DISCUSSION PAPER INEXTIA LN EMPLOYMENT Graham Pyatt Fetruary 1984 D a v a l o ~ Raaaarch Depattmeat t Eerioomiu and Rasurch Staff World Bank Tha vim prancnted >are are chore ?f the author, and they should not ba itateqxoted a8 reflecting thore of the World Bank Draft for discussion and comment. Please do not quote without permission of the author. INERTIA N EMPLOYMENT- Graham Pyatt Development Research Department February 1984 World Bank GPD/R-191/02.22.84 Summary This paper a r i s e s from an attempt to provide a theoretical basis for the observed duality and inertia i n labor markets. l":e standard treatment of short-run equilibrium of the f i m is modified by distinguishing the firm's inherited or ex ante labor force from the external pool of potential new recruits. Nev recruits are assumed t o be less efficient than established vorkers within the firm, and t h i s difference is shorn t o modify standard results in interesting ways. In particular, it is shown that the actual level of employment w i l l be inert relative t o changes i n product demand and that, i f existing workers are risk averse, then the firm may find itself operating r-inefficiently. 1. Introduction 11 This paper is concerned with the dcro-economic foundations of . . employment determination a t arr eleawntary level. and, i n particular, with an explanation a s to why employment m y be insensitive t o changes i n product market demand. It focuses on the short-ruu equilibrium of a firm. Assumptions about production technology and'product d&nd are more or less standard, while the assumed objective of the f i m is t o maximize profits with given capital stock. Eiowsver, rather than complete the standard model by assuming an exogenous supply of homogeneous labor, two types of labor a r e distinguished here, viz, the existing workers currently-cmploytd-kj the-ff., and others, external t o the firm, who might be recruited. The former correspond to the firm's internal labor supply, the l a t t e r t o its external SUPP~Y* h crucial assumption i n the analysis allows that internal labor is potentially cheaper than external labor of equal efficiency. Such a differential could a r i s e for a variety of reasons. For example, the firm's activity may require s k i l l s vhich a r e not readily available externally, so that new recruits would either have t o be trained, implying training costs, or simply given t i m e t o acquire such s k i l l s on the job, with an implied differential +n efficiency for the period of learning (by doing). An alternative type of explanation would be that a firm's current level of - .c -11 I am indebted to a number of current and former colleagues. includin~ participants of the Waraick Economics Seminar, for comments on an e a r l i e r draft. Special thanks a r e due t o Walter Galenson, Chal Sussangkarn rind, w i n particular, Clive Bell, while I r e ~ e i nwholly responsible, of coufse, for the views expressed here, which should not be attributed t o the World Bank or any of its affiliates. employment ~ x h a u s t e dits l o c a l labor market, necessitating a premium (removal cost) t o a t t r a c t labor f r o a elsewhere i f employment were t o expand. O r search costs may be involved i n finding nev r e c r u i t s of the same quality a s existing workers. In e i t h e r case, it can be noted, there is no permanent difference between i n t e r n a l and external labor. The d i s t i n c t i o n is transient. New r e c r u i t s , following t t a i n i n g o r 'relocation, a r e indistinguishable from established workers. The difference, i f any, maintains only within the (short-run) period over which t r a i n i n g or learning takes placz. Eovever, within t h i s period, there may well be d i f f e r e ~ c e s ,and the potential consequence which matters here i.s t h a t , per efficiency u n i t , the. supply 'price of external labor may d i f f e r f r o a the reservation wage of wist?.ng workers,. - The reservation wage of the existinq workers is t o be defined - r e l a t i v e t~ t h e i r current employment. If t h i s reservation wage' is w - , then the actual wage, v, mus; exceed w i f existing vorkers a r e t o prefer retaining rn t h e i r present jobs t o seeking some alternative. In t h i s sense, w measures rn the attractiveness of a l t e r n a t i v e employment. Therefor2 v is sensitive to the a v a i l a b i l i t y of a l t e r n a t i v e employment a s v e l l 6s t o wage r a t e s outside the firm and the l e v e l of unemployment benefits under s o c i a l security programs. The supfiy price o r opportunity cost t o the firm of external'labor is . denoted This is also measured i n efficiency units so tha;, while the _reservation wage per man may be the same f a t internalzand external labor, an!? 9 greater efficiency of the f o r s e r within the ff7.i w i l l * imply t h a t - > . The condition > 6 e s s e n t i a l l y defines tb circumstances vhich a r e - explored in t h i s paper. In the l i m i t , as (-:,;;;) -t 1, tl-- r e s u l t s to be presented will reduce t o those of the standard textbock case in which internal. and exten.al labor a r e indistinguisable. It can be argued that to make the d i s t i n c t i o n is a useful g:n:r?lifation, since it moves i n the d i r e c t i o n of greater realism f o r the moa.:l. The analysis assumes t h a t preferences of e x i s t i n g workers can be expressed collectively, while those of external labor, i n . i u d i n g any who might be recruited, a r e of no consequence i n the short run, given t h a t t h e i r reservation wage requirements a r e s a t i s f i e d i f the f i n i n f a c t o f f e r s them jobs. Accordingly, f o r the short-run determination of wages and ~mployment, it is only the respective preferences of the firm and its existin3 employees t h a t a r e of consequence, with the former being measured by p r o f i t s and. the l a t t e r by a c o l l e c t i v e u t i l i t y function which has a s its acguments the wage v and the proportion of e x i s t i n g workers retained by the firm. Given these assumptions, the present analysis goes only part-way towards a theory of employment contracts, since a complete specification of bargaining between the firm and its existing workers turns out t o be . unnecessary t o the derivation of some interesting results. A f i r s t s t e p is t o explore the potential f o r mutual s e l f - i n t e r e s t a s between the firm and its i n t e r n a l labor force. An agreement on employment and vages under which both t h e firm and its existing employees a r e better-off than they would be without each other (i.e., the firm having t8 r e c r u i t an e n t i r e l y new labour force, and the e x i s t i n g labour force a l l having t o find new jobs), is described as an attractive contract. - It is shown in Section 3 below that such contracts w i l l be possible when w_ exceeds G, t h a t is, when the supply price of external labor exceeds the r e s h a t i o n wage of i n t e r n a l labour a s . previously discussed. Next, beyond the notion of agreements o r contracts which a r e a t t r a c t i v e , is the q ~ e s t i o nof efficiency: an agreement is defined a s being e f f i c i e n t i f it maxisizes the preference function of one party f o r a given s a t i s f a c t i o n l e v e l of the other. Efficient agreements a r e discussed i n Section 4 of the paper, which contains the main results. While actual arrangements in a labor market can perhaps be expected t o r e s u l t i n cutcomes which a r e a t t r a c t i v e i n the sense previously discussed, there can be less confidence i n assuming that they w i l l be efficient. How- ever, i f indeed they a r e not e f f i c i e n t , then it follows that ( a t l e a s t ) cne party could be b e t t e r off without any l o s s t o the other. Such circumstances would seem t o suggest pressures f o r i n s t i t u t i o n a l change i n labor mdrkets, so as to encourage efficient agreements to be reached. The set of e f f i c i e n t contracts is necessarily a subset of those t h a t a r e attractive. The s i z e , and hence the characteristics of t h i s subset w i l l depend oa how the preferences of the firm and its existing workers a r e specified. Here, throughout, the former a r e measured by profits. The l a t t e r a r e considered a t two levels. A t the f i r s t level, the welfare of existing workers is not specified beyond assuming t h a t it is p o s i t i - ~ e l yrelated t o the number of existing workers retained by the firm and the wage it pays them. On t h i s basis, some . I interesting r e s u l t s a r e obtained. Attractiveness and 'efficiency of a contract imply that existing workers vill enjoy goodwill i n the sense of having f i r s t - choice 04available jobs. One of three regimes w i l l maintain. The firm say * r e t a i n a U existing workers and, i n addition, h i r e some recruits; or it may - simply r s a i n a l l existing workers; or some of the existing workers may be l a i d off. But hiring aid f i r i a g v i l l not co-exist. Also, the s i z e of the existing ex ante labor force w i l l have no e f f e c t on the actual l e v e l of employment i f the ex ante o r inherited labor force is below a c r i t i c a l size. Otherr-ise, with the ex ante labor force above the c r i t i c a l level, the existecce of t h i s inherited labor force implies a higher level of actual employment than would othervise m i n t a i n . These r e s u l t s suggest that :he assumptions s e t out i n Section 2 below imply a iabor market which is i n e r t i n the sense of being resistant t o dovnward movements i n the l e v e l of employment. This implication scmes out more forcefully when the specification of preferences f o r existing vorkers is tightened. This is achieved here by assuming that the collective preferences of existing workers a r e measured by t h e i r expected u t i l i t y , being Sefined as an appropriately weighted average of :he u t i l i t y of t h e i r resatvatlo2 wage and - that of the wage paid by the F i n n t o those it reteins. Given t h i s strongcr statement of preferecces it is alnost, but nor. quite, possible t o make exact statements about what the level of emplopcnt w i l l be, independent of the precise wage or, indeed, of the specific' bargaining process. The condition that contracts should be a t t r a c t i v e and e f f i c i e n t is now sufficient to go a long way towards d e t e m ~ n i n gthe level of employment. Specifically, it can be noted-that the lowest levels of employ- ment are l e a s t likely. when the u t i l i t y f u ~ c t i o nof existing workers is l i n e a r i n t h e i r wage. This 3s because, i n t h i s case, tne trade-off between wages and employment which is acceptable t o existing workers w i l l be most sensitive t o the l a t t e r . It w i l l be shown that, in this limiting case' the a c t u a l l e v e l of '4 employment dl1 be given by the cedian of three nuabers, _denoted here by - 1 - 0 Ti, b0 and %. Of these, no is the actual number of e x i s t i e elrployees, while - I and a r e such that m (z) = and m (G) = W , w:.ere in (n) is the marginal revenue product schedule f o r labor. Necessarily, therefore, with ii > G, it follows that < %. The result: n = nedian (G, no, 2) describes an i n e r t or 'sticky' labor market. If no < ?i (implying m (no) > c), the firm hires extra labor up . t o the point t h a t m (n) = Similarly, i f , m (no) < z, the f inn wj 11 lay- off members of its existing labor force i n s u f f i c i e n t number t o r a i s e m . 2 3 ,the N (n) t o w But f o r a l l intermediate cases, with m (no) firm w i l l neither h i r e nor f i r e , and employment w i l l be stationary a t the l e v e l , no, inherited from the past. History matters, therefore, i n t h i s model. An implication of the atove r e s u l t is t h a t fluctuations i n product demand do not necessar.ily produce fluctuations i n employment. In general, cycies i n product demand w i l l e i t h e r have no influence on employment, or they v i l l r e s u l t in a cyclic response which is both lagged and damped r e l a t i v e t o the product demand cycle. This r e s u l t is a particular aspect of the more general i s p l i c a t i o n of n = median (G, no, , t o the e f f e c t t h a t attractiveness and efficiency c r i t e r i a may w e l l yield short-run labor market equilibria i n which there is a f a i l u r e of the market t o adjust l e v e l s of esployment i n response t o changes i n product markets. The level of 2mployment is no longer exactly determined i n a l l cases when employees' preferences a r e such that U O ' (.) o, i.e. < when workers0 L u t i l i t y is a non-linear function of the vage. The e f f e c t , i f any, of non- l i n e a r i t y is f o r employment to be higher i n those cases were n would other- N wise be n. Accordingly, introducing r i s k aversion among existin* workers i n the form U"(.) < o potentially implies feyer lay-offs. Risk aversion there- for* adds t o the i n e r t i a of employment levefs. J u s t haw aiuch it may add is not determined here. But it is sho- that it may be enough t o imply a level . c; employment a t which the marginal revenue product of labour ~ o u l dbe negative i f t h e firm was t o operate a t f u l l technological efficiency. Of course, i n euc! a s i t u a t i o n , t h e firm w i l l choose t o operate within its tech- nological f r o n t i e r , simply producing enough output t o maximize t o t a l revenue. I n t h i s way, the present analysis allows the intriguing prospert t h a t x-inefficiency may characterize an a t t r a c t i v e and (distributionally) e f f i c i e n t determination of emplqymc.nt and wages a t the level of the firm. This analysis has antecedents in the e a r l y l i t e r a t u r e on aployment i n e r t i a , notably O i [1962], a d more generally with theories of segmented labour markets as usefully reviewed i n Cain [1976]. However, O i ' s forzulaiion of the problem is s p e c i f i c a l l y dependent on t r a i n i n g costs and-regards wag- a s E:ogenous. Here, wages f o r e x i s t i r ~ gworkers a r e clearly endogenous-, ' ( a l t h o ~ g hthe analysis is not pushed t o the point at which they a r e completely determined), while e x p l i c i t training costs a r e not necessary t o t h e argument, which can be .;ustained on the basis of learning-bydoing alone. Indeed, while the r e s u l t s obtained a r e t h o u ~ h tt o be of more general relevance, the i n i t i a l motivation 'n undertaking the work reported here was t o explore the linkages between duality in wage systems of developing countries and leaming-by-doing a s an important c h a r a c t e r i s t i c of t h e i r industrialization processes. While the wage implications of the assumptions made here remain t o be f u l l y developed, i n the'anklysis presented, wages f o r tetained eaployees a r e i n f a c t bounded, with the reservation wage defining the lower bond. Clearly, then, the& is scope within the framework for a dual wage syutem t o maintain. _ ! - Finally, the present analysis has some lnplications for i n s t i t u t i o n a l f o r d i n the labor market. - Collective bargaining by existing workers is taken ) w a s given, but there are no r e s t r i c t i v e practices, only efficiency differences, t o explain > 2. However, it can be c ~ t e dt h a t e s s e n t i a l l y the same model - would r e s u l t i f there were no efficiency differences betoeen i n t e r n a l and external labor, and instead t h e former were ayle, through r e s t r i c t i v e - practices, t o r a i s e the supply price, - w, of external labor r e l a t i v e t o the reservation wage, w. Accordingly, these r e s t r i c t i v e practices which a r e reasonably captured by the assumption > 3 lead t o the same model and nence the same conclusions. Moreover, it can now be seen that Fn s i t u a t i o n s i n which there a r e efficiency d i f f e r e n t i a l s between established workers and new r e c r u i t s , the f o r s a l unionization of labor may be only a cosmetic change, having no e f f e c t , of i t s e l f , on t h e l e v e l and s t a b i l i t y of employment t h a t would otherwise maintain. While the present paper does not go f a r into the q u e s t i o P , o f - m e determination, these investigations a r e enough t o show that the wage v i l l only coincidentally be equal t o the marginal revenue product of labor under e f f i c i e n t agreements. Evidentally, therefore, t h e mutual i n t e r e s t of the firm and its employees i n reaching e f f i c i e n t contracts vill lead t o i n s t i t u t i o n a l forms i n the labour market i n support of behaviour which is d i f f e r e n t from p r o f i t maximization by the firm, given the wage rate, a s normally assumed. 2. Basic Assumptions L The analysis is b u i l t oo a s e t of f i v e assumptibns which a r e described i n t h i s section, together with notation and soae prelimicary - results. 2.1 Product ion Technology The output l e v e l of t h e firm i e d e n o t e d . Implicitly, c a p i t a l is fixed and it is assumed t h a t raw material inputs a r e s t r i c t l y compliinentary. Output therefore depends only on the level o i employment,' n ,and on techno- logical efficiency. The specification of t h i s relatfonship is: Assumption 1: Output, 1, is bounded via employment, a, a s A 1 q < q(n) for a l l n > o, where q(n) is such that ( i ) for n = o , q(n) = o ; and ( i i ) for a l l n > o , i8(n) > o > ;"(a). The function q(n) therefore describes the technological l i m i t s on output. It is assumed to be twice differentiable and characterized by diainishing . returns with respect t o the variable input, n 2.2 Product Demand It is convenient to define product demand in terms of a rev- function, v(q), which is net of raw material costs. Hence v(q) is value added, and specified as: Assumption 2: Xet output or value added of the firm is denoted by v(q) such that . (1) for q = o, v(q) = o; and < o and v8(q) < o> ( i i ) for q > o, v"(q) dnpending on q q*. The assumption v"(q) < o i m p l i & simply that marginal revenue diminishes a s q increases. Beyoud this, the formulation assumes that 'there exists a level of output, q , a t which marginal (net) revenue is zero. * No loss of aerier- < i a l i t y is involved since - MY be arbitrarily large. - L Assumyions 1 a 2 can be co6bined t o yield the following &terneat about the marginal revenue product of labor: Result 1: If ;(a) * v [:(n)] and o(n) = ;'(n) then, from assumptions 1 and 2 it follows that: . A (i) for n = o, v(n) = o and m(n) = v(n)/n; and (ii) for n > o , mf(n) = ;"(n) < o while m(n)>o , < * depending on n,n< * , where n is such that * qin ) = q * , * . i.e. m(n ) = o Proof is standard and therefore omitted. The result s t a t e s that the marginal *. revenue product of labor diminishes, and that it has value zero when n = n It should 'be noted that the marginal (net) revenue product of labor is defined only when the firm is fully exploiting its technological capability, i.e., A when q q(n). 2.3 Preferences of Current Employees and 'rbor Supply The firm's labor input, n , is a measure of employment m efficiency units, taking the efficjency of an existing worker a s numeraire. If t h i s t o t a l input is made up of e units supplied by existing workers and r units .from new recruits, then n = e + r ; and the total wage b i l l is now w e + r , where w is the wage paid to those of the existing w~rkerswho - are retained by the f i n , and w is similarly the wage per efficiency unit of new recruits. Hence, if new r e c ~ u i t sare half as efficient as existing - workers, then w is twice the wage per man actually paid to new recruits. If there are no existing mrkers, then e ,the number of those retained, must be such that e &no. S u i l a r l y r efficiency units of new labor are hired, - and r 2 0; and no e of the existing units are laid of:, - where no - e ) 0. w The notation introduced above contributes to the definition of labor supply a s follows: Assum~tion3: _L-- (a) Phternal labor is i n perfectly e l a s t i c supply a t a wage per - efficiency unit of w ; and ) The collective preferences of the no r !sting workers a r e given by . (1) U (e, w) such that aulae, aulaw > o; or ( i i ) u (w) = [e U(w) + (no - e l ~ ( G ) l / n ~ such t h a t U8(-) > o > U"(*). Part (a) of t h i s assumption is straightforward. Part Cb) c a l l s f o r some. explanatinn. It has two versions, either of which fapLieo tkoh actsting workers, .IS a group, have a collective preference function. Under b ( i ) t h i s is characterized as being an increasing function of e (which is the number of existing workers offered continuing employment by the f i&),and w, (which is the wage they are paid). The a l t e r n a t i v e provided by b ( i i ) defines U(o) a s . a weighted average of U(w) and u(;) Since U8(w) > 0 -> U"(w) by as- sumption, U( .) can be interpreted as a u t i l i t y function. Hence U(w) is the level of u t i l i t y t o be aseociated with being offered continuing employ- ment. If u(;) is similarly defined as the level of utility contingent on . I being laid-off, and :.f rhe e jobs offered t o the no members of the existing labor force a r e allocated a t random, then it follows that U(o) is the - expected i t i l i t y of an existing member of the labor force, and can there- fore be interpreted a s the mean equivalent wage for such a person. -.. o Similarly, ... * W can be-thought of as the mean equivalent wage contingent on being laid-off and, a s such, dl1 be a measure of the a l t e r n a t i v e prospects for e i t h e r a new CI job or being dependent on s o c i a l security. Only i f w is greater than w w i l l the firm's ex ante o r e x i s t i n g labor force prefer continuing employment 15 with the firn;. Accordingly, w aay be referred t o a s the reservation wage, r e l a t i v e t o current employment, of existing workers. A s such w i l l be - . r e l a t e d t o v In the simplest scenario, i n t e r n a l and external workers w i l l have the same reservaticn .-age per m a , so t h a t the r a t i o ;/3 measures the r e l a t i v e l y greater efficiency, v i t h i n the firm, of its e x i s t i n g workers, by - - - v i r t u e of t h e i r firm-specific s k i l l o r some other comparative advantage. It can be noted for future reference that: Result 2: Assumption 3 b(i1) is a special case of 3b(i) which requirea w > z. - 2.4. The Firm's Objective Given production technology and product demand, the firm is assumed t o seek maximum profits. Thus Assumption 4: The objective of the firm is t o maximize short-run p r o f i t s , denoarcl by n ,where e -- employment of existing workers, w vage paid to existing workers, r -- employment in efficiency unit$ df new r e c r u i t s , and ;; wage paid (per efficiency unit) to new recruits. '9 C This leads d i r e c t l y t o t h e folloving standard result: - Result 3: I f the firm is r e s t e e t e d t o its external supply of labor, (1.e. e = o), which is perfectly elastic ac wage r a t e & > o , then the firm w i l l s e t output q - and employment n = n such that: ( i ) i f ;h(01,thedo o = q ; and I - ( i i ) i f w C m(o) ,then q = where n - * is such that m(ii) = w . and o c ii o , the convexity of the relationship between r(z) and w follows directly from the diminishing llrarginal revenue product for labor established in W u l t f(fi). - V Figure 1 The relationshi? between ptofics, n(i) and the external supply price of labor,- w, when the firm is precluded from utilizing internal labor, (e = 01. 9 3. Attractive Agreements 3.1 Definition The assumptions introduced in the previous section involve five endogenous variables, e, rmn, v and q . Without necessarily implying that any or a l l of these variables are the subject of collective bargaining, a determination of them can be referred to a s a-contract or agreement between the firm and its existing employees. Such an agreement can be described as attractive i f it implies that the firm makes more profit than it would by simply relying on the external Labor market; and also that current employees are better off than i f they were t o do likewise. These considerations a r e formalized in the following definitioa: Definition: An agreemznt (e, r, n, v, q) is attractive i f and only ff both - (a) r = ~ ( q ) we- 3 r > n(c) > o ; and (b) u > G t o r oon-negative values of e,r, and q such that . o < c < nom q < ;(a) and e + r = n 3.2 Existence . I Given the definition of an attractive agreement and the assumptions previously s e t out, the conditions under which such agreements can 'exist may be ex?lored. It is useful to do so in terms of some further notation, which - is introduced in Result 4: " * - Result 4: ~~t v*(n) = r ~ (j(n), v(n 1) x and * y(n) - n(w)}/n; {V (n) then * min(n,ni') a i (a) v (n) = /om(v)dv > v(n) > r(q) and (b) dv*(n)/dn = max(m(n), o) ; and - w n dn(n,n*) ( i i ) (a) ny(n) + I m(v)dv so that n (b) Y(;) = min(w,m(o)) > y(n); (c) y(n) + o a s n + , 0 - 0and : : (d) mx(o,m(n)) y(n) and dy(n)/dn :a. o according a s n * 4 The functaon v (n) introduced i n Result 4 is simply v(n) . hen - * A n < n* ,and otherwise has the constant value v(n j. Hence, unlike v(n), * *. v (n) does not diminish a s a function of n when n exceeds n Result * 4(i)(a) and (b) formally express these properties of v (n) and some implica- tions. Their proof is obvious and therefore omitted. . * Result 4(ii)(a) follows directly from the definitions of y(n), v (n) (as in Result 4(i)(a)), and n(w). Result 4(ii)(b) follows from it. If - - > m(o), tht 9 n o and the integral measuring y(n) takes the limiting - value m(o). .Uternatively, if :; d o ) , then < - - > - n o. Now m(ii) w by definition of n ,?Result 3 ( i i ) ) , while m(v) m(z) for a l l - v > ,R e s u l t i . Henee Eii)= ii > y(n) for n > E . Similarly, m(v) > m(;) for a l l v < z, so y > y (n) for a l l n < E. To * * establish 4(ii)(c) note that, as n + , min (n, n ) + n so that ny(n) . -c constant: hence *y(n) + o Finally, it is easily shokn that * n2~ ' ( n )= vm(v)l . )- m d The second part of Result . 11 n 4(ii)(d) follows d i r e c t l y from t h i s , given t h a t m'(v) o Xoreover, since \ ny'(nj = max (0, ~ ( n ) ) y(n), the f i r s t part of Fesult 4(ii)(d) can be - obtained from the second. The function y(n) is illustrated i n Figure 2 for the alternative cases i; > m(o) and t < m(o) . The graph of m(n) versus - n is also shown i n the figure, and t h i s defines a level of employment, n ,such that - . m(z) = w This dl1 be referred to subsequently. Bere it can be noted t h a t the importance of the function y(n) derives largely frem the follovimg re! u l t : Result 5: A necessary condition f o r an agreement (e, r, n, w, q) t o be a t t r a c t i v e is t h a t y(n) > w > G. The condition w > is evident from the definition of an attractive agreemen: provided above. The additional requirement y(n) > w can be estab- * lished a s follows, By definition, n = v n - * u , while v (n) )v(q) . from Result 4(i)(a), But prof its, u , are given by r = v(q) - - we Hence, by s u t s t i t u t i o n , e(y(n) - w) + r (y(n) - 2 r - n(;) , Now n > n(c) is necessary, by definititn, for an attractive agreement, as is e > o and r .' o Since y(n) C from Result 4(ii)(b) , follows it that y(n) > w is a necessary condition for an agreement to be attractive as 9 - L s t a t e d i n Result 5. p i s condition imposes an upper bound on the wage, w ,_ t o be paid t o e x i s t i n e workers, and t h i s bound is a function of n , the S m aggregate level of employment a s shown i n Figure 2, - 18- Case (a) : 3 2 m(o) Case (b) : 3 < m(o) Figure 2 Restrictions on the Set of 4ttractive Contracts Following on from Result 5, the following theorem can now be proved. Theorem 1: Attractive agreements exist i f and only i f both min(3, m(o)) > and no > o . Since e , the number of existing employees retained, must be such that e < no , it follows that > 0 < no is necessary for e 0 ,i.e. the i n i t i a l number of employees must be positive. Next, from Result 4(ii)(b), min(v, d o ) ) > y(n), while y(n) > W > is necessary for an attractive agreement from Result 5. Hence mln(3, m/o) > is a necessary condition for an attractive agreement, s-s stated in the theorem, and the two sides of t h i s . inequality provide, respectively, upper and lower bounds for y(n) and w Figure 2 is dram on the assumption that the condition min(z, d o ) ) > 3 is satisfied. To show that these same conditions are sufficient, l e t e = min(n, no) by assumption. If o < no, then there exists n such that o < n C no and, for such n, e = n > 0 = r , which is consistent with an attractive - contract. In addition, l e t q q*(n) by assumption, where q*(n) is simply' * * * i(n) provided n C n ; and q (n) = q i f n > n Then * * v(q) = v (n) = ny(n) + a(;) where . v (n) and y(n) are defined in Result - I 4. Hence prof its, a , a r e given by v (q) + we+ ik = n (ii) + n (y(n) w). - Accordingly, a > r (3) and w - > G, a s required by an efficient contract, i f - y (n) > w > w, i.eU if the necessary condition required by Result 5 is 6 satisfied. Since p i n (w, m (01) p (n), from Result 4 ( i i ) (b), it follovs that if atin (;, m g ) ) > w, then Y (n) > w > i3 possible. Hence th. condl- tions specified fn the theorem are sufficient for a t t t a c t i - ~ econtracts to exist. Indeed, the argument shows that sll points (n, w) within the areas M C - - of Figure 2 are consistent v i t h attractive agreements when q - q*(n), - e min(n, no) and given that t n e. The following corollary of the theorem can be established by refer- ence t o Figure 1: Corollary 1: Given no > o ,a necessary and sufficient condition f o r . t h e existence of attractive agreements is that u (G) > r (a, The corollary states simply tlut, i f a firm has an existing labor force, then both the Ģinn and these employees can potentially reach an attractive' agree- ment i f , pursuing prof it maximization and w i n g only exte'tnal labor, the firm CI would prefer t o be faced by perfectly e l a s t i c labor supply a t a wage of w rather than 3 , It can be noted (from Result 3) that, in the former case the firm would employ efficiency units of labour, where ';i is defined by rn (z) ..) = w. 3.3 Some Implications: Theorem 1 establishes that, under certain conditions, there is,scope for both the firm and its current employees (the ex ante labor force) t o gain . from an appropriate selection of values for the variables e, r, n, wand q . * Result 5 provides some conditions which such apropriate values clust s a tisf y , - > w - including the condition v > G. Xov, if the difference between - and derives entirely from efficiency differences betwen new recruits and rL established workers, a s has previously been suggested as q p o s s i b i l i t y , then s * i f It measures the relative inefficiency of new recruits ibfollows that - A Hence the condition > W > can b written as Z/X > w > which implies that established workers are t o be paid more than their reservation Jage but l e s s than t h e i r g r e a t e r efficiency r e l a t i v e t o external labor would justify. A system of payment based simply on efficiency differences (a piece- r a t e system: is ruled out, tkorefore, not necessaL.ily by t h e firm, but by t h e potential mutual concern of the firm and its employees t o secure an a t t r a c t i v e agreement. Similarly, if the firm was t o pay the same sage per man t o both 5 new r e c r u i t s and established workers, then t h i s vdge would have t o exceed t o r e t a i n the l a t t e r , and w u l d be unnecessarily extravagant with respect t o the former. The condition > w > 3 therefore implies giving some recogni- t i o n , but less than f u l l recognition, t o the greater efficiency of established workers. The greater efficiency of established workers generates a rent. The condition 3 > w > implies t h a t t h i s is t o be shared i n some way between these workers and t h e firm, by paying a premium o r d i f f e r e n t i a l t o established workers, but one t h a t is less than a f u l l reZlect3on of t h e i r efficiency difference. 4. E f f i c i e n t Agreements 4.1 Definition: From t h i s point on, it w i l l be assumed t h a t the conditions required by Theorm 1 a r e s a t i s f i e d so t h a t a t t r a c t i v e agreements e x i s t and it is therefore up t o the labor market, however constituted, tp make a detersination of the p a r t i c u l a r agreement t o be adopted. Depending ca how the labor market works, a range of outcomes is p o s s i l l e , and these uay or may not be e f f i c i e n t according t o the following definition: - Definition: An agreement (e, r n, v, q) is e f f i c i e n t i f it is b a t t r a c t i v e and, i n addition, there is no other a t t r a c t i v e agreement which exiqtieg ~ r k e r sprefer while the firm is worse off; or which the firm prefers while the existing workers are not worse off. The s e t of a t t r a c t i v e agreements which a r e e f f i c i e n t constitute the contract set. Acrual contracts can be expected t o correspond t o points i n t h i s set. If they do not, then it is l i k e l y that the market mechanisms/ i n s t i t u t i o n s which brought them about w i l l be under some pressure t o change. 4.2 Weak Conditions f o r an E f f i c i e n t Contract Assumption 3(b)(i) pravides a weak formulation of the preferences of t h e ex ante labor force. kcordingly, the set of contracts which a r e e f f i - cient r e l a t i v e t o assumptioa 3(b)(i) is only weakly restricted. Ir i.deffmd by Theorem 2: Theorem 2: Under assumptions 1 t o 4, a contract (e, r, n, w, q) is attractive and efficient only if: r = n - e = max (n -- no, 0); and ( i l l ) if no < z, then n = n; and Lf nG > z, then e i t h e r (a) n = no and m (no) > w; o r (b) < n . c 'no a ~ wd > a (n) - - To prove the theorem, it can be noted t h a t the collective preferences - 9 6 of existing workers a r e given by assumption 3 (b)(i) a s * - U(e, w) ,while r = v (q) - we - b measurer the preferences of the firm. Since the ftrrner depend only on e and w ,efficiency requires that the l a t t e r must be a maximum with respect t o q , a and r , subject to given v and e = n - r. Given t h a t w < w is necessary from Result 5, this implies (i)and * - - - ( i i j of the theorem, and hence that o = v (n) + (& w) min (n, no) wn, . and U = U (min (n, no), 2 W) It now follows that if n no, U "5 independent of n and A increases a s r! increases i f m (n) > w. Similarly, i f n < no, U increases with n , (since n = e and aU/ae > o by assumption), while n increases v i t h a i f m (a) > w. Accordingly, if > no then, for given w, U increases as n increases t o no, and is constant thereafter; while n increases a s n increases t o no, (pro-~idedthz; w < v), and - continues increasing with n u n t i l n = n, declining thereafter. Since - .. w < is necessary for an attractive contract it follows that, when n0 < , t o be both e f f i c i e n t and a t t r a c t i v e , a contract must be such t h a t - n = n, thus establishing the f i r s t ?art of ( i i i ) of the theorem. - It remains t o consider the case no> z. In t h i s case, a s n rises t o n , vith w given, U increases, and continues t o rise u n t i l n = n0 being constant , - subsequently. P r o f i t s n a l s o increase a s n rises -initially to n , (provided w < i),and p r o f i t s =st decline i f a increases beyond no - because if n > no when no > z, then 18 (n) < m (no) < P (z) = W. Between these two p o s s i b i l i t i e s , v i t h ?in w, or by ii < n c no with w > m(n). These possible c h a r a c t e r i s t i 3 correspond t o the remaining p o s s i b i l i t i e s under - ( i i i ) of Theorem 1. r, The f i r s t part of Thzorem 1, t h a t q = q* (a), s t a t e s simply that i f n exceeds n*, then output is held back t o level q* t o avoid the. reduc- * tion i n revenue, v (q), othervise entailed i n producing more than q . By 1 implication, i f n > n , then * q < q (a) and the firm is producing v i t h i n its technological limits. I n t h i s sense, Theorem 1 does not exclude the pos- s i b i l i t y t h a t x-ineffl.ciency can co-exist with d i s t r i b u t i v e or contractual - efficiency. Under ( i i ) of Theorem 1, e min (n, no), which im?lies t h a t existing workers g e t f i r s t choice of available jobs. Contractual efficiency excludes the possibility that the firm w i l l siaultaneously h i r e new labour and lay-off new recruits. I f no no, irrespective of the precise level .& of wages paid t o the existing labour force. . @ 'dhen no > n, the employatent level is not so preciseiy determined, - and it may depend t o some extent on the wage. Specifically, for - no n, Theorem 2 implies that e i t h e r (a) n = no, with the uagf! a t some level i n the Lnterval y (no) > w > G; or (b) n and w a r e such $hat the point (n, w) w pies within that part of the area ADC in Figure 2 which-also s a t i s f i e s n w > z, or < n +'and y (n) > w > max (Z, m (n)). Evidently, tne conditions r e s t r i c t the s z t of possible outcomes, especially as they relate t o employment, and it is only incidentally that the wage, w, w i l l be equal t o labour's marginal revenue product, m (n). Finally it can be noted that while Figure 2 shows y (n*) > Gs * t h i s is not necessary. Now y (2)= { ;(n*) +r (3) } / , - n so that the * . . previous inequality can be written as v (a ) - *wa > r Hence: Result 6:- Theorem 2 does not exclude the possibility that an efficient contract can imply x - incf f iciency i f - * > n - S { ; - n * } > o . This is evidently stronger that the condition - ~r(I;) (-:I IT > o provided by .Corollary 1 Lor the existence of attractive contracts. 4.3 Stronger Conditions for an Efficient Contrac; The results in Theorem 2 are obtained on the basis of a definition of efficiency which involves the weak specification of preferences for existing workers as i n ,Qssumptioa 3(b)(i). Strengthening this as in 3 (b)(ii) implies further restrictions for contractual efficio,ncy, and hence a more restricted s e t of efficient contracts: Theotem 3: If the preferences of existing workers 'can be expressed simply in terms of their mean equivalenr income, w , then the conditions of Theorem 2 for an attractive contract to be efficient are strengthened i n the case no > ii t o the extent that i f n < no, then ma% (0, m (n)) = w - - {U (w) U (G)}/u'(w) and n a E. To prove t h i s rasult, it can be noted that Theorem 1 determines - - q and e a s q = q*(n) end e m h (n, no), while either n max (n0 n) , or n < no. - Hence either u is uniquely deeermined or . U(w) { n U(v) + (no - n) U(~))/nOand * -- v*(n) - wn. Developing t h i s l a t t e r case, profits are constant if ndw - {max (0, m(n)) v) dn. Given this restriction, it can readily be shown that - U - nodU/dn = [U (<) (v)] + {max (0, m (n)) v} U'(v), implying t h a t - - max (0, rn(n)) = w - - [Nu) U(;)]/U'(W) is a necessary condition for aa efficient agreement a s stated in Theorem 3. - - - The relationship max (0, a(n)) w [ ~ ( w ) u(;)]/u'(v) is graphed - i n Figure 3. For n > 2,aux (0, 140)) is zero so that v has some constant value such that w U'(u) - u(;). U(w) Since U'(w) > o r U"(w) by - - assumption, the expression v - - [ ~ ( w ) u(;)]/u'(w) has the sign of (G -w) , Since the expression has value when W W, it must therefore be zero for some value of w i n u c r r a of 5 as shown in the figure. For values of n l e s s * than n , max (0, m(n)) drcreaaes as n increases. Hence for ii < n < n*, Fige r 3 skwg a monotoaic increasing relationship between - - z. n and w such that i f n then w For n > 5, max (0, m(n)) > n(s); and U"(w) < 0 - - \ implies that w [ ~ ( w ) u(;)]/u'(w) - < G. Since m(n) = G, it follows that i ? the relationship betme1 w and a is nor defined for n < 5. Figure 3 To explore further the implications of Theorem 3, it can be noted - - that if existing workers are cot risk averse, so that U"(.) - - = 0 , then ;. v -[ ~ ( w ) U(Z)]/IJO(u) - w for a l l Hence Theorem 3 implies that n n i f n < no. The folloving corollary of Theorem 3 is therefore established: Corollary 3: If the preferences of existing workers depend only on their expected wage ex ante, 1.e. on (e/no) w + {l-(e/no)l Z, - then a necessary condition for an efficient - contract is that n median (s, no, n). This simple result has some interesting fmplicatioas. First, it implies that there can be no indeterminacy in the level of employment f o r an . . efficient contracc unler8 there is an element of risk aversion on the part of workers, 1.e U"(.) < 0. Similarly, a distributionally efficient contract cannot kt x-inefficient ualess existing workers are risk averse. And finally, there is a degree of inertia fn employment relative to fluctuations i n product demand which is independeat of risk aversion on the parc of workers. Each of these l a s t points requires some elaboration. From Theorems 2 and 3, when no ...< > an attractive and efficient contract must imply n a C no and m(n) = w - [U(v)- U(;;))/U'(W). In - % addition, from Result 5, y(n) > w . Since y(n) w implies a monotonic i decreasing relationship betveen n and w ,(see Figure 2), while 9 - m(n) = v - (U(* u(;)I/u'(w) implies a monotonic non-decreasing reGtion- L ship (see Figure 3), these two equations can be satisf led simultaneousfy only a t a unique point, which can be denoted , 1 , where > 2 and > From the diagrams, it is relatively straightforward to see that , 1 must exist, i.e., that the NO functional relationships between n and w s u s t have a point of intersection in the region n > z, v > G- The point (;, G) i e of some interest because it can-be shown that, f o r an efficient contract, n C d n (no, z) i.e., that is an upper bound on the level of employment, irrespective of how large the I n i t i a l endowment of - labour, no, might be. This follows from the fact that i f n > z, then - - (n, W) must be such that both m(n) v [U(V) u(~)]/u'(w) and y(n) > w, * If is an upper bound on n , it now follows that > n is a necessary condition f o r a distributively efficient contract t o be compatible with x-inefficiency, In other words: Corollary 4: A necessary condition for a distributively efficient contract to imply technological x-inef ficiency -is that y ( d ) > v* when w* is such that * * * v UO(w ) = U(w ) ~ ( 3 )i.e, - - * dlog (U(w) u(Z)]/dlog w *. 1 f o r w = w Pinally, Figure 4 develops the Implication of Corollary 3 f o r the sensitivity of employment t o fluctuations i n product demand, In the figure, product demand is'assumed t o follow some cycle over a series of short runs, . 8 - c which is reflected in cycles for n and via movements i n the marginal . revenue product rchedule m(n) Boycver, cycles i n and do not 8 necessarily imply any change in the employment level n : i f no is ' i n i t i a l l y between the levels A and B in Figure 4(a)-, then the i n i t i a l value of n = no w i l l maintain throughout. Alternativ&ly, if no is i n i t i a l l y a t point C, say, then the rule n = median (z, no, z) implies that n dl1 follow the path C D E F C ,at which point it reaches level 8 , and w i l l remain there subsequently. An alternative scenario is provided by Figure 4(b). Bere again, there is some uncertainty about the i n i t i a l path . . - - for n Starting at C , for example, n w i l l follow the path C D E F G ~t G, n0 n, and n w i l l remain a t no even though f a l l s away, until point is reached. A t H , no = and f a l l s , bringing n d o n t o . point I Employment now remains constant at t h i s level until J , at vhich point it increases again, etc. The implication of Figme 4 is that employment will be iaert relative t o product demand. After an i n i t i a l s e t t l i n g d o n , cycles i n product demand, may produce a damped response in employment or no response at all. And if there is a response, then t h i s w i l l appear a s a lagged response, with the product demand cycle leading the cycle i n eiployment. . . Case a ~YAF-W- IX -53. A - ' - - - - - - - - - - - - - - - - - n tice Case b Figure 4 Comparative Static Effects of Cyclical Movements i n Product Demand on the Level of Employment Cairr, i,. G. (1976): "The Challenge of Segrented Labour Narket Theories to Orthodox Theory: A Survey,"Journal of Economic L iteratiire, Vol. 14. O i ,W. -1. (1962): "Labor a Quasifiso,d Factor," Journal of Political Economy, Pol. 70.