Search Frictions, Real Rigidities, and Inflation Dynamics

CARLOS THOMAS

Search Frictions, Real Rigidities, and Inflation

Dynamics

The literature on New Keynesian models with search frictions in the labormarket commonly assumes that price setters are not actually subject to suchfrictions. Here, I propose a model where firms are subject both to infrequentprice adjustment and search frictions. This interaction gives rise to realprice rigidities, which have the effect of slowing down the adjustment of theprice level to shocks. This has a number of consequences for equilibriumdynamics. First, inflation becomes less volatile and more persistent. Moreimportantly, the model’s empirical performance improves along its labormarket dimensions, such as the size of unemployment fluctuations and therelative volatility of the two margins of labor.

JEL codes: E32, J60Keywords: search and matching, real rigidities, New Keynesian

Phillips curve, labor market fluctuations.

THE SEARCH AND matching model has become a popular treat-ment of labor market dynamics in New Keynesian models of the monetary transmis-sion mechanism.1 One of the main advantages of this kind of framework is that itmakes it possible to analyze the joint dynamics of unemployment and inflation in arelatively simple way. Due to search frictions in the labor market, it takes time forunemployed workers to find jobs. This, together with recurrent job destruction, givesrise to unemployment in equilibrium. On the other side of the labor market, searchfrictions imply that firms must spend time and resources before they can find suitableworkers. To the extent that firms have monopoly power on the goods they sell, thisnaturally raises the question as to how pricing decisions are affected by the fact thatfirms cannot costlessly and instantaneously adjust the size of their workforce.

I am very grateful to Kosuke Aoki, Nobu Kiyotaki, Michael Krause, Jordi Gali, Chris Pissarides, KevinSheedy, Tommy Sveen, Francesco Zanetti, and anonymous referees for their comments and suggestions.The views expressed here are those of the author and do not necessarily represent the views of Banco deEspana.

CARLOS THOMAS is at DG Economics, Statistics and Research, Banco de Espana(E-mail: [email protected]).

Received October 10, 2008; and accepted in revised form February 11, 2011.1. For a simple exposition of the search and matching model, see Pissarides (2000, ch. 1).

Journal of Money, Credit and Banking, Vol. 43, No. 6 (September 2011)C© 2011 The Ohio State University

1132 : MONEY, CREDIT AND BANKING

In fact, the existing literature has paid very little attention to the latter question. Thereason is an assumption commonly made in previous studies, namely that the firmssetting prices are different from the firms that are subject to search frictions (see Walsh2005, Trigari 2009, Christoffel and Linzert 2005, Andres, Domenech, and Ferri 2006,Blanchard and Galı 2010, Barnichon 2008, Thomas 2008). These two subsets of firmsare sometimes called “retailers” and “producers,” respectively, whereby the latter sellan intermediate good to retailers at a perfectly competitive price. This assumptionis very convenient, as it allows one to disentangle forward-looking vacancy-postingand pricing decisions and thus simplify the analysis. However, it eliminates fromthe outset the possibility of analyzing the effect of search frictions on the pricingdecisions of individual firms. The aim of this paper is to build a model where pricesetters do face such frictions and analyze the resulting implications for equilibriumdynamics. In particular, I consider a framework in which firms reset their prices atrandom intervals. In order to meet a sudden change in demand for its product, eachfirm can immediately adjust the number of hours worked by its employees. However,in order to adjust employment, the firm must first incur the cost of posting vacanciesand then wait for the latter to be filled.

I find that the interaction of price-setting decisions and search frictions slows downthe adjustment of the price level to shocks. The reason is the following. Due to searchfrictions, firms’ short-run marginal costs depend on the cost of increasing productionalong the intensive margin of labor. Wage bargaining between the firm and its workersimplies that the latter must be compensated for the disutility of work. If the latteris realistically convex in hours worked, firms’ marginal cost curves become upwardsloping.2 This gives price-setting firms an incentive to keep their prices in line withthe overall price level. That is, search frictions give rise to real rigidities in prices,using Ball and Romer’s (1990) terminology. For example, suppose that following anaggregate shock that decreases marginal costs, each price setter considers a certainreduction in its nominal price. Given the prices of other firms, the reduction in thefirm’s nominal price represents a reduction in its real price. This leads the firm toanticipate stronger sales and therefore higher marginal costs for the duration of theprice contract. As a result, the firm ends up choosing a smaller price cut than the oneinitially considered. Because all price setters follow the same logic, real rigiditiesslow the adjustment of the overall price level in response to the same fluctuationsin average real marginal costs. This effect is reflected in a flatter slope of the NewKeynesian Phillips curve (NKPC).

The real rigidity mechanism just described is absent in New Keynesian search-and-matching models with a producer–retailer structure, because each retailer’s marginalcost (the price of the intermediate good) is independent of its own output. Interest-ingly, I show that the log-linear equilibrium conditions in a producer–retailer modelwith identical preferences and technology are exactly the same as in the modelwith real rigidities, except for the slope of the NKPC. This allows me to use the

2. The notion that the marginal wage (and hence the marginal cost of production) is increasing inaverage hours per employee goes back to Bils (1987). See also Rotemberg and Woodford (1999).

CARLOS THOMAS : 1133

producer–retailer model as a “control” for isolating the effect of real rigidities onequilibrium dynamics. Two main results arise. First, inflation becomes less volatileand more persistent for a given frequency of nominal price adjustment. Through thisendogenous mechanism, real rigidities help the model match inflation dynamics with-out the need of assuming unrealistically long price contracts. This is a well-knownproperty of real rigidity mechanisms, as exemplified by models with firm-specificcapital (Sveen and Weinke 2005, Woodford 2005, Altig et al. 2011) and models withindustry-specific labor markets (Woodford 2003).

Second, and perhaps more importantly, real rigidities improve the empirical perfor-mance of the New Keynesian search-and-matching model along those labor marketdimensions that the standard New Keynesian model is not designed to address. For aplausible calibration, and relative to the producer–retailer specification, I show thatthe model with real rigidities comes closer to U.S. data in two important dimensions:the volatility of unemployment (both in absolute terms and relative to output), andthe volatility of the extensive margin of labor (employment) relative to the intensivemargin (hours per employee).3 At the heart of these results is again the interactionbetween infrequent price adjustment and search frictions, together with the long-termnature of employment relationships in this framework. Once the firm sets its price,its output is demand determined and its revenue is independent of its number ofemployees. Therefore, job creation decisions are driven, not by the marginal revenueproduct of new hires (as in flexible price models), but by the fact that hiring additionalworkers allows the firm to satisfy its future demand with less hours per employee andthus reduce its wage bill. This implies that, when firms expect hours per employeeto be higher in future periods, they have an incentive to create more jobs. By makingthe price level more sluggish, real rigidities amplify fluctuations in aggregate demandand hence in hours per employee; as a result, job creation becomes more volatile andso does unemployment. Crucially, since what matters for job creation is the entireexpected path of hours per employee (due to the ongoing nature of jobs), a certainincrease in the volatility of hours per employee produces a more than proportionalincrease in the volatility of employment; the same effect implies that the increase in(un)employment volatility takes place also relative to output. Notice that these la-bor market effects are indirect, as they operate through the increased sluggishness ofprices produced by the flattening of the NKPC. I conclude that integrating search fric-tions and staggered price adjustment at the firm level has important payoffs in termsof labor market fluctuations, while increasing only slightly the model’s complexity.

Other papers have departed from the producer–retailer assumption in the contextof New Keynesian models with search and matching frictions. A notable example is

3. This part of the analysis has some connection with the literature on unemployment volatility insearch models initiated by Shimer (2005). The latter author emphasized the inability of the canonicalsearch and matching model to produce realistic unemployment fluctuations (the so-called “unemploymentvolatility puzzle”). An extensive literature has subsequently suggested a number of mechanisms aimed atamplifying unemployment fluctuations in search models. While this paper is not aimed at addressing theunemployment volatility puzzle, it does illustrate that real rigidities provide an amplification mechanismin the context of New Keynesian models with search frictions.


Krause and Lubik (2007). These authors assume quadratic costs of price adjustment,rather than staggered price adjustment. As a result, price decisions are symmetricacross firms. Since all real prices are one, the real rigidity effect is absent in sucha framework.4 More closely related is the independent work of Sveen and Weinke(2009) and Kuester (2010). Sveen and Weinke identify a real rigidity mechanismsimilar to the one analyzed here. Our papers differ mainly in focus: whereas Sveenand Weinke emphasize the implications of strategic complementarities in price settingand of real wage stickiness for inflation dynamics, I stress the consequences of realprice rigidities for the cyclical behavior of the labor market. Kuester’s model featuresfirm–worker pairs where both nominal prices and wages are bargained in a staggeredfashion.5 This gives rise to real rigidities in prices as well as in wages, the latter effectamplifying fluctuations in unemployment. The mechanism is therefore different fromthe one presented here, which does not rely on staggered wage bargaining. Our papersalso differ in terms of scope. Kuester incorporates a number of additional frictions(habit formation, price indexation, and wage indexation) and estimates the modelusing a number of U.S. macro-economic time series. Instead, I compare two calibratedsearch and matching models where monopolistic competition and staggered pricesetting are the only additional frictions, with the purpose of isolating the effect of thereal price rigidities on equilibrium dynamics.

This paper is also related to previous analyses of how the specificity of labor cangive rise to real rigidities in New Keynesian models. In particular, Woodford (2003, ch.3) considers a setup of industry-specific labor markets where firms in each industryhire labor at that industry’s perfectly competitive wage. This generates upward-sloping marginal cost curves at the industry level and hence real rigidities. This paperconsiders instead a framework in which the search frictions that characterize the labormarket give rise endogenously to long-run employment relationships, thus makinglabor specific to each firm.

There also exists a parallel between the real rigidity mechanism studied in thispaper and the one arising in models of firm-specific capital (Sveen and Weinke2005, Woodford 2005, Altig et al. 2011). Here, the employment stock plays ananalogous role to the capital stock in the latter class of models, namely that ofbeing a firm-specific endogenous state variable. This implies similar complicationsin firms’ price-setting decisions, because the latter interact with forward-lookinghiring decisions (here) or investment decisions (in firm-specific capital models) in anontrivial manner. In order to solve for pricing decisions, I use Woodford’s (2005)solution method, which he develops in the context of a model with firm-specificcapital. Regarding the nature of the real rigidity mechanism, in both cases the latterarises because the firm’s marginal cost curve becomes upward sloping, although fordifferent reasons. In the case of firm-specific capital, marginal costs slope upward

4. Krause and Lubik (2007) focus their analysis on the relevance of real wage rigidity for inflationdynamics.

5. Notice that, in the model proposed here, firms are assumed to employ many workers, which realis-tically allows them to change both margins of labor over time.


due to decreasing marginal returns to labor for a given capital stock. Here, it is dueto increasing marginal disutility of labor and hence increasing marginal wages.

The remainder of the paper is organized as follows. Section 1 presents the model.Section 2 shows how to solve for individual pricing decisions in this framework, usingWoodford’s (2005) methodology. I then derive the NKPC and analyze the effect ofreal rigidities on equilibrium dynamics. Section 3 calibrates the model and providesa quantitative assessment of the theoretical mechanisms. Section 4 concludes.

1. THE MODEL

I now present a New Keynesian model with search and matching frictions inthe labor market. The model therefore brings together two frameworks that havebecome the standard for analyzing the monetary transmission mechanism and thecyclical behavior of the labor market, respectively. The main difference with respectto previous models of this type is that I do not separate the firms making the pricingdecisions from the firms that face search frictions. Instead, I consider a single setof firms that set prices and post vacancies in a labor market characterized by searchfrictions.

1.1 The Matching Function

The search frictions in the labor market are represented by a matching function,m(vt, ut), where vt is the total number of vacancies and ut is the total number ofunemployed workers. Normalizing the labor force to 1, ut also represents the un-employment rate. The function m is strictly increasing and strictly concave in botharguments. Assuming constant returns to scale in the matching function,6 the match-ing probabilities for unemployed workers, m(vt, ut)/ut = m(vt/ut, 1), and for vacancies,m(vt, ut)/vt = m(1, ut/vt), are functions of the ratio of vacancies to unemployment,also known as labor market tightness. I denote the latter by θ t ≡ vt/ut. In what follows,I let p(θ t) ≡ m(θ t, 1) denote the matching probability for unemployed workers. Thelatter is an increasing function of θ t: in a tighter labor market, job seekers are morelikely to find jobs. Similarly, I let q(θ t) ≡ m(1, 1/θ t) denote the matching probabilityfor vacancies. The latter is decreasing in θ t: firms are less likely to fill their vacanciesin a tighter labor market.

1.2 Households

In the presence of unemployment risk, we may observe differences in consumptionlevels between employed and unemployed consumers. However, under the assump-tion of perfect insurance markets, consumption is equalized across consumers. This

6. See Petrongolo and Pissarides (2001) for empirical evidence of constant return to scale in thematching function for several industrialized economies.


is equivalent to assuming the existence of a large representative household, as in Merz(1995). In this household, a fraction nt of its members are employed in a measure-onecontinuum of firms. The remaining fraction ut = 1 − nt search for jobs. All memberspool their income so as to ensure equal consumption across members. Householdwelfare is given by

Ht = u(ct ) − nt b −∫ 1

0nit

h1+η

i t

1 + ηdi + βEt Ht+1, (1)

where nit and hit represent the number of workers and hours per worker, respectively,in firm i ∈ [0, 1], b is labor disutility unrelated to hit (forgone utility from homeproduction, commuting time, etc.), and

ct ≡(∫ 1

0c

γ−1γ

i t di

) γ

γ−1

is the Dixit–Stiglitz consumption basket, where γ > 1 measures the elasticity ofsubstitution across differentiated goods. Cost minimization implies that the nominalcost of consumption is given by Ptct, where

Pt ≡(∫ 1

0P1−γ

i t di

) 11−γ

is the corresponding price index. The household’s budget constraint is given by

Mt−1 + Rt−1 Bt−1 + Tt

Pt+

∫ 1

0nitwi t (hit )di + �t ≥ ct + Bt + Mt

Pt, (2)

where Mt−1 and Bt−1 are holdings of money and one-period nominal bonds, re-spectively, Rt−1 is the gross nominal interest rate, Tt are net cash transfers from thegovernment, wit(hit) is the wage income earned by workers in firm i as a function ofhours worked, and �t = ∫ 1

0 �i t di are aggregate real profits, which are reverted tohouseholds in a lump-sum manner.

Employed members separate from their jobs at the exogenous rate λ, whereas un-employed members find jobs at the rate p(θ t). Therefore, the household’s employmentrate evolves according to the following law of motion,

nt+1 = (1 − λ)nt + p(θt )(1 − nt ). (3)


It is useful at this point to find the utility that the marginal worker in firm i contributesto the household. Equations (1), (2), and (3), together with nt = ∫

10nitdi, imply that

∂ Ht

∂nit= u′(ct )wi t (hit ) − b − h1+η

i t

1 + η− p(θt )

∫ 1

0

v j t

vtβEt

∂ Ht+1

∂n jt+1d j

+ (1 − λ)βEt∂ Ht+1

∂nit+1,

(4)

where p(θ t)vjt/vt is the probability that an unemployed member is matched to firm j ∈[0, 1]. The right-hand side of equation (4) consists of the real wage (in utility units),minus labor disutility and outside opportunities, plus the continuation value of thejob.

I assume the existence of a standard cash-in-advance (CIA) constraint on thepurchase of consumption goods.7 Assuming that goods markets open after the closingof financial markets, the household’s nominal expenditure in consumption cannotexceed the amount of cash left after bond transactions have taken place,

Pt ct ≤ Mt−1 + Tt − Bt . (5)

Cash transfers are given by Tt = Mst − Ms

t−1, where Mst is exogenous money supply.

The growth rate of money supply, ζ t ≡ log (Mst /Mt−1

s), follows an AR(1) process,ζ t = ρmζ t−1 + εm

t , where εmt is an iid shock. Assuming that the net nominal interest

rate (i.e., the opportunity cost of holding money) is always positive, equation (5)holds with equality. In equilibrium, money demand equals money supply, Mt = Ms

t ,which implies Mt−1 + Tt = Mt. Combining this with (5) and the fact that bonds arein zero net supply (Bt = 0), I obtain

Pt ct = Mt . (6)

1.3 Firms

The value of firm i ∈ [0, 1] in period t is given by

Vit = Pit

Ptyd

it − wi t (hit )nit − χ

u′(ct )vi t + Etβt,t+1Vit+1,

where Pit and ydit are the firm’s nominal price and real sales, respectively, vit are

vacancies posted in period t, χ is the utility cost for the management of posting a

7. I use the CIA specification for aggregate demand in order to simplify the exposition of the mainmechanisms. The second subsection of Section 3.3 considers alternative demand specifications, such asmoney in the utility function (MIU) or a Taylor rule for the nominal interest rate. All the main qualitativeresults regarding the effects of real rigidities on inflation and labor market volatility are invariant to thesealternative specifications.


vacancy, and β t,T ≡ βT−tu′(cT )/u′(ct) is the stochastic discount factor between periodst and T ≥ t. Due to imperfect substitutability among individual goods, the firm facesthe following demand curve for its product,

ydit =

(Pit

Pt

)−γ

yt , (7)

where aggregate demand is given by yt = ct. The firm’s production technology isgiven by

ysit = At nit hit ,

where At is an exogenous labor productivity process. The log of the latter, at ≡ log At,follows an AR(1) process, at = ρaat−1 + εa

t , where εat is an iid shock. Once the firm

has chosen a price, it commits to supplying whatever amount is demanded at thatprice, ys

it = ydit. This requires the following condition to hold at all times,(

Pit

Pt

)−γ

yt = At nit hit . (8)

In each period, the individual firm posts a number vit of vacancies. Assuming thatfirms are large, λ and q(θ t) are the fraction of workers that separate from the firmand the fraction of vacancies that the firm fills, respectively. Due to the time involvedin searching for suitable workers and (possibly) training them, new hires becomeproductive in the following period. Therefore, the firm’s workforce, nit, is given atthe start of the period. The law of motion of the firm’s employment stock is given by

nit+1 = (1 − λ)nit + q(θt )vi t . (9)

Let mcit and φit denote the Lagrange multipliers with respect to constraints (8) and(9), respectively. Therefore, mcit represents the real marginal cost of production. Thefirm chooses the state-contingent path {hit, vit, nit+1}∞

t=0 that maximizes

E0

∞∑t=0

β0,t

⎧⎪⎪⎨⎪⎪⎩(Pit/Pt )1−γ yt − wi t (hit )nit − χvi t/u′(ct )

+ mcit [At nit hit − (Pit/Pt )−γ yt ]

+φi t [(1 − λ)nit + q(θt )vi t − nit+1]

⎫⎪⎪⎬⎪⎪⎭ .

The first-order conditions are given by

mcit = w′i t (hit )

At, (10)

χ

u′(ct )= q(θt )φi t , (11)


φi t = Etβt,t+1 [mcit+1 At+1hit+1 − wi t+1(hit+1) + (1 − λ)φi t+1] . (12)

According to equation (10), the real marginal cost is given by the ratio betweenthe real marginal wage, w′

it(hit), and the marginal product of labor, At. Intuitively,since employment is predetermined, the firm needs to raise hours per employee inorder to increase production. This comes at a marginal cost of w′

it(hit) per employee.Equation (11) says that the marginal cost of posting a vacancy must equal the proba-bility that the vacancy is filled times the expected value of an additional worker in thefollowing period. The latter, from equation (12), is given by the expected marginalreduction in the firm’s cost, minus the expected wage to be paid to the new hire, plusher continuation value for the firm.

Wage bargaining. I assume Nash wage bargaining between the firm and each indi-vidual worker. Both the firm and the worker enjoy an economic surplus from theiremployment relationship. The worker’s surplus in consumption units, which I denoteby Sw

it ≡ (∂Ht/∂nit)/u′(ct), is given by equation (4) divided by u′(ct), that is,

Swi t = wi t (hit ) − b + h1+η

i t /(1 + η)

u′(ct )− p(θt )

∫ 1

0

v j t

vtEtβt,t+1Sw

j t+1d j

+ (1 − λ)Etβt,t+1Swi t+1. (13)

On the firm’s side, the surplus obtained from the marginal worker is given by

S fit = mcit At hit − wi t (hit ) + (1 − λ)Etβt,t+1S f

it+1. (14)

The term mcitAthit is the marginal increase in costs that the firm would have to incurif the employee walked away from the job. Since the firm is demand constrained, itwould have to make up for the lost production, Athit, by raising working hours forall other employees, which comes at a cost of mcitAthit. Therefore, the contributionof the marginal worker to flow profits is given, not by the marginal revenue productof the worker (as in standard real business cycle [RBC] models), but by the marginalreduction in the wage bill.8

Let ξ denote the firm’s bargaining power. Nash bargaining implies the followingsurplus-sharing rule,

(1 − ξ )S fit = ξ Sw

i t . (15)

8. This result is analogous to the one in Woodford’s (2005) model of firm-specific capital, where themarginal contribution of capital to flow profits is given by the marginal reduction in the wage bill, ratherthan the marginal revenue product of capital.


Combining this with (13) and (14), I obtain the following wage agreement,

wi t (hit ) = (1 − ξ )mcit At hit

+ ξ

[b + h1+η

i t /(1 + η)

u′(ct )+ p(θt )

∫ 1

0

v j t

vtEtβt,t+1Sw

j t+1d j

]. (16)

Therefore, the worker receives a weighted average of her contribution to cost reductionand the opportunity cost of holding the job (the sum of labor disutility and outsideoptions). It is possible to simplify the equation (16). Notice first that, from equations(12) and (14), it follows that φit = Etβ t,t+1Sf

it+1. This, together with equations (11)and (15), implies that∫ 1

0

v j t

vtEtβt,t+1Sw

j t+1d j = 1 − ξ

ξ

∫ 1

0

v j t

vtEtβt,t+1S f

jt+1d j = 1 − ξ

ξ

χ

q(θt )u′(ct ).

Inserting this into equation (16), and using the fact that p(θ t)/q(θ t) = θ t, I finallyobtain

wi t (hit ) = (1 − ξ )

[mcit At hit + χ

u′(ct )θt

]+ ξ

[b + h1+η

i t /(1 + η)

u′(ct )

]. (17)

Vacancy posting. Combining the first-order conditions (11) and (12), and the realwage schedule, equation (17), I obtain the following expression for the vacancy-posting decision,

χ

q(θt )= βEt

{ξ

[u′(ct+1)mcit+1 At+1hit+1 − b − h1+η

i t+1

1 + η

]

− (1 − ξ )χθt+1 + (1 − λ)χ

q(θt+1)

}. (18)

The real wage schedule, equation (17), implies the following real marginal wage,

w′i t (hit ) = (1 − ξ )mcit At + ξ

hη

i t

u′(ct ).

Using this to substitute for w′it(hit) in equation (10), I can express the real marginal

cost in terms of the marginal rate of substitution between consumption and labor,

mcit = hη

i t/u′(ct )

At. (19)


Using this in equation (18), I finally obtain

χ

q(θt )= βEt

[ξ

(η

1 + ηh1+η

i t+1 − b

)− (1 − ξ )χθt+1 + (1 − λ)

χ

q(θt+1)

]. (20)

According to equation (20), the firm’s incentives to hire are driven by fluctuationsin the expected path of hours per employee. Intuitively, if the firm expects hours tobe higher in the future, it also expects larger reductions in its wage bill from havingadditional workers. This leads the firm to post more vacancies today, up to the pointin which the expected marginal benefit of hiring equals its marginal cost, χ /q(θ t).9

Pricing decision. As is standard in the New Keynesian literature, I use the Calvo(1983) model of staggered price setting. Each period, a randomly selected fraction δ

of firms cannot change their price. Therefore, δ represents the probability that a firmis not able to change its price in the following period. At any time t, the part of thefirm’s value that depends on its current price is given by

E0

∞∑T =t

δT −tβt,T

{(Pit

PT

)1−γ

yT − mciT | t

(Pit

PT

)−γ

yT

}, (21)

where the subscript T | t denotes period T values conditional on the firm not havingreset its price since period t. Therefore, mciT | t is the firm’s real marginal cost inperiod T conditional on the price Pit being still in place. When a firm has the chanceto reset its price, it chooses Pit so as to maximize (21). The first-order condition isgiven by

Et

∞∑T =t

δT −tβt,T Pγ

T yT

(P∗

i t

PT− γ

γ − 1mciT | t

)= 0, (22)

where P∗it is the pricing decision. Using the expression for real marginal cost, equa-

tion (19), and the fact that hours must adjust in order for the firm to meet demand,hit = yit/(Atnit), I can express mciT | t as a function of the firm’s output in period T ,

mciT | t =(

yiT | t

AT niT | t

)η 1

u′(cT )AT, (23)

where yiT | t = (P∗it/PT )−γ yT . Equation (23) implies that, under the assumption of

convex labor disutility (η > 0), the firm’s marginal cost curve is an increasingfunction of its own output level.

9. Notice that, if one were to assume instantaneous hiring instead of time-to-hire (hence replacingequation (9) by nit = (1 − λ)nit−1 + q(θ t)vit), the resulting job creation condition would feature currenthours, rather than expected hours (as in equation (20)). Under the maintained assumption of linear hiringcosts, hours per worker would then be equalized across firms, hit = ht for all i. Since marginal costs underinstantaneous hiring would still be given by equation (19), the latter would also be equalized across firms,mcit = mct for all i, thus implying the absence of real rigidities.


2. LOG-LINEAR EQUILIBRIUM DYNAMICS

Following standard practice in the New Keynesian literature, I now perform alog-linear approximation of the equilibrium conditions around a zero-inflation steadystate. This will allow me to obtain the law of motion of inflation, also known as the“New Keynesian Phillips curve.” At this point, I assume the following functionalforms for the utility function and the matching function,

u(c) = c1−σ−1

1 − σ−1,

m(v, u) = ςvεu1−ε,

where σ , ς > 0 and ε ∈ (0, 1). In terms of notation, I use “hats” to denote log-deviations of a certain variable from its steady-state value, and “tildes” to denotelog-deviations of that variable from its cross-sectional average.

2.1 Relative Dynamics of the Firm

Log-linearization of the firm’s pricing decision, equation (22), yields

log P∗i t = (1 − δβ)Et

∞∑T =t

(δβ)T −t [mciT | t + log PT ]. (24)

Equation (23) implies that the real marginal cost in period T ≥ t of a firm that hasnot changed its price since period t can be expressed as

mciT | t = mcT + η(yiT | t − yT ) − ηniT | t , (25)

where

yiT | t = yT − γ (log P∗i t − log PT ) (26)

and mcT is the average real marginal cost. Notice that a firm’s relative marginal costis decreasing in its relative stock of workers, niT | t . Having more workers allows thefirm to produce a certain amount of output with a smaller number of hours per worker,which reduces the marginal labor disutility of its workers and hence the marginal realwage. I now combine (24), (25), and (26) to obtain

(1 + ηγ ) log P∗i t = (1 − δβ)Et

∞∑T =t

(δβ)T −t [mcT + (1 + ηγ ) log PT − ηniT | t ].

(27)

This expression for a firm’s pricing decision is very similar to the one produced by astandard New Keynesian model (see Walsh 2003, ch. 3). The only difference is thepresence of the Et niT | t terms, which reflect the fact that a firm’s marginal cost is


decreasing in its stock of workers. These additional terms complicate the analysis inthe following way. In order to determine log P∗

it, we need to compute the expectedpath of niT | t . The latter, however, depends on the firm’s current and future expectedvacancy-posting decisions, which in turn depend on the price chosen today. Solvingfor the firm’s pricing decision therefore requires that one considers the effect of afirm’s relative price on the evolution of its relative employment stock.

With this purpose, I now follow Woodford’s (2005) method to solve for the firm’srelative dynamics.10 I start by noticing that, in a log-linear approximation, the firm’spricing decision is a linear function of the state of the economy and its individualstate, ni t . On the other hand, since price setters are randomly chosen, their averageemployment stock coincides with the economy-wide average employment stock.Therefore, it is plausible to guess that a firm’s pricing decision, relative to the averagepricing decision, is proportional to its relative employment stock,

log P∗i t = log P∗

t − τ ∗ni t . (28)

I now log-linearize the vacancy-posting decision, equation (20), and rescale theresulting expression by u′(c)y/n to obtain

sv

λ(1 − ε)θt = βEt

[ξ

η

μhi t+1 +

(1 − λ − 1 − ξ

1 − εp(θ )

)sv

λ(1 − ε)θt+1

], (29)

where sv ≡ [χ /u′(c)]v/y is vacancy-posting costs over GDP in the steady state and μ

≡ γ /(γ − 1) is the monopolistic markup.11 Notice that the only idiosyncratic term inequation (29) is Et hit+1. The latter depends on Pit (by affecting the firm’s demand int + 1 should it not reset its price) as well as on its stock of workers at the beginningof t + 1. It is now possible to obtain the following result.12

PROPOSITION 1. Let relative pricing decisions be given by equation (28), up to alog-linear approximation. Then, the relative employment stock of any firm evolvesaccording to

nit+1 = −τ n (log Pit − log Pt ) , (30)

where

τ n = γ δ

1 − γ (1 − δ)τ ∗ . (31)

Intuitively, firms with a higher price in the current period also expect to have ahigher price in the next period, which means that they also expect lower demand.

10. Woodford (2005) develops his method in the context of a model where capital, rather than labor,is specific to each individual firm.

11. In the derivation of equation (29), I use the steady-state relations hη = u′(c)mcA (equation (19) inthe steady state), mc = 1/μ (derived from equation (22)) and h = y/(An). I have also used the fact that, inthe steady state, q(θ )v = λn.

12. The proofs of all propositions are in the Appendix.


Anticipating this, such firms post a number of vacancies that leaves them with asmaller workforce than the average firm in the following period. Proposition 1 allowsme to write

Et

∞∑T =t

(δβ)T −t niT | t = ni t + δβEt

∞∑T =t

(δβ)T −t niT +1|t

= ni t − δβτ n Et

∞∑T =t

(δβ)T −t (log P∗i t − log PT ). (32)

Using (32) in equation (27), I can write the firm’s pricing decision as

(1 + φ) log P∗i t = (1 − δβ)Et

∞∑T =t

(δβ)T −t [mcT + (1 + φ) log PT ] − (1 − δβ)ηni t ,

(33)

where φ ≡ ηγ − δβητ n. Averaging (33) across price setters, and using the fact thatthe latter are randomly chosen, I obtain

(1 + φ) log P∗t = (1 − δβ)Et

∞∑T =t

(δβ)T −t [mcT + (1 + φ) log PT ]. (34)

Substracting (34) from (33) yields (1 + φ)(log P∗i t − log P∗

t ) = −(1 − δβ)ηni t . Thisis consistent with my initial guess, equation (28), only if

τ ∗ = (1 − δβ)η

1 + ηγ − δβητ n. (35)

Therefore, if relative pricing decisions and relative employment stocks are to havea solution, the latter is given by equations (28) and (30), respectively, where theparameters τ* and τ n must satisfy equations (31) and (35). The following resultestablishes that such a solution exists and is unique.

PROPOSITION 2. The firm’s relative employment stock evolves according to equa-tion (30), where the parameter τ n > 0 is given by equation (31). A price setter’s pricedecision, relative to the average price decision, is given by equation (28), where

τ ∗ = −b − √b2 − 4ac

2a> 0,

a ≡ (1 + ηγ )γ (1 − δ) > 0,

b ≡ − [1 + γ (2 − δ − δβ)η] < 0,

c ≡ (1 − δβ)η > 0.


2.2 Real Rigidities and Inflation Dynamics

I am now ready to discuss the presence of real rigidities in this framework and howthey affect inflation dynamics. The average pricing decision, equation (34), can bewritten as

(1 + φ) log P∗t = (1 − δβ)[mct + (1 + φ) log Pt ] + δβEt (1 + φ) log P∗

t+1. (36)

In the Calvo model of staggered price setting, the law of motion for the price level isgiven by P1−γ

t = δP1−γ

t−1 + (1 − δ)(P∗t )1−γ . The latter admits the following log-linear

approximation, log P∗t − log Pt = [δ/(1 − δ)]π t, where π t ≡ log (Pt/Pt−1) is the

inflation rate. Combining this with (36), I obtain the following NKPC,

πt = κmct + βEtπt+1, (37)

where

κ ≡ (1 − δβ)(1 − δ)

δ

1

1 + φ, (38)

φ ≡ ηγ − δβητ n. (39)

The parameter φ has two components, ηγ and δβητ n. The term ηγ reflects theexistence in this framework of strategic complementarities in price setting, alsoknown as real rigidities after Ball and Romer (1990). This mechanism has the effectof slowing down the adjustment of the overall price level in response to fluctuationsin average real marginal costs. To see this, take a price setter that is consideringa reduction in its price. Given the prices of other firms, a reduction in the firm’snominal price represents also a reduction in its real price. This increases its sales(with elasticity γ ) and therefore, given its employment stock, the required amountof hours per worker. This increases the firm’s marginal costs through the increase inworkers’ marginal disutility of labor (with elasticity η). The anticipated rise in itscurrent and future expected marginal costs leads the firm to choose a smaller pricecut than the one initially considered. Therefore, the fact that some firms keep theirprices unchanged leads price setters to change theirs by little, hence the “strategiccomplementarity” in price setting. Equivalently, price setters have an incentive tokeep their prices in line with the overall price level, hence the “real rigidity” in prices.Because all price setters follow the same logic, the price level and therefore inflationbecome less sensitive to changes in average real marginal costs.

The term δβητ n reflects the fact that the position of a firm’s marginal cost curvedepends on its stock of workers, by affecting how many hours per worker are neededto produce a certain amount of output. This has the effect of accelerating priceadjustment. To see this, take the same firm considering a price cut. From Proposition1, today’s price cut leads the firm to expect a larger relative employment stock and,by equation (25) a lower marginal cost in future periods. Holding everything else


constant, this would lead the firm to choose an even larger price cut than initiallyconsidered. It is possible to show, however, that this latter effect is dominated by thereal rigidity effect. Using the definition of τ n, equation (31), I can write

ηγ − δβητ n = ηγ − δβη

(γ δ

1 − γ (1 − δ)τ ∗

)= ηγ

(1 − δ2β

1 − γ (1 − δ)τ ∗

).

The latter expression is positive only if the expression in brackets is. The Appendixshows that τ* must be smaller than 1/γ in order for the model to have convergentdynamics. This implies that

1 − δ2β

1 − γ (1 − δ)τ ∗ > 1 − δ2β

1 − γ (1 − δ) 1γ

= 1 − δβ > 0.

It follows that φ > 0. Therefore, the real rigidities in price setting that arise undersearch frictions unambiguously flatten the NKPC.

It is interesting to note the parallelism between the real rigidity mechanism justexplained and the one arising in models of firm-specific capital (Sveen and Weinke2005, Woodford 2005, Altig et al. 2011). In the present framework, the firm’s em-ployment stock plays an analogous role to the capital stock in models of firm-specificcapital, namely, that of being a firm-specific endogenous state variable. In the latterclass of models, firms are able to adjust production in the short run by varying theirlabor input; here, they adjust production by varying the number of hours workedby their employees. In both cases, real rigidities arise because the firm’s marginalcost curve becomes upward sloping. In the case of firm-specific capital, this is dueto decreasing marginal returns to labor for a given capital stock, although such aneffect may be reinforced by the existence of industry-specific labor markets (see, e.g.,Woodford, 2005). Here, it is due to increasing marginal disutility of labor and henceincreasing marginal wages.13

13. It is interesting to compare the slope of the NKPC in this framework with the one that arises in amodel of firm-specific capital. As shown by Woodford (2005), the slope coefficient in such a model canbe expressed again as in equation (38), with the parameter φ replaced by

φ f sk ≡(

η

α+ 1 − α

α

)γ − δβ

1 − δβλ

(η

α+ 1 − α

α− η

)τ ,

where 1 − α is the capital elasticity in a Cobb–Douglas production function, and both τ and λ are functionsof structural parameters. The interpretation of φfsk is analogous to that of φ: the first component, (η/α +(1 − α)/α)γ , captures the real rigidity effect, where decreasing returns to labor (α < 1) add to convexlabor disutility (η > 0) as a source of strategic complementarities. The second term captures the fact thatthe position of the marginal cost curve depends on the capital stock. While a comparative analysis offirm-specific capital and search frictions as sources of real rigidities would be an interesting exploration,such a comparison is, however, beyond the scope of this paper.


2.3 Aggregate Equilibrium

Equilibrium in the search model with real rigidities is characterized by the AR(1)processes for exogenous money growth and labor productivity, together with thefollowing six equations,

πt = κmct + βEtπt+1, (E1)

mct = ηht + σ−1 yt − at , (E2)

yt = yt−1 + ζt − πt , (E3)

sv


[ξ

η

μht+1 +

(1 − λ − 1 − ξ

1 − εp(θ )

)sv

λ(1 − ε)θt+1

], (E4)

ht = yt − at − nt (E5)

nt+1 = (1 − λ − p(θ ))nt + λεθt . (E6)

Equation (E2) is obtained by log-linearizing (19), averaging across all firms andusing the fact that ct = yt . Equation (E3) is obtained by log-linearizing (6), takingfirst differences and using again ct = yt . Equation (E4) is obtained by averagingequation (29) across firms.14 Equation (E5) is the log-linear version of the firm’sproduction function, after averaging across firms. Finally, equation (E6) is the log-linear approximation of equation (3), where I also use the steady-state condition λn= p(θ )(1 − n). This log-linear representation allows to understand easily the effectof shocks on the economy. In response to a positive monetary shock (an increasein ζ t in equation (E3)), aggregate demand increases, which puts upward pressureon real marginal costs and inflation. To the extent that the increase in demand ispersistent, firms anticipate longer hours per employee in the future. In order to avoidlarge pay rises for existing employees, firms post more vacancies. This results in atightening of the labor market (equation (E4)) and an increase in total employment(equation (E6)). In response to a positive productivity shock (an increase in at), realmarginal costs fall and so does inflation. For a constant level of nominal GDP, thefall in prices produces an expansion in aggregate demand (equation (E3)). The effecton employment is, however, ambiguous. If the expansion in output is strong enoughrelative to the increase in at, firms expect hours per employee to be higher, whichleads them to post more vacancies and thus increase employment. If the increase inoutput is weak enough, the opposite will be true.

14. As shown in the Appendix, Et hit+1 can be written as Et ht+1 − γ δ Pit − [1 − γ (1 − δ)τ ∗] ni t+1,which averages to Et ht+1.


Comparison to a search model with a producer–retailer structure. Most of theliterature on New Keynesian models with search and matching frictions separatesvacancy-posting and pricing decisions by assuming a producer–retailer structure,in which the former are subject to search frictions and the latter to staggered pricesetting. While simplifying the analysis, this assumption eliminates from the outsetthe possibility of analyzing price-setting decisions in an environment in which pricesetters cannot adjust employment costlessly and instantaneously. In such models,producers produce a homogenous intermediate good that is sold to retailers at aperfectly competitive real price. We may denote the latter by mct. Each retailerthen transforms the intermediate good into a differentiated final good using a lineartechnology. Therefore, mct represents the real marginal cost common to all retailers.

It is relatively straightforward to construct a model with this kind ofproducer–retailer structure that is otherwise equivalent to the model presented inSection 1. In particular, I may assume that household preferences and the productionfunction of producers are the same as in the model with real rigidities. The Appendixderives the equilibrium conditions in such a model. Once the producer–retailer modelis log-linearized, the NKPC is given by

πt = κpr mct + βEtπt+1, (40)

where

κpr ≡ (1 − δβ)(1 − δ)

δ> κ.

Therefore, the NKPC in the model with a producer–retailer structure is steeper thanin the model with real rigidities. Because retailers can buy as much intermediate inputas they need at the perfectly competitive price mct, their pricing decisions have noeffect on their own marginal costs. As a result, the real rigidity effect disappears.

As shown in the Appendix, the remaining log-linear equilibrium conditions in theproducer–retailer model are exactly the same as in the model with real rigidities,equations (E2)–(E6). The producer–retailer model thus serves as a “control” thatallows me to isolate the effect of real rigidities in models with search frictions andstaggered price setting. One important dimension of this comparison is the differencein inflation dynamics between both models. Real rigidities have the property ofslowing down the adjustment of the price level in response to different shocks, for agiven degree of nominal price rigidity (δ). As a result, the inflation response to shockswill be smaller on impact and more persistent in the model with real rigidities. Thiseffect is shared in general by models that incorporate a real rigidity mechanism,such as the New Keynesian model with firm-specific capital introduced by Sveen andWeinke (2005) and Woodford (2005), and further analyzed by Altig et al. (2011).15

15. As argued by Woodford (2005) and Altig et al. (2011), estimates of New Keynesian models (orNKPC) that abstract from real rigidities imply average durations of nominal price contracts that are toolong when compared with actual micro data. These authors propose firm-specific capital, and the resultingreal rigidities, as a way for econometric estimates to imply more realistic durations of price contracts.


More importantly, this comparison allows to measure the extent to which realrigidities affect the model’s behavior along those labor market dimensions that thestandard New Keynesian model is not designed to address, such as unemployment,employment, and hours per employee. Take for instance a positive money growthshock. Ceteris paribus, the slower response of the price level in the presence ofreal rigidities leads to a larger increase in aggregate demand. To the extent that theeconomic expansion is expected to persist, firms expect larger increases in hours peremployee. Since the latter are the driving force of job creation, real rigidities generatea larger rise in job creation and therefore a larger drop in unemployment. The nextsection will show that real rigidities also amplify the unemployment response toproductivity shocks. Therefore, real rigidities amplify the unconditional fluctuationsin employment and unemployment. Furthermore, because employment relationshipshave a long-term nature in this framework, firms base their hiring decisions on theentire expected path of hours per employee, such that a small increase in the latter’svolatility is enough to generate a large increase in employment volatility. Therefore,real rigidities also have the property of increasing the volatility of employmentrelative to that of hours per employee. The same effect implies that the amplificationof (un)employment fluctuations takes place not just in absolute terms, but also relativeto those in output. I now turn to the quantitative assessment of these mechanisms.

3. QUANTITATIVE ANALYSIS

3.1 Calibration

Following most of the literature on search and matching models, I calibrate themodel to monthly U.S. data. As in most of the RBC literature, I set the discountfactor to 4% per quarter, or β = 0.991/3. I also choose a standard value for theintertemporal elasticity of substitution, σ = 1 . It is customary to set 1/η on the basisof micro estimates of labor supply elasticities. According to Card’s (1994) review ofthe empirical literature, this elasticity is surely no higher than 0.5. I therefore set η to2 as a conservative choice.16

Regarding the New Keynesian side of the model, following the evidence in Bilsand Klenow (2004), I assume that firms change prices every 1.5 quarters, or 4.5months, which implies δ = (4.5 − 1)/4.5 = 0.78. As in Woodford (2005), I choose amonopolistic markup of μ = 1.15, which implies an elasticity of substitution amongdifferentiated goods of γ = μ/(1 − μ) = 7.67. Given the values of β, η, γ , and δ, theparameters governing relative firm dynamics are given by τ* = 0.08 and τ n = 6.90.From (39), the parameter measuring (net) real rigidity equals φ = 4.63. From (38),the slope of the NKPC equals κ = 0.011. This compares with a slope of κpr = 0.064in the producer–retailer model.

16. Notice that a lower labor supply elasticity (i.e.,, a higher η) would actually strengthen the realrigidity effect by reducing further the slope of the inflation equation. It is in this sense that η = 2 is chosenas a conservative calibration.


The parameters that describe the labor market are calibrated as follows. FollowingShimer (2005), I set the monthly separation probability, λ , to 0.035. Shimer (2008)and Pissarides (2009) calculate an average job-finding probability and an averagevacancies-to-unemployment ratio of 0.286 and 0.72, respectively, for the UnitedStates. I therefore target the steady-state values p(θ ) = 0.30 and θ = 0.72. Theelasticity of the matching function with respect to vacancies, ε, is set to 0.6, followingthe evidence in Blanchard and Diamond (1989). This together with p(θ ) = ςθε andthe targets for p(θ ) and θ imply ς = 0.365. Following standard practice in theliterature, I set the bargaining power parameter, ξ , equal to ε. Notice, however, that ξ

does not affect the slope of the NKPC. Finally, following Andolfatto (1996), Gertlerand Trigari (2009), and Blanchard and Galı (2010), I target a steady-state ratio ofvacancy-posting costs to GDP, sv, of 1%. With log utility, this requires setting theutility cost of posting a vacancy to χ = 0.133.17 One can then use the steady-statejob creation condition to solve for the fixed component of labor disutility, obtainingb = 0.563.18

The shock parameters are calibrated as follows. Following Shimer (2005), the pa-rameters of the labor productivity process are calibrated in a model-consistent way.Aggregating equation (8) across firms, we have that At = yt�t/(ntht), where ht ≡n−1

t

∫nithitdi are average hours per employee and �t ≡

∫(Pit/Pt)−γ di is a price disper-

sion term that experiences second-order fluctuations around one (Woodford 2003, ch.6). We thus have log At = log yt − log (ntht), up to a first-order approximation. UsingBLS data for real output (yt) and total hours (ntht) to construct a quarterly series forlog labor productivity, I obtain an autocorrelation coefficient and a standard deviationfor the corresponding monthly process of ρa = 0.86 and σ a = 0.56%, respectively.19

As in Krause and Lubik (2007), I set the autocorrelation coefficient of the moneygrowth process to 0.49 on a quarterly basis, or ρm = 0.491/3 on a monthly basis.Finally, the standard deviation of money growth shocks, σ m, is calibrated to matchthe standard deviation of real output in the data. Table 1 summarizes the calibration.

3.2 Impulse Responses

In order to illustrate graphically the effects of real rigidities on equilibrium dynam-ics, I now compare the economy’s response to shocks in the model with and withoutreal rigidities.

17. From the definition of sv, we have that χ = svyu′(c)/v = sv/v = sv/(θu), where I have used u′(c) =1/c, c = y, and v = θu. The values of p(θ ) and λ imply a steady-state unemployment rate of u = λ/(λ +p(θ )) = 0.1045, with together with sv = 0.01 and θ = 0.72 implies χ = 0.133.

18. In using equation (20) in the steady state to solve for b, I also use the fact that (γ − 1)/γ = mc =hηy/A = h1+ηn, which implies h = [(γ − 1)/(γ n)]1/(1+η), where n = 1 − u = 0.8955.

Notice that there are no unemployment benefits in the model (consumers are perfectly insured as aresult of the large household assumption). However, the steady-state ratio between the real value of totallabor disutility, [b + h1+η/(1 + η)]/u′(c), and worker output, y/n, can be interpreted as an (endogenous)counterpart to the unemployment payoff parameter in the standard search and matching model. Under mycalibration, the latter ratio equals 0.794, which is in between the values of the above-mentioned parameterused by Shimer (0.40) and Hagedorn–Manovskii (0.955), and similar to the value 0.71 used by Hall andMilgrom (2008) and Pissarides (2009).

19. See Section 3.3 for details about the data sources, the sample period, and the detrending procedure.


TABLE 1

PARAMETER VALUES

Value Description Target/source

β 0.991/3 Discount factor Standardσ 1 Intertemporal elast. of subs. Standardη 2 Convexity of labor disutility 1/η = 0.5δ 0.78 Fraction of sticky prices 1/(1 − δ) = 4.5 monthsγ 7.67 Elasticity of demand curves γ /(γ − 1) = 1.15λ 0.035 Job separation rate Shimer (2005)ς 0.365 Scale of matching fct. p(θ ) = 0.30, θ = 0.72ε 0.6 Elasticity of matching fct. Blanchard–Diamond (1989)ξ 0.6 Firm’s bargaining power ξ = εχ 0.133 Vacancy-posting cost [χ /u′(c)]v/y = 0.01b 0.56 Labor disutility parameter Steady-state JC conditionρa 0.86 AC of productivity shock Dataσ a 0.56% SD of productivity shock Dataρm 0.491/3 AC of monetary shock Krause–Lubik (2007)σ m 0.47% SD of monetary shock Output volatility

Reduced-form parameters

φ 4.63 Net real rigidityκ 0.011 Slope of NKPC, model with real rigiditiesκpr 0.064 Slope of NKPC, producer–retailer model (no real rigidities)sv 0.01 Vacancy-posting costs/GDP

Monetary shocks. Figure 1 displays the impulse responses of prices, inflation,output, and both labor margins to a one-standard-deviation positive shock to moneygrowth. The real rigidity mechanism slows down the adjustment of the price level.This is reflected in an inflation response that is both smaller on impact and morepersistent afterward. Given the exogenous expansion in nominal GDP, the moresluggish price adjustment leads to a larger expansion in output in the model withreal rigidities. As a result, the dynamic path of hours per employee experiences alarger increase. Since hiring incentives are driven by fluctuations in expected hoursper employee, job creation increases more strongly under real rigidities, which leadto a larger employment expansion.

Productivity shocks. Figure 2 displays the economy’s response to a one-standard-deviation positive shock to labor productivity. Again, the inflation response is moremuted and more persistent under real rigidities. Since nominal GDP remains un-changed in this simulation, the extra sluggishness in the price level leads to a weakerexpansion in aggregate demand. In both models, the expansion in aggregate demandis not strong enough to compensate for the fact that firms now need less labor toproduce the same output. As a result, total hours worked fall, the adjustment beingshared by both labor margins (hours per employee and employment).20 However, the

20. Such an effect of productivity shocks on total hours is consistent with a large body of empiricalevidence starting with Galı (1999). It is also consistent with the predictions of estimated medium-scaledynamic stochastic general equilibrium (DSGE) models of the U.S. economy, such as Smets and Wouters’(2007).


0 5 10 15 20 250

0.5

1

1.5

2

2.5price level

real rigidities

no real rigidities

0 5 10 15 20 250

0.1

0.2

0.3

0.4inflation

0 5 10 15 20 250

0.2

0.4

0.6

0.8output

0 5 10 15 20 250

0.1

0.2

0.3

0.4hours per employee

0 5 10 15 20 250

0.1

0.2

0.3

0.4employment

0 5 10 15 20 250

0.2

0.4

0.6

0.8total hours

FIG. 1. Impulse Responses to a Positive Monetary Shock.

0 5 10 15 20 25

-0.25

-0.2

-0.15

-0.1

-0.05

price level

real rigidities

no real rigidities

0 5 10 15 20 25-0.15

-0.1

-0.05

0

0.05inflation

0 5 10 15 20 250

0.1

0.2

output

0 5 10 15 20 25-0.6

-0.4

-0.2

0

0.2hours per employee

0 5 10 15 20 25-0.15

-0.1

-0.05

0

0.05employment

0 5 10 15 20 25-0.6

-0.4

-0.2

0

0.2total hours

FIG. 2. Impulse Responses to a Positive Productivity Shock.


TABLE 2

BUSINESS CYCLE STATISTICS

Model with real rigidities Model without real rigidities

U.S. data Monetary Productivity Both Monetary Productivity Both

σ (yt), % 2.03 2.00 0.41 2.05 0.66 0.70 0.95σ (ut), % 10.70 9.90 2.24 10.25 2.99 0.93 3.14σ (ut)/σ (yt) 5.28 4.95 5.46 5.00 4.53 1.33 3.31σ (vt)/σ (yt) 6.52 5.86 9.72 6.09 6.30 2.88 4.90σ (θ t)/σ (yt) 11.59 9.88 12.67 10.06 9.52 3.40 7.11σ (nt)/σ (yt) 0.62 0.58 0.64 0.58 0.53 0.15 0.39σ (ht)/σ (yt) 0.35 0.44 1.29 0.50 0.50 0.53 0.52σ (ntht)/σ (yt) 0.86 1.00 1.79 1.05 1.00 0.64 0.84σ (ht)/σ (nt) 0.57 0.76 2.03 0.87 0.95 3.40 1.35

ρ(wht , yt) 0.54 0.99 0.24 0.97 0.99 0.82 0.88

σ (π qt ), % 0.74 1.45 0.16 1.47 1.79 0.38 1.83

weaker output increase under real rigidities produces a stronger fall in total hours.Notice that the drop in the dynamic path of hours per employee is just slightly largerwhen real rigidities are present. However, because of the long-term nature of jobs inthis framework, such a drop is enough to generate a substantially larger drop in jobcreation and therefore in employment.

3.3 Labor Market Volatility

I now analyze the extent to which real rigidities affect the empirical performance ofthe New Keynesian model with search and matching frictions regarding labor marketaggregates. The second column of Table 2 displays a set of business cycle statistics inthe United States. I use seasonally adjusted data on real output in the nonfarm businesssector (yt, in model notation), hours of all persons in the nonfarm business sector(ntht, where ht ≡ n−1

t

∫ 10 nit hit di are average hours per employee), total nonfarm

payroll employment (nt), number of unemployed (ut), average hourly earnings inthe private sector (wh

t ≡ (nt ht )−1∫ 1

0 nitwi t (hit )di), and quarter-on-quarter inflationin the Consumer Price Index (πq

t ≡ log Pt − log Pt−3), all of them from the Bureau ofLabor Statistics; I also use the Conference Board’s Help-Wanted Advertising Indexas a proxy for vacancies (vt). The sample runs from 1964:Q1 to 2008:Q2. Sinceoutput and total hours are only available quarterly, and the data on employment,unemployment, inflation, hourly wages, and vacancies are monthly, I take quarterlyaverages of the latter five series. I then use the identities ht = (ntht)/nt and θ t = vt/ut

in order to obtain quarterly series for average hours per employee and labor markettightness, respectively. All data except inflation are logged and Hodrick-Prescott-filtered with a conventional smoothing parameter (1,600). The remaining columns of


Table 2 display the corresponding statistics generated by the model with and withoutreal rigidities.21

Two main messages can be extracted from Table 2. First, real rigidities amplifythe size of fluctuations in employment and unemployment, both in absolute termsand relative to those of output.22 For instance, the relative unconditional standarddeviation of unemployment is 95% of its data counterpart, compared to 63% in themodel without real rigidities. Similarly, relative employment volatility is 94% of thecorresponding value in the data, which contrasts with 63% in the absence of realrigidities. As can be seen in the fifth row, the effect of real rigidities operates throughthe amplification of fluctuations in labor market tightness, which is the single drivingforce of (un)employment fluctuations. Both models, however, overstate the relativestandard deviation of hours per employee by a factor of about one and a half. As aresult, the relative standard deviation of total hours is somewhat higher in the modelwith real rigidities than in the data.

Second, real rigidities increase the volatility of employment relative to that of hoursper employee. According to the model without real rigidities, the intensive marginof labor is about one-third more volatile than the extensive margin. This is clearlyat odds with the data, according to which the intensive margin is roughly half asvolatile as the extensive one. By amplifying employment fluctuations substantially,real rigidities manage to make employment more volatile than hours per employee,even though the ratio of standard deviations is still 30 percentage points higher thanin the data.23

Finally, it is interesting to discuss the implications of real rigidity for the cyclicalbehavior of inflation and real wages. As shown by the last line of the table, bothmodels overpredict the volatility of quarter-on-quarter inflation. This is not surprising,however, given the model’s simplicity and the fact that the shock processes have been

21. Model moments are calculated as follows. I simulate 624 months of artificial data. I take quarterlyaverages and discard the first 30 observations so as to eliminate the effect of initial conditions, whichleaves me with 178 observations (the sample size). I then calculate the relevant second moments. I repeatthis operation 200 times and finally take averages for each vector of moments.

22. Notice that amplification holds unconditionally, as well as conditional on each type of shock(to money growth and productivity), reflecting our earlier analysis of impulse responses. Most of theamplification in absolute volatility, σ (u), is due to monetary shocks, whereas most of the amplification inrelative volatility, σ (u)/σ (y), is due to productivity shocks.

23. While the paper’s focus is to understand the effects of search frictions and the resulting realrigidities on equilibrium dynamics, it is also interesting to ask how other model features (such as two labormargins, monopolistic competition and sticky prices, and monetary shocks in addition to productivityshocks) contribute to (un)employment volatility. First, it can be shown that, in a stripped down version ofthe model with flexible prices and without variations in hours per worker, unemployment is invariant toproductivity shocks. This neutrality result is analogous to the one found by Blanchard and Galı (2010) andfurther analyzed by Shimer (2010). Second, the latter result continues to hold after introducing variationsin the intensive margin. Third, introducing monopolistic competition and sticky prices (while abstractingfrom real rigidities and monetary shocks) produces a standard devation of unemployment of 0.93%, asshown in the second-to-last column of Table 2. Fourth, adding monetary shocks raises unemploymentvolatility to 3.14% (last column). Finally, adding real rigidities increases unemployment volatility to10.25%. It is therefore clear from this analysis that real rigidities are the most important factor contributingto unemployment fluctuations.


calibrated to match the cyclical volatility of labor productivity and output.24 Therelevant insight is that for a given frequency of price adjustment calibrated to matchmicro data, real rigidities work toward reducing inflation volatility and hence bringit closer to the data. The flip side of this coin is that, if one were to estimate bothmodels with macro data, the average duration of price contracts implied by the realrigidity specification would be shorter and hence closer to the micro data.25 Similarly,both models overpredict the cyclicality of average real wages, as measured by theircontemporaneous correlation with output. This is of course a consequence of wagesbeing perfectly flexible in the model. While introducing some form of wage rigiditywould improve the model’s performance along this dimension, the assumption ofwage flexibility allows me to isolate the effects of real rigidities in the absence of anyother amplification mechanisms.

Robustness: The great moderation period. As shown by a large number of studies,the United States seems to have experienced a significant break in the size of itsaggregate fluctuations starting approximately in 1984 (see, e.g., Kim and Nelson1999, McConnell and Perez-Quiros 2000). To the extent that such a break has affectedthe size of fluctuations in labor market aggregates relative to those of output, thebaseline results shown in Table 2 may be affected. For this reason, I now calculate thesame set of statistics as in Table 2 using the sample period 1984:Q1-2008:Q2, andcompare them with the corresponding statistics generated by the models. The resultsare displayed in Table 3.26

A quick comparison between Tables 2 and 3 reveals that the labor market hasactually become more volatile in relative terms in the Great Moderation period. Thatis, the decline in output volatility that has been extensively documented seems notto have been accompanied by a proportional decline in the volatility of the labormarket. As a result, both models now find it harder to match the observed labormarket volatility. However, the model with real rigidities again performs much betterin this regard. For instance, the relative standard deviation of unemployment is now71% of that in the data, compared to 41% when real rigidities are abstracted from.The corresponding figures for the relative standard deviation of employment are 76%and 44%, respectively. This allows the model with real rigidities to come closer tothe data also in terms of the volatility of total hours.

Also, the intensive margin of labor seems to have become more volatile relativeto the extensive margin in the Great Moderation period, although the former is stillless than three-quarters as volatile as the latter. Both models actually hit the target for

24. Another way to interpret these results is that the data seem to be calling for an even longer averageduration of price contracts than the one assumed in the baseline calibration (1.5 quarters).

25. See Altig et al. (2011) and Woodford (2005) for early discussions on the effects of real rigiditieson the inference of price adjustment frequencies.

26. The quarterly series of labor productivity since 1984 now implies an autocorrelation coefficientand standard deviation of the corresponding monthly process of ρa = 0.84 and σ a = 0.45, respectively.The standard deviation of the shock to money growth is recalibrated to match again the standard deviationof output (1.13% in 1984–2008), yielding σ m = 0.26%.


TABLE 3

BUSINESS CYCLE STATISTICS, GREAT MODERATION PERIOD (1984–2008)

Model with real rigidities Model without real rigidities

U.S. data Monetary Productivity Both Monetary Productivity Both

σ (yt), % 1.13 1.10 0.28 1.13 0.37 0.50 0.62σ (ut), % 8.07 5.45 1.67 5.67 1.68 0.70 1.81σ (ut)/σ (yt) 7.12 4.95 5.97 5.02 4.53 1.39 2.92σ (vt)/σ (yt) 9.18 5.90 11.00 6.32 6.31 3.10 4.50σ (θ t)/σ (yt) 15.79 9.90 14.08 10.20 9.52 3.63 6.36σ (nt)/σ (yt) 0.78 0.58 0.70 0.59 0.53 0.16 0.34σ (ht)/σ (yt) 0.56 0.44 1.51 0.56 0.50 0.59 0.56σ (ntht)/σ (yt) 1.22 1.00 2.04 1.09 1.00 0.70 0.82σ (ht)/σ (nt) 0.71 0.76 2.17 0.96 0.95 3.60 1.64

ρ(wht , yt) 0.11 0.99 0.16 0.94 0.99 0.78 0.84

σ (π qt ) 0.38 0.79 0.12 0.80 0.99 0.29 1.03

the relative standard deviation of hours per employee. However, the low volatility ofemployment in the model without real rigidities leads to the implausible predictionthat hours per employee are about two-thirds more volatile than employment. Incontrast, the model with real rigidities predicts slightly smaller fluctuations in theintensive margin than in the extensive one.

Robustness: Alternative demand specifications. The analysis so far has assumed asimple CIA specification for aggregate demand, in order to simplify the explanationof the main mechanisms at play. I now check the robustness of the baseline results toalternative specifications for aggregate demand. First, I consider a “money in the util-ity function” specification, in order to introduce an interest-sensitive money demandequation. Following Krause and Lubik (2007), I assume that households enjoy a utilityflow χ log (Mt/Pt) from their holdings of real money balances, Mt/Pt ≡ mt. The first-order condition for money, Mt, can be expressed as χ /mt = u′(ct)(Rt − 1)/Rt, which to-gether with the standard consumption Euler equation, u′(ct)/Pt = βRtEtu′(ct+1)/Pt+1,represent the new aggregate-demand block (replacing equation (6)). Log-linearizingthe latter two equations and the identity mt/mt−1 = (MtPt−1)/(PtMt−1) = exp (ζ t)/π t,and using ct = yt , we obtain

mt = σ−1 yt − β

1 − βRt ,

yt = Et yt+1 − σ (Rt − Etπt+1), (41)

mt = mt−1 + ζt − πt ,


TABLE 4

BUSINESS CYCLE STATISTICS, ALTERNATIVE DEMAND SPECIFICATIONS

Money in the utility function Taylor rule

U.S. data Real rigidities No real rigidities Real rigidities No real rigidities

σ (yt), % 2.03 2.02 1.22 2.03 1.26σ (ut), % 10.70 8.08 2.83 6.54 2.29σ (ut)/σ (yt) 5.28 4.00 2.32 3.22 1.82σ (vt)/σ (yt) 6.52 5.90 4.39 5.48 3.71σ (θ t)/σ (yt) 11.59 8.58 5.54 7.32 4.52σ (nt)/σ (yt) 0.62 0.47 0.27 0.38 0.21σ (ht)/σ (yt) 0.35 0.67 0.71 0.73 0.70σ (ntht)/σ (yt) 0.86 1.05 0.91 1.01 0.84σ (ht)/σ (nt) 0.57 1.44 2.64 1.93 3.30

ρ(wht , yt) 0.54 0.97 0.92 0.97 0.93

σ (π qt ), % 0.74 0.93 1.37 0.61 1.13

respectively, which replace equation (E3) in the log-linear representation of the model.The third and fourth columns of Table 4 report unconditional second moments in theMIU specification, for the case both with and without the real rigidity mechanism.Notice first that, relative to the results from the baseline CIA specification in Ta-ble 2, the MIU specification generates less labor market volatility. This reflects thefact that part of the variation in real money balances is now absorbed by nominalinterest rates and not just by output, hence dampening the effects on job creation and(un)employment. However, I find again that the version with real rigidities performsbetter in terms of labor market volatility.27 Relative unemployment volatility in themodel equals 76% of the empirical measure, compared to 44% in the model withoutreal rigidities; the volatility of vacancies, labor market tightness, and employment arealso better captured when real rigidities are at work. Both models now counterfactu-ally predict larger fluctuations in employment than in hours per employee, althoughthe model with real rigidities again performs better in this respect.

Second, following most of the recent New Keynesian literature, I consider a cash-less environment in which the central bank directly sets the nominal interest rate. Inparticular, I assume that the monetary authority follows a standard Taylor rule,

Rt = ρR Rt−1 + (1 − ρR)(φππt + φy�yt ) + εRt , (42)

where ρR, φπ, and φy capture the degree of interest-rate smoothing and the responseto inflation and output growth, respectively, and εR

t is an iid shock that replaces themonetary shock εm

t . Equation (42), together with (41), replace now equation (E3)

27. Real rigidities amplify employment and unemployment fluctuations also conditional on each typeof shock (monetary and productivity shocks). Conditional results are available upon request.


as the aggregate-demand block of the log-linearized model. The last two columnsof Table 4 report unconditional second moments generated by the Taylor rule spec-ification with and without real rigidities.28 Overall, this specification produces lesslabor market volatility than the MIU or CIA specifications. Following for instancea shock to the policy rate, the systematic component of the Taylor rule moves inthe opposite direction, hence dampening the effects of the exogenous policy impulseon the economy. Once again, however, real rigidities act toward increasing labormarket volatility and hence improve the model’s performance. For instance, the rel-ative volatilities of unemployment and employment are both 61% of their respectiveempirical counterparts, compared to 34% in the absence of real rigidities. Indeed,the weaker inflation response under real rigidities dampens the fluctuations in thesystematic component of the policy rule. Following, for example, a negative shockto the policy rate, the actual rate falls by more, hence amplifying the expansion inoutput and employment. Following a positive productivity shock, the policy rate fallsby less and output increases by less; as a result, labor demand falls by more.

4. CONCLUSION

This paper has studied the effect of search and matching frictions in the labormarket on firms’ pricing decisions, in a model where price setters are actually subjectto such frictions. In doing so, it departs from most of the literature on New Keynesianmodels with search and matching frictions, which separates the firms making thepricing decisions from the firms that face search frictions. The framework presentedhere therefore helps understand how price decisions are made in a context in whichfirms cannot costlessly and immediately adjust employment.

The main theoretical result is that search frictions give rise to real rigidities (or“strategic complementarities”) in price setting. This mechanism leads each individualprice setter to make smaller price changes in response to the same macro-economicfluctuations. On the aggregate, real rigidities slow the adjustment of the overall pricelevel. This is reflected in a smaller sensitivity of inflation to average real marginalcosts, that is, in a flatter NKPC. The increased sluggishness in the price level makesinflation less volatile and more persistent for a given average frequency of priceadjustment, a feature that is common to other real rigidity mechanisms such as firm-specific capital. More importantly, real rigidities improve the model’s performancealong those labor market dimensions that the standard New Keynesian model is notdesigned to address. In particular, real rigidities bring both the size of unemploymentfluctuations and the relative volatility of the two labor margins closer to the data.The corollary is that having firms make both hiring and pricing decisions within

28. For the purpose of this exercise, I set the policy rule coefficients to standard values: ρR = 0.8,φπ = 1.5, and φy = 0.5. The standard deviation of the interest rate shock, σ (εR

t ) = 0.37%, is chosen tomatch the standard deviation of output in the data. Results conditional on each shock type (monetary andproductivity) are available upon request.


the popular New Keynesian search-and-matching model does not merely representan increase in its realism but can also help it match those labor market facts thatconstitute its raison d’etre.

APPENDIX

A.1 Proof of Proposition 1

From equation (29) in the text, I can write the firm’s vacancy-posting decision as

sv


[η

μ(hi t+1 + ht+1) +

(1 − λ − 1 − ξ

1 − εp(θ )

)sv

λ(1 − ε)θt+1

],

(A1)

where hi t+1 = hi t+1 − ht+1 is the firm’s relative number of hours per worker. Hoursper worker admit the exact log-linear representation hi t = yd

i t − at − ni t . Therefore,I can write hi t = yd

i t − ni t . This becomes

hi t = −γ Pi t − ni t (A2)

once I use the fact that ydi t = −γ Pi t . The firm’s expected relative price is given by

Et Pit+1 = δEt (log Pit − log Pt+1) + (1 − δ)Et (log P∗i t+1 − log Pt+1)

= δEt (Pi t − πt+1) + (1 − δ)Et

(log P∗

i t+1 − log P∗t+1 + δ

1 − δπt+1

)= δ Pi t − (1 − δ)τ ∗ni t+1. (A3)

In the second equality, I have used the fact that log P∗t+1 − log Pt+1 = [δ/(1 − δ)]π t+1,

where π t+1 ≡ log (Pt+1/Pt) is the inflation rate. In the third equality, I have usedlog P∗

i t+1 − log P∗t+1 = −τ ∗ni t+1. Using (A2) and (A3), expected relative hours are

given by

Et hit+1 = −γ Et Pit+1 − ni t+1

= −γ δ Pi t − [1 − γ (1 − δ)τ ∗]ni t+1. (A4)

This implies that Et hit+1 averages to zero. Averaging (A1) across all firms andsubstracting the resulting expression from (A1) yields Et hit+1 = 0. Combining thiswith (A4), I finally obtain

ni t+1 = − γ δ

1 − γ (1 − δ)τ ∗ Pi t .


A.2 Proof of Proposition 2

Using (31) in the text to substitute for τ n in (35), I obtain the following equationfor τ*,

τ ∗ = (1 − δβ)η

1 + ηγ − δβη

(γ δ

1 − γ (1 − δ)τ ∗

) .

This can be written as

a(τ ∗)2 + bτ ∗ + c = 0, (A5)

where

a ≡ (1 + ηγ )γ (1 − δ) > 0, (A6)

b ≡ − [1 + γ (2 − δ − δβ)η] < 0, (A7)

c ≡ (1 − δβ)η > 0. (A8)

The quadratic equation (A5) has two solutions. The latter are real numbers if andonly if b2 − 4ac > 0. Using the definitions of a, b, and c, the inequality b2 − 4ac >

0 can be written as

[1 + γ (2 − δ − δβ)η]2 > 4(1 + ηγ )γ (1 − δ)(1 − δβ)η.

After some algebra, it is possible to express the latter inequality as

1 + (ηγ )2 δ2(1 − β)2 + 2ηγ δ [1 − δβ + β(1 − δ)] > 0,

which holds for any δ ∈ [0, 1) and β ∈ [0, 1). Equation (A5) has therefore two realsolutions, given by

(τ ∗1 , τ ∗

2 ) =(

−b − √b2 − 4ac

2a,−b + √

b2 − 4ac

2a

).

It is possible to show that the solutions for both τ* and τ n have to be positive. To seethis, define

τ n1 (τ ∗) ≡ 1 + ηγ − (1 − δβ)η/τ ∗

δβη,

τ n2 (τ ∗) ≡ γ δ

1 − γ (1 − δ)τ ∗ .


The function τ n1(τ*) is obtained by solving for τ n

1 in equation (35) in the text. Thesolutions for τ n and τ* are given by the two points of intersection of both functionsin (τ*, τ n) space. Both functions are strictly increasing in τ*. For τ* < 0, τ n

1(τ*) >

(1 + ηγ )/δβη and τ n2(τ*) < γδ. Since (1 + ηγ )/δβη > γ δ, there can be no solution

for τ* < 0. For τ* > 0, the signs of τ n1 and τ n

2 coincide only in the nonempty interval((1 − δβ)η/(1 + ηγ ), γ −1(1 − δ)−1), in which both functions are strictly positive.

Finally, it is possible to show that τ ∗2 implies explosive dynamics. To see this,

notice that a firm’s relative price and employment stock evolve according to[Et Pit+1

ni t+1

]=

[δ + (1 − δ)τ ∗τ n 0

−τ n 0

] [Pi t

ni t

].

This system implies convergent dynamics only if the eigenvalues of the 2 × 2 matrixare inside the unit circle. These eigenvalues are 0 and δ + (1 − δ)τ*τ n. Since δ + (1− δ)τ*τ n > 0 (as a result of both τ* and τ n being positive), a nonexplosive solutionmust satisfy δ + (1 − δ)τ*τ n < 1, or equivalently τ*τ n < 1. Using equation (31) inthe text, this requires in turn

τ ∗ <1

γ. (A9)

I now define F(τ*) ≡ a(τ*)2 + bτ* + c, where a, b, and c are given by equations(A6), (A7), and (A8), respectively. Since F(τ*) is a convex function, it follows thatF(τ*) < 0 ⇔ τ* ∈ (τ ∗

1, τ 2*), where τ ∗1, τ 2* are the two roots of F(τ*). Evaluating F

(·) at 1/γ , I obtain

F

(1

γ

)= (1 + ηγ )

1

γ(1 − δ) −

[1

γ+ (2 − δ − δβ)η

]+ (1 − δβ)η

= − δ

γ< 0.

It follows that τ ∗1 < 1/γ < τ 2*, which means that τ ∗

2 violates (A9) and thereforeimplies explosive dynamics. As emphasized by Woodford (2005), in order for alog-linear approximation around the steady state to be an accurate approximation ofthe model’s exact equilibrium conditions, the dynamics of firms’ relative prices andemployment stocks must remain forever near enough to the steady state. Since τ ∗

2violates this condition, I set τ* equal to τ ∗

1.

A.3 A Search Model with a Producer–Retailer Structure

Consider an economy where technology and preferences are the same as in themodel presented in Section 1, but with a different goods market structure. In particular,a continuum of identical producers produce a homogenous intermediate good thatis sold to retailers at the perfectly competitive price mct. Profits of an individual


producer are given by

�i t = mct At nit hit − wt (hit )nit − χ

u′(ct )vi t + Etβt,t+1�i t+1.

The surplus of worker and firm are given, respectively, by

Swi t = wt (hit ) − b + h1+η

i t

/(1 + η)

u′(ct )− p(θt )

∫ 1

0

v j t

vtEtβt,t+1Sw

j t+1d j

+ (1 − λ)Etβt,t+1Swi t+1,

S fit = mct At hit − wt (hit ) + (1 − λ)Etβt,t+1S f

it+1. (A10)

Hours per employee are chosen in a privately efficient way, that is, so as to maximizethe joint match surplus, Sw

it + Sitf . The resulting first-order condition is given by

mct At = hη

i t

u′(ct ), (A11)

which implies that hours are equalized across firms, hit = ht. Since all producersbehave symmetrically, I can drop the subscript i. Log-linearization of equation (A11)produces equation (E2) in the text.

Nash bargaining implies that (1 − ξ )log Sft = ξ log St

w. The solution for the realwage is given by

wt (ht ) = (1 − ξ )mct At ht + ξ

[b + h1+η

t

/(1 + η)

u′(ct )+ p(θt )Etβt,t+1Sw

t+1

]

= (1 − ξ )

[mct At ht + χ

u′(ct )θt

]+ ξ

b + h1+ηt

/(1 + η)

u′(ct ). (A12)

The first-order conditions with respect to vacancies and employment are given byequations (11) and (12) in the text, without i subscripts. Combining the latter with(A11) and (A12), we have that the job creation condition is given again by equa-tion (20), without i subscripts, which becomes (E4) when log-linearized.

Finally, retailers buy the intermediate input at the real price mct and transform itinto differentiated final goods with a linear technology. Therefore, mct is also the realmarginal cost of retailers and is independent of their pricing decisions. The optimalpricing decision common to all price-setting retailers is given by

Et

∞∑T =t

δT −tβt,T Pγ

T yT

(P∗

t

PT− γ

γ − 1mcT

)= 0.

Log-linearizing the previous equation and combining it with π t = [(1 − δ)/δ](log P∗t

− log Pt), I obtain equation (40) in the text.


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