Wage Dynamics and Labor Market Dynamics Istvan Konya Hungarian National Bank

Michael Krause Deutsche Bundesbank

June 6, 2008 Abstract In this paper we examine the ability of the search and matching model to fit important statistics of the labor market. Following Shimer (2005), we first present stylized facts for many Euro Area economies. Then we show that, similarly to the U.S., the basic model is unable to replicate the relative volatility of unemployment found in the data. Second, we present the full, general equilibrium version of the model, and discuss some simple extensions suggested by the literature that may help solving the ”unmployment volatility puzzle”. Finally, we calibrate and simulate the model to see if these extensions are consistent with the Euro Area stylized facts. An important diﬀerence from previous studies is that in addition to unemployment, we also examine the behavior of wages. Our main finding is that a simple form of real wage rigidity goes a very long way towards solving the puzzle, while other explanations fail to match the low volatility of wages.

1

Introduction

Understanding the determinants of real wages is crucial for understanding the dynamics of both the labor market and inflation. A tight labor market is likely to translate into higher wage and cost pressures that accelerate inflation. The responsiveness of wages to aggregate labor market conditions itself feeds back on firms’ hiring behavior and thus unemployment. The state of the labor market is therefore important for monetary policy makers’ assessment of inflationary pressures. In this paper, we analyze real wage dynamics from the perspective of search and matching models (Mortensen and Pissarides, 1994) of equilibrium unemployment. Such models have become the standard for addressing problems of aggregate unemployment. Furthermore, since the model has 1

been integrated in New Keynesian sticky price models, it potentially helps understand the link between labor costs and inflation. Our focus on real wage rigidity is also motivated by findings of Shimer (2005) and Hall (2005), who have shown that the standard search and matching framework can not account for the volatility of unemployment. This is because wages in the model are perfectly flexible and respond instantaneously to aggregate labor market conditions.1 This cyclical flexibility of wages mitigates firms’ incentives to create jobs in an upswing, leading to an implausibly low cyclicality of vacancies posted and of unemployment. Thus, most attempts to improve the performance of the model have centered around the assumptions on wage determination, in particular, to introduce real wage rigidity. The literature has often avoided taking literally the implications of the search and matching model for the real wage, because wage determination is modeled in a fairly stylized manner. In particular, the cyclicality of currently paid wages in the model appears at odds with the data. We therefore explore mechanisms that make wages more rigid, and discuss their ability to bring wage dynamics as well as employment dynamics more in line with the data. We also discuss the role of diﬀerent cyclical responsiveness of the wages of new hires and existing employees, where the former is likely to be more volatile than the latter.2 In the remainder of the paper, we first establish basic facts about the relative volatilities of vacancies, unemployment, real wages, and productivity in Euro Area data. The regularities identified by Shimer (2005) for U.S. data also extend to European data: unemployment and vacancies are an order of magnitude more volatile than real wages, productivity, or output. In Section 3, we present the core search and matching model of the labor market, and show that the model does not match the labor market facts. In Section 4, explores the reasons for this shortcoming, and illustrates in a simple manner the solutions that have been proposed in the literature. Section 5 then presents the full, real business cycle version, of the model, and conducts a quantitative business cycle analysis. Section 6 concludes. 1

Nonetheless, wages are rigid in the sense that they do not fall so as to align the returns to working to workers’ outside options. 2 See Pissarides (2007).

2

2

Wage and labor market dynamics in the Euro Area

We characterize wage and labor market dynamics in eight selected countries of the Euro Area, namely Austria, Belgium, Finland, France, Germany, Italy, Netherlands, and Spain. These countries represent variation in size and institutional setups in the Euro Area. The focus is on the dynamics of hourly (or weekly) real wages, vacancies, unemployment, labor productivity, and output. The data are taken from various sources, and have been compiled by Andrew McCallum of the ECB for the European System of Central Banks "Wage Dynamics Network". Table 1: Standard deviations of key labor market variables

Austria Belgium Finland France Germany Italy Netherlands Spain

y/n

u

v

v/u

w

0.03 0.04 0.09 0.02 0.04 0.09 0.03 0.05

0.19 0.24 0.30 0.12 0.37 0.10 0.23 0.17

0.29 0.43 0.41 0.32 0.37 0.42

0.44 0.69 0.71 0.44 0.43 0.42

0.02 0.02 0.06 0.03 0.03 0.32 0.03 0.05

In all countries, the relative percentage standard deviations of vacancies and unemployment are much larger than those of productivity or wages. The volatility of wages is similar to that of productivity (except Italy). Also, vacancies are much volatile than productivity, and typically also more volatile than unemployment. As a result, the vacancy-unemployment ratio is highly volatile in all countries, and this is the key variable in the search and matching model. The wage rate, on the other hand, tends to be somewhat less volatile than labor productivity, although there is a fair amount of variation across countries.3 As a particular example to these patterns, we plot Germany on Figure 1. As discuss above, unemployment is highly volatile, while wages and labor productivity are roughly equally volatile. These, and all the numbers 3

And, of course, the wage data are not necessarily comparable across countries.

3

Table 2: Relative standard deviations

y/n Austria Belgium Finland France Germany Italy Netherlands Spain

1 1 1 1 1 1 1 1

u

v

v/u

w

6.38 9.55 14.39 0.78 6.88 12.07 19.53 0.54 3.49 4.82 8.31 0.71 6.95 18.32 25.41 1.86 9.46 9.59 11.02 0.66 1.15 4.70 4.61 3.54 9.31 1.03 3.33 0.96

in Table 1, are HP filtered variables using a smoothness coeﬃcient of 100000, which follows Shimer (2005). The results are qualitatively similar if we used λ = 1, 600, as typical for quarterly series, instead.

-1

-.5

0

.5

Figure 1: Germany

1975q1

1980q1

1985q1

1990q1 time

Wages Labor productivity

1995q1

2000q1

2005q1

Unemploy ment

We see that the European data confirm the basic regularities established by Shimer (2005) for U.S. data. Key labor market variables are highly volatile relative to productivity, except for the wage rate which is somewhat less volatile. If exogenous productivity shocks are a significant driving force of 4

aggregate fluctuations, then any model of the labor market should generate substantial propagation to these shocks. However, as Shimer showed, the search and matching model is unable to replicate this pattern. Rather, the response of unemployment is far too low, whereas real wages are relatively to volatile. The next sections explore this in detail.

3

The business cycle model with search frictions

The central elements of the search and matching model are costly search in the labor market, where new matches between workers and firms are formed according to an aggregate matching function. This is analogous to a production function with unemployed, searching workers, and vacant jobs as inputs. The output is a flow of new matches. The fact that search is costly generates a surplus arising from a match between worker and firm, which must be divided according to some bargaining protocol. A typical assumption is that the spoils from the match are shared according to the Nash bargaining solution, which determines a wage that maximizes the joint surplus of worker and firm.4 We begin by characterizing labor market frictions, then show the hiring behavior of firms in the frictional labor market, and finally discuss the representative household. We do this last because it allows us to highlight a number of subtle issues related to risk sharing. When we embed the model into a real business cycle model, we assume that the economy consists of two sectors: an intermediate goods sector that produces a homogenous output using labor as the only input, and a final goods sector in which monopolistically competitive firms use the intermediate sectors output to produce diﬀerentiated products sold to households. Even though most of our analysis is in real terms, this allows us to model demand shocks in terms of markup variations.

3.1

Labor market frictions

Matching frictions are modelled by means of a matching function, which combines a measure of the stock of searching workers, or the unemployment rate, with the measure of searching firms, or vacancies posted. ¯ tϑ u1−ϑ , mt = mv t 4

References to Andolfatto, Merz, Den Haan et al., Pissarides, Mortensen-Pissarides.

5

where ut = 1 − nt with nt equal to employment, and vt is the measure of vacancies. The labor force is normalized to one. Existing employment relationships separate at a — for now — exogenously given job destruction rate ρ. Thus, aggregate employment evolves according to nt = (1 − ρ) nt−1 + mt−1 . Given the constant returns assumption, the match flow mt can be expressed as a function of labor market tightness, θt = vt /ut , namely mt = q(θt ) vt mt = = θt q(θt ). ut

qt = st

Then, mt = vt q(θt ). Here we entertain the one job-one worker assumption, so that there is only one vacancy posted per firm that want to find a worker. Alternatively, we could assume large firms which each post a number of vacancies, taking as given the aggregate match probability, q(θt ). Then the evolution of firm level employment would be written as nit = (1 − ρ) nit−1 + vit−1 q(θt−1 ). where each firm fills a number of vacancies in proportion to the total vacancies posted: vit /vt mt . This construction will be discussed further when the joint price setting and employment decision is introduced in a later chapter.

3.2

Firms

Firms in the intermediate sector hire workers in the frictional labor market, using a production function ytw = At hαt , where output depends on the amount of hours worked per worker, ht , and aggregate productivity At . Output is sold at a price xt to the final goods producers specified later. Posting vacancies to find workers comes at a utility cost κ from search services provided by the aggregate household (described later). Thus, in terms of goods, a firm has to spend κt = κ/λt for a vacancy, where λt is the households marginal utility of wealth. Notice that these costs are time varying and may depend on business cycle conditions. The present value of a job filled with a worker is then given by Jt = xt At hαt − wt ht + (1 − ρ) Et β t+1 Jt+1 where wt is the wage paid to the worker, and β t+1 = βλt+1 /λt is the households stochastic discount factor with which firms discount their profits. The 6

present value of a vacancy is given by Vt = −κt + Et β t [qt Jt+1 + (1 − qt ) Vt+1 ] There is free entry into vacancy posting, so firms will enter until the cost of creating a vacancy κt equals the benefit of doing so. Therefore Vt ≡ 0, which implies a condition for the cost and benefit of posting vacancies: κt = Et β t+1 Jt+1 qt Thus the value of a job can also be written as Jt = xt At hαt − wt ht + (1 − ρ)

κt qt

Inserting this in the above condition yields a job creation condition, familiar from equilibrium unemployment theory (see Pissarides, 2000). ∙ ¸ κt+1 κt α . = Et β t+1 xt+1 At+1 ht+1 − wt+1 ht+1 + (1 − ρ) qt qt+1 The expected cost of posting a vacancy, κt divided by the probability of filling the vacancy, qt , must equal the expected benefit of having this vacancy filled. This is the expected flow profit plus the present value of the job one period hence, given that the job survives job destruction with probability 1−ρ. This later part is reflected by the expected search costs by virtue of the free entry condition.

3.3

Households

Before we can consider wage determination via bargaining, we need to determine which type of objective function the worker maximizes when bargaining. Here simplifying assumptions are made, which allow an analytically tractable solution of the search and matching framework. A key assumption is that of risk sharing. One assumption is that workers are part of a large family that eﬀectively erases all idiosyncratic income risk arising from the loss of employment. In that case, the family receives its members’ income and distributes it equally among them (this follows Merz, 1995). Another assumption is that of perfect capital markets, in which workers obtained contingent claims before their individual employment status was known. These claims pay out when workers are unemployed, so that there is complete risk sharing among workers (Andolfatto, 1996). The key implication is that all workers can be treated equally and no individual asset positions need to be tracked. 7

Either of the two risk-sharing assumptions allow us to write the aggregate households optimization problem as " # ∞ 1+φ 1−σ X − 1 h c max E0 βt t − χnt t 1 − σ 1+φ t=0

subject to

Bt Bt−1 − = wt nt ht + (1 − nt ) b + π t − ct − Tt . Pt Rt Pt where Bt are the households bond holdings (equal to zero in equilibrium), Pt is the price level, and Rt the nominal interest rate. These variable are included here with an eye on the inclusion of nominal stickiness later on. Consumption is ct , profits of the monopolistic competitors is π t , and Tt is a lump-sum tax. The large household earns wage income as well as income from unemployment benefits b paid to the non-employed members by a government. The (non-labor) first-order conditions are as follows: = λt , c−σ t λt = βRt Et λt+1

Pt . Pt+1

The wage and hours decisions are not part of the household’s maximization problem, but are taken jointly by workers and firms each period. We describe this mechanism below. The marginal value of an employed worker to the aggregate household’s welfare is Wt = wt ht −

χ h1+φ t + Et β t+1 [(1 − ρ) Wt+1 + ρUt+1 ] λt 1 + φ

while the value of an unemployed worker to household welfare is Ut = b + Et β t+1 [st Wt+1 + (1 − st ) Ut+1 ] This allows us to write the surplus relative to unemployment that an employed worker generates as Wt − Ut = wt ht −

χ h1+φ t − b + Et β t+1 (1 − ρ − st ) (Wt+1 − Ut+1 ) , λt 1 + φ

which is used below. 8

3.4

Wage and hours bargaining

As in Trigari (2006), we assume that workers and firms bargain over wages and hours worked each period. The bargaining outcome is chosen to maximize the Nash product so as to maximize the joint surplus of the match. The result is a surplus sharing rule and a condition on the optimal choice of hours.5 Formally, the bargaining outcome is a solution to the following problem: max (Wt − Ut )η Jt1−η , wt ,ht

where the parameter η captures the bargaining power of workers. The firstorder conditions can be written as: ηJt = 1 (1 − η) (Wt − Ut ) αAt xt hα−1 − wt t = 1. φ χht /λt − wt From the second condition, we can calculate the hours equation as = χhφt /λt . αAt xt hα−1 t Thus the marginal value product of an extra hour worked equals its marginal disutility for a worker, which implies that hours are chosen eﬃciently in this setup.6 Using again the first condition, and the equations for Jt and Wt − Ut , one can derive the wage equation, Ã ! 1+φ χ ht wt = η (At xt hαt + κt θt ) + (1 − η) +b . λt 1 + φ The wage is a combination of the opportunity cost of working for workers and the value of the job for firms, including the savings of not having to post a new vacancy, reflected in κt θt . 5

An alternative assumption is that once wages are chosen, firms pick hours to maximize profits (right-to-manage bargaining, Trigari 2006). This solution is however not paretooptimal. We leave the discussion of RTM to Chapter 2 of this survey. 6 Note that in the case of linear utility, the bargaining problem would also be the solution of the individual worker’s problem. However, here, workers are assumed to bargain in the interest of the aggregate family or household. In the case of contigent claims, one would have to presume that enforcable contracts make sure the worker bargains such that the two bargaining conditions are met.

9

The job creation condition, the stochastic discount factor, the evolution of employment, and the wage equation allow a complete characterization of the labor market. As shown in Pissarides (2000), and many related papers, the model is well suited to analyze labor market issues such as eﬀects of unemployment insurance on the labor market, the role of firing costs and taxation, and other determinants of equilibrium unemployment. However, as Shimer (2005) and Costain and Reiter (2004) have shown, the model in its basic form cannot explain the dynamics of the labor market well. In particular, it cannot generate a suﬃcient volatility of unemployment and job finding rates in response to ‘plausible’ changes in productivity.

3.5

Final goods

We assume that the final consumption good is produced by retailers who buy the intermediate good and diﬀerentiate it. The market structure for retailers is monopolistic competition, as in the sticky price version of the model. Households view the retail goods as imperfect substitutes, through a symmetric CES sub-utility function, with time-varying elasticity of substitution t : ∙Z 1 ¸ t /( t −1) t −1 ct (i) t di ct = 0

Maximizing utility for a given level of consumption ct yields as set of demand functions for each variety i : ¸− ∙ pt (i) t ct (i) = ct , Pt where Pt is the associated minimum expenditure price index: Pt =

∙Z

1

1−

pt (i)

0

t

di

¸1/(1− t )

.

Given the demand function, retailers follow monopolistic practise and optimally mark up prices above marginal costs. This yield the following pricing policy for each good i : pt (i) = (1 + μt ) Pt xt , where μt = 1/( t − 1) is the (net) markup of nominal price over nominal marginal cost Pt xt , with xt the real price of the intermediate good. We include a time-varying markup for two reasons. One is that it serves as a proxy for endogenous markup movements when there is nominal price 10

stickiness. Thus our computational experiments below are to some extent suggestive of the behavior in the presence of price stickiness. The other reason is that markup shocks are a plausible alternative driving force of labor market dynamics, as suggested by Rotemberg (2006) and Krause, LopezSalido, and Lubik (2008). In contrast to productivity shocks, (negative) markup shocks are expansionary while at the same time lower the marginal revenue of firms, thus reducing upward wage pressures in the search model with Nash bargaining. With sticky prices, the prices level reacts sluggish to changes in real marginal costs, and a forward looking component is introduced in the price setting equation. We implement price stickiness in the Calvo (1983) fashion. That is, only a random fraction of firms is allowed to change prices. The end-result of this consideration is the New Keynesian Phillips curve : π t = Et βπt+1 +

(1 − β)(1 + γβ) xt γ

where π is inflation and γ is the probability of not adjusting the price.

3.6

Equilibrium

Market clearing on the markets for bonds, labor and goods imply bond holding Bt = 0, as all households are equal, and At nt ht = ct . That is, all production is consumed. In symmetric equilibrium, individual prices p(i) are equal and hence Pt = pt (i). Consequently, from the above pricing relationship, we have that xt =

1 . 1 + μt

This is the familiar condition that real marginal costs are equal to the inverse of the (gross) markup. For given costs, an increase in the markup makes firms want to raise their price, thus moving up their demand curves. The resulting fall in production depresses demand for the intermediate input, which in turn lowers its price.

4

The volatility ‘puzzle’

We use a simplified version of the model to illustrate the diﬃculty of the search and matching model to match labor market data, and well as the 11

proposed solutions to it. For this, ignore for the time being the role of hours (setting α = χ = 0) and the endogeneity of the discount rate (setting σ = 1, so that λt ≡ 1, κt = κ, and β t = β). The job creation condition and wage equation become, respectively: ∙ ¸ κ κ = (1 − ρ)βEt xt+1 At+1 − wt+1 + (1) q(θt ) q(θt+1 ) and wt = η (xt At + θt κ) + (1 − η) b.

Again, the first condition shows that the marginal cost of producing a new job next period, must equal the expected return of having a job filled. Incentives to post vacancies thus depend on the expected values of marginal revenue, wages, and the continuation value of the match. A persistent increase in At will lead to a rise in the right-hand side, which induces firms to post more vacancies. As they do so, labor market tightness, θt = vt /ut , rises, and thus q(θt ) falls. The left hand side rises until equality is restored. How much labor market tightness rises depends on how responsive the return to posting a vacancy is to changes in its determinants. As the second equation shows, wages respond to changes in xt , At , and θt . In particular, changes in θt = vt /ut immediately translate into changes in wages. Then, from the job creation condition above, one can see that as long as marginal revenue and labor market tightness are expected to be high in the future, expected wages will be higher, and thus incentives to post vacancies will be lower. Unemployment will not respond much. On the other hand, if wages are constant, the incentive to post vacancies is not reduced by rises in wages. Then subsequent declines in unemployment will be larger for a given shock. This is essentially Shimer’s point. Therefore, as wages become less responsive to xt At and θt , incentives to post vacancies are higher. To show these arguments formally, insert the wage equation into the job creation condition (1), to obtain ∙ ¸ κ κ = (1 − ρ)βEt (1 − η)xt+1 At+1 − ηθt+1 κ − (1 − η)b + q(θt ) q(θt+1 ) and then log-linearize and multiply with q(θ)/κ: ∙ ¸ ³ ´ q(θ) (1 − ρ)β b b b Et (1 − η)x x bt+1 + At+1 + [ϑ − ηs] θt+1 θt = ϑ κ 0

(2)

(θ) where ϑ = − qq(θ) θ > 0, so that κ/q(θt ) ≈ ϑκ/q(θ)b θt . Since u bt is predeterbt corresponds to a change in vbt . Note now mined, any change in b θt = vbt − u

12

that the size of the response of vt depends on (1 − η), that is, how much workers share the increased marginal revenue of the firm. Second, the degree to which vacancies depend on future expectations on b θt+1 depends on [ϑ − ηs] , which depends on the elasticity of the matching function, the bargaining power of workers, and the probability of finding a job, st . Third, the response of the incentives depends on the steady-state value of q(θ)/κ. This will be further explored below. We can see now that the response of vacancies depends on how wages are set. Because for a positive η, the term ϑ − ηs can be small. In fact, for plausible calibrations, it may be very close to zero. For example, it is exactly zero for values not uncommon in the literature, bargaining power η = 0.5, elasticity of the matching function ϑ = 0.4, and steady-state worker finding rate s = θq(θ) = 0.8. Thus wage flexibility mutes labor market dynamics, as incentives to post vacancies are reduced. Only for η = 0 would the full incentives maintain. Equivalently, note that if one iterates condition (2) forward, one can see that ϑ − ηs = 0 implies that any influence of future bt+1 on current period’s vacancy posting would be shut values of x bt+1 and A down. Only the next period would matter. Additionally, q(θ)/κ aﬀects the responsiveness of vacancies. Its steadystate value is related to the size of the unemployment benefit b and the marginal revenue x, from the steady-state job creation condition: κ [ηBs + 1] = (1 − η)B[x − b], q(θ)

(3)

where B = (1 − ρ)β/(1 − (1 − ρ)β). With ρ = 0.1, and β = 0.98, we have that B ≈ 7.48. If we assume that b is about half of x, and marginal revenue x = 0.9, then κ/q(θ) = (1 − η)B[x − b]/ [ηBs + 1] = 1.68/3.99 = 0.42. The inverse is thus q(θ)/κ = 2.38. Multiplied by (1 − η)x = 0.45, we see that vacancies change about one-to-one (or 1.07) to changes in xt+1 or At+1 . There is basically no propagation in the search and matching model. How can the ‘puzzle’ be solved that the data show volatilities of a order of magnitude above productivity? The proposal by Hall (2005) and Shimer (2005) is to impose real wage rigidity. Eﬀectively, this amounts to setting η = 0 in the linearization above. Then ∙ ¸ ´ q(θ) ³ b b b ϑθt = (1 − ρ)βEt x x bt+1 + At+1 + ϑθt+1 κ θt . Of and future expected values of b θt+1 determine the current response of b course, by iterating forward, b θt can be shown to depend on all future values bt+1 . of x bt+1 and A 13

Alternatively, rather than assuming that wages are rigid, we can boost the volatility of b θt by assuming a higher q(θ) . Going back to the steady-state κ job creation condition (3), we see that the closer the outside option b is to marginal revenue x, the smaller must the left-hand side of the equation be. This is only obtained by a smaller κ, which in turn raises q(θ) . In eﬀect, we κ b bt+1 by bt+1 and A can obtain an arbitrarily large response of θt to changes in x assuming an arbitrarily small x − b. This is the point made by Hagedorn and Manovskii (2008). However, apart from raising issues which level of workers’ outside options is reasonable, raising the unemployment benefit to get the dynamics right makes steady-state unemployment too responsive to changes in benefits. This finding goes back to Costain and Reiter (2004). Suppose we want to calibrate a search and matching model with parameters that imply some observed longrun unemployment rate, for example 10 percent, and assume a job destruction rate of 10 percent. The equation for the evolution of employment nt = (1 − ρ) nt−1 + mt−1 ⇐⇒ ρn = m ⇐⇒ ρ(1 − u) = θq(θ)u implies u=

ρ ρ + θq(θ)

(4)

For this to hold, it must be that θq(θ) = mθ ¯ 1−ϑ = 0.9, where we use the parameterization above. Rewrite (3) as ηBθκ + m1¯ θϑ κ = (1 − η)B[x − b], and insert the implied m, ¯ so that θκ =

(1 − η)B (x − b) ηB + 1/0.9

Now if we choose a high responsiveness of job creation to productivity shocks, x − b would have to be small, or b relatively high. Thus θκ must be small in steady state. Further assuming that we calibrate m ¯ to some value, say, 0.9, then steady state θ = 1. Thus κ must be small if b is high. Consider a reduction in unemployment benefits from such a high level, and the response of endogenous variables, given the parameters just chosen. Linearized: (1 − η)B b b b θ=− b ηB + 1/0.9 κ

Thus, given the parameter values, the lower is κ, the stronger is the response of θ to a change in b. Given equation (4), we see that this implies a larger response of unemployment. Turning back to real wage rigidity, a simplified representation has been used by Krause and Lubik (2007) and Christoﬀel and Linzert (2006), following Hall (2005). The real wage is assumed to be driven by the Nash 14

bargain that would have been agreed in the absence of rigidity, and some wage norm. The latter may be given by some internal equity constraint, or some unmodelled firm level agreement. In any case, the wage follows et , wt = γwtN + (1 − γ) w

e is the wage norm. The norm where wtN is the Nash-bargained wage and w may be last period’s wage w et = wt−1 , or the steady state wage, w et = w. ¯ Under the assumption of a steady state wage norm, the linearized job creation condition becomes ∙ ¸ ³ ´ q(θ) (1 − ρ)β b bt+1 + [ϑ − γηs] b (5) Et (1 − γη)x x bt+1 + A θt+1 θt = ϑ κ

The more rigid the wage becomes, that is, γ becoming smaller, the more responsive is vacancy posting to both changes in current marginal product and to future changes in labor market tightness. The steady state of the model is not aﬀected by this extension, since wN = w, e in the steady state. Finally, let’s consider a proposal by Hall and Milgrom (2008), who have argued that the relevant threat point of a worker is not his or her outside option, but the value of continuing negotiations. The possibility of delaying negotiations gives the worker bargaining power. The consequence is that labor market conditions do not explicitly enter the wage, at least do not aﬀect it much cyclically. We can approximate this point by writing the wage as: wt = η (xt At + θκ) + (1 − η) b. In this case, even though the steady-state wage depends labor market tightness, the cyclical response of the wage is independent of it. Eﬀectively, we remove ηs from ϑ − ηs, so that future expectations of θt fully enter the linearized job creation condition. But productivity changes continue to aﬀect wages.

5 5.1

Quantitative analysis Calibration

We choose parameters partly based on independent information or calibrate them to be consistent with certain labor market facts. The intertemporal elasticity of consumption is set to σ = 1.5, the discount factor is β = 0.992, and the disutility of labor has elasticity φ = 2. These values are common in the literature and found close to estimates of general equilibrium models. 15

For the production side, the elasticity of output with respect to hours is set to α = 0.66, which is taken to correspond to the labor share in aggregate production functions. The average (gross) markup of the monopolistic competitors is assumed to be 1.1, which implies the real price of the intermediate good of x = 1/1.1 ≈ 0.91. In the sticky price version of the model, we assume a probability of not adjusting prices of γ = 0.75, and simple Taylor rule with interest rate smoothing parameter δ r = 0.85, and coeﬃcient on inflation of δ π = 1.5. We take from European data an average unemployment rate of 9 percent, and a job finding rate of 0.7. The job destruction rate is 3% per quarter, which is lower than in the U.S. These values imply a steady state labor market tightness θ. For lack of independent information we calibrate the bargaining power η = 0.5, a value commonly used in labor market studies that employ the Nash bargaining assumption. The elasticity of the matching function with respect to unemployment is 1 − ϑ = 0.6. With these values, we can calculate steady-state labor market tightness, and the implied scale parameter of the disutility of labor, χ = αn−σ x. Assuming a unemployment benefit of b = 0.65, we obtain the implied cost of filling a vacancy, κ, and the steady state wage w. The persistence of markup and technology shocks is assumed to be 0.95.

5.2

Simulation results

In this section we illustrate the mechanisms discussed in the previous sections by simulating the various special cases. We use the European calibration discussed in Chapter 2 of this survey, and summarized above. Our goal is to see whether and how the various simple extensions of the baseline model help resolving the unemployment-volatility puzzle quantitatively. Table 3 presents the results. We report seven model specifications. Model 1 is the baseline Mortensen-Pissarides (1994) framework with linear utility, no intensive margin (hours), and flexible prices and wages. We can see the quantitative failure of the model also on European data: while labor market tightness is quite a bit more volatile than labor productivity, the diﬀerence is far from what we see in the data. The volatility of the wage rate, however, is matched very well. Models 2 adds the hours margin and concave utility to the baseline, while Model 3 adds sticky prices. As the results show, these modifications do not really improve the model’s ability to explain relative volatilities. Hours and concavity help make θ more volatile, but wage volatility also rises. On the other hand, the latter is dampened by adding sticky prices, but at the expense of reducing the variance of θ. 16

Table 3: Simulated relative volatilities

Model y/n 1 2 3 4 5 6 7 8

1 1 1 1 1 1 1 1

u 1.39 1.67 1.31 3.17 5.68 5.36 0.56 5.27

v

v/u

w

h

1.53 2.69 0.94 0 1.87 3.25 1.17 0.22 1.33 2.50 0.96 0.33 4.32 6.55 1.83 0.60 6.51 11.13 1.21 0.40 6.07 10.47 1.06 0.38 1.05 1.33 1.13 0.17 5.98 10.31 1.06 0.38

Model 4 extends Model 3 further by adding two more shocks, a monetary policy shock and a price markup shock. For this exercise, we calibrate the productivity shock to have a variance and persistence which is around the average of the 8 countries in Table 1 (σ a = 0.03, ρa = 0.95). We set the standard deviation of the mark-up shock to be 0.01 (while the steady state markup is 1.1), and we choose the standard deviation of the monetary policy shock to be 25 basis points on a quarterly basis. Not surprisingly, the model’s performance improves significantly, although the volatility of the wage rate increases too much. Both shocks move the labor market indicators in the same direction, and are able to generate the increase in volatility even if introduced alone (not shown). Of course, the importance of the two new shocks depends on their selected volatilities, which requires more careful calibration. Model 5 returns to model 2 and explores the argument put forth by Hagedorn and Manovskii (2008), see also in Section 4. We set the replacement rate to a higher value, bu = 0.74. With this choice of b, we are able to match labor market volatilities quite well. Notice that this value is well below unity, which was suggested by Hagedorn and Manovskii (2008). The reason for this is that our model includes hours choice and hence disutility of labor is explicitly taken into account. Thus when workers decide about taking a job, they need to be compensated not only for lost unemployment income, but also for lost leisure. In our calibration the value of leisure is about 0.2, yielding an upper limit of the replacement rate that is consistent with a positive job surplus of about 0.8. This exercise highlights the crucial importance of the replacement rate,

17

but in an unexpected way. It shows that reasonable values for b in Europe are in a range where the model is sensitive to its choice (recall that in our baseline calibration bu = 0.65, which is already relatively high when hours are included in the model). Thus the careful calibration of this parameter is very important. Model 6 introduces a form of real wage rigidity (discussed in Section 4) into the framework of model 2, where . As Table 3 shows, this simple extension goes a very long way towards explaining the volatility puzzle. Unemployment, vacancy, and labor market tightness volatilities increase significantly, and are roughly in line with the data. Wage volatility, on the other hand, remains subdued, and also close to that seen in the data. While not surprising qualitatively, the quantitative success of such a simple modification is very encouraging. Models 7 and 8 continue with forms of real wage rigidity, using two different types of wage norms. We thus modify the wage equation to wt = γwtN + (1 − γ) w, e

e is the wage norm. In Model 7, where wtN is the Nash-bargained wage and w the norm is last period’s wage w e = wt−1 . In model 8, the wage norm is the steady state wage, w e = w. ¯ The numbers reported in Table 3 are calculated using a value of γ = 0.5. Model 7 performs poorly. While the volatility of the real wage declines so do the volatilities of unemployment, vacancies, and labor market tightness. With model 8, on the other hand, we are able to replicate the relative volatilities quite well. Thus, similarly to model 6, decoupling the wage rate from current labor market conditions accounts well for the quantitative features of European labor markets. Moreover, using lower values for γ in model 8 we are able to increase the unemployment and vacancy volatilities even further, as can be seen in Table 4. Intuitively, if we associate lower values of γ with a more rigid labor market, the model predicts that the volatilities of u and v are higher and the volatility of the wage rate is lower the lower γ is. Inspecting Table 2, this pattern seems to be present in the data. We see the highest relative volatilities in France, Belgium and Austria, while Finland has the lowest volatilities among the countries we have data for. Of course, our data is only indicative of this pattern, and it should be combined with institutional variables to test this hypothesis more thoroughly. We leave this issue for future research.

18

Table 4: Simulated relative volatilities

Value for γ 0.1 0.3 0.5 0.7 0.9

5.3

y/n 1 1 1 1 1

u

v

v/u

w

h

11.50 13.19 22.56 0.86 0.65 9.42 10.77 18.45 0.92 0.56 5.27 5.98 10.31 1.05 0.38 3.51 3.97 6.86 1.12 0.30 2.2 2.48 4.30 1.16 0.24

New Hires versus Incumbent Workers

An issue that has attracted considerable attention is whether wages of new workers are less rigid than those of incumbent workers. Note that while the search and matching model implies volatile wages for both incumbent and new workers, only the flexibility of new hires’ wages matters for job creation. The initial split of the surplus is what aﬀects the incentives of firms to post vacancies. Once jobs are filled, how exactly the revenue is allocated between worker and firms is immaterial as long as no party has an incentive to separate. This would be the case if the previously agreed wage were either too high for the firm to earn any profits, or too low for the worker to earn more than the outside option (taking account of present values of course). In all the cases we considered thus far, real wage rigidity aﬀected always both old and new workers. Pissarides (2007) argues that the search and matching model is perfectly consistent with the volatility of new hires’ wages. In other words, wage rigidity is not necessarily the solution to the unemployment volatility puzzle uncovered by Shimer (2007). In contrast, authors that have alluded to internal equity constraints that force new hires’ wages to be in line with incumbents’ wages, argue that such wages are in fact rigid. An example is Gertler and Trigari (2006), who assumed staggered wage setting at the firm level of large employers, where new workers enter the existing wage bargain. This would also be in line with Bewley’s findings.

19

6

Conclusions

This paper explores the real wage and labor market dynamics in the business cycle version of the search and matching model. In the search and matching model, search by firms and workers is subject to search frictions, and thus are a costly process. Search also gives rise to rents which firm and worker need to split by an appropriately determined wage. The details of wage determination however play a crucial role in how the labor market responds to shocks. This survey is also designed to set the stage for a comprehensive study of the implications of labor search for real marginal cost dynamics and thus inflation. We show first that wage and labor market dynamics in European Union data are similar to those found in U.S. data.7 The baseline search and matching model as analyzed by Shimer (2005) and Hall (2005) thus will have similar diﬃculty matching European data. The literature has proposed a number of changes in the calibration and setup of the search and matching model to resolve this puzzle, and we illustrate and survey these in a unified manner. Qualitatively, all these changes should increase the volatility of vacancies and unemployment, but it is not clear whether this bears out quantitatively. In our quantitative analysis we show that the nature of shocks (monetary and markup) help bring the model closer to the data by a large margin. But this comes at the price of a real wage volatility that is too large. Raising the outside option (unemployment benefits) of workers also has the predicted eﬀect on labor market volatilities, but wage volatility is too large. Interestingly, the proposal by Hall and Milgrom (2008) appears help the model match the data surprisingly well, while at the same time not requiring subtle arguments about how workers’ outside options actually ought to be quantified.8 Similarly, reducing the responsiveness of the real wage by including a steady state wage norm in the wage equation also helps the model match labor market volatilities. The advantage of this approach is that the model contains a parameter that measures the extent of wage rigidity, helping us relate the quantitative findings to individual European countries. Finally, adding some form of wage rigidity circumvents the Costain and Reiter critique that search models with high unemployment benefits predict excessively large responses of steady-state unemployment to reduction in such benefits. 7 8

See, for example, Shimer (2005). This is the debate spurred by Hagedorn and Manovskii (2008).

20

References [1] Andolfatto, David (1996). "Business Cycles and Labor Market Search". American Economic Review, 86(1): 112-132 [2] Blanchard, Olivier, and Jordi Gali (2008). "Labor markets and monetary policy: a new Keynesian model with unemployment", NBER Working Paper No. 13897 [3] Christoﬀel, Kai, and Tobias Linzert (2006). "The role of real wage rigidity and labor market frictions for unemployment and inflation dynamics". Bundesbank Discussion Paper No. 11 [4] Costain, James, and Michael Reiter (2006). "Business Cycles, Unemployment Insurance, and the Calibration of Matching Models". Universitat Pompeu Fabra Working Paper No. 872 [5] Hagedorn, Markus, and Iourii Manovskii (2008). "The Cyclical Behavior of Equilibrium Unemployment and Vacancies Revisited". American Economic Review, forthcoming [6] Hall, Robert (2005). "Employment Fluctuations with Equilibrium Wage Stickiness". American Economic Review, 95(1): 50-65 [7] Hall, Robert, and Robert Milgrom (2008). "The limited influence of unemployment on the wage bargain". American Economic Review, forthcoming [8] Krause, Michael, David Lopez-Salido, and Thomas Lubik (2008). "Inflation dynamics with search frictions: a structural econometric analysis". Journal of Monetary Economics, forthcoming [9] Krause, Michael, and Thomas Lubik (2007). "The (ir)relevance of real wage rigiditiy in the New Keynesian Model with Search frictions". Journal of Monetary Economics, 54(3): 706-727 [10] Merz, Monika (1995). "Search in the labor market and the real business cycle". Journal of Monetary Economics, 36: 269-300 [11] Mortensen, Dale, and Eva Nagypal (2006). "More on Unemployment and Vacancy Fluctuations". Review of Economic Dynamics, 10(3): 327-347 [12] Mortensen, Dale, and Christopher Pissarides (1994). "Job Creation and Job Destruction in the Theory of Unemployment". Review of Economic Studies, 61(3): 397-415 21

[13] Pissarides, Christopher (2000). Equilibrium Unemployment Theory. MIT Press [14] Pissarides, Christopher (2007). "The Unemployment Volatility Puzzle: Is Wage Stickiness the Answer?". CEP Discussion Paper No. 839, London School of Economics [15] Rotemberg, Julio (2006). "Cyclical Wages in a Search-and-Bargaining Model with Large Firms". Working paper, Harvard Business School [16] Shimer, Robert (2005). "The Cyclical Behavior of Equilibrium Unemployment and Vacancies", American Economic Review, 95(1): 25-49 [17] Shimer, Robert (2005). "The Cyclicality of Hires, Separations, and Jobto-Job Transitions". Federal Reserve Bank of St. Louis Review, 87(4): 493-507 [18] Trigari, Antonella (2006). "The role of search frictions and bargaining for inflation dynamics". IGIER Working Paper No. 304

22

Michael Krause Deutsche Bundesbank

June 6, 2008 Abstract In this paper we examine the ability of the search and matching model to fit important statistics of the labor market. Following Shimer (2005), we first present stylized facts for many Euro Area economies. Then we show that, similarly to the U.S., the basic model is unable to replicate the relative volatility of unemployment found in the data. Second, we present the full, general equilibrium version of the model, and discuss some simple extensions suggested by the literature that may help solving the ”unmployment volatility puzzle”. Finally, we calibrate and simulate the model to see if these extensions are consistent with the Euro Area stylized facts. An important diﬀerence from previous studies is that in addition to unemployment, we also examine the behavior of wages. Our main finding is that a simple form of real wage rigidity goes a very long way towards solving the puzzle, while other explanations fail to match the low volatility of wages.

1

Introduction

Understanding the determinants of real wages is crucial for understanding the dynamics of both the labor market and inflation. A tight labor market is likely to translate into higher wage and cost pressures that accelerate inflation. The responsiveness of wages to aggregate labor market conditions itself feeds back on firms’ hiring behavior and thus unemployment. The state of the labor market is therefore important for monetary policy makers’ assessment of inflationary pressures. In this paper, we analyze real wage dynamics from the perspective of search and matching models (Mortensen and Pissarides, 1994) of equilibrium unemployment. Such models have become the standard for addressing problems of aggregate unemployment. Furthermore, since the model has 1

been integrated in New Keynesian sticky price models, it potentially helps understand the link between labor costs and inflation. Our focus on real wage rigidity is also motivated by findings of Shimer (2005) and Hall (2005), who have shown that the standard search and matching framework can not account for the volatility of unemployment. This is because wages in the model are perfectly flexible and respond instantaneously to aggregate labor market conditions.1 This cyclical flexibility of wages mitigates firms’ incentives to create jobs in an upswing, leading to an implausibly low cyclicality of vacancies posted and of unemployment. Thus, most attempts to improve the performance of the model have centered around the assumptions on wage determination, in particular, to introduce real wage rigidity. The literature has often avoided taking literally the implications of the search and matching model for the real wage, because wage determination is modeled in a fairly stylized manner. In particular, the cyclicality of currently paid wages in the model appears at odds with the data. We therefore explore mechanisms that make wages more rigid, and discuss their ability to bring wage dynamics as well as employment dynamics more in line with the data. We also discuss the role of diﬀerent cyclical responsiveness of the wages of new hires and existing employees, where the former is likely to be more volatile than the latter.2 In the remainder of the paper, we first establish basic facts about the relative volatilities of vacancies, unemployment, real wages, and productivity in Euro Area data. The regularities identified by Shimer (2005) for U.S. data also extend to European data: unemployment and vacancies are an order of magnitude more volatile than real wages, productivity, or output. In Section 3, we present the core search and matching model of the labor market, and show that the model does not match the labor market facts. In Section 4, explores the reasons for this shortcoming, and illustrates in a simple manner the solutions that have been proposed in the literature. Section 5 then presents the full, real business cycle version, of the model, and conducts a quantitative business cycle analysis. Section 6 concludes. 1

Nonetheless, wages are rigid in the sense that they do not fall so as to align the returns to working to workers’ outside options. 2 See Pissarides (2007).

2

2

Wage and labor market dynamics in the Euro Area

We characterize wage and labor market dynamics in eight selected countries of the Euro Area, namely Austria, Belgium, Finland, France, Germany, Italy, Netherlands, and Spain. These countries represent variation in size and institutional setups in the Euro Area. The focus is on the dynamics of hourly (or weekly) real wages, vacancies, unemployment, labor productivity, and output. The data are taken from various sources, and have been compiled by Andrew McCallum of the ECB for the European System of Central Banks "Wage Dynamics Network". Table 1: Standard deviations of key labor market variables

Austria Belgium Finland France Germany Italy Netherlands Spain

y/n

u

v

v/u

w

0.03 0.04 0.09 0.02 0.04 0.09 0.03 0.05

0.19 0.24 0.30 0.12 0.37 0.10 0.23 0.17

0.29 0.43 0.41 0.32 0.37 0.42

0.44 0.69 0.71 0.44 0.43 0.42

0.02 0.02 0.06 0.03 0.03 0.32 0.03 0.05

In all countries, the relative percentage standard deviations of vacancies and unemployment are much larger than those of productivity or wages. The volatility of wages is similar to that of productivity (except Italy). Also, vacancies are much volatile than productivity, and typically also more volatile than unemployment. As a result, the vacancy-unemployment ratio is highly volatile in all countries, and this is the key variable in the search and matching model. The wage rate, on the other hand, tends to be somewhat less volatile than labor productivity, although there is a fair amount of variation across countries.3 As a particular example to these patterns, we plot Germany on Figure 1. As discuss above, unemployment is highly volatile, while wages and labor productivity are roughly equally volatile. These, and all the numbers 3

And, of course, the wage data are not necessarily comparable across countries.

3

Table 2: Relative standard deviations

y/n Austria Belgium Finland France Germany Italy Netherlands Spain

1 1 1 1 1 1 1 1

u

v

v/u

w

6.38 9.55 14.39 0.78 6.88 12.07 19.53 0.54 3.49 4.82 8.31 0.71 6.95 18.32 25.41 1.86 9.46 9.59 11.02 0.66 1.15 4.70 4.61 3.54 9.31 1.03 3.33 0.96

in Table 1, are HP filtered variables using a smoothness coeﬃcient of 100000, which follows Shimer (2005). The results are qualitatively similar if we used λ = 1, 600, as typical for quarterly series, instead.

-1

-.5

0

.5

Figure 1: Germany

1975q1

1980q1

1985q1

1990q1 time

Wages Labor productivity

1995q1

2000q1

2005q1

Unemploy ment

We see that the European data confirm the basic regularities established by Shimer (2005) for U.S. data. Key labor market variables are highly volatile relative to productivity, except for the wage rate which is somewhat less volatile. If exogenous productivity shocks are a significant driving force of 4

aggregate fluctuations, then any model of the labor market should generate substantial propagation to these shocks. However, as Shimer showed, the search and matching model is unable to replicate this pattern. Rather, the response of unemployment is far too low, whereas real wages are relatively to volatile. The next sections explore this in detail.

3

The business cycle model with search frictions

The central elements of the search and matching model are costly search in the labor market, where new matches between workers and firms are formed according to an aggregate matching function. This is analogous to a production function with unemployed, searching workers, and vacant jobs as inputs. The output is a flow of new matches. The fact that search is costly generates a surplus arising from a match between worker and firm, which must be divided according to some bargaining protocol. A typical assumption is that the spoils from the match are shared according to the Nash bargaining solution, which determines a wage that maximizes the joint surplus of worker and firm.4 We begin by characterizing labor market frictions, then show the hiring behavior of firms in the frictional labor market, and finally discuss the representative household. We do this last because it allows us to highlight a number of subtle issues related to risk sharing. When we embed the model into a real business cycle model, we assume that the economy consists of two sectors: an intermediate goods sector that produces a homogenous output using labor as the only input, and a final goods sector in which monopolistically competitive firms use the intermediate sectors output to produce diﬀerentiated products sold to households. Even though most of our analysis is in real terms, this allows us to model demand shocks in terms of markup variations.

3.1

Labor market frictions

Matching frictions are modelled by means of a matching function, which combines a measure of the stock of searching workers, or the unemployment rate, with the measure of searching firms, or vacancies posted. ¯ tϑ u1−ϑ , mt = mv t 4

References to Andolfatto, Merz, Den Haan et al., Pissarides, Mortensen-Pissarides.

5

where ut = 1 − nt with nt equal to employment, and vt is the measure of vacancies. The labor force is normalized to one. Existing employment relationships separate at a — for now — exogenously given job destruction rate ρ. Thus, aggregate employment evolves according to nt = (1 − ρ) nt−1 + mt−1 . Given the constant returns assumption, the match flow mt can be expressed as a function of labor market tightness, θt = vt /ut , namely mt = q(θt ) vt mt = = θt q(θt ). ut

qt = st

Then, mt = vt q(θt ). Here we entertain the one job-one worker assumption, so that there is only one vacancy posted per firm that want to find a worker. Alternatively, we could assume large firms which each post a number of vacancies, taking as given the aggregate match probability, q(θt ). Then the evolution of firm level employment would be written as nit = (1 − ρ) nit−1 + vit−1 q(θt−1 ). where each firm fills a number of vacancies in proportion to the total vacancies posted: vit /vt mt . This construction will be discussed further when the joint price setting and employment decision is introduced in a later chapter.

3.2

Firms

Firms in the intermediate sector hire workers in the frictional labor market, using a production function ytw = At hαt , where output depends on the amount of hours worked per worker, ht , and aggregate productivity At . Output is sold at a price xt to the final goods producers specified later. Posting vacancies to find workers comes at a utility cost κ from search services provided by the aggregate household (described later). Thus, in terms of goods, a firm has to spend κt = κ/λt for a vacancy, where λt is the households marginal utility of wealth. Notice that these costs are time varying and may depend on business cycle conditions. The present value of a job filled with a worker is then given by Jt = xt At hαt − wt ht + (1 − ρ) Et β t+1 Jt+1 where wt is the wage paid to the worker, and β t+1 = βλt+1 /λt is the households stochastic discount factor with which firms discount their profits. The 6

present value of a vacancy is given by Vt = −κt + Et β t [qt Jt+1 + (1 − qt ) Vt+1 ] There is free entry into vacancy posting, so firms will enter until the cost of creating a vacancy κt equals the benefit of doing so. Therefore Vt ≡ 0, which implies a condition for the cost and benefit of posting vacancies: κt = Et β t+1 Jt+1 qt Thus the value of a job can also be written as Jt = xt At hαt − wt ht + (1 − ρ)

κt qt

Inserting this in the above condition yields a job creation condition, familiar from equilibrium unemployment theory (see Pissarides, 2000). ∙ ¸ κt+1 κt α . = Et β t+1 xt+1 At+1 ht+1 − wt+1 ht+1 + (1 − ρ) qt qt+1 The expected cost of posting a vacancy, κt divided by the probability of filling the vacancy, qt , must equal the expected benefit of having this vacancy filled. This is the expected flow profit plus the present value of the job one period hence, given that the job survives job destruction with probability 1−ρ. This later part is reflected by the expected search costs by virtue of the free entry condition.

3.3

Households

Before we can consider wage determination via bargaining, we need to determine which type of objective function the worker maximizes when bargaining. Here simplifying assumptions are made, which allow an analytically tractable solution of the search and matching framework. A key assumption is that of risk sharing. One assumption is that workers are part of a large family that eﬀectively erases all idiosyncratic income risk arising from the loss of employment. In that case, the family receives its members’ income and distributes it equally among them (this follows Merz, 1995). Another assumption is that of perfect capital markets, in which workers obtained contingent claims before their individual employment status was known. These claims pay out when workers are unemployed, so that there is complete risk sharing among workers (Andolfatto, 1996). The key implication is that all workers can be treated equally and no individual asset positions need to be tracked. 7

Either of the two risk-sharing assumptions allow us to write the aggregate households optimization problem as " # ∞ 1+φ 1−σ X − 1 h c max E0 βt t − χnt t 1 − σ 1+φ t=0

subject to

Bt Bt−1 − = wt nt ht + (1 − nt ) b + π t − ct − Tt . Pt Rt Pt where Bt are the households bond holdings (equal to zero in equilibrium), Pt is the price level, and Rt the nominal interest rate. These variable are included here with an eye on the inclusion of nominal stickiness later on. Consumption is ct , profits of the monopolistic competitors is π t , and Tt is a lump-sum tax. The large household earns wage income as well as income from unemployment benefits b paid to the non-employed members by a government. The (non-labor) first-order conditions are as follows: = λt , c−σ t λt = βRt Et λt+1

Pt . Pt+1

The wage and hours decisions are not part of the household’s maximization problem, but are taken jointly by workers and firms each period. We describe this mechanism below. The marginal value of an employed worker to the aggregate household’s welfare is Wt = wt ht −

χ h1+φ t + Et β t+1 [(1 − ρ) Wt+1 + ρUt+1 ] λt 1 + φ

while the value of an unemployed worker to household welfare is Ut = b + Et β t+1 [st Wt+1 + (1 − st ) Ut+1 ] This allows us to write the surplus relative to unemployment that an employed worker generates as Wt − Ut = wt ht −

χ h1+φ t − b + Et β t+1 (1 − ρ − st ) (Wt+1 − Ut+1 ) , λt 1 + φ

which is used below. 8

3.4

Wage and hours bargaining

As in Trigari (2006), we assume that workers and firms bargain over wages and hours worked each period. The bargaining outcome is chosen to maximize the Nash product so as to maximize the joint surplus of the match. The result is a surplus sharing rule and a condition on the optimal choice of hours.5 Formally, the bargaining outcome is a solution to the following problem: max (Wt − Ut )η Jt1−η , wt ,ht

where the parameter η captures the bargaining power of workers. The firstorder conditions can be written as: ηJt = 1 (1 − η) (Wt − Ut ) αAt xt hα−1 − wt t = 1. φ χht /λt − wt From the second condition, we can calculate the hours equation as = χhφt /λt . αAt xt hα−1 t Thus the marginal value product of an extra hour worked equals its marginal disutility for a worker, which implies that hours are chosen eﬃciently in this setup.6 Using again the first condition, and the equations for Jt and Wt − Ut , one can derive the wage equation, Ã ! 1+φ χ ht wt = η (At xt hαt + κt θt ) + (1 − η) +b . λt 1 + φ The wage is a combination of the opportunity cost of working for workers and the value of the job for firms, including the savings of not having to post a new vacancy, reflected in κt θt . 5

An alternative assumption is that once wages are chosen, firms pick hours to maximize profits (right-to-manage bargaining, Trigari 2006). This solution is however not paretooptimal. We leave the discussion of RTM to Chapter 2 of this survey. 6 Note that in the case of linear utility, the bargaining problem would also be the solution of the individual worker’s problem. However, here, workers are assumed to bargain in the interest of the aggregate family or household. In the case of contigent claims, one would have to presume that enforcable contracts make sure the worker bargains such that the two bargaining conditions are met.

9

The job creation condition, the stochastic discount factor, the evolution of employment, and the wage equation allow a complete characterization of the labor market. As shown in Pissarides (2000), and many related papers, the model is well suited to analyze labor market issues such as eﬀects of unemployment insurance on the labor market, the role of firing costs and taxation, and other determinants of equilibrium unemployment. However, as Shimer (2005) and Costain and Reiter (2004) have shown, the model in its basic form cannot explain the dynamics of the labor market well. In particular, it cannot generate a suﬃcient volatility of unemployment and job finding rates in response to ‘plausible’ changes in productivity.

3.5

Final goods

We assume that the final consumption good is produced by retailers who buy the intermediate good and diﬀerentiate it. The market structure for retailers is monopolistic competition, as in the sticky price version of the model. Households view the retail goods as imperfect substitutes, through a symmetric CES sub-utility function, with time-varying elasticity of substitution t : ∙Z 1 ¸ t /( t −1) t −1 ct (i) t di ct = 0

Maximizing utility for a given level of consumption ct yields as set of demand functions for each variety i : ¸− ∙ pt (i) t ct (i) = ct , Pt where Pt is the associated minimum expenditure price index: Pt =

∙Z

1

1−

pt (i)

0

t

di

¸1/(1− t )

.

Given the demand function, retailers follow monopolistic practise and optimally mark up prices above marginal costs. This yield the following pricing policy for each good i : pt (i) = (1 + μt ) Pt xt , where μt = 1/( t − 1) is the (net) markup of nominal price over nominal marginal cost Pt xt , with xt the real price of the intermediate good. We include a time-varying markup for two reasons. One is that it serves as a proxy for endogenous markup movements when there is nominal price 10

stickiness. Thus our computational experiments below are to some extent suggestive of the behavior in the presence of price stickiness. The other reason is that markup shocks are a plausible alternative driving force of labor market dynamics, as suggested by Rotemberg (2006) and Krause, LopezSalido, and Lubik (2008). In contrast to productivity shocks, (negative) markup shocks are expansionary while at the same time lower the marginal revenue of firms, thus reducing upward wage pressures in the search model with Nash bargaining. With sticky prices, the prices level reacts sluggish to changes in real marginal costs, and a forward looking component is introduced in the price setting equation. We implement price stickiness in the Calvo (1983) fashion. That is, only a random fraction of firms is allowed to change prices. The end-result of this consideration is the New Keynesian Phillips curve : π t = Et βπt+1 +

(1 − β)(1 + γβ) xt γ

where π is inflation and γ is the probability of not adjusting the price.

3.6

Equilibrium

Market clearing on the markets for bonds, labor and goods imply bond holding Bt = 0, as all households are equal, and At nt ht = ct . That is, all production is consumed. In symmetric equilibrium, individual prices p(i) are equal and hence Pt = pt (i). Consequently, from the above pricing relationship, we have that xt =

1 . 1 + μt

This is the familiar condition that real marginal costs are equal to the inverse of the (gross) markup. For given costs, an increase in the markup makes firms want to raise their price, thus moving up their demand curves. The resulting fall in production depresses demand for the intermediate input, which in turn lowers its price.

4

The volatility ‘puzzle’

We use a simplified version of the model to illustrate the diﬃculty of the search and matching model to match labor market data, and well as the 11

proposed solutions to it. For this, ignore for the time being the role of hours (setting α = χ = 0) and the endogeneity of the discount rate (setting σ = 1, so that λt ≡ 1, κt = κ, and β t = β). The job creation condition and wage equation become, respectively: ∙ ¸ κ κ = (1 − ρ)βEt xt+1 At+1 − wt+1 + (1) q(θt ) q(θt+1 ) and wt = η (xt At + θt κ) + (1 − η) b.

Again, the first condition shows that the marginal cost of producing a new job next period, must equal the expected return of having a job filled. Incentives to post vacancies thus depend on the expected values of marginal revenue, wages, and the continuation value of the match. A persistent increase in At will lead to a rise in the right-hand side, which induces firms to post more vacancies. As they do so, labor market tightness, θt = vt /ut , rises, and thus q(θt ) falls. The left hand side rises until equality is restored. How much labor market tightness rises depends on how responsive the return to posting a vacancy is to changes in its determinants. As the second equation shows, wages respond to changes in xt , At , and θt . In particular, changes in θt = vt /ut immediately translate into changes in wages. Then, from the job creation condition above, one can see that as long as marginal revenue and labor market tightness are expected to be high in the future, expected wages will be higher, and thus incentives to post vacancies will be lower. Unemployment will not respond much. On the other hand, if wages are constant, the incentive to post vacancies is not reduced by rises in wages. Then subsequent declines in unemployment will be larger for a given shock. This is essentially Shimer’s point. Therefore, as wages become less responsive to xt At and θt , incentives to post vacancies are higher. To show these arguments formally, insert the wage equation into the job creation condition (1), to obtain ∙ ¸ κ κ = (1 − ρ)βEt (1 − η)xt+1 At+1 − ηθt+1 κ − (1 − η)b + q(θt ) q(θt+1 ) and then log-linearize and multiply with q(θ)/κ: ∙ ¸ ³ ´ q(θ) (1 − ρ)β b b b Et (1 − η)x x bt+1 + At+1 + [ϑ − ηs] θt+1 θt = ϑ κ 0

(2)

(θ) where ϑ = − qq(θ) θ > 0, so that κ/q(θt ) ≈ ϑκ/q(θ)b θt . Since u bt is predeterbt corresponds to a change in vbt . Note now mined, any change in b θt = vbt − u

12

that the size of the response of vt depends on (1 − η), that is, how much workers share the increased marginal revenue of the firm. Second, the degree to which vacancies depend on future expectations on b θt+1 depends on [ϑ − ηs] , which depends on the elasticity of the matching function, the bargaining power of workers, and the probability of finding a job, st . Third, the response of the incentives depends on the steady-state value of q(θ)/κ. This will be further explored below. We can see now that the response of vacancies depends on how wages are set. Because for a positive η, the term ϑ − ηs can be small. In fact, for plausible calibrations, it may be very close to zero. For example, it is exactly zero for values not uncommon in the literature, bargaining power η = 0.5, elasticity of the matching function ϑ = 0.4, and steady-state worker finding rate s = θq(θ) = 0.8. Thus wage flexibility mutes labor market dynamics, as incentives to post vacancies are reduced. Only for η = 0 would the full incentives maintain. Equivalently, note that if one iterates condition (2) forward, one can see that ϑ − ηs = 0 implies that any influence of future bt+1 on current period’s vacancy posting would be shut values of x bt+1 and A down. Only the next period would matter. Additionally, q(θ)/κ aﬀects the responsiveness of vacancies. Its steadystate value is related to the size of the unemployment benefit b and the marginal revenue x, from the steady-state job creation condition: κ [ηBs + 1] = (1 − η)B[x − b], q(θ)

(3)

where B = (1 − ρ)β/(1 − (1 − ρ)β). With ρ = 0.1, and β = 0.98, we have that B ≈ 7.48. If we assume that b is about half of x, and marginal revenue x = 0.9, then κ/q(θ) = (1 − η)B[x − b]/ [ηBs + 1] = 1.68/3.99 = 0.42. The inverse is thus q(θ)/κ = 2.38. Multiplied by (1 − η)x = 0.45, we see that vacancies change about one-to-one (or 1.07) to changes in xt+1 or At+1 . There is basically no propagation in the search and matching model. How can the ‘puzzle’ be solved that the data show volatilities of a order of magnitude above productivity? The proposal by Hall (2005) and Shimer (2005) is to impose real wage rigidity. Eﬀectively, this amounts to setting η = 0 in the linearization above. Then ∙ ¸ ´ q(θ) ³ b b b ϑθt = (1 − ρ)βEt x x bt+1 + At+1 + ϑθt+1 κ θt . Of and future expected values of b θt+1 determine the current response of b course, by iterating forward, b θt can be shown to depend on all future values bt+1 . of x bt+1 and A 13

Alternatively, rather than assuming that wages are rigid, we can boost the volatility of b θt by assuming a higher q(θ) . Going back to the steady-state κ job creation condition (3), we see that the closer the outside option b is to marginal revenue x, the smaller must the left-hand side of the equation be. This is only obtained by a smaller κ, which in turn raises q(θ) . In eﬀect, we κ b bt+1 by bt+1 and A can obtain an arbitrarily large response of θt to changes in x assuming an arbitrarily small x − b. This is the point made by Hagedorn and Manovskii (2008). However, apart from raising issues which level of workers’ outside options is reasonable, raising the unemployment benefit to get the dynamics right makes steady-state unemployment too responsive to changes in benefits. This finding goes back to Costain and Reiter (2004). Suppose we want to calibrate a search and matching model with parameters that imply some observed longrun unemployment rate, for example 10 percent, and assume a job destruction rate of 10 percent. The equation for the evolution of employment nt = (1 − ρ) nt−1 + mt−1 ⇐⇒ ρn = m ⇐⇒ ρ(1 − u) = θq(θ)u implies u=

ρ ρ + θq(θ)

(4)

For this to hold, it must be that θq(θ) = mθ ¯ 1−ϑ = 0.9, where we use the parameterization above. Rewrite (3) as ηBθκ + m1¯ θϑ κ = (1 − η)B[x − b], and insert the implied m, ¯ so that θκ =

(1 − η)B (x − b) ηB + 1/0.9

Now if we choose a high responsiveness of job creation to productivity shocks, x − b would have to be small, or b relatively high. Thus θκ must be small in steady state. Further assuming that we calibrate m ¯ to some value, say, 0.9, then steady state θ = 1. Thus κ must be small if b is high. Consider a reduction in unemployment benefits from such a high level, and the response of endogenous variables, given the parameters just chosen. Linearized: (1 − η)B b b b θ=− b ηB + 1/0.9 κ

Thus, given the parameter values, the lower is κ, the stronger is the response of θ to a change in b. Given equation (4), we see that this implies a larger response of unemployment. Turning back to real wage rigidity, a simplified representation has been used by Krause and Lubik (2007) and Christoﬀel and Linzert (2006), following Hall (2005). The real wage is assumed to be driven by the Nash 14

bargain that would have been agreed in the absence of rigidity, and some wage norm. The latter may be given by some internal equity constraint, or some unmodelled firm level agreement. In any case, the wage follows et , wt = γwtN + (1 − γ) w

e is the wage norm. The norm where wtN is the Nash-bargained wage and w may be last period’s wage w et = wt−1 , or the steady state wage, w et = w. ¯ Under the assumption of a steady state wage norm, the linearized job creation condition becomes ∙ ¸ ³ ´ q(θ) (1 − ρ)β b bt+1 + [ϑ − γηs] b (5) Et (1 − γη)x x bt+1 + A θt+1 θt = ϑ κ

The more rigid the wage becomes, that is, γ becoming smaller, the more responsive is vacancy posting to both changes in current marginal product and to future changes in labor market tightness. The steady state of the model is not aﬀected by this extension, since wN = w, e in the steady state. Finally, let’s consider a proposal by Hall and Milgrom (2008), who have argued that the relevant threat point of a worker is not his or her outside option, but the value of continuing negotiations. The possibility of delaying negotiations gives the worker bargaining power. The consequence is that labor market conditions do not explicitly enter the wage, at least do not aﬀect it much cyclically. We can approximate this point by writing the wage as: wt = η (xt At + θκ) + (1 − η) b. In this case, even though the steady-state wage depends labor market tightness, the cyclical response of the wage is independent of it. Eﬀectively, we remove ηs from ϑ − ηs, so that future expectations of θt fully enter the linearized job creation condition. But productivity changes continue to aﬀect wages.

5 5.1

Quantitative analysis Calibration

We choose parameters partly based on independent information or calibrate them to be consistent with certain labor market facts. The intertemporal elasticity of consumption is set to σ = 1.5, the discount factor is β = 0.992, and the disutility of labor has elasticity φ = 2. These values are common in the literature and found close to estimates of general equilibrium models. 15

For the production side, the elasticity of output with respect to hours is set to α = 0.66, which is taken to correspond to the labor share in aggregate production functions. The average (gross) markup of the monopolistic competitors is assumed to be 1.1, which implies the real price of the intermediate good of x = 1/1.1 ≈ 0.91. In the sticky price version of the model, we assume a probability of not adjusting prices of γ = 0.75, and simple Taylor rule with interest rate smoothing parameter δ r = 0.85, and coeﬃcient on inflation of δ π = 1.5. We take from European data an average unemployment rate of 9 percent, and a job finding rate of 0.7. The job destruction rate is 3% per quarter, which is lower than in the U.S. These values imply a steady state labor market tightness θ. For lack of independent information we calibrate the bargaining power η = 0.5, a value commonly used in labor market studies that employ the Nash bargaining assumption. The elasticity of the matching function with respect to unemployment is 1 − ϑ = 0.6. With these values, we can calculate steady-state labor market tightness, and the implied scale parameter of the disutility of labor, χ = αn−σ x. Assuming a unemployment benefit of b = 0.65, we obtain the implied cost of filling a vacancy, κ, and the steady state wage w. The persistence of markup and technology shocks is assumed to be 0.95.

5.2

Simulation results

In this section we illustrate the mechanisms discussed in the previous sections by simulating the various special cases. We use the European calibration discussed in Chapter 2 of this survey, and summarized above. Our goal is to see whether and how the various simple extensions of the baseline model help resolving the unemployment-volatility puzzle quantitatively. Table 3 presents the results. We report seven model specifications. Model 1 is the baseline Mortensen-Pissarides (1994) framework with linear utility, no intensive margin (hours), and flexible prices and wages. We can see the quantitative failure of the model also on European data: while labor market tightness is quite a bit more volatile than labor productivity, the diﬀerence is far from what we see in the data. The volatility of the wage rate, however, is matched very well. Models 2 adds the hours margin and concave utility to the baseline, while Model 3 adds sticky prices. As the results show, these modifications do not really improve the model’s ability to explain relative volatilities. Hours and concavity help make θ more volatile, but wage volatility also rises. On the other hand, the latter is dampened by adding sticky prices, but at the expense of reducing the variance of θ. 16

Table 3: Simulated relative volatilities

Model y/n 1 2 3 4 5 6 7 8

1 1 1 1 1 1 1 1

u 1.39 1.67 1.31 3.17 5.68 5.36 0.56 5.27

v

v/u

w

h

1.53 2.69 0.94 0 1.87 3.25 1.17 0.22 1.33 2.50 0.96 0.33 4.32 6.55 1.83 0.60 6.51 11.13 1.21 0.40 6.07 10.47 1.06 0.38 1.05 1.33 1.13 0.17 5.98 10.31 1.06 0.38

Model 4 extends Model 3 further by adding two more shocks, a monetary policy shock and a price markup shock. For this exercise, we calibrate the productivity shock to have a variance and persistence which is around the average of the 8 countries in Table 1 (σ a = 0.03, ρa = 0.95). We set the standard deviation of the mark-up shock to be 0.01 (while the steady state markup is 1.1), and we choose the standard deviation of the monetary policy shock to be 25 basis points on a quarterly basis. Not surprisingly, the model’s performance improves significantly, although the volatility of the wage rate increases too much. Both shocks move the labor market indicators in the same direction, and are able to generate the increase in volatility even if introduced alone (not shown). Of course, the importance of the two new shocks depends on their selected volatilities, which requires more careful calibration. Model 5 returns to model 2 and explores the argument put forth by Hagedorn and Manovskii (2008), see also in Section 4. We set the replacement rate to a higher value, bu = 0.74. With this choice of b, we are able to match labor market volatilities quite well. Notice that this value is well below unity, which was suggested by Hagedorn and Manovskii (2008). The reason for this is that our model includes hours choice and hence disutility of labor is explicitly taken into account. Thus when workers decide about taking a job, they need to be compensated not only for lost unemployment income, but also for lost leisure. In our calibration the value of leisure is about 0.2, yielding an upper limit of the replacement rate that is consistent with a positive job surplus of about 0.8. This exercise highlights the crucial importance of the replacement rate,

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but in an unexpected way. It shows that reasonable values for b in Europe are in a range where the model is sensitive to its choice (recall that in our baseline calibration bu = 0.65, which is already relatively high when hours are included in the model). Thus the careful calibration of this parameter is very important. Model 6 introduces a form of real wage rigidity (discussed in Section 4) into the framework of model 2, where . As Table 3 shows, this simple extension goes a very long way towards explaining the volatility puzzle. Unemployment, vacancy, and labor market tightness volatilities increase significantly, and are roughly in line with the data. Wage volatility, on the other hand, remains subdued, and also close to that seen in the data. While not surprising qualitatively, the quantitative success of such a simple modification is very encouraging. Models 7 and 8 continue with forms of real wage rigidity, using two different types of wage norms. We thus modify the wage equation to wt = γwtN + (1 − γ) w, e

e is the wage norm. In Model 7, where wtN is the Nash-bargained wage and w the norm is last period’s wage w e = wt−1 . In model 8, the wage norm is the steady state wage, w e = w. ¯ The numbers reported in Table 3 are calculated using a value of γ = 0.5. Model 7 performs poorly. While the volatility of the real wage declines so do the volatilities of unemployment, vacancies, and labor market tightness. With model 8, on the other hand, we are able to replicate the relative volatilities quite well. Thus, similarly to model 6, decoupling the wage rate from current labor market conditions accounts well for the quantitative features of European labor markets. Moreover, using lower values for γ in model 8 we are able to increase the unemployment and vacancy volatilities even further, as can be seen in Table 4. Intuitively, if we associate lower values of γ with a more rigid labor market, the model predicts that the volatilities of u and v are higher and the volatility of the wage rate is lower the lower γ is. Inspecting Table 2, this pattern seems to be present in the data. We see the highest relative volatilities in France, Belgium and Austria, while Finland has the lowest volatilities among the countries we have data for. Of course, our data is only indicative of this pattern, and it should be combined with institutional variables to test this hypothesis more thoroughly. We leave this issue for future research.

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Table 4: Simulated relative volatilities

Value for γ 0.1 0.3 0.5 0.7 0.9

5.3

y/n 1 1 1 1 1

u

v

v/u

w

h

11.50 13.19 22.56 0.86 0.65 9.42 10.77 18.45 0.92 0.56 5.27 5.98 10.31 1.05 0.38 3.51 3.97 6.86 1.12 0.30 2.2 2.48 4.30 1.16 0.24

New Hires versus Incumbent Workers

An issue that has attracted considerable attention is whether wages of new workers are less rigid than those of incumbent workers. Note that while the search and matching model implies volatile wages for both incumbent and new workers, only the flexibility of new hires’ wages matters for job creation. The initial split of the surplus is what aﬀects the incentives of firms to post vacancies. Once jobs are filled, how exactly the revenue is allocated between worker and firms is immaterial as long as no party has an incentive to separate. This would be the case if the previously agreed wage were either too high for the firm to earn any profits, or too low for the worker to earn more than the outside option (taking account of present values of course). In all the cases we considered thus far, real wage rigidity aﬀected always both old and new workers. Pissarides (2007) argues that the search and matching model is perfectly consistent with the volatility of new hires’ wages. In other words, wage rigidity is not necessarily the solution to the unemployment volatility puzzle uncovered by Shimer (2007). In contrast, authors that have alluded to internal equity constraints that force new hires’ wages to be in line with incumbents’ wages, argue that such wages are in fact rigid. An example is Gertler and Trigari (2006), who assumed staggered wage setting at the firm level of large employers, where new workers enter the existing wage bargain. This would also be in line with Bewley’s findings.

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6

Conclusions

This paper explores the real wage and labor market dynamics in the business cycle version of the search and matching model. In the search and matching model, search by firms and workers is subject to search frictions, and thus are a costly process. Search also gives rise to rents which firm and worker need to split by an appropriately determined wage. The details of wage determination however play a crucial role in how the labor market responds to shocks. This survey is also designed to set the stage for a comprehensive study of the implications of labor search for real marginal cost dynamics and thus inflation. We show first that wage and labor market dynamics in European Union data are similar to those found in U.S. data.7 The baseline search and matching model as analyzed by Shimer (2005) and Hall (2005) thus will have similar diﬃculty matching European data. The literature has proposed a number of changes in the calibration and setup of the search and matching model to resolve this puzzle, and we illustrate and survey these in a unified manner. Qualitatively, all these changes should increase the volatility of vacancies and unemployment, but it is not clear whether this bears out quantitatively. In our quantitative analysis we show that the nature of shocks (monetary and markup) help bring the model closer to the data by a large margin. But this comes at the price of a real wage volatility that is too large. Raising the outside option (unemployment benefits) of workers also has the predicted eﬀect on labor market volatilities, but wage volatility is too large. Interestingly, the proposal by Hall and Milgrom (2008) appears help the model match the data surprisingly well, while at the same time not requiring subtle arguments about how workers’ outside options actually ought to be quantified.8 Similarly, reducing the responsiveness of the real wage by including a steady state wage norm in the wage equation also helps the model match labor market volatilities. The advantage of this approach is that the model contains a parameter that measures the extent of wage rigidity, helping us relate the quantitative findings to individual European countries. Finally, adding some form of wage rigidity circumvents the Costain and Reiter critique that search models with high unemployment benefits predict excessively large responses of steady-state unemployment to reduction in such benefits. 7 8

See, for example, Shimer (2005). This is the debate spurred by Hagedorn and Manovskii (2008).

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[13] Pissarides, Christopher (2000). Equilibrium Unemployment Theory. MIT Press [14] Pissarides, Christopher (2007). "The Unemployment Volatility Puzzle: Is Wage Stickiness the Answer?". CEP Discussion Paper No. 839, London School of Economics [15] Rotemberg, Julio (2006). "Cyclical Wages in a Search-and-Bargaining Model with Large Firms". Working paper, Harvard Business School [16] Shimer, Robert (2005). "The Cyclical Behavior of Equilibrium Unemployment and Vacancies", American Economic Review, 95(1): 25-49 [17] Shimer, Robert (2005). "The Cyclicality of Hires, Separations, and Jobto-Job Transitions". Federal Reserve Bank of St. Louis Review, 87(4): 493-507 [18] Trigari, Antonella (2006). "The role of search frictions and bargaining for inflation dynamics". IGIER Working Paper No. 304

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