the government shock is the samein each model.

It is evident that under the government shock the finance premium behaves more coun-tercyclically the higher the monitoring imperfections or riskiness of entrepreneurs’ activity.The nominal interest rate is more dampened under higher imprefections.

5 Conclusion

In this paper, we have studies credit and labor market frictions embedded in a conventionalNew Keynesian model: a labor search problem in the labor market and a costly state ver-ification problem in the credit market. The first friction allows for endogenous equilibriumunemployment while the latter introduces riskiness of capital management. The claim ofthis paper is that credit markets may play an important role for the dynamics of the labormarket. To illustrate the above proposition, we present a simple framework to analyze thejoint behavior of access to credit and labor market prices. We consider the responses of un-employment in the economies to a monetary shock, a technology shock and a governmentconsumptions shock. Differences in credit markets can play role for propagating shocks thatoriginate outside the financial sector that and in turn increase the persistence and impact oflabor market variables.

19

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21

Technical Appendix

A Analysis

A.1 Collecting equations

The equations characterizing the equilibrium are:

Marginal consumption:

λt = (ct − %ct−1)−1 − β% (ct+1 − %ct)−1 , (43)

Aggregate demand:

yt = ct + dt + ς

(1− ιι

)(nwEt+1 − wE

)+ it + gt + φΥvt, (44)

Aggregate supply:

yt = kαt x1−αt − φW , (45)

Euler equation:

λt = βEt

{λt+1

RNt

πt+1

}, (46)

Real interest rate:

rt = Et

{RNt

πt+1

}, (47)

Labor demand:

mct (1− α)

(ktxt

)α= pWt , (48)

Capital demand:

mctα

(xtkt

)1−α

= rKt , (49)

Capital producer’s supply:

qt =

(1− ϑ

(itkt− δ))−1

, (50)

Entrepreneurial demand for capital (Debt contract):

0 = Et

{[1− Γ (ωt+1)]

rFt+1

rt+ λωt+1

[(Γ (ωt+1)− µG (ωt+1))

rFt+1

rt− 1

]}, (51)

Zero profit condition on financial intermediaries:

0 = (Γ (ωt+1)− µG (ωt+1))rFt+1

rt−(

1−nwEt+1

qtkt+1

), (52)

Entrepreneurial law of motion:

22

nwEt+1 = ι(rFt − rt−1 − µG (ωt+1) rFt

)qt−1kt + wE + ιrt−1nw

Et , (53)

Entrepreneurial return on capital:

rFt+1 = Et

{rKt+1 + (1− δ) qt+1

qt

}, (54)

Capital law of motion:

kt+1 = (1− δ) kt + it −ϑ

2

(itkt− δ)2

kt, (55)

Phillips curve:

Pt =(ζ (Pt−1)1−ε + (1− ζ) (P o

t )1−ε) 11−ε , (56)

P ot

Pt= (1 + µP ) Et

∑∞

j=0 (βζ)j Λt,t+j

(1

Pt+j

)1−επt,t+jmct+jyt+i∑∞

j=0 (βζ)j Λt,t+j

(1

Pt+j

)1−εyt+j

, (57)

Unemployment:

ut = 1− nt, (58)

Labor agency’s production:

xt = ntτAt h

νt , (59)

Matching function:

lt = luψMt v1−ψMt , (60)

Employment law of motion:

nt+1 = (1− χ) (nt + lt) , (61)

Job creation condition:φΥ

qt= (1− χ) Et

{βΛt,t+1

[pWt τ

At h

νt − wt+1 +

φΥ

qt+1

]}, (62)

Wage schedule:

wt = η

(pWt τ

At h

νt + φΥ

vtut

)+ (1− η)

(bU +

aLλt

h1+σLt

1 + σL

), (63)

Labor supply (Hour condition):

pWt νhν−1t =

aLhσLt

λt, (64)

Taylor rule:

log(RNt

)= (1− ρm) log

(π

β

)+ ρm log

(RNt−1

)+ (1− ρm)

(γπ log

(Et {πt+1}

π

)+γy4

log

(yty

))+ εmt . (65)

23

A.2 Model Steady State

Compute steady state employment from labor law of motion equation (2):

N =(1− χ) f

1− (1− χ) (1− f)

U = 1−NL = fU

V =L

s.

We normalize the technology shock τa to one. Fix a range for the values of rK . Themarkup equation (17) for the marginal costs implies:

mc =1

1 + µP= 1− 1

ε

The Euler equation (??) and the definition of real interest rate R provide:

R =1

β

The steady state price of capital from (20):

q = 1

Calculate the gross rate of return RF for entrepreneurs from (22):

RF = rK + (1− δ)

Compute X/K from the cost minimization problem of the production firm:

X

K=

(rK

αmc

) 11−α

Solve the optimal debt contract equation:{[1− Γ (ω)]

RF

R+

Γω (ω)

Γω (ω)− µGω (ω)

[(Γ (ω)− µG (ω))

RF

R− 1

]}= 0

for ω. Then, use the first-oder condition for optimal debt level to find:

nwE

K= 1− RF

R[Γ (ω)− µG (ω)]

Compute nwE from the law of motion of entrepreneur’s net worth:

24

nwE =wE

1− ι [RF −R− µG (ω)RF ]K

nwE− ιR

It is straighforward to solve for G (ω), K, X, I, d and pW :

G (ω) = Γ (ω)− ω (1− F (ω))

K =K

nwEnwE

X =X

KK

h =

(X

N

) 1ν

using labor production function (29)

I = δK, using capital law of motion equation (21)

d = µGRFK

pW = mc (1− α)

(X

K

)−α.

Compute fixed costs in such a way that production firms make zero profits in equilibrium:

φ = KαX1−α (1−mc)

Then:

Y = KαX1−α − φ

Estimate the vacancy cost parameter:

Φυ =Φssυ Y

V

Compute consumption:

C = (1− φg)(Y −Θ

1− ιι

(nwE − wE

)− d− ΦυV

)− I

Compute government consumption:

C =φg

1− φg(C + I)

Estimate the marginal utility of consumption (6):

λ =

(1− βb1− b

)1

C

Compute again h from the hour sharing condition (38):

25

h =

(λpWν

aL

) 1−ν+σL+1

Compute the monthly wage per worker from the job creation condition (33):

w = pWhν − ΦυV

(1− χ) βL(1− (1− χ) β)

Express B from the wage equation (37):

B =wt

1− η− η

1− ηpWt h

ν − aLλt

(h1+σLt

)(1 + σL)

+η

1− ηΦυftst

Finally, compute the rK , over the range specified at the start of the algorithm, till thevalue of h from the hour sharing condition (38) above coincides with the initial value of hestimated from the labor production function (29) .

A.3 Log-Linearization

Variables21 variables Yt, Ct, It, λt,mct, Kt, ht, πt, r

Kt , R

Nt , Rt, R

Ft , qt, ωt, nw

Et , p

Wt , ht, Lt, Vt, Ut, Nt with

21 equations.Marginal Consumption

λt = − (1 + βb2)

(1− βb) (1− b)Ct +

b

(1− βb) (1− b)Ct−1 +

βb

(1− βb) (1− b)Ct+1

Aggregate Demand

Yt =

C

YCt +

δK

YIt +

g

Ygt +

d

Y

(RFt + qt−1 + Kt

)+Θ

(1− ιι

)nwE

YnwEt+1 +

µGωωRF qK

Yˆωt +

ΦvV

YYt

Aggregate Supply

Yt =KαX1−α

YτAt + α

KαX1−α

YKαt + (1− α)

KαX1−α

YX1−αt

Euler Equation

λt = Et

{λt+1 + Rn

t − πt+1

}Real Interest Rate

Rt = Et

{Rnt − πt+1

}Demand for Labor meets Supply

26

mct + αKt − αXt = pWt

Capital Demand

mct + (1− α) Xt − (1− α) Kt = rKt

Capital Producers’ Optimality Condition (Capital Supply)

qt = ϕ(It − Kt

)where ϕ = − ξ

′′(·)

ξ′ (·)IK

= ϑ IK

measures the elasticity of the price of capital with respect to theinvestment-capital ratio.

Entrepreneurial Demand for Capital (Debt Contract)

0 = Et

{λzpRF

t+1 − λzpRt − [1− Γ]RF

R

[Γωωω

Γω− (Γωω − µGωω)λzpω

Γω

]ˆωt+1

}λzp = Γω

Γω−µGω denotes the steady state value of the multiplier on the bank zero-profit condi-tion in the Lagrangian representation of the problem solved by the optimal contract. Here,Γωω and Gωω denote the second derivative of Γ and G with respect to ω, evaluated in steadystate.

Zero Profit on Financial Intermediaries

−qt−1 − kt + nwEt =

(qK

nwE− 1

)(RFt − Rt−1

)+

(qK

nwE− 1

)(Γω − µGω) ω

Γ− µGˆωt

Entrepreneurial Law of Motion

nwEt+1 = ι

RF qK − dnwE

RFt +

(1− qK

nwE

)RRt−1

+

(RF −R

)qK − d

nwE

(qt−1 + Kt

)− dGωω

GnwEˆωt +RnwEt

Return on Capital

RFt+1 =

rK

qRFrKt+1 +

(1− δ)RF

qt+1 − qt

Capital Law of Motion

Kt+1 = (1− δ) Kt + δIt

A Phillips curve can be derived from the wholesale sector optimization problem for prices,where 1 − ζ is the probability of adjusting prices and 1 + µ is the price mark up in steady

27

state:

πt = βEt {πt+1}+(1− βζ) (1− ζ)

ζmct.

Unemployment

Ut = −NUNt

Labor Production

Xt = Nt + τAt + νht

Matching Function

Lt = ψUt + (1− ψ) Vt

Employment Law of Motion

Nt+1 = (1− χ) Nt + χLt

Job Creation Condition

Vt − Lt = Et

{λt+1 − λt

}+ (1− χ) β

pWhνL

ΦυVEt

{pWt+1 + τAt+1 + νht+1

}(1− χ) βEt

{− wL

ΦυVwt+1 + Vt+1 − Lt+1

}Wage Schedule

wt = η

(pWhν

w

)(pWt + τAt

)+

(ηpWhνν

w+ (1− η)

aLh1+σLt

λw

)ht

−(

(1− η)aLh

1+σLt

(1 + σL)λw

)λt + η

(ΦυV

Uw

)(Vt − Ut

)Hour Condition

pWt = (σL − ν + 1) ht − λt

Taylor Rule

Rnt = ρmR

nt−1 + (1− ρm) γπEt {πt+1}+

γy (1− ρm)

4Yt + εmt

Law of motion of the shocks:

28

τat = ρaτat−1 + εat εat

iid∼ N (0, σ2a)

gt = (1− ρg) g + ρggt−1 + εgt εgtiid∼ N

(0, σ2

g

)τmt = εmt εmt

iid∼ N (0, σ2m)

B Analytical expressions for the idiosyncratic entrepreneurs’shock

The idiosyncratic productivity disturbance ωt (b) has a log-normal distribution:

logωt ∼ N(−0.5σ2

σ, σ2σ

).

Given this distributional assumption, it is convinient to express the default threshold inthe standardized form:

zt =log ωt + 0.5σ2

σ

σσ.

In the text F (·) and nf (·) denote cumulative distribution function and standard normaldensity function. Respectively, we can express the expected gross share of profits to the lenderΓ (ω), capital production value in case of default G (ω) and their derivatives as follows:

Γ (ωt+1) = ωt+1 [1− F (zt+1)] + Ft (zt+1 − σσ)

G (ωt+1) = F (zt+1 − σσ)

Γω (ωt+1) = 1− F (zt+1)

Gω (ωt+1) =nf (zt+1)

σσ

Γωω (ωt+1) = −nf (zt+1)

ωt+1σσ

Gωω (ωt+1) = − ztσσ

nf (zt+1)

ωt+1σσ,

where F (z) quantifies the probability of default, and the expected realization of the produc-tivity disturbance in the event of default is given by F (z − σσ).

C Tables

D Figures

29

10 20 30−0.8

−0.6

−0.4

−0.2

Output

perc

ent d

evia

tion

from

ss

benchmarkbenchmark + mu=0.16benchmark + F(omegabar)=0.02

10 20 30

2

4

6

8

10

x 10−3Nominal Int. Rate (annual)

perc

ent p

oint

10 20 30−0.4

−0.3

−0.2

−0.1

0Inflation (annual)

perc

ent p

oint

10 20 30−0.4

−0.3

−0.2

−0.1

Net Worth

perc

ent d

evia

tion

from

ss

10 20 30

−0.2

−0.1

0

Premium (annual)

perc

ent p

oint

10 20 30−0.6

−0.4

−0.2

0Investment

perc

ent d

evia

tion

from

ss

10 20 30

−1

−0.8

−0.6

−0.4

−0.2

Employee Hour

perc

ent d

evia

tion

from

ss

Response to a shock in monetary policy

10 20 300

0.2

0.4

0.6Unemployment

perc

ent d

evia

tion

from

ss

Figure 1: Response to a shock in monetary policy

30

10 20 30

−0.1

0

0.1

0.2

0.3

Output

perc

ent d

evia

tion

from

ss

benchmarkbenchmark + mu=0.16benchmark + F(omegabar)=0.02

10 20 30

−0.035

−0.03

−0.025

−0.02

−0.015

Nominal Int. Rate (annual)

perc

ent p

oint

10 20 30

−0.3

−0.2

−0.1

Inflation (annual)

perc

ent p

oint

10 20 30−0.08

−0.06

−0.04

−0.02

0

0.02

Net Worth

perc

ent d

evia

tion

from

ss

10 20 30

−0.08−0.06−0.04−0.02

00.020.04

Premium (annual)

perc

ent p

oint

10 20 30

0

0.05

0.1

0.15

0.2

Investment

perc

ent d

evia

tion

from

ss

10 20 30

−0.8

−0.6

−0.4

−0.2

Employee Hour

perc

ent d

evia

tion

from

ss

Response to a shock in the technology neutral process

10 20 30−0.4

−0.2

0

Unemployment

perc

ent d

evia

tion

from

ss

Figure 2: Response to a shock in the technology neutral process

31

10 20 30

0.020.040.060.08

0.10.12

Output

perc

ent d

evia

tion

from

ss

benchmarkbenchmark + mu=0.16benchmark + F(omegabar)=0.02

10 20 300

0.5

1

1.5

2

x 10−3Nominal Int. Rate (annual)

perc

ent p

oint

10 20 300

0.01

0.02

0.03

0.04Inflation (annual)

perc

ent p

oint

10 20 300

5

10

x 10−3 Net Worth

perc

ent d

evia

tion

from

ss

10 20 30

−2

0

2

4

x 10−3Premium (annual)

perc

ent p

oint

10 20 30

−0.02

−0.01

0

Investmentpe

rcen

t dev

iatio

n fr

om s

s

10 20 30

0.05

0.1

0.15

Employee Hour

perc

ent d

evia

tion

from

ss

Response to a shock in government consumption

10 20 30−0.3

−0.2

−0.1

0Unemployment

perc

ent d

evia

tion

from

ss

Figure 3: Response to a shock in government consumption

32