WAGE SETTING ACTORS, STICKY WAGES, AND OPTIMAL MONETARY POLICY

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1 WAGE SETTING ACTORS, STICKY WAGES, AND OPTIMAL MONETARY POLICY Miguel Casares D.T.007/0

2 Wage Setting Actors, Sticky Wages, and Optimal Monetary Policy Miguel Casares y Universidad Pública de Navarra February 007 Abstract Following Erceg et al. (000), sticky wages are generally modelled assuming that households set wage contracts à la Calvo (983). This paper compares that sticky-wage model with one where wage contracts are set by rms, assuming exible prices in any case. The key variable or wage dynamics moves rom the marginal rate o substitution (households set wages) to the marginal product o labor ( rms set wages). Optimal monetary policy in both cases ully stabilizes wage in ation and the output gap ater technology or preerence innovations. However, nominal shocks make the assumption on who set wages relevant or optimal monetary policy. JEL codes: E4, E3, E5. Keywords: Wage-setting households, Wage-setting rms, Optimal monetary policy. Introduction In a highly in uential article, Erceg et al. (000) show how sticky wages can be modeled in an optimizing ramework by giving households a xed probability à la Calvo (983) to optimally reset their wage contract. Thus, each household (or union as a group o households) owns some di erentiated labor service and may decide the nominal wage associated with its labor supply. This line o research started in 006 while I was a visiting ellow at the Federal Reserve Bank o St. Louis. I eel completely grateul to the Federal Reserve Bank o St. Louis or its kind invitation and hospitality, and I also thank Bill Gavin, James Bullard, and Ben McCallum or insightul comments and suggestions or this paper. Financial support was provided by the Spanish Ministry o Education and Science (Postdoc Fellowships Program and Research Project SEJ /ECON). y Departamento de Economía, Universidad Pública de Navarra, 3006, Pamplona, Spain. [email protected] (Miguel Casares). The assumption o providing households with market power to set wages had already been taken in Blanchard (986) and Rankin (998).

3 The optimal wage can be reset only when receiving a market signal that arrives with a constant probability. As a result, the dynamics o either wage in ation or the real wage can be ormulated in a single equation. Fluctuations o wage in ation (and the real wage) are governed by the gap between the aggregate marginal rate o substitution o households and the real wage. This sticky-wage structure is becoming very popular among New Keynesian researchers in recent times (Amato and Laubach, 003; Smets and Wouters, 003; Woodord, 003; Giannoni and Woodord, 004; Christiano et al., 005; Levin et al., 005; Casares, 007). As one alternative sticky-wage variant, this paper examines the implications o moving the decision-making on the nominal wage rom households to rms. Thus, the nominal wage contract may also be the one that maximizes pro t o a monopsonistically competitive rm subject to a labor supply constraint. 3 Nominal rigidities can be readily introduced as Calvo-style contracts where rms are the wage setting actors. In turn, the paper shows how uctuations o wage in ation are explained by a orward-looking equation that depends on the gap between the marginal product o labor and the real wage. The consequences o nominal rigidities on the optimal design o monetary policy were rst examined assuming that prices were sticky (Rotemberg and Woodord, 997; Clarida et al., 999). Such analysis was extended to the case o economies where both prices and wages were sticky by Erceg et al. (000). 4 Our contribution on this regard is to derive the optimal monetary policy in an optimizing macro model where the only source or nominal rictions is wage stickiness. 5 In that respect, the analysis will distinguish the implications o our two variants on wage setting actors or optimal monetary policy within a general equilibrium economy with exible prices, and sticky wages. Using the targeting rules approach introduced by Svensson (999a, 999b) and Woodord (999), the optimal monetary policy can be obtained by solving a central bank optimizing program subject to a set o model equations. Furthermore, the central-bank objective unction can be written as an approximation to social welare based on the average utility value. This paper also Analogously, a number o papers use Taylor (980) staggered wage contracts set by households in a very similar optimizing ramework. As representative examples, see Ascari (000), Huang and Liu (004), and Huang et al. (004). 3 Early contributions by Azariadis (975), Hoehn (988), and Flodén (000) also put the wage setting decision on pro t-maximizing rms. 4 See Woodord (003, ch. 6), and Amato and Laubach (004) or discussions on optimal monetary policy under other model settings. 5 A number o empirical papers recently argue that prices are not as sticky as generally assumed (Golosov and Lucas, 003; Bils and Klenow, 004) and others put emphasis on sticky wages as the source o nominal rigidities in the economy (Christiano et al., 005; Levin et al., 005).

4 compares the welare-theoretic loss unction or the central bank in each variant o our sticky-wage model. 6 We nd that both cases agree on having variability o wage in ation and the output gap as the only monetary policy targets in a sticky-wage, exible-price economy. However, their optimal policy reaction is distinct. I households set wages the rate o wage in ation must all when there is a positive change in the output gap whereas the reaction must be o opposite sign in the case o rms acting as wage setters. The rest o the paper is organized as ollows. The sticky-wage model where households set wages is described in Section. The case where rms are wage setting actors is introduced in Section 3 as another sticky-wage variant. Section 4 is devoted to the theoretical monetary policy analysis as the optimal monetary policy rules are computed and discussed. The analysis is completed in Section 5 with simulation exercises such as impulse-response unctions, calculation o secondmoment statistics, and variance decomposition. Finally, Section 6 reviews the main conclusions o the paper. Sticky wages set by households Since the well-known paper by Erceg et al. (000), sticky wages have been typically incorporated in the New Keynesian model by allowing households to set the nominal wage in the labor market. Hence, each household owns a di erentiated labor service that supplies at her speci c nominal wage. In this setup, rms demand bundles o labor services to be employed in their production processes. A labor bundle is obtained using the aggregation scheme rst described by Dixit and Stiglitz (977) Z n t = 0 h =( h ) (n t (h)) ( h )= h dh ; () where the time period is indicated in the subscript o the variables, h > :0 is a constant parameter, and n t (h) is the labor service provided by the h-th household. A competitive labor agency assembles di erentiated labor services rom households to get labor bundles that will sell to rms. maximum-pro t condition or such labor agency leads to the ollowing demand unction (regarding the h-th labor service) 7 The Wt (h) h n t (h) = n t ; () W t 6 To derive the welare-theoretic loss unctions, we borrow the computational methodology used by Erceg et al. (000) and Woodord (003, ch. 6). 7 See Erceg et al. (000) or details. 3

5 in which W t (h) is the nominal wage set by the h-th household, W t is the Dixit-Stiglitz aggregate nominal wage, and h gives the elasticity o substitution across labor services. Also as in Erceg et al. (000), let us assume that a separable CRRA utility unction ranks the preerences o the h-th representative household over consumption, c t, and the supply o labor, n t (h), U t (h) = exp( t) (c t ) (n t (h)) + ; (3) + where t is an AR() shock on consumption preerence, t = t + " t with " t N(0; " ). Units o consumption do not reer to the type o household because there are complete nancial markets that ensure the same consumption across di erentiated households. Intertemporal utility is maximized subject to a budget constraint and a demand or labor constraint as displayed in the ollowing optimizing program Max E t j U t+j (h) W t+j (h) s:to : n t+j (h) = c t+j + ( + R t+j ) B t++j(h) P t+j P t+j Wt+j (h) h and to : n t+j (h) = n t+j or j = 0; ; ; ::: W t+j B t+j (h) P t+j or j = 0; ; ; ::: The budget constraint is expressed in real magnitudes; labor real income may be spent on units o consumption and on net purchases o bonds. 8 I nominal wages can be optimally reset every period, we would obtain, in period t, this rst order condition with respect to W t (h) & t n t (h) P t + h t (h) n t(h) W t (h) = 0; that includes the Lagrange multipliers on the budget constraint, & t, and on the labor-demand constraint t (h). The rst order conditions or consumption and labor imply the ollowing values o the multipliers 9 & t = U ct and t (h) = W t (h) U nt(h) + & t ; (4) P t which can be substituted into the nominal wage optimality condition to yield W t (h) P t = h h U nt(h) U ct : (5) The optimal nominal wage sets the real wage as a mark-up over the marginal rate o substitution (mrs) between disutility o labor and utility o consumption. 8 Regarding notation, P t is the aggregate price level in period t, B t+(h) is the nominal amount o bonds purchased in period t to be reimbursed in period t +, and R t is the nominal interest rate attached to such purchase. 9 For simplicity in notation, U ct represents the consumption marginal utility and U nt (h) the marginal disutility o labor. 4

6 Following the assumption by Calvo (983), nominal rigidities on wage setting can be easily introduced assuming that households only reset optimally the wage contract in states o nature that arrive with a constant probability. Thereore, households are not able to lay out the optimal wage contract with probability. As in Casares (007), the raction o households that cannot set the optimal wage will apply the ollowing stochastic indexation rule W t (s) = ( + ss + t ) W t (s) (6) reerred to some s-th suboptimal household. The indexation actor in (6) depends on the steadystate rate o in ation, ss, and also on the stochastic element, t, which ollows the AR() process t = t + " t with " t N 0; ". The t term can be interpreted as a cost-push shock that will ultimately a ect the rate o economy-wide wage in ation (as it will be shown below). Incorporating sticky wages à la Calvo, the nominal wage optimality condition becomes j E t j j k= 4& ( + 3 ss + t+k ) n t+j (h) t+j + h t+j (h) n t+j(h) 5 = 0; (7) W t (h) P t+j whereas the rst order condition on the desired supply o labor or period t + j turns out to be 0 j k= t+j (h) ( + ss + t+k ) W t (h) nt+j (h) + & t+j A : (8) P t+j Substituting (7) into (8), and then loglinearizing around steady state yield cw t (h) = ( )E t j j ( P b t+j + dmrs t+j (h)) E t j j t+j ; (9) j= where W c t (h) and P b t+j represent the log o the optimal nominal wage and the log o the aggregate price level, and dmrs t+j (h) represents the log o the mrs, i.e. dmrs t+j (h) = U b nt+j (h) bu ct+j. Equation (9) can be transormed to present the optimal wage contract depending exclusively on aggregate magnitudes (see Appendix A or the proo) cw t (h) = W c t + ( ) E t j j ( dmrs t+j + h bw t+j ) + E t j j W t+j t+j ; (0) j= with W c t denoting the log o the aggregate nominal wage, dmrs t the log o the aggregate mrs, and bw t the log o the aggregate real wage, bw t = W c t Pt b. In addition, W t is the notation or the rate o wage in ation, W t = W c t Wt c. The optimal wage contract has three determinants in (0): the aggregate nominal wage, the current and uture expected gaps between the aggregate mrs and the 5

7 real wage, and the expected uture rates o wage in ation once the indexation shock is deducted. The relative wage taken rom (9) can be replaced by c W t (h) c Wt = W t t as obtained by loglinearizing the aggregate nominal wage ( c W t = ( ) c W t (h)+ c W t + t ). Then, by computing E t W t+ and calculating W t E t W t+, we get to the ollowing wage in ation equation W t = E t W t+ + ( )( ) ( dmrs t bw t ) + ( ( + h ) ) t : () Wage in ation dynamics evolve depending on three arguments: expected wage in ation, E t W t+, the gap between the mrs and the real wage, dmrs t bw t, and the exogenous indexation shock, t. This is the wage in ation equation commonly used in the New Keynesian model with sticky wages (Erceg et al., 000; Woodord, 003, ch. 3; Smets and Wouters, 003; Christiano et al., 005) with the novelty o the indexation shock, t. With respect to prices, we suppose that they ully adjust every period to clear the labor market. The exible-price assumption guarantees that the labor market is always in equilibrium. Thus, rms can choose their demand or labor bundles that implies maximum pro t. In turn, the exible-price condition yields bw t = d mpl t ; () where d mpl t is the log o the marginal product o labor. Combining () and (), it is obtained W t = E t W t+ + ( )( ) ( dmrs t mplt d ) + ( ( + h ) ) t : (3) For illustrative purposes, we would rather present the wage in ation equation as a unction o the output gap, ey t, which is de ned as the di erence between the log o current output and the log o potential output ey t = by t by t : (4) As suggested by Woodord (003), potential (natural-rate) output is the amount o output obtained in a perect-competition economy with both exible prices and wages. I that is the case, it is wellknown that loglinear uctuations on the mrs must be equal to those on the marginal product o labor, dmrs t = d mpl t, since both variables coincide with the real wage (Erceg et al., 000). 0 As standard, let us assume that output is produced using a Cobb-Douglas production technology F (z t ; n t ) = [exp(z t )n t ] ; (5) where 0 < <, and z t is an AR() technology shock z t = z z t + " z t with " z t N(0; " z). The condition dmrs t = mpl d t or one economy with the utility unction () and the production unction 0 This can be easily veri ed by setting a null Calvo probability or non-optimal wage contracts ( = 0:0). 6

8 (5) leads to the ollowing equation or potential output uctuations by t = ( )++ [( + ) z t + t ] ; (6) which implies dmrs t d mplt = ( )++ ey t : The last result can be inserted in (3) to yield W t = E t W t+ + h ey t + ( ) t ; (7) ( )( ) ( )++ with h = (+ h ). The model with sticky wages set by households implies a positive relationship between wage in ation and the output gap. As we just showed above, a positive output gap means that the mrs is higher than the marginal product o labor, and also higher than the real wage (recall the exible-price condition bw t = d mpl t ). Under that circumstance, households wish to work less hours and those that are able to reset their labor contract will choose a higher nominal wage that deliver less hours when entering (). As a result, the aggregate nominal wage and the rate o wage in ation will rise. 3 Sticky wages set by rms This section introduces another way o modeling sticky wages assuming that rms are the wage setting actors instead o households. An economy with wage setting rms requires some degree o labor demand di erentiation to claim heterogeneity on wage contracts. Thus, in somehow a symmetric manner to the model with household setting wages, each rm acts as a monopsonistically competitor in the labor market because it is the only employer or a di erentiated type o labor service. Meanwhile, households disutility o labor depends on how many bundles o labor services they supply, obtained rom the Dixit-Stiglitz aggregator n t = " Z 0 # n t () + d + (8) where > 0:0, and n t () represents the type o labor service supplied to the -th rm. The instantaneous utility unction (3) can be rewritten or this model variant as ollows U t = exp( t)c t n + t + ; (9) The presence o E t W t+ in (7) makes all expected uture output gaps also a ect positively or the determination o current wage in ation. 7

9 where n t is given by (8). Unlike the model with household setting wages, households are identical and they decide the same amounts or consumption and labor services since they act both as price h R i takers and wage takers. The aggregate nominal wage is de ned by W t = 0 W t() + + d so that the amount o nominal labor income obtained rom all di erentiated labor services supplied is the same as the income obtained by supplying bundles o labor, R 0 W t()n t ()d = W t n t. Taking this into account or the kind o budget constraint presented in the previous section, we can compute the ollowing household s rst order conditions with respect to the number o labor bundles, n t, and the supply o the -th type o labor service, n t () n t n t (n t ()) n t & t W t P t = 0; & t W t () P t = 0: The value o the Lagrange multiplier, & t, obtained rom the rst equation can be substituted out onto the second equation to yield Wt () n t () = n t ; (0) which represents the supply o labor constraint aced by the -th rm when setting its nominal wage (analogous to () or the case o households setting wages). Note that the elasticity o substitution o the household across di erentiated labor services is now positive at the constant parameter. W t Turning to the rms optimizing behavior on setting wages, let us suppose that they produce output using the Cobb-Douglas technology (5). Assuming that rms seek to maximize intertemporal pro ts, the optimizing program or the -th representative rm can be written as P Max E t F j W t+j () (z t+j ; n t+j ()) n t+j () P t+j Wt+j () s:to n t+j () = n t+j or j = 0; ; ; ::: W t+j With no wage rigidity, the optimality condition on W t () is n t () P t n t () ' t () = 0; () W t () where ' t () is the Lagrange multiplier o the supply o labor constraint in period t. The value o ' t () is de ned by the rst order condition on the demand or labor n t () ' t () = W t() P t mpl t (); () 8

10 where mpl t () denotes the marginal product o the -th type o labor. Combining the last two expressions and rearranging terms, we obtain W t () = + P t mpl t (): The optimal wage contract is a raction o the market-valued marginal product o labor. Thereore, the value o the wage contract is obtained by applying the mark-down, o labor productivity. +, over the market value Introducing both wage rigidities à la Calvo and the wage indexation rule (6) adapted to the rm, the optimality condition () or the wage contract set in period t changes to P j E t j j k= 4 ( + 3 ss + t+k ) n t+j () n t+j () + ' t+j () 5 = 0; W t () P t+j where inserting () in the place o ' t (), and ' t+j () = (j k= (+ss + t+k ))W t() P t+j mpl t+j () or uture ' t+j () terms conditional to no optimal wage resetting, we nd (ater loglinearizing) cw t () = ( )E t j j ( P b t+j + mpl d t+j ()) E t j j t+j : (3) j= The log o the optimal wage contract depends on current and expected uture values o the log o the price level and the log o the speci c marginal product o labor. Using some algebra, the optimal wage can also be written in terms o aggregate magnitudes as ollows (see Appendix A or the proo) cw t () = W c t + ( ) ( + ) E t j j mplt+j d bw t+j + E t j j W t+j t+j : (4) The interpretation o (4) is straightorward. The optimal nominal wage contract departs rom the aggregate nominal wage whenever current or expected uture labor productivity values are above the real wage, d mpl t+j than the innovation on the indexation rate, W t+j bw t+j > 0:0, and also i the rate o wage in ation is expected to be higher j= t+j > 0:0. In order to nd the rate o wage in ation equation or this economy, we can combine (4) with the implied relationship rom the Calvo-type aggregation scheme, W t = (c W t () c Wt ) + t ; to reach W t = E t W t+ + ( ) ( ) ( + ) dmplt bw t + ( ) t : (5) The rate o wage in ation is a purely orward-looking variable whose quarter-to-quarter uctuations are governed by the gap between the aggregate marginal product o labor and the real wage, as 9

11 well as by the innovation on the wage indexation rule. When households set wages (equation ), wage in ation was reacting to the gap between the household s marginal rate o substitution and the real wage. Hence, the labor productivity becomes the driving variable i rms become the wage setting actors while that was the marginal rate o substitution when households set wages. Recalling the exible-price scenario introduced above, prices will entirely adjust as needed to clear the labor market. It implies bw t = dmrs t ; (6) since the supply o labor bundles decided by households is optimal when their mrs equates the real wage. This exible-price condition (6) can be plugged into (5) to obtain W t = E t W t+ + ( ) ( ) ( + ) dmplt dmrs t + ( ) t ; (7) and then substituting dmrs t d mplt = ( )++ ey t, it yields with = ( )( ) (+ ) W t = E t W t+ ey t + ( ) t ; (8) ( )++. Remarkably, the sticky-wage model where rms are wage setting actors delivers a negative relationship between the output gap and wage in ation, i.e., the opposite sign to that obtained in the sticky-wage model where households set wages. When the output gap is positive, the marginal product o labor alls below the real wage and rms wish to hire less labor. Thus, rms that can decide on a new wage contract will drop the value o the nominal wage in order to reduce the level o labor implied by (0). Besides their opposite sign, the numerical values o the slope coe cients are also di erent. Thus, the ratio o the output gap slopes in the wage in ation equations (7) and (8) is h = + + h. 4 Optimal monetary policy The optimal monetary policy can be derived or the two variants o the sticky-wage model with exible prices discussed above. The two cases only di er on who are the wage setting actors, either households (as commonly assumed in the New Keynesian literature) or rms. Following Woodord (003, ch. 6), optimal monetary policy is obtained or some given model rom the rst order conditions that maximize a measure o social welare approximated rom the utility unction o that model. Such approximation consists o writing the average value o the utility unction across households as a second-order expression that depends on a ew aggregate variables o the model and also on the underlying structure o that model (price/wage rigidities, 0

12 technology, distorting taxes,...). Since the second-order terms typically have a negative impact on the measure obtained or social welare, the central-bank objective unction is presented as a loss unction to be minimized. The Appendix B o this paper shows that the period loss unction o the model where households set wages is L h t = W t + h (ey t ey ) ; (9) ( )( ) ( )++ where h = (+ h ) and ey denotes the e cient level o the output gap in steady h ( ) state. 3 Thereore, welare losses depend on the weighted sum o variabilities o wage in ation and the output gap relative to its e ciency level. For the case with rms setting wages, the welare-theoretic loss unction is (see also Appendix B or the proo) L t = W t + (ey t ey ) ; (30) ( )( ) ( )++ with = (+ ) : Coincidently, social utility o households is also damaged by ( ) variability o wage in ation and the output gap when rms are wage setting actors. Thereore, the case o having either rms or households setting the wage contracts does not a ect the de nition o the targeting variables on optimal monetary policy because they are wage in ation and the output gap in either way. However, the relative weight o the output gap is not exactly de ned in (9) as in (30) which may lead to distinctive policy reactions that will be examined in Section 5. 4 For a long-run commitment to an optimal plan, we borrow the timeless perspective criterion, proposed in Woodord (999, page 8) and Woodord (003, pages ). Thus, the optimal plan or monetary policy in the case o having households setting wages is obtained by solving the ollowing optimizing program: Min E t P j W t+j + h (ey t+j subject to the all-time sequence o wage in ation equations ey ) W t+j = E t+j W t+j+ + h E t+j ey t+j + ( )E t+j t+j j = :::; ; ; 0; ; ; ::: The rst order conditions with respect to wage in ation and the output gap in period t are W t ' h t + ' h t = 0, and h (ey t ey ) h ' h t = 0; The reader can veriy that this welare-theoretic loss unction is the particular exible-price case o that derived by Woodord (003, pages ). 3 The e cient output gap is the amount produced i markets were perectly competitive with no distortion (see Woodord, 003, pages ). 4 Obviously, the di erent model structure may also have e ects on the design o optimal monetary policy.

13 in which ' h t is the Lagrange multiplier in period t. Taking ' h t = h h (ey t ey ) obtained rom the second equation, together with its lagged expression or ' h t equation result in where h h = h (, and inserting both into the rst W t = h h (ey t ey t ) ; (3) ). According to the time-consistent plan (3), optimal monetary policy requires that the rate o wage in ation responds in the opposite direction to the rst-di erence o the output gap. This policy resembles the "leaning against the wind" recommendation obtained by Clarida et al. (999) and Woodord (003, ch. 7) or a New Keynesian ramework with sticky prices à la Calvo. I the output gap is increasing rom the previous period the central bank must adjust the nominal interest rate to turn the wage in ation downwards by a actor o h ( ). Hence, the optimal monetary policy requires anticyclical reactions o wage in ation relative to the output gap. Let us check i the alternative sticky-wage speci cation leads to the same condition or the optimal monetary policy behavior. With rms setting wages, the central-bank optimizing program is Min E t P j W t+j + (ey t+j subject to the all-time sequence o wage in ation equations ey ) W t+j = E t+j W t+j+ E t+j ey t+j + ( )E t+j t+j j = :::; ; ; 0; ; ; ::: The rst order conditions on wage in ation and the output gap or period t are W t ' t + ' t = 0, and (ey t ey ) + ' t = 0; where ' t is now the Lagrange multiplier. Inserting ' t = (ey t ey ) and its lagged expression in the wage in ation optimality condition, it yields W t = (ey t ey t ) : (3) with = ( ). Comparing (3) with (3), it seems that the optimal monetary policy is opposite depending on the wage setting assumption. Thus, i rms are wage setting actors, the central bank will pursue a monetary policy that adjusts wage in ation on the same direction to the change in the output gap. It could be said that wage in ation must be procyclical since it should react with the same sign to changes in the output gap. The reaction actor is ( or the variant with households setting wages. ) which mimics that obtained

14 What are the consequences o applying the optimal monetary policy or our two variants o a sticky-wage model? In the case where households set wages, the central-bank optimality condition (3) and its expected next period s expression can be substituted in the wage in ation equation (7) to yield (ater little algebra) ey t = h h h (+)+ h E t ey t+ + ey t ( ) h h t i : (33) The only source o variability or the output gap in (33) is the wage indexation shock t. particular, a positive indexation shock brings in a negative output gap. Real-side shocks such as technology or preerence innovations have no impact on the output gap. 5 The nominal interest rate would adjust as necessary to bring current output back at its potential level. Thereore, the output gap will have no variability under the optimal policy in the absence o indexation shocks. Moreover, wage in ation would also stay constant as implied by equation (7). The optimal monetary policy would be ully e cient because the loss unction would value zero at all times. In The presence o (nominal-side) wage indexation shocks is necessary to evaluate optimal policy tradeo s in our exible-price model with sticky wages set by households. This result was already pointed out by Taylor (979) and Clarida et al. (999) as the lack o an output-in ation variability tradeo in models without nominal shocks (also called cost-push shocks). Turning to the case o rms acting as wage setters, the optimality condition (3) can be combined with the wage in ation equation (8) to obtain ey t = (+)+ h E t ey t+ + ey t + ( ) t i : (34) As in the case where households set wages, the evolution o the output gap also depends here exclusively on wage indexation shocks, t. Nevertheless, the output gap becomes positive in (34) ater a positive realization o t. Another implication o (34) is that both technology and preerence shocks are also neutralized by the optimal monetary policy and leave no e ect on the output gap. With no change in the output gap, those real-side shocks have no impact on wage in ation as implied by (8). Thereore, the model with rms setting wages also needs (nominal) indexation shocks or evaluating tradeo s between output gap and wage in ation variabilities. Without such nominal perturbations, optimal monetary policy achieves ull stabilization in both model variants. 5 A preerence shock can be considered a real shock because it shits the aggregate labor supply curve. 3

15 5 Model simulations with alternative wage setting actors Table summarizes the key equations that show why the assumption on who are the wage setting actors is relevant or a sticky-wage model. There are three aspects o the model a ected. First, wage in ation dynamics are di erent because its relationship to the output gap may be positive (households set wages) or negative ( rms set wages). Secondly, the optimal monetary policy implies a central-bank reaction o opposite sign: the rate o wage in ation must decrease when there is a positive change in the output gap (households set wages) whereas wage in ation must be raised in reaction to that output gap change ( rms set wages). At the third level o distinction shown in Table, the exible-price assumption gives rise to di erent market-clearing conditions or the labor market. Either the (log o the) real wage must be equal to the (log o the) marginal product o labor (households set wages) or the (log o the) real wage equates the (log o the) marginal rate o substitution ( rms set wages). 6 So ar, we have shown that having either households or rms as wage setters plays a role on the dynamic behavior o wage in ation and also on the design o the optimal monetary policy. Is this also in uential on the short-run uctuations o other key macro variables such as output or (price) in ation? To answer this question we need to build the rest o the model. For the demand sector, let us introduce the ollowing IS equation by t = E t by t+ (R t E t t+ ) + t; (35) which can be obtained by combining the rst order conditions on consumption and bonds on the household optimizing program presented in Section. As typical in a New Keynesian ramework, output uctuations in (35) are demand-determined in response to changes in expected output, the real interest rate (with a negative impact), and the consumption preerence shock. Finally, the de nition o the economy-wide real wage, w t = Wt P t, implies that the rates o price and wage in ation are related as ollows t = W t bw t + bw t : (36) Summarizing, we have two variants or a exible-price model with sticky wages à la Calvo, and optimal monetary policy: - A model where households are wage setting actors: The three equations on the let column o Table, plus common equations (4), (6), (35), and (36) comprise a set o seven equations that provide solution paths or seven variables by t, t, W t, bw t, R t, ey t, and by t. 6 The loglinearized marginal rate o substitution displayed in Table can be obtained by using the speci cation or the utility unction (9), the Cobb-Douglas technology (5), and the market-clearing condition by t = bc t. 4

16 - A model where rms are wage setting actors: The three equations on the right column o Table, plus common equations (4), (6), (35), and (36), comprise a set o seven equations that provide solution paths or seven variables by t, t, W t, bw t, R t, ey t, and by t. For illustrative purposes, we will simulate these two sticky-wage variants by analyzing impulseresponse unctions and some business cycle statistics obtained rom them. On that regard, some numerical values need to be assumed on the parameters o the models (see Table ). Repeating the baseline quarterly calibration chosen by Erceg et al. (000), we have = 0:99, = :5, = :5, h = 4:0, = 0:3, and wage contracts last on average or one year, = 0:75. The household s elasticity o substitution across labor services in the model where wages are decided by rms is set at = 4:0 to have it equal to that elasticity or the other model variant. As or the stochastic elements o the model, some reasonable numbers are arbitrarily assigned. 7 Thus, the standard deviations on the innovations o the real-side shocks are those that yield a standard deviation o potential output around.8%, and 3/4 o uctuations o potential output are driven by technology shocks. 8 The innovations on the wage indexation shock have a standard deviation that provides signi cant reactions o wage in ation in the impulse-response analysis (around 0.5% in annualized terms). As common in the literature, the coe cient o autocorrelation o the technology shock is very high ( z = 0:95) whereas preerence shocks have less inertia ( = 0:80). The wage indexation shock also has a moderate coe cient o autocorrelation ( = 0:80) as assumed by Woodord (003, page 496) or his analogous cost-push shock. As a result o our numerical selection, the slope coe cients in the wage in ation equations o Table are rather distinctive. I rms are wage setting actors, wage in ation is much more sensitive to the output gap because the slope coe cient is = 0:59, more than three times that slope when households set wages ( h = 0:050). A similar result is obtained when comparing the relative weight o the output gap variability in the central-bank loss unction. Stabilizing the output gap appears to be more important or social welare i rms set wages ( = 0:057 versus h = 0:08). The business cycle properties o the two variants presented above can be examined by describing the reactions to a technology shock, z t, a consumption preerence shock, t, and an indexation shock, t. The technology shock represents a supply-side shock because it gives rise to output uctuations due to exogenous changes in productivity. Meanwhile, the preerence shock represents a demand-side shock since the desired level o consumption is exogenously altered as a result o a change in its marginal utility. The indexation shock can be viewed as one example o a cost-push shock used in the literature on optimal monetary policy to analyze preerences on macroeconomic 7 The aim o this paper is to show theoretical aspects without any particular empirical t to some actual economy. 8 The latter was veri ed in the long-run variance decomposition. 5

17 stabilization. What are the in uence o these shocks in the model variants? Is the assumption on who set wages critical or the business cycle patterns in sticky-wage models with exible prices and optimal monetary policy? We will try to give answers to these questions by computing impulseresponse unctions and selected statistics rom each model variant. Throughout Figures -3, impulse response unctions are plotted as ractional deviations rom steady state or output, and the real wage, while annualized departures rom their steady-state levels or price in ation, wage in ation, and the nominal interest rate. The shocks are normalized by their standard deviations provided in Table. Technology shocks Figure shows the responses to a technology shock in the sticky-wage model where either households set wages or rms do it. Interestingly, all variables react in the same way with either households or rms setting wages as identically displayed in Figure. At rst, the optimal monetary policy is able to ully stabilize both wage in ation and the output gap under either case. 9 A zero output gap implies that current output is at its potential level (by t = by t ), which yields a 0.5% sudden increase that incorporates long inertia. Putting the zero output gap di erently, uctuations o the aggregate marginal product o labor (o rms) are the same as those o the aggregate marginal rate o substitution (o households). Accordingly, the real wage reports in both cases an upward response consistent with the productivity hike (see Figure ). Meanwhile, there is a sharp one-time in ation drop, deeper than % in annualized terms, required to clear the labor market. Finally, it is worth commenting on the optimal reaction o the nominal interest rate to a technology shock. As displayed in Figure, the nominal interest rate must be lowered at the time o the shock in order to help current (demand-determined) output to catch up with potential output and close down the output gap. This monetary policy ease has two characteristics: rst, it represents a weak reaction because the interest-rate cut is only by 6 (annualized) basis points at most. The second characterizing aspect is that the nominal interest rate returns very slowly to its steady-state level. As shown in Figure, hal o the initial interest-rate cut still remains in place teen quarters ater the shock. Summarizing, a supply-side technology shock that raises productivity brings about long-lasting increases in current output and the real wage, a one-time drop in in ation, and keep wage in ation and the output gap unchanged. This result is observed when applying the optimal monetary policy based on a gentle and persistent interest-rate cut. These responses are obtained independently 9 This result was already discussed at the end o Section 4. 6

18 rom the assumption on who set wages in the labor market. Preerence shocks Figure displays how a demand-side shock on consumption preerence has the same impact on the two sticky-wage variants. Since this is another example o a real-side shock, the optimal monetary policy also achieves ull stabilization or wage in ation and the output gap in both cases. Thus, output and potential output equally rise; a 0.4% increase at the time o the shock that ades out ater 8-0 quarters. Meanwhile, the real wage drops due to the reduction in the mrs (or in the marginal product o labor). In ation also reacts equally in both sticky-wage variants as it rises by more than 0.5% at the quarter o the shock although it returns to its steady state rate in the next quarter. The optimal policy requires a strong monetary tightening: the nominal interest rate must be raised by 57 (annualized) basis points at the quarter o the shock (see Figure ). The return o the nominal interest rate to its steady-state level should take 8-0 quarters. Thereore, the response o the nominal interest rate to a preerence shock is larger, less persistent, and with opposite sign to that observed ater a technology shock. This contractionary monetary policy is the necessary central-bank action to pull down current output onto its potential (natural-rate) level and thereore eliminate the output gap. Indexation shocks Unlike the two real-side shocks examined above, a (nominal) indexation shock allows di erent responses across our two variants or a sticky-wage model. As a matter o act, the reactions di er signi cantly in terms o both output and price in ation. Hence, output slightly goes up i rms set wages whereas it drops by a much greater extent i households set wages (see Figure 3). Why do we obtain such di erentiated output reactions? Since the wage indexation shock does not a ect potential output, the response o the output gap is the same as the response o output in both models. I rms are wage setting actors, the negative relationship between wage in ation and the output gap in (8) makes the central bank to increase the output gap in order to reduce wage in ation as indicated by the optimal policy condition (3). When households are wage setters, the relationship between wage in ation and the output gap is positive in (7) and the optimal monetary policy (3) will create a negative output gap to stabilize wage in ation. The optimal response o the nominal interest rate is also sensitive to the assumption on who set wages. As shown in Figure 3, the nominal interest rate drops around 50 basis points i rms set wages whereas there is much larger interest-rate cut (50 basis points) when households set wages. 7

19 Summarizing, the implementation o optimal monetary policy in a sticky-wage model with exible prices and alternative wage setting actors leads to the same responses to real-side shocks but to signi cantly di erent reactions to nominal shocks. Table 3 provides a selection o second-moment statistics computed rom the two sticky-wage models. Output, the nominal interest rate, and, especially, the output gap are more volatile in the model with households setting wages as much higher standard deviations were ound. This is a consequence o their larger reactions observed ater a wage indexation shock. Wage in ation is much more stable than price in ation in both model cases. 0 Regarding inertia, output, the real wage, and the output gap show long time persistence, characterized by high coe cients o autocorrelation in Table 3. The nominal interest rate moves along with moderate inertia when applying the optimal monetary policy in both models (coe cients o autocorrelation around ). The exible-price assumption leads to coe cients o autocorrelation o in ation close to zero. The only discrepancy between the two model variants in terms o autocorrelations is that the coe cient on wage in ation is somewhat higher i households set wages. As or the coe cients o correlation with output, Table 3 reports that price in ation, wage in ation, and the nominal interest rate are basically acyclical in the two variants, with coe cients close to zero in all the cases. The real wage is clearly procyclical only in the model with rms setting wages. Finally, the variance decomposition in the long-run (00 periods ahead) is displayed in Table 4. Technology shocks are the source o most uctuations in output, the real wage, and price in ation in both sticky-wage cases. Meanwhile, preerence shocks are the driving orce or changes in the nominal interest rate also in both variants, though especially in the model where rms set wages. Wage indexation shocks explain all the variations in the two targeting variables o monetary policy: wage in ation and the output gap. This result was anticipated above in the impulse-response unctions analysis where the impact o technology and preerence shocks on these variables were ully neutralized by optimal monetary policy. Indexation shocks also determine a signi cant raction o long-run uctuations on output and the nominal interest rate in the model where households are wage setting actors, and on price in ation and the real wage in the variant where rms set wages (around 0. in all cases). 0 This result should be expected or exible-price models where price in ation variability is not a targeting variable or the central bank. 8

20 6 Conclusions The assumption on who set wages is not trivial or an optimizing macroeconomic model with sticky wages. I households may decide on the value o nominal wage contracts, the optimal contract will depend on the di erence between the marginal rate o substitution and the real wage. By contrast, i rms can set the wage contract, they will determine its value looking at the di erence between the marginal product o labor and the real wage. In addition, we have shown that, in a sticky-wage, exible-price economy with Calvo-type wage contracts, the rate o wage in ation is orward-looking on the output gap with a sign de ned by who are the wage setting actors. Hence, when households set wages the relationship is o positive sign whereas when rms set wages that relationship turns negative. In addition, the assumption on wage setting actors distinguishes the analytical value o the output gap slope in the wage in ation equation h = + + h. Finally, optimal (welare-theoretic) monetary policy must minimize, in both variants or a stickywage model, a weighted sum o variabilities o the output gap and the rate o wage in ation. The central bank optimal plan leads to a di erent targeting rule on each case: wage in ation must react either positively ( rms set wages) or negatively (households set wages) to the output gap. Despite their di erent targeting rule, optimal monetary policy in both cases achieves ull stabilization o the two targeting variables in the presence o either technology or preerence shocks. Nevertheless, the assumption on who act as wage setters is relevant or the optimal interest-rate responses to (nominal) wage indexation shocks. Such nominal shocks also give rise to distinctive second-moment statistics. 9

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22 Hoehn, J G. (988). Procyclical real wages under nominal-wage contracts with productivity variations, Federal Reserve Bank o Cleveland Review 4, No. 4, -3. Huang, K.. D., and Z. Liu (00). Staggered price setting, staggered wage setting, and business cycle persistence, Journal o Monetary Economics 49, Huang, K.. D., Z. Liu, and L. Phaneu (004). Why Does the Cyclical Behavior o Real Wages Change Over time?, American Economic Review 94, Levin, A. T., A. Onatski, J. C. Williams, and N. Williams (005). Monetary Policy Under Uncertainty in Micro-Founded Macroeconometric Models, NBER Working Papers, No. 53. Rankin, N. (998). Nominal rigidity and monetary uncertainty, European Economic Review 4, Rotemberg, J. J., and M. Woodord (997). An optimizing-based econometric ramework or the evaluation o monetary policy, NBER Macroeconomics Annual 997, B.M. Bernanke and J.J. Rotemberg, eds., MIT Press, Cambridge, MA, Smets, F., and R. Wouters (003). An estimated Dynamic Stochastic General Equilibrium model o the Euro area, Journal o the European Economic Association, Svensson, L. E. O. (999a). Price Level Targeting vs. In ation Targeting: A Free Lunch?, Journal o Money, Credit, and Banking 3, Svensson, L. E. O. (999b). In ation Targeting as a Monetary Policy Rule, Journal o Monetary Economics 43, Taylor, J. B. (979). Estimation and Control o a Macroeconomic Model with Rational Expectations, Econometrica 47, Taylor, J. B. (980). Aggregate dynamics and staggered contracts, Journal o Political Economy 88, -4. Woodord, M. (999). Commentary: How Should Monetary Policy Be Conducted in an Era o Price Stability?, in New Challenges or Monetary Policy: A Symposium Sponsored by the Federal Reserve Bank o Kansas City, Federal Reserve Bank o Kansas City. Woodord, M. (003). Interest and prices: Foundations o a theory o monetary policy, Princeton University Press, Princeton, NJ.

23 APPENDI A. Derivation o the equation or uctuations on the optimal nominal wage depending on aggregate variables. i) Model with wages set by households. Let us start rom repeating equation (9) rom the text cw t (h) = ( )E t j j ( P b t+j + dmrs t+j (h)) E t j j t+j ; (A) j= where dmrs t+j (h) represents uctuations on the mrs or the households who decide the optimal contract in period t and will not adjust it in uture periods. From the de nition o the (aggregate) real wage, we can substitute b P t+j = c W t+j bw t+j = c W t + P j k= W t+k bw t+j into (A) to obtain cw t (h) = ( )E t j j ( W c t bw t+j + dmrs t+j (h)) + E t j j W t+j t+j : (A) j= Recalling our utility unction speci cation (3), the (log-linearized) relationship between the householdspeci c mrs and the aggregate mrs is dmrs t+j (h) = dmrs t+j + (bn t+j (h) bn t+j ): (A3) Now we wish to write the relative mrs as a unction o the relative wage. Using () in loglinear terms or the t + j period conditional to the indexation rule (6), it yields bn t+j (h) bn t+j = h ( c W t (h) + j t+k Wt+j c ); (A4) where it should be noticed that the conditional wage contract evolves as c W t (h) + P j k= t+k in any k= t + j uture period. Combining (A3) and (A4), it is obtained dmrs t+j (h) = dmrs t+j h ( c W t (h) + where inserting W c t+j = W c t + P j k= W t+k leads to j t+k Wt+j c ); k= dmrs t+j (h) = dmrs t+j h ( c W t (h) c Wt j k= W t+k t+k ): (A5) Finally, the substitution o (A5) into (A) yields cw t (h) = ( )E t j j ((+ h ) W c t + dmrs t+j bw t+j hwt c (h)+(+ h )E t j j W t+j t+j : j=

24 Putting terms together, we can write a dynamic equation or the wage contract that depends entirely on aggregate magnitudes cw t (h) = W c t + ( ) E t j j ( dmrs t+j + h which is equation (0) in the main text o the paper. ii) Model with wages set by rms. bw t+j ) + E t j j W t+j t+j ; In section 3, we obtained equation (3) or uctuations on the optimal nominal wage contract set in period t cw t () = ( )E t j j ( P b t+j + mpl d t+j ()) j= E t j j t+j ; where d mpl t+j () denotes the marginal product o labor in case o no optimal wage adjustment over uture periods. The (log-linear) Cobb-Douglas production technology (5) can be used to nd the relative relationship between the marginal product o labor and demand or labor j= (A6) dmpl t+j () d mplt+j = (bn t+j () bn t+j ): (A7) Loglinearizing the labor supply constraint (0) or period t + j and using the wage indexation rule (6), we get bn t+j () = bn t+j + c Wt () + The combination o (A7) and (A8) leads to which can be inserted in (A6) to reach cw t () = (! j t+k Wt+j c : (A8) k= dmpl t+j () = d mpl t+j c Wt () +! j t+k Wt+j c ; k= )E t j j Pt+j b + mpl d t+j cwt () E twt+j c Terms on the aggregate price level are dropped by inserting b P t+j = c W t+j cw t+j = W c t + P j k= W t+k to nd ( + ) W c t () = ( )E t j j ( mpl d t+j ( + ) E t j j t+j : j= bw t+j. Next, we can use bw t+j +( + ) W c t )+( + ) E t j j W t+j t+j ; which can be also written as our equation (4) o the text cw t () = W c t + ( ) ( + ) E t j j mplt+j d bw t+j + E t j j W t+j t+j : j= j= 3

25 APPENDI B. Derivation o the welare-theoretic loss unction or the variants o the stickywage model o the paper. All the approximations taken here are based on second-order Taylor expansions used in Erceg et al. (000, pages 307-3), and Woodord (003, pages ). i) Model where households set wages. For convenience, let us de ne V t = exp( t )(ct) and S t (h) = (3) such that (n t(h)) + + rom the utility unction U t (h) = V t S t (h): (B) Ater dropping out constant and exogenous terms, the second-order Taylor approximation o V t yields V t ' U c [c t c] + U cc [c t c] + U c [c t c] [exp( t ) ] : By inserting the equilibrium condition c t = y t, using U cc = t, it is obtained V t ' U c [y t y] A second-order Taylor expansion or yt y is Uc c, and the approximation [exp( t) ] ' U c y [y t y] + U c [y t y] t : (B) y t y ' + by t + by t ; (B3) which implies the second-order approximations [y t y] ' y by t + by t and [y t y] ' y by t ; (B4) that can be substituted in (B) to reach V t ' yu c by t + by t yu cby t + yu c by t + by t t : Next, the term yu c by t t is dropped or being o third order and terms are grouped to obtain the nal second-order approximation o V t V t ' yu c by t + ( )by t + by t t : (B5) Let us turn to the second term in (B), S t (h) = becomes (n t(h)) + +. Its second-order Taylor approximation S t (h) ' U n [n t (h) n] + U nn [n t (h) n] ; (B6) 4

26 where constant and exogenous terms were taken out. Using U nn = Un n and the approximations analogous to (B4) or [n t (h) n] and [n t (h) n] in (B6), it yields S t (h) ' nu n bn t (h) + ( + )bn t (h) : (B7) Integrating the household-speci c utility unction (B) over all di erentiated households, we obtain the aggregate utility unction U t = Z 0 Z U t (h)dh = V t S t (h)dh; that represents the social utility unction to be maximized by the central bank. Here, we need to aggregate the labor disutility across households. Using (B7), it yields Z 0 0 (B8) S t (h)dh ' nu n E h bn t (h) + ( + ) h(e h bn t (h)) + var h bn t (h)i ; (B9) where, ollowing Woodord (003, page 694), we used the de nitions E h bn t (h) = R 0 bn t(h)dh and var h bn t (h) = R 0 bn t (h)dh (E h bn t (h)) : A Taylor series approximation to () implies the ollowing equation or aggregate labor uctuations bn t ' E h bn t (h) + h var h bn t (h); h which can be used to eliminate both E h bn t (h) and (E h bn t (h)) rom (B9) and thus to obtain (ater dropping terms o order higher than two) Z S t (h)dh ' nu n bn t + ( + )bn t + ( + )var hbn t (h) : (B0) 0 Now, both (B5) and (B0) can be substituted in the social utility unction (B8) to get U t ' yu c by t + ( )by t + by t t nu n bn t + ( + )bn t + ( + )var hbn t (h) : (B) The (log-linear) Cobb-Douglas production unction (5) relates labor and output in the ollowing way bn t = ( ) by t z t, which implies bn t = ( ) by t + z t ( ) by t z t. Taking into account these two relationships, (B) can be rewritten in terms o output uctuations and the dispersion o di erentiated labor services U t ' yu c by t + ( )by t + by t t nu n ( ) by t + ( + ) ( ) by t ( ) by t z t + where exogenous terms were dropped. h h (B) ( + h )var hbn t (h) ; Following Woodord (003, chapter 6), the steady-state solution o the model with zero in ation implies the relationship nu n = yu c ( )( y ) where 5

27 y represents the inverse o the steady-state markup o the real wage over the mrs. This result can be used in (B) to obtain U t ' yu c y by ( )++ t by t + by t ( t + ( + )z t ) ( )( + h )var hbn t (h) ; (B3) where the terms y by t ; y by t z t, and y var h bn t (h) were neglected as justi ed by Woodord (003, pages ). Again,. ollowing Woodord (003, page 395), the di erence between the output gap and its e ciency level is ey t ey = by t by t ey ; (B4) where ey is obtained as the ractional di erence in steady-state between the level o output produced in a perectly competitive economy and the level o potential output obtained in a monopolistically competitive economy, i.e. ey = log y. For the model described in the text, we have y ey = $ log( y ) ' $ y where $ completely depends on the values o parameters regarding preerences and technology $ = ( )++. Computing the square o the e cient output gap de ned in (B4) and using the approximation ey = $ y, the square output uctuations, by t, can be written as ollows by t = (ey t ey ) + by t by t + $ y by t $ y by t by t ($ y ) ; (B5) where potential output uctuations are by t = $ ( t + ( + )z t ) as in equation (6) o the text. Inserting (B5) into (B4), we obtain (ater dropping exogenous terms) U t ' yu c $ (ey t ey ) + ( )( + h )var hbn t (h) : (B6) By loglinearizing (), the variance on di erentiated labor services can be expressed in terms o wage dispersion var h bn t (h) = h var h c W t (h): (B7) Applying the results o Woodord (003, page ) to our Calvo-style sticky-wage structure, we have which implies var h c Wt (h) ' var h c Wt (h) + W t j E t var hwt+j c (h) ' In this particular sticky-wage model, y = h. ( ) ( ) ; j E t W t+j (B8) 6

28 Inserting (B7) in (B6), it is obtained U t ' yu c $ (ey t ey ) + ( ) h ( + h )var hwt c (h) : (B9) Finally, let us assume that the central bank maximizes the intertemporal (social) utility unction P j E t U t+j. Combining (B9), and (B8), it yields j E t U t+j ' yu c j E t $ (ey t+j ey ) + ( ) h( + h ) ( ) ( ) W t+j : (B0) Rearranging terms in (B0), we have j E t U t+j ' h j E t W t+j + $ ( ) ( ) ( ) h ( + h ) (ey t+j ey ) ; with h = yuc ( ) h (+ h ) ( )( ), which implies that the central-bank loss unction or period t is L h t = W t + h (ey t ey ) ; with h = ( )( ) ( )++ (+ h ) h as de ned in equation (9) o the text. ( ) ii) Model where rms set wages. In this case, the household utility unction and the social utility unction are the same because both consumption and the supply o bundles o labor services are identical across households. Recalling (9), the social utility unction in period t is in this model U t = exp( t) (c t ) (n t ) + + = V t S t : (B) As shown above, the two terms o (B) can be approximated by the Taylor series expansions V t ' yu c by t + ( )by t + by t t ; and (Ba) S t ' nu n bn t + ( + )bn t : (Bb) The term on disutility rom the supply o labor bundles, S t, will be a ected by the degree o wage dispersion through its impact on uctuations o market-clearing labor, bn t. Thus, a Taylor series expansion on (0) yields bn t ' E bn t () + + var bn t (); (B3) 7

29 where E bn t () and var bn t () respectively denote the expected value o the demand or labor and its variance computed across the di erentiated rms. Recalling the log-linear Cobb-Douglas technology (5) or the speci c -th rm, we obtain which, aggregating over the space, implies bn t () = ( ) by t () z t ; E bn t () = ( ) E by t () z t ; and (B4) var bn t () = ( ) var by t (): Aggregate output in this economy with di erentiated rms is y t = R 0 y t()d, which leads to the Taylor series approximation by t ' E by t () + var by t (): Substituting (B4) into (B3) and then the value o E by t () implied by (B5), it is obtained bn t ' ( ) by t var by t () z t + + var bn t (); where inserting var by t () = ( ) var bn t () rom (B4) results in bn t ' ( ) by t ( ) var bn t () z t + + Putting terms together and dropping the exogenous variable, we get bn t ' ( ) by t + + var bn t (): The substitution o (B6) into (Bb) yields S t ' nu n ( ) by t + ( + )var bn t () + ( + )bn t var bn t (): ; (B5) (B6) where using bn t = ( ) by t ( ) by t z t rom the log-linear Cobb-Douglas technology (5) results in S t ' nu n ( ) by t + ( + )var bn t () + ( + )( ) by t ( + )( ) by t z t : (B7) The steady-state relationship nu n = yu c ( )( y ), which was already used above, still holds in this model because it keeps the same preerences and technology. Plugging that into (B7), it yields S t ' yu c ( y ) by t + ( )( + )var bn t () + ( + )( ) by t ( + )by t z t : (B8) Nevertheless, the value o y is now y = ( + ). 8

30 Using (Ba) or V t and (B8) or S t, the social utility unction can be approximated by U t ' yu c y by t $ by t + by t ( t + ( + )z t ) ( )( + )var bn t () ; recalling $ = ( )++ and where the terms yby t ; y by t z t, and y var bn t () were dropped as we did in the model where households set wages. The de nition o the e cient output gap was used above to obtain (B5) which can be inserted in the last expression to nd (ater dropping exogenous terms and using by t = $ ( t + ( + )z t )) U t ' yu c $ (ey t ey ) + ( )( + )var bn t () : (B9) By loglinearizing (0), the variance on di erentiated labor services can be expressed in terms o wage dispersion which leaves (B9) as ollows U t ' yu c var bn t () = var c W t (); $ (ey t ey ) + ( )( + )var Wt c () : Accordingly, the central-bank intertemporal (social) utility unction P j E t U t+j becomes j E t U t+j ' yu c j E t $ (ey t+j ey ) + ( )( + )var Wt+j c () : (B30) Again, we can use Woodord (003, pages ) to have which implies var c W (h) ' var c Wt () + W t j E t var Wt+j c () ' ( ) ( ) ; j E t W t+j : (B3) Combining (B30) and (B3), it is obtained j E t U t+j ' yu c j E t $ (ey t+j ey ) + ( )( + ) ( ) ( ) which can be reorganized as ollows j E t U t+j ' j E t with = yuc t is with = ( W t+j ; W t+j + $ ( ) ( ) ( )( + ) (ey t+j ey ) ; ( )(+ ) ( )( ). Our last result implies that the central-bank loss unction or period )++ ( )( ) ( )(+ ) L t = W t + (ey t ey ) ; as de ned in equation (30) o the text. 9

31 Table. Key equations or a exible-price, sticky-wage model with optimal monetary policy Households set wages Firms set wages Wage in ation: W t = E t W t+ + hey t + ( ) t W t = E t W t+ ey t + ( ) t Opt. monetary policy: W t = h ( ) (ey t ey t ) W t = ( ) (ey t ey t ) ( by t z t t Flexible prices: bw t = by t + z t bw t = )+ Table. Numerical values o parameters Utility unction = 0:99 = :5 = :5 = 0:80 " = 0:040 Production unction = 0:3 z = 0:95 " z = 0:0085 Wage stickiness and Dixit-Stiglitz elasticities = 0:75 h = 4:0 = 4:0 Wage indexation rule ss = 0:0 = 0:80 " = 0:004 Slope coe cients o the wage in ation eq. h = 0:050 = 0:59 Output gap weights or monetary policy design h = 0:08 = 0:057 Table 3. Business cycle statistics. by W bw R ey Annualized standard deviations (%): Households set wages: Firms set wages: Coe cients o autocorrelation: Households set wages: Firms set wages: Correlation with output: Households set wages: Firms set wages:

32 Table 4. Long-run variance decomposition. by W bw R ey Households set wages: Technology shocks, z t Preerence shocks, t Indexation shocks, t.8 < Firms set wages: Technology shocks, z t Preerence shocks, t Indexation shocks, t

33 Figure : Impulse-response unctions under optimal monetary policy. technology shock. One standard deviation 3

34 Figure : Impulse-response unctions under optimal monetary policy. consumption preerence shock. One standard deviation 33

35 Figure 3: Impulse-response unctions under optimal monetary policy. (nominal) wage indexation shock. One standard deviation 34