In ation and Labor Market Dynamics Revisited

Transcription

1 In ation and Labor Market Dynamics Revisited Tommy Sveen Research Department, Norges Bank and CREI Lutz Weinke Department of Economics, Duke University April 27, 27 Abstract Firms adjust labor both at the intensive and at the extensive margin (See, e.g., Hansen and Sargent 988). Moreover, employment adjustment is not frictionless (See, e.g., Mortensen and Pissarides 994). What does this imply for in ation dynamics? To address this question we develop a New Keynesian (NK for short) model featuring two margins of labor adjustment as well as a simultaneous price setting and employment decision at the rm level. Otherwise our model is similar to the one developed in Blanchard and Galí (26). We nd that the presence of an empirically plausible labor adjustment decision at the rm level rationalizes strategic complementarities in price setting which help explain in ation dynamics. Prepared for the Symposium "The Phillips Curve and the Natural Rate of Unemployment" to be held at the Kiel Institute for the World Economy, 3-4 June 27. We thank seminar participants at Norges Bank, Universitat de Girona and Universitat Pompeu Fabra for useful comments on earlier versions of this paper. We are especially thankful to Stephanie Schmitt-Grohé, Martín Uribe, Jordi Galí and Thijs van Rens. The views expressed in this paper are those of the authors and should not be attributed to Norges Bank.

2 Introduction Firms adjust labor both at the intensive and at the extensive margin (See, e.g., Hansen and Sargent 988). Moreover, employment adjustment is not frictionless (See, e.g., Mortensen and Pissarides 994). What does this imply for in ation dynamics? To address this question we develop a New Keynesian (NK for short) model featuring two margins of labor adjustment as well as a simultaneous price setting and employment decision at the rm level. of labor market frictions per se for in ation dynamics. 2 Our main focus is the role Given that focus it is natural to take into account, in the context of our model, that price setting and employment decisions are typically made simultaneously at the rm level. 3 This allows us to capture the strategic complementarities in price setting implied by the restrictions on employment adjustment. 4 Our motivation for allowing for two margins of labor adjustment in our model is twofold. First, we note that the standard NK model assumes changes at the intensive margin only (See, e.g., Galí 23). On the other hand the Mortensen and Pissarides (994) model as well as the monetary models proposed by Walsh (25), Blanchard and Galí (26) and Krause and Lubik (27) feature only an employment margin. The fact that these extreme and polar assumptions have been adopted in most of the related existing literature 5 might be surprising, given that in the data the cyclical uctuations in total hours result from both changes in employment and changes in hours. More importantly, labor Otherwise our model is similar to the one developed in Blanchard and Galí (26). 2 In the context of monetary models labor market frictions are typically combined with other real rigitities. See, e.g., Walsh (25), Trigari (26) and Krause and Lubik (27). 3 Most of the related existing literature has made the assumption that price setting and hiring decisions take place in di erent sectors. A notable exception is Krause and Lubik (27) who propose a sticky price model à la Rotemberg with only an employment margin for labor adjustment. Kuesters (27) proposes a Calvo-style model featuring one-worker rms whithin which a simultaneous negotiation over product price and wage takes place. 4 See, e.g., Woodford (23, Ch. 3) on the importance of strategic complementarities for understanding in ation dynamics. 5 A notable exception is Trigari (26). She allows for two margins of labor adjustment but abstracts from the simultaneity of price setting and employment decisions. Another recent paper which introduces two margins of labor adjustment in a monetary model is Barnichon (26). He considers a simultaneous price setting and employment problem in a Calvo-style model but, in contrast to our paper, the real wage in his model evolves independently of monetary policy. 2

3 adjustments at the two margins have di erent implications for the determination of the marginal cost, and hence for in ation dynamics: adjustments at the hours margin are current-looking, while employment adjustment introduces a forwardlooking element in the determination of the marginal cost. Ignoring one margin could therefore imply misleading results, as far as in ation dynamics are concerned. We use the resulting framework as our baseline model and compare it to an alternative speci cation featuring a Walrasian labor market. Three sets of results emerge. First, our model implies reasonable variability of key labor market variables. Second, the restrictions on employment adjustment rationalize strategic complementarities in price setting which help explain in ation dynamics. Third, the presence of two margins of labor adjustment is of critical importance for the implied in ation dynamics. Let us put our results into perspective. Krause and Lubik (27) emphasize the way in which the presence of an employment margin only changes the determination of the marginal cost with respect to the standard NK model featuring only an hours margin. We show that the empirically plausible assumption of two margins of labor adjustment implies marginal cost dynamics which are similar to the ones implied by the standard NK model. Trigari (26) argues that labor market frictions per se do not have a quantitatively important e ect on in ation dynamics. 6 We con rm her result if we change our baseline model in such a way that price setting and employment takes place in di erent sectors. Our main result shows, however, that this simpli cation is not innocuous for in that case important strategic complementarities are assumed away. Strategic complementarities in price setting resulting from labor market frictions are also not present in the models proposed by Chéron and Langot (2), Christo el and Linzert (25), Galí and Blanchard (26), and Krause and Lubik (27). This motivates our revisiting of in ation and labor market dynamics. 6 Interestingly, if only an employment margin is assumed then labor market frictions combined with consumption habit and policy inertia reduce the in ation impact and amplify the real impact of a nominal interest rate shock. This is shown in Walsh (25). In the present paper we are mainly concerned, however, with isolating the role of labor market frictions for in ation dynamics. 3

4 Along the way we also analyze the role of real wage rigidity for in ation dynamics. It is shown that this feature is almost irrelevant. The last result is in line with the ndings in Krause and Lubik (27). We note, however, that real wage rigidity reduces strategic complementarities in price setting which is an interesting aspect of the economic mechanism underlying the irrelevance of that feature for in ation dynamics. The remainder of the paper is organized as follows. Section 2 outlines our model. In Section 3 the results are presented and interpreted. Section 4 concludes. 2 The Model 2. Households There is a continuum of households and they are assumed to have access to a complete set of nancial markets. We follow Merz (995), Andolfatto (996) and Trigari (26) in assuming that each household is a large family consisting of a continuum of members with names on the unit interval. In equilibrium some members are unemployed while others work for rms. Each member has the following period utility function U (C t ; H t ) = ln C t which is separable in its two arguments C t and H t. 7 H+ t + ; () The former denotes a Dixit- Stiglitz consumption aggregate while the latter is meant to indicate hours worked. Throughout the analysis the subscript t is used to indicate that a variable is dated as of that period. Parameter is a scaling parameter whose role will be discussed below and can be interpreted as the inverse of the (aggregate) Frisch labor supply 7 Combining the assumptions of large families, complete markets and separable utility implies that employment heterogeneity does not translate into consumption heterogeneity. 4

5 elasticity. The consumption aggregate reads Z C t C t (i) di ; (2) where is the elasticity of substitution between di erent varieties of goods C t (i). The associated price index is de ned as follows Z P t P t (i) di ; (3) where P t (i) is the price of good i. Requiring optimal allocation of any spending on the available goods implies that consumption expenditure can be written as P t C t. Households are assumed to maximize expected discounted utility X E t k= k U (C t+k ; H t+k ) ; (4) where is the subjective discount factor. The maximization is subject to a sequence of budget constraints which take the following form P t C t + E t fq t;t+ D t+ g D t + P t W t H t N t + BU M t + T t ; (5) where Q t;t+ denotes the stochastic discount factor for random nominal payments and D t+ gives the nominal payo associated with the portfolio held at the end of period t. We have also used the notation W t for the real wage and T t for lump-sum transfers including dividends resulting from ownership of rms as well as lump-sum taxes. The unemployment bene t for unemployed household members is denoted by B, while N t gives the fraction of employed household members and Ut M N t is period unemployment. The consumer Euler equation implied by this structure takes the following stan- 5

6 dard form Q t;t+ = Ct C t+ Pt P t+ : (6) Let us nally note that there exists a simple relationship between the gross nominal interest rate, R t, and the stochastic discount factor: E t fq t;t+ g = R t. 2.2 Firms There is a continuum of rms and each of them is the monopolistically competitive producer of a di erentiated good. Each rm i is assumed to maximize its market value subject to constraints implied by the demand for its good, the production technology it has access to, the law of motion of its employment and a Calvo type restriction on price adjustment. Importantly, we assume that each rm takes the relationship between hours hired and its wage as given. Wages are determined as the outcome of a bargain between a rm and its workers which is assumed to take place after price setting and hiring. We will discuss the details of that bargain below. Since rms are assumed to satisfy demand at the posted price 8 hours hired by each rm are determined and the only object of the negotiation is therefore the wage. Let us now be more speci c about a rm s constraints. Technology is assumed to take the following simple form Y t (i) = Z t N t (i) H t (i) ; (7) where Z t indicates total factor productivity, N t (i) is the number of employed workers in rm i and H t (i) denotes average hours worked. We assume that production is linear in total hours Ht tot (i) N t (i) H t (i). The law of motion of employment is given by N t (i) = ( s) N t (i) + L t (i) ; (8) 8 This is rationalized by the assumption of monopolistic competition in the goods market. 6

7 where parameter s denotes the separation rate and L t (i) is meant to indicate the newly hired workers of rm i. where We assume the following employment adjustment costs Lt Nt (i) G t (i) G A L t (i) + G F N t (i) ; (9) U t N t (i) Lt G A U t Lt U t # ; G F () = G F () = ; and G () N : The de nition of function G t () re ects our assumption that there are two distinct types of adjustment costs. The rst term is meant to capture the hiring cost strictly speaking. That cost depends on aggregate labor market conditions, as parametrized by # and. We have also used the de nition U t ( s) N t to denote the fraction of the labor force that is searching for a job at the beginning of period t. The second term in the de nition of function G t () measures the additional cost associated with integrating the newly hired worker into the existing workforce of the rm. Parameter N > measures that cost in the log-linear approximation to the equilibrium dynamics to which we will restrict attention. The role of employment adjustment costs has been emphasized a lot in the literature on labor market dynamics. 9 In the context of our model that feature allows us to obtain a reasonable split of variations in total hours between the two margins of adjustment, as we will discuss below. Cost minimization on the part of households implies that demand for good i is given by Pt (i) Y t (i) = Y t ; () 9 See, e.g., Nickell (986), Hammermesh and Pfann (996) and Cooper and Willis (22). P t 7

8 where Y t denotes aggregate output which is de ned in the following way Z Y t Y t (i) di : Finally, the Calvo restriction on price adjustment states that each period a lottery takes place and with probability ( ) a rm gets to re-optimize its price while with probability the rm posts its last period s price. Since households are assumed to be the ultimate owners of the rms in the economy rms use the stochastic discount factor to discount future pro ts. s.t. A rm s problem therefore reads max X k= E t < : Q Y 4 t+k (i) P t+k (i) = 5 t;t+k P t+k [W t+k (i) N t+k (i) H t+k (i) + G t+k (i)] ; Y t+k (i) = Pt+k (i) Y t+k; P t+k Y t+k (i) = Z t+k N t+k (i) H t+k (i) ; N t+k (i) = ( s) N t+k (i) + L t+k (i) ; 8 < Pt+k+ (i) with prob. ( ) P t+k+ (i) = : P t+k (i) with prob. : The rm s problem implies a standard rst order condition for price setting where X k= k E t fq t;t+k Y t+k (i) [P t (i) P t+k MC t+k (i)]g = ; () denotes the frictionless markup and MC t (i) is meant to indicate the real marginal cost at the rm level. Under the Calvo assumption prices are set in a forward-looking manner or, more precisely, a new price is chosen in such a way that over its expected lifetime the average markup is equal to its desired frictionless 8

9 value. The real marginal cost is determined in the following way MC t (i) = W t (i) + H t t(i) Y t(i) : (2) H t(i)n t(i) As usual, the shadow value of relaxing the technological constraint implied by the production function can be interpreted as the marginal cost. With wage bargaining the rm takes rationally into account that a marginal change in hours implies a change in its real wage. This is re ected in the H t(i) cost equation. write term in the marginal Combining the rst order conditions for employment and for hiring allows us to Ht W t (i) H t (i) + G A + G Nt (i) F U t N t (i) = MC t (i)y t (i) =N t (i) + ( s) E t Q R Ht+ t;t+g A +E t Q R t;t+ G F Nt+ (i) Nt+ (i) N t (i) N t (i) U t+ G F Nt+ (i) N t (i) : (3) where Q R t;t+ P t+ P t Q t;t+ is the real stochastic discount factor. The last equation has an intuitive interpretation. The left hand side gives the cost associated with hiring one additional worker. That cost includes both a wage payment and adjustment costs. The right hand side gives the bene t from hiring one additional worker, i.e. the marginal savings in the cost of using hours associated with having an additional worker in place as well as expected reductions in future adjustment costs. 9

10 2.3 Wage Negotiation The household s value of a match with rm i is given by H t (i) fw + t (i) = W t (i) H t (i) C t (4) + h +E t nq R t;t+ ( s) W f t+ (i) + s F t+wt+ f + ( F t+ ) U e io t+ : where f W t R f W t (i) Lt(i) L t di denotes the average value of a match and F t Lt U t is the job- nding probability. The value of a match with rm i consists of three elements. First, the real wage income resulting from working H t (i) hours at the wage W t (i). Second, the associated disutility of supplying labor (expressed in units of consumption). Third, the expected discounted value of continuing the match in the next period or of searching for a job. The value of being unemployed after hiring has taken place is given by n eu t = B + E t Q R t;t+ hf t+wt+ f + ( F t+ ) U e io t+ ; (5) which equals the unemployment bene t and the expected discounted value of looking for a job in the next period. We follow Blanchard and Galí (26) in assuming that newly hired workers become productive instantaneously. This implies that the value of a match for rm i corresponds to the cost of hiring a worker ej t = G A (F t ) ; (6) which is independent of the rm. The value of an open vacancy for rm i is zero, given our assumptions. The wage is chosen in such a way that the Nash product is maximized, which

11 implies the following rst order condition ( ) J e t = fwt (i) e Ut ; (7) where ( ) denotes the weight of workers in the bargain. Next, we substitute for e J t, e U t and f W t (i) in the last equation. Noting that f W t (i) is equal across rms allows us to nd the wage resulting from the bargain in the following way where W t (i) = C H t(i) + t + + t ; (8) H t (i) t B + G A (F t ) E t Q R t;t+ [( s) ( F t+ ) G A (F t+ )] : (9) For future reference let us rewrite the marginal cost in the following way MC t (i) = C th t (i) N t (i) Y t (i) =H t (i) = MRS t (i) Y t (i) =H t (i) N t (i) ; (2) where MRS t (i) denotes the marginal rate of substitution of consumption for leisure, which is common to all workers hired by rm i. Finally, it is assumed that all markets clear. 2.4 Some Linearized Equlibrium Conditions In what follows we consider a log-linear approximation to the equilibrium dynamics around a zero in ation steady state. Unless stated otherwise lower case letters denote the log-deviation of the original variable from its steady state value. The consumption Euler equation reads c t = E t fc t+ g (r t E t t+ ) ; (2)

12 where parameter denotes the household s time preference rate. Up to the rst order aggregate production is given by y t = z t + n t + h t : (22) Aggregating the linearized law of motion of rm-level employment results in n t = ( s) n t + sl t ; (23) Linearized unemployment reads u t = ( s) N U n t ; (24) where we have used the notation that a variable without a time subscript denotes the steady state value of that variable. Period unemployment is given by u M t = N U M n t: (25) Aggregating and linearizing the rst order condition for rm-level employment implies n t = E t fn t+ g + N Y N (mc t + y t n t ) W H (w t + h t ) (26) F # [#f t ( s) E t f#f t+ + (r t t+ + )g] : The following relationships holds true f t = l t u t : (27) 2

13 The real wage is given by w t = H+ C + CH + W H c t + W H h t + W H t; (28) where t = ( ) R [#f t ( s) E t f( F ) (r t E t t+ ) + ( F ) #f t F f t+ g] : (29) The real marginal cost reads mc t = c t + ( + ) h t y t + n t : (3) In the Appendix the following in ation equation is derived t = E t t+ + mc t ; (3) where parameter is computed numerically using the method outlined in Woodford (25). Finally, let us state the exogenous driving forces. Technology is assumed to follow a stationary AR() process z t = z z t + e zt : (32) and monetary policy is assumed to take the form of a Taylor rule r t = r r t + ( r ) [ + t + y y t ] + e rt ; (33) where e rt denotes an iid shock to monetary policy. 3

14 2.5 Calibration Let us now discuss the values which we assign to the model parameters in most of the quantitative analysis that we are going to conduct. The period length is one quarter. We let be :99, which implies a steady state real interest rate of about 4 per cent. We follow Golosov and Lucas (27) and set the elasticity of substitution between goods,, to 7. This implies a steady-state mark-up of about 2 per cent. Our baseline value for the Calvo parameter for price setting,, is 2=3, which is consistent with the recent empirical nding of Nakamura and Steinsson (26) that rms change their prices on average every third quarter. As far as monetary policy is concerned we set y and to :5 and :5, respectively, as originally suggested by Taylor (993) and the parameter measuring interest rate smoothing, r, is set to :9. These parameter values are reasonable given the empirical results in, e.g., Clarida et al. (2). The estimates reported by MaCurdy (98) on the labor supply elasticity center around :5, which is the value we choose for =. Below we compare our model to a benchmark with a Walrasian labor market with only an hours margin for labor adjustment and we correspondingly set the labor supply elasticity to unity in the latter framework. We follow Shimer (25) in setting steady state period unemployment to :57 and the quarterly job- nding rate to :7. Given our model this implies a separation rate of about :5 and steady-state search unemployment of about :2. Following Hall (25) the unemployment bene t, B, is set to 4% of steady state labor income. The employment adjustment cost parameter, N, is set to 2 which is in line with the estimates reported in Cooper and Willis (22). In order to calibrate the P We compute the quarterly rate as :34 3 ( :34) j, where :34 is the corresponding j= monthly rate reported by Shimer. The values used in the literature range from.7 (Merz 995) to.5 (Andolfatto 996). 4

15 parameters in the hiring cost function, and #, we follow Blanchard and Galí (26) and use a simple relationship between the hiring cost model and the Mortensen and Pissarides (994) (MP for short) model. given by!v U In the latter, the matching function is, where V denotes the vacancy to unemployment ratio, is the elasticity of the matching function and! is a constant. Moreover, let cc be the real cost of posting vacancies. The cost of hiring an additional worker is therefore cc! (V=U). In our set-up it is equal to L U # =! # (V=U) #, where we have used that matches in the MP model are given by L =!V U. We therefore use # = and = cc! +#. Hall (25) estimates to be :765 and we correspondingly set # = :765 :765 = :37. In order to satisfy the Hosios (99) condition we choose equal to :765. Further we follow Hall (25) and set the steady state vacancy to unemployment ratio to :539. Given the elasticity of the matching function and the steady-state search unemployment, this allows us to pin down the value of!. The rst-order condition for employment evaluated in steady state and the wage equation imply two conditions to pin down the steady state wage income W H and the vacancy cost cc. Last, we use to pin down hours in steady state to =3 of available time. We set the standard deviation of the monetary policy shock to :2. Walsh (25) argues that this value is consistent with estimated Federal Reserve reaction functions. Moreover, the coe cient of autocorrelation in the process of technology, z, is assumed to take the value :95, as in Erceg et al. (2) and Walsh (25). The standard deviation of the productivity shock is set to :8542, in which case the baseline model matches the standard deviation of GDP of :69% in the data. 3 Results We start by analyzing the labor market dynamics implied by our model. The results are shown in Table. 5

16 Table : Data 2 and Model statistics Variable U.S. Data Model Std Std. Rel. to Empl. Std Std. Rel. to Empl. GDP :69% :82 :69% :29 Hours :42=:5% :294=:357 :6% :4378 Employment :43% :3956% The variability of employment and hours is reasonably well in line with the data. In order to illustrate how labor market frictions help explain in ation dynamics we analyze next the dynamic consequences of a 25 basis point increase in the nominal interest rate. The results are shown in Figure. [Figure about here] Speci cally, we compare our baseline model to an alternative speci cation featuring a Walrasian labor market, i.e. in the latter model it is assumed that there exists only an hours margin for labor adjustment coupled with exible wages and perfect competition in the labor market. This is a common modeling choice in the New Keynesian literature. To make the comparison meaningful we assume that the labor supply elasticity,, is set to one in the Walrasian model, as in Galí (23). Four aspects of that comparison are worth highlighting. First, wage bargaining implies a muted response of the average real wage. The reason is the wedge between the marginal rate of substitution of consumption for leisure and the real wage which is implied by the surplus sharing between rms and their workers. Interestingly, the corresponding average real marginal cost schedules display the opposite pattern: the average real marginal cost moves more in our baseline model that it is the case in the Walrasian benchmark economy. Intuitively, the marginal cost is linked to the marginal rate of substitution of consumption for leisure in both models, but the total e ect of the relatively smaller hours adjustment in the presence of a relatively smaller labor supply elasticity in the baseline model implies a di erence in 2 See, Bachmann (26). The second numbers in the second row include unpaid hours. 6

17 the respective marginal cost schedules. This is our second nding. Third, and most importantly, the in ation response is muted in our baseline model with respect to its counterpart in the Walrasian economy. A price setting rm internalizes the consequences of that decisions for the marginal cost it faces over the expected lifetime of the chosen price. This economic incentive makes the price setter relatively reluctant to change its price in response to a change in the current (or future expected) average real marginal cost. Our forth observation regards the fact that variation in total hours (and hence, given our production function, output) is similar in our baseline model and in the Walrasian benchmark economy, but, by construction, only in our baseline model the labor market variables display the empirically plausible use of the two margins. Our main result in the above comparison is that the strategic complementarities in price setting implied by the restrictions on employment adjustment are quantitatively important. To emphasize that point somewhat more we show next how the picture changes if these strategic complementarities are taken away by assuming that price setting and employment takes place in di erent sectors. The results are shown in Figure 2. [Figure 2 about here] In that case the two models imply similar dynamics. This con rms the result in Trigari (26) according to which labor market frictions per se do not matter for in ation dynamics. Our point is that the simplifying assumption of separating price setting and employment decisions is not inconsequential for the results obtained. Finally, we want to stress that the discipline imposed by the labor market facts helps explain in ation dynamics. If it is assumed, counterfactually, that labor adjustment takes place at the extensive margin only then it follows that in ation is more volatile in our model that in the Walrasian benchmark case: the conclusion which obtains in the baseline model is reversed! This is shown in Figure 3. [Figure 3 about here] 7

18 The reason for the last result is that the determination of the marginal cost changes dramatically if there is only an employment margin. This has been recently discussed in Krause and Lubik (27). Restricting attention to an employment margin only is therefore another simpli cation which is not innocuous for the conclusions regarding in ation and output dynamics. In that context it is also interesting to analyze how real wage rigidity a ects in ation dynamics. At this step we follow Krause and Lubik (27) in assuming a wage norm in the spirit of Hall (25), i.e. rm i s real wage is given by W t (i) = W + ( ) W n t (i) ; where W n t (i) is the bargained real wage and W is the wage norm. We set to :5 and W equal to the steady state real wage. Our nding is that real wage rigidity is almost irrelevant for in ation dynamics. This is illustrated in Figure 4. [Figure 4 about here] The last result is in line with the ndings in Krause and Lubik (27) 3, but the economic mechanism at work is somewhat di erent. One aspect worth highlighting is the fact that real wage rigidity reduces the strategic complementarities in price setting implied by our model. 4 Conclusion We make some progress in understanding the role of labor market frictions for in ation dynamics. Our work shows the importance of taking into account the fact that rms adjust labor input both at the intensive and at the extensive margin. That 3 Trigari (26) shows that the assumptions regarding the bargaining mechanism are important for the assessment of the role of real wage rigidity for in ation dynamics. Christo el and Linzert (25) generalize and con rm her result. Interestingly, the bargaining mechanism assumed in the present paper takes the form under which real wage rigidity would matter for in ation dynamics in the context of their models. This is therefore another instance in which it plays a role for the results obtained whether price setting and employment decisions take place simultaneously or not. 8

19 assumption combined with a simultaneous price setting and employment decision at the rm level helps explain in ation dynamics. Earlier results according to which labor market frictions per se are irrelevant for in ation dynamics are therefore an artefact of the commonly used simpli cation that employment and price setting take place in di erent sectors. In the present paper the modeling of the labor market is relatively simple. In work in progress we extend that work in order to contribute to the strand of literature which asks how monetary settings help explain labor market dynamics. 9

20 Appendix: Price Setting and Employment We posit rules for price setting and for employment. Since our model features a convex employment adjustment cost we have bn t (i) = bp t (i) + 2 bn t (i) ; and bp t (i) = bp t + bn t (i) : Our goal is to nd conditions for the unknown coe cients in the rules. To this end we rst consider the linearized equation for the relative to average employment at the rm level where N N Y bn t (i) = E t fbn t+ (i)g + b h t (i) ;. Combining the last equation with the rm s demand constraint + and with its production function results in + + bn t (i) = E t fbn t+ (i)g + bn t (i) bp t (i) Our next goal is to nd a condition on the unknown coe cients in the employment rule. Using the above rules as well as the Calvo assumption allows us to write bn t (i) = ( ) bp t (i) + bn t (i) ; which imposes the following two constraints on the undetermined coe cients and 2 in the employment rule 2 = ( ) ; 2

21 and = 2 : Clearly, the last two conditions depend on the unknown parameter from the pricing rule. Next we turn to the linearized price setting equation to nd a condition for this parameter. We can write the newly set price chosen by rm i as follows bp t (i) = X () j E t t+j + ( ) X () k mc t+j + j= ( ) X () k E t bn t+j (i) : + j= j= Using the employment rule as well as the Calvo assumption we nd after some algebra X () k E t bn t+j (i) = j= 2 2 bn t (i) + ( ) ( 2 ) bp t (i) ( ) ( 2 ) X () j E t t+j : j= Combining the last two equations and invoking the Calvo assumption, i.e. noting that the average value of bn t (i) is zero in the group of time t price setters we can impose the following condition on the unknown parameter in the pricing rule = ( ) ( ) + ( )( 2 ) : The average newly set price reads bp t = X () j E t t+j + j=! X () k mc t+j ; j= 2

22 where! [ + ] ( 2) + : ( 2 ) Solving the last equation forward and invoking the linearized price index gives t = E t f t+ g + mc t ; where ( ) ( )! : Next we impose stability. Invoking once more the pricing and employment rules, as well as the de nition of the price index we obtain E tbp t+ (i) E t bn t+ (i) = A 2 4 bp t (i) bn t (i) where A 4 ( ) 5. Stability requires that both roots of matrix ( ) + 2 A are inside the unit circle. For candidate parameter values which satisfy the stability requirement we therefore solve the following system ( ; 2 ) = = 2 ( ) ( 2 ) ( + ) ; 2 2 ; 3 5 ; = ( + ) ( ) : This pins down the coe cients ( ; 2 ; ). 22

23 References Andolfatto, David (996): Business Cycles and Labor Market Search, American Economic Review 86(), Bachmann, Rüdiger (26): A Tale of Two Margins, mimeo. Barnichon, Régis (26): Productivity, Aggregate Demand and Unemployment Fluctuations, mimeo. Blanchard, Olivier, Jordi Gali (26): A New Keynesian Model with Unemployment, mimeo. Chéron, Arnaud, François Langot (2): The Phillips and Beveridge Curves revisited, Economic Letters 69, Christo el, Kai, Tobias Linzert (25): The Role of Real Wage Rigidity and Labor Market Frictions for Unemployment and In ation Dynamics, European Central Bank Working Paper No Clarida, Richard, Jordi Galí, and Mark Gertler (2), Monetary Policy Rules and Macroeconomic Stability: Evidence and some Theory, Quarterly Journal of Economics 5, Cooper, Russel, Jonathan L. Willis (22): The Cost of Labor Adjustment: Inferences from the Gap, Federal Reserve bank of Kansas City Working Paper, RWP 2-. Galí, Jordi (23): New Perspectives on Monetary Policy, In ation, and the Business Cycle, in: Advances in Economic Theory, edited by: M. Dewatripont, L. Hansen, and S. Turnovsky, vol. III, 5-97, Cambridge University Press. Golosov, Mikhail, Robert E. Lucas Jr. (27): Menu Costs and Phillips Curves, Journal of Political Economy 5, Hall, Robert E. (25): Employment Fluctuations with Equilibrium Wage Stickines, American Economic Review 95(),

24 Hamermesh, Daniel S., Gerard A. Pfann (996): Adjustment Costs in factor Demand, Journal of Economic Literature 34, Hansen, Gary D., Thomas J. Sargent (988): Straight Time and Overtime in Equilibrium, Journal of Monetary Economics 2, Hosios, Arthur (99): On the E ciency of Matching and Related Models of Search and Unemployment, Review of Economic Studies 57 (2), Krause, Michael U., Thomas A. Lubik (27): The (ir)relevance of real Wage Rigidity in the New Keynesian Model with Search Frictions, Journal of Monetary Economics, forthcoming. Kuester, Keith (27): Real Price and Wage Rigidities in a Model with Matching Frictions, European Central Bank Working Paper No. 72. Macurdy, Thomas E. (98): An Empirical Model of Labor Supply in a Life-Cycle Setting, Journal of Political Economy 89(6), Merz, Monica (996): Search in the labor Market and the Real Business Cycle, Journal of Monetary Economics 36(2), Mortensen, Dale T., Pissarides, Christopher A. (994): Job Creation and Job Destruction in the Theory of Unemployment, Review of Economic Studies 6(3), Nakamura, Emi, Jón Steinsson (26): Five Facts About Prices: A Reevaluation of Menu Cost Models, mimeo. Nickell, Stephen. (986): Dynamic Models of Labor Demand, in Handbook of Labor Economics, ed. by Orley Ashenfelter, and Richard Layard. Amsterdam: North Holland. Shimer, Robert (25): The Cyclical Behavior of Equilibrium Unemployment and Vacancies, American Economic Review 95(), Taylor, John B. (993): Discretion versus policy rules in practice, Carnegie- Rochester Conf. Ser. Public Pol. 39 (993),

25 Trigari, Antonella (26): The Role of Search Frictions and bargaining for In ation Dynamics, IGIER Working Paper No. 34. Walsh, Carl E. (25): Labor Market Search, Sticky Prices, and Interest Rate Policies, Review of Economic Dynamics 8, Woodford, Michael (23): Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press. Woodford, Michael (25): Firm-Speci c Capital and the New Keynesian Phillips Curve", International Journal of Central Banking 2,

26 Output 5 Unemployment.4 Nominal Interest Rate.5.5 Hiring Cost Walrasian Inflation Marginal Cost Real Wage Employment Hours Total Hours Figure I: Shock to Monetary Policy Rule 26

27 .5.5 Output Hiring Cost Walrasian 5 5 Unemployment Nominal Interest Rate Inflation Marginal Cost Real Wage Employment Hours Total Hours Figure II: Two-Sector Model: Shock to Monetary Policy Rule 27

28 .5.5 Output Hiring Cost Walrasian 3 2 Unemployment Nominal Interest Rate Inflation Marginal Cost Real Wage Employment Hours Total Hours Figure III: Only Employment Margin: Shock to Monetary Policy Rule 28

29 Output 5 Unemployment.4 Nominal Interest Rate.5.5 Baseline RWR Inflation Marginal Cost Real Wage Employment Hours.5.5 Total Hours Figure IV: Real Wage Rigidity: Shock to Monetary Policy Rule 29