Employment, Wages and Optimal Monetary Policy

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Employment, Wages and Optimal Monetary Policy Martin Bodenstein and Junzhu Zhao 2017-091 Please cite this paper as: Bodenstein, Martin, and Junzhu Zhao (2017). “Employment, Wages and Optimal Monetary Policy,” Finance and Economics Discussion Series 2017-091. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2017.091. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Employment, Wages and Optimal Monetary Policy Martin Bodenstein Junzhu Zhao Federal Reserve Board National University of Singapore August 7, 2017 Abstract We study optimal monetary policy when the empirical evidence leaves the policymaker uncertain whether the true data-generating process is given by a model with sticky wages or a model with search and matching frictions in the labor market. Unless the policymaker is almost certainaboutthesearchandmatchingmodelbeingthecorrectdata-generatingprocess, thepolicymaker chooses to stabilize wage in(cid:13)ation at the expense of price in(cid:13)ation, a policy resembling the policy that is optimal in the sticky wage model, regardless of the true model. This (cid:12)nding re(cid:13)ects the greater sensitivity of welfare losses to deviations from the model-speci(cid:12)c optimal policy in the sticky wage model. Thus, uncertainty about important aspects of the structure of theeconomydoesnotnecessarilytranslateintouncertaintyaboutthefeaturesofgoodmonetary policy. JEL classi(cid:12)cations: E52 Keywords: optimalmonetarypolicy, optimaltargetingrules, searchandmatching, stickywages, model uncertainty (cid:3) The views expressed in this paper are solely the responsibility of the authors and should not be interpretedasre(cid:13)ectingtheviewsoftheBoardofGovernorsoftheFederalReserveSystemoranyotherperson associatedwiththeFederalReserveSystem. WearegratefultoJohnHam,SereneTan,ChristopherGust, Robert Tetlow, Roger Farmer, and Albert Marcet for helpful comments and suggestions. (cid:3)(cid:3) Contactinformation: MartinBodenstein(correspondingauthor),E-mailmartin.r.bodenstein@frb.gov; Junzhu Zhao, E-mail junzhuzhao@gmail.com 1

1 Introduction Macroeconomic models built to capture the cyclical properties of employment and wages often feature either nominal wage rigidities or search and matching frictions in the labor market. We study optimal monetary policy when the policymaker is uncertain which of these two approaches is the correct data-generating process. Unless the policymaker is almost certain about the search and matching model being the correct data-generating process, the policymaker chooses to stabilize wage in(cid:13)ation at the expense of price in(cid:13)ation, a policy resembling the policy that is optimal in the sticky wage model, regardless of the true model. This (cid:12)nding re(cid:13)ects the greater sensitivity of welfare losses to deviations from the model-speci(cid:12)c optimal policy in the sticky wage model. Our analysis features two New Keynesian (NK) models that are identical except for thedetailsofthelabormarket. FollowingCalvo(1983), pricesarestickyasretail(cid:12)rmssell differentiated goods that are priced using staggered contracts. In the sticky wage model, we assume that in addition wages for differentiated labor varieties are set in a staggered fashion, see Erceg, Henderson, and Levin (2000). The empirical NK literature has largely relied on similar settings to generate empirically plausible labor market dynamics in monetary models.1 In sharp contrast to the sticky wage model, nominal wages are (cid:13)exible in the search and matching model. Although common in other areas of macroeconomics, the models with search and matching frictions in the labor market pioneered by Diamond (1982), Mortensen (1982), and Pissarides (1985) are rarely considered in monetary economics.2 Wholesale (cid:12)rms post vacancies and workers search for jobs. When a (cid:12)rm and a worker are matched, they (Nash) bargain over the terms of employment (wages and hours worked). The details follow Faia (2009) and Ravenna and Walsh (2011) with the important difference that individual hours worked of an employed worker are elastic in our setup to maintain comparability with the sticky wage model. Furthermore, by modeling explicitly the opportunity costs of employment, we improve the model’s ability to capture the empirical patterns of unemployment and vacancies.3 1Fine illustrations of this approach are Christiano, Eichenbaum, and Evans (2005) and Smets and Wouters (2007). 2 Notable exceptions are Krause and Lubik (2007), Ravenna and Walsh (2008), and Christiano, Eichenbaum, and Trabandt (2013). An approach combining the search and matching framework with staggered multi-period wage contracts is due to Gertler and Trigari (2009). 3Shimer (2005) argues that search and matching models cannot generate labor market movements that are in line with the empirical evidence for plausible parameter choices|a view subsequently challenged by other authors. Our approach builds on Hagedorn and Manovskii (2008), but models the opportunity costs of employment explicitly. 2

At the core of our analysis lies the idea that policymakers can formulate multiple models that provide a good approximation to the true data-generating process given the available empirical evidence against which these models are assessed. We implement this view in our setup by showing that both the sticky wage and the search and matching model can generate dynamics that are in line with evidence from structural vector autoregressions for reasonable parameter values under the assumption that monetary policy follows an estimated interest rate rule.4 Putting other empirical evidence aside, this exercise leaves the policymaker with two plausible descriptions of the data-generating process and uncertainty about the true data-generating process. Without being able to settle on a unique model of the economy, conducting optimal monetary policy is complicated by the fact that the two models have vastly different normative implications. For each model, we consider the optimal monetary policy under commitment from the timeless perspective when the policymaker’s preferences coincide with those of the representative household as in Woodford (1999). In the model with search and matching frictions, the optimal policy keeps price in(cid:13)ation under tight control while nominal wages display large movements. Although the search and matching process leads to inefficient allocations in our setting, monetary policy cannot correct the underlying distortions in the labor market. Thus, the optimal monetary policy addresses exclusively the dynamic distortions stemming from sticky prices in the product market. Low in(cid:13)ation reduces the differences in relative prices across product varieties and the associated inefficient shifts in relative demand. The degree of in(cid:13)ation stabilization is only constrained by the possible trade-off between in(cid:13)ation and resource utilization (as measured by the output gap). By contrast, in the NK model with sticky nominal wages, the optimal policy needs to strike a balance between price and wage in(cid:13)ation. Similar to the product market, wage in(cid:13)ation distorts relative real wages and labor demand; price in(cid:13)ation supports the adjustment of real wages under staggered nominal wages. The near complete stabilization of wages re(cid:13)ects the high welfare costs associated with even minor relative wage differences in empirical sticky wage models. Given these normative differences between the two models, how important is it for the policymaker to know which one represents the true data-generating process? Using the concept of optimal targeting rules as in Giannoni and Woodford (2016), we show that transplanting the optimal targeting rule from one model into the other results in welfare 4Using the formulation of the search and matching model in Hall and Milgrom (2008), Christiano, Eichenbaum, and Trabandt (2013) report a similar (cid:12)nding. 3

losses that are orders of magnitudes larger than the welfare costs of business cycles in Lucas(2003).5 Thelackofrobustnessoftheoptimaltargetingrulesisfarfromsymmetric. The optimal targeting rule derived from the search and matching model stabilizes price in(cid:13)ation and induces excessive movements in wage in(cid:13)ation when applied to the sticky wage model; the resulting welfare costs are ten times larger than in the opposite case and re(cid:13)ect the high welfare costs of relative wage differences in empirical sticky wage models. Excessive stabilization of wage in(cid:13)ation in the search and matching model under the targeting rule that is optimal in the sticky wage model induces relatively small welfare losses. The lack of robustness of the optimal targeting rules makes it unattractive to resolve model uncertainty via standard model selection exercises prior to the evaluation of monetary policy. Instead of opting for a speci(cid:12)c model based on inconclusive empirical evidence we allow the policymaker to incorporate model uncertainty as a component in the evaluation of policy as advocated in Brock, Durlauf, and West (2007). We show that in this case, the policymaker selects a policy that resembles the optimal targeting rule derived in the sticky nominal wage model unless the policymaker is very certain about the search and matching model being the correct data-generating process. Building on the ideas developed in Levin, Wieland, and Williams (2003), we arrive at this conclusion under two approaches of deriving the optimal monetary policy under model uncertainty. Under the model averaging approach, the policymaker chooses a policy|implemented through an interest rate or a targeting rule|that minimizes the expected loss for a given probability distribution of the policymaker over the relevant reference models. When the policymaker adopts a minmax strategy, the policy minimizes the maximum expected loss. This approach does not require the policymaker to specify a probability distribution over models. Re(cid:13)ecting the lack of robustness of policies that are (close to) optimal in the search and matching model, the optimal policy under model uncertainty mimics the optimal targeting rule derived in the sticky wage model under both approaches and stabilizes wage in(cid:13)ation at the expense of price in(cid:13)ation, unless the policymaker attaches a low (or, in the case of the minmax strategy, zero) probability on the sticky wage model being the true model. Thus, uncertainty about the true model does not necessarily translate into uncertainty about the features of good monetary policy. 5The optimal targeting rule speci(cid:12)es the variables|including the relative importance and the dynamic structure ofeachvariable|inasingletargetcriterionthatseekstoimplementtheoptimalmonetarypolicy. Inotherwords, the optimal targeting rule is a commitment to a certain relationship between the model variables. 4

Throughout our analysis, we assume that the policymaker adopts preferences over economic outcomes that are consistent with those of the households in the reference models. If in departure from the microeconomic foundations of the models, the policymaker’s preferences do not re(cid:13)ect those of the households it is possible to derive policies that the policymaker deems (close to) optimal for both models. However, such policies are not robust from the perspective of households. Assigning arbitrary preferences to the policymaker as often the case in the literature is not innocuous. The remainder of the paper proceeds as follows. Section 2 discusses related literature. We present the NK models with search and matching frictions and sticky nominal wages, respectively, in Section 3. In Section 4, we discuss the details of our empirical strategy to parameterize the two models. Section 5 derives optimal targeting rules for each model and assesses their robustness across models. Optimal policy under model uncertainty is discussed in Section 6. Concluding remarks are offered in Section 7. 2 Related literature Our approach is closest to Levin and Williams (2003) and Levin, Wieland, and Williams (2003) which also study robust monetary policy with competing reference models. Other related papers include Cogley and Sargent (2005) and Svensson and Williams (2005), but we abstract from the learning dynamics featured in these contributions. In sync with our conclusions, these works recommend policymakers not to tailor policies towards a model withrecommendationsthatarenotrobusttomodelmisspeci(cid:12)cationanduncertaintyeven if the model is considered quite likely to be (close to) the correct data-generating process.6 Yet,ouranalysisdiffersfromallthesecontributionsalongimportantdimensions. First, werestrictattentiontomicrofoundedmodelsandexcludemacro-econometricmodelsfrom the set of reference models. Thus, we can consider objective functions of the policymaker that are consistent with the preferences of the economic agents in the underlying reference models and that re(cid:13)ect the policymaker’s probability distribution over the models. The aforementioned contributions assume that the policymaker’s preferences are independent of the reference models, an approach we show to sometimes falsely suggest the existence 6Research on model uncertainty and policy evaluation has taken several directions. One direction is to assume a given baseline model and consider all models within a given distance as in Hansen and Sargent (2007), Tetlow and von zur Muehlen (2001), and Giannoni (2002). The second approach, takenin this paper, does not require the models tobeclosetoeachother. AnotherrecentexampleofthisapproachisTaylorandWieland(2012). Inadditiontomodel uncertainty, data uncertainty and parameter uncertainty are other areas of concern for policymakers. 5

of robust policies. Second, we parameterize the models to (cid:12)t the same empirical evidence under empirical interest rate rules before deriving the optimal monetary policy. In Cogley and Sargent (2005) and Svensson and Williams (2005), model parameters are estimated conditional on the policymaker setting policy to maximize a given quadratic objective; no two models (cid:12)t the data equally well over a given historical episode in their works and the ranking of the models according to the quality of (cid:12)t switches between episodes. In Levin and Williams (2003) and Levin, Wieland, and Williams (2003) the models are not parameterized using the same empirical evidence. Third, labor market aspects are at the core of our analysis and we stress the importance of smoothing wage in(cid:13)ation at the expense of price in(cid:13)ation as a general principle of robust optimal monetary policy. These considerations are ruled out in the earlier contributions by the choice of models. Sensitivity analysis suggests that our results survive if the policymaker’s preferences resemble those of earlier studies and are common across models as long as the policymaker has sufficient dislike for price in(cid:13)ation. 3 Two competing models of the labor market The two reference models of the policymaker build on the New Keynesian (NK) model withstickynominalprices; themodelsdifferwithregardtothedetailsofthelabormarket. The (cid:12)rst model features search and matching frictions in the labor market as in Diamond (1982), Mortensen (1982), and Pissarides (1985). Each worker negotiates the terms of employment with the matched (cid:12)rm. While the real wage may adjust slowly to shocks, the nominal wage is fully (cid:13)exible. By contrast, the second model introduces sticky nominal wages as in Erceg, Henderson, and Levin (2000). Unlike NK models with Walrasian labor markets, these two models (cid:12)t well the impulse responses of labor market variables derived from structural vector auto-regressions (SVAR) for reasonable parameter choices. We provide brief model descriptions in the main text and refer to Appendix A for details.7 3.1 NK model with search and matching frictions Households are modeled as in Andolfatto (1996) and Merz (1995). At any point in time n agents of the household are employed (w) and 1 (cid:0) n agents are unemployed (u). t t 7We chose not to include the model by Gertler and Trigari (2009) which merges the ideas of the search and matching framework with those of nominal rigidities in wage setting. As shown in Thomas (2008), the optimal policy recommendations derived from this hybrid framework resemble those of the sticky wage model. 6

As in Walsh (2005) and Christiano, Eichenbaum, and Trabandt (2013), we assume that each household member has the same concave preferences over consumption and that the household provides perfect consumption insurance. The household maximizes the inter-temporal utility of the members [ ] ∑1 (c (cid:0)(cid:22)c )1(cid:0)(cid:27) (h )1+ϕ E (cid:12)t t t(cid:0)1 (cid:0)n ϕ t (1) 0 1(cid:0)(cid:27) t 0 1+ϕ t=0 subject to the budget constraint B Pr T R B c + t+1 ⩽ [w h n +bu(1(cid:0)n )]+ t + t + t(cid:0)1 t : (2) t t t t t P P P P t t t t E is the expectations operator conditional on all the information available up to period 0. 0 (cid:12) is the time discount factor. Consumption is denoted by c , and the hours worked by the t n employed household members are measured by h . Unemployed household members do t t not experience disutility from working. The real wage is given by w and unemployment t bene(cid:12)ts are measured by bu. Bond holdings B , taxes and transfers T , and pro(cid:12)ts Pr t t t are measured in nominal terms and are converted into real units through division by the price level P . R is the nominal interest rate on bonds. We denote by (cid:21) the Lagrange t t t multiplier attached to the budget constraint when solving the household’s problem. As in Walsh (2005) we assume that total consumption c consists of a manufactured good cm t t and home production bu(1(cid:0)n ), i.e., c = cm +bu(1(cid:0)n ). This assumption guarantees t t t t that it is in principle possible under the conditions in Hosios (1990) for the outcomes of the search and matching process to be efficient.8 The labor market features search and matching frictions. Firms post vacancies v . t The share of agents searching for jobs is measured by u . New matches m between (cid:12)rms t t and agents are formed according to the matching function m = (cid:31)u(cid:16)v1(cid:0)(cid:16) (3) t t t 8If unemployment bene(cid:12)ts are modeled as tax-(cid:12)nanced, imposing the conditions in Hosios (1990) is not sufficient to achieve efficiency for bu > 0. The exact way of modeling unemployment bene(cid:12)ts is of little consequence for us as for empirical reasons we are not interested in parameterizations that satisfy the conditions in Hosios (1990). However, the modeling choice matters in our companion paper Bodenstein and Zhao (2016) from which we draw in this paper. 7

while employment n evolves according to t n = (1(cid:0)(cid:26))n +m (4) t t(cid:0)1 t where (cid:26) is the exogenous rate at which existing matches break up. The number of job seekers in period t follows u = 1(cid:0)n +(cid:26)n = 1(cid:0)(1(cid:0)(cid:26))n : (5) t t(cid:0)1 t(cid:0)1 t(cid:0)1 Wholesale (cid:12)rms employ labor to produce the good yw which is sold at the competitive t market price Pw. To hire workers, wholesale (cid:12)rms have to (cid:12)rst post a vacancy at the cost t (cid:20)v. These (cid:12)rms maximize pro(cid:12)ts subject to the law of motion for employment and the production technology ( ) ∑1 Pw W max E (cid:12)t(cid:21) t yw (cid:0) t n h (cid:0)(cid:20)vv fnt;y t w;vt g1 t=0 0 t=0 t P t t P t t t t s:t: n = (1(cid:0)(cid:26))n +q v t t(cid:0)1 t t yw = a n h (6) t t t t where (cid:12)rms take the probability of (cid:12)lling an open vacancy q = mt as given. Total factor t vt productivity a follows a standard AR(1) process t log(a ) = (cid:26) log(a )+"a (7) t a t(cid:0)1 t with normally distributed innovations "a (cid:24) N(0;(cid:27)2). t a When an agent and a (cid:12)rm are matched, they engage in Nash bargaining over wages and hours worked. The solution to the bargaining problem is obtained from maxJ1(cid:0)(cid:24)H(cid:24) (8) t t wt;ht where (cid:24) stands for the bargaining powerof the worker. The marginal valueof employment to the (cid:12)rm J is given by the period pro(cid:12)t of the additional worker, i.e., the excess of the t marginal product over the real wage payment, plus the continuation value if the match 8

survives into the next period ( ) Pw W (cid:21) J = t a (cid:0) t h +(1(cid:0)(cid:26))E (cid:12) t+1 J : (9) t t t t t+1 P P (cid:21) t t t The marginal value of employment to the household H satis(cid:12)es t ( ) W ϕ h1+ϕ (cid:21) H = t h (cid:0)bu (cid:0) 0 t +(1(cid:0)(cid:26))E (cid:12) t+1 (1(cid:0)s )H (10) t t t t+1 t+1 P 1+ϕ (cid:21) (cid:21) t t t W and consists of the increase in household income t h (cid:0)bu of having an additional houset P t hold member employed over the monetary equivalent to compensate the now employed ϕ h1+ϕ member for the loss of leisure time 0 t as well as the continuation value if the 1+ϕ (cid:21) t match survives into the next period. Retail prices experience nominal rigidities. Retail (cid:12)rms produce differentiated goods using wholesale goods as the sole input. The optimization problem of retail (cid:12)rm i consists of two parts. The cost minimization problem min Pwyw(i) t t y t w(i);yt(i) s:t: y (i) = yw(i): (11) t t delivers an expression for the retailer’s real marginal costs mc t Pw mc = t : (12) t P t Retailer i adjusts its price P (i) each period with the (cid:12)xed probability 1(cid:0)(cid:24)p. For (cid:12)rms t that do not re-optimize their price in a given period, prices will be updated as a weighted average of (cid:5) = Pt , the nominal price in(cid:13)ation in the previous period, and (cid:5) (cid:22) , the steady t Pt(cid:0)1 state in(cid:13)ation rate ( ) P (i) = P ~ (i) (cid:5) (cid:19)p(cid:5) (cid:22)1(cid:0)(cid:19)p : (13) t+1 t t Retail (cid:12)rm i sets its price to maximize [( ( ) )] ∑1 ∏s (cid:21) maxE ((cid:24)p(cid:12))s t+s (1+(cid:28)(cid:22)p)P ~ (i) (cid:5)(cid:19)p (cid:5) (cid:22)1(cid:0)(cid:19)p (cid:0)MC y (i) P~ t(i) t s=0 (cid:21) t t l=1 t+l(cid:0)1 t+s t+s 9

(cid:21)p 0 ( )1 P ~ (i) ∏ s (cid:5)(cid:19)p (cid:5) (cid:22)1(cid:0)(cid:19)p (cid:0) (cid:21)p (cid:0)1 B t t+l(cid:0)1 C s:t: y t+s (i) = B @ l=1 C A y t+s (14) P t+s where the subsidy (cid:28)(cid:22)p offsets distortions due to monopolistic competition in the steady state. We introduce a markup shock directly into the (cid:12)rst order condition of the retailer which under a linear approximation of the model is equivalent to variations in (cid:28)(cid:22)p or (cid:21)p. In choosingitsprice, the(cid:12)rmtakesintoaccountthedemandcurveforitsdifferentiatedgood. This demand curve is derived from the pro(cid:12)t maximization problem of the producers of the (cid:12)nal composite consumption good y which is produced from the differentiated goods t according to [∫ ] 1 (cid:21)p 1 y t = y t (i) (cid:21)pdi : (15) 0 The term (cid:21)p refers to the elasticity of substitution between the retail varieties. y is (cid:21)p(cid:0)1 t used for consumption cm and to cover the costs of posting vacancies v . t t 3.2 NK model with sticky nominal wages The model with sticky nominal wages differs from the search and matching model with regardtothelabormarketdetails. Inthismodel,allhouseholdmembersareemployedand nominal wages are set in staggered contracts following Calvo (1983). Each household j choosesconsumptionandassetholdingsbymaximizingtheinter-temporalutilityfunction [ ] ∑1 (c (j)(cid:0)(cid:22)c (j))1(cid:0)(cid:27) h (j)1+ϕ E (cid:12)t(cid:0)t0 t t(cid:0)1 (cid:0)ϕ t (16) t0 1(cid:0)(cid:27) 0 1+ϕ t=t0 subject to the budget constraint P c (j)+B (j) = (1+(cid:28)(cid:22)w)W (j)h (j)+R B (j)+Pr (j)+T (j): (17) t t t+1 t t t(cid:0)1 t t t Labor bundlers package the differentiated labor services supplied by households and sell the aggregate labor service at the nominal wage W . The labor bundling technology t 10

satis(cid:12)es [∫ ] 1 (cid:21)w 1 h t = h t (j) (cid:21)wdj (18) 0 where the term (cid:21)w measures the elasticity of substitution between differentiated labor (cid:21)w(cid:0)1 services. The bundler’s demand for variety j of labor services is given by [ ] (cid:0) (cid:21)w W (j) (cid:21)w(cid:0)1 t h (j) = h : (19) t t W t Eachhouseholdj suppliesdifferentiatedlaborservicesh (j)andsetsawagerateunder t monopolistic competition. The household can readjust its wage with (cid:12)xed probability 1(cid:0)(cid:24)w each period. If the household cannot reoptimize its wages, wages will increase by a weighted average of past in(cid:13)ation and the steady state in(cid:13)ation rate according to ( ) W (j) = W ~ (j) (cid:5)(cid:19)!(cid:5) (cid:22)(1(cid:0)(cid:19)!) : (20) t+1 t t A reoptimizing household chooses its wage as the solution to the following problem [ ] ∑1 (c (cid:0)(cid:22)c )1(cid:0)(cid:27) ϕ maxE ((cid:24)w(cid:12))s t+s t(cid:0)1+s (cid:0) 0 h (j)1+ϕ W~ t(j) t s=0 1(cid:0)(cid:27) 1+ϕ t+s s:t: P c +B = (1+(cid:28)(cid:22)w)W (j)h (j)+R B +Pr +T t+s t+s t+s+1 t+s t+s t+s(cid:0)1 t+s t+s t+s (cid:21)w ( ) (cid:0) W t+s (j) (cid:21)w (cid:0)1 h (j) = h t+s t+s W t+(s ) ∏s W (j) = W ~ (j) (cid:5)(cid:19)w (cid:5) (cid:22)1(cid:0)(cid:19)w (21) t+s t t+l(cid:0)1 l=1 where the subsidy (cid:28)(cid:22)w is set to eliminate the labor supply distortions arising from monopolistic competition in the steady state. Wholesale (cid:12)rms purchase aggregate labor services h from the labor bundler. Ret tail (cid:12)rms purchase the wholesale good, differentiate it, and set prices using staggered contracts, just as in the NK model with search and matching frictions. 11

3.3 Linearized models We brie(cid:13)y turn to the log-linear approximation of the two models around their respective steady states for the baseline speci(cid:12)cation without indexation of prices and wages ((cid:19)p = (cid:19)! = 0) and consumption habits ((cid:22) = 0); we display the core equations only. The details for the search and matching model are provided in Appendix B. For the sticky wage model, we refer the reader to Erceg, Henderson, and Levin (2000). Our linear search and matching model resembles the one in Ravenna and Walsh (2011) with two important exceptions: (i) the steady state is inefficient as we do not impose the conditions stated in Hosios (1990), (ii) the individual labor supply is elastic. Under the (cid:12)rst assumption, the (cid:13)exible price economy is not efficient which in turn complicates (cid:12)nding the second order approximation to the preferences of the representative household conducted later. The second assumption yields an additional equation not present in the linearized model of Ravenna and Walsh (2011). The linearized search and matching model reduces to three equations (excluding a description of monetary policy) in four endogenous variables [( ) ] ϖyss (cid:25) = (cid:12)E (cid:25) +(cid:20)p ϕ+(cid:27) y^ (cid:0)(1+ϕ)a^ (cid:0)((cid:18) n^ +(cid:18) n^ ) + ^ (cid:18) (22) t t t+1 1(cid:0)(cid:20)c t t 1 t 2 t(cid:0)1 p;t 1(cid:0)(cid:20)c y^ = E y^ (cid:0) (i (cid:0)E (cid:25) +((cid:18) (cid:0)ϕ)(E n^ (cid:0)n^ )+(cid:18) (n^ (cid:0)n^ )) (23) t t t+1 (cid:27)ϖyss t t ( t+1 1 ) t t+1 t 2 t t(cid:0)1 (cid:27)ϖyss (cid:13) E n^ +(cid:13) n^ +(cid:13) n^ = 1+ϕ+ y^ (cid:0)(1+ϕ)a^ 1 t t+1 2 t 3 t(cid:0)1 1(cid:0)(cid:20)c t t ( ) (1(cid:0)(cid:26))(cid:12) (cid:20)v (cid:0) (1(cid:0)(cid:24)q (cid:18) ) (i (cid:0)E (cid:25) ): (24) ss ss t t t+1 (cid:23) q ss Pricein(cid:13)ation(cid:25) , outputy^, employmentn^ , andthenominalinterestratei areexpressed t t t t in deviations from their steady state values; \hatted" variables are in log-deviations. The ^ exogenous shocks to technology (a^ ) and markups ((cid:18) ) follow standard AR(1) processes. t p;t Equations(22)and(23)aretheNKPhillipscurveandtheaggregatedemandrelationship, respectively. In contrast to standard NK models without search and matching frictions in the labor market, the level of employment n^ enters these equations. The third equation, t which can be traced back to the Nash bargaining over real wages relates the evolution of employment to the other variables. This model reduces to the standard NK model if each household member is employed at every point in time which, among other assumptions, requires that vacancy posting 12

costs are set to zero. Absent posting costs, (cid:20)c assumes the value of 0, ϖyss = 1, and n^ = 0 t for all t. Equation (24) is dropped due to the lack of wage bargaining. Alternatively, the model in Ravenna and Walsh (2011) with inelastic individual labor supply emerges in the limit as ϕ approaches in(cid:12)nity implying lim (cid:18) = ϕ and lim (cid:13) = ϕ. Equation ϕ!1 1 ϕ!1 2 (24) converges to n^ = y^ (cid:0) a^ which simply describes the production technology when t t t hours worked are (cid:12)xed. Using this result, equations (22) and (23) can be written in terms of in(cid:13)ation, the nominal interest rate, and employment. Notice, that the search and matching model with (cid:12)xed hours worked does not nest the standard NK model. The sticky nominal wage model features NK Phillips curves for prices and wages (cid:25) = (cid:12)E (cid:25) +(cid:20)p(w^ (cid:0)a^ )+ ^ (cid:18) (25) t t t+1 t t ( p;t ) 1+ϕ (cid:25)w = (cid:12)E (cid:25)w +(cid:20)w((cid:27) +ϕ) y^ (cid:0) a^ (cid:0)(cid:20)w(w^ (cid:0)a^ ) (26) t t t+1 t (cid:27) +ϕ t t t 1 y^ = E y^ (cid:0) (i (cid:0)E (cid:25) ) (27) t t t+1 t t t+1 (cid:27) w^ = w^ +(cid:25)w (cid:0)(cid:25) (28) t t(cid:0)1 t t Equation (28) describes the evolution of the real wage. If wages are fully (cid:13)exible, i.e., (cid:20)w (cid:0)! 1, the model reduces to the standard NK model since equation (26) reduces ( ) to w^ (cid:0) a^ = ((cid:27) +ϕ) y^ (cid:0) 1+ϕa^ and equation (25) can be written in its standard t t t (cid:27)+ϕ t formulation. Absentpriceindexation,in(cid:13)ationisaforward-lookingphenomenoninbothmodelsand can be expressed as the discounted present value of real marginal costs. Although real wages are not the exclusive determinant of real marginal costs, the real wage dynamics shape the dynamics of real marginal costs importantly. Thus, the dynamics of price in(cid:13)ation depend on the adjustment process for real wages and in turn on the impact that price in(cid:13)ation has on the adjustment of real wages. Monetary policy in(cid:13)uences the interplay between these variables. 4 Empirical Strategy At the core of our analysis lies the idea that the policymaker can formulate multiple models that provide a good approximation to the true data-generating process given the available empirical evidence against which the models are assessed. The large variety of business cycle models found in the academic literature (that all try to explain similar 13

aspects of the data) and the diverse set of models used within central banks in practice support this view. To arrive at a setting in which the policymaker considers multiple models for the purpose of policymaking, we search for parameterizations of the two models introduced in the previous section that match the same empirical evidence. Given the number of free parameters in most theoretical models, it is basically impossible to reduce the set of candidate models to a single one and obtain model certainty. This section discusses the criteria by which we judge the empirical performance of the sticky wage model and the search and matching model. 4.1 Estimation We estimate selected parameters of the two models using impulse response function matching introduced by Rotemberg and Woodford (1997). We focus on the case of the neutral technology shock; the empirical impulse responses are taken from Christiano, Eichenbaum, and Trabandt (2013).9 For each model, we divide the parameters into two groups: calibrated and estimated parameters. The values assigned to the (cid:12)rst group of parameters are taken from the literature. The parameters in the second group shape the dynamics of the model importantly; with clear evidence hard to come by for these parameters, we allow the data to determine their values. In this part of our analysis, monetary policy is assumed to follow a simple rule for the nominal interest rate as commonly found in the literature and in central bank analysis. In detail, we assume i = (cid:26) i +(cid:26) (cid:25) +(cid:26) x (29) t R t(cid:0)1 (cid:25) t x t where (cid:25) refers to price in(cid:13)ation in deviation from its long-run target value, and i denotes t t the short term nominal interest rate in deviation from steady state. The output gap is x . t The coefficients (cid:26) and (cid:26) govern the degree of interest rate smoothing and the reaction R (cid:25) of the nominal interest rate to current price in(cid:13)ation, respectively. In what follows, we 9Estimating a structural vectorautoregressive(SVAR)model, Christiano, Eichenbaum, and Trabandt (2013)identify shocks to monetary policy, as well as neutral and investment-speci(cid:12)c technology shocks. Our focus on the neutral technology shock is determined by certain features of our analysis: (i) we abstract from capital (and thus the investment-speci(cid:12)cshock)tosimplifythederivationofoptimaltargetingrulesinthenextsection,(ii)monetarypolicy shocks of the kind identi(cid:12)ed in SVAR models play no role in our subsequent analysis of optimal monetary policy. 14

abstract from the output gap by setting (cid:26) = 0. When we included the output gap in the x estimation, our results hardly changed. For each model, conditional on the calibrated parameters (cid:0)c, we search over the remaining parameters | collected in the vector (cid:0) | to minimize the distance between the impulse response functions generated from the model, denoted by G((cid:0);(cid:0)c)model, and the empirical impulse response functions from the SVAR in Christiano, Eichenbaum, and Trabandt (2013), denoted by G: ( ) ( ) ( ) (cid:0) ^ = argmin G(cid:0)G((cid:0);(cid:0)c)model ′ (cid:9)0 (cid:0)1 G(cid:0)G((cid:0);(cid:0)c)model : (30) (cid:0) As customary, the diagonal weighting matrix (cid:9)0 is obtained from the empirical variancecovariance matrix of the empirical impulse response functions (cid:9) by setting all off-diagonal ^ elements in (cid:9) to zero. The estimate (cid:0) minimizes the objective function in (30). Before reporting the results of the estimation, we review the values assigned to some key parameters collected in (cid:0)c. The parameter values are recorded in Table 1. To the extent appropriate, we assign identical values to the parameters both models have in common. In particular, we set the labor supply elasticity equal to 1=2, implying ϕ = 2, in line with the results reported by Smets and Wouters (2007). Hours worked are assumed to be 1=3 in the steady state. The parameter (cid:24)p which governs the degree of nominal price rigidities is (cid:12)xed at 0:75. The markup for prices is set at 20 percent in the steady state implying (cid:21)p equal to 1:2. Inthestickywagemodel,wealsoneedtospecifytheparametergoverningthestickiness ofnominalwages(cid:24)w andthesteadystatewagemarkup(cid:21)w. AsinChristiano,Eichenbaum, and Trabandt (2013), we set (cid:24)w = (cid:24)p and (cid:21)w = (cid:21)p. Parameters speci(cid:12)c to the model with search and matching frictions are chosen as follows. The breakup probability of a match with (cid:26) = 0:1 implies a quarterly separation rate of 10 percent which is in line with the estimate in Shimer (2005) of 3.4 percent per month. The parameter (cid:16) in the matching function is set at 0:54, just in the range of plausible values between 0.5 and 0.7 reported in Pissarides and Petrongolo (2001). We target a vacancy (cid:12)lling rate in the steady state q of 0:7 following Ramey, den Haan, and ss Watson (2000). The unemployment rate in the steady state is set at 0:055, the average US unemployment rate over the period from 1951Q1 to 2008Q4 reported by Christiano, Eichenbaum, and Trabandt (2013). We assume that the costs associated with posting and (cid:12)lling vacancies are proportional to the number of posted vacancies. Relative to output 15

these costs amount to (cid:17) = 0:0066. s We estimate the remaining parameters for each model by matching their impulse responsestoaneutraltechnologyshocktothecorrespondingSVARestimatesinChristiano, Eichenbaum, and Trabandt (2013). We include the (cid:12)rst 15 periods after the shock. While we (cid:12)x the persistence of the technology shock at (cid:26) = 0:9999, we estimate the standard a deviation of the shock (cid:27) . Furthermore, for each model we estimate the coefficients in the a interest rate rule, (cid:26) and (cid:26) , the degree of internal consumption habits (cid:22), and the degree R (cid:25) of price indexation (cid:19)p. In the search and matching model, we also estimate the replacement ratio ru. Finally, we estimate two versions of the sticky wage model. In the (cid:12)rst one we abstract from wage indexation, i.e., (cid:19)! is (cid:12)xed at 0, and in the second one we estimate the degree of wage indexation. For the sticky wage model, the estimation includes the impulse response functions for output, in(cid:13)ation, the short-term interest rate as measured by the federal funds rate, hours worked, real wages, and consumption. In the case of the search and matching model, we also include the responses of the unemployment rate, vacancies, and the job (cid:12)nding rate. Table 2 summarizes the estimates. Figure 1 shows the resulting impulse responses for the two models absent wage indexation in the sticky wage model and Figure 2 plots the responseswhenthedegreeofwageindexationisestimated. Inboth(cid:12)gures, thetheoretical modelsmatchtheempiricalresponseswell. Withtheexceptionofhoursworked,themodel responses lie within the con(cid:12)dence bands of the empirical responses and the responses are reasonably close to the SVAR point estimates and to each other. The sticky wage model with indexation, estimated at (cid:19)! = 1, provides a better (cid:12)t to the data than the model without wage indexation according to the value of our criterion function, see bottom of Table 2. Most of the difference in (cid:12)t stems from the model’s implications for hours worked. However, some authors have expressed skepticism regarding the presence of wage indexation in the data, see Levine, McAdam, and Pearlman (2012) and Christiano, Eichenbaum, and Trabandt (2013). The estimates for the coefficients in the policy rule, the variance of the technology shock, price indexation, and consumption habits are almost identical across models. The estimated policy rule features a high degree of interest rate smoothing and the implied long-run response of the interest rate to in(cid:13)ation is just strong enough to satisfy the Taylor principle, e.g., for the search and matching model this value is 1:0003 = (1 (cid:0) 0:8555)=0:1445. By coincidence, the estimated simple interest rate rules are close to 16

identical across models. When forcing the simple rule to coincide across models the estimates of the remaining parameters stayed stable. For all model speci(cid:12)cations, price indexation is estimated to be zero. Overall, our estimates resemble those in Christiano, Eichenbaum, and Trabandt (2013) despite the greater simplicity of our models.10 4.2 The elasticity of labor market tightness We also estimate the replacement ratio ru in the search and matching model. At a value of 0:5345 our point estimate is well below the implausibly high estimate in Christiano, Eichenbaum, and Trabandt (2013) for the search and matching model with Nash bargaining. The subsequent discussion explains how this difference across models related to our decision of modeling the disutility from labor explicitly. The responses of unemployment and vacancies are important dimensions to judge the performance of the search and matching model. The unemployment rate (and thus the number of job seekers u ) drops signi(cid:12)cantly after rising initially and vacancies v increase t t strongly over the medium term. Both in the data and the model the directions and the magnitudes of these responses imply a strong response of labor market tightness (the ratio of un(cid:12)lled vacancies to job seekers). ^ As shown in Appendix B.1, labor market tightness (cid:18) (expressed in log deviation from t steady state) is approximately proportional to (the log-deviations from steady state of) the marginal product of labor, hours worked, and real marginal costs in our model: ^ (cid:18) = v^ (cid:0)u^ t t t ( ) (cid:25) 1 [( ) 1+ ϕ ϕ mpl ss h ss mc ss ] m d pl +h ^ +mcc (31) (cid:7) ϕ (cid:0)ru mpl h mc +ru(1(cid:0)(1(cid:0)(cid:26))(cid:12)) (cid:20)v t t t 1+ϕ ss ss ss qss where (1(cid:0)(cid:26))(cid:12)(cid:24)q (cid:18) (1(cid:0)(cid:16)) ss ss (cid:7) = (cid:16) + : (32) [1(cid:0)(1(cid:0)(cid:26))(cid:12)(1(cid:0)(cid:24)q (cid:18) )] ss ss 10Christiano, Eichenbaum, and Trabandt (2013) include investment, capacity utilization, and the relative price of investment in the set of impulse responses and they require their models to also match the empirical responses to monetary policy and investment-speci(cid:12)c technology shocks. Our estimate of no consumption habits contrasts with theirestimateof(cid:22)lyingbetween0:7and0:8,adifferencethatstemsfromtheinclusionofthemonetarypolicyshockin Christianoetal. Their(recursivelyidenti(cid:12)ed)monetarypolicyshockinducesahump-shapedresponseinconsumption, a feature not shared by the neutral technology shock. Habit persistence is key to match the consumption response to the monetary policy shock. When we (cid:12)xed (cid:22) at a strictly positive value the parameters reported in Table 2 changed marginally. 17

(cid:7) lies in the interval [(cid:16);1], where (cid:16) is often set around 0:5 (in our case 0:54). Abstracting from the disutilityof workingfor employedworkers(i.e., ϕ ! 1 ), Shimer (2005) argues that standard search and matching models cannot reproduce the strong response of labor market tightness relative to the movements in the marginal product of labor found in the empirical evidence for plausible parameter choices, in particular for the replacement ratio ru. According to Shimer, a strongly pro-cyclical real wage dampens the responses of vacancies and unemployment resulting in a much muted response of labor market tightness vis-a-vis the data.11 Numerous authors have offered approaches to resolve this issue: Hall (2005) and Shimer (2005) propose real wage rigidities; Hagedorn and Manovskii (2008) argue in favor of high opportunity costs of employment; Hall and Milgrom (2008) suggest departures from Nash bargaining over wages; Petrosky-Nadeau and Wasmer (2013) analyze the role of (cid:12)nancial frictions. Our framework avoids the criticism in Shimer (2005) by modeling the disutility from working explicitly building on the ideas in Hagedorn and Manovskii (2008). With a labor supply elasticity of 0:5, i.e., ϕ = 2, the value for ru required to match the empirical evidence on unemployment, vacancies, and labor market tightness drops from almost 1 to near 0:5. 4.3 Additional shocks In addition to technology shocks, our model features markup shocks. Unfortunately, we are not aware of a broadly accepted scheme to identify markup shocks using SVAR analysis. We assume that each economy is subject to purely transitory markup shocks. The standard deviation of the markup shock is set at 0:0135 in the sticky wage model. The standard deviation of the markup shock in the search and matching model of 0:0104 minimizes the distance between the impulse responses for the markup shocks in the two models given the remaining parameters in Tables 1 and 2. The smaller value of the standard deviation in the search and matching model re(cid:13)ects the stronger impact of an equal-sized markup shock on output and in(cid:13)ation in the search and matching model compared to the sticky wage model. An alternative approach to ours would employ full information estimation of each 11For our parameterization, the steady state values of the marginal product of labor mpl and marginal costs mc ss ss are 1, and hours worked h are 1=3, implying h mpl mc = 1=3. With the term (1(cid:0)(1(cid:0)(cid:26))(cid:12)) (cid:20)v assuming the value 0:0024, the elasticity s o s f labor market tightn s e s ss ca s n s be ss raised to its value in the data by choosin qs g s ru sufficiently close to 1. In a setting similar to ours, Christiano, Eichenbaum, and Trabandt (2013) estimate ru to be 0:88. 18

model as in Smets and Wouters (2007). While delivering an empirical speci(cid:12)cation of each shock, full information estimation comes at the cost of requiring additional modeling features and assumptions about a range of stochastic disturbances that are of varying economic plausibility. 5 Optimal policy and robustness As both models match the SVAR evidence well for reasonable parameters under the estimated simple interest rate rule, the policymaker has little guidance for choosing one modelovertheother. Inaddition, thetwomodelshavecon(cid:13)ictingnormativeimplications, makingitpotentiallyworthwhileforthepolicymakertoshyawayfromselectingonemodel prior to the evaluation of monetary policy: in the search and matching model, the optimal monetary policy seeks to stabilize price in(cid:13)ation at the expense of wage in(cid:13)ation whereas the optimal monetary policy in the sticky wage model seeks the opposite outcome. Thus, if the policymaker formulates policies on the assumption that the search and matching model is the true data-generating process when in fact the sticky wage model constitutes the true process, the policymaker implements a policy that may result in big welfare losses. Instead, it appears suitable to make model uncertainty a component of the policy evaluation. One way for the policymaker to address the tensions between the normative recommendations of the models for the purpose of (cid:12)nding an optimal policy under model uncertainty is to choose a policy that minimizes an expected loss function under a probability distribution Ω over the candidate models. Let the loss function be of the general form t { } ∑1 E (cid:12)tL(y~;Ω ) (33) 0 t t t=0 where y~ is a vector de(cid:12)ned to include the variables of all reference models including t their leads and lags when appropriate. The loss function depends explicitly on the policymaker’s probability distribution over models to re(cid:13)ect the fact that the policymaker’s preferences over economic outcomes may differ across models. In principle, the policymaker updates Ω over time in response to the observed economic outcomes under the t policymaker’s choices. The modelling of this updating process can take on different degrees of complexity with no learning about the true data-generating process and Bayesian 19

optimal learning de(cid:12)ning the range of possible setups. In this paper, we consider the following scenarios to provide insight into the policymaker’s decision problem under model uncertainty: 1. the policymaker knows the correct model and implements a policy that is optimal given the preferences of the representative household (standard assumption as in Woodford (2003)), 2. the policymaker selects policy on the assumption that model 1 is true, when in fact model 2 is true and the other way around (this approach requires the use of optimal targeting rules), 3. the policymaker has a time-invariant probability distribution over models and (cid:12)xes a policy rule at the beginning of time (as in Levin, Wieland, and Williams (2003), Brock, Durlauf, and West (2007), and Taylor and Williams (2010)) consistent with a weighted-averageoverthepreferencesoftherepresentativehouseholdinthereference models, 4. the policymaker chooses a policy rule that minimizes the maximum welfare losses of the representative household given the reference models (as in Levin, Wieland, and Williams (2003)). We discuss the (cid:12)rst two scenarios in this section, and the remaining two in Section 6. 5.1 Optimal policy and optimal targeting rule To implement scenarios 1 and 2, we follow Svensson and Woodford (2004), Giannoni and Woodford (2003, 2016) who advocate for the use of optimal targeting rules to characterize theoptimalmonetarypolicy. Theoptimaltargetingrulespeci(cid:12)esthevariables|including the relative importance and the dynamic structure of each variable|in a single target criterion that seeks to implement the optimal monetary policy. Optimal targeting rules can be computed for any preferences assigned to the policymaker. However, we adopt the approach in Woodford (2003) and assume the preferences of the policymaker to coincide with those of the representative household in the model. Therefore, obtainaing optimal targeting rules in our settings requires to: (1) derive the objective function of the policymaker as a purely quadratic approximation to the prefer- 20

ences of the representative household;12 (2) obtain the (cid:12)rst order conditions associated with the policymaker’s problem of optimizing the (quadratic) objective function subject to the (linear) equations that describe the behavior of the private sector using the Lagrangian approach; (3) combine the (cid:12)rst order conditions to obtain a single equation without Lagrange multipliers; this targeting rule describes the relationship between the endogenous and exogenous variables under the optimal policy. Importantly, the evolution of the economy that is consistent with the targeting rule is unique. The optimal targeting rule implements the optimal monetary policy in the model from which it is derived, in contrast to simple instrument rules with optimally chosen coefficients. However, similar to instrument rules, the optimal targeting rule is expressed in terms of economically relevant model variables only; instrument rules prescribe how to adjust the policy instrument (such as the short term interest rate) in response to variables such as in(cid:13)ation, output, etc., and targeting rules describe how to adjust a target variable (for example, price in(cid:13)ation) to output, wage in(cid:13)ation and other variables. Finally, the evolution of the optimal target criterion does not necessarily involve the policy instrument of the central bank directly. Optimal targeting rules are therefore ideally suited to investigate the robustness of the optimal policy implied by one model in a different model as in our second scenario. The optimal targeting rule for the NK model with sticky wages is easily obtained from the linear quadratic approximation of the original model. Employing results from Woodford (2003) for the policymaker’s objective function and the linear approximation of the model’s structural equations, the policymaker’s problem is to ∑1 min E (cid:12)tLsw (34) f(cid:25)t;(cid:25)w t ;y^t;it;w^t g1 t=0 0 t=0 t s:t: equations (25) to (28) and pre-commitments for period 0 (timeless perspective). Absent consumption habits, price and wage indexation, the loss function consistent with the second order approximation of household preferences satis(cid:12)es ( ) (cid:27) +ϕ 1+ϕ 2 1+(cid:18)p 1+(cid:18)w Lsw = y^ (cid:0) a^ + (cid:25)2 + ((cid:25)w)2: (35) t 2 t (cid:27) +ϕ t 2(cid:18)p(cid:20)p t 2(cid:18)w(cid:20)w t 12We follow Woodford (1999) and Benigno and Woodford (2012) in adopting the concept of \optimality from the timelessperspective"|anecessaryassumptiontoobtainthecorrectlinearquadraticapproximationtoour(nonlinear) model. 21

Giannoni and Woodford (2003) show that the (cid:12)rst order conditions associated with the problem in (34) can be recombined to obtain the optimal targeting rule ( ) ( ) 1+(cid:18)p 1+(cid:12) +(cid:20)p 1+(cid:18)w 0 = (cid:25) +x (cid:0)x + (cid:25)w +x (cid:0)x (cid:18)p t t t(cid:0)1 (cid:20)w (cid:18)w t t t(cid:0)1 ( ) ( ) (cid:12) 1+(cid:18)w 1 1+(cid:18)w (cid:0) (cid:25)w +x (cid:0)x (cid:0) (cid:25)w +x (cid:0)x (36) (cid:20)w (cid:18)w t+1 t+1 t (cid:20)w (cid:18)w t(cid:0)1 t(cid:0)1 t(cid:0)2 with x = y^ (cid:0) 1+ϕa^ . See Appendix D for details of the derivation. Absent sticky wages, t t (cid:27)+ϕ t i.e., (cid:20)w (cid:0)! 1, equation (36) reduces to the optimal targeting rule in the standard NK model with (cid:13)exible wages 1+(cid:18)p (cid:25) +x (cid:0)x = 0 (37) (cid:18)p t t t(cid:0)1 which suggests lowering the target value for price in(cid:13)ation in the current period below its long run value when the growth rate of the output gap is positive. When wages are sticky, the targeting rule also features the position of wage in(cid:13)ation relative to the output gap (over the near past and future). For stickier nominal wages and thus a (cid:13)atter NK Phillips curve for wages ((cid:20)w (cid:0)! 0), the policymaker is less concerned with deviations of pricein(cid:13)ationfromitslong-runtarget. Inthelimitingcaseof(cid:13)exibleprices,i.e., (cid:20)p (cid:0)! 1, the target value for wage in(cid:13)ation in the current period is set below its long run value when the growth rate of the output gap is positive. Thusfar, derivationsoftheoptimaltargetingrulesinmodelswithsearchandmatching frictions are absent from the literature. Blanchard and Gal(cid:19)(cid:16) (2010), Thomas (2008), and Ravenna and Walsh (2011) derive purely quadratic objectives for the policymaker from household preferences under the assumption that the search and matching process does not induce inefficiencies as in Hosios (1990). None of these papers derives the implied optimal targeting rule. Furthermore, if the Hosios condition is not imposed, even the (cid:12)rst step of obtaining an explicit second order approximation to the preferences of the representative household is missing in the literature. To derive a purely quadratic objective for the policymaker in the presence of a distorted steady state for the search and matching model, we employ the numerical approach described in Bodenstein, Guerrieri, and LaBriola (2014) which is consistent with the theoreticalresultsinBenignoandWoodford(2012). AppendixCshowsthatthepolicymaker’s (period) loss function consistent with a second order approximation to the preferences of 22

the representative household can be written as Ls&m = P (cid:25)2 +P y^2 +P n^2 +P n^2 +P n^ y^ +P y^n^ t (cid:25);(cid:25) t y;y t n;n t n(cid:0);n(cid:0) t(cid:0)1 y;n t t y;n(cid:0) t t(cid:0)1 ^ ^ +P n^ n^ +P n^ a^ +P n^ (cid:18) +P y^a^ +P y^(cid:18) : (38) n;n(cid:0) t t(cid:0)1 n;a t t n;p t p;t y;a t t y;p t p;t This formulation of the loss function is already simpli(cid:12)ed to include those variables only that enter the linear model in equations (22)-(24). We cannot obtain closed form expressions for the composite coefficients in (38), but our approach provides numerical values based on the underlying deep parameters of the model. The (linear-quadratic) problem of the policymaker is ∑1 min E (cid:12)tLs&m (39) f(cid:25)t;it;n^t;y^t g1 t=0 0 t=0 t s:t: equations (22) to (24) and pre-commitments for period 0 (timeless perspective). Rearranging the (cid:12)rst order conditions associated with the optimization problem in (39) delivers the optimal targeting rule in the search and matching model ^ ^ ϖ n^ +ϖ n^ +ϖ n^ +ϖ y^ +ϖ y^ +ϖ a^ +ϖ (cid:18) +ϖ (cid:25) +ϖ (cid:25) +ϖ P 1 t 2 t(cid:0)1 3 t+1 4 t 5 t+1 6 t 7 p;t 8 t 9 t+1 10 t(cid:0)1 +ϖ y^WA +ϖ n^WA +ϖ a^WA +ϖ ^ (cid:18) WA +ϖ P ^WA = 0 (40) 11 t 12 t 13 t 14 p;t 15 t where we de(cid:12)ne (cid:25) = P ^ (cid:0)P ^ (41) t t t(cid:0)1 y^WA = (cid:12) y^WA +y^ (42) t (cid:14) t(cid:0)1 t n^WA = (cid:12) n^WA +n^ (43) t (cid:14) t(cid:0)1 t P ^WA = (cid:12) P ^WA +P ^ : (44) t (cid:14) t(cid:0)1 t a^WA = (cid:12) a^WA +a^ (45) t (cid:14) t(cid:0)1 t ^WA ^WA ^ (cid:18) = (cid:12) (cid:18) +(cid:18) (46) p;t (cid:14) p;t(cid:0)1 p;t ^ In the steady state, the price level grows with the steady state in(cid:13)ation rate. The term P t denotes deviations of the price level from this growth path. Compared to the sticky wage model, the optimal targeting rule in the search and matching model involves weighted in(cid:12)nite-moving averages of output, employment, the price level, and the shocks. The 23

^ presence of the markup shock (cid:18) in the targeting rule is solely due to our decision not to p;t impose the efficiency condition by Hosios (1990)|a decision that was needed to obtain a good (cid:12)t of the search and matching model to the data in Section 4.13 To be able to exchange the optimal targeting rules between models requires expressing the targeting rules in variables that are common across models. For example, when implementing the rule in equation (40) in the sticky wage model we (cid:12)rst substitute out for employment n^ in terms of hours worked, output, and technology using the aggregate t production function and de(cid:12)ne the price level in the sticky wage model. Similarly, a de(cid:12)nition of wage in(cid:13)ation is added to the search and matching model when solving that model under the optimal targeting rule derived in the sticky wage model, equation (36). 5.2 Robustness of optimal targeting rules We assess the robustness of the optimal targeting rules across the two models starting with the search and matching model. Figure 3 depicts the case of the technology shock ^ a^ , and Figure 4 shows the case of the price markup shock (cid:18) . Under the technology t p;t shock, the optimal monetary policy as implemented by the targeting rule in equation (40) for the search and matching model calls for almost full stabilization of price in(cid:13)ation. No meaningful trade-offs arise as the welfare-relevant gaps move in the same direction: the technology shock exerts downward pressure to prices, and upward pressure on output and employment with the expansions being held back by sticky nominal prices. An interest rate cut reduces the downward pressure on prices and speeds up the expansion in output and employment. As a result, the real variables follow closely their paths in a real economy without sticky prices. These (cid:12)ndings are reminiscent of the standard NK model with (cid:13)exible wages (and no search and matching frictions in the labor market) in which the optimal monetary policy fully stabilizes in(cid:13)ation and closes the output gap. In fact, for the parameters in Table 1 and 2, the weights ϖ with j = 1;:::;15 in equation (40) j are such that the optimal targeting rule from the standard NK model (without search and matching frictions) displayed in equation (37) is a close approximation to the optimal targeting rule derived from the search and matching model.14 Notably, the labor market adjusts almost instantly to the shock in sharp contrast to 13In the sticky wage model the markup shock does not enter equation (36) since the steady state is assumed to be efficient; otherwise the markup shock would appear in the targeting rule as well. See also Benigno and Woodford (2005). 14See Appendix C and the discussion in Bodenstein and Zhao (2016). 24

theempiricalresponsesinFigure1. Undertheoptimalpolicytherealwageadjustsswiftly facilitated by a pronounced spurt in wage in(cid:13)ation. The movements in wages re(cid:13)ect the persistent jump in the marginal value of employment to the (cid:12)rm that gives rise to the front-loaded response in vacancies and the fall in unemployment. In contrast to the rule in equation (40), the optimal targeting rule derived from the sticky wage model given in equation (36) places greater emphasis on stabilizing wage in(cid:13)ation and less emphasis on price in(cid:13)ation. Applied to the search and matching model, this second rule keeps nominal wages basically constant. Yet, the persistent rise in technology pressures real wages to rise and thus price in(cid:13)ation must fall below its target value to facilitate at least gradual adjustment in the real wage. As (cid:12)rms and households cannot reap all the bene(cid:12)ts of improved technology and higher real wages immediately, vacancy postings, employment, and unemployment display inertia relative to the optimal responses. Adjustments in output and consumption are consequently delayed, as well. As is apparent from Figure 3, the targeting rule that is optimal in the sticky wage model does not induce the optimal responses in the search and matching model after a technology shock. In the case of the markup shock, similar differences emerge between the two policy rules in the search and matching model. With the exception of price in(cid:13)ation, all other variables react more strongly to the shock under the optimal policy, equation (40). As the markup shock induces a trade-off between variables, price in(cid:13)ation is not fully stabilized under the optimal policy to temper the (cid:13)uctuations in the other variables. Again, when the targeting rule derived from the sticky wage model is imposed instead, i.e., the rule in equation (36), wage in(cid:13)ation is almost fully stabilized at the expense of higher price in(cid:13)ation. The responses of all other variables are greatly muted compared to the optimal policy when the policymaker follows equation (40). The lack of robustness of the targeting rules across models also applies to the model with sticky wages. Figures 5 and 6 plot the responses in the sticky wage model to technology and markup shocks, respectively, for the two targeting rules displayed in equations (36) and (40). The optimal policy under sticky nominal wages, implemented through equation (36), stabilizes wage in(cid:13)ation in response to both shocks. This policy avoids welfare-costly wage dispersion in the sticky wage model, whereas price in(cid:13)ation induces movements in the real wage that in turn facilitate the adjustment process for all other variables under the optimal policy. By overly stabilizing price in(cid:13)ation, the targeting rule 25

derived in the search and matching model (40) allows more wage in(cid:13)ation than is optimal in the sticky wage model and in turn causes hours worked, output, and consumption, in particular, to exceed the optimal responses. To sum up, the targeting rule (40), which is optimal in the search and matching model, favours stabilizing prices over stabilizing wages irrespective of the model in which the rule is implemented. The targeting rule (36), which is optimal in the sticky wage model, favours stabilizing wages over stabilizing prices irrespective of the model under consideration. Exchanging targeting rules between the models induces welfare losses that are orders of magnitudes larger than the welfare costs of business cycles in Lucas (2003). For the sticky wage model the welfare loss (measured in CEV) is considerably higher than for the search and matching model (1.3033 versus 0.1133) re(cid:13)ecting the high welfare costs associated with even minor relative wage differences in the sticky wage model. The lack of robustness of the optimal targeting rules also applies when nominal wages are indexed to past in(cid:13)ation in the sticky wage model as in the (cid:12)nal column of Table 2.15 5.3 Sensitivity Thus far, we have derived optimal targeting rules for each model assuming that the policymaker adopts the preferences of the representative household. To distinguish the role of preferences from the remaining model features for our results, we consider the case that the policymaker’s preferences are identical across models and are given by the widely-used simple quadratic loss function Lsql = (cid:25)2 +(cid:21) x2. The parameter (cid:21) governs t t x t x the relative importance of stabilizing price in(cid:13)ation versus the output gap. We consider the case of (cid:21) = 0:0429 under which the policymaker places high emphasis on price x in(cid:13)ation similar to Ls&m, and the case of (cid:21) = 1 which implies a low emphasis on price t x in(cid:13)ation as under Lsw.16 Figure 7 shows the impulse responses under the targeting rules t derived from the simple loss functions. In the search and matching model, the optimal policy consistent with preferences Lsql t 15Forthecaseoffullindexation,thefocusofoptimalmonetarypolicyinthestickywagemodelshiftsfromsmoothing wagein(cid:13)ationtosmoothingthedifferencebetweenwagein(cid:13)ationandlaggedpricein(cid:13)ation,i.e.,(cid:25)w t (cid:0)(cid:25) t(cid:0)1 . Thewelfare lossofimplementinginthestickywagemodelwithfullindexationthe(unchanged)targetingrulethatisoptimalinthe searchandmatchingmodelrisesto1:9728whenmeasuredasCEV.Theoverallwelfarelossinthesearchandmatching model rises to 0:1680 when policy follows the optimal targeting rule derived in the sticky wage model. Appendix E provides more details on this case. 16The choice (cid:21) = 0:0429 is consistent with the weight on the output gap in the loss function derived for the x standard NK model with (cid:13)exible wages under the parameters in Tables 1 and 2. The alternative speci(cid:12)cation of (cid:21) =1 is popular in the literature. x 26

for(cid:21) = 0:0429resemblestheoptimalpolicyderivedunderpreferencesLs&m|thetoprow x t of panels in the (cid:12)gure. However, with the exception of price in(cid:13)ation all variables react by less to the markup shock than in Figure 4, indicating that under this parameterization of Lsql the policymaker prefers price in(cid:13)ation to bear more of the burden of adjustment than t in our original case. When imposing onto the search and matching model the optimal targetingrulederivedinthestickywagemodelunderpreferencesLsql,thesamequalitative t differences emerge as in Figure 4 despite the fact that the policymaker’s preferences are now constant across models. Similarly, in the sticky wage model the gaps between the impulse responses under the two targeting rules derived for preferences Lsql remain large t albeitsmallerthaninFigure6. Evenwhenthepolicymaker’spreferencesareheldconstant across models, the optimal targeting rules are not necessarily robust. However, if the policymaker assigns even lower relative importance to price in(cid:13)ation, the optimal targeting rules are robust. The lower two rows of panels in Figure 7 show the impulse responses for (cid:21) = 1. In both the search and matching model and the sticky x wage model, the gaps between the impulse responses generated by the optimal targeting rules derived for (cid:21) = 1 are minor. The robustness of optimal targeting rules is therefore x sensitive to the preferences assigned to the policymaker.17 It is important to realise, though, that the robustness of the optimal targeting rules under (cid:21) = 1 only applies from x the viewpoint of the policymaker with preferences Lsql = (cid:25)2 +x2. For the representative t t t householdsintherespectivemodelsthesepoliciesaresuboptimalashouseholdpreferences continue to be given by Ls&m and Lsw. t t 6 Robust policy Having established, that the reference models differ with respect to their normative recommendations and that the welfare costs of applying the wrong policy to a given model can be sizeable, we move on to scenarios 3 and 4 outlined in Section 5 and make model uncertainty a direct component of the evaluation of monetary policies. 17Using a very different set of models, Levin and Williams (2003) also (cid:12)nd that optimal targeting rules are more likely to be robust if the policymaker focuses less on price stability. 27

6.1 Methodology To obtain optimal policies under model uncertainty, we need to specify the policymaker’s preferences and possibly a probability distribution over models. Furthermore, we need to make assumptions about the implementation of monetary policy. 6.1.1 Objective function of the policymaker We begin with specifying the preferences of the policymaker. Previous works on model uncertainty separate the policymaker’s preferences from the underlying models by assuming a simple quadratic loss function that punishes deviations of in(cid:13)ation and the output gap from their respective long-run target values. However, as discussed in the previous section, the model-consistent preferences over outcomes are given by the function Ls&m t in the search and matching model and by the function Lsw in the sticky wage model, t respectively. Thus, we propose to better align the preferences of the policymaker with those of the representative household. We present results for a model averaging approach and a minmax approach. Under model averaging, the policymaker’s preferences over economic outcomes are a weighted average over the preferences of the representative household in the reference models. The weights are taken to coincide with the policymaker’s probability distribution over models. We assume the probability distribution over models to be time-invariant. With these assumptions in place, the policymaker’s preferences are ( ) ( ) ∑1 ∑1 Lav((cid:2)) = ! E (cid:12)tLs&m((cid:2)) +(1(cid:0)!) E (cid:12)tLsw((cid:2)) (47) 0 t 0 t t=0 t=0 where (cid:2) indicates the monetary policy common to both models as speci(cid:12)ed below. Levin and Williams (2003) and Taylor and Williams (2010) refer to this approach as Bayesian strategy. The model averaging approach can be interpreted literally as the case of a single policymaker assigning a probability distribution over the reference models based on statistical analysis. Forexample,Levine,McAdam,andPearlman(2012)translateposterioroddsratios of models estimated with full information Bayesian techniques into the policymaker’s probability distribution over models. The policymaker then engages in model averaging. In this sense, the policymaker’s model is a weighted average across the reference models. An alternative interpretation is related to decision-making in committees. Each member 28

of the committee selects a single model that re(cid:13)ects her/his views over the economy. The optimal policy under uncertainty is not optimal from any individual member’s point of view, but it produces outcomes that might be acceptable to all members. When the policymaker pursues the minmax approach, the policymaker’s loss under policy (cid:2) is the maximum welfare loss across the two reference models {( ) ( )} ∑1 ∑1 Lminmax((cid:2)) = max E (cid:12)tLs&m((cid:2)) ; E (cid:12)tLsw((cid:2)) : (48) 0 t 0 t t=0 t=0 This policymaker is concerned with avoiding the worst case scenario of setting a policy that could result in large welfare losses under any circumstances. In comparison to the model averaging approach, the minmax approach allows the optimal policy under model uncertainty to be greatly in(cid:13)uenced by the model which displays greater sensitivity of welfare losses to deviations from its model-speci(cid:12)c optimal policy even if that model is considered unlikely. The results under the minmax approach are independent from the policymaker’s probability distribution over models as long as non-zero probability is attached to each model. 6.1.2 Formulating policy We assume that monetary policy follows a simple instrument rule of the type i = (cid:26) i +(cid:26) (cid:25) +(cid:26)w(cid:25)w +(cid:26) x : (49) t R t(cid:0)1 (cid:25) t (cid:25) t x t { } The vector (cid:2) = (cid:26) ;(cid:26) ;(cid:26)w;(cid:26) collects the coefficients of the rule. According to the R (cid:25) (cid:25) x rule, the nominal interest rate is adjusted in response to movements in price and wage in(cid:13)ation, as well as the output gap. Furthermore, the rule allows for interest rate inertia. We search for the (non-negative) values of the parameters in (cid:2) that minimize the welfare loss of the policymaker under the given objective as in Levin and Williams (2003) and Levin, Wieland, and Williams (2003).18 Using simple rules limits the number of coefficients to be determined and thus facilitates computations and transparency. Despite the small number of variables included in 18Foragivenparameterizationoftherule,wecomputetheunconditionalwelfarelossforeachmodeltoapproximate the conditional welfare as the discount factor is close to 1 and to eliminate the potential impact of arbitrary initial conditions. Following Benigno and Woodford (2012) we include a correction term to account for the fact that the rule may violate the pre-commitment conditions imposed in deriving the loss functions. Rules that lead to equilibrium indeterminacy are discarded. 29

the simple rule, these rules can provide high-quality approximations to the optimal policy. We con(cid:12)rm that absent model uncertainty the simple rule (49) can indeed approximate the optimal policy with a high degree of accuracy for each reference model. In addition to simple instrument rules with optimized coeffients, we also consider a simple targeting rule of the form ( ) ( ) 1+(cid:18)p 1+(cid:12) +(cid:20)p 1+(cid:18)w 0 = (cid:26) (cid:25) +x (cid:0)x + (cid:25)w +x (cid:0)x tr (cid:18)p t t t(cid:0)1 (cid:20)w (cid:18)w t t t(cid:0)1 ( ) ( ) (cid:12) 1+(cid:18)w 1 1+(cid:18)w (cid:0) (cid:25)w +x (cid:0)x (cid:0) (cid:25)w +x (cid:0)x (50) (cid:20)w (cid:18)w t+1 t+1 t (cid:20)w (cid:18)w t(cid:0)1 t(cid:0)1 t(cid:0)2 with the policymaker choosing the parameter (cid:26) given the values for the remaining patr rameters. This simple targeting rule can approximate well the optimal targeting rule from the search and matching model for large values of (cid:26) under our parameterization as tr discussed earlier and it replicates the optimal targeting rule from the sticky wage model for (cid:26) = 1 regardless of the parameterization. In implementing each of these two rules tr we de(cid:12)ne the output gap as the difference between actual output and the output that would have prevailed absent nominal rigidities.19 6.2 Monetary policy rules under model uncertainty Table 3 reports in Panel (a) the optimal simple rules under the benchmark parameterization of the search and matching model and the sticky wage model. For the model averaging approach, we consider multiple speci(cid:12)cations of the policymaker’s probability distribution with !, the probability that the policymaker assigns to the search and matching model being the true data-generating process, ranging from 0 to 1. We refer to the optimal simple rule associated with a given probability distribution as the \!-optimal simple rule." Under the minmax strategy, the policymaker’s probability distribution over models is irrelevant. Welfare is reported in terms of consumption equivalent variations (CEV). In Panel (b), we report the (cid:12)ndings when the policymaker follows the simple targeting rule. Finally, for comparison, the table repeats in Panel (c) the welfare implications of implementing the optimal targeting rules derived in the previous section across models. In Panel (a), we distinguish three regions for the probability ! under model averaging: 19This implementation differs from Section 5 where we abstain from de(cid:12)ning output gaps in the implementation. The differences in outcomes under the two approaches are minor for targeting rules. 30

low (! (cid:20) 0:2), intermediate (0:3 (cid:20) ! (cid:20) 0:8), and high (! (cid:21) 0:9). The !-optimal simple rule varies distinctly across these regions. In the (cid:12)rst region with little probability weight on the search and matching model, the nominal interest rate responds primarily to wage in(cid:13)ation in line with optimal policy prescriptions of the sticky wage model. In the second region, the rule responds to wage and price in(cid:13)ation with the coefficients assigned to the two variables being of similar magnitude. It is only in the third region that the !-optimal simple rule displays signi(cid:12)cant interest rate inertia. The coefficient on wage in(cid:13)ation basically drops to zero whereas the nominal interest rate responds to price in(cid:13)ation. With the policymaker assigning a high probability to the search and matching model, the importance of wage in(cid:13)ation stabilization fades. Consequently, in the sticky wage model the welfare loss (relative to the optimal monetary policy in that model) under the !-optimal simple rule is larger for higher values of ! and the welfare loss in the search and matching model is reduced. To illustrate the dynamic implications of the various optimal simple rules, we plot in Figure 8 the impulse responses of output, price and wage in(cid:13)ation in both models to the technology shock and the markup shock for ! = 0;0:2;0:3;0:8;0:9;1. In the sticky wage model, the !-optimal simple rules with ! < 0:9 induce impulse responses (bottom two rows of panels) that are reasonably close to those under ! = 0 (the optimal simple rule if the policymaker is certain about the sticky wage model being the true data-generating process). For the search and matching model (top two rows of panels), !-optimal simple rules with ! < 0:9 induce responses that differ noticeably from those under ! = 1 (the optimal simple rule if the policymaker is certain about the search and matching model beingthetruedata-generatingprocess).20 Thus,thepolicymakereffectivelybiasespolicies towards the optimal policy in the sticky wage model for the low and the intermediate region of ! even though the optimally chosen parameters differ across the two regions. The welfare losses induced by the !-optimal simple rules reported in Table 3 con(cid:12)rm this conclusion from a normative perspective. When moving from the 0:8-optimal simple rule to the 0:9-optimal simple rule the CEV value for the sticky wage model goes from negligible to 0:2. While the welfare losses in the search and matching model are generally small, the CEV value is practically zero under the 0:9-optimal simple rule. 20For the search and matching model, the Euclidean distance betweenthe impulse responses of price in(cid:13)ation, wage in(cid:13)ation and output to the price markup shock for the rules ! =0:8 and ! =0:9 measured against the case of ! =1, respectively, drops from 0.0347 to 0.019. For the sticky wage model, by contrast, the Euclidean distance between the impulse responses of price in(cid:13)ation, wage in(cid:13)ation and output to the price markup shock for the rules ! = 0:8 and ! =0:9 measured against the case of ! =0, respectively, more than doubles from 0.0116 to 0.0245. 31

Thereasonfortheapparentbiasoftheoptimalpolicyundermodeluncertaintytowards the sticky wage model lies in the high welfare costs associated with even minor relative wage differences in the sticky wage model. The desire to avoid bad economic outcomes caused by bad monetary policy is even more explicit when the policymaker adopts a minmax strategy. In this case, the optimal simple rule coincides with the 0-optimal simple rule, which in turn mimics the optimal targeting rule derived in the sticky wage model.21 To complement the (cid:12)ndings for the optimal simple rules, Panel (b) reports the optimal parameterization of the simple targeting rule proposed in equation (50). In the case of model averaging, when the policymaker holds the sticky wage model reasonably likely, the coefficient (cid:26) is set near 1 and the rule allows for wage in(cid:13)ation to be a primary concern tr of monetary policy. Only for ! close to 1 does the policymaker switch to stabilizing price in(cid:13)ation aggressively: (cid:26) exceeds 6e+05 for ! = 1, but it assumes a value around 15 for tr ! = 0:9. It is in the interval ! 2 [0:9;1] that the welfare loss in the search and matching model under the simple targeting rule drops to almost zero, whereas the welfare loss in the sticky wage model soars.22 Under the minmax approach, the policymaker chooses (cid:26) = 1 and implements the optimal targeting rule of the sticky wage. tr 6.3 Sensitivity of results 6.3.1 Functional form of the policy rule In principle, the simple rule in equation (49) allows the policymaker to respond to the lagged value of the nominal interest rate, price and wage in(cid:13)ation, and the output gap. However, the response coefficient on each variable is assigned the value of zero for some !; the patterns of zeroes de(cid:12)ne the three distinct regions of the !-optimal simple rules in Table 3 for the model averaging approach. To assess the sensitivity of our (cid:12)ndings to the functional form of the simple rule, Table 4 reports optimal simple rules that, in comparison to (49), are restricted not to respond to either the lagged interest rate, the output gap, price in(cid:13)ation, or wage in(cid:13)ation, respectively.23 21The 0-optimal simple rule is close to but not identical to the optimal targeting rule derived in the sticky wage model, as the functional form of the simple rule in equation (49) is not quite (cid:13)exible enough. 22By construction, the simple targeting rule is biased in favor of the sticky wage model and somewhat less (cid:13)exible than the optimal simple rule. The value of the objective function assumes larger values than in the case of optimal simple rules for high values of !. 23ThepresenceofthreedistinctparameterregionsinTable3Panel(a)undermodelaveragingsuggeststheexistence of multiple local optima. In computing restricted optimal simple rules we can also con(cid:12)rm that the !-optimal simple 32

Absent interest rate smoothing (Case I in Table 4), the optimal simple rule changes only for ! (cid:21) 0:9 compared to Table 3. The response coefficient for price in(cid:13)ation becomes very large to compensate for the lack of interest rate smoothing in the rule, but overall welfare and welfare in the search and matching model deteriorate nevertheless. In its eagernessto(cid:12)ghtpricein(cid:13)ation,therulefor! = 1and(cid:26) = 0isparticularlyunattractive, R as it induces welfare losses in the sticky wage model that by far exceed the corresponding loss in Table 3 Panel(a). Eliminating the output gap from the list of response variables (Case II) affects the computations of the optimal simple rules only for ! (cid:20) 0:2. These restricted rules respond towagein(cid:13)ationbymorethaninTable3|theoptimizerreachestheupperboundof100| where the !-optimal simple rule responded importantly to the output gap for ! (cid:20) 0:2. The overall welfare loss is higher mostly because the restricted rules perform worse in the sticky wage model. Moredramaticchangesintheoptimalsimplerulesappeariftherulesarerestrictednot to respond to price in(cid:13)ation or wage in(cid:13)ation (Case III). Setting (cid:26) = 0 leads to higher (cid:25) response coefficients for wage in(cid:13)ation and, depending on the value of !, the output gap or interest rate smoothing. The deterioration in overall welfare is borne by the search and matching model; welfare in the sticky wage model improves for most values of ! and never declines. Finally, when eliminating the policymaker’s ability to respond to wage in(cid:13)ation directly, welfare losses increase in both the sticky wage and the search and matching model for most values of ! (Case IV). The form of the simple rule in this (cid:12)nal case coincides with the speci(cid:12)cation adopted in our estimation. Even more so, under ! = 0:6 and ! = 0:7, the restricted optimal simple rules feature parameter values that are close to the values retrieved in our estimation: the interest rate smoothing coefficient lies around 0:8 and the short-run coefficient assigned to price in(cid:13)ation lies between 0:1 and 0:2. If we interpret the estimated simple rules obtained in Section 4 (which basically coincide for the two models) as arising from optimal policy considerations under model uncertainty|where the policymaker intentionally excludes a direct response to wage in(cid:13)ation|U.S. policymakers assign probability 0:6 to 0:7 to the search and matching model being the true data-generating process. rules are indeed globally optimal. 33

6.3.2 Shock persistence and consumption habits Thus far, we have assumed that the markup shock is transitory and that households do not experience habit persistence in consumption. As our estimation strategy is silent on the parameterization of the markup shock, we also explore the possibility of a mildly persistent markup shock ((cid:26) = 0:2). In a second alternative, we investigate the impact of u habit persistence in consumption ((cid:22) = 0:6) on our results.24 Table 5, summarizes in Panel (a) the results for the case of mildly persistent markup shocks. Overall, the results are similar to those in Table 3, if not stronger. Under model averaging, the !-optimal simple rule is biased towards improving the outcomes in the sticky wage model: the welfare loss (measured in CEV) in the sticky wage model is smaller than in the search and matching model as long as ! (cid:20) 0:8 and negligible for ! (cid:20) 0:4 (compared to ! (cid:20) 0:2 in Table 3). The minmax strategy continues to pick the !-optimal simple rule for ! = 0. Our results also withstand the introduction of habit persistence as shown in Panel (b) of Table 5. Yet, in the presence of this real rigidity the bias of the optimal policy under model uncertainty towards the sticky wage model is slightly less pronounced. 6.3.3 Simple loss function Lastly, we return to the role of the policy objective function in obtaining robust monetary policy under model uncertainty. We replace the preferences of the policymaker under model averaging in equation (47) with ( ) ( ) ∑1 ∑1 Lsql((cid:2)) = ! E (cid:12)tLsql;s&m((cid:2)) +(1(cid:0)!) E (cid:12)tLsql;sw((cid:2)) (51) 0 t 0 t t=0 t=0 where Lsql;i((cid:2)) = ((cid:25)i) 2 + (xi) 2 is the value of the simple loss function for model i = t t t fs&m;swg. As in Section 5.3, but in contrast to our baseline case, the preferences of the policymaker are thus assumed to be independent of the reference models. Table 6 summarizes the optimal simple rules derived under the simple loss function. Although the parameterization of the !-optimal simple rule varies strongly with !, the induceddynamicsinthetwoeconomieshardlydifferacrosstherules(notshown)|similar 24The impulse response function matching in Section 4 (cid:12)nds no support for consumption habits in contrast to Christiano, Eichenbaum, and Trabandt (2013). In part, this result emerges as we exclude monetary policy shocks from the empirical analysis. Monetary policy shocks induce a pronounced hump-shaped response of consumption and output in the SVAR. One way to capture this feature is to introduce habit persistence in consumption. 34

to Figure 7 when we analyzed the robustness of the optimal targeting rules under this simple loss function ((cid:21) = 1). Further evidence along these lines stems from the obserx vation that the welfare losses in each model under the model-speci(cid:12)c true loss functions (as opposed to the simple loss function) are highly stable across values of !; the low CEVs computed for the sticky wage model indicate that all the rules resemble closely the optimal monetary policy in the sticky wage model (under the true loss function of the sticky wage model). The results in Table 6 could be viewed as evidence for the existence of a monetary policy rule that is (more or less) robust to the extent of model uncertainty. Yet, this conclusion is true only from the perspective of the policymaker whose preferences are described by the simple loss function. From the perspective of the representative household with preferences Ls&m and Lsw the results are suboptimal: the simple rules are unnecest t sarily biased towards the sticky wage model when ! is close to 1 compared to Table 3. To the extent that the welfare implications of mircofounded models are of interest, this result discourages the use of arbitrary loss functions for policy analysis. 7 Conclusion We contrast the optimal monetary policy recommendations in two models that the policymaker views as good approximations of the true data-generating process. The models differ with regard to the details of the labor market. The (cid:12)rst model follows the search and matching literature in assuming that workers have to search for jobs and (cid:12)rms post vacancies. For a worker to become employed and for the production of goods to occur, the worker has to be matched with a (cid:12)rm. The two parties then negotiate over the real wage which, as a result, may experience substantial inertia. In the second model, nominal wages are rigid as the result of staggered (nominal) wage contracts. We apply impulse response function matching to estimate key parameters of the models. While the two models produce very similar impulse responses for common variables under the estimated policy rules, the responses differ importantly when monetary policy is chosen optimally. Under sticky wages without indexation, the optimal policy induces little variation in nominal wages; the dynamics of the real wage are determined by the adjustment in prices.25 In the search and matching model, it is optimal to stabilize prices 25When wages are fully indexed, the optimal policy smoothes the difference between wage and past price in(cid:13)ation. 35

and to allow for substantial real wage adjustment brought about by changes in nominal wages. We(cid:12)llagapintheliteraturebyderivingtheoptimaltargetingruleforasearchand matching model|a single target criterion that seeks to implement the optimal monetary policy. We investigate the performance of each economy under the optimal targeting rule derived in the other model. In particular, the optimal targeting rule derived for the search and matching model is not robust, in the sense that it induces large welfare losses in the sticky wage model. While the optimal targeting rule derived for the sticky wage model also alters the dynamics in the search and matching model relative to the optimal monetary policy in that model, the welfare consequences are less dramatic. Given the models’ sensitivity to the optimal targeting rules, we compute optimal simple rules and simple targeting rules when the policymaker considers both the sticky wage and the search and matching model to be good candidates for the true data-generating process. Applying a model averaging and a minmax approach to obtain the optimal parameterization of each rule, we (cid:12)nd that unless the policymaker places high probability weight on the search and matching model the resulting policies are biased towards stabilizing wage in(cid:13)ation, the key feature of the optimal monetary policy in the sticky wage model. 36

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Table 1: Calibrated Parameters Description Parameter Search and Matching Sticky Wage discount factor (cid:12) 0.99 0.99 exogenous seperation rate (cid:26) 0.1 matching function share of unemployment (cid:16) 0.54 steady state in(cid:13)ation rate (cid:25)(cid:22) 1 1 Calvo price stickiness (cid:24)p 0.75 0.75 steady state price markup (cid:21)p 1.2 1.2 Calvo wage stickiness (cid:24)w - 0.75 steady state wage markup (cid:21)w - 1.2 invers consumption elasticity (cid:27) 1 1 inverse labor supply elasticity ϕ 2 2 hiring (cid:13)ow cost / output (cid:17) 0.0066 s steady state unemployment rate u~ 0.055 ss steady state vacancy (cid:12)lling rate q 0.7 ss steady state working hour h 1/3 1/3 ss Shock Process technology shock: AR (cid:26) 0.9999 0.9999 a markup shock: AR (cid:26) 0 0 u markup shock: Std (cid:27) 0.0104 0.0135 u Implied Deep Parameter Value hiring (cid:12)xed cost (cid:20)(cid:22) 0 hiring (cid:13)ow cost (cid:20)v 0.0154 unemployment bene(cid:12)t bu 0.1769 worker’s share of surplus (cid:24) 0.7438 matching efficiency (cid:31) 0.6625 scaling of working hour disutility ϕ 27.8940 27 0 Note: Table1summarizestheparametersandcalibrationtargetsfortheNKmodelwithsearchandmatchingfrictions and the NK model with sticky wages. 41

Table 2: Estimated Parameters Description EstimatedParameter Search StickyWage StickyWagewithIndexation interestratesmoothing (cid:26) 0.8555 0.8379 0.8895 R [0.0294] [0.0450] [0.0260] weightsonin(cid:13)ation (cid:26) 0.1445 0.1622 0.1105 (cid:25) [1.5e-05] [3.12e-05] [2.4e-05] stdtechnologyshock (cid:27) 0.0031 0.0033 [0.0031] a [0.0002] [0.0002] [0.0002] habitpersistence (cid:22) 0 0 0 [0.5148] [0.4394] [0.7734] replacementratio ru 0.5345 - - [0.0185] - priceindexation (cid:19)p 0 0 0 [0.3123] [0.3204] [0.2714] wageindexation (cid:19)w - - 1 - - [0.1656] MinimumDistanceEstimator Description Search StickyWage StickyWagewithIndexation criterionvalue(9variables) 124.8128 - criterionvalue(6variables) 99.6490 136.0783 77.2143 Note: The top panel of Table 2 summarizes the estimated parameters for the NK model with search and matching frictions and the NK model with and without wage indexation. The parameters are estimated using impulse response function matching under neutral technology shocks. The empirical impulse responses against which the performance of the theoretical models is assessed are taken from the SVAR estimation in Christiano, Eichenbaum, and Trabandt (2013). The numbers in the square bracket are the standard deviations of the estimates. The lower panel provides the value of the criterion function (30) at the minimum. 42

Table 3: Optimal Simple Rules and Optimal Simple Targeting Rules Panela: OptimalSimpleRules Coefficients WelfareLoss Approach Prior (cid:26) (cid:26) (cid:26)w (cid:26) Objective Ls&m((cid:2)(cid:3)) CEVs&m((cid:2)(cid:3)) Lsw((cid:2)(cid:3)) CEVsw((cid:2)(cid:3)) R (cid:25) (cid:25) x t t (0;1) 0 0 66.6844 2.3852 3.1047 2.1568 0.1094 3.1047 0.0010 (0:1;0:9) 0 0 61.5860 2.0019 3.0099 2.1566 0.1092 3.1048 0.0010 (0:2;0:8) 0 0 56.4038 1.6763 2.9151 2.1565 0.1091 3.1048 0.0011 (0:3;0:7) 0 0.6240 0.5226 0 2.8139 2.1028 0.0554 3.1186 0.0149 (0:4;0:6) 0 0.6368 0.5160 0 2.7123 2.1025 0.0551 3.1188 0.0151 ModelAveraging (0:5;0:5) 0 0.6558 0.5131 0 2.6106 2.1022 0.0548 3.1190 0.0153 (0:6;0:4) 0 0.7005 0.5158 0 2.5088 2.1014 0.0540 3.1199 0.0162 (0:7;0:3) 0 0.8135 0.5231 0 2.4067 2.0994 0.0520 3.1240 0.0202 (0:8;0:2) 0 1.1725 0.5245 0 2.3031 2.0920 0.0446 3.1475 0.0438 (0:9;0:1) 0.8177 0.8860 0 0 2.1870 2.0623 0.0149 3.3098 0.2061 (1;0) 0.9366 2.1197 0 0 2.0477 2.0477 0.0003 4.0851 0.9814 Minmax N.A. 0 0 66.6844 2.3852 3.1047 2.1568 0.1094 3.1047 0.0010 Panelb: OptimalSimpleTargetingRules Coefficients WelfareLoss Approach Prior (cid:26) Objective Ls&m((cid:2)(cid:3)) CEVs&m((cid:2)(cid:3)) Lsw((cid:2)(cid:3)) CEVsw((cid:2)(cid:3)) tr t t (0;1) 1.0000 3.1037 2.1602 0.1128 3.1037 0.0000 (0:1;0:9) 1.0847 3.0094 2.1601 0.1127 3.1037 0.0000 (0:2;0:8) 1.2319 2.9150 2.1600 0.1126 3.1038 0.0000 (0:3;0:7) 1.4166 2.8206 2.1598 0.1124 3.1038 0.0001 (0:4;0:6) 1.6773 2.7262 2.1595 0.1121 3.1040 0.0003 ModelAveraging (0:5;0:5) 2.0417 2.6317 2.1591 0.1117 3.1043 0.0006 (0:6;0:4) 2.6065 2.5371 2.1585 0.1111 3.1050 0.0013 (0:7;0:3) 3.5951 2.4423 2.1576 0.1102 3.1068 0.0031 (0:8;0:2) 5.7812 2.3471 2.1557 0.1083 3.1129 0.0091 (0:9;0:1) 15.0510 2.2499 2.1495 0.1021 3.1534 0.0497 (1;0) 6.1979e+05 2.0488 2.0488 0.0014 4.3001 1.1964 Minmax N.A. 1.0000 3.1037 2.1607 0.1133 3.1037 0.0000 Panelc: OptimalTargetingRules WelfareLoss OptimalTargetingRule Ls&m CEVs&m Lsw CEVsw t t s&m 2.0474 0.0000 4.4070 1.3033 sw 2.1607 0.1133 3.1037 0.0000 Note: Table 3 reports the optimal parameterizations of the simple rule in (49) in Panel (a) and simple targeting rule (50) in Panel (b) when the policymaker has two reference model, the NK model with search and matching frictions (s&m) and the NK model with sticky wages and no indexation (sw). The model is parameterized as in Tables 1 and 2. Under model averaging, the policymaker minimizes the expected loss given a probability distribution (prior). Undertheminmaxstrategy,thepolicymakersearchesforapolicyrulethatminimizesthemaximumloss. \Objective" measuresthevalueofthepolicymaker’sobjectivefunctionattheoptimum. ThecolumnsLs&m((cid:2)(cid:3))andLsw((cid:2)(cid:3))give t t the value of the expected loss in each model, the columns CEVs&m((cid:2)(cid:3)) and CEVsw((cid:2)(cid:3)) translate these losses into consumption equivalent variations. Panel (c) displays the welfare costs of implementing the optimal targeting rules in each model. 43

Table 4: Restricted Optimal Simple Rules RestrictedOptimalRule WelfareLoss ModelAveraging Prior (cid:26) (cid:26) (cid:26)w (cid:26) Objective Ls&m((cid:2)(cid:3)) Lsw((cid:2)(cid:3)) R (cid:25) (cid:25) x t t (0;1) 0 0 65.8388 2.3084 3.1047 2.1567 3.1047 (0:1;0:9) 0 0 60.6858 1.9798 3.0099 2.1566 3.1048 (0:2;0:8) 0 0 55.4967 1.6522 2.9151 2.1565 3.1048 (0:3;0:7) 0 0.6230 0.5235 0 2.8139 2.1028 3.1186 (0:4;0:6) 0 0.6369 0.5160 0 2.7123 2.1025 3.1188 CaseI: (0:5;0:5) 0 0.6561 0.5133 0 2.6106 2.1022 3.1190 Nointerestratesmoothing (0:6;0:4) 0 0.7006 0.5159 0 2.5088 2.1014 3.1199 (0:7;0:3) 0 0.8136 0.5231 0 2.4067 2.0994 3.1240 (0:8;0:2) 0 1.1724 0.5245 0 2.3031 2.0920 3.1475 (0:9;0:1) 0 2.2488 0.4734 0 2.1922 2.0730 3.2651 (1;0) 0 51.5995 5.2078 0.5170 2.0485 2.0485 20.0584 (0;1) 0 0 100 0 3.1059 2.1556 3.1059 (0:1;0:9) 0 0 100 0 3.0108 2.1556 3.1059 (0:2;0:8) 0 0 100 0 2.9158 2.1556 3.1059 (0:3;0:7) 0 0.6242 0.5224 0 2.8139 2.1028 3.1186 (0:4;0:6) 0 0.6389 0.5167 0 2.7123 2.1025 3.1188 CaseII: (0:5;0:5) 0 0.6564 0.5130 0 2.6106 2.1022 3.1190 Nooutputgap (0:6;0:4) 0 0.7012 0.5160 0 2.5088 2.1014 3.1200 (0:7;0:3) 0 0.8179 0.5238 0 2.4067 2.0993 3.1242 (0:8;0:2) 0 1.1725 0.5245 0 2.3031 2.0920 3.1475 (0:9;0:1) 0.8177 0.8860 0 0 2.1870 2.0623 3.3098 (1;0) 0.9366 2.1197 0 0 2.0477 2.0477 4.0851 (0;1) 0 0 66.0161 2.3594 3.1047 2.1567 3.1047 (0:1;0:9) 0 0 61.0171 1.9932 3.0099 2.1566 3.1048 (0:2;0:8) 0 0 56.0134 1.6655 2.9151 2.1565 3.1048 (0:3;0:7) 0 0 93.0510 2.1243 2.8203 2.1563 3.1048 (0:4;0:6) 0 0 93.5958 2.3938 2.7254 2.1564 3.1048 CaseIII: (0:5;0:5) 0.5709 0 5.0689 0.2221 2.6309 2.1554 3.1063 Nopricein(cid:13)ation (0:6;0:4) 0.3093 0 20.0193 0.4160 2.5357 2.1559 3.1055 (0:7;0:3) 0.4090 0 20.0150 0.2744 2.4407 2.1556 3.1060 (0:8;0:2) 0.5257 0 20.0071 0.1165 2.3457 2.1554 3.1067 (0:9;0:1) 0.6127 0 19.9914 0 2.2505 2.1553 3.1075 (1;0) 0.9269 0 19.3246 0 2.1552 2.1552 3.1076 (0;1) 0 1.0001 0 12.0434 3.1107 2.2814 3.1107 (0:1;0:9) 0.0587 0.9414 0 2.3607 3.0267 2.2598 3.1119 (0:2;0:8) 0.9900 0.0113 0 0.0011 2.9370 2.1308 3.1386 (0:3;0:7) 0.9900 0.0112 0 7.5200e04 2.8357 2.1232 3.1410 (0:4;0:6) 0.9900 0.0111 0 5.1800e04 2.7335 2.1178 3.1439 CaseIV: (0:5;0:5) 0.9900 0.0110 0 3.1900e-04 2.6304 2.1130 3.1478 Nowagein(cid:13)ation (0:6;0:4) 0.9079 0.1172 0 0 2.5243 2.1024 3.1572 (0:7;0:3) 0.8947 0.2260 0 0 2.4176 2.0912 3.1790 (0:8;0:2) 0.8669 0.4487 0 0 2.3061 2.0769 3.2226 (0:9;0:1) 0.8177 0.8861 0 0 2.1870 2.0623 3.3099 (1;0) 0.9366 2.1204 0 0 2.0477 2.0477 4.0856 Note: Table4reportstheoptimalsimplerulesunderthemodelaveragingapproachsimilartoTable3whenrestricting the rule not to respond to one of the variables in (49) at the time. See also footnote Table 3. 44

Table 5: Sensitivity of Optimal Simple Rules Panela: persistentmarkupshock(cid:26) =0:2 u OptimalSimpleRule WelfareLoss Approach Prior (cid:26) (cid:26) (cid:26)w (cid:26) Objective Ls&m((cid:2)(cid:3)) CEVs&m((cid:2)(cid:3)) Lsw((cid:2)(cid:3)) CEVsw((cid:2)(cid:3)) R (cid:25) (cid:25) x t t (0;1) 0 0 66.6812 1.6100 3.2524 2.2305 0.1520 3.2524 0.0014 (0:1;0:9) 0 0 62.1310 1.3100 3.1501 2.2302 0.1518 3.2524 0.0014 (0:2;0:8) 0 0 57.0042 1.0663 3.0479 2.2300 0.1516 3.2524 0.0015 (0:3;0:7) 0 0 51.9820 0.8663 2.9457 2.2299 0.1514 3.2524 0.0015 (0:4;0:6) 0 0 47.4445 0.6725 2.8434 2.2297 0.1513 3.2525 0.0016 ModelAveraging (0:5;0:5) 0 0.6436 0.5084 0 2.7087 2.1448 0.0663 3.2726 0.0217 (0:6;0:4) 0 0.6743 0.5110 0 2.5958 2.1441 0.0656 3.2735 0.0226 (0:7;0:3) 0 0.7419 0.5165 0 2.4827 2.1424 0.0639 3.2768 0.0259 (0:8;0:2) 0 0.9590 0.5250 0 2.3683 2.1361 0.0576 3.2973 0.0463 (0:9;0:1) 0.8744 0.5682 0 0 2.2419 2.0993 0.0208 3.5253 0.2744 (1;0) 0.9789 1.4644 0 0 2.0785 2.0785 0.0000 4.7228 1.4719 Minmax N.A. 0 0 66.6812 1.6100 3.2524 2.2305 0.1520 3.2524 0.0014 Panelb: habitpersistence(cid:22)=0:6 OptimalSimpleRule WelfareLoss Approach Prior (cid:26) (cid:26) (cid:26)w (cid:26) Objective Ls&m((cid:2)(cid:3)) CEVs&m((cid:2)(cid:3)) Lsw((cid:2)(cid:3)) CEVsw((cid:2)(cid:3)) R (cid:25) (cid:25) x t t (0;1) 0 0 68.1424 1.4345 3.1519 2.1796 0.1347 3.1519 0.0007 (0:1;0:9) 0 0.6175 1.3949 0 3.0546 2.1689 0.1240 3.1530 0.0018 (0:2;0:8) 0 0.6561 1.3497 0 2.9561 2.1678 0.1229 3.1532 0.0020 (0:3;0:7) 0 0.7064 1.3276 0 2.8575 2.1664 0.1215 3.1537 0.0025 (0:4;0:6) 0 0.7754 1.3139 0 2.7586 2.1646 0.1197 3.1547 0.0035 ModelAveraging (0:5;0:5) 0 0.8786 1.3099 0 2.6594 2.1621 0.1172 3.1568 0.0056 (0:6;0:4) 0.9882 1.4557 0 0.2369 2.5576 2.1205 0.0756 3.2132 0.0620 (0:7;0:3) 0.9198 2.4368 0 0.3228 2.4461 2.1042 0.0593 3.2438 0.0926 (0:8;0:2) 0.8178 4.0965 0 0.4114 2.3289 2.0867 0.0418 3.2976 0.1464 (0:9;0:1) 0.6763 6.9621 0 0.4346 2.2020 2.0677 0.0228 3.4110 0.2598 (1;0) 0 41.4453 1.6503 0 2.0465 2.0465 0.0016 5.0518 1.9006 Minmax N.A. 0 0 68.1424 1.4345 3.1519 2.1796 0.1347 3.1519 0.0007 Note: Table5reportstheoptimalparameterizationsofthesimplerulein(49)whenthepolicymakerhastworeference model,theNKmodelwithsearchandmatchingfrictions(s&m)andtheNKmodelwithstickywagesandnoindexation (sw)fortwoalternativespeci(cid:12)cationsofthemodel. ThemodelisparameterizedasinTables1and2,withtheexception that we raise the persistence of the price markup shock from zero to (cid:26) =0:2 in Panel (a), and we raise the degree of u habitpersistencefromzeroto(cid:22)=0:6(Panelb). Undermodelaveraging,thepolicymakerminimizestheexpectedloss given a probability distribution (prior). Under the minmax strategy, the policymaker searches for a policy rule that minimizesthemaximumloss. \Objective"measuresthevalueofthepolicymaker’sobjectivefunctionattheoptimum. The columns Ls&m((cid:2)(cid:3)) and Lsw((cid:2)(cid:3)) give the value of the expected loss in each model, the columns CEVs&m((cid:2)(cid:3)) t t and CEVsw((cid:2)(cid:3)) translate these losses into consumption equivalent variations. 45

Table 6: Optimal Simple Rules under a Simple Loss Function Lsql t OptimalSimpleRule WelfareLoss Approach Prior (cid:26) (cid:26) (cid:26)w (cid:26) Objective Ls&m((cid:2)(cid:3)) CEVs&m((cid:2)(cid:3)) Lsw((cid:2)(cid:3)) CEVsw((cid:2)(cid:3)) R (cid:25) (cid:25) x t t (0;1) 0 0 0 41.8350 0.0120 2.2892 0.2418 3.1106 0.0069 (0:1;0:9) 0.4376 0 4.0245 20.1245 0.0118 2.2708 0.2234 3.1104 0.0067 (0:2;0:8) 0.9269 0 4.1139 20.1418 0.0117 2.2703 0.2229 3.1107 0.0070 (0:3;0:7) 0.9999 0 4.0938 19.9256 0.0116 2.2702 0.2228 3.1107 0.0070 (0:4;0:6) 0.9999 0 2.1613 10.2326 0.0114 2.2697 0.2223 3.1108 0.0071 ModelAveraging (0:5;0:5) 0.9999 0 1.3662 6.2492 0.0113 2.2691 0.2217 3.1109 0.0072 (0:6;0:4) 0.9999 0 0.8275 3.5735 0.0112 2.2681 0.2207 3.1111 0.0074 (0:7;0:3) 0.9999 0 0.4872 1.9215 0.0110 2.2665 0.2191 3.1114 0.0077 (0:8;0:2) 0.9999 0 0.2646 0.9015 0.0109 2.2636 0.2162 3.1123 0.0086 (0:9;0:1) 0.9999 0 0.1128 0.3014 0.0107 2.2582 0.2108 3.1148 0.0111 (1;0) 0.9999 0.05 0 0.1003 0.0104 2.2501 0.2027 3.1236 0.0199 Minmax N.A. 0 0 0 41.8350 0.0120 2.2892 0.2418 3.1106 0.0069 Note: Table6reportstheoptimalparameterizationsofthesimplerulein(49)whenthepolicymakerhastworeference model,theNKmodelwithsearchandmatchingfrictions(s&m)andtheNKmodelwithstickywagesandnoindexation (sw). In contrast to Table 3, the policymaker’s preferences are described by the simple loss function of the form L = (cid:25)2 + x2 in both models. The model is parameterized as in Tables 1 and 2. Under model averaging, the t t t policymaker minimizes the expected loss given a probability distribution (prior). Under the minmax strategy, the policymaker searches for a policy rule that minimizes the maximum loss. \Objective" measures the value of the policymaker’s objective function at the optimum, i.e., the simple loss function. The columns Ls&m((cid:2)(cid:3)) and Lsw((cid:2)(cid:3)) t t give the values of the expected loss in each model from the perspective of the representative household, the columns CEVs&m((cid:2)(cid:3)) and CEVsw((cid:2)(cid:3)) translate these losses into consumption equivalent variations. 46

Figure 1: Impulse response function matching under neutral technology shock Note: Figure 1 depicts the impulse responses to a neutral technology shock in the search and matching model (blue) and the sticky wage model (red). The solid black lines show the point estimates of the empirical impulse responses along with the 90% con(cid:12)dence interval, the grey shaded area. In(cid:13)ation rates and the federal fund rate are annualized. 47

Figure 2: Impulse response function matching under neutral technology shock with wage indexation in the sticky wage model Note: Figure 2 depicts the impulse responses to a neutral technology shock in the search and matching model (blue) and the sticky wage model (red). The solid black lines show the point estimates of the empirical impulse responses along with the 90% con(cid:12)dence interval, the grey shaded area. In(cid:13)ation rates and the federal fund rate are annualized. 48

Figure 3: Targeting rules in the search and matching model: neutral technology shock opt. tar. rule s&m opt. tar. rule sw 1. Individual Hour Worked 2. Output 3. Consumption 0.4 -0.01 0.32 -0.02 0.35 0.3 -0.03 0.28 -0.04 0.3 -0.05 0.26 -0.06 0.24 0.25 -0.07 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 4. Price Inflation 5. Wage Inflation 6. Real Wage 0 1.2 0.3 -0.05 1 0.25 -0.1 0.8 -0.15 0.6 0.2 -0.2 0.4 0.15 -0.25 0.2 0 0.1 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 7. Marginal Cost 8. Nominal Interest Rate 9. Job Finding Rate 0 0 0.4 -0.05 -0.02 -0.04 0.3 -0.1 -0.06 0.2 -0.08 -0.15 -0.1 0.1 -0.2 -0.12 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 10. Unemployment Rate 11. Employment 12. Total Vacancy 1.6 0.1 -0.02 1.4 0.08 1.2 -0.04 1 0.06 0.8 -0.06 0.04 0.6 -0.08 0.4 0.02 0.2 -0.1 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Note: Figure 3 plots the impulse responses in the search and matching model to a neutral technology shock when policy follows the optimal targeting rule from the search and matching model (purple) and the sticky wage model (yellow). 49

Figure 4: Targeting rules in the search and matching model: price markup shock opt. tar. rule s&m opt. tar. rule sw 1. Individual Hour Worked 2. Output 3. Consumption -0.2 -0.5 -0.4 -0.5 -1 -0.6 -0.8 -1.5 -1 -1 -2 -1.5 -1.2 -2.5 -1.4 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 4. Price Inflation 5. Wage Inflation 6. Real Wage 3 10 -0.5 2.5 5 -1 2 -1.5 1.5 0 -2 1 -5 -2.5 0.5 -3 0 -10 -3.5 -0.5 -4 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 7. Marginal Cost 8. Nominal Interest Rate 9. Job Finding Rate -1 3 -2 -2 2 -4 -3 1 -6 -4 0 -8 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 10. Unemployment Rate 11. Employment 12. Total Vacancy 1.4 0 1.2 -0.2 -5 1 -0.4 -10 -0.6 0.8 -15 -0.8 0.6 -20 -1 0.4 -1.2 -25 0.2 -1.4 -30 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Note: Figure 4 plots the impulse responses in the search and matching model to a price markup shock when policy follows the optimal targeting rule from the search and matching model (purple) and the sticky wage model (yellow). 50

Figure 5: Targeting rules in the sticky wage model: neutral technology shock opt. tar. rule sw opt. tar. rule s&m 1. Hour Worked 2. Output 0.4 0.7 0.3 0.6 0.2 0.5 0.1 0.4 0 0 5 10 15 20 0 5 10 15 20 3. Consumption 4. Nominal Interest Rate 0.3 0.7 0.2 0.6 0.1 0.5 0 -0.1 0.4 -0.2 0 5 10 15 20 0 5 10 15 20 5. Price Inflation 6. Wage Inflation 0.25 0 -0.05 0.2 -0.1 0.15 -0.15 0.1 -0.2 -0.25 0.05 -0.3 0 0 5 10 15 20 0 5 10 15 20 7. Real Wage 8. Marginal Cost 0 0.3 -0.05 0.25 -0.1 0.2 -0.15 0.15 -0.2 0.1 0 5 10 15 20 0 5 10 15 20 Note: Figure5plotstheimpulseresponsesinthestickywagemodeltoaneutraltechnologyshockwhenpolicyfollows the optimal targeting rule from the sticky wage model (blue) and the search and matching model (yellow). 51

Figure 6: Targeting rules in the sticky wage model: price markup shock opt. tar. rule sw opt. tar. rule s&m 1. Hour Worked 2. Output 0 0 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -6 -6 0 5 10 15 20 0 5 10 15 20 3. Consumption 4. Nominal Interest Rate 0 8 -1 6 -2 -3 4 -4 2 -5 0 -6 0 5 10 15 20 0 5 10 15 20 5. Price Inflation 6. Wage Inflation 4 3 0 2 -0.2 1 -0.4 0 -0.6 -1 -0.8 0 5 10 15 20 0 5 10 15 20 7. Real Wage 8. Marginal Cost 0 0 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -1 0 5 10 15 20 0 5 10 15 20 Note: Figure 6 plots the impulse responses in the sticky wage model to a price markup shock when policy follows the optimal targeting rule from the sticky wage model (blue) and the search and matching model (yellow). 52

Figure 7: Targeting rules with simple loss function: price markup shock Search and Matching Model: Simple Loss Function with λ = λ∗ x Price Inflation Wage Inflation Output 4 3 -0.2 tar. rule s&m 2 -0.4 tar. rule sw 2 -0.6 0 -0.8 1 -2 -1 -4 -1.2 0 -1.4 -6 -1.6 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Sticky Wage Model: Simple Loss Function with λ = λ∗ x Price Inflation Wage Inflation Output 4 0 0 3 -0.5 tar. rule sw 2 tar. rule s&m -0.1 -1 -0.2 -1.5 1 -0.3 -2 0 -2.5 -0.4 -1 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Search and Matching Model: Simple Loss Function with λ = 1 x Price Inflation Wage Inflation Output 4 3.5 -0.02 3 tar. rule s&m 3 tar. rule sw 2.5 -0.04 2 2 -0.06 1.5 1 1 -0.08 0.5 -0.1 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Sticky Wage Model: Simple Loss Function with λ = 1 x Price Inflation Wage Inflation Output 4 0.08 -0.02 3 tar. rule sw tar. rule s&m 0.06 -0.04 2 -0.06 0.04 1 -0.08 0.02 -0.1 0 0 -0.12 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 ( ) Note: Figure 7 compares the performance of optimal targeting rules derived from the loss function (cid:25)2+(cid:21) x2 for t x t both the search and matching model and the sticky wage model in response to a price markup shock. In the upper six (cid:3) panels, it is (cid:21) =(cid:21) =0:0429; in the lower six panels it is (cid:21) =1. x x 53

Figure 8: Impulse responses under optimal simple rules Search and matching model under technology shock Price Inflation Wage Inflation Output 0 0.4 1.2 -0.05 1 0.35 -0.1 0.8 -0.15 0.6 0.3 -0.2 0.4 0.2 -0.25 0.25 0 -0.3 0 5 10 15 0 5 10 15 0 5 10 15 Search and matching model under price markup shock Price Inflation Wage Inflation Output 10 3 2.5 -0.5 5 2 -1 1.5 0 1 -1.5 -5 0.5 -2 0 -10 -2.5 -0.5 0 5 10 15 0 5 10 15 0 5 10 15 Sticky wage model under technology shock Price Inflation Wage Inflation Output 0 0.3 1 -0.05 0.25 -0.1 0.2 0.8 -0.15 0.15 0.6 -0.2 0.1 -0.25 0.05 0.4 -0.3 0 0 5 10 15 0 5 10 15 0 5 10 15 Sticky wage model under price markup shock Price Inflation Wage Inflation Output 4 0.2 0 3 0 -1 2 -2 Prior (0,1) -0.2 Prior (0.2,0.8) -3 1 Prior (0.3,0.7) Prior (0.8,0.2) -0.4 -4 Prior (0.9,0.1) 0 Prior (1,0) -5 -1 -0.6 0 5 10 15 0 5 10 15 0 5 10 15 Note: Figure 8 compares the performance of the search and matching and the sticky wage model under !-optimal simple rules (0;1), (0:2;0:8), (0:3;0:7), (0:8;0:2), (0:9;0:1), and (1;0) for the neutral technology shock and the price markup shock. 54