Frictional Labor Markets, Bargaining Wedges, and Optimal Tax-Rate Volatility

Frictional Labor Markets, Bargaining Wedges, and Optimal Tax-Rate Volatility David M. Arseneau Federal Reserve Board Sanjay K. Chugh University of Maryland First Draft: January 2008 This Draft: April 18, 2008 Abstract We re-examine the optimality of tax smoothing from the point of view of frictional labor markets. In frictional labor markets, unlike in Walrasian labor markets, wages can play various roles and can be determined in various ways. Our first main result is that if wages are determined by ex-post Nash bargaining, tax smoothing in the face of business cycle shocks is not optimal. Quantitatively, in what has emerged as the standard DSGE labor search and bargaining model, the optimal labor tax rate is one to two orders of magnitude more volatile than in standard Ramsey models. Tax volatility partially offsets cyclical wedges due to bargaining. Our second main result is that if wages are instead posted before workers and firms meet and search is directed by wages that is, labor markets are governed by competitive search equilibrium the optimality of tax smoothing is restored because bargaining wedges are absent. Our results are robust to a number of alternative features of the environment that govern the the severity of search frictions. Thus, our main conclusion is that one has to accept a largely Walrasian, or competitive, view of wage determination in order to accept the prescription of tax smoothing. Keywords: labor market frictions, optimal taxation JEL Classification: E24, E50, E62, E63 We thank seminar participants at the University of Delaware and the Federal Reserve Board s International Finance Workshop for comments, and especially David Stockman and Andrea Raffo for helpful discussions. views expressed here are solely those of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. An early draft of this paper circulated under the title Tax Smoothing May Not Be As Important As You Think. E-mail address: david.m.arseneau@frb.gov. E-mail address: chughs@econ.umd.edu. The 1

Contents 1 Introduction 4 2 Baseline Model 7 2.1 Production......................................... 7 2.2 Households......................................... 9 2.3 Wage Bargaining...................................... 10 2.4 Government......................................... 11 2.5 Matching Technology.................................... 12 2.6 Private-Sector Equilibrium................................ 12 3 Ramsey Problem in Baseline Model 13 4 Optimal Taxation in Baseline Model 14 4.1 Parameterization and Solution Strategy......................... 14 4.2 Main Result: The Optimality of Tax Volatility..................... 15 4.3 Recovering Tax Smoothing in the Baseline Model.................... 19 5 Alternative Views of the Labor Market 21 5.1 Instantaneous Hiring (Model 2).............................. 22 5.1.1 Modifications of the Model............................ 22 5.1.2 Optimal Taxation................................. 23 5.2 Endogenous Labor Force Participation.......................... 25 5.2.1 Baseline Timing Assumption (Model 3)..................... 25 5.2.2 Instantaneous Hiring (Model 4).......................... 27 5.2.3 Optimal Taxation................................. 27 5.3 Competitive Search Equilibrium (Model 5)....................... 31 5.3.1 Modifications of the Model............................ 31 5.3.2 Static Tax Wedge................................. 32 5.3.3 Optimal Taxation................................. 33 6 Summary and Discussion 35 7 Conclusion 39 A Nash Bargaining in Model with Standard Timing 41 B Nash Bargaining in Model with Instantaneous Hiring 43 2

C Elasticity of Market Tightness to Labor Tax Rate 45 D Derivation of Implementability Constraint 47 E Dynamic Bargaining Power Effect 48 3

1 Introduction We re-examine the optimality of tax smoothing from the point of view of frictional labor markets. Since Barro s (1979) partial-equilibrium intuition, Lucas and Stokey s (1983) general-equilibrium analysis, and continuing through to today s quantitative DSGE models used to study optimal fiscal policy, the prescription that governments ought to hold labor tax rates virtually constant in the face of aggregate shocks is well-known to macroeconomists. We show that this cornerstone optimal-policy prescription and the intuition underlying it depend crucially on a Walrasian view of labor markets. If one instead takes what has emerged as the standard search and bargaining view of labor markets, tax-smoothing ceases to be important for empirically-relevant labor market parameters. If wages are determined in bilateral bargaining after workers and firms meet, not only is tax-smoothing unimportant, but purposeful tax-rate volatility is actually welfare-enhancing by partially offsetting cyclical bargaining-induced wedges. Our baseline search and bargaining environment is identical to the one that has come into widespread use in recent DSGE modeling efforts. We quantitatively demonstrate the optimality of labor tax-rate volatility in this environment. In an effort to recover tax smoothing, we then incrementally alter the environment in a number of ways, each of which in principle reduces the severity of search and bargaining frictions. In particular, we change the timing of labor market flows, introduce a labor-force participation margin, and allow for wages to be determined in a competitive fashion, rather than through ex-post bilateral bargaining. Changing the timing of labor market flows and allowing a labor-force participation choice each, as well as together, only modestly reduces the degree of optimal tax-rate volatility. However, allowing for competitive determination of wages using Moen s (1997) concept of competitive search equilibrium, in which ex-ante posted wages direct search activity is a critical change in market structure that reinstates the optimality of tax smoothing. For the competitive search economy, we show analytically that it is only labor taxes that create relatively-standard static wedges between marginal rates of substitution between consumption and leisure and corresponding marginal rates of transformation. The usual reasons for tax smoothing that wedges between this MRS and MRT should be kept (nearly) constant over time then apply. As a methodological by-product of our analysis, we are, to the best of our knowledge, the first to provide a simple MRT interpretation for a labor-search model, one that differs from the notion of MRT in standard neoclassical model of the labor market. If wages are determined by ex-post bilateral bargaining, proportional labor taxes affect the labor market in a dramatically different way. We identify two distinct roles played by the labor tax in our bargaining environments: the usual static role, in which a positive labor tax wedge is needed period-by-period in order to raise revenue for the government, and a novel dynamic role, in which 4

changes in tax rates affect private-sector search activity. Thus, for any given level of the labor tax rate in period t, the change in the tax rate is a distinct lever that the Ramsey government can use as a way of directing labor-market outcomes over the business cycle. This dynamic role of labor taxes which we refer to as a dynamic bargaining power effect only arises in the environment with bilateral bargaining and can only be revealed in an explicitly dynamic analysis. The key difference between the two market structures, which drives the stark difference in optimal-policy prescriptions, is the fundamental forces underlying wage determination. In the competitive search economy, posted wages allow unemployed individuals to optimally direct their search activity in the labor market, which in turn generates competition amongst wage-setting firms. This competition ensures private-sector labor-market activity is efficient up to the static tax wedge over the business cycle, much like in standard Walrasian-based Ramsey models; hence the prescription to smooth tax rates over time. In contrast, if wages are determined by ex-post bilateral bargaining, such competitive forces are absent. We show that ex-post bargaining in and of itself introduces a time-varying wedge between the notions of MRS and MRT that we develop. One interpretation of these bargaining-induced wedges is the holdup problems inherent in the bilateral monopoly of ex-post bargaining, a feature of bargaining that Caballero (2007) emphasizes. A related interpretation of these wedges is that they reflect incomplete-contracting problems. Regardless of the interpretation of the wedge, in order to partially offset these cyclical bargaining-induced wedges, the Ramsey government exploits the dynamic bargaining power effect we identify, and optimal labor tax rates are volatile. Our results show that bargaining frictions can dramatically alter conventional thinking regarding one of the most basic optimal-policy prescriptions. One has to accept a competitive view of wage determination one devoid of bargaining frictions in order to accept the prescription of tax smoothing. The conclusion that the optimality of tax smoothing depends critically on the market structure that determines real wages in and of itself may not be surprising. What is surprising are the quantitative magnitudes we find. In our baseline search and bargaining model, optimal tax rate volatility is between 1.5 percent and 7 percent for empirically-relevant calibrations. For comparison, the Ramsey literature s conventional tax-smoothing result entails optimal tax rate volatility near 0.1 percent or less for example, see the overview in Chari and Kehoe (1999). These numbers all refer to the standard deviation of the level of the Ramsey-optimal labor tax rate, and all of them refer to fluctuations around average tax rates in the range 20-60 percent. Thus, labor tax rates are one to two orders of magnitude more volatile in our baseline search and bargaining model than in simple Walrasian-based Ramsey environments. Our results suggest that optimal-policy theorists perhaps ought not to be so enchanted with the cornerstone Ramsey dynamic tax-smoothing result. 5

Werning (2007) recently shed new analytical insight on the cornerstone Ramsey dynamic taxsmoothing result. By bringing distributional issues between different types of consumers to the foreground, Werning (2007) connects results in the Ramsey literature to insights from the new dynamic public finance literature, which emphasizes tradeoffs between distributions and efficiency. Just like the standard Ramsey intuition for tax smoothing, however, Werning s (2007) analysis and insights depend as, indeed, do many results in the new dynamic public finance literature on fundamentally spot views of all markets. Labor markets indeed, perhaps goods markets and capital markets as well seem to be ill-characterized as spot markets. Given the broad macro literature s recent embrace of labor search models, we think it worthwhile to begin exploring how such models might change conventional thinking regarding optimal policy. Our conclusions that the details of market structure and the precise notion of equilibrium in search models can matter for issues in macroeconomics connect with those of Rocheteau and Wright (2005). Rocheteau and Wright (2005) show that in search-based monetary models, the precise notion of equilibrium can matter a great deal for efficiency. Like us, they show that whether frictional markets are best characterized by a bargaining equilibrium or a competitive search equilibrium can have quite important implications for policy, in their case monetary policy. They do not study Ramsey-optimal policy or consider stochastic environments, but some of the essence of our results echoes their insights. Finally, we also note that tax volatility in our model is not driven by any incompleteness of government debt markets, which is a well-understood point in Ramsey models since Aiyagari, Marcet, Marimon, and Sargent (2002). Rather, our point of departure from a standard Ramsey setup is that we model labor trades as governed by primitive search and matching frictions. While search-based DSGE models have become commonplace in recent years, their ability to shed new insights on optimal fiscal policy has not yet been much explored. 1 Our model features no capital accumulation, in order to highlight the dynamics of labor taxes. We see no reason why our basic intuition and results would not extend to the classic case of Ramsey taxation of both labor and capital income. The rest of our work is organized as follows. Section 2 lays out the baseline model, which hews very closely to the recent vintage of DSGE labor-search models, in which we study optimal fiscal policy. Section 3 lays out the Ramsey problem in our baseline model. Section 4 presents our main result that tax smoothing is unimportant in the presence of search frictions in the labor market. The rest of our analysis, in Section 5, explores modifications to the baseline search model that may be important in recovering the optimality of tax smoothing; we in turn allow for match formation 1 To our knowledge, the only work investigating aspects of optimal fiscal policy in labor-search dynamic general equilibrium models is Domeij (2005) and our own previous work, Arseneau and Chugh (2006, 2007); in none of these is the focus explicitly on tax smoothing. 6

to occur instantly, an endogenous labor force participation decision, and for wages to be determined in a competitive manner, rather than through bargaining. Section 6 provides several alternative ways to understand our results, and Section 7 concludes. 2 Baseline Model We establish our main result in a simple model featuring labor search and matching frictions. As many other recent studies have done, we embed the Pissarides (2000) textbook search model into a dynamic stochastic general equilibrium framework. We present in turn the choice problems of the representative firm, the representative household, the determination of wage payments, the actions of the government, and the definition of equilibrium. As an aid to understanding the timing of events in our model, Figure 1 accompanies the ensuing description of the environment. 2.1 Production The production side of the economy features a representative firm that must open vacancies, which entail costs, in order to hire workers and produce. The representative firm is large in the sense that it operates many jobs and consequently has many individual workers attached to it through those jobs. The firm requires only labor to produce its output. The firm must engage in costly search for a worker to fill each of its job openings. In each job k that will produce output, the worker and firm bargain over the pre-tax real wage w kt paid in that position. Output of any job k is given by y kt = z t, which is subject to a common technology realization z t. Any two jobs k a and k b at the firm are identical, so from here on we suppress the second subscript and denote by w t the real wage in any job, and so on. Total output of the firm thus depends on the technology realization and the measure of matches n f t that produce, y t = z t n f t. (1) The total real wage bill of the firm is the sum of wages paid at all of its positions, n f t w t. The firm begins period t with employment stock n f t. Its future employment stock depends on its current choices as well as the random matching process described below. With probability k f (θ), taken as given by the firm, a vacancy will be filled by a worker. Labor-market tightness is θ v/u, where u denotes the number of individuals searching for jobs. Matching probabilities depend only on market tightness given the Cobb-Douglas matching function we will assume. Wages are determined through bargaining, as we describe below. In the firm s profit maximization problem, the wage-setting protocol is taken as given. 2 The firm thus chooses vacancies to post 2 This assumption is without loss of generality in the standard Pissarides-type model because even if the firm 7

v t and future employment stock n f t+1 to maximize discounted nominal profits starting at date t, { ]} E t β t Ξ t 0 [z t n f t w tn f t γv t. (2) t=0 where Ξ t 0 is the period-0 value to the representative household of period-t goods, which we assume the firm uses to discount profit flows because households are the ultimate owners of firms. 3 In period t, the firm s problem is thus to choose v t and n f t+1 to maximize (2) subject to a sequence of perceived laws of motion for its employment level, n f t+1 = (1 ρx )(n f t + v tk f (θ t )). (3) Firms incur the real cost γ for each vacancy created, and job separation occurs with exogenous fixed probability ρ x. Note the timing of events embodied in the law of motion (3) and shown in Figure 1. Period t begins with a stock n t that is used for period-t production. Following production, a measure ρ x of jobs end and the labor matching process occurs. A measure ρ x of newly-formed matches are also destroyed before ever becoming productive, thus determining the new employment stock n t+1. The firm s first-order conditions with respect to v t and n f t+1 yield the standard job-creation condition ( )] γ k f (θ t ) = (1 γ ρx )E t [Ξ t+1 t z t+1 w t+1 + k f, (4) (θ t+1 ) where Ξ t+1 t Ξ t+1 0 /Ξ t 0 is the household discount factor (again, technically, the real interest rate) between period t and t + 1. The job-creation condition states that at the optimal choice, the vacancy-creation cost incurred by the firm is equated to the discounted expected value of profits from a match. Profits from a match take into account future marginal revenue product from the match, the wage cost of the match, and the asset value of having a pre-existing relationship with an employee in period t + 1. This condition is a free-entry condition in the creation of vacancies and is a standard equilibrium condition in a labor search and matching model. believed it could opportunistically manipulate the wages it paid by under- or over-hiring, the fact that labor s marginal product is independent of total employment prevents such opportunistic manipulation of wage-bargaining sets. Thus, in the standard exogenous-productivity Pissarides model, holdup problems are in principle present, but there is no lever by which firms can strategically react to them. If firm output exhibited diminishing marginal product in its total employment level, then the firm would have an incentive to over-hire. See, for example, Cahuc, Marque, and Wasmer (2008) and Krause and Lubik (2006). 3 Technically, of course, it is the real interest rate with which firms discount profits, and in equilibrium the real interest rate between time zero and time t is measured by Ξ t 0. equilibrium result too early, we skip this intermediate level of notation and structure. Because there will be no confusion using this 8

Fraction ρ x of new matches separate yields Bargaining occurs Production (using n (i.e., asset values t employees), goods Search and matching n defined here) t markets and asset in labor market n t+1 = (1-ρ x )(n t + m(u t, v t )) markets meet and clear Period t-1 Period t Period t+1 Employment Aggregate separation occurs state (ρ x n realized t employees separate) Figure 1: Timing of events in baseline model. 2.2 Households There is a representative household in the economy. Each household consists of a continuum of measure one of family members. In our baseline model, there is no labor-force participation decision, hence each member of the household either works during a given time period or is unemployed and searching for a job. There is a measure n h t of employed individuals in the household and a measure u h t = 1 n h t of unemployed individuals. We assume that total household income is divided evenly amongst all individuals, so each individual has the same consumption. The representative household thus maximizes expected lifetime discounted utility E 0 β t u(c t ) (5) t=0 subject to the sequence of flow budget constraints c t + b t = n h t (1 τt n )w t + (1 n h t )χ + R t b t 1 + d t, (6) where χ is the flow of unemployment benefits each unemployed individual receives, (1 τt n )w t is the after-tax wage rate each employed individual earns, d t is firm dividends received lump-sum by the household, and b t 1 is the household s holdings of a state-contingent one-period real government bond at the end of period t 1, which has gross payoff R t at the beginning of period t. Important to note is that the government is able to issue fully state-contingent debt; thus, none of our optimal policy results will be driven by an inability on the part of the government to use debt as a shock absorber. Incompleteness of government debt markets can be an important driver of results in 9

Ramsey models see, for example, Aiyagari, Marcet, Marimon, and Sargent (2002). Finally, because this is a Ramsey-taxation model, no lump-sum taxes or transfers exist. With no labor-force participation decision, the only equilibrium condition stemming from household optimization is the usual consumption-savings condition, u [ (c t ) = E t βu ] (c t+1 )R t+1. (7) In equilibrium, firms discount profit flows according to Ξ t+1 t = βu (c t+1 )/u (c t ). 2.3 Wage Bargaining We assume period-by-period Nash negotiations over the wage payment. Important to note in Figure 1 is the point in time at which bargaining occurs. As is standard in this class of models, period-t asset values are defined at the point in time at which period-t bargaining occurs, which is after labor-market matching (of period t 1) has taken place. We think this timing is natural because negotiations over the wage can take place only after the two parties meet. With ex-post wage negotiation, each vacancy opening in which a firm invests is thus undertaken without perfect knowledge of the wage that the firm will negotiate with the worker that fills that position. The same is true for search activity on the part of individuals: while engaged in the search process, individuals do not know the wage they might receive if search is successful. Caballero (2007) interprets the inefficiencies that arise in such an environment as holdup problems. In their survey of labor-search theory, Mortensen and Pissarides (1999) interpret these inefficiencies as due to incomplete-contracting problems. ex-post bargaining arise naturally in search environments. 4 Regardless of the precise interpretation, inefficiencies due to Here, we simply present the Nash wage-bargaining outcome; a detailed derivation is presented in Appendix A. Assuming that η is a worker s Nash bargaining power and 1 η a firm s Nash bargaining power, the Nash wage outcome is given by [ w t = η z t + γ ] ( (1 η)χ (1 ρ x )(1 k h ) [ ( (θ t )) k f + (θ t ) 1 τt n η 1 τt n E t Ξ t+1 t (1 τt+1) n z t+1 w t+1 + γ k f (θ t+1 ) (8) The wage outcome (8) is a variation of the typical Nash wage solution in labor-search models. The first term on the right-hand-side of (8) is standard: it represents the firm s flow return from production and the asset value from the formation of an additional job-match. This term essentially is the upper bound of the wage-bargaining set. The second term on the right-hand-side also seems relatively standard: it represents the outside option (χ) of a worker appropriately deflated by the 4 When we change the timing of markets in Section 5, the precise point in time at which bargaining occurs will change, resulting in a slight change in the wage bargaining outcome, but bargaining-induced frictions will still be present. )]. 10

take-home rate 1 τ n t. This term essentially is the lower bound of the wage-bargaining set. From the point of view of the Ramsey planner in our environment, the possibility that τ n, and hence the lower end of the wage-bargaining set, can be purposefully manipulated over time is important. Finally, the third term on the right-hand-side of (8) also shows how variations in labor tax rates affect outcomes. It reveals that both current (period-t) and expected future (period-t + 1) labor tax rates affect the current wage bargain. Intuitively, the third term reflects part of the continuation value of a match. A similar term arises in a standard labor search model; what is different, here, however, is that this component of the continuation value is influenced by changes in tax rates. By cyclically altering ex-post wage-bargaining sets, tax rate variations influence the ex-ante incentives for parties to invest in the costly matching process in the first place. Thus, tax-rate variability has the potential to mitigate inefficiencies stemming from bargaining over the business cycle. We could also describe the effects of policy by saying that tax-rate variations alter effective (i.e., inclusive of tax) bargaining shares. Arseneau and Chugh (2007) dubbed variations in effective bargaining shares stemming from cyclical policy changes a dynamic bargaining power effect, and this effect is also present in our model here. The dynamic bargaining power effect is the source of our optimal-policy results. 5 If labor taxes were constant at τt n = τ n t in which case the dynamic bargaining power effect of course disappears the wage outcome would be w t = η(z t + γθ t ) + (1 η)χ 1 τ. 6 In this case, the presence of the labor tax only changes firms effective bargaining power in a static manner: τ n > 0 causes (1 η)/(1 τ n ) > 1 η. This kind of static bargaining wedge underpins the results in Arseneau and Chugh (2006); in our model here, this static effect would be unable to correct variations in the severity of bargaining-induced wedges along the business cycle. If furthermore τ n t = 0 t, the bargained wage collapses to the even more familiar w t = η(z t + γθ t ) + (1 η)χ, which can be found in, for example, Pissarides (2000, p. 17). 2.4 Government The government finances an exogenous stream of spending {g t } by collecting labor income taxes and issuing real state-contingent debt. The period-t government budget constraint is τ n t w t n t + b t = g t + R t b t 1 + (1 n t )χ. (9) 5 In standard models, the third term on the right-hand-side of (8) can be simplified using the job-creation condition and then condensed with the first-term on the right-hand-side of (8); these standard manipulations yield a wage equation that depends on only period-t variables. However, the presence of the stochastic tax rate τ n t+1 prevents such simplification here and means that wage-setting is explicitly forward-looking. 6 To obtain [ this, note from)] the job-creation condition (4) that γ = (1 k f (θ t ) ρx γ )E t Ξ t+1 t (z t+1 w t+1 +. k f (θ t+1 ) 11

As we noted when we presented the problem of the representative household, the fact that the government is able to issue fully state-contingent real debt means that none of our results are driven by incompleteness of debt markets. We include payment of unemployment benefits as a government activity for two reasons. First, we think it empirically descriptive to view the government as providing such insurance. 7 Second, from a technical standpoint, including (1 n t )χ in the government budget constraint means that χ does not appear in the economy-wide resource constraint (presented below). In DSGE labor-search models, it is common to include unemployment benefits in the household budget constraint but yet exclude them from the economy-wide resource constraint see, for example, Krause and Lubik (2007) or Faia (2007). In such models, the government budget constraint is a residual object due to the presence of a lump-sum tax. In contrast, we rule out lump-sum taxes in order to conduct our Ramsey analysis and thus cannot treat the government s budget as residual. A Ramsey problem requires specifying both the resource constraint and either the government or household budget constraint as equilibrium objects, and this requires us to take a more precise stand on the source of unemployment benefits than usually taken in the literature. To make our model setup as close as possible to existing ones, we must assert that payment of unemployment benefits is a government activity. 2.5 Matching Technology In equilibrium, n t = n f t = n h t, so we now refer to employment simply as n t. Matches between unemployed individuals searching for jobs and firms searching to fill vacancies are formed according to a constant-returns matching technology, m(u t, v t ), where u t is the number of searching individuals and v t is the number of posted vacancies. A fraction ρ x of matches that produce in period t are exogenously destroyed before period-t + 1, and a fraction ρ x of newly-formed matches in period t are destroyed before ever becoming productive. This timing in DSGE labor-search models is conventional. The evolution of aggregate employment is thus given by n t+1 = (1 ρ x )(n t + m(u t, v t )). (10) 2.6 Private-Sector Equilibrium A symmetric private-sector equilibrium is made up of endogenous processes {c t, w t, n t, θ t, u t, R t } t=0 that satisfy the job-creation condition (4), the consumption-savings optimality condition (7), the equilibrium wage condition (8), the law of motion for the aggregate stock of employment (10), the 7 Notwithstanding the home-production explanation often offered for χ in labor-search models. 12

resource constraint on the total size of the labor force n t + u t = 1, (11) and the aggregate resource constraint of the economy c t + g t + γu t θ t = z t n t. (12) In (12), total costs of posting vacancies γu t θ t are a resource cost for the economy; note that we have made the substitution v t = u t θ t, eliminating v t from the set of endogenous processes. As we discussed above, unemployment benefits χ do not absorb any part of market output. The private sector takes as given stochastic processes {z t, g t, τt n } t=0. 3 Ramsey Problem in Baseline Model In standard Ramsey models with flexible prices, a well-known result is that all equilibrium conditions, apart from the resource frontier, can be encoded by a single, present-value implementability constraint (PVIC) that must be respected by Ramsey allocations. In more complicated environments, such as Schmitt-Grohe and Uribe (2005), Chugh (2006), and Arseneau and Chugh (2007), it is not always possible to construct such a single constraint, meaning that, in principle, all of the equilibrium conditions must be imposed explicitly as constraints on the Ramsey problem. Our model presents an environment in which it is instructive to construct a PVIC but nonetheless leave some equilibrium conditions as separate constraints on the Ramsey planner. As we show in Appendix D, we can construct a PVIC starting from the household flow budget constraint (6) and using the standard household optimality conditions (7); the PVIC for our model is given by E 0 β t { u (c t ) [c t (z t τ n w t ) n t (1 n t )χ + γv t ] } = u (c 0 )R 0 b 1. (13) t=0 But we refrain from substituting other equilibrium conditions into the PVIC and instead impose them separately as explicit constraints on the Ramsey optimization. The Ramsey problem is thus to choose state-contingent processes {c t, w t, n t, τ n t, θ t, u t } t=0 to maximize (5) subject to the PVIC (13), the job-creation condition (4), the Nash wage outcome (8), the aggregate law of motion for the employment stock (10), the restriction on the size of the labor force (11), and the aggregate resource constraint (12). Note that in our formulation we leave the policy process {τt n } as an explicit object of Ramsey optimization. The labor tax rate cannot be eliminated from the set of Ramsey choice variables because it appears in the time t 1 forward-looking wage expression (8). Because τ n t cannot be eliminated through simple static conditions as in basic Ramsey models, we view it as a fundamental part of the allocation. 13

4 Optimal Taxation in Baseline Model We characterize both the Ramsey steady state and dynamics numerically. Before presenting quantitative results, we describe our baseline parameterization. 4.1 Parameterization and Solution Strategy Well-understood since Hagedorn and Manovskii (2007) is that the (net-of-tax) social flow gain from employment, z χ 1 τ, is important for dynamics in search models. In Appendix C, we derive n the steady-state elasticity of labor-market tightness to the labor tax rate, which shows that the social flow gain from unemployment is likely to be important for the optimality of tax smoothing as well. 8 Because we normalize the steady-state level of technology to z = 1 and τ n is of course endogenous under the Ramsey policy, it is thus χ that is crucial. We report and discuss our main dynamic results for calibrations of χ such that unemployment benefits constitute 40 percent and 95 percent of after-tax real wages. The 95-percent replacement rate corresponds to the Hagedorn and Manovskii (2007) hereafter, HM calibration, while the 40-percent replacement rate is the Shimer (2005) value. To gain deeper understanding of how the model works, we also document results for alternative values of unemployment benefits, corresponding to 70 percent, 90 percent, and 99 percent replacement rates. The rest of our calibration is relatively standard in this class of models. We assume a quarterly subjective discount factor β = 0.99 and instantaneous household utility u(c) = ln c. The matching function is Cobb-Douglas, m(u, v) = ψu ξu t v1 ξu t, with ξ u = 0.4, in line with the evidence in Blanchard and Diamond (1989), and ψ set so that the quarterly job-finding rate of a searching individual is 60 percent in the model with a 50-percent replacement rate. The resulting value is ψ = 0.66, which we hold constant as we vary the replacement rate as well when we consider extensions of our model along several dimensions. Given our Cobb-Douglas specification, this corresponds to calibrating k h (θ) = 0.60 in the Ramsey equilibrium with a 40-percent replacement rate. The fixed cost of opening a vacancy is set so that posting costs absorb 5 percent of total output in the Ramsey equilibrium with a 40-percent replacement rate. We fix the Nash bargaining weight at η = 0.40 so that it ostensibly satisfies the well-known Hosios (1990) condition η = ξ u for static search efficiency. We use this as our main guidepost, although we point out that because τ n 0 in the Ramsey equilibrium, the naive calibration η = ξ u actually does not deliver search efficiency in the steady state. Rather, the proper setting must also take into account the steady-state distortionary tax rate. 9 Our main focus, though, is not on steady- 8 Specifically, see expression (75) in Appendix C. 9 Recall we pointed out when we discussed wage determination that if τ n t = 0 t that the Nash bargain collapses to the usual one in Pissarides (2000), which is the basis for the standard Hosios (1990) condition. 14

state holdup and other inefficiencies, but rather, as we mentioned above, possible inefficiencies in the cyclical fluctuations of search behavior; setting η = ξ u does not obscure this. Arseneau and Chugh (2006) provide an extensive analysis of how labor taxation drives a steady-state wedge in the usual Hosios efficiency condition. Finally, the exogenous productivity and government spending shocks follow AR(1) processes in logs, ln z t = ρ z ln z t 1 + ɛ z t, (14) ln g t = (1 ρ g ) ln ḡ + ρ g ln g t 1 + ɛ g t, (15) where ḡ denotes the steady-state level of government spending, which we calibrate in our baseline model with a 40-percent replacement rate to constitute 17 percent of steady-state output in the Ramsey allocation. The resulting value is ḡ = 0.07, which we hold constant across all experiments and all specifications of our model. The innovations ɛ z t and ɛ g t are distributed N(0, σɛ 2 z) and N(0, σɛ 2 g), respectively, and are independent of each other. We choose parameters ρ z = 0.95, ρ g = 0.97, σ ɛ z = 0.006, and σ ɛ g = 0.027, consistent with the RBC literature and Chari and Kehoe (1999). Also regarding policy, we assume that the steady-state government debt-to-gdp ratio (at an annual frequency) is 0.5, in line with evidence for the U.S. economy and with the calibrations of Schmitt-Grohe and Uribe (2005) and Chugh (2006, 2007). To study dynamics, we approximate our model by linearizing in levels the Ramsey first-order conditions for time t > 0 around the non-stochastic steady-state of these conditions. We use our approximated decision rules to simulate time-paths of the Ramsey equilibrium in the face of a complete set of TFP and government spending realizations, the shocks to which we draw according to the parameters of the laws of motion described above. Our numerical method is our own implementation of the perturbation algorithm described by Schmitt-Grohe and Uribe (2004). As is common when focusing on asymptotic policy dynamics, we assume that the initial state of the economy is the asymptotic Ramsey steady state, thus adopting the timeless perspective. As we mentioned above, we assume throughout, as is also typical in the literature, that the firstorder conditions of the Ramsey problem are necessary and sufficient and that Ramsey allocations are interior. We conduct 5000 simulations, each 200 periods long. For each simulation, we then compute first and second moments and report the medians of these moments across the 5000 simulations. 4.2 Main Result: The Optimality of Tax Volatility Table 1 presents our main result. Our focus is on the dynamics of the optimal labor tax rate, displayed in the first row of each panel. The top panel shows that at the Shimer calibration (in which the usual Hosios parameterization η = ξ u is satisfied and unemployment benefits constitute 15

40 percent of after-tax wages on average), the optimal labor tax rate is extremely volatile, with a standard deviation of 7 percent around a mean of about 16 percent. 10 To appreciate how startlingly volatile this tax rate is, compare our result with that of Chari and Kehoe (1999, p. 1710), which has become the quantitative benchmark for modern stochastic Ramsey models. Chari and Kehoe (1999) report a standard deviation of τ n of 0.10 percentage point around a mean of 23.87 percentage points. That is, in their benchmark Ramsey model, the optimal labor tax rate remains between 23.77 percent and 23.97 percent two-thirds of the time in the face of aggregate business cycle shocks. A long line of other studies for example, Schmitt-Grohe and Uribe (2005), Siu (2004), and Chugh (2006, 2007) have confirmed this result. Werning (2007, p. 944) confirms such magnitudes analytically. A large part of the DSGE labor search literature has embraced something close to the Shimer calibration, so we think it worthwhile to first discuss the intuition behind our results for this case. The basic reason that tax smoothing is not part of the optimal policy prescription is that labor tax rates affect allocations in a very different way than they do in a Walrasian labor market. In the standard frictionless view, the labor tax rate is the only potentially time-varying component of wedges between labor demand and labor supply. Time-variation in the labor wedges caused by a positive average tax rate is avoided by the Ramsey planner because total discounted deadweight losses over time are convex in labor wedges this is Barro s (1979) basic intuition, one that has carried over to general-equilibrium Ramsey environments. In contrast, in our search and bargaining model, the labor tax rate can play a quite different role. As we described in Section 2.3, a dynamic bargaining power effect arises through which variations in tax rates, because they affect ex-post wage-bargaining sets, can affect parties ex-ante search incentives. By cyclically altering ex-ante search incentives, tax-rate volatility can ease variations in the magnitude of bargaining-induced inefficiencies along the business cycle. Or, in language similar to Mortensen and Pissarides (1999) or Boone and Bovenberg (2002), because tax rates are known before any private-sector agents make their search decisions (even though pre-tax wages are not), labor-tax policy can be thought of as easing the incomplete contracting problems inherent in an environment of ex-ante search but ex-post bargaining. In Section 6, we develop this idea further, along with several other, related, ways that one can understand optimal tax-rate volatility in our model. The simplest way to intuitively understand the result is that ex-post bargaining in and of 10 In the interest of documenting the properties of our model, we also present our model s dynamics for several other salient variables, but we do not dwell on them. The one aspect of our results to which we draw the reader s attention is that tax-rate volatility does not stem from unreasonably-high aggregate volatility of our model: the volatility of total output is fairly stable, between about 1.3 and 1.7 percentage points, as unemployment benefits rise from 40 percent to 95 percent of after-tax wages, well within the range of calibrated models and in line with estimates of U.S. GDP volatility. 16

Variable Mean Std. Dev. Auto corr. Corr(x, Y ) Corr(x, Z) Corr(x, G) Shimer calibration (η = 0.4, ue benefits 40 percent of after-tax wage) τ n 0.1625 0.0701 0.5327 0.6832 0.6443-0.2310 gdp 0.8446 0.0148 0.9293 1.0000 0.9955-0.0757 c 0.6727 0.0071 0.9465 0.9302 0.9104-0.3861 N 0.8439 0.0028 0.9529 0.9109 0.8685-0.1779 w 0.8685 0.0161 0.9101 0.9341 0.9065-0.2095 θ 0.8579 0.0159 0.9061 0.9846 0.9815-0.1935 v 0.1339 0.0011 0.3292 0.2128 0.2946-0.0476 u 0.1561 0.0028 0.9529-0.9109-0.8685 0.1779 Exact HM calibration (η = 0.052, ue benefits 95 percent of after-tax wage) τ n 0.6114 0.0166 0.3609-0.4796 0.5212-0.1520 gdp 0.4620 0.0065 0.5225 1.0000 0.1104 0.6938 c 0.3759 0.0036 0.0657 0.5506 0.2557-0.1656 N 0.4618 0.0087 0.7886 0.6314-0.6702 0.5208 w 0.9620 0.0375 0.3730-0.4716 0.5335-0.1516 θ 0.0400 0.0037 0.2384-0.3502-0.3276 0.2456 v 0.5382 0.0087 0.7886-0.6314 0.6702-0.5208 Modified HM calibration (η = 0.4, ue benefits 95 percent of after-tax wage) τ n 0.5929 0.0141 0.9385 0.3569 0.8450-0.2484 gdp 0.4714 0.0060 0.8495 1.0000 0.6825 0.4735 c 0.3843 0.0031 0.8074 0.5700 0.7829-0.4018 N 0.4711 0.0050 0.9715 0.2274-0.5064 0.6149 w 0.9610 0.0191 0.9530 0.4357 0.8987-0.2198 θ 0.0425 0.0010 0.8579-0.1250-0.5414 0.6274 v 0.0225 0.0005 0.7874-0.2521-0.4209 0.4741 u 0.5289 0.0050 0.9715-0.2274 0.5064-0.6149 Table 1: Baseline model results at the Shimer calibration, the Hagedorn and Manovskii (HM) calibration, and a modified HM calibration. 17

itself gives rise to a time-varying wedge in the labor market that tax-rate volatility is designed to offset. We defer a more complete discussion of this idea to Section 6 because it will be easier to understand this channel once we have presented all of our models and results. By now, it is well-known that under the Shimer parameterization, the standard labor search model does not deliver the large degree of volatility in unemployment observed in the U.S. data. One resolution to this puzzle that has received a fair amount of attention is the HM calibration, which contends that for studying aggregate cyclical dynamics, one needs to model the flow gain from employment as being much smaller than a typical calibration assumes. With the HM calibration, the standard search and matching model generates dynamics that are consistent with the cyclical behavior of unemployment and vacancies. In light of this, we think it reasonable to test the robustness of our main result to the HM calibration. Another motivation to quantitatively assess our model with the HM calibration is the analytics presented in Appendix C, which suggests that the sensitivity of the labor market to tax rates becomes large as the flow gain from employment becomes small. The middle panel of Table 1 shows that even at the HM calibration of very low worker bargaining power (η = 0.052) and a 95-percent replacement rate, we still find substantial tax rate volatility a standard deviation of 1.6 percent, while lower than under the Shimer calibration, is still an order of magnitude larger than Chari and Kehoe (1999). The HM calibration assigns a larger allocative role to the (after-tax) real wage because the flow gain of employment is relatively small at the margin, in the spirit of an RBC model. This larger allocative role of the after-tax real wage is reflected in the drop in tax rate volatility to 1.6 percent from the 7 percent under the Shimer calibration, but we would still not call this tax smoothing in the sense understood in the Ramsey literature. The HM calibration differs from the Shimer calibration in two respects: it features both a higher replacement rate and a lower bargaining power for workers. In the lower panel of Table 1, we keep η fixed at the Shimer value but use HM s 95-percent replacement rate. At 1.4 percent, tax-rate volatility under this modified HM calibration is quite similar to that under the exact HM calibration. Quantitatively, then, optimal tax-rate variability is much more sensitive to the flow gain from employment than to η. To keep the number of models and cases we analyze further below manageable, we fix attention from here on to the parameterization η = ξ u = 0.40. Thus, when we speak of the HM calibration from here on, really what we refer to is a modified calibration that retains HM s 95-percent replacement rate but fixes η = ξ u. In summary, then, we find it makes no difference for our main qualitative results whether one employs the Shimer calibration or the HM (or, at least, HM-style) calibration, although it does matter quantitatively. Hence, we have our main result tax smoothing is not an important 18

optimal policy prescription in the presence of search and bargaining frictions in the labor market as typically modeled. 4.3 Recovering Tax Smoothing in the Baseline Model A conclusion of the preceding analysis is that the most important parameter governing the degree of tax smoothing is unemployment benefits or, equivalently, the flow gain from employment. We thus present in Table 2 Ramsey dynamics for several other replacement rates, holding, as justified at the close of the previous section, η fixed at 0.4. Reading down Table 2, the average labor tax rate rises sharply as unemployment benefits rise, while tax-rate volatility declines. The rise in the average tax rate reflects two forces: a sharp fall in the labor tax base and the larger unemployment benefits that the government must finance. First, note that the employment stock n is only about 60 percent as large at a 95-percent replacement rate as it is at a 50-percent replacement rate. That n/ χ < 0 is quite intuitive: the larger is the flow benefit from unemployment, the less incentive to work. More precisely, because our model (so far) lacks a labor force participation margin, the mechanism that leads to n/ χ < 0 is that the Nash-bargained wage rises close to the marginal product of labor as χ rises, which diminishes firms incentives to create vacancies. The resulting lower job-finding rate for individuals, k h (θ), leads to fewer jobs in equilibrium. For a given level of government spending, the Ramsey government must impose a higher τ n as the tax base shrinks. Recall also from (9) that we have assumed that the government bankrolls unemployment benefits. Larger (exogenous) χ, ceteris paribus, also requires a higher average tax rate; coupled with lower (endogenous) n, the average tax rate unambiguously rises as unemployment benefits become more generous. Costain and Reiter (2007) have recently argued that solutions, such as the HM calibration, to Shimer s (2005) volatility puzzle lead to quite unrealistic predictions regarding policy. One could view the implausibly high average tax rates our Ramsey model predicts as replacement rates become very large as being subject to their critique. We do not view this as a damning criticism of our results for two reasons. First, our work is more a theoretical exploration what sorts of labor-market models and mechanisms are important for the cornerstone Ramsey result of taxsmoothing rather than a data-matching exercise. Second, as Rogerson, Visschers, and Wright (2007) counter, one could introduce other realistic features, such as home production, to circumvent the Costain and Reiter (2007) critique. The exact quantitative effects of introducing such a feature or features into our model is an open question that we leave for future research. As unemployment benefits rise, the decline in optimal labor tax rate volatility our model predicts reflects the fact that the range of wage-bargaining outcomes acceptable to both firms and workers shrinks because the upper end of the bargaining interval (the marginal product of labor) is fixed 19

Variable Mean Std. Dev. Auto corr. Corr(x, Y ) Corr(x, Z) Corr(x, G) ue benefits 50 percent of after-tax wage τ n 0.1853 0.0482 0.9645 0.8048 0.7493-0.2427 gdp 0.8288 0.0147 0.9306 1.0000 0.9952-0.0753 c 0.6662 0.0071 0.9497 0.9298 0.9096-0.3870 N 0.8281 0.0029 0.9546 0.9149 0.8720-0.1701 w 0.8782 0.0161 0.9719 0.9593 0.9311-0.2156 θ 0.7083 0.0128 0.9002 0.9855 0.9871-0.1845 v 0.1217 0.0010 0.4030 0.2553 0.3403-0.0529 u 0.1719 0.0029 0.9546-0.9149-0.8720 0.1701 ue benefits 70 percent of after-tax wage τ n 0.2551 0.0344 0.9747 0.8564 0.8097-0.2406 gdp 0.7761 0.0139 0.9316 1.0000 0.9952-0.0648 c 0.6366 0.0070 0.9484 0.9239 0.9082-0.3915 N 0.7755 0.0028 0.9534 0.9211 0.8791-0.1148 w 0.9026 0.0171 0.9747 0.9502 0.9217-0.2109 θ 0.4068 0.0062 0.8474 0.9658 0.9830-0.1194 v 0.0913 0.0008 0.4277 0.3441 0.4294-0.0428 u 0.2245 0.0028 0.9534-0.9211-0.8791 0.1148 ue benefits 90 percent of after-tax wage τ n 0.4436 0.0213 0.9684 0.7655 0.8486-0.2479 gdp 0.6119 0.0092 0.9021 1.0000 0.9727 0.0899 c 0.5094 0.0051 0.9166 0.8366 0.8988-0.4139 N 0.6115 0.0020 0.9366 0.2467 0.0382 0.6593 w 0.9427 0.0189 0.9726 0.8378 0.9102-0.2171 θ 0.1098 0.0012 0.6541 0.1373 0.0276 0.6016 v 0.0427 0.0004 0.4514 0.0233 0.0125 0.3306 u 0.3885 0.0020 0.9366-0.2467-0.0382-0.6593 ue benefits 95 percent of after-tax wage (modified HM calibration) τ n 0.5929 0.0141 0.9385 0.3569 0.8450-0.2484 gdp 0.4714 0.0060 0.8495 1.0000 0.6825 0.4735 c 0.3843 0.0031 0.8074 0.5700 0.7829-0.4018 N 0.4711 0.0050 0.9715 0.2274-0.5064 0.6149 w 0.9610 0.0191 0.9530 0.4357 0.8987-0.2198 θ 0.0425 0.0010 0.8579-0.1250-0.5414 0.6274 v 0.0225 0.0005 0.7874-0.2521-0.4209 0.4741 u 0.5289 0.0050 0.9715-0.2274 0.5064-0.6149 ue benefits 99 percent of after-tax wage τ n 0.9308 0.0029 0.1000-0.3384 0.5232-0.2075 gdp 0.1409 0.0052 0.8465 1.0000 0.0027 0.8979 c 0.0694 0.0013-0.1621 0.1939 0.1373-0.2415 N 0.1409 0.0055 0.8783 0.9173-0.3548 0.8432 w 0.9878 0.0246 0.2465-0.3086 0.6188-0.2040 θ 0.0025 0.0004-0.0417-0.0341-0.1710 0.3929 v 0.0022 0.0003-0.0656-0.0750-0.1550 0.3550 u 0.8591 0.0055 0.8783-0.9173 0.3548-0.8432 Table 2: Baseline model results at various replacement rates. 20