Partial adjustment without apology Robert G. King Boston University, Federal Reserve Bank of Richmond and NBER Julia K. Thomas1 University of Minnesota, Federal Reserve Bank of Minneapolis and NBER March 01, 2004 1Please direct correspondence to Julia Thomas, Department of Economics, University of Minnesota, 271 19th Avenue South, Minneapolis, MN 55455; tel: 612 626 9675; email: [email protected]. We thank seminar participants at the Oxford Meetings of the Society of Eco- nomic Dynamics and the University of Virginia for their comments. All remaining errors are our soleresponsibility. Theviewsexpressedinthispaperarethoseoftheauthorsanddonotnecessarily reflect the views of the Federal Reserve Bank of Richmond or Minneapolis, or the Federal Reserve System. Abstract Many kinds of economic behavior appear to be governed by discrete and occasional indi- vidualchoices. Despitethis, econometricpartialadjustmentmodelsperformrelativelywell at the aggregate level. Analyzing the classic employment adjustment problem, we show how discrete and occasional microeconomic adjustment is well described by a new form of partial adjustment model that aggregates the actions of a large number of heterogeneous producers. We begin by describing a basic model of discrete and occasional adjustment at the micro level,whereproductionunitsareessentiallyrestrictedtoeitheroperatewithafixednumber of workers or shut down. We show that this simple model is observationally equivalent at the market level to the standard rational expectations partial adjustment model. We then construct a related, but more realistic, model that incorporates the idea that increases or decreases in the size of an establishment’s workforce are subject to fixed adjustment costs. In the market equilibrium of this model, employment responses to aggregate disturbances include changes both in employment selected by individual establishments and in the mea- sure of establishments actively undertaking adjustment. Yet the model retains a partial adjustment flavor in its aggregate responses. Moreover, in contrast to existing models of discrete adjustment, our generalized partial adjustment model is sufficiently tractable to allow extension to general equilibrium. 1 Introduction Inmanycontexts,actualfactordemandsclearlyinvolvecomplicateddynamicelements absent in static demand theory. For example, empirical studies of the market demand for labor typically find that lags, either of demand or of the determinants of demand, con- tribute substantially to the explanation of employment determination. The most frequent rationalization of such lags is that individual plants face marginal costs of adjustment that are increasing in extent of adjustment, leading them to choose partial adjustment toward the levels suggested by static demand theory. Many empirical studies also indicate, however, that the partial adjustment model is inconsistent with the behavior of individual plants or firms. For example, Hamermesh (1989) shows that individual plants undertake discrete and occasional workforce adjust- ments rather than the smooth changes implied by partial adjustment. Nonetheless, the model continues to be a vehicle for applied work, essentially because it is a tractable way of capturing some important dynamic aspects of market demand. It is frequently thus employed in an apologetic manner, with the researcher suggesting that it is a description of market, rather than individual, factor demand.1 In fact, Hamermesh (1989) finds that a labordemandaggregate, madeupofjustsevenplants, appearsatleastaswelldescribedby the partial adjustment model as by positing a representative firm that adjusts in a discrete and occasional manner. In this paper, we develop several models which embody the idea that individual pro- duction units adjust in a discrete and occasional manner, yet have the property that there is smooth adjustment at the aggregate level. In the first of these models, there is an exact observationalequivalenceattheaggregatelevelbetweenthestandardrationalexpectations 1See, for example, Kollintzas (1985). 1 model of partial adjustment (Sargent (1978)) and our model of an industry’s employment demand. More generally, this model illustrates key features of our modeling approach. Specifically, individual units face differing fixed costs of adjustment, so that the timing of their adjustments is occasional and asynchronized. Nevertheless, aggregation across plants leads to a smooth pattern of industry labor demand that is well-approximated by the standard partial adjustment model. Thus, the structural features that result in grad- ual adjustment (distributed lags) also imply that individual plants base their employment decisions on expectations of future wages and productivities (distributed leads). Our subsequent models provide a microeconomic foundation for the variety of plant- level adjustment examined in the empirical work of Caballero and Engel (1992, 1993) and Caballero, Engel, and Haltiwanger (1997). There, individual production units are assumed to adjust employment probabilistically, and that the probability of adjustment is a function of the difference between a target level of employment and actual employment. Aggregating from such adjustment hazard functions, which are their basic unit of analysis, they examine the implications of the resulting state-dependent adjustment behavior for aggregateemploymentdemanddynamics. Intheabsenceofamicroeconomicfoundationfor suchprobabilisticadjustment,CaballeroandEngel(1993,p. 360,paragraph2)explainthat they “trade some deep parameters for empirical richness.” In contrast, we explicitly model the plant’s adjustment decision as a generalized (S,s) problem and derive the adjustment hazard functions that are the starting point of previous research.2 We summarize some of the key stylized facts uncovered in the empirical literature and require that our theoretical models be consistent with them. One such finding is that an importantroutebywhichaggregateshocksaffectaggregateemploymentisbychangingthe 2Generalized (S,s) models were first studied by Caballero and Engel (1999) to explain the observed lumpiness of plant-level investment demand. 2 fraction of plants that choose to adjust. Accordingly, we move from our initial model to develop a generalized partial adjustment model in which the aggregate adjustment rate is anendogneousfunctionofthestateoftheeconomy. Indoingso, werelaxtheobservational equivalence to the traditional partial adjustment model where aggregate adjustment rates are invariant to changes in economic policy. Nonetheless, impulse responses establish that our generalized model retains the basic features of gradual partial adjustment. Another distinguishing feature of our theoretical approach is that it is feasible to undertake gen- eralized (S,s) analysis within a general equilibrium framework, so that the influence of aggregate shocks on equilibrium adjustment patterns may be systematically studied. The organization of this discussion is as follows. In section 2, we present the standard partial adjustment model, and in section 3 we discuss the evidence on microeconomic adjustment patterns that this standard model fails to reproduce. In section 4, we present the simplest form of a gradual adjustment market labor demand model consistent with discrete and occasional adjustment choice at the plant level. We show that this model is observationally equivalent to the standard partial adjustment model, given suitable choice of parameters. However, this preliminary model is limited in that individual production unitsundertakeonlyasingleadjustmentdecision,whichcorrespondstoenteringproductive activity with a fixed level of employment. In section 5, building on the model of entry described above, we develop a model that is consistent with the observation that plants hire varying amounts of labor, with these quantities adjusted at discrete and occasional times. We use this model to illustrate a hedging effect on the demand for labor that arises when a plant recognizes that there may be future departures from the workforce prior to its next employment adjustment. Next, in section 5.1, we endogenize the timing of employment changes by assuming that each plant faces a fixed cost of adjustment that is random across both time and plants. The 3 resulting generalized (S,s) model allows us to examine the influence of deep parameters on the adjustment process. Moreover, with a large number of plants, the model is sim- ilar to the traditional partial adjustment model in that it yields a smooth market labor demand. We illustrate the properties of this generalized partial adjustment model using a series of numerical examples.3 A distinguishing feature of our model, over and above its consistency with the microeconomic evidence on employment adjustment, is that it is able to reproduce the sharp changes in market employment demand found in the data during episodes involving large changes in productivity.4 We also illustrate how our framework may be tractably embedded within a fully specified general equilibrium macroeconomic model. Section 6 provides a brief conclusion. 2 The standard partial adjustment model The standard partial adjustment model relates current employment, N , to target or t desired employment, N , through t∗ N N =κ[N N ], (1) t t 1 t∗ t 1 − − − − 3Our generalized partial adjustment model is distinguished from earlier generalized cost of adjustment models, as summarized, extended and critiqued in Mortensen (1973), in that it suggests very different dynamicsattheestablishment-level. Nonetheless,becausethefinalmodelthatwepresentisessentiallyone withmanydynamicallyrelatedfactordemands,itiscapableofgeneratingsomeoftheaggregatedynamics that motivated researchers in this earlier area. For example, under unrestricted parameters, interrelated factordemandmodelswerefoundtobeconsistentwithoscillatoryapproachestothelong-runposition. The generalizedstochasticadjustmentmodelthatwedevelopcanalsogeneratesuchrichdynamics,althoughit does not do so under the parameters selected here. 4This is because the economywide rate of adjustment implied by our model varies with aggregate con- ditions. The traditional model under-predicts employment changes during such episodes precisely because the adjustment rate there is constant. 4 with 0 < κ < 1 being the fraction of the gap N N that is closed in the period. t∗ t 1 − − This specification implies the influence of past actual or desired employment on current employment, ∞ N =κN +(1 κ)N =κ (1 κ)jN . (2) t t∗ t 1 t∗ j − − − − j=0 X As shown by Sargent (1978), this empirical partial adjustment model may be derived as the solution to a firm’s dynamic profit maximization problem under the assumption that there are quadratic costs of adjusting the workforce. To develop this standard partial adjustment model, assume that the firm’s workforce declines, due to quits or mismatches, at the rate d [0,1), in the absence of any costly employment-adjusting action. If e is the t ∈ number of employees hired at time t, t =0,..., then N =(1 d)N +e (3) t t 1 t − − and the cost of actively adjusting the workforce is Ξ(e ). The quadratic cost assumption t is that B Ξ(e )= e2, (4) t 2 t where B >0 is a cost parameter. Equation (4) captures the idea that the firm’s marginal adjustment cost is rising in the extent of employment adjustment.5 Let A represent a productivity shift term, and let W represent the real wage at time t t t, t = 0,1,.... Both A and W are serially correlated random variables, known at the t t beginning of period t. The flow profit of the firm at time t, π , is output f(N ,A ) less t t t adjustment costs, Ξ(e ), and the wage bill, W N . t t t π =f(N ,A ) Ξ(e ) W N . t t t t t t − − 5This same idea is incorporated in alternative adjustment cost functions that are used in applied work. 5 Discounting future earnings by the constant factor, β (0,1), the firm solves the following ∈ optimization problem: ∞ max E βt f(N ,A ) Ξ(e ) W N (A ,W ) t t t t t 0 0 {Nt,et}∞t=0 ÃXt=0 h − − i¯¯ ! ¯ subject to (3), N given. ¯ 1 ¯ − Let v represent the current-value multiplier associated with (3). Then the efficient t choice of labor requires: ∂f(N ,A ) t t =W +v βE v (1 d), (5) t t t t+1 ∂N − − t while the corresponding condition for gross employment changes, e , is t ∂Ξ(e ) t v = . (6) t ∂e t Using (3), (4) and (6) to simplify (5), we have ∂f(N ,A ) t t =W +B(N (1 d)N ) β(1 d)E B(N (1 d)N ). (7) t t t 1 t t+1 t ∂Nt − − − − − − − Assuming that the production function is quadratic or, more generally, approximating it using a second-order Taylor expansion, 1 1 f(N ,A ) f +f N + f N2+f A N +f A + f A2, (8) t t ≡ n t 2 nn t na t t a t 2 aa t and defining Φ= B−fnn+β(1−d)2B, we can rewrite (7) as (1 d)B − W f A f t na t n βE (N ) ΦN +N = − − . (9) t t+1 t t 1 − − (1 d)B − 6 Note that Φ > 0, since f < 0 is required by concavity of the production function. This nn is sufficient to ensure that the second-order stochastic difference equation (9) has two real roots, µ (0,1 d) and µ >[β(1 d)] 1, that jointly solve 1 2 − ∈ − − 1 µ µ = 1 2 β Φ µ +µ = . 1 2 β The firm’s target employment in (1) is then given by N = µ1 E ∞ ( 1 )j f A W (A ,W ) + fn , (10) t∗ B(1 d)(1 µ ) µ na t+j − t+j ¯ t t 1 1  − − 1 hXj=0 2 ³ ´¯¯ i − µ2  ¯  and its adjustment rate is ¯ ¯ κ =1 µ . (11) 1 − Our expression for target employment illustrates Sargent’s (1978) result that the pres- ence of lags in employment, as in (1) under rational expectations, implies leads. Expec- tations of future wages and productivity influence the current employment target since its choice, given adjustment costs, will in part determine future employment. This presence of expectational leads dampens the response of current employment to changes in current wage and productivity and yields smooth, gradual changes in employment over time. Cur- rent employment, N , is directly related to lagged, N . Moreover, the other determinant t t 1 − of current employment, target employment N , is a discounted sum of future wages and t∗ productivities, and, as such, is only partly determined by current wage and productivity. 7 3 Disconcerting evidence While the traditional partial adjustment model offers a tractable framework within whichtostudygradualaggregatelaboradjustment,thereisconsiderableempiricalevidence tosuggestthatthemodelisnot consistentwiththebehaviorofindividualproductionunits. This evidence also suggests a number of stylized facts about individual and aggregate adjustment, which this section summarizes. Stylized fact 1: Adjustment at the plant level is discrete, occasional and asynchronous. Hamermesh (1989) examines monthly data on output and employment between 1983 and 1987 across seven manufacturing plants. For each plant, output fluctuates substantially over the sample. Employmentexhibitslong periods of constancy broken by infrequent, but large, jumps at times roughly coinciding with the largest output fluctuations. Hence, the plant data are not consistent with the smooth employment adjustment that would arise from convex adjustment costs. Stylized fact 2: Aggregates exhibit smooth and partial adjustment. Hamermesh (1989) alsoexaminesthebehavioroftheaggregateofhissevenmanufacturingplants. Hefindsthat fluctuations in aggregate employment across plants resembles the dynamics of aggregate output and appears consistent with smooth adjustment behavior of aggregates. More specifically, Hamermesh argues that the standard partial adjustment model works quite well at the aggregate level, even though it does not describe the behavior of individual production units.6 6In particular, he compares log likelihood values from the estimation of a smooth adjustment model based on quadratic adjustment costs with those from a lumpy adjustment fixed-cost alternative. Forplant leveldata,thelattermodelachievesmuchlargerlikelihoodvalues,indicatingthatlumpyadjustmentbased onfixedcostsbetterdescribestheplantleveldata. Further,theswitchingmodelestimatesofthepercentage ‘disequilibrium’ required to induce adjustment are large. This indicates that plants vary employment with 8