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The labour market over the business cycle: can theory fit the facts?

We examine the ability of six labour market models to account for the business cycle behaviour of UK labour markets when embedded in a stochastic growth model. We assess the models in terms of: (i) their ability to mimic general business cycle correlations and volatility (ii) their success at explaining the persistence of labour market fluctuations and (iii) whether they can explain why the growth and speed of adjustment of labour market variables changes between periods of expansions and contractions. The main success of the models is their ability broadly to account for business cycle correlations and comovements and the variations in employment/unemployment growth rates between expansions and contractions. However, there are three main failures: (i) the models tend to produce insufficiently volatile employment and unemployment fluctuations (ii) they tend to produce too strong a correlation between wages and employment and (iii) most of them generate only brief temporary deviations in unemployment in response to shocks rather than the protracted dynamics of the data.

I. INTRODUCTION

The real business cycle (RBC) paper of Kydland and Prescott (1982) heralded a methodological revolution in macroeconomics, whereby numerical simulation of stochastic dynamic general equilibrium models became an established means of evaluating macroeconomic models. By expressing models in terms of the primitives of an economy (technology, preferences, and market structure), this approach has delivered the form of macroeconomic modelling advocated in the seminal paper of Lucas (1976). This approach offers both detailed microeconomic foundations of business cycle phenomena and a platform from which to construct optimal economic policy. Since its original emphasis on productivity shocks, the RBC literature has developed in numerous directions, e.g. incorporating money (i.e. Cooley and Hansen, 1989; Fuerst, 1992), extending to an open economy (Backus et al., 1992), introducing government expenditure and taxation (Braun, 1994; McGrattan, 1994), as well as models of sticky wages and prices (Cooley and Hansen, 1995; Yun, 1996). While these contributions have met with varied empirical success, there now exists a wide range of basic general equilibrium models with which to view macroeconomic phenomena.

One particularly active area of research has focused on labour markets (see, inter alia, Hansen, 1985; Hansen and Sargent, 1988; Bencevenga, 1991; Benhabib et al., 1991; Christiano and Eichenbaum, 1992; Cho and Cooley, 1994; Danthine and Donaldson, 1990; Boldrin and Horvath, 1995; Merz, 1995). The reasons for this focus are twofold.

(i) Fluctuations in employment seem fundamentally connected to the business cycle. For both the USA and UK, the cyclical volatility of hours worked is of the same magnitude as the volatility in output, suggesting that `an understanding of aggregate labour market fluctuations is a prerequisite for understanding how business cycles propagate over time' (Kydland, 1994). Further, in terms of the policy debate the main social cost of business cycles is invariably seen as fluctuations in unemployment.

(ii) At the heart of the basic RBC model (see King et al., 1988) is a neoclassical model of the labour market which essentially explains employment fluctuations via intertemporal substitution. In other words, employment varies because of changes in relative wages between periods. According to this mechanism, when wages are high, agents are prepared to work harder, while when wages are low, labour supply declines. The difficulties faced by this model in explaining observed employment fluctuations have been well documented (i.e. Barro and King, 1984; Mankiw et al., 1985), but as yet no consensus has emerged regarding an alternative model. The purpose of this paper is to examine a sample of these alternative aggregate labour market models and see the extent to which they can satisfactorily account for certain features of UK data.

While other studies (e-.g. Hansen and Wright, 1992; Burgess, 1993; Fairise and Langot, 1994) have compared alternative labour market models, two features distinguish our approach.

(i) We use UK data to evaluate the competing labour market models. This is important because many of the models we examine have been constructed in order to explain US data, but according to the methodology outlined in Prescott (1986), these models (suitably calibrated) should also account for UK data. Given the extreme behaviour of UK labour markets, this represents a substantial challenge.

(ii) Our analysis differs in the way it assesses the performance of these labour market models. Within the RBC literature the customary way of evaluating models is to compare standard deviations and cross-correlations constructed from the data with the same measures constructed from simulations of the model. However, there' are numerous other important and well documented statistical features of the labour market which these theoretical models should account for. For instance, Blanchard and Summers (1986) show that European unemployment is highly persistent, so that shocks to unemployment have very long-lasting if not indefinite effects. Further, numerous authors, e.g. Neftci (1984), Stock (1989), and Acemoglu and Scott (1994), document that the behaviour of the labour market changes over the business cycle. In particular, they find that the growth of labour market variables differs between expansions and contractions, and also that their persistence changes -- shocks tend to have longerlasting impacts in recessions. We therefore examine the ability of our theoretical models to explain both the persistent and asymmetric nature of labour market fluctuations.

The structure of the paper is as follows. Section II outlines the key stylized facts about the cyclical behaviour of the UK labour market that we want our theoretical models to replicate. Section III then discusses the various alternative models we examine, while section IV examines the simulation properties of these models and their ability to mimic the data. Section V concludes.

II. UK LABOUR MARKETS AND THE BUSINESS CYCLE

(i) Cyclical Stylized Facts

Table 1 documents some basic stylized business cycle facts about the UK labour market. In doing so it follows the majority of the RBC literature and quotes standard deviations and cross-correlations for the cyclical component of UK labour market variables (the data are explained in an appendix). In other words, our main interest is in the volatility of different macroeconomic variables over the business cycle and also in whether groups of macroeconomic variables move in tandem. In order to focus on the business cycle component we first detrend the data using the Hodrick-Prescott (HP) filter (Hodrick and Prescott, 1980). Both the accuracy of this choice of filter (see Harvey and Jaeger, 1993; King and Rebelo, 1993; Cogley and Nason, 1995a) and the relevance of this set of statistics used to define the business cycle (see Watson, 1993; Hansen and Heckman, 1996) have been the subject of much criticism. However, in order to enable easy comparison with the rest of the literature, we follow standard practice in applying the HP filter.

[TABULAR DATA NOT REPRODUCIBLE IN ASCII]

From Table 1 we stress the following business cycle facts concerning the UK labour market.

(i) Whole-economy total hours worked is at least

as volatile as output (as measured by GDP). (ii) Changes in total hours are split approximately

equally between average hours worked and

employment. (iii) All the employment measures are strongly

procyclical (i. e. employment increases and

decreases at the same time as output). (iv) Employment tends to lag, while average hours

tend to lead output over the business cycle. (v) Real wages are barely correlated with

employment or output. (vi) Unemployment and vacancies display the most

variability over the business cycle. (vii) Unemployment is strongly counter cyclical and

tends to lag output, whereas vacancies are

procyclical and tend to lead output. (viii) There is a strong negative relationship between

unemployment and vacancies over the

business cycle (the so-called `Beveridge curve').

It is this core set of facts that we will use to examine the performance of our various candidate theoretical models.

(ii) Persistence in the Labour Market

While these stylized facts are illuminating, there are many other important features of the labour market which are ignored in Table 1. One of these is the belief that labour market fluctuations are extremely persistent -- that is, shocks to employment or unemployment tend to have long-lasting effects. For instance, Blanchard and Summers (1986) argue that European unemployment is characterized by significant hysteresis. As a consequence, negative shocks to unemployment do not lead to merely temporary changes in the unemployment rate but tend to produce permanent increases.

To measure the persistence of UK labour market variables we use the unit root test of Cochrane (1988). The intuition behind this test is relatively simple. If a variable is a random walk (that is, the best forecast of the variable next period is equal to its current value, so that any shocks today have a permanent influence on future values), then the further ahead in time one goes, the more uncertainty there is about the level of the variable. In particular, the variance of two-period changes in the variable should have twice the variance of the one-period change, the variance of the three-period change should be three times the variance of the one-period change, etc. Cochrane (1988) therefore proposes a test of a unit root by examining how the variance of the k period difference changes relative to k times the variance of one-period changes.(2) If Cochrane's test has a value near to zero, then the variable tends to show only temporary changes in response to a shock, or, in other words, permanent shocks are not very important. However, if the test statistic is near to one, then persistent random-walk-type behaviour is very important and shocks have a long-lasting effect. If the test statistic goes above one, then there is even more persistence in the variable and the effect of the shock gets amplified over time. An alternative interpretation of Cochrane's test, which we shall sometimes use, is that it measures the relative importance of permanent shocks in contributing to the volatility of a variable -- the nearer to zero is the test statistic, the less important are permanent shocks.

Figure 1 shows the results of the Cochrane test for our set of UK labour market variables. The finding of Blanchard and Summers regarding the persistence of UK unemployment is immediately obvious -- shocks to unemployment have a very longlasting impact, with no evidence that eventually things return to their previous equilibrium. The evidence from vacancies also suggests that these contain important permanent shocks. However, unlike unemployment, the value of the Cochrane test for vacancies does decline after around 10 quarters, suggesting that, while shocks to vacancies have a long-lasting effect, they are not as persistent as shocks to unemployment. The same comment holds for the unemployment-vacancy ratio -- in other words, over the course of the cycle, the Beveridge curve changes position and shows no tendency to return. Like unemployment, employment shows strong evidence of permanent shocks affecting the labour market and this naturally feeds through into the total hours series. Average hours and average earnings show less in the way of persistence, although after 30 quarters the test statistic suggests that even these series reflect important random-walk-type behaviour. Overall, the picture that emerges is not one of temporary fluctuations in the labour market, but rather fluctuations which are characterized by highly persistent changes or regime shifts.

(iii) Asymmetric Behaviour in the Labour Market

Another distinguishing feature of the labour market is its non-linear behaviour over the business cycle. Burgess (1993) stresses that, in order to explain cyclical labour market phenomena, it is crucial to understand how hiring costs change over the cycle with the tightness of the labour market, and, indeed, secularly with legislative changes. Neftci (1984), Stock (1989), and Acemoglu and Scott (1994) all document evidence that reveals that the stochastic properties of the labour market change over the business cycle. To investigate this issue we estimate equations of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [S.sub.t] is a variable which takes the value 1 when the economy is in recession and zero otherwise, and [x.sub.t] denotes a particular labour market variable (such as employment, unemployment, average earnings, etc.), all detrended using the HP filter. If the growth own is lower in recessions than expansions, then we should find K, to be significantly negative (we shall call this mean asymmetry). If the persistence (or the speed at which the labour market responds to exogenous disturbances) of [x.sub.t] also varies, then [rho.sub.1], should also be significantly different from zero. (If [rho.sub.1], is negative, then labour market dynamics slow down in recession, while if it is positive, the labour market works more quickly. We shall call evidence that [rho.sub.1] does not equal zero dynamic asymmetry.) The economic hypothesis here is whether shocks feed through the labour market quicker in economic downturns than in upswings. This idea captures the fact, for instance, that unemployment tends to rise more quickly in recessions than it declines in expansions. To detect labour market asymmetries in this way relies upon a particular choice of [S.sub.t] or, equivalently, a particular definition of the UK business cycle. Acemoglu and Scott (1994) use a variety of different measures, but in this paper we use the HP filter to define our cyclical indicator in order to provide consistency with our earlier results. We use the HP filter to define a cyclical component of GDP, [y.sup.c.sub.t]=[y.sub.t] -[y.sup.HP.sub.t] where [y.sub.t] denotes GDP and [y.sup.HP.sub.t] is the estimate of the trend from the HP filter. We then define [S.sub.t] = 1 [ify.sup.c.sub.t] < 0 and [S.sub.t] = 0 otherwise. Therefore our recessions are defined as periods when output is below trend, and expansions where output is above its trend.

We test for mean asymmetry by examining whether [Nu.sub.1] = 0 and for dynamic asymmetry by whether [Rho.sub.1] = 0 (in cases where we use more than one lag in the dependent variable, then the test is for all the interactive terms between the recession dummy and lags). Table 2 shows the results of testing for labour market asymmetries using (1) and quotes the p-values for the significance of the asymmetries. An asterisk denotes that we find evidence in favour of asymmetries at the 5 per cent significance level. We also show the estimated conditional growth rate of the variables in expansions and contractions [Nu.sub.E] and [Nu.sub.C] respectively) and estimates of the persistence(3) in expansions and contractions ([Rho.sub.E] and [Rho.sub.C] respectively).
Table 2
Business Cycle Asymmetries in the UK Labour Market

Variable            Mean        [Nu.sub.E]   [Nu.sub.C]    Dynamic
asymmetries                                  asymmetries

Total hours         0.018(*)       0.003       -0.004      0.044(*)
Employment          0.468          0.001        0.000      0.858
Average hours       0.008(*)       0.000       -0.002      0.659
Average earnings    0.544          0.006       -0.006      0.527
Unemployment        0.057         -0.002        0.006      0.527
Vacancies            .009(*)       0.014       -0.006       .001(*)
U/V                 0.026(*)      -0.008        0.011      0.001(*)

Variable            [Rho.sub.E]   [Rho.sub.C]
asymmetries

Total hours            0.016        -0.345
Employment             0.737         0.734
Average hours         -0.038        -0.558
Average earnings       0.846         0.785
Unemployment           0.846         0.785
Vacancies              0.510         0.818
U/V                    0.643         1.021




Notes: The table shows the p-value of a restriction test for excluding the recession dummy. Mean asymmetry denotes a test for whether the growth rate differs over the business cycle and dynamic asymmetry denotes a test for whether dynamics differ over the business cycle. An asterisk denotes the recession dummy is significant and there are business cycle asymmetries. All tests reported are from the optimal lag length AR model, where the lag length was chosen using Schwarz Information Criteria. The columns [Nu.sub.E] and [Nu.sub.C] denote the mean growth rate of the variable in expansion and contraction periods respectively, and [Rho.sub.E] and [Rho.sub.C] denote the persistence of the variable in expansion and contraction periods respectively.

Although the evidence is not as pervasive as that outlined in Acemoglu and Scott (1994),(4) there is still evidence that the behaviour of the labour market varies over the business cycle. The asymmetries are all of a form that accords with intuition: growth in total hours is positive in expansions but negative in recessions, average hours contract in recessions but not in expansions, vacancies rise in expansions and decline in contractions, and the UV curve moves in during an expansion and shifts out in recessions.

III. SOME LABOUR MARKET MODELS

In this section we describe the basic structure and intuition of six alternative models of the labour market which we then embed in a stochastic dynamic general equilibrium model. The choice of these particular six models is inevitably arbitrary and suffers from sins of omission. However, we choose these models in the belief that they reflect a broad spectrum of views regarding the operation of the labour market, e.g. Walrasian and non-Walrasian, models of instantaneous market adjustment and more sluggish adjustment, models with fully competitive markets versus imperfect competition models. We are also aware that we have chosen relatively simple models, each of which deviates in a limited number of ways from the textbook neoclassical paradigm. Because a successful labour market model is likely to contain several such deviations, the models here are inevitably going to have the appearance of being `straw men'. However, our hope is that even if none of the models can successfully match the various features of the labour market noted above, we can arrive at some suggestions regarding the most promising direction for future research.

(i) Basic Neoclassical Labour Market

We take as our basic neoclassical model that of King et al. (1988). At the heart of this model is a representative agent who each period decides how many goods and services to consume and how much in the way of labour services to provide. The agent makes these choices by trying to maximize the present discounted value of future utility. Normalizing the time endowment to 1 this can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [c.sub.t] denotes consumption at time t, [n.sub.t] denotes hours worked so that 1 - [n.sub.t] denotes time consumed as leisure and [Beta] is the discount factor. These consumption and leisure choices are made subject to an economy-wide feasibility constraint, namely

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [k.sub.t] denotes the capital stock, [Nu] = 1 - [Delta.sub.k] (where [Delta.sub.k] is the depreciation rate of capital) denotes the proportion of the capital stock which does not depreciate during the period, and [y.sub.t] denotes output. The second equality defines our production function, where we assume output is produced by a constant-returns Cobb-Douglas function subject to a random technology term, [Theta.sub.t], as in Kydland and Prescott (1982). This problem yields the well-known first-order conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [U.sub.jt] denotes the derivative of the utility function with respect to j at time t (e.g. the marginal utility of leisure or consumption) and MPK and MPL refer to the marginal products of capital and labour respectively.

The first equation says that the consumer sets their marginal rate of substitution for leisure and consumption equal to the real wage rate (which, owing to our assumption of competitive labour markets, equals the marginal product of labour). Therefore, if the real wage rate increases, then by the standard properties of the utility function [U.sub.ll], [U.sub.CC] < 0), this must be met by some combination of increased consumption and less leisure. The second equation is the intertemporal equation which relates consumption growth to the net return on capital.

In this simple model the source of economic fluctuations is variations in the technology term, [Theta.sub.t], which causes fluctuations in employment by affecting the MPL, or labour demand, as shown in Figure 2. A good (bad) productivity shock causes an increase (decrease) in labour demand and assuming a constant labour supply curve brings about some combination of increased (decreased) wages and employment. The extent to which wages and/or employment benefit from this increased labour demand depends on the slope of the labour supply curve and, as stressed in Lucas and Rapping (1969), this depends on the intertemporal substitution of labour supply. If there are large temporary changes in real wages, and agents are prepared willingly to substitute labour supply between time periods, then the labour supply curve will be flat (i.e. very elastic) and productivity shocks will bring about large changes in employment but relative modest increases in wages. By contrast, if agents are unwilling to rearrange their work effort intertemporally, then the labour supply curve will be near vertical and wages rather than employment will increase in response to higher labour demand.

In summary, this model is based solely on productivity shocks leading to significant shifts in labour demand which are translated into employment movements according to the willingness of agents to reallocate their work effort between time periods.

(ii) Indivisible Labour Model

A common criticism of the basic model outlined above is that to generate the observed combination of large cyclical movements in employment and acyclical wages (as seen in Table 1) it is necessary to assume a large intertemporal elasticity of substitution. However, a wide body of empirical evidence (see the survey by Pencavel (1986)) based on both microeconomic and aggregate data suggests that male labour supply is fairly inelastic -- in other words agents do not alter their labour supply by much when wages change. According to this evidence the above model is unlikely successfully to account for observed UK business cycle employment fluctuations as the labour supply curve should be nearly vertical.

This observation motivates the model of Hansen (1985) and Rogerson (1988), who suggest a model of the labour market which can generate a highly elastic aggregate labour supply curve irrespective of the labour supply elasticity of individual agents. The means of achieving this is to assume that all employment fluctuations occur at the extensive as opposed to the intensive margin. That is, all employment fluctuations occur as a result of changes in the number of people employed, as opposed to variations in the average number of hours worked per person employed. While Table 1 shows this is not a good approximation for the UK, it is a more reasonable assumption for the USA.

The Hansen-Rogerson model assumes that all employed individuals work a fixed shift length, [h.sub.0], but that only a proportion, [Omega.sub.t], of individuals work in any time period, so that aggregate hours worked in the economy is [n.sub.t] = [h.sub.0][Omega.sub.t]. All those agents who are not employed (1 - [Omega.sub.t] of them) consume their time endowment totally as leisure. To justify this result that agents work either zero or [h.sub.0] hours, Hansen assumes that working involves some fixed costs (such as commuting time) which means that agents either prefer to work a fixed shift or not at all. In other words, it is not optimal for each agent to work [Omega.sub.t] [h.sub.0] hours.

To determine who is employed in any one period, Hansen and Rogerson assume the (stylized) existence of a lottery, where every individual has a [Omega.sub.t] probability of obtaining work. The effect of the lottery is that it convexifies individuals' preferences so that the utility function of the representative agent is linear in employment, e.g. the marginal utility of leisure is the same irrespective of how many hours the representative agent works. As a result, the labour supply curve of the representative agent is highly elastic, as large employment fluctuations bring about no changes in the marginal utility of leisure. Therefore, by assuming labour is indivisible, the Hansen model can generate a flat labour supply curve and potentially explain the observed combination of large fluctuations in employment and the absence of cyclical variation in wages.

(iii) Labour Hoarding

The Hansen-Rogerson model assumes that firms can only increase total hours worked by increasing employment; it rules out any adjustment along the intensive margin. In contrast, Burnside et al. (1993) place the main focus of their model on this intensive margin, i.e. average hours. As in the indivisible labour model, they assume that individuals work a fixed shift length, [h.sub.0]. However, what is important for the production function is not the length of the shift that individuals work, but how much effort they contribute during their shift. In other words, actual labour services supplied by an individual are given by [e.sub.t][h.sub.0], where [e.sub.t] denotes effort and which we normalize to lie between zero and one. If individuals work at their full capacity, then [e.sub.t] = 1, while if they put in no effort [e.sub.t] = 0. Burnside et al. motivate this effort variable in terms of labour hoarding or, in other words, labour utilization. When [e.sub.t] = 1 workers are fully utilized and there is no labour hoarding, but as [e.sub.t] tends to zero labour hoarding rises. Denoting [N.sub.t] as the number of individuals employed, the production function is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From the consumer's perspective, leisure is given by T - [Xi] - [e.sub.t][h.sub.0], where T is their time endowment, [Xi] is the same fixed cost to working (i.e. commuting time) as in the indivisible labour model, and the final term shows that the disutility from working depends on how much effort people contribute during their shift.

With the exception of the introduction of an effort variable, this model is the same as for our previous models. However, Burnside et al. make one additional change which enables the labour-hoarding variable to play a substantive role. They assume that firms have to choose employment [N.sub.t] before they are aware of the value of [Theta.sub.t], and once they are aware of the productivity shock they then have to choose [e.sub.t]. In other words, firms make their decision about hiring or firing workers based on their expectation of what the productivity shock will be. Once they discover the true value of the productivity shock they cannot immediately adjust their work-force -- hiring and firings have to be done next period. However, while they cannot adjust the extensive margin in response to productivity shocks, they can alter the intensive margin by persuading workers to contribute more effort. Thus they adjust their input of labour services by varying [e.sub.t] or, in other words, by varying labour hoarding, as shown in Figure 3. For instance, if there is a bad productivity shock the firm would want to reduce employment ([N.sub.t]) but cannot do so this period and so instead it lowers [e.sub.t]. However, come the end of the period it can adjust the level of employment so that, in essence, the Burnside et al. model is one where the firm faces infinite adjustment costs to changing employment within the current period but no adjustment costs at the end of the period.

(iv) Search

The above variants of the neoclassical model are Walrasian in that they try and explain labour market fluctuations by the intersection of time-varying labour supply and labour demand curves. A popular model of the labour market which dispenses with this Walrasian perspective is the search model (for a textbook treatment see Pissarides (1990) and for an application to the RBC literature see Merz (1995)). The basic insight of search models is that increases in employment cannot be achieved instantaneously but instead it takes time to match up unemployed workers with firms who are advertising vacancies and therefore employment increases only slowly. Further, the speed at which the unemployed find jobs depends on how many people are competing for the same vacancy, so that the speed at which the labour market adjusts will vary over the cycle.

The key analytical concept in search models is the matching function which determines the way in which individuals seeking work (the unemployed) combine with firms offering vacancies to produce new hires. Following the UK evidence of Jackman et al. (1989), we write this matching function as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [m.sub.t] denotes new hires, [v.sub.t] denotes the number of vacancies available, [u.sub.t] denotes the number of individuals unemployed, and A is a parameter which reflects the efficiency with which the labour market clears. The matching function is analogous to a production function, but in this case the inputs are individuals looking for work and firms looking for workers, and the output is the number of new hires, i.e. the number of vacancies filled during the period. While (6) explains inflows into employment, we also need to explain inflows into unemployment. Mortensen and Pissarides (1994) offer an endogenous model of job destruction, but, for ease of analysis, we follow the standard search model and assume that every period a proportion ([Delta]) of the work-force enters into unemployment. The maximization problem of the representative agent is as outlined in (1) except that employment now evolves subject to the law of motion

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the consumer takes the level of vacancies as given.

In order to determine employment and output we need to know the level of vacancies. We assume that having a vacancy involves a cost (per vacancy) of [a.sup.t] and that firms choose their employment sequence to maximize the present discounted value of their profits. Current period profits are defined as [Theta.sub.t][k.sup.Alpha.sub.t-1][h.sup.1-Alpha.sub.t] - [w.sub.t] [h.sub.t] - [r.sub.t][k.sub.t-1] - [a.sub.t][v.sub.t], where [w.sub.t] and [r.sub.t] denote the real wage and rental price of capital respectively. In choosing their optimal level of employment for current and future periods, firms have to balance the fact that hiring may take several periods (so that it may run the risk of having insufficient employment during a high productivity period) against the fact that unfilled vacancies incur a cost per period.

A major difference in the search model compared to our previous labour market models is the determination of wages. In the previous models factor markets are competitive and so labour is paid its marginal product. However, in the case of search this is no longer the case. Because both the firm and the worker cannot guarantee immediately to find a new employee/job, each side can try and extract rents from the match (the rents being the difference between what the matched job and worker produce compared to what they would earn elsewhere allowing for the fact that for a period the firm will have an unfilled vacancy and the worker will be unemployed). The distribution of these rents will depend on the monopoly power of the firm and worker in the bargain. However, assuming some monopoly power on the part of labour, the consequence is that wages reflect both the marginal product of labour and also the surplus that is created by the worker and the firm.

The introduction of search into the RBC framework has three main implications.

(i) Because it takes time for a firm to fill a vacancy or a worker to find a new job, the search-based model alters the cyclical dynamics of employment. In response to a shock it now takes employment several periods to respond; so that employment dynamics are more spread out over the cycle rather than responding rapidly to productivity shocks.

(ii) Wages no longer reflect just the marginal product of labour and so they will no longer reflect so directly changes in productivity.

(iii) The previous models we have outlined have offered a model of employment but not explicitly modelled unemployment. Instead, unemployment is simply the opposite of employment. By contrast, the search model contains a detailed structure which explains how vacancies and unemployment vary period to period. Figure 4 shows how search models explain unemployment and vacancies. The UV (or Beveridge) curve shows how the matching function relates unemployment and vacancies together. The higher the level of unemployment then the lower the number of unfilled vacancies on the market -- if lots of people are competing for any jobs that are available, then a large number of vacancies will be filled. By contrast, the VS curve describes how the number of vacancies supplied by firms depends on the level of unemployment. If unemployment is very high, then a firm knows that it will fill its vacancy quickly and so the expected cost of announcing a vacancy is low. As a consequence, firms open a large number of vacancies and so the VS curve is upward sloping. Labour market equilibrium is where the UV and VS curves intersect to determine equilibrium unemployment and vacancies. As outlined in Pissarides (1990), the VS curve shifts in response to aggregate shocks and so traces out the UV curve so that unemployment and vacancies are negatively correlated over the business cycle.

(v) Gali's Imperfect Competition Model

Gali (1995) also allows for non-competitive factor markets but, in addition, he introduces monopoly power on the part of firms in the product market. He then uses this market power to introduce a concept of `involuntary unemployment' in an attempt to make RBC models make contact with more Keynesian notions of unemployment. In what follows we give a brief overview of Gali's model.

The production side of the economy consists of intermediate industries which provide inputs used by the final goods producing firms. Each intermediate industry has some market power, as different intermediate goods are not perfect substitutes in production of the final goods and, as a result they price their products above marginal costs.

The consumption side of the economy is as before, but with one substantial exception. In previous models consumers chose their labour supply, taking as given the wage rate. In contrast, Gali assumes that labour supply decisions are undertaken by trade unions who set wages and that, at these wages, firms then choose their employment. At the beginning of each period the trade unions negotiate anew wage and firms choose new employment levels. When the trade union negotiates it takes account of how setting a higher wage will lead to lower labour demand so that the first-order condition for labour supply becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [eta] denotes the wage elasticity of labour demand with respect to wages at time t. Combining the first-order conditions of firms and consumers gives an equilibrium sequence for employment and wages, which we denote {[n.sup.*.sub.t]} and {[w.sup.*.sub.t]} respectively. This equilibrium wage sequence reflects the fact that the trade union knowingly accepts lower employment as a necessary cost of achieving a higher wage. However, in the absence of trade union power, consumers would want to provide a higher level of employment at this equilibrium wage. Gali uses this insight to define a concept of involuntary unemployment. He computes the employment sequence In,**) which consumers would provide given a wage sequence N%') and in the absence of trade unions, and then defines involuntary unemployment as [n.sup.**.sub.t] -- [n.sup.*.sub.t], as we show in Figure 5. Gali terms this a `Keynesian' model, in the sense that its concept relies on imperfect competition and the notion that, under monopoly power, prices will always be too high and output too low relative to the social optimum.

(vi) Distortionary Taxes

All of the above models have relied upon one shock (random variations in productivity) to produce employment fluctuations. The first-round effects of these productivity shocks is to shift the labour demand curve rather than the labour supply curve. Increasingly, a number of models have been developed which rely upon other shocks as well as productivity to explain business cycles (i.e. moretary policy shocks, see inter alia, Cooley and Hansen (1989) and Benhabib and Farmer (1994)). Another important additional source of labour market fluctuations is variation in labour and capital taxes, which affects the real wage and real interest rate received by individuals and so influences the equilibrium sequence of output and employment. Using models calibrated to US data, Braun (1994) and McGrattan (1994) show how allowing for variation in labour and capital tax rates can improve the performance of the basic RBC model.

Introducing taxes and government expenditure does not alter the economy-wide budget constraint, but it does alter the utility function where consumption is now defined as a weighted average of personal consumption and government expenditure. However, the introduction of taxes does alter the relative prices consumers face, so that the first-order conditions become

[U.sub.lt] = [U.sub.cf][MPL.sub.t]( -- [tau.sup.l.sub.t]

[U.sub.et] = [E.sub.t] [Beta](1 + ([MPK.sub.t+1] - [delta])

(1 -- [tau.sup.k.sub.t]))[U.sub.ct] + 1

where [[tau.sup.l.sub.t] and [tau.sup.k.sub.t] denote the marginal rate of tax on labour and capital income respectively. In other words, what matters for labour supply and consumption is the after-tax wage and the net real interest rate. Equation (9) makes clear that variations in tax rates will, for a given gross wage, lead to shifts in the labour supply curve. That is, for a given gross wage, an increase in labour taxes reduces the net wage received by the consumer and so reduces the number of hours they are prepared to work. Therefore, in this model, business cycle fluctuations are caused by productivity shocks shifting the labour demand curve and changes in taxes shifting the labour supply curve. The advantage of this is that, assuming that both labour demand and labour supply curves move together over the business cycle (as shown in Figure 6), it will generate large changes in employment but very little fluctuation in real wages -- exactly what we observe in the UK data.

IV. MODEL PROPERTIES

A key component of the methodological innovations made in Kydland and Prescott (1982) is the use of model calibration and simulation to assess theoretical models. This process involves first choosing values for key model parameters, such as the degree of risk aversion, the persistence of productivity shocks, etc., and then using numerical techniques to solve the model before performing simulations to analyse the properties of these `laboratory' economies.(5) If the simulations are to have any relevance to understanding UK business cycles, the calibration process is critical. It is obviously important to choose the model parameters so that they accord with estimates using UK data. Our use of six models means that we have a large number of parameters to calibrate and space constraints prevent us from detailing all these calibration decisions.(6) However, common to all of our simulations are the utility function and production function, and so it is important to detail our calibrations of these key features. We assume that the utility function is given by the constant relative risk aversion form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where we use the panel data estimates of Attanasio and Weber (1993) to set [tau] = 1, so that utility is logarithmic. We use the results of Alogoskoufis (1981) to set (1 - [Phi])/[Phi] = 2. For the production function we use the results of Holland and Scott (1996) who, using UK National Accounts, calculate [Alpha] = 0.4436 and specify the random technology shock to follow the process:

ln [theta.sub.t] = ln [theta.sub.t-l] + [Mu] + [epsilon.sub.t]

where [Mu] = 0.00207, F-, is white noise, and the standard deviation of [epsilon] is 0.009253. In other words, the technology process follows a random walk and technology shocks are both highly persistent and very volatile. These values were used for all models with the exception of the model with stochastic taxes. As explained in the Data Appendix, the only data available on tax rates are annual and so we converted all of these quarterly calibrations into their annual equivalents.

(i) Stylized Facts

Tables 3 and 4 show the results relating to labour market volatility from simulating each model 1,000 times (each simulation for 150 periods) and then averaging across all simulations. Our focus is on the labour market variables, so we shall not comment on the mixed performance of the models in explaining consumption and investment volatility. Concentrating on real wages we can see that the models do fairly well (and the tax model extremely well) in accounting for wage variability. However, with the exception of the tax model, the performance regarding employment and unemployment volatility is far more disappointing. While there are significant differences between the performance of the various models, what is most noticeable is that these differences are insignificant compared to the differences between the data and the model simulations. Employment and unemployment are an order of magnitude more volatile in the data than any of the models can explain. The one exception to this is the performance of the tax model (calibrated on annual data and shown in Table 4) which manages almost exactly to match the volatility of employment in the data. The tax model is also the best at explaining unemployment volatility, but even then it can only account for around 6 per cent of the observed volatility in unemployment.
Table 3
Business Cycle Facts and Simulations

            Consumption  Investment  Employment  Wages  Unemployment

Data           0.97        2.47        1.11      0.68      8.43
Lucas/Rapping  0.38        1.42        0.225     0.507     0.118
Hansen         0.834       3.05        0.357     0.312     0.203
Labour
  hoarding     0.315       1.61        0.422     0.486     0.249
Keynesian
  model        0.325       1.422       0.254     0.475     0.309
Search         0.871       1.476       0.220     0.892     0.131




Notes: The table shows standard deviations of the detrended variables listed in the column headings,divided by standard deviations of detrended output, either in the data (first row) or from the simulations (the remaining rows).
Table 4
Business Cycle Facts and Simulations for Annual Tax Model

                   Std dev.     -3         -2         1         0

Consumption          1.35      -0.054     0.083     0.249     0.532
                    (1.13)    (-0.055)   (0.26)    (0.62)    (0.880)
Investment           2.58       0.492     0.575     0.632     0.614
                    (4.63)     (0.02)    (0.370)   (0.74)    (0.93)
Employment           1.07       0.410     0.499     0.582     0.631
                    (1.19)    (-0.57)     4.28)    (0.23)    (0.74)
Wages                0.95      -0.096    -0.02      0.069     0.259
                    (0.91)     (0.22)    (0.24)    (0.31)    (0.45)
Correlation with
  employment
 Wages                         -0.356    -0.430    4.505     -0.575
                               (0.23)    (0.21)    (0.29)    (0.34)

                   Std dev.      1          2         3

Consumption         1.35       0.614      0.646     0.610
                   (1.13)     (0.750)    (0.391)   -0.051)
Investment          2.58       0.315      0.090     0.078
                   (4.63)     (0.72)     (0.29)    -0.14)
Employment          1.07       0.284      0.033    -0.144
                   (1.19)     (0.94)     (0.73)    (0.30)
Wages               0.95       0.441      0.560     0.59
                   (0.91)     (0.53)     (0.41)    (0.15)
Correlation with
  employment
 Wages                       -0.191       0.062     0.229
                             (0.28)      (0.07)    -0.17)




Notes: The first column of statistics shows the standard deviation of the cyclical component of the variable listed in the first column divided by the standard deviation of the cyclical component of output. The remaining columns show the correlation between the cyclical component of output and the cyclical component of variable listed in first column. The columns show the correlation of [X.sub.t+j] with output at time t, where j is given by the colunm heading, e.g. -- 3 means how current output is correlated with the variable three periods ago. Therefore the left-hand side of the table focuses on whether or not the variable leads output over the business cycle.

Not surprisingly, the Lucas/Rapping model performs worst of all in generating employment and unemployment volatility. This is due to two factors: (i) our calibration of the utility function implies only a modest willingness to substitute employment intertemporally; and (ii) our calibration of the productivity shock implies that shocks are permanent, and not temporary, so there is very little expected variation in relative wages between time periods. As a consequence, not only are agents not very willing to substitute labour intertemporally, there is also very little incentive to do so. The volatility of employment and unemployment improves substantially under both the indivisible labour and labour-hoarding assumptions, although by nowhere near enough to account for the data. The Keynesian model produces very little employment volatility but, with its definition of involuntary unemployment, does best of all the quarterly models at explaining unemployment. Most surprising of all, is the poor performance of the search model in accounting for employment volatility. Given our parameterization of the model, most of the volatility in the search model feeds through into wages rather than employment.

Figure 7 shows how the correlations between wages, employment, and output produced with our simulated models compare with those from the data. Focusing on the employment/output correlations we see that all of the models perform reasonably well in producing procyclical employment movements closely correlated with output. In particular, the models do extremely well in explaining the correlation of output with lagged employment, although they do less well in explaining the high correlation of employment with lagged output. However, Figure 7(b) reveals the crucial failing in all of these models-whereas, in the data, wages show little correlation with employment, in all of the simulations (with the exception of the tax model) wages are strongly procyclical. THis result is due to the fact that underlying all of these models is a single source of business cycle fluctuations -- random technology shocks. Positive technology shocks increase labour productivity, boost the demand for labour, and so lead to higher employment but also higher wages.

As a consequence, wages are strongly procyclical in clear contradiction to the data. This result holds for all models with just the single source of uncertainty, regardless of whether the labour market is perfectly competitive or Walrasian. The only model which does not display a strong correlation between wages and output is the tax model, in which both the labour supply and demand curves shift in response to tax and productivity shocks. However, even in this case the model still predicts a small but significant positive relationship between wages and output. Also revealing is Figure 7(c) which shows the correlation between wages and employment. As for the wage-output correlation, most of the models suggest a strong positive correlation rather than the zero correlation observed in the data. However, while the tax model performs well in explaining the volatility of employment and the correlation of wages with output, Figure 7(c) shows that this comes at a cost. The movements in the labour supply curve required to generate this employment volatility are so large in our simulations that there is a strong negative correlation between wages and employment, which is as much a contradiction to the data as the positive correlation generated by the other models.

(ii) Persistence

Figure 8 shows our measure of persistence for unemployment, real wages, and employment for both the data and our simulated models. The variable whose persistence the models most successfully explain is real wages, although, even in this case, no one model clearly outperforms others. The search and Gali models perform best in explaining persistence over the first four quarters, and the tax model comes closest to matching the long-run persistence in real wages. In general, most of the models rely on permanent productivity shocks and Figure 8(b) suggests that as a result the models tend to generate too much persistence in real wages.

As was the case for the volatility of employment and unemployment, although there are important differences between the various simulated models, Figures 8(a) and 8(c) show that the most marked differences are between the data and all the various models. None of the models generates the extreme persistence in unemployment that characterizes UK labour markets. As documented by Cogley and Nason (1995b), the RBC model's combination of persistent productivity shocks and capital accumulation provides very little persistence to output and employment fluctuations over and above that inherited from the productivity shock. While fluctuations in UK unemployment display no evidence of mean reversion, the picture that emerges from most of our simulated models is of small and temporary deviations of unemployment away from its average value.

However, while no one model comes close to matching the data, the search model clearly outperforms all the various other models. The search model is the only one which generates a value greater than I for the Cochrane test at any horizon. Whereas the Cochrane statistic declines continuously for all the other models, for the search model the test statistic actually increases for the first 8 quarters, proving that. search does provide a substantial increase in persistence.(7)

(iii) Cyclical Asymmetries

Our final criterion for assessing our six models is their ability to explain the fact that the growth and persistence of employment and unemployment varies over the cycle. Tables 5 and 6 show the results from our simulations. They show the average p-values of testing for the insignificance of dummies which allow the mean and persistence of each variable to alter in recessions compared to expansions. A p-value of less than 0.05 suggests the stochastic properties of the variable do change between stages of the business cycle.

Table 5 shows that, with one exception, the models do extremely well in capturing shifts in the rate of change of employment and unemployment over the business cycle. However, Table 6 shows that while the models have some successes, they are generally less successful in generating changes in the persistence of employment and unemployment over the cycle. As we found in the previous section, the models tend not to be able adequately to capture the dynamics of the labour market -- either the overall persistence of the market or the way in which persistence varies between expansions and contractions.

V. CONCLUSION

Our examination of six different labour market models has revealed some successes and some failures. Overall, the models tend to produce fluctuations which are characterized by relatively small unemployment fluctuations consisting of limited and temporary movements of unemployment away from an average rate. On the success side, we have the ability of the models broadly to replicate the cross-correlations of employment and output over the business cycle and the ability to mimic the way in which the growth of labour market variables changes over the cycle. However, there were three distinct failures:

(i) the models tend not to generate sufficient volatility

in either employment or unemployment; (ii) they predict too strong a correlation between

wages and employment; (iii) they cannot explain the observed persistence of

UK unemployment and employment, nor the

extent to which this persistence varies over the

cycle.

The extent of the model failures along these three dimensions is sufficiently large to suggest that none of the basic labour market models that we have outlined can be thought of as adequate to capture UK labour market dynamics. This has wider ramifications in that it suggests that using any of these labour market models to examine the welfare implications of alternative monetary and fiscal policy rules could be misleading.

Given the simple nature of these models, the finding of these inadequacies is hardly surprising, although the extent to which the differences between the data and the model simulations swamp the differences between the wide variety of models we examine is. However, the main aim of the paper was not to expose limitations but to investigate potential areas for future development. We believe that our results are informative in this direction. Examining the three main failures of the various models, we find two particular specifications which fared reasonably well -- the tax model, in explaining employment volatility and in producing a non-positive correlation between employment and wages, and the search model, in generating additional persistence in employment and unemployment.

All of our models contain a very volatile and highly persistent stochastic productivity term. Our simulations suggest that, while this alone is capable of generating appropriate amounts of output volatility, it is not sufficient to generate the observed volatility in employment. Only when we supplement the productivity shock with an additional disturbance can we produce enough employment variability. Given the variety of different labour market models we examined, this seems to suggest strongly that, in order to account for UK labour market behaviour, we have to introduce an additional source of disturbance into the stochastic growth model. Further, our simulations suggest that, while allowing for varying tax rates offers some scope for improvement, salvation probably involves another source of uncertainty.

Our tax simulations also reveal that the strong positive correlation between wages and employment can be rectified relatively easily by introducing an additional source of fluctuation which shifts the labour supply curve. In our simulations, introducing stochastic taxes actually took the adjustment too far in the opposite direction and produced a negative correlation, but the general point is clear -- adding additional disturbances can help make employment/unemployment more volatile and lead to a zero correlation between wages and employment. What these simulations do not tell us is which additional shock to include, and there is a wide choice -- anything from incorporating preference shifts to sticky prices and monetary shocks would help to improve the performance of the models. Perhaps the most striking failure of all was the inability of the models to generate enough persistence in employment/unemployment. The only model which helped extend the time it took unemployment to adjust to a productivity shock was the search-based model, but even this improvement left the models well short of the data.

One way to generate additional persistence would be simply to add costs of adjustment for the firm in changing employment. In particular, the Burnside et al. (1993) model could be extended to have hiring and firing costs so that, instead of having an infinite current period adjustment cost but a zero cost at the end of the period, firms always faced some positive finite adjustment cost. However, the relative success of the search model suggests that it may be profitable to investigate additional sources of propagation within this framework. One obvious way would be to allow for the job destruction rate (the inflow into unemployment) to vary endogenously as in Mortensen and Pissarides (1994). However, we feel that a more likely means of matching the persistence in the search model with that in the data is to move away from the representative agent model. In our simulations, an unemployed individual swiftly moves back into the work-force after a few periods, even allowing for the additional persistence provided by the search mechanism. However, most empirical analysis of the labour market stresses the importance of heterogeneities such as skill, education, age, sex, race, and region. Incorporating these heterogeneities into the model would significantly decrease the flexibility of the labour market and the search process and make unemployment and employment fluctuations much more persistent. Adding additional features, such as skill diminution as unemployment continues, would improve the model performance yet further.

To summarize, we have found that the basic RBC models we have examined fail in three important ways to mimic the behaviour of the UK labour market over the business cycle. Further, we found that these deficiencies were shared across a very wide range of different labour market models and not just a few categories. This suggests that to explain the UK business cycle, an increased in model complexity is required. In particular, we identify adding additional non-productivity-based sources of uncertainty and moving away from the representative agent paradigm as potentially the most rewarding ways of proceeding.

DATA APPENDIX

The labour market data that we use to present 'stylized facts' in this paper comes from two main sources: Labour Market Trends and The Labour Force Survey, both published by the Office for National Statistics (ONS). All data are for the UK (unless otherwise stated) and are seasonally adjusted. With the exception of the tax data all the data are quarterly and cover the period 1976Q2-1996Q2, apart from the vacancy data, which are only available from 1980QI. The tax data are annual and cover the period 1949-96.

Employment is defined as 'Employees in employment' (ONS code BCAJ), and Total hours is 'Total actual weekly hours (Great Britain)' taken from the Labour Force Survey. This series is only available quarterly since 1992QI and is available annually from 1984. This annual number represents the results of a first-quarter survey, and to arrive at a quarterly series for this period we interpolated. To arrive at a longer run of data we then regressed the ratio of average hours worked in non-manufacturing to average manufacturing hours on a constant, the growth of GDP, the share of manufacturing in employment, the share of part-time workers in employment, and a time trend for the period over which we have data and then backcast. We also calculated stylized facts and persistence using other measures of total hours (e.g. manufacturing) and found our main results to be robust. Unemployment is 'Claimant unemployment' (ONS code BCJD), Vacancies are 'Vacancies at jobcentres' (ONS code DPCB), while Real wages are 'Index of whole economy average earnings (1990=100, Great Britain)' (ONS code DNHS). To arrive at real wages we deflated by the RPI excluding mortgage interest payments (ONS code CHMK). Output is measured by GDP excluding oil and gas extraction (ONS code CKJL).

Additionally, the following data were used to calibrate the tax and government-spending shock process in the model of Braun (1994). Government spending -- general government final consumptions (ONS code DIAT). Taxes on labour income -- the marginal tax rate faced by a worker on average earnings equals the basic rate of income tax plus the marginal national insurance contribution faced by such a worker divided by 1 + the marginal national insurance contribution faced by their employer. Taxes on capital income -- the marginal tax rate faced by a basic rate taxpayer who buys equity at the beginning of the year and sells it at the end of the year having made a return on his holding. For details see King and Fullerton (1984). We are very grateful to Mark Robson at the Bank of England for the calculation of these rates.

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(1) We are grateful to John Muellbauer, two anonymous referees, and seminar participants at the SEDC meetings in Mexico City, 1996 and the CEPR Unemployment Dynamics Conference, Cambridge 1996. The views expressed in this paper are those of the authors and not the institutions to which they are affiliated.

(2) More specifically the test is [MATHEMATICAL EXPRESSION OMITTED].

(3) We define persistence here as simply the sum of the coefficients on all the lagged dependent variable terms in (1).

(4) We attribute this to the fact that our use of the HP filter does not produce as reliable a measure of the UK business cycle as those used by

(5) The details of these numerical techniques are inevitably complex and beyond the scope of this paper. For reference we use the productivity-enhancing activity(PEA) approach of den Haan and Marcet (1990) to solve all our models, except for the search model which we solved via quadratic approximation around the steady state. To assess the accuracy of our solutions we used the test of den Haan and Marcet (1994). A full collection of GAUSS codes is available from the authors on request.

(6) All of our parameter choices are taken from UK employment studies and, where possible, we choose consensus estimates. A detailed list of choices and sources is available on written request.

(7) The search model also matches extremely well the volatility of vacancies, as well as replicating the Beveridge curve.
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Title Annotation:United Kingdom labor market and business cycles
Author:Millard, Stephen; Scott, Andrew; Sensier, Marianne
Publication:Oxford Review of Economic Policy
Date:Sep 22, 1997
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