Non-Walrasian Labor Markets and Real Business Cycles 1

Transcription

1 Review of Economic Dynamics 4, (2001) doi: /redy , available online at on Non-Walrasian Labor Markets and Real Business Cycles 1 Marco Maffezzoli Istituto di Economia Politica, Università Bocconi, Via Sarfatti 25, Milan, Italy marco.maffezzoli@uni-bocconi.it Received June 6, 2000 The standard real business cycle literature mainly focuses on Walrasian models designed to fit the U.S. institutional framework. Differences between the United States and Europe, mostly evident in the labor market, suggest that a purely Walrasian model may be inappropriate for the study of European business cycles. I present a stochastic version of the dynamic generalequilibrium modelof Daveri and Maffezzoli (2000, A Numerical Approach to Fiscal Policy, Unemployment and Growth in Europe, Econometrics and Applied Economics Working Paper , IEP, Università Bocconi), where unemployment is generated by monopolistic unions, and calibrate it to reproduce several long-run features of the Italian and U.S. economies. This framework is then compared with an indivisible labor model built on Hansen (1985, Journal of MonetaryEconomics 16, ) and Rogerson and Wright (1988, Journal of MonetaryEconomics 22, ). I focus on the impulse response functions, the standard business cycle statistics, and the ability to reproduce the cyclical components of the main macroeconomic variables. The main results are as follows: (i) the impulse response functions of the monopoly union (MU) modelshow a higher degree of overa lpersistence; (ii) the business cycle statistics are similar; (iii) the MU model enjoys a statistically significant advantage in reproducing the Italian business cycle, but not that of the United States. Journal of Economics Literature Classification Numbers: E32, E24, J23, J Academic Press KeyWords: real business cycle; general equilibrium; trade union; indivisible labor; unemployment. 1 The author is heavily indebted to F. Daveri for many fruitful discussions. He owes thanks also to G. Aldashev, T. Boeri, A. Cuñat, C. A. Favero, V. Galasso, C. Häfcke, G. Hansen, M. Marcellino, R. Perotti, G. Tabellini, and two anonymous referees for their useful suggestions. Finally, he greatly benefitted from comments of seminar participants at Bocconi University and the University of Brescia. The usual disclaimers apply /01 $35.00 Copyright 2001 by Academic Press All rights of reproduction in any form reserved. 860

2 non-walrasian labor markets and rbcs INTRODUCTION The realbusiness cycle (RBC) modelwas introduced by Kydland and Prescott (1982) and Long and Plosser (1983). For nearly two decades, it has been considered the profession s workhorse for the study of business cycles in market economies. The RBC literature has focused on purely competitive models, designed to fit the U.S. institutional framework. However, the institutionaldifferences between the United States and Europe, mostly evident in the labor market, suggest that a Walrasian model may be inappropriate for the study of business cycles in European countries, as advocated by Gali (1996). The RBC methodology has been extended by several authors to incorporate non-walrasian features. 2 My aim is to contribute to this increasing literature by presenting a non-walrasian model designed to better fit the European labor market institutions, and the Italian ones in particular. Even though the Italian labor market became more flexible during the 1980s and the 1990s, as described by Bertola and Ichino (1995), it still features a pervading overall rigidity. Its structure is far from the highly competitive one of the U.S. labor market. A striking difference concerns the role and importance of trade unions. Union density and coverage rates, reported in Table I, have been consistently higher in Italy over the last 20 years. The density rate decreased significantly in both countries during the 1980s; however, a considerable drop in the coverage rate followed in the United States only. The coverage rate in Italy remained above 80% and may have been even higher (almost close to one among formally employed workers), as suggested by the OECD (1997). This empiricalevidence strongly suggests 2 For a survey on non-walrasian models, see Danthine and Donaldson (1995). Hall (1998) suggests that realfrictions are essentialto explaining employment fluctuations in the modern DSGE framework. Merz (1995), Andolfatto (1996), Gomes et al. (1997), and Den Haan et al. (2000) incorporate labor-market search into the RBC framework. TABLE I Union Density and Coverage Rates Density (%) Coverage (%) Italy USA Source: OECD (1997).

3 862 marco maffezzoli that the role of trade unions should be taken into account if the Italian business cycle is to be studied. 3 Following Daveri and Maffezzoli (2000), I develop a DSGE model with endogenous growth and equilibrium unemployment, generated respectively by learning-by-doing and monopolistic unions, and calibrate it to reproduce several long-run features of the Italian economy. 4 The resulting toy economy is employed to answer a specific question, namely whether the monopoly union (MU) model exhibits a comparative advantage over a version of the Hansen Rogerson Wright (HRW) indivisible labor model in explaining the Italian business cycle. 5 To answer this question, three different comparison metrics are used. First, I compare the impulse response functions, highlighting the different transmission mechanisms. Second, I compare the standard business cycle statistics under the H-P filter. Third, I feed both models with the estimated shocks to productivity and evaluate their relative abilities to reproduce the cyclical components of the main Italian macroeconomic series. The simulated series for all main variables tend to be more persistent in the MU modelthan in the HRW model; productivity shocks in the former have longer-lasting effects on employment. The two models generate similar business cycle statistics: only the relative volatility of employment is significantly higher and closer to the observed one in the HRW model. The difference in persistence is reflected in the correlation of Total Factor Productivity (TFP) with lags and leads of the other variables: the MU model displays higher correlations of current TFP with the other variables leads. The MU model reproduces the observed Italian cyclical components consistently better than its alternative: the former generates correlation coefficients between the observed and simulated series for output, consumption, investment, and employment that are, respectively, 8, 6, 4, and 26 points higher (see Table V). These differences are crucial: the dimension in which the MU model clearly outperforms the HRW (i.e., the dynamics of employment) depends directly on the assumptions characterizing it. 3 As far as the role of unions is concerned, Italy is by no means an exception among the main European countries. At the beginning of the 1990s, the coverage rates ranged from 68% in Spain to 92% in France (OECD, 1997). 4 Gali (1995) and Bénassy (1997) build DSGE models on similar lines: the former focuses on the dynamics of U.S. unemployment, the latter on the persistence of unemployment. Palokangas (2000) discusses at length the role of trade unions in macroeconomic modeling. 5 Rogerson (1988) shows that employment lotteries can improve welfare when labor is indivisible and markets are competitive. Hansen (1985) introduces this framework in the RBC literature. King and Rebelo (2000) incorporate the more general Rogerson and Wright (1988) setup in their benchmark RBC model.

4 non-walrasian labor markets and rbcs 863 By considering the simulated series as competing forecasts and applying the Diebold Mariano testing framework, 6 I provide a formalstatistical argument in favor of the MU model. The prediction mean squared error (PMSE) generated by the MU model is significantly lower, at least for output, investment, and employment. In other words, its overall prediction accuracy seems significantly higher. These results suggest that the MU model is actually better suited to replication of the Italian business cycle. I conjecture that this advantage derives from a labor-market structure better tailored to the Italian institutional framework. If this is the case, the HRW model should perform relatively better if the focus moves to a country with a more competitive labor market, such as the United States. I calibrate the toy models to the U.S. economy and replicate the previously described set of experiments. The results show that the MU model actually loses any advantage: the competing models predictive power is not statistically different. To summarize, we conclude that introducing non-walrasian elements into the RBC framework may help to improve its explanatory power, if we focus on economies characterized by pervading realrigidities, such as, for instance, the European ones. My result suggest that these gains may be substantial. The structure of the paper is as follows. Section 2 describes both the HRW and the MU models in detail, Section 3 outlines the calibration procedure, and Section 4 presents our results. Concluding remarks are in Section SET-UP In this section I present a Walrasian model, built on Hansen (1985) and Rogerson and Wright (1988), and a stochastic version of the non-walrasian model developed by Daveri and Maffezzoli (2000), which extends the wellknown monopoly union model into a full-fledged DSGE framework The Hansen Rogerson Wright Model 7 The economy is populated by many identical, infinitely living firms and households, each of measure zero. Firms produce a homogeneous consumption good, combining capitalservices and labor through a constant- 6 Diebold and Mariano (1995) develop a formal test to evaluate the relative forecast accuracy of competing econometric models. In particular, their approach tests the null hypothesis of equalprediction mean squared errors against a two-sided alternative. 7 The HRW setup is extensively described in King and Rebelo (2000, pp ). Mulligan (2001) discusses the aggregate implications of indivisible labor.

5 864 marco maffezzoli returns-to-scale technology subject to persistent but stationary productivity shocks. Households own both factors of production, physicalcapitaland labor, and they sell each period the factor services to the firms. Firms and households behave as price-takers on all markets. Under these assumptions, firms can be safely aggregated into a representative firm. Endogenous growth is introduced in the framework by assuming that human capital accumulates through a simple learning-by-doing process: I interpret this production of knowledge as an economy-wide externality Firms The representative firm produces a homogeneous consumption good, using a constant-returns-to-scale technology summarized by the CES production function Y t = a t αk η t + n t H t η 1/η (1) where η<1 and α>0. In my notation, Y t represents the aggregate percapita output level, K t the aggregate per-capita levelof physicalcapital services, n t the employment rate, H t the aggregate per-capita human capital stock, and a t the aggregate level of TFP. The product n t H t corresponds to labor in efficiency units. The elasticity of substitution between physical capital and efficient labor equals ξ 1/ 1 η. The representative firm maximizes the expected discounted flow of future profits, [ ] max = E t R s n s K s a s t Y s W s n s r s K s (2) s=t s=t where R s t is a time-varying market discount factor, given the sequence of wage rates, W s a s s=t ; rentalrates, r s a s s=t ; and human capitalstocks, H s a s s=t. The first-order conditions are the usualones, W t = a t αk η t + n t H t η 1 η /η n η 1 t H η t (3) r t = a t αk η t + n t H t η 1 η /η αk η 1 t (4) The TFP level follows an exogenous stochastic Markov process, approximated by the stationary AR(1) process ln a t+1 =ρ ln a t +ɛ t (5) where 0 <ρ<1 is the persistence parameter and ɛ t N 0 σ 2 is the iid innovation. Finally, human capital evolves according to H t+1 = H t + ϕn t H t (6)

6 non-walrasian labor markets and rbcs 865 where ϕ>0 is a simple scale parameter. In as much as Eq. (6) is linear in human capitalstock, the economy willbe characterized by endogenous growth as long as n t > 0 t Households Each household member chooses between working a fixed number of hours (normalized to one) and not working at all. As the individual labor supply is indivisible, the choice set is not convex; it may be convexified by introducing employment lotteries, i.e., efficient contracts on which individuals agree in equilibrium. By entering a lottery, an individual chooses to work a fraction ñ of her days and to remain unemployed the other 1 ñ days; the allocation of individuals to work or leisure is completely random, and the lottery s outcomes are independent over time. Prior to the lottery draw, the expected intratemporal utility function is (I abstract from time indexes) [ 1 µ [ 1 µ ñ C e v 0 ] + 1 ñ C u v 1 ] (7) where C e is the consumption level of employed individuals (a tilde identifies individualvariables), Cu is the consumption level of idle individuals, ñ is the ex ante probability of being employed at the individual level, and v is the utility of leisure. Note that v 0 and v 1 are strictly positive constants; therefore, we can define v 0 v 0 and v 1 v 1. If asset markets are complete, individuals can perfectly insure themselves against the idiosyncratic risk of being unemployed. 9 Under perfect risk sharing the marginalutilities of consumption are equalacross employed and unemployed individuals. Anticipating this equilibrium outcome, and defining the average consumption level as C =ñ C e + 1 ñ C u, we can rewrite the instantaneous utility function (7) as ) 1 µ /µ ] µ C [ñ 1 µ v1 + 1 ñ ( (8) v 0 8 Barro and Sala-i-Martin (1995, pp ) show that a Ramsey Cass Koopmans model with a CES production function generates endogenous growth if the elasticity of substitution is sufficiently high (greater than one). This possibility is avoided by assuming an elasticity of substitution lower than one. 9 Alternatively, we may assume the existence of a competitive insurance market characterized by a zero-profit condition for the firms offering insurance; in this case, the relative price of consumption under employment and unemployment will be n/ 1 n. See Hansen (1985, Appendix) for details.

7 866 marco maffezzoli The employed and unemployed individuals can then be aggregated into a stand-in representative household, whose intertemporal utility function is { [ Cs U t = E t β s t φ ñ s ] } 1 µ (9) s=t 1 µ where [ ( ) 1 µ /µ ] µ/ 1 µ v1 φ ñ t ñ t + 1 ñ t (10) The function φ n can be interpreted as the disutility of employment for the representative household. The elasticity of φ with respect to its argument is defined as ξ φ φ n n/φ n. Under perfect risk sharing allindividualincomes are pooled, so that our stand-in representative household is subject to a dynamic constraint, K t+1 = 1 δ K t + r t Kt +ñ t W t C t (11) where δ is the depreciation rate of physicalcapital. The representative household maximizes the intertemporal utility function (9), under the dynamic constraint (11), with respect to { Cs } ñ s K s+1 a s s=t, taking the sequence of contingent prices r s W s a s s=t as given.10 The HRW modelhas two relevant properties. First, the stand-in representative individualhas an infinite λ-constant elasticity of labor supply, i.e., her labor supply is infinitely elastic for any given shadow value of installed physical capital. This result does not depend on the elasticity of the individual labor supply. Second, choosing to bear some uncertainty by entering the employment lottery is a way to substitute consumption for leisure; thus, the possibly positive equilibrium unemployment rate is Pareto-efficient Equilibrium In equilibrium, individual 11 and aggregate per capita quantities coincide. In particular, the aggregate employment rate equals the ex ante probability of being employed at the individual level. A recursive dynamic equilibrium for the HRW economy is then defined as a set of contingent prices, r s W s a s s=t ; individual probabilities of being employed, consumption levels, and physicalcapitalstocks, ñ s C s K s a s s=t ; v 0 10 Under my assumptions, the first-order and transversality conditions are jointly necessary and sufficient to solve the representative household s problem. 11 By individualquantities I now mean the quantities chosen by the representative household.

8 non-walrasian labor markets and rbcs 867 aggregate employment rates, consumption levels, and physical capital stocks, n s C s K s a s s=t ; human capitalstocks, H s a s s=t ; such that the individualquantities ñ s C s K s a s s=t solve the representative household optimization problem for the given sequences r s W s a s s=t ; the aggregate quantities n s K s a s s=t solve the representative firm s problem for the given sequences r s W s H s a s s=t ; the individualand aggregate quantities are consistent, ñ t a t = n t a t, C t a t = C t a t, and K t a t = K t a t ; the consumption good market clears; the human capital accumulation equation holds. By imposing the equilibrium conditions on the first-order and transversality conditions, I obtain a non-linear system of stochastic difference equations defining the full set of feasible equilibrium allocations. Note that, as the employment rate is strictly positive in equilibrium, the system is nonstationary and therefore has to be normalized with regard to H t The Monopoly Union Model As in the HRW model, the individual labor supply is indivisible; if employed, household members work a fixed shift, selling their labor endowment to only one firm at a time. Each firm is endowed with a pool of households from which it can hire. Each pool contains infinitely many households, so that the individual household member is again of measure zero. 12 All household members in a particular pool remain price-takers in the capital market. However, by clearly identifying their job provider, they perceive the scope for extracting some producer surplus by organizing themselves into a firm-specific trade union. 13 The unions negotiate the wage on behalf of their members. They act as monopolists on the labor market and maximize the welfare function n t W t + 1 n t W t (12) 12 Given constant returns to scale, firms and household pools may be better interpreted as industries or sectors. 13 I do not model explicitly the process of union formation. Horn and Wolinsky (1988) is a classical reference. Westermark (1999) shows that highly substitutable workers have an incentive to form a union when contracting over their wage, while complementary workers do not. My assumption of identical households is equivalent to perfect substitutability.

9 868 marco maffezzoli where n t is the employment rate at the pool level and W t is the unions reservation wage, taking the conditionallabor demand function of the representative firm and the reservation wage as given. Once the wage has been set, firms are free to choose the employment rate along their labor demand function. 14 Though large at the firm or sector level, the unions are small at the economy-wide leveland thus take the rentalprice of physicalcapital as given, as the household members do. Equation (12) implies that unions are risk-neutral and maximize the members average wage bill. 15 The reservation wage can be broadly interpreted as the disutility of employment perceived by the unions; it encompasses the value assigned to spare time enjoyed by union members and any unemployment subsidy possibly paid by the government and financed via lump-sum taxation. 16 For the sake of simplicity, I consider the sequence of reservation wages W s s=t as completely exogenous. The unions can be aggregated into a representative union facing the representative firm: they jointly determine the wage rate and the aggregate unemployment rate. The allocation of union members to work or leisure is completely random and independent over time. The ex ante probability of being employed at the individual level, then, corresponds to the aggregate employment rate. Although being employed or unemployed is unknown ex ante, the unemployed may feel uneasy ex post and leave the union. To tackle this critical issue, the union pursues a redistributive goal, acting as a substitute for competitive insurance markets. In particular, the union offers its members the possibility of insurance against the risk of being unemployed; insurance is supplied under a zero-profit condition and is therefore actuarially fair. 17 In equilibrium, the marginal utility of consumption will again be equalized 14 I adapt to my needs the well-known monopoly union model, introduced by Dunlop (1944) and Oswald (1982), which is a particular case of the more general sequential bargaining framework proposed by Manning (1987). 15 Farber (1986) discusses different unions objective functions from both a theoreticaland an empirical point of view. I adopt the simplest functional form, following Palokangas (2000, pp ), which corresponds to the risk-neutral analogue of the utilitarian utility function in Oswald (1982). Similar welfare functions have been used by Anderson and Devereux (1988) and Pissarides (1998) among others. 16 In this framework, any unemployment subsidy would have no clear ex post redistributive role. However, we may argue that the government wants to guarantee ex ante a certain degree of income redistribution, independently from the equilibrium outcome. 17 Some empiricalevidence on the redistributive role of trade unions is described in Pencavel (1985, pp ). Horn and Svensson (1986) discuss the issue from a theoreticalpoint of view. My extreme assumption serves the technicalpurpose of keeping the heterogeneity among employed and unemployed household members at a tractable level. A similar approach is followed in Bénassy (1997).

10 non-walrasian labor markets and rbcs 869 across employed and unemployed household members; therefore, they can be safely aggregated into a single representative household The Wage Equation The firm s choice of capitalservices adds an extra dimension to the firm union interaction, often neglected in the theoretical literature on trade unions. Anderson and Devereux (1988) discuss the issue in detail: I extend one of their equilibrium concepts and assume that employment rates, wages, and demands for physicalcapitalservices are determined in a non-cooperative dynamic game between the union and the firm. The strategic variables are, respectively, the contingent wage rates for the union and the contingent demands for capitalservices for the firm. The scope for pre-commitment is barred on both sides. In other words, I assume that neither the union nor the firm can credibly commit, respectively, to a sequence of future wage rates and demands for capitalservices. The union maximizes the expected discounted flow of future average labor incomes, { } max E t R s W s a s t n sw s + 1 n s W s (13) s=t s=t subject to the conditionaldemand for labor, defined by (3), and given the sequences K s a s s=t H s a s s=t, and W s a s s=t. The sequences of employment rates, wage rates, and demands for capitalservices n s a s s=t W s a s s=t, and K s a s s=t that jointly solve problems (13) and (2) for the given sequences of contingent rental rates, human capitalstocks, and reservation wages r s a s s=t H s a s s=t, and W s a s s=t form a Nash equilibrium for the game. The union and the firm effectively solve a sequence of independent games. This is due to the fact that (i) the services of physicalcapitaland labor are to be purchased in each period; (ii) pre-commitment is ruled out; (iii) the union takes the rentalrate as given. Hence, the union fails to internalize the dynamic consequences of today s wage setting on capital accumulation. I restrict my attention to Markov strategies depending only on the current endogenous and exogenous state variables. Given my assumptions, the employment rate, the wage rate, and the demand for physicalcapitalservices are jointly determined in each period by Eqs. (3) and (4), together with a t αk η t + n t H t η 1 η /η ηn η 1 t H η t + 1 η a t αk η t + n t H t η 1 η /η 1 n 2 η 1 +1 t H 2η t = W t (14) Condition (14) obtains from rewriting the union s welfare function as a t αk η t + n t H t η 1 η /η n η t H η t + 1 n t W t (15)

11 870 marco maffezzoli and partially deriving it with respect to n t. 18 In turn, we may rewrite the corresponding first-order condition as ( ) at n t H η [ ( ) t at n η + 1 η t H η ] t n = W t Y t Y t (16) t Y t Equation (16), being highly non-linear, has no closed-form solution, but implicitly defines the monopoly employment rate. The corresponding wage obtains from the firm s conditionallabor demand function Households If the union offers actuarially fair insurance, households will again perfectly share the risk of being unemployed. Following the line of reasoning outlined in Section 2.1.2, we obtain the representative household s intertemporalutility function, { [ Cs U t = E t β s t φ n s ] } 1 µ (17) 1 µ s=t where φ n is defined as in (10). Note that n t represents the aggregate employment rate and is no longer an individual decision variable. The representative household faces the following dynamic constraint: K t+1 = 1 δ K t + r t Kt + n t W t C t (18) The representative household maximizes (17) subject to (18) with respect to C s K s+1 a s s=t, taking the sequences of contingent prices r s W s a s s=t and aggregate employment rates n s a s s=t as given Equilibrium A recursive dynamic equilibrium for the MU economy is defined as a set of contingent prices, r s W s a s s=t ; individual consumption levels and physical capital stocks, { Cs } K s a s s=t ; aggregate consumption levels and physical capital stocks, C s K s a s s=t ; 18 Once the labor demand function has been taken into account, the choice of the control variable is irrelevant. 19 Note that the production function tends to a Cobb Douglas for η 1. The first-order condition (16) would collapse to n t = s 2 Y N t/ W t, where s N is the labor share in income. In this particular case, the employment rate is simply proportional to the current output level.

12 non-walrasian labor markets and rbcs 871 employment rates, n s a s s=t ; human capitalstocks, H s a s s=t a s s=t ; and reservation wages, W s such that the individualquantities C s K s a s s=t solve the representative household optimization problem for the given sequences r s W s n s a s s=t ; the aggregate quantities n s K s a s s=t solve the representative firm s problem for the given sequences r s W s H s a s s=t ; the employment rates n s a s s=t solve the union s optimization problem for the given sequences r s W s K s H s a s s=t ; the individualand aggregate quantities are consistent, C t a t = C t a t and K t a t = K t a t ; the consumption good market clears; the human capital accumulation equation holds. As before, I obtain a non-linear system of stochastic difference equations by imposing the equilibrium conditions on the first-order and transversality conditions. Assuming that the employment rate is strictly positive in equilibrium, I normalize the system with regard to H t. Furthermore, the reservation wage is assumed to grow steadily at the endogenous rate, so that its normalized value remains constant over time, w t = w. 20 The union s intratemporal problem is graphically described in Fig. 5. We can rewrite the union s normalized first-order condition as mr a k n w a k n +w a k n n = w (19) The monopoly employment rate is determined by the intersection of the horizontalline corresponding to the current reservation wage the union s fixed marginalcost with the downward-sloping curve denoted mr the union s marginalrevenue. 21 The wage rate is then determined by the conditionallabor demand curve. The aggregate wage-setting function is positively sloped, and the value of this slope is completely unrelated to the elasticity of the individual labor 20 This specification is not the only possible one. We may assume, for instance, that the government pays an unemployment subsidy equal to a constant share of current income. However, shocks to total factor productivity would immediately affect the reservation wage and therefore would not influence the employment rate on impact. 21 Being w n < 0 by assumption, mr n <w n n>0.

13 872 marco maffezzoli supply, as in the HRW framework. The two models address the same problem, i.e., they generate an elastic aggregate wage-setting function compatible with an inelastic individual labor supply. 22 For a sufficiently low reservation wage, the union chooses to employ all of its members. 23 Otherwise, the employment rate remains inside the unit interval. In our simplified framework, the reservation wage is just the union s preference parameter and can therefore be calibrated to reproduce the observed steadystate employment rate. This guarantees that, for small deviations around the steady state, the employment rate remains inside the unit interval. Rogerson (1988) shows that, in a static indivisible labor model, a competitive equilibrium exists where employed and unemployed individuals choose different consumption levels. Furthermore, he shows that the introduction of employment lotteries allows the economy to reach a Pareto-efficient equilibrium that strictly dominates the previous one. This implies that the HRW model should generally strictly dominate the MU, since the latter is sub-optimal. 24 However, if employment lotteries were ruled out, the competitive model would not necessarily Pareto-dominate the MU, since the firm union bargaining process, together with the union s redistributive role, may give access to equilibrium allocations that are otherwise unavailable. 3. CALIBRATION Being highly non-linear, neither normalized system has a closed-form solution. To study their stochastic properties, I apply the solution procedure of King et al. (1988). In other words, I assume certainty equivalence, loglinearize both systems around their steady states and solve them with the Blanchard and Kahn (1980) algorithm. To proceed in the analysis, then, a sensible parameterization is needed. It turns out that the dynamics of the two models depend on the same set of parameters. In particular, it depends on the elasticity of intertemporal substitution, µ; the intertemporaldiscount factor, β; the elasticity of the disutility of employment, ξ φ ; the elasticity of substitution between capital and labor, ξ; the relative weight of capital in the CES production function, α; the depreciation rate, δ; the scale parameter in the human capital accumulation equation, ϕ; the persistence parameter ρ; and the standard deviation of productivity shocks, σ. 22 The household production framework developed by Benhabib et al. (1991) and the search-theoretic framework discussed in Hall (1998) feature the same property. Note that my approach is compatible with a search-theoretic model of the labor market. 23 This minimum reservation wage is defined as w min = w 1 +w In fact, the MU model may be calibrated to reproduce any competitive allocation, and thus is, in principle, compatible with Pareto-efficient allocations.

14 non-walrasian labor markets and rbcs 873 I assume that µ = 2 β = 0 99, and ξ = The first two figures are standard in the literature. I borrow the elasticity of substitution between capitaland labor from Pissarides (1998) and perform an extensive sensitivity analysis on this parameter. The preference parameter, ξ φ ; the technological parameter, α; the depreciation rate, δ; and the scale parameter, ϕ are calibrated to reproduce the observed long-run employment rate, investment share, quarterly capitaloutput ratio, and quarterly growth factor in Italy. All of them are obtained from the recently revised ISTAT series, 25 spanning the period from 1982:1 to 1998:4, expressed in constant 1995 prices. Output is approximated by realgdp, investment by realgross fixed investment, capitalby the real totalnet capitalstock, and employment by standard units of labor 26 ;all variables are normalized with respect to the working-age population. The employment and growth rates, the investment share, and the capital output ratio are respectively equal to n = 58% γ= 0 48% s i = 20%, and k/y = The implied values for the calibrated parameters are ξ φ = 0 76 α= 1 15 δ= 1 05%, and ϕ = The persistence parameter and the standard deviation of the productivity shocks can be estimated by fitting an AR(1) modelon an empirical measure of quarterly TFP. To obtain the latter, I start by constructing an approximated quarterly series for the physicalcapitalstock, since only annual observations are available. For each year, given the observed initial physicalcapitalstock and the quarterly figures for gross fixed investment, I numerically solve for the quarterly depreciation rate that generates the next-year observed initialcapitalstock through the permanent inventory method. Then, the yearly series for physical capital, the quarterly series for gross fixed investment, and the previously obtained yearly series for the quarterly depreciation rate jointly imply an approximated series for the quarterly physicalcapitalstock The Italian National Accounts series have recently been revised and reorganized according to the SEC95 standard. The revised quarterly series are currently available from 1982:I to 1999:4. Annual observations for the net capital stock are currently available only for the period from 1982 to The standard unit of labor series combines raw employment data with national accounting information to obtain the equivalent number of workers effectively employed for a standard unit of time. 27 The preference parameter ξ φ is calibrated to make the HRW model reproduce the observed long-run consumption share and the implied labor share in income. 28 The implied depreciation rates vary slightly over time and increase significantly at the beginning of the 1990s. However, this variability reflects a change in the composition of physical capital or simply measurement errors rather than the effect of variable capital utilization, since the observed series was constructed assuming asset-specific constant depreciation rates.

15 874 marco maffezzoli Residual Actual Fitted FIG. 1. Output and TFP for Italy. I estimate the initiallevelof human capital, H 0, by running a non-linear regression on ln Y t = 1 t ] } {αk η ln η η t + [n t H γ j + ε t (20) where all parameters are set to their calibrated values, and γ t = ϕn t. Seasonally adjusted quarterly ISTAT series from 1982:1 to 1998:4, expressed in constant 1995 prices, are used. Output is approximated by realgdp and employment by the standard units of labor; again, both series were normalized with respect to the working-age population. The residuals generated by this procedure, plotted in Fig. 1, represent my proxy 29 for the logarithm of TFP. By fitting an AR(1) model, I obtain estimates for the persistence parameter and the standard deviation of productivity shocks. The resulting figures are ρ = 0 87 and σ = Note that there is no evidence against the null hypothesis of these productivity shocks being realinnovations. The Jarque Brera test generates a P value equal to 0.86, so that the null hypothesis of normality cannot be rejected. Furthermore, the Breusch Godfrey serial correlation LM test (two lags included) generates a P value equal to 0.41, so that also the null hypothesis of serial j=0 29 The models at hand do not explicitly incorporate variable capacity utilization and labor dynamics along the intensive margin. The residuals estimated with our procedure are only a proxy for an empirically based measure of TFP; however, they are consistent with the theoreticalframework. We may agree with Robert Solow, and consider them as a measure of our ignorance.

16 non-walrasian labor markets and rbcs 875 uncorrelation also cannot be rejected. Finally, the ADF statistic (again, two lags included) equals 4 25, while the 1% critical value is 2 60; the null hypothesis of a unit root can be safely rejected. 30 To summarize, my benchmark parameterization is the following: µ = 2 β = 0 99 ξ φ = 0 76 ξ = 0 70 α = 1 15 δ = 1 05% γ = 0 48% ϕ = ρ = 0 87 σ = RESULTS In this section, I evaluate the relative performance of my competing models, using three different but related comparison metrics. First, the impulse response functions are discussed in detail. Second, the simulated standard business cycle statistics are compared with the Italian ones. Third, both models are fed with the estimated innovations to TFP, and the resulting simulated cyclical components are compared with the observed ones Impulse Response Functions Figures 2 and 3 show the impulse responses expressed in percentage deviations from the steady state to a 1% positive productivity shock of the output, consumption, and investment levels; the employment, rental, and wage rates; and the physicalcapitalstock, respectively, for the HRW and MU models. All variables, except the employment and rental rates, are normalized with respect to the human capital stock The HRW Model Focusing on Fig. 2, we observe that, on impact, the persistent but transitory shock to TFP translates into a non-negligible increase in the representative household s permanent income; therefore, both consumption and investment increase on impact for the standard intertemporalconsumption smoothing argument. The wage, rental, and employment rates increase significantly, too. However, while the employment and rental rates converge rapidly to their steady-state values, the wage rate s deviation remains persistently high. The joint dynamics of employment and wages can be better understood from a partial equilibrium point of view. Following King and Rebelo (2000, p. 49), the representative household s labor supply can be characterized 30 I repeated the procedure for different values of the elasticity of substitution between labor and capital, using the corresponding calibrated values for the other parameters, and obtained similar results.

17 876 marco maffezzoli % dev. from s. s y c i n Quarters r w k FIG. 2. The HRW impulse responses. by combining the log-linearized first-order conditions for consumption and labor to obtain ŵ t = ˆλ t /µ, where ŵ t is the normalized wage rate, expressed in percentage deviations from the steady state, and ˆλ t is the normalized shadow value of installed capital (and the marginal utility of consumption), expressed again in percentage deviations. As stressed in Section 2.1.2, the labor supply curve is infinitely elastic for a given value of the costate variable. Figure 4 describes how the competitive labor market reacts to the productivity shock. For a given value of λ, the labor supply curve is horizontal at the equilibrium wage rate. The employment rate is simply determined at the intersection of the labor supply and demand curves (point a). On impact, the shock shifts the demand curve to the right: if the value of λ were to remain constant, only the equilibrium employment rate would increase (point b). However, the shock raises consumption, too, and therefore reduces ceteris paribus its marginal utility; this implies a less than proportional upward shift in the labor supply curve. Both the employment and the wage rate, then, increase on impact (point c). After the initial shock, TFP converges quickly to its steady-state value, and this shifts ceteris paribus the labor demand curve back to the left.

18 non-walrasian labor markets and rbcs 877 % dev. from s. s y c i n Quarters r w k FIG. 3. The MU impulse responses. However, the labor demand and supply curves depend on the endogenous state variables, too. Physicalcapitalaccumulation shifts ceteris paribus again the labor demand curve to the right and the labor supply curve upward, via its effect on the shadow value of installed capital. Human capital accumulation, fostered by the increase in employment, has two effects: first, it shifts the demand curve leftward via its effect on the normalized FIG. 4. A shock to TFP in the HRW model.

19 878 marco maffezzoli capitalstock; second, it shifts the supply curve downward via its effect on the normalized Lagrange multiplier. As a result, the negative effect of TFP strictly dominates: the employment rate decreases monotonically and converges quickly to its steady-state value. The wage rate, instead, deviates persistently from the steady state, being strictly linked to the marginal utility of consumption. The latter will certainly be lower than in steady state as long as the deviations of consumption and employment remain strictly positive. The consumption level deviates persistently from its steady-state value, again for the standard consumption smoothing argument, while the employment rate converges monotonically from above. Therefore, the marginalutility of consumption remains persistently lower than in the steady state, and the wage rate higher The MU Model Comparing Figs. 2 and 3, we note that, on impact, the reaction of employment is slightly less pronounced in the MU model. However, its deviation from the steady state is clearly more persistent: after 60 quarters the employment rate is still 0.4 percentage points higher. 31 In the HRW model, instead, the employment rate converges back to the steady state in 20 quarters only. A more persistent deviation of employment is reflected into more persistent deviations of output, consumption, and investment. As a consequence, the dynamics of physicalcapitalis more pronounced and persistent, too; after 60 quarters, the physicalcapitalstock is still0.40 points higher in the MU model, while only 0.20 points higher in the competing one. The dynamics of the wage and rental rates are clearly different. On impact, the wage rate increases more in the MU model, then converging monotonically to its steady-state value. In the HRW model, the wage rate follows instead a hump-shaped path. The rental rate converges monotonically from above in the MU model, while it turns negative at some point and then converges from below in the HRW. To conclude, the simulated series for output, consumption, investment, employment, and physicalcapitalgenerated by the MU modeltend to be more persistent than the corresponding series generated by the HRW model; the only exception is the wage rate. Thanks to the particular internalpropagation mechanism, described below, productivity shocks in the MU model have a longer-lasting effect on almost all aggregate variables. Figure 5 describes how the union and the firm jointly react to the productivity shock. On impact, both the conditionallabor demand curve and the marginalrevenue curve mr shift to the right. The shift in the mr 31 The relationship between imperfect competition in the labor market and unemployment persistence has already been stressed by Bénassy (1997).

20 non-walrasian labor markets and rbcs 879 FIG. 5. A shock to TFP in the MU model. curve, however, is less than proportional. The normalized reservation wage remains constant by assumption, and therefore both the wage and the employment rates increase. The TFP level returns quickly to its steady-state level and shifts ceteris paribus the labor demand and marginal revenue curves back to the left: the employment and wage rates decrease. The accumulation of physical capital, instead, shifts ceteris paribus again both curves to the right, further increasing the employment rate. Output, investment, and physical capitalfurther increase, too, and therefore employment, and so on. If the growth rate of human capitalwere to remain constant, this virtuous circle would eventually drive the system to a new steady state, characterized by a higher employment rate. This is due to the fact that the union s firstorder condition does not depend on the shadow value of installed physicalcapital. 32 However, human capital accumulation quickens as long as the employment rate deviates positively from its steady-state value. This shifts ceteris paribus the labor demand curve to the left and therefore decreases the employment and wage rates over time. During the initialphase of the transition, the effect of physicalcapital accumulation dominates that of human capital and dampens the decrease in employment implied by the fast convergence of TFP. At some point, however, the effect of human capital accumulation remains the only relevant one and drives the system back to its initialsteady state. 33 As stressed by Hall(1998), persistence is a criticalproperty of the data that the standard neoclassicalmodelcan hardly reproduce. In the HRW 32 More precisely, because a wedge between the marginal productivity of labor and its marginal rate of substitution is allowed. 33 If the value of ϕ, the scale parameter in the human capital accumulation equation, is small enough, the increase in the employment rate has a negligible effect on the accumulation of human capitaland therefore on the endogenous growth rate. In this case, the normalized system converges back to the initial steady state at an extremely low speed; it behaves like a non-stationary system, and the numericalalgorithm may failto provide a stationary solution.

21 880 marco maffezzoli model, the persistence in all impulse responses depends directly on the properties of the exogenous driving process. In the MU model, instead, there is an endogenous mechanism that independently introduces persistence in employment and consequently in the other main variables Business Cycle Statistics Table II summarizes the standard business cycle statistics for the main Italian macroeconomic variables and compares them with the simulated ones generated by the competing models under the benchmark parameterization. The table is based on seasonally adjusted quarterly ISTAT series from 1982:1 to 1998:4, expressed in constant 1995 prices. Output is approximated by GDP, consumption by the households final consumption expenditure, investment by the gross fixed investment, and employment by the standard TABLE II Business Cycle Statistics: Monopoly Union vs. Hansen Rogerson Wright Std. Dev. % Volatility Italy MU HRW Italy MU HRW Y C I K N A Autocorrelation Comovement Y C I K N A Notes: The observed business cycle statistics are based on seasonally adjusted quarterly ISTAT series from 1982:1 to 1998:4, expressed in constant 1995 prices. Y: GDP over workingage population; C: households final consumption expenditure over working-age population; I: gross fixed investment over working-age population; N: standard units of labor over workingage population; A: total factor productivity. All variables have been transformed in logarithms and H-P filtered. The reported statistics are respectively the standard deviation, the relative standard deviation with respect to output, the autocorrelation coefficient, and the correlation coefficient with output. The simulated business cycle statistics are based on 1000 simulations over a 68-quarter horizon. The monopoly union (MU) and Hansen Rogerson Wright (HRW) models share the same benchmark parameterization. The simulated series have been H-P filtered for comparison purposes. All figures are averages across simulations.

22 non-walrasian labor markets and rbcs 881 units of labor. All variables were normalized with respect to the workingage population, transformed in logarithms, and H-P filtered. The simulated statistics are based on a set of 1000 simulations over a 68-quarter horizon the size of our sample. All simulated series have been H-P filtered for comparison purposes. The reported statistics are the standard deviation, the relative volatility (the ratio between the standard deviation of each variable and the standard deviation of output), the autocorrelation coefficient, and the correlation coefficient with output. We note that consumption in Italy is relatively more volatile than employment, and that investment is more persistent than both output and consumption. Furthermore, the correlation of employment with output is positive but relatively low. Turning to the simulations, we note that the standard deviation of output is higher than the observed one, respectively, 1.14 for the MU model and 1.24 for the HRW, against The models internal propagation mechanism enhances the effect of productivity shocks. Both models fail to reproduce the high relative volatility of consumption and employment. The simulated volatilities are equal on average, respectively, to 0.42 and 0.67 for the MU model and to 0.42 and 0.75 for the HRW model, while the observed volatilities are 0.94 and As expected, the more pronounced reaction of employment in the HRW model is directly reflected into a higher relative volatility. Note, furthermore, that even the rank is not reproduced, since consumption is more volatile in the data, while employment is more volatile in the simulations. However, both models reproduce the relative volatility of investment almost perfectly. The simulated autocorrelation coefficients are consistently lower than the observed ones. Note that investment is the more persistent variable in the data, but the less persistent one in the simulations, while the remaining variables namely consumption, employment, and output preserve their rank. All of the simulated series are highly procyclical; employment, in particular, is almost perfectly correlated with output. The last result is not surprising, since the dynamics of employment is mainly driven by shifts in the labor demand curve, which in turn are strictly related to the dynamics of output. This sharply contrasts with the observed correlation of employment with output, equalto only In the previous section I concluded that productivity shocks have more persistent effects in the MU model. This should be reflected in the simulated business cycle statistics, and particularly in the correlations of TFP with the lags and leads of the other simulated variables. Table III summarizes the correlation coefficients between current TFP and different lags and leads of output, consumption, investment, and employment; all variables have been H-P filtered. As expected, the MU model generates higher