The labour market in an equilibrium business cycle model

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/222441896 The labour market in an equilibrium business cycle model Article in Journal of Monetary Economics · January 1983 DOI: 10.1016/0304-3932(83)90017-X · Source: RePEc CITATIONS READS 13 11 1 author: George Alogoskoufis Tufts University 71 PUBLICATIONS 1,282 CITATIONS SEE PROFILE All content following this page was uploaded by George Alogoskoufis on 12 February 2014. The user has requested enhancement of the downloaded file. Journal of Monetary Economics 1i (2983) ll?-128. North-Holland Publishing Company zyxwvutsrqponmlkjihg THE LABOUR MARKET IN AN EQUILIBRIUM BUSINESS CYCLE MODEL George S. ALOGOSKOUFIS* zyxwvutsrqponmlkjihgfedcbaZYXWVUT London School of Economics, London W C2A 2AE. UK This paper extends the basic, multimarket model of Lucas (1972), to explicitly consider the labour market. It builds on an important distinction between the product wage, entering the decision function of firms, and the real wage, entering the decision function of workers. Because of the unobservability of the price level workers make forecasting errors in trying to calculaz their real wage, despite having rational expectations. This gives rise to a Phillips curve. The major new result of the paper is the demonstration that wages are less variable than prices. which offers an equilibrium interpretation of wage stickiness. 1. Introduction The Friedman-Phelps tradition of :,$e Phillips curve emphasizes the informational problems c:sused by the physical separation of individual firms and markets, and the need for agents to form expectations about unobservable economy wide variables. The most popular model in this tradition today is the model of Lucas (1972, 1973), extended subsequently by Barro (1976) and others. These models embody the natural rate hypothesis, which amounts to asserting that agents’ decisions depend only on relative prices. They also impose the rational expectations hypothesis to arrive at a set of propositions about the role of monetary policy that deny any systematic influence of anticipated money growth on real variables. This paper presents an extension of the Lucas and Barro models by introducing an explicit labour market. ILucas and Barro consider an economy where only one non-durable commodity is traded. In the model of this paper we consider a similar economy where both labour services and a non-durable commodity are traded between firms and workers, via a medium of exchange called money. The paper builds on an important distinction in the wage variable that enters the respective decision functions of firms and workers. In particular. *This paper is adapted from chapter 2 of the author’s Ph.D. thesis. The uuthor gratefull! acknowledges the comments and suggestions of George Akerlof, Steve Nickel1 and Ch,ris Pissarides who have acted as thesis supervisors. The comments of the following are ailso acknowledged: Morley Ashenfclter, Richard Layard, John Nissim, Rafl’u Repullo. Ton! Zabalra. un anonymous referee and purticipants at various seminurs at LSE and the Centre for Lahour Economics. Sole :responsibility for remaining shortcomings rests with the author. firms are interested in their producv wug~, which is the nominal wage they paqf d~~~t~~ by the price of their output. Workers, however, are interested in if rt*tiEwuge, which is the nominal wage defiated by the econlomy wide use of a one period lag in the flow of information across markets, the ttr price level is not currently observable. Workers have to form teti;~ about it, which are assumed to be formed rationally. Firms rve ~JICprice for their otrtput and they do not have an inference nominal wage in each market is detlermined by the equilibrium of ad aaid supply for Jabour, and so it depends on tastes, technology, the ice and the expectations of workers, about t!ze price level. The analysis confirms the anaJytica1 results derived by the one commodity f Lucas, and leads to a new result concerning the variability of Equilibrium nominal wages are shown to display lower equilibriuni prices. Thus, the alleged ‘stickiness’ of nominal dative to prices is shown not to be necessarily inconsistent with the rium approach to macroeconomics. Jt arises naturally in the model tk inability of workers to observe their current real ::++e, ;as current ation about the price level is inadequate. It has to be emphasized that xpioit available information. The model also implies a live c-ovariance between employment and real wages. This is the familiar ation of static neoclassical models. Note, however, that, as Sargent s shown. a dynamic neoclassical model is not inconsistent with the haviour of real wages over the cycle. re# of the paper is organized as foJlows: section 2 presents the model. 3 derives rational expectations equilibrium and its properties. The ult about the Bower variability of wages than prices is also Section 4 contains conclusions. P a large number of identical competitive firms distributed over Z nf markers. Each firm produces a non-storable commodity Y&), =zf t -.7 . -. 2. In wh%t follows we shall refer to firm z, as a typical firm in of information within a market is instantaneous, but it lags one m~rk~ts, This set up is analytically convenient, and yields the extraction problems emphasized by Lucas (1972, 19731,as to o&rye economy wide variables currently. nly factor of production, and the production technology is The de c isio n func tio ns o f firm s and workers in each market following: zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA YXZ) = 00 + are the O<U,< 1, (1-a&l,(z), zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (1) All variables are in logarithms. P,(Z) is average output in market 3, n,(z) is average employment, w,(z) is the nominal wage, p,(z) the product price, and pp = Ep, 1Z,(z) is the current expectation of workers about the price level. w,* and p: are expectations about wages and prices formed at the end of period t .- 1, where all markets had a common information set. w: -pF is defined as the normal real wage, and n: as normal employment, which is the level of employment at the normal real wage. We can think of these normal variables, as expectations based on the information set available at the end of period t - 1. Superscript d stands for a demand function, and s for a supply function. (1) can be obtained by taking the logarithm of a Cobb-Douglas production function. (2) is obtained from the first order condition that the marginal product of labour equals the product wage when firms maximize profits.’ (3) rests on the intertemporal substitution of leisure hypothesis of Lucas and Rapping (1969). Labour supply deviates from normal, as the perceived current real wage deviates from normal.2 It is important to note an important distinction between (2) and (3). Firms are interested in t.heir procfuct wage, i.e., the wage the pay deflated by the ‘Consider the following production function: Y,(z)=(A;‘(1 - zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO CJ~))NJZ)’ -“ :, A L- 0. zyxwvutsrqponmlkjihgfedcbaZYXW 0 < u2 < I. In log-linear form it yields (l), where a O=ln 4 -ln( 1-a,). The first order condition for the maximization of profills is: .~N,(z)-~~= H$z)/P,(z). In log-linear form it is, In .4 - zyxwvutsrqponmlkjihgfedcbaZ IJ,~ ,(:) = w,(z) -p,(z), which can be rewritten as n:‘(z)= -a; ‘(a, + w,(z)-pJz)). where -11, =In A. Assume there exists some normal lejvel of employment r$ and a normal real wage ~,:--p:. Then they must - lJ2 ‘(H.,(_‘)- fPI(=i- ~~~+p:I, satisfy IIF= -a; ‘(a, -+w: -p:). We can then write. &l-n:= zyxwvutsrqponmlkjihgfedcbaZYXWVUTS whiclh immediately gives (2). ‘The log-linear version of the Lucas-Rapping labour supply function. ignoring assets and the discount factor, is n;(z);=!+,+ b,(wJz) -pf)b,(wf - p1* ). b,>O, i=O. 1.2. Subtracting b,(n: -p:), we get: nxz)=b,+(b,-b,)(w: -pF)+b,(w,(z)-p:-wF+pf). This is the same as (34 with $*==b, +(b, - b,)(wf -pr). Note that since we have another equation for n: (footnote 1). we cau determine w: --p: by the equality of the right-hand sides. It is given by, H.: --p: =(a, -a,B,)/(a,(b, -b;)+ 1). The normal real wage in the model is constant. and depends on the fixed parameters that characterize the technology of firms and the tastes of workers. SC>is normal employment. ‘f%is is not surprising since the model abstracts from factors accounting for serial persistence, such as inventory holdings [Blinder and Fischer (1978)], multiperiod lags in the acquisition of information [Lucas (1975)J convex adjustment costs [Sargent (1’379)], or shocks with a permanent and a transitory component [Brunner. Cukierman and Meltzer (19X0)]. G.S. Alogwhoufii, Labour m&et in equilibrium business cycle model product, whereas workers are interested in their reuE wage, the t, deflated by the price level. To rationalize this distinction, we that goods differ in their consumption characteristics, and that m&e their purchases t’rom a ‘supermarket’ at the end of period, current market clearing prices. These are determined at the beginning 1 when firms sdi their output to the 6supermarke:f’.3 for commodities in each market is assumed proportional forthcoming: 14) is ahe average stock of money forthcoming in market z. ney stock consists of a deterministic component that can be using the infomation set available at the end of period t- 1, te white noise shock and a market specific white noise shock. ven by (0, c:), eZY N(O,crz), and m,*= Em,(z)Ii,_1 = Em,ll, _ I . m, is the y wide average money stock4* 5 mode4 is now completely descsibed. (4) and (1) are the demand and for output, and (2) and (3) is the demand and supply for labour e can now proceed to impose rational expectations equilibrium. assumption about the flow GCinformation across markets, workers observe the price level when they make their labour supply decision. they have a genuine inference problem. On the other hand, firms variables enting their decision functions. us first consider the determination of expected prices under the ioa that normal employment is n,* and the money supply is aa m,‘. marginai productivity of iabour, firms make positive are distributed to workers who own identical portfolios economy. No Scapitalmarket operates. A less restricted set of talmdet can be found in Barro (1980). created by the lack of double: coincidence of wants Of course its role could be played by a ‘clearing zyxwvutsrqpo m o de l is by no means a full description of a essential features of such an economy. to be any rule. For example, Barro 11976) uses m,_ , as the pply. As all results can be proven for a general specification G.S. Alogoskoujis. Lahour market in equilibrium business cy cle model By the commodity market equilibrium I21 condition, * zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA +(l ---a&z,*. Pi = -ua,+m: (6) It is easy to see from (6), that the elasticity of expected prices with respect ta expected money is unity. We can next consider rational expectations equilibrium. The details of imposing the equilibrium conditions in the commodity and labour markets are in the appendix. Assuming’that workers derive an optimal forecast of the currently unobservable price level, by using all information up to the end of period t- 1, plus an observation of their current market specific price, we get the following reduced form price equation:” Pi@)= P: + 8= l+n,b, ’ (4 + G), 1 +b,[l -d(l -at)] (7) a,Z/(a,2 i- a,2). 8 is the proportion of demand variance in an individual market which is attributable to aggregate demand variance. It enters (7) through the optimal signal extraction of workers. The price equation can be decomposed into a part due to aggregate factors, and another due to market specific factors. They are respectively defined as, Pt=Pt*fYut, rz = ye,, (8) where l+a2b1 Y=l+b,[l-B(l-a,)] (9) <l ’ , We can therefore rewrite the price equation as in Lucas (1973), (10) where “The details of the derivatifon of this equation are in the appendix. !22 G.S. Alogoskot&s, Labour market in equilibriuwJ businexs cycle model It is easy to see that PF- Pf = Jet - Pa 1Md = 8(Pl(z) - P3. (12) Before going on to examine the equilibrium wage distribution, we derive some comparative statics properties of the price level, using (8). First, the elasticity of the price level with respect to anticipated money is unity, ++ldM I “,,@ = @p,@pf)@p,* /iii%,*) = 1. (13) Second, the elasticity of the price level with respect to unanticipated is fess than one, which immediately indicates that unanticipated affects output and employment, money money Third, the partial derivative of y with respect to the share of demand var&& that is attributed to aggregate factors, is positive, zyxwvutsrqponmlkjihgfedcbaZYXWVU d2P, h,ze b*(l -a2KI+a,h) m'= , (1 +b,[l-e(l -a2)])2 >() - (15) The higher the fraction of demand variations that is attributed to regate factors, the higher the influence of unanticipated monetary growth on the price level. All these results are in accord with the one commodity business cycle models of Lucas (1973) and Barre (1976). The equilibrium condition in labour markets gives the following wage equation? Corisider first the divergence crf the firms’ product zyxwvutsrqponmlkjihgfedcbaZYXWVUT wa g e fro m no rm a l. It is w&l to , (17) from eq. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (A.12) in the appendix. by the use of (16). (17) implies that in equilibrium, product wages are lower than normal in markets where p,(z)-pjf: >O, and higher than normal in markets where pXz)-p,* ~0. Labour demand rires or falls correspondingly. These movements in product wages are accompanied by opposite movements in perceived real wages. By ( 12) and (16) the divergence of perceived real wages from normal is equal to :,“h(PM - P3. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON (18) Mz)--w,*)--_(P:--P,*)= * 2 I Since 1 - 0>0, perceived real, wages move in an opposite direction from product wages. Therefore, as equilibrium product wages move in an opposite direction from perceived real wages, equilibrium labour demand and supply move together. Following an unexpected increase in demand, nominal wage changes are such, that the product wage falls, and perceived real wages rise. For unexpected decreases in nominal demand, the opposite apply. To derive the distribution of the aggregate wage level, we average (16) over markets. We get w,= NJ,*+- I+ a,!@ * +* b (Pt-P3 3 1 (19) From (19), variations in the aggregate wage level take place according to a normal distribution, with mean w,* and variance, The variance of wages is smaller than the variance of prices, because the term in brackets is less than unity. (20) offers an equilibrium interpretation of the relative stickiness of wages vis-a-vis prices and is the major new result of this paper.8 The intuitive explanation for this result is the following: Workers do not observe the price level, but only their market specific prices. When unanticipated inflation occurs, prices in all markets are higher than the case would have been otherwise. Workers in each and every market, optimally attribute some of the: ‘rise they observe, to market specific factors. As a result. they demand less than a proportional rise in their nominal wages. Firms arc prepared, because of favourabfe demand, to offer them a higher increase than that which would induce them to offer their normal labour supply. “The variance is not of course in general an appropriate measure of variability. In our case where distributions are normal it is, however, a suitable measure. See Rothschild and Stiglitz (1970). C&S. Abgoskoujis, Labour marker in equilibrium hslness cycle model I24 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA uently, they work more than normal. In a symmetrical way, with ted deflation, prices in all markets are lower than the case would otherwise. Workers are not willing to accept a proportional wage in each market, they attribute some of the fall they observe, to market faors. Therefore, they again demand less than a proportional fall in inal zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA wages, Firms are again prepared, becsuse of unfavourable d~~nd, ta offer them a higher decrease than that which would induce them to oXJer their normal labour su,pply. Consequently they work less than ~~~al~ Et is the same intuition that is implicit in Friedman (1968) and Lucas 61973).it can be algebraically demonstrated by considering, fr fl9). (21) is positive because both terms in the right-hand side are -five; it is less than unity because both terms are less than unity; and it is aller than t~p,/ih, because the first term in the product of the right-hand e is less than unity. The fact that wages respond less than prices to tc shocks, induces fluctuations in real wages. Had there been no sion between aggregate and relative demand shocks, i.e., O= 1, the real would be constant. Tu derive a Phillips curve, we can substitute (17) in the demand for labour 1. Koie that (17) is the equilibrium wage equation, and we could as well titute in the supply of labour equation. nJz)=n,*+- 1-s az zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (P,(Z5- P3, 42q where B=( 1 +rz,b,B)/( I+ azbl)= b,/dP. Clearly, (i-6)/a, =(bp- 6,,)/u2bp > 0. Averaging over markets, we get an aggregate employment function, (23) 123) is analogous to the aggregate supply function of Lucas (1973). However, it is derived, as implicit in the analysis of Lucas (1973) or Barro ~~76~from a formal explicit modelof the labour market9 784)has demonstrated that the auction market outcome is only efficient if firms ve the dirlnte degree of relative risk aversiort. The auction market outcome is t Enthis model. as we hue not assumed that the tastes of firms and workers towards To derive the covatiance of employment and the real wage. we subtract + and p, -p: from both sides of 619). This gives us w,- - p,- w,*+p:= - a2h,(! - 8) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK , +c r h- - (Pc - P:). 2 (24) t Using (23) and (24), (2.5) Thus, the present model imphes a negative association between employment and real wages. However, it should be born in mind that normal wages and employment have been assumed constant and we have not been concerned with factors accounting for serial persistence. As Sargent (1978) has shown, a dynamic neoclassical model is not inconsistent with the actual behaviour of real wages over the cycle. 4. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA C o nc lusio ns This paper extends the basic multimarket, stochastic equilibrium model of Lucas (1973) to explicitly incorporate a labour market. The model is built around the important distinction between the product wage, entering the decision function of firms, and the real wage entering the decision function of workers. The real wage is currently unobservable because the price level is only known with ? one period lag.” Workers are assumed to form rational expectations about the price level. However, they cannot distinguish perfectly between relative and absolute price shifts as they do not have enough information. The end result is that wages fluctuate less than prices. This endogenous sluggishness of nominal wages is the major new result of this paper. The sluggishness of wages is not imposed as in Phelps (1969). The labour market is an auction market. It does not rest on expectational non-neutralities, either. It is only due to the unobservability of the price level. That is what allow:; for purely monetar) shocks to have real effects, such as fluctuations in real wages. employment and output. Thl.: implied covariance between real wages and employment is the familiar negative one of static neoclassical models. “It has been argued that, whereas workers in modern economies do not even know the price of their product, they know the prices or a host of other prodUs which they encounter in their daily purchases. However, as long as workers do not fully observe the price level. the qualitative implications of this paper are not modified by this objection. If more prices are being observed. workers get a better estimate of the price level and the employmsnt-unanticipated intlation tradeoff gets steeper. ‘This has been shown by Cukierman {I9791 for the one commodity model. G.S. Alogoskoujis, L&our market in equilibrium businw zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH cycle model 526 iw: A De&a- of rational expectations equilibrium commodity market equilibrium condition gives p&S= -~ao+m~+u,+e,-(l -a&(z). (A.1) Subtracting the relevant ex ante condition, eq. V99 pfjz)= pf + u, + 6zz- (1 - a&(z) - n,*). (A.21 &2) states that the deviation of market prices from what had been . is equal to the money supply shocks, dampened &y the deviations ayment from normal. Employment, however, is an endogenous rhMe simultaneously determined with prices. The h&our market equilibrium condition gives (A.3) ate that, if pf = PAZ),then, w,(z)-p,(z) = W:-p,*, and the product and real are equal at their constant normal level. Substltuiing the equilibrium wage equation in the demand for labour, eq. CL gives (A.41 Substitalting for n,(z), in tA.2) from (A.4), and solving for p,(z), (A.9 To arrive at a price equati’on in terms of predetermined and exogeno?Js aCab& we have to substitute for p: in terms of such variables. It is here c the rational exmtations hypothesis is imposed. t method to solve for rational expectations equilibrium is the of undetermined coeffncients. we postulate an equation for &z), depending only on predetermined ous variables, which in this model are p,*, u,, e,. Such an equation P&HhPi*+wl+q3ez, ng the undetermined coefficients. (A.61 G.S. Alogosko~fis, Labour market in eyuilihriunr zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR hrlsirwss c pde rnm d d 127 Averaging (A.6) over markets we get an equation for the price level, (A.71 Since we have assumed that e, has a zero mean, upon averaging over a large number of markets, this term disappears. Imposing. the assumption that individuals have rational expectations amounts to requiring that they know the zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG q’s. The expected price level given market specific information is P~=EP, Ih@)=q,p:+q,E~,I (A.8 l,(z). The information set is assumed to contain, in addition to r’,_ , , an observation of the market clearing price in the specific market. Since individuals know p: from the end of the previous period, the additional information contributed by zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA p&j is an observation of the sum q+,+ q3eE. Since both random variables are normal, the conditional expectation of U, is given by the regressiou of U, on the aforementioner? sum. This is a simple application of the signal extraction problem [Sargent ( 1979)], The 8 coeficien- measures the fraction of total price variance produced by aggregate demand variance, The remaining fraction, 1-0, measures the fraction attributable to market specific variance. Substituting (A.9) in (4.8) gives (A.10) Su‘bstituting (A.lO) in (A.5) gives 1 +a,b, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA (1 -adb, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML P,(z)=-- 1 +b (Pt*+“,+e,) + CqIpI*+~(q,u,$-q,e,)l. (A.11) z +b zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK 1 1 We now have two equations for p,(z), (A.6) anti (A.1 I), both in terms of exogenous variables. By comparing coefficients, 41 = 1, 1 +a2b, -‘* = 1 cb,[lH(1--rr,)] 43=Yz. l G.S. Alogoskoujis. Labour zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA market in equilibrium business cycle model f28 By substituting for tF’ceq’s in (A.@,eq. (7) is obtained. To obtain the wage equation, we first note that the labour market aquilibrium condition, (AL+),can be written as (A.12) By substituting for p;-- ,pf from i It?), eq. (16) is obtained. riadih C.. 19711,Escalator clauses and the allocation; of cyclical risks, Journal of Economic xpectations and the role of monetary policy, Journal of Monetary market in an equilibrium business cycle model. Econometrica 48, cr, A.!%.and S. Fischer. 1978, Inventories, rational expectations and the business cycle, MIT A. Gukiermsn and AH. 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