2-.: ?,... ,,.LC! :I. :;;.;;her:, ar.8 Joseph E. S t i c 1 i . t ~ WAGE RIGIDITY, IMPLICIT CONTRACTS, AVD ECONOMIC EFFICIEKCY: ARE MARKET WGES TOO FLEXIBLE?* David M. Kewbery Develophent Research Department The World Bank and Joseph E. S t i g l i t z Department of Economics Princeton University August 1982 Revised June 1983 . &Preliminary and Confidential: Not f o r quotation o r a t t r i b u t i o n wtthout p r i o r ~leciilci~l~e i r ~ uL I Z dutnars. ;'ie;js exp;~,,cld d : ~ :i;c;e of Th,e ~ a r h c r - ,J ~ Z33 - not necessarily r e f l e c t those o a t h e World aank o r Princeton University. lie a r e indebted t o Richard Arnott,$ostas Azariadis, Sandy Grossman, Oliver h u t * and Arthur Hosios. a m WAGE RIGIDITY, IbIPLICIT CONTRACTS, AND ECONOYIC EFFICIENCY: ARE MARKET WAGES TOO FLEXIBLE? Abstract The a n a l y s i s of i m p l i c i t c o n t r a c t s between risk-averse workers and risk-neutral firms must recognize t h a t ( i ) the c o n t r a c t is i m p l i c i t , not e x p l i c i t , ( i i ) it may only be conditioned on observable v a r i a b l e s , and (iii) . t h e r e a r e limits t o c o n t r a c t complexity. If c o n t r a c t s a r e p e r f e c t l y f l e x i b l e than n e i t h e r t h e c o n s t r a i n t of e n f o r c e a b i l i t y nor o b s e r v a b i l i t y r e s u l t i n unemp:oyment. However, even with p e r f e c t e n f o r c e a b i l i t y and f l e x i b i l i t y , l i n i t a t i o n s on con2racc f l e x i b i l i t y may generate unemployment. F i n a l l y , e v e n \ with f l e x i b l e enforceable c o n t r a c t s anti no unemployment t h e market e q u i l i b r i u m is inefficient. We c o n s t r u c t some simple general equilibrium models and explore t h e consequences of r e s t r i c t i o n s , on t h e s e t of f e a s i b l e c o n t r a c t s , at t h e same time comment!ag on t h e present s t a t e of i m p l i c i t c o n t r a c t theory. The i m p l i c i t contract theory of wages argues t h a t workers a r e -re r i s k averse than firms, and t h a t firms w i l l therefore find it less c o s t l y to h i r e labour i f they provide some degree of income insurance against fluctuations i n demand which would otherwise lead t o f l u c t u a t i o n s i n t h e demand p r i c e of labour. Wages w i l l be less f l e x i b l e than i n t h e Walrasian model, and, i n extreme cases, w i l l be rigid. Since t h e r e is a ~ i d e s p r e dand l ~ n gstanding b e l i e f t h a t wage r i g i d i t i e s give rise t o unempl~yment, it is natural t o conclude t h a t the r i s k s h i f t i n g aspect of implicit ccrrltracts fs responsible f o r unemployment, and t h a t these r i s k s h i f t i n g b e n o f i t s o u t w d g h the costs of unemployment. In the strong form 04 chis argument, it is claimed t h a t t h e resulting l e v e l s of unemployment a r e constrained P a r e t o e f f i c i e n t , where the c o n s t r a i n t is t h a t the absent income insurance markets which give rise t o the demand f o r reduced wage f l u c t u a t i o n s remain absermt,- There a r e f i v e important problems with t h i s theory. F i r s t , if: s t a t e s of nature zre observable t o both employers and employees, then tbis axplanation of unemployment f a i l s , for the r e s u l t i n g equilibrium is iderarr+cal t o one i n which t h e firm always pays Valrasian wages, and the inaividual. purchases income insurance from an insurance company. The equilibrium W d e n t a i l f u l l emp?.ope;lt, and t h e income insurance w i l l , i f anything, red- the v a r i a b i l i t y i n demand. -11 Second, the standard theory -- a s w e l l a s some of the more recent developments - f a i l s t o d i s t i n g u i s h between i m p l i c i t and e x p l i c i t contracts. The f a c t t h a t contracts a r e i m p l i c i t has several important .- consequences. Thus e x p l i c i t contracts can relp on t h i r d party enforcemegt through the law courts, but i m p l i c i t contracts cannot. Instead, t h e f i r m slst -9 - - 11 For e a r l i e r discussions on t h i s point, see S t i g l i t z (1977) and Akerlad and --- Miyazaki (1983). The ~ k e r r o fand Miyazaki paper, l i k e t h a t of much 4 rhe implicit contract l i t e r a t u e , does not allow f o r any v a r i a b i l i t y i n 8 3 ~ d) ~ number of hours worked by individuals, and thus cannot address some d the 0 c e n t r a l questions a t issue. have an incentive t o honour the i m p l i c i t c o n t r a c t , and t h e most obvious such i n c e n t i v e is the preservarion of its reputation as an employer. 1/ It is important t o recognise t h a t there are l i m i t s t o t h e e n f o r c e a b i l i t y of c o n t r a c t s from 'L-0ths i d e s of t h e market. Firms have an incentive t o c a t waees more than the workers expected i n bad times, whilst workers may have a n i n c e n t i v e t o q u i t i n good times even though the firm had continued t o pay them throughmt a s l a c k period. (See, e.g. Holmstrom, 1983). The r e p u t a t i o n mechanism may work t o prevent the f i r s t , but not t h e second. Third, while e x p l i c i t c o n t r a c t s can i n p r i n c i p l e contain e l a b o r a t e and c o . q l l c a t e d provisi,ons which coq&$.>e enforced, it seems unlikely t h a t L I n _ . i m p l i c i t c o n t r a c t s relying on reputation can be very complicated. The r e l a t i o n s h i p bztween what the p a r t i e s t o t h e i m p l i c i t c o n t r a c t can observe and what consequences follow must be c l e a r t o both s i d e s , f o r , by d e f i n i t i o n , t h e y do not have a w r i t t e n s e t of r u l e s which s p e c i f i e s consequences f o r each individual case. Another way t o make t h e same point is t h a t only very simple P ' . r e l a t i o ~ s h i p scan be i d e n t i f i e d econometrically from p a s t , limited, observations, and more complex r e l a t i o n s h i p s would r e q u i r e more d a t a t o d i s t i n g u i s h from alternatives than is l i k e l y t o be a v a i l a b l e . Thus i m p l l c i t c o n t r a c t s must have a simple form, and t h i s w i l l g r e a t l y l i m i t t h e i r a b i l i t y t o provide both f u l l employment and adequate insurance., 31 Fourth, what terms of the contracr are enforceable depends both on . what is observable and whL?t the enforcement mechanism is. Clearly, t h e terms of any contract can only be contingent on v a r i a b l e s observable by both s i d e s of a c o n t r a c t , but i f the contract is t o be enforced by appeal t o a t h i r d - 21 Myopic s e l f - i n t e r e s t1s another possible enforcement mechanism - workerr may respond d i r e c t l y t o an u n j u s t i f i e d drop i n t h e l r wages i n such a way a s t o reduce t h e firm's p r o f i t s . The e f f i c i e n c y wage hypothesis argues i d ' ~ A A r r i r aK iiAc1l kdgrr dre C U L -- e l t I l t 2 C L O X n u t r i t i o n a l reasons ( B l i s s and S t e r n , 1978), o r because of a drop i n morale. See Calvo a979 ), Shapiro and S t i g l i t z (forthcoming), S t i g l i t z 5 (1976), Arnot Hosics and S t i g l i t z (1980), Weiss (1980), Moore (1982). For a survey some e f f i c i e n c y wage models, see S t i g l i t z (1983). -3/ The p o t e n t i a l importance of r e s t r i c t i o n s w a s a l s o noted by Akerlof and bliyazaki (1980). The r e s t r i c t i o n s which they focused upon, however, w e r e markedly d i f f e r e n t from those which we consider below. party, the observations must a l s o be v e r i f i a b l e , which is 3 much stronger requirement. Much information is t r a n s i e n t , and may be readily observable by workers and t h e i r employer a t the relevant rnonent, but because it is hard to record, may be imrnedi i c e l y l o s t and hence unavailable f o r v e r i f i c a t i o n . Reputation mechanisms, r~;t the other hand, do not require the sane kind of v e r i f i a b i l i t y . On the other hand, i f a contract is t o be enforced by a reputation mechanism then additional c o n s t r a i n t s on t h e s e t of f e a s i b l e contracts a r e required: the firm must be b e t t e r off maintaining its reputation than it would be "breaking" the i m p l i c i t contract. The e a r l i e r i m p l i c i t contract l i t e r a t u r e assumed t h a t t h e s t a t e of nature was both observable and v e r i f i a b l e , and thus contracts could be w r i t t e n which were contingent upon t h e s t a t e of nature. The more recent l i t e r a t u r e &/ recognises t h a t much of the information about t h e s t a t e of nature is n e i t h e r observable nor v e r i f i a b l e , but f a i l s t o give a convincing account of what information is both available and can be used i n t h e design of an enforceable i m p l i c i t contract. I n p a r t i c u l a r , it f a i l s t o introduce i n t o the analysis information which is plausibly a v a i l a b l e and v e r i f i a b l e ( l i k e the aggregate u-employment r a t e , the price l e v e l , product prices charged by o t h e r f i r a s , etc.) which would be introduced i n any e x p l i c i t contract which attempted t o maximise the expect?d ( u t i l i t y o f ) p r o f i t s of t h e firm, s u b j e c t t o the workers receiving a p a r t i c u l a r l e v e l of expected u t i l i t y , and s u b j e c t t o various self-selection constraints. (But see Grossman, Hart and Maskin, 1982). In t h i s paper, we s h a l l argue t h a t while t h e s t a t e of nature may n o r be d i r e c t l y observable, the probability d i s t r i b u t i o n of wages and hours associated with any firm can be observed. While such informaticn might be i r r e l e v a n t f o r t h e s t r u c t u r e of one period c o n t r a c t s which were t o be enforced s o l e l y by l e g a l l e a n s , it is c r i t i c a l f o r multi-period contracts, and t h e fact . e @ r - 3a/ See, e .g. A rdriadis and S t i g l i t z (19837, ilart (1983), Azariadis (1983), Grossman and Hart (1981, 1982), Chari (1983), Green and Kahn (1983) and Cooper (1982). that most implicit contracts are by necessity enforced through a reputation mechanism makes these contracts, implicitly, multi-period contracts. -I 4 Fifth, and finally, 5/ the implicit contract literature is concerned with explaining unemployment; unemployment is a macro-econonic phenomenon, yet ith a few exceptions, the implicit contract literature has analyzed only partial equilibrium models. It is only within a model of the whole economy w that one can assess claims about the effect of alternative contractual arrangements on the equilibrium level of employment. Noreover, the frequently expressed claims about the optimality of contract equilibrium / are usually nothing more than the restatement of the problem of contract equilibrium: the firm maximizes its expected (utility of) proflt, subject to being able to obtain workers, i.e. giving them a given level of expected utility (subject to whatever informational or other constraints that are imposed). The solution to this problem is obviously efficient in a partial equilibrium sense, e.g. given the probability distribution of prices, the probability distribution of wages qffered by other firms, etc. However, the relevant question is whether - 41 This is a second sense in which the recent literature on implicit contracts with asymetric information is internally inconsistent the fact that such contracts are enforced by reputation makes them multi-period coct-racts,while these papers characterize the optimal one period ccatract. However, with asymetric information, the optimal multi-period contract may look markedly different. For instance, if the rate of discount is zero, and there were an infinite number of periods, equilibrium will be close to that which would have prevailed with symmetric verification. See below, Section 2.4. -51 We should emphasize that this list of deficiencies with the implicit contract literature is not meant to be exhaustive. Two other problems should be mentioned. A theory of unemployment must not only explain the aggregate number of hours worked, and why this is less than that . associated with full employment: but it aiso needs to explain the pattern I of unemployment, why, in partlcuiar, thlire are layoffs rather than work sharing. The standard arguments provided by the tradit-ionalimplicit c o n t r a c t l i t e r a t u r e a r e cons1s t e n t n n l y wi r h t h e r e h ~ l work s h a r i n r , ; n ~ they do not explain layoffs. This tact, j.n itself, should make-shese theories suspect. Second, the fact that a worker is laid off do& not mean that he is'unemployed. With costless labor mobility, a worker who is laid off will bl immediately rehired by some other firm. Thus, S theory of unernployment'must not only explain why workers are laid off, Kut also explain why other firms do not immediately rehire them. (Hosios, and Arnott, Hosios, and Stiglitz have recently constructed such models.) - 61 Cf Barro (1977). the market equilibrium is (constrained) Pareto e f f i c i e n t when a l l firms sign such contracts, and prices and wages-are determined endogenously. Does t h e r e e x i s t , f o r instance, a set of taxes acd suka,sidiss which can generate a Pareto improvement t o t h e economy? We show t h a t , i n general, the equilibrium generated by i m p l i c i t contracts is not Pareto e f f i c i e n t . The objective of t h i s paper then is t o construct some simple,general equilibrium models within which we can explore the consequences of a l t e r n a t i v e assumptions about t h e nature of the set of f e a s i b l e contracts. 8/ We s h a l l examine three aspects of contract design: what is observable :which dete-&&&s t h e events upon whkh t h e contract can be contigent); what is enforceable; and what l i m i t s the complexity of admissible contracts. While the t r a d i t i o n a l i m p l i c i t contract l i t e r a t u r e assumed p,rfect. observability, perfect enforcea5ility, and perfect f l e x i b i l i t y , we s h a l l argue t h a t p l a u s i b l e assumptions on observability and enforceability combined with perfect f l e x i b i l i t y gives rise t o f u l l employment. Consequently we argue t h a t a c e n t r a l explanation of unemployment is the limited f l e x i b i l i t y ailowed by implicit contracts. Before beginning our a n a l y s i s , we should explain what we mean by unemployment. The standard d e f i n i t i o n focuses on t h e individuals' perceptions: i f t h e individual is a c t i v e l y seeking a s u i t a b l e job and is unable t o obtain one, then he is unemployed. The requirement t h a t t h e job be s u i t a b l e f o r the individual is obviously important, f o r there mav be a job opening f o r a dishwasher, but the f a c t t h a t a trained engineer, who cannot get - - - 7/ Taking the set of insurance markets and the a v a i l a b l e information a s given. For a discussion of notions of constrained opiimality, see Diamond (1967), &d Newbery and S t i g l i t z (1982). -8/ One such attempt, from a very d i f f e r e n t perspective than t h a t reported here, is t h a t of Hall and Lazear- They assume t h a t workers cannot confirm t h p ;tl:r7 C T ~ n c l t 3 , , - ~ ,(;;,niv r.,q.- r i - 7 1~ ? r q < ~ f ~ ct vt f v i ..,i:hi~ t l j n f i r ~ . ?Xt!f . they a l s o assume t h a t the firm cannot write a contract with wages contingent on hours o r employment levels. The only f e a s i b l e c o n t r a c t s are those w i t h completely r i g i d wages. 1; a sense, this is an extreme case of t h e model t h a t we consider i n ~ e c t i o n ? ~where we postulate t h a t t h e r e is ~ , a simple r e l a t i o n s h i p between t h e wage paid and c e r t a i n macro-economic variables- ;There a r e , of course, a number ~f other important d i f f e r e w e s between the two models including the f a c t t h a t they model only the l a b o r market, not the general equilibrium of the economy.) a job a s an engineer, does no?. take t h i s job does not v i t i a t e the f a c t t h a t ne is involuntarily unemployed, though some economists would suggest that it does. The point is tha: t h e r e a r e individuals with t h e same s k i l l s and q u a l i f i c a t i o n s who have jobs and he is w i l l i n g t o take an identica! job but he cannot obtain one. The d i f f i c u l t y w'ih t h i s approach is t h a t it r e s t s on t h e individual's perception of h i s q c a l i f i c a t i o n s , and an individual yay have an overly i n f l a t e d view of h i s a b i l i t i e s . To avoid t h i s d i f f i c u l t y we employ a more technical d e f i n i t i o n : we say t h a t an i n d i v i d u a i is ( p a r t i a l l y ) unemployed i f h i s marginal rate of s u b s t i t u t i o n between consumption and l e i s u r e d i f f e r s from h i s marginal r a t e of transformation; hence i f the wage is equal t o the marginal r a t e of transformation, and he would l i k e t o work more a t t h e going wage, we say he is ( p a r t i a l l y ) unemployed. The d e f i n i t i o n d o e s not sag anything about whether t h e l e v e l of employment, i n t h e given s i t u a t i o n , is g r e a t e r o r l e s s than it would be i n t h e corresponding Walrasian economy (with perfect infotmation, perfect insurance markets, etc.) For t h a t requires a camp.-rison of general e q u i l i b r i i ~s o l u t i o n s , which t a k e s us beyond t h e scope of t h i s paper. Nor do we d i r e c t l y address t h e philosophical question of whether a n individual who signs a c o n t r a c t which has t h e provisEon t h a t with some p r o b a b i l i t y he w i l l be l a i d o f f , is v o l u n t a r i l y o r i n v o l u n t a r i l y unemployed when l a i d o f f . He has v a l u n t a r i l y signed a c o n t r a c t . which ha^ assigned t o o t h e r s t h e r i g h t t o make a d e c i s i o n , which he would n o t v o l u n t a r i l y make himself. Our d e f i n i t i o n does not, i n i t s e l f , say anything about t h e e f f i c i e n c y of t h e equili5rium. Since we a r e describing market s t r u c t u r e s in . I which there a r e , f o r instance, r e a l c o s t s associated with information, t h e s e c o s t s need t o be taken i n t o account. An equilibrium i n an economy with c o s t l y -- . - - ? R : - , ~ , T . : ; , ! .:(>q ;<,;-,2,;, .. . . .... , xi;:>, p d z i a & - . . - : . , , > ( ! I . . . - .ci ... . 3 : - . . , - < , . - . . frd::: : ~ i : ~ i :d l J:: d~ j r~ c ; ~ I :O:F 1 F I Y ~ C ! c n e f f i L r n t r: c: - technology: ! ' C = min (Nth, K/v) where C is the output of the consumption goods industry, Nc is ~aploymenti n the industry, h is the hours worked, K is the c a p i t a l stock i n the induszry, and v is the fixed capital-output r a t i o in the sector. Capital must ezrn an expected return of r. Our concern is with the design of t h e optimal contract i n t h e non- traded sector. For s i m p l i c i t y , we assume workers i n t h e export s e c t o r are s e l f -employed, and thus recei;e a wage of (per u n i t of labor supplied). A t the beginnicg of t h e period workers are f r e e l y mobile, but once they have decided whether t o work i n t h e export o r no11-traded goods s e c t o r , they are p e r f e c t l y immobile. We s h a l l be concerned with interior equilibria i n which there are workers i n both the non-traded and he expnrted goods sector. 411 i n d i v i d u a l s are identical; they have a standard u t i l i t y functicn defined over t h e i r consumption of t h e imported consumption good, the non-craded consumption good, and l e i s u r e (work); it is more conven::nt i f we represent i t by t h e i n d i r e c t l y u t i l i t y function, V = V(q, w), giving u t i l i t y as a function of the wage received, w, and t h e p r i c e of the non-traded consumption good, q, provided t h a t workers a r e not constrained i n the amount of l a b o r which they can supply. I f they a r e constrained, then t h e i r u t i l i t y can be represented by a modified i n d i r e c t u t i l i t y function V = V(q, y, h ) , giving u t i l i t y a s a function of t h e p r i c e of the non-traded consumption good, t h e i r income, y, and t h e number of hours which they work. I n equilibrium, there w i l l be some r e l a t i o n s h i p between t h e p r i c e o f t h e export good ( t h e exogenous source of uncertaiaty i n t h i s model) and t h e p r i c e of the non-traded consumption good, which we denote by q = (p; Q ) , where Q is a vector of market parameters, t o be s p e c i f i e d l a t e r . ..." . Since f o r workers"in tKe export s e c t o r (subscript x) - ** w e can write t h e exptcted u t i l i t y of workers i n t h e export s e c t o r by 0 The equilibrium i m p l i c i t contract i n t h e on-.traded consumption goods s e c t o r is "signed" before t h e state is known. How the c o n t r a c t is formulated depends, of course, on what assumptions concerning o b s e r v a b i l i t y , e n f o r c e a b i l i t y , and f l e x i b i l i t y we make. We begin our a n a l y s i s by a l t e r i n g only one of t h e three assumptions of the standard implicit contract l i t e r a t u r e ; w e s h a l l assume t h a t the s t a t e of narure is p e ~ f c c t l yobservable, and t h a t the contract can be a s complicated a function of the s t a t e of n a t u r e a s desired. However, we s h a l l assume ;hat the consract is not enforceable as an e x p l i c i t contract. \Je s h a l l assume that the firm anriDunces a wage a s a functiorl of t h e s t a t e of nature (p) a t the beginning of the production p c r f d , - and then, once p becomes known, it commits i t s e l f t o pay a partic.:ilaiS wage, though not necessarily the oue implied by its e a r l i e r announc~mcnt. The ftrn can break its i r l p l i c i t contract i n two d i f f e r e n t ways -- it can lay off workers when the contract c a l l s f o r t h e i r r e t e n t i o n , o r pay them a iower wage than the contract implies. The problem of enforcement is differenc i n each case. The second kind of contract breach -paying t h e wrong wage -- can be avoided by the following self-enforcement mechanism: i f the worker r e c e i v e s a wage lower than the i m p l i c i t l y agreed wage, then he s h i r k s , and h i s labour prodactivity drops by more than the s h c r t f a l l i n wages, thus discouraging the firqn from deviating from its implicit wage schedule. -91 On the other hand, i f it lays off workers t h a t it had promSsed not t o lay o f f , those l a i d off workers cannot r e t a l i a t e by shirking. ( I t is, of course, possible t h a t retained workers s h i r k i n synpathy, but we ignore t h i s possibility.) Pie implication Is that the contract can specify the number of -91 The following argument demonstrates t h a t it is not i g t h e i n t e r e s t of t h e firm t o deviate from its wage schedule. Suppose the worker e i t h e r provides the correct l e v e l of e f f o r t , e, or zero e f f o r t . I f the f i r m c a n subsequently observe but not v e r i f y the worker'c e f f o r t , tben it cannot maKe r n e c u r r r n c wdgr C U L ~ ~ A A A ~V ZA . i> ~~ r ~-:LO .;'~ : . zIy -, . -- F ~ ii however, make future ezployment o r future wages contingent upon observed l e v e l s of e f f o r t i n e a r l i e r periods,-and the worKer can make h i s l e v e l of e f f o r t contingent upon the behaviou of t h e firm i n previous periods, There is a perfect equilibrium i n w &ch f i m s respond t o shirking by paying a zero wage (or f i r i n g the worker), and i n which workers respond to contract breach by f i r n s by providing zero e f f o r t . Given t h a t the worker believes t h a t once he has shirked, the firm w i l l pay him zero wages f r o m tl en on, it pays the worker t o s h i r k from then 2n; and given t h a t t h e worker s h i r k s from then on, it pays the firm t o pay the worker only zero wages. Similarly, given t h a t the firm has broken its c o n t r r c t and that it expects the worker t o shirk i n subsequent periods, the worker expect9 t h e firm w i l l pay him a zero wage i n subsequent periods; and given t h a t the worker believes t h a t t h e firm w i l l be paying him a zero wage i n subsequent periods, i t pays him t o s h i r k t h i s period. \lours worked and income paid a s a function of t h e s t a t e , p, but t h a t t h e set of (hour, income) combinations which a r e admissable a r e those f o r which it is i n t h e i n t e r e s t s of t h e firm t o r e t a i n t h e worker, i.e. f o r which t h e v a l u e of :he worker'? marginal product exceeds t h e wage paid. - 101 There is m o t h e r argument f o r t h e asymetric treatment of t h e e n f o r c e a b i l i t y of wage promists and lay-off promises when t h e enforcement mechanism is reputation. A firm which l o s e s its r e p u t a t i o n w i l l have to pay workers higher wages i n t h e f u t u r ? i n order t o r e c r u i t them. But t h e f i r m w i l l only l o s e its r e p u t a t i o n i f it is observed v i o l a t i n g t h e i m p l i c i t c o n t r a c t , and informatiori about wages may -he more r e a d i l y obtainable t h a n information about l a y o f f s . Individuals a r e l a i d off f o r a v a r i e t y of reasons. The worker may not be a b l e t o d i s t i n g u i s h i n d i v i d u a l s who have been l a i d off because t h e firm is reducing its employment from indivduals who h a v e been l a i d off f o r some o t h e r reasons (e.g. because they a r e bad workers.) Since firms a r e always e x p a n d i ~ ~some operations, contract-ing o t h e r s , t h e g wor!cer would f i n d it d i f f i c u l t t o v e r i f y compliance f o r even a n e x p l i c i t c o n t r a c t provision t h a t s p e c i f i e d the f r a c t i o n of t h e employees t h a t would be l a i d off i n s,me s t a t e of nature. Clearly, v e r i f y i n g such a provision in an i m p l i c i t c o n t r a c t is even more d i f f i c u l t . I n a formal sense, then, t h e c o n t r a c t s which we ar-alyze i n s e c t i o n 2.2, where, i n some s t a t e s of n a t u r e , t h e e n f o r c e a b i l i t y c o n s t r a i n t is binding, can be thought of a s t h e op2imal -101 li workers can never receive a wage i n excess of the value of the maiginil product, a n d ~ i ft h e competitive equilibrium c o n t r a c t is t o have t h e property ( f o r r i s k n e u t r a l firms) t h a t t h e expected wage equal t h e value of t h e i r expected marginal product, then does t h i s imply t h a t t h e wage - must be equal. t o the value of the margin21 producr i n every state o f ..., [Idcure i .> L i , c Ld;:zL:-;LJ,: L j r L c ; ; L s ,--,.,..~. . ~~., . < . .i .. in :? + , It-;lkc? f hPo APrnr!p.t r h e .r '9 d i f f e r e n c e between t h e ex a n r e and ex post marginal products. Before thc indvidua; is hired, we compare the p r o f i t s of the firm with and without + the i n d i v i d u a l , takinc; i n t o account a l l of t h e a n c i l i a r y expenditu'res c required f o r him t o be prod!::.: ively employed, e.g. s p e c i f i c crainlLtrg, machines, e t c . AfLer he has been h i r e d , a t l e a s t some cf t h e s e a r i c i l i a r y c o s t s a r e sunk, and t h u s a r e i r r e l e v a n t f o r t h e d e c i s i o n about whether po lay o f f an individual. Although we could have couched our a n a l y s i s i n terms of s p e c i f i c t r a i n i n g c o s t s , we s h a l l assume t h a t m c h i n d i v i d u a l requires a f i x e d amount of c a p i t a l t o be employed; t h i s is t h e sunk c o s t which r e s u l t s i n t h e r e being a d i s t i n c t i o n between ex post and ex ante marginal p r o d u c t i v i t i e s . c o n t r a c t cor.tingent on t h e unobservability of t h e layoff r a t e . 11!Workers of course know t h a t t h e behavior cf t h e firm w i l l be a f f e c t e d by t h e wages s p e c i f i e d i n t h e co.itract ( t h i s is t h e standard moral hazard problem) and it is this which motivates them t o prefer contracts in which the wage does n o t f a l l belou, c2:. value of che marginal product. Thus, ra summarize, t5e c c n t r a c t w i l l be chosen t o maximize p r o f i t s , subject t o two c o n s t r a i n t s : ( i ) it must be p o s s i b l e f o r the f i r m t o h i r e workers, i.e., t h e expected u t i l i t y of workers u n d ~ t h e c o n t r a c t must be at least equal t o Wx; and ( i i ) t h e c o n t r a c t must s a t i s f y the e n f o r c e a b i l i t y d c o n s t r a i n t s j u s t discussed. I n equilibrium t h e number of firms i n t h e industry w i l l be such t h a t ( t h e maximized value of expected n e t ) p r o f i t s im t h e industry ere driven t o zero. For purposes of comparison with t h e constraj-ned s o c i a l optimum, St is more convenient i f we analyze the dual of the problem that we have just formulated; i.e., consider t h e c o n t r a c t which maximizes the expected u t i l i t y of workers, s u b j e c t t o a c o n s t r a i n t on p r o f i t s (which, i n equilibrium, is simply the zero p r o f i t c o n s t r a i n t ) , and subject t o t h e e n f o r c e a b i l i t y c o n s t r a i n t . Thus, t h e i m p l i c i t c o n t r a c t -121 subject t o h(p)q(p; i2) ,y(p) , ( t h e e n f o r c e a b i l i t y c o n s t r a i n t ) - , t K/v ,Nch(p), (production e f f i c i e n c y - see equation (1)) qnrl - E { ~ ( P ) ~ ( Pfl); - b ( p ) } ,rK/Nc r (zero p r o f i t c o n s t r a i n t ) -111 Thus, it should be no&d t h a t an i m p l i c i t assumption of t h e recent l i t e r a t u r e on i m p l i c i t c o n t r a c t s with asymetric information is t h a t the lay-off r a t e is observable. -121 Note t h a t t h i s formulation is s u f f i c i e n t l y g e c e r a l t o admit both l a y o f f s and work-sharing. Layoffs e n t a i l h = 0, with some probability. Complete work sharing e n t a i l s , i n any s t a t e of nature, h being the same f o r all individuals. where each of the competitive firms takes a s given the relationship betweem the s t a t e (p) and the market eqsilibrium price ( q ) , which is given by t h e market equilibrium equaticn that i n each s t a t e demand equal supply:- 131 where Dc and Dx a r e per capita demands. Demand is a function of the market price, q, and i n addition, a vector of market parameters -- the price of tie export good (equal t o the income of export workers), the income of workers i n the non-traded sector, the number of hours they work, and the a l l o c a t i c n of labor between the two sectors. (In the special madel of Part 111, the demand depends only on some of these parameters.) h e demand curve can easily be derived from the behavior of workers i n each ~f the two indusv -ies. (See Appendix 1). Thus f o r workers i n the export industry, the demand (per worker) of the non-traded good is given by Dx = -V / V where VI is the marginal 9 I u t i l i t y of !ncome. In the non-traded sector demand per worker is A A D = -V C 9/V Y where hats distinguish the non-traded sector. ' Variations i n the price of the export crop give rise t o v a r i a c t a a s i n the demand f o r the non-traded :onsumption good. These variations, i n Uurra, lead t o variations i n the equilibrium price, hours, output and possibly employment i n the non-traded consumption goods sector. The ( r a t i o n a l expectations) market equilibrium may. now be analyzed. Given a function q(p; 0) and an allocation of labor between t k tm sectors, Nc, the maximization problem (4) yields an equilibrium contra-t I I {h(p), y(p)} and an equilibrium level of expected u t i l i t y for consumptio~llgood wol-kers (subscripted c ) , W {q(p) }. The market equilibrium equation (8) then C - y i e l d s an equilibrium price function q+0 , i.e. capital is fully employed, then h is given by h.= h = K/vNC. The margiaal product of labor is not well defined; it is clax zero for increasss in labor, q for decreases in labor. Equation (13) says that the marginal rate of substitution is greater than zero but less than q.) Onc might be mislead by these results into believing that the economy is efficient. After all, the labor market contracts have workers working at an efficient level and the marginal rate of substitution equals the margina? rate of transformation. But the contracts have been drawn up with each firm taking the price distributions as given; and these price distributions depend on the allocation of labor between the two sectors and on the level of investment in the non-traded sector. In the next section, we show that these decisions will not be made correctly in a market economy. We summarize the result of this section in - Proposition 1. An equilibrium implicit contract wit.h or without the enforceability constraint entails full employment. -141 2.3 h e Efficiency of Market Equilibrium -7 We now consider the optimal policy of a government which controls the non-traded goods sector. To make the comparison with the market Z equilibrium as clear as possible, we shall assume that the government faces the same enforceability constraints as does the private sector, that it cannot directly control labor allocation, and that as a result, it must equate the I . level of welfare in the two sectors. (There is no compelling reason why any I i of these constraints should be binding upon the government; if they are not, I the case for the non-opiimality of the market equilibrium is even stronger.) I The government's maximization problem can thus be summarized by the SNc 1 + N-N C m A(P) {hq- Y} + V(P) {h/v- rich } + ~~{hq y - - rK/Nc} + ( Nc {;- V}] , - -111 This result parallels that of Akerlof and Hiyazaki (1980) and Stiglitz (1977), but they did not allow workers to vary the number of hours that they worked, nor did they impose the enforceability constraint. where the control variables are {y,h,s,~=,K}, and where q is a function of the control variables given by (8). Differentiating (14), we obtain The competitive market will be efficient if there exists a value of a for which (15) and (16) are consistent with (11) and (12) when s = 0- (As the distributional weight a varies, the social optimum will trace out the set of efficient allocations. We are adopting the ssse strategy to explore the efficiency of competitiva equilibrium that we used in Newbe~yand Stiglitz, 1982.) If aL/aq = 0, then the two solutions coincide, but in general, aL/aq # 0 and hence the market equilibril!m will be inefficient. It is fairly straightforward to show that essentially the same conditions are required for market efficiency as those established by Newbery and Stiglitz (1982), namely, that risk markets must be redundant. To show this we examine the conditions needed for aL/dq = 0: Substitute for ( A + p) from (15) and use Roy's formula and (8) to give - * N = (N- 5)Dx C {(l + E) V y - V I {a- -5-1) (18) C But a i s chosen to make aL/as = Oat s = 0, i.e. Substitute (19) i n t o (18): A Equation (20) says that aL/ aq = 0 i f and only i f the r a t i o of the marginal A u t i l i t i e s of income i n the t w c sectors, V /V is constant across states of Y 1' nature (i.e. a s p and hence q varies). This is a very stringent requirement, equivalent t o A A A Even i f V = V = 0, the remainlng terms w i l l i n general only be zero i f t h e YP Iq agents are r i s k neutral, for h a f f e c t s the income of the non-traded good workers, whilst p = wx d i r e c t l y a f f e c t s the income of the a g r i c u l t u r a l workers. These r e s u l t s can be summarized a s follows. Proposition 2. An equilibrium implicit contract equilibrium is i n generrl i n e f f i c i e n t unless agents a r e r i s k neutral. 2.4 Reputation and the Enforceability Constraint In the preceding sections we assumed t h a t wages were constrained t o - being l e s s than or equal t o the value or the m d r g ~ u d i~ c u . ~ L LCE~ .Icibor. 9 Otherwise, r ~ e argued, the firm would have an incentive t o renLege on the e (implicit) dbntract. This assumed that there was no p e n a l t y g n reneging. In * - f a c t , it is plausible t o assume t h a t any firm that reneged on0its contract could only obtain workers i n the future a t a higher wage; there is a penalty associated with paying a wage d i f f e r e n t from the promised or of laying off a worker or making him work more than one promised. The penalty associated with reneging on a contract ("losing one's reputation") i 3 J(p,r2), where p continues t o denote the s t a t e and where Q is a vector of market parameters, J represents the reduction i n the present discounted value of p r o f i t s from a l o s s of reputation. The firm considering laying off a worker that i t has promised t o pay a wage of w(p), which exceeds the value of h i s marginal product, has a one-time gain of w - qh. Thus, enforceability requires t h a t w - qh < J(p, Q) for a l l p. d The analysis of the market equilibrium is i d e n t i c a l t o that discussed e a r l k r (with the straightforward modification of the Lagrangian (lo)), and we a g a i n find that there w i l l be f u l l employment. It w i l l still be true that the market equilibrium w i l l not be Pareto e f f i c i e n t , but nnw there is an additional source of inefficiency. The m a g p i - tude of the penal-y ; r i l l i n general be a function of the wage, and the employ- ment policies of ,111 other firms. I f that is the case, any single firm ignores its effect 3n the set of enforceable contracts, but the government, i n deciding or. the wage, employment policy, w i l l take t h i s e f f e c t i n t o accouct. - 15/ Imperfect observability. So f a r , we have assumed that the s t a t e of nature is perfectly observable t o both firms and workers. Assume, hovrever, t h a t the s t a t e of nature is not observable, but the probability d i s t r i b u t h n of the s t a t e s of nature is known. Suppose, f o r example, that it is known that 50 percent of the time, p is high, and 50 percent of *the t i m e p is low. aben a contract which specified that 50 percent of the time the wage should be vl dild JU ::-.: . ( J Y c w~ ~ ~ ~ L L ~ L LU: L L ~ . L I a 1 - I 7.7, 1 1 1i)J,. i y thp waqp that L would have prevailed with perfect observability of the s t a t e of nature, w w l d have the firm paying the high vsge i> the good s t a t e and the low wage i n the 8 bad s t a t e : the r e s t r i c t i o n on the probability d i s t r i b u t i o n of s t a t e s f o r c e s - 15/ For a formal analysis of t h i s point, i n a different. context, see Shaptro and S t i g l i t z (1982). t h e f i r m t o be honest. 16/g/Wc: can thus reformulate our problem crf determining t h e optimal i m p l i c i t c o n t r a c t , by posing an additional set of c o n s t r a i n t s ; i n t h e c a s e of two equally l i k e l y s t a t e s these c o n s t r a i n t s c a n be w r i t t e n - 181 Constraint (23) s a y s t h a t t h e p r o f i t s when t h e firm annouces t h a t it is state 1 when it is s t a t e 1 (and, therefore, t h a t it is s t a t e 2 when it is state 2) exceeds t h e p r o f i t s which would accrue i f t h e f i r m announced t 5 a t it were -16/ This is not q u i t e t r u e i f t h e r e i s a p o s i t i v e i n t e r e s t r a t e ; it may pay t h e firm t o announce t h a t it is a bad s t a t e when it is i n f a c t e good s t a t e , and some t i m e l a t e r , announce t h a t it is a good s t a t e , when it is i n f a c t a bad s t a t e ; such l i e s reduce t h e undiscounted value of p r d f i t s , but s i n c e t h e f i r m may g a i n i n t h e e a r l i e r period, and t h e subsequent l o s s e s i n t h e l a t e r period a r e discounted, such lies may i n c r e a s e the present discounted value of t h e firm's p r o f i t s . 'Inus, i.f t h e r e is a p o s i t i v e i n t e r e s t r a t e , the c o n t r a c t s must be chosen t o impose a s u f f i c i e n t l y l a r g e penalty on l y i n g t h a t l y i n g never occur. - 17/ ~ o t i c 6the d i f f e r e n c e i n information assumptions between our a n a l y s i s and t h a t standard i n t h e i m p l i c i t contract l i t e r a t u r e with asymetric information. In both it is assumed t h a t t h e worker knows t h e p r o b a b i l i t y r ? f ~ t r i b - ~ t t rofo the s t a t e s of n a t u r e , and t h a t t.h e a c t u a l s t a t e of nature ) . is not observable. dut 111 C L L ~ a;diru~z.2 - I O J P I A . - l F r v - - - ' ; r \ n ~ , fy is assumed t h a t workers know t h e firm's p r o f i t function, w h i l s t not b e i n g a b l e t o observe p r o f i t s ( o r else t h e s t a t e of t h e world could be deduced, and we w o ~ l dbe back i n t h e f u l l information, f u l l employmeht case,) On the o t h e r hand, we assume t h a t workers do not know t h e prof?t f u n c t i o n , o r t h e l e v e l of p r o f i t s but do know t h e p r o b a b i l i t y d i s t r i b u t i o n of wages and hours. W e would argue t h a t t h i s assumption about the o b s e r v a b i l i t y of t h e p r o f i t function and the non-observability of p r o f i t s is implausible, and is better replaced by the assumption that workers can observe the p r o b a b i l i t y d i s t r i b u t i o n of wages and hours. - 18/ This condition w i l l obviously be s a t i s f i e d i n t h e f i r s t b e s t c o n t r a c t , provided only t h a t t h e marginal r a t ? of s u b s t i t u t i o n between consamption and l e i s u r e does not vary g r e a t l y with t h c r e l a t i v e p r i c e of consumption goods. state 1, when it is s t a t e 2, and conversely. The presence of t h i s s e l f - 191 s e l e c t i o n c o n s t r a i n t leves unaltered a l l of our e a r l i e r conclusions.- 111. IMPERFECT WAGE INDEXING We noted e a r l i e r that i f wages could be s e t a s any a r b i t r a r y function of t h e state of nature ( t h e p r i c e of t h e export good), then, even i f t h e r e is a n e n f o r c e a b i l i t y c o n s t r a i n t l i m i t i n g t h e 1.osses which employers are w i l l i n g t o undertake on t h e i r employees, and workers a r e r i s k a v e r s e , t h e optimal i m p l i c i t c o n t r a c t nevertheless involves f u l l employment ( i n t h e s e n s e - defined above). Wages may, i n f a c t , be p a r t i a l l y r i g i d , but they are never s o r i g i d a s t o give rise t o unemployment. These c o n t r a c t s e f f e c t i v e l y require not only t h a t wages be indexed on the exogenous sources of u n c e r t a i n t y i n t h e economy 201 but t h a t t h e index formula can take on any form. I n general, t h e r e l a t i o n s h i p between y (and h ) and p w i l l be highly nonlinear. I n t h i s s e c t i o n , we show t h a t r e s t r i c t i n g t h e c o n t r a c t t o a l i n e a r index function leads t o a n optimal degree of wage f l e x i b i l i t y which w i l l , i n general, e n t a i l a f i n i t e degree of unemployment. Since we a r e only concerned t o demonstrate t h e p o s s i b i l i t y of unemployment, t h e model is chosen t o be t h e simplest analytica?.ly soluble g e n e r a l equilibrium model we could devise. Rather than model t h e p a r t i c u l a r form unemployment takes (work-sharing versus l a y o f f s ) we s h a l l f u r t h e r s i m p l i f y by . assuming t h a t i f there is a constraint on the demand f o r labor, there is p e r f e c t work sharing. I I - . i y , . < , r,d c,up,~asi;~ L;-~'IL ;:.? !:.L.?C.; ,:(;-.:: ?,.!': ,.l j , : , . ~ ~ c : ~ : . - , d , . .~ i r ~t'it-: r e ~ p n r l i t e r a t u r e on asse&etric information c o n t r a c t s are n e c e s s a r i l y e x p l i c i t c o n t r a c t s , s i n c e a e y a r e one period c o n t r a c t s and cannot be enforced by !! reputation. We s uld a l s o -. ' n t out t h a t i f firms are r i s k n e u t r a l , and workers have u t i l y functiorrs which a r e separable i n consumption and l e i s u r e , then t h e assymetric information equilibrium w i l l be c h a r a c t e r i z e d by overemployment, not underemployment. If firms a r e r i s k n e u t r a l and individuals have u t i l i t y functions which are (a monotone transform) of t h e form C + b V) ( h ) , then t h e r e is n e i t h e r underemployment nor overemployment. Only i f firms a r e s u f f i c i e n t l y r i s k averse, w i l l t h e r e be underemployment. (See Azariadis and S t i g l i t z , 1983). -201 If there a r e endogenous v a r i a b l e s t h a t a r e monotonically r e l a t e d t o t h e exogenous v a r i a b l e s , c l e a r l y the c o n t r a c t can be indexed on t h e s e endogenous variables. We assume t h a t a l l workers a r e identical and have the u t i l i t y function: u 1-0 = 10g(2 -h) + D log c + (1 - log m - log{gO(l - 8) 1 (24) where, for simplicity, we have assumed that the individual dces not consume the exported good, but does consume an imported good, m, the price of which we take t o be our numeraire. If the individual's income is y (wh!.ch, i n the absence of lumpsum transfers w i l l be wh), then he w i l l spead a f r a c t i o n B of h i s income on t h e non-traded consumption good, and 1 - 0 on the imported good, so that h i s indirect u r i l i t y function w i l l be, substituting i n (24): A V ( ~ , Y,h) = log (2-h) + log y - B log q. If there a r e no constraints on labour supply, so that the worker can choose h freely, then he w i l l maximize hfs stility vhen h = 1. Since h i s unconstrained labour supply is independent of prices and wage r a t e s , it is easy t o i d e n t i f y f u l l employment i n t h i s model a s h = 1. In the unconstrained case the worker's indirect u t i l i t y function is just - A V = V = log w 0 log q. This u t i l i t y function has properties which greatly simplify the analysis. In p d r r ~ c u ~ d cA;= , jdJ,;L :c -;L~; :r 1"' i+I/ v f ~ i i . q7,irlf?r~a n n l y - i f s simple, whilst it continues t o give a simple indirect u t i l i t y func'zion i f the worker is constrained i n the labour market. (These usefulZproperties account f o r *its widespread use in the analysis of constrainezmarkets, e.g., Malinvaud [1977].) Finally, it generates unit price e l a s t i c demands which can be perfectly aggregated across individuals of d i f f e r i n g incomes. Thus, i f t o t a l wage income i n the economy is Y, the t o t a l demands for non-traded and traded gooA.s are The c r i t i c a l assumption which gellerates unemployment is the r e s t r i c t i o n on the s e t of admissible contracts. There is a single, exogenous source of randomness i n our economy, and hence it is natural t o index wages on p. But the s e t of contracts is r e s t r i c t e d t o those i n which wages a r e a l i n e a r function of price: - w(s)-- w P(S)- -P), = ( l - p ) ( - Here the export price is p(s) i n s t a t e s, p is its mean value, whilst i s the nean non-traded sector (or urban) wage, and p is a measure of wage r i g i d i t y . (Equivalent, 1 - p measures wage f l e x i b i l i t y . ) If p = 0 , then urban wages w i l l be identical t o a g r i c u l t u r a l wages (from (2)) and hence, obviously, u t i l i t y levels w i l l be equated, a s required by (3). If p = 1 ,urban money wages are perfectly inflexible, and always equal t o G. In Figure 1 t h e l e v e l of unemployme~tgenerated in:-state s by a given urban wage w(s) is shown t o be determined by the demand schedule (whit.. i t s e l f depends on the s t a t e s ) and t h e supply schedule, vhich, given the enforceability constraint, is equal t o the wage cost i n low demand s t a t e s . Figure 1 shows the equilibrium f o r a representative firm with capacity employment of one man. We w i l l consider how the econoply equilibrates for any given choices of p, the degree of r i g i d i t y i n equation (28). There are two p o s s i b t l i t i e s , Either the range of values of p is such that unemployment of t k 2ype shown i n - Figure 1 never occurs, or peri2dic unemployment d r e crle sdmd, ~L.L L L I ~ivfa L: '4 - *- the equations w i l l d i f f e r . We consider f i r s t the simpler case of f u l l employment. FIGURE 1 Supply and Demard f o r Non-Traded Goods 3.1 Full Employment Equilibrium. Tn i n w n f i u :, we -?lve c x p l l c i t l y f o r t h e equilibrium v a l u e s of thr economy. If t h e r e is f u l l employment, we have already argued chat rl =i. Since t h e u t i l i t y function of (25) is separable, t h e condition t h a t workers* expected u t i l i t y be t h e same i n non-traded a n d*e x p o r t s e c t o r becomefi (using 28) E log ;{P + (l-p)p(s)/p } = E log w(s) = E log p(s). (29) Once p has been fixed, t h i s f i x e s w a s a f u n c t i o n of the d i s t r i b u t i o n of export p r i c e s , p(s). Knowing w, one can s o l v e f o r a l l t h e o t h e r v a r i a b l e s in the model. Welfare Analysis a t F u l l Employment. I f the p o s s i h i i i t y of unemployment can be ignored, t h e r e is an i n t u i t i v e argument why the o~timum degree of wage f l e x i b i l i t y , 1 - p* , should be roughly 0 , the share of t h e traded good i n consumers' ex2enditure. The argument is t h a t t h e non-traded good price responds t o income, which f l u c t u a t e s with the prl:e oZ exports. Urban workers w i l l therefore want higher money wages when non-traded goods a r e expensive. If t h e i r income r i s e s by 0 times the rise ?n exporr: p r i c e s , and if non-traded goods prices rose a s xuch a s export prices, then t h e i r r e a l purchasing power would be maintained, and hence r e a l income stabil.Lzed. This argurent is e s s e n t i a l l y c o r r e c t , and i n Appendix 1 t h e optimum degree of wage f l e x i b i l i t y is shown t o be where r is the required average r a t e of return of (7), and v is the c a p i t a l - output r a t i o of (lj. Expected welfare increases a s r e a l incomc becomes mre s t a b l e , acd a graph of expected welfare, W, against wage r i g i d i t y , p, is a s s h w n i n F i g u r e 2, which is drawn on the assum~ti.ont h a t t h e r e is nc unemployment - an assumption which is explured below. The form of the dependence of welfare on wage f l e x i b i l i t y is e a s i l y analyzed once it is G p r e c i a t e d t h a t since all'workers enjoy t h e same average l e v e l of u t i l i t y i n equilibrium, it is only necessary t o study t h e average q-;,?l~~ r ?of -In n z r l r , ! ?t:~rrl! vqrker . Fror?. equation ( 2 5 ) this i s '? I The e f f e c t of wage r i g i d i t y on welfare can be fol~ndby d i f f e r e n t i a t i n g (31) t o t a l l y with respect t o p: Expected Welfare Degree of wage 1 Rigidity, P FIGURE 2 Benefits of \gage Rigidity (without unemployment) The algebraic d e t a i l s of evaluating equation (32) a r e provided i n Appendix h, Qhich shows that -. riw ; ( > - < - L'/J 0 and .+ C, dp I - dp lpl 0 a s shown. However, a s p is inc-sed Zowards 1, so the variable cost of 0 production of ;he non-traded good rises r e l a t i v e t o the demand price in t h e A lowest demand s t a t e of the world, u n t i l a t some c r i t i c a l value p > 0, unemployment f i r s t occu* i n t h i s s t a t e , a s shown i n Figure 1. For P >p, the equations on vhich the formulae were derived become invalid, and - - A if p* > p, it becomes necessary to ask what effect unemplc:;:~ent has cm the desirable level of wage rigidity. The results of this section may now be summarized: Proposition -3. If the variability if- price is sufficiently small that there is always full employaent, then there is an optimal degree of wage flexibility, approximately equal to the share of the non-traded good 5.n consumption. 3.2 Equilibrium with Periodic Unemployment. Suppose that there are a finite number, S, of values that the export price may take, each occurring with strictly positive probability, ...< with p(1) < p(2) < p(s). Pssuming that rv > 0, it follows thah if the degree of rigidity, p, is high enough, but not too high, there will PP unemployment in state 1, but in no other state, as shown in Figure 1. We A denote by p the critical value of pat when ~mploymentfirst appears, In Appendix 1 we show that wheze a is defined as the proportional downside range of p. For the optimum degree of wage rigidity to imply periodic - A unemployment, p < p* , where p* is approximately 1 6. Clearly, this fs possihle if fl (the consumer's expenditure share on the non-traded gocd), is small, and a is large. !lelfare Analysis with Unemployment. - * Suppose that, ignoring unemployment, the desired level of wage A ri:: d l t - r f c, p* ,f \,t h e level at which unemployment first occurs, so that welfare increases as p is increased to p, as show in Figfre 3 below, The crucial question is what happened to the slope dWldp for values of A # above p.. s-.. p FIGURE 3 Welfare with Unemployment Two possibilities are shown. Curve A has P+ in which case the optimum degree of wage rigidity implies some unemployment, whilst curve B has 0 A implying that the optimum degree of wage rigidity is p, the point at which unemployment is just about the occur. To establish which case occurs it is necessary to compute dWldpI Whilst it is difficult in general to I P+' sign dW/dpI it can be shown that the possibility of unemployment makes I * ' wage rigidity uniformly less attractive, in the following sense: That is, ingle fi in Figure 3 is positive, 2nd curves A and B both lie below - the dotted curve, which ignores the unemployment constraint, as shown in Figure 3. There is one condition in which it is possible to sign dW/dpI for I*' it is obvious that unemployment is never desirable in a two-state world. If there are only two s tes of the world, there need only be two wages; equivalently, a linear schedule r~hichfixes the two parameters p and w allows wages to be any desired function of the states. This explains the special results obtained in Newbery and Stiglitz [1981, Ch. 261, in which unemployment . I was never optimal. However, the two state case is very misleading, for any realistic description of uncertainty would recognize many more than two states. The next section shows that with four states and linear contracts (which are now really restrictive) periodic unemployment may be implied by the optimal contract. . 3.3 The Desirability of Intermittent Unemployment If the implicit wage contract is constrained to be a simple linear function.of~the state (the export price) as in (28), then, provided there are more states than parameters in the wage function, it is possible that the best - such contract will generate periodic unemployment. One such example has a 1 symmetri four point distribution tor expoit prices: ~(2)= 1 - 012 - n p(3) = 1 + a/2 with probability o < n < V 2 112- n ' - where, for convenience, the mean export price, p = Ep(s), is taken a s unity. The model can now be solved once 0, rv/y, a and JI a r e specified, where f3 is the share of non-traded goods i n consumption, and rv/F is, roughly, the average mark up on prime costs i n the non-traded good sector. Table 1 summarizes the main r e s u l t s , and is t o be interpreted a s follows. Column 5 gives the coefficient of variation (CV) of export prices implied by a and JI, and column 10 measures the welfare of the economy assuming p = 0 , that is, with no wage r i g i d i t y i n the urban sector. The measure is taken a s t h a t r i s k l e s s income which yields the same u t i l i t y a s the expected u t i l i t y , expressed a s a percentage of the income which would be received i f export prices were perfectly stabilized a s their mean price. Thus, looking across row A, with za wage r i g i d i t y , but a CV of export prices of 35%, the economy achieves 94.81% of t.he riskless level; o r , put another way, the cost of the export price i n s t a b i l i t y is 5.19% of average income. (The method of calculating these welfare measures is explained i n Appedix 2.) Column 6 gives the value of p a t which welfare is maximized, ignoring the p o s s i b i l i t y of unemployment, while column 11 gives the welfare i n t h a t case. Again, looking across row A, welfare rises t o 95.462, or the loss is reduced t o A 4.54%. However, the c r i t i c a l value of p for which unt?mployment occurs, p, %s given i n column 7, welfare there is given i n column 1 2 , and given the parameters of Table 1, p* cannot be achieved without une~ployment. The optimum wage r i g i d i t y allowing f o r unemploymnt, is given i n column 8, the associated level of unemployment i n s t a t e 1 is given i n column 9, while t h e I e welfare associated with p is given i n column 13. Except for cas'e D, where u A the optimum l e v e l of unemployment is zero (i.e., pU = p), the l a s t column st r ~ c c i yexceeds c o i~ m n'-" . . , : + ii;-.rn+ I ~ P ~ h o x ni C r r t i ~ 3 l . u The table showsdwhat should be i n t u i t i v e l y c l e a r from the argument, 0 A s the r i s k of unemploymbt f a l l s (as JI f a l l s ) so the optimum Cegree of wagE - r i g i d i t y r i s e s and unemproyment rises. (Average unemployment, Jb*, may rise and then f a l l . ) A sa increases, the optimum l e v e l of unempl~ymenti n s t a t e 1 a l s o r i s e s (compare A and E, B and F). A s the p r o f i t mark up f a l l s (rv/p) t h e optimum level of unemployment r i s e s because it is more l i k e l y t h a t unemployment w i l l t,ccur for given wage r i g i d i t y (compare u*, f o r F and G.) As the share of non-traded goods, f3 , f a l l s , so optimal unemployment 'U ' Table 1: SENSITIVITY ANALYSIS OF OPTIMAL WAGE RIGIDITY Welfare, W, a s a function of r i g i d i t y , p ,or nside Probability Profit C-Good CVof C r i t i c i a l Optimum a s a function of unemployment, u, expressed ange of Worst Statc Mark Up Share r'rlce, p Rigidity Rigidity Unemployment a s equivalent rioklese income (X) a,@F' 4 1 0.5 0.05 0.25 0.2 29 0.8 0.32 0.57 16 96.55 97 .OO 96.84 96.88 0 -5 0.05 0.25 0.1 29 0.9 0.29 0.60 20 96.13 96.40 96.28 96.31 Computer Simulation rises, though the effect of the optimum r i g i d i t y pU, , is ambiguous. The re-son f o r t h i s l a s t e f f e c t becomes c l e a r i n Figure 3, A s - * 0 f a l l s , p* rises, but p f a l l s , moving P t o a s t e e p e r p a r t of t h e graph and r a i s i n g (dW/dpl. Thus, it may be d e s i r a b l e t o increase wage r i g i d i t y t o t h e point a t which periodic unemployment ensues, and t h i s is t h e more l i k e l y t h e s m a l l e r t h e r i s k of low demand ( n ) , t h e g r e a t e r t h e d i s p e r s i o n of export p r i c e (a), t h e smaller t h e p r o f i t share i n t h e non-traded good s e c t o r ( r v l p ) , and t h e smaller is t h e f r a c t 4 0 n of income spent on t h e non-traded A Simple Model of I n f l a t i o n and t h e P h i l l i p s Curve So f a r we have assumed t h a t urban money wages a r e a t least p a r t i a l l y . f l e x i b l e . The standard Keynesian assumption is, however, t h a t they are sticky downwards, though t h e r e is nothing t o prevent urban money wages being b i d up i n periods of high demand. For real wages t o be f l e x i b l e i n such an economy, it is necessary t o devalue when export prices are low i n order t o raise the domestic p r i c e l e v e l and hence reduce real wages, If t h e government is w i l l i n g t o do t h i s , then a l l traded goods p r i c e s w i l l rise by the extent of the devaluation ( f o r a small country), and the economy can behave e x a c t l y as before i n r e a l terms. I n t h e simplest case, suppose t h e r e a r e only two states I ! of the world, with e i t h e r a high export p r i c e o r low price. Then, l e t t i n g wh denote wages i n t h e good state, wm i n t h e bad s t a t e , traded goods p r i c e s w i l l ' rise by a factor wh/wm whenever world prices f a l l . The p r i c e Level w i l l follow a simple random walk with drif.- a t rate per period. The g r e a t e r t h e r a t i o wh/wm, the higher w i l l be t h e average rate of i n f l a t i o n O. In o t h e r words, the more f l e x i b l e r e a l wages are, o r t h e inore responsive they a r e t o demand, t h e higher t h e r a t e of i n f l a t i o n , bet t h e l o w e r the l e v e l of urban unemployment. Conversely, t h e more r i g i d r e a l wages are, t h e higher t h e r a t e of unemployment, but t h e p r i c e l e v e l now does not need c o be raised so much when export prices f a l l , so i n f l a t i o n is lower on average. In short, there is a trade-off between the average r a t e of i n f l a t i o n , @, - and the average r a t e of urban unemployment u = h. For example, i f there a r e just two s t a t e s i n which p = 1 +- a with equal probability, then, i f a = 0.5 = B , rv/- = 0.25, the average rate of P i n f l a t i o n f a l l s from 28% p.a. a t zero unemployment t o 8.7% p.a. a t a n average level of urban unemployment of 10%. and zero i n f l a t i o n a t the maximum f e a s i b l e - level of average urban unemployment, u = 15% (where wh = wm). The trade-off . is essentially linear over the whole range with slope about -1.9. CONCLUSIONS The c e n t r a l thesis of t h i s paper has been that the analysis of the consequences of implicit contracts between r i s k averse workers and r i s k neutral firms must take i n t o account three c e n t r a l aspects of contract ' design: (a) the contracts a r e implicit, rather than e x p l i c i t , and t h i s r e s t r i c t s the terms of the contract which can be enforced; (b) the terms of the contract can only be made contingent upon variables which a r e observable; and (c) there are limitations on the degree of complexity of contracts; the provisions of even e x p l i c i t contracts a r e seldom contigent upon a l l t h e observable, potentially relevant, variables i n the economy. In p a r t i c u l a r , i n labor contracts, we seldom see the kind of complex indexing of c o n t r a c t s on publicly available price and quantity data t h a t an economic theory which ' 'ignored the "complexity constraint" would predict. Our analysis has three c e n t r a l r e s u l t s . F i r s t , with perfectly c10.<151a - - q ~ ~ r r ~? nr ltt h~n,r t h e ponstraints on enforceability nor t h e limitations on observability lead t o unemployment i n long term c o n t r a c t a a l arrangements. Second, even with perfect enforceabiliy and observabilfty, limitations on the f l e x i b i l i t y of contracts may lead !o unemployment. Third, even when firms sign optimal contracts with t h e i r workers, i.e. c o n t r a c t s which, given marke; parameters such a s wage and price d i s t r i b u t i o n s and t h e level of unemployment, maximize the expected p r o f i t s of the firm, subject to the worker a t t a i n i n g a given level of expected u t i l i t y , the market e q u i l i b r i l a is not Pareto efficient. The source of the inefficiency, the f a i l u r e of each f i r m t o take i n t o account the changes i n r i s k s which face consumers and workers in other firms, resulting from the changed probability distribution of prices and emplopent, is quite distinct from the kinds of inefficiencies which have been noted elsewhere in the literature.- 211 Our analysis shares one deficiency with that of most of the other literature on implicit contracts: it explains underemployment but not unemployment; it is consistent with work-sharing, but not with lay-offs. There are two sorts of arguments why firms prefer to lay off workers rather than to engage in work sharing. The first identifies some non-convexity in the ?reduction process, which implies, for instance, that it is more efficient to have one worker work 40 hours a week, rather than two workers work 20 hours eacn. But this is only an explanation for the time unit of work sharing: tf there n machines and 2n workers, obviously, they should not rotate every 30 seconds; the optimal rotation may be a week, or a month, but it unlikely that the non-convexities are such that no rotation is desirable. The second we refer to broadly as "efficiency wage arguments". The wage (or more generally, the terms of the contract) affects the net productivity of the labor force (either by affecting the quality mix of those who apply or of those who leave; the quit rate and hence the turnover costs, or the effort level.) The problem is that efficiency wage considerations can lead to unemrloyment, even without implicit contracts. Does that mean that the cheory of implicit contracts is redundant? We think not. We suspect that there are circumstances in which, in the absence of implicit contracts, efficiency wage . t considerations by themselves would not lead to unemplbyment. Implicit contracts explain the presence of unemployment, efficiency wage considerations - explain the .@rm whicil tlie u~lc;n;lzyncat t :tkes. 1 - - 211 E.g. the search externalities associated with the work of Hosios (1981) and Diantmd (1981), the effort incentive externalities, associated with the work c' Shapiro and Stiglitz (1982), the more general moral hazard externalities, (Arnott and Sttglitz, 1982) or the informal externalities (Greenwald and Stiglitz, 1983). - 22/ An exception would arise if there were signlficant learning by dolng effects. Appendix 1 Derivation of Formulae of Part I11 Full Employment Equilibrium In t h i s case e f f i c i e n c y (1) and rhe zero p r o f i t c o n s t r a i n t (7) together with h = 1 ( f u l l employment) and y = w imply a 1 Eq - Ew = rK/Nc = rv. The market clearing price f o r consumption goods can be derived by e q u a t i o g the demand schedule (27) t o supply: - where Y is aggregate income and w = p . This may be s u b s t i t u t e d i n ( A l l to X give GJven ;, since and r v a r e exogenous, t h e a l l o c a t i o n of labour between I t s e c t o r s , Nx/N,, is then determined. Equations (A3) and (A4) can be s o l d $or: q i n each s t a t e , s: The Optimal Degree of Wage R.igidi& Assum.:ng Full Employment . Equation (32) can be solved using (A5) where, i n equilibrium, is a function of p from (29): -E d d lop W(S) = -E log p(s) = o dp dP confirming the intuition that increased wage rigidity (higher p ) generates - risk benefits which perct a lower average wage, w. Equation (A7) can be substituted in (A6) and then into (32) to sive the effect on average welfare, which will depend on the value of p. At p = 0 we have: demonstrating that it is desirable to offer some income insurance by increasing wage rigidity starting from completely flexible wages. However, at P = 1, corresponding to complete rigidity, The graph of welfare against p thus looks like Figure 2, with the optimum * degree of wage rigidity, p , . strictly between 0 and 1. At the optimum . dW/dp = 0, or, from (A6) where Equation (A8) can be solved for p once the distribution of p (or u) i s specified, and can be approximately solved by expanding about u - 0 and ignoring hiber terms than u2 t o give (30). Equilibrium With Periodic Unenploynent .- If workers are constrained t o work h < 1 i n s t a t e 1 and there is perfect work sharing, then urban u t i l i t y is given by (25). The unemployment rate is then 1 - h 5 u, and the urban wage which equates expected u t C i t i e s is given (compare (29)) by 2, 2, E l o g ( l -u )w = E l o g ( l - u )w{p + (1-p)p(s)/p} = E log 5. (A9) The allocation of labour between the sectors is found a s follows. In s t a t e 1 there is unemployment and hence excess capacity, a s shown i n Figure 1. Hence the price f a l l s t o the variable c o s t , q(1) = w(1). Labour i n p u t is hfic = (1-u)Nc, where Nc is the number of urban workers. Equation (A3) gives the non-traded goods price i n s t a t e 1 as: If we define the ( p r o p o r t i ~ r ~ a ldownside range or p as a t h e n ) - We can now calculate the c r i t i c a l value of r i g i d i t y , p, a t which unemployment f i r s t appears i n s t a t e 1. From equations (A2) and (All) i f u = 0 which gives (34). The average r a t e of p r o f i t must again be rv per unit of output but t h i s time profit is only positive i n s t a t e s s = 2, 3, ..., S. Thus (A4) becomes where n(s) is the probability of s t a t e s occurring. If II (=n(l)) is the probability of s t a t e 1 occurring, then, from (A22) and (A23) Substituting these and (All-12) i n t o (A14) gives Equation (A9) can be written Equations (A15) and (A16) can then be s ~ l v e df o r w and u given p and t h e 1) probability d i s t r i b u t i o n of export prices, p(s). For the numerical solutioms A -presented i n Table 1 p is f i r s t calculated from (34) and (29) (by an '?iterative algorithm). Unernploymen'r, us is increased irum zzro i n 2;: st-.;s, -and f o r each step, p is found from (A15) and (A16) (again by ar Lterative t *algorithm). Finally, welfare is evaluated for each pair (us p(u)} i o l o c a t e - I) the optimum degree of r i g t d i t y and associated unemployment rate. Appendix 2 Welfare Measures We would l i k e a monetary measure or' the costs of export price i n s t a b i l i t y and of the benefits (or costs) of wage rigidity. One natural method is t o take as benchmark the welfare enjoyed i f export prices a r e stabilized a t t h e i r mean, but workers pay a lump suln equal t o a f r a c t i o n L of t h e i r income, w, t o enjoy the advantages of s t a b i l i z e d prices. Normalize so t h a t the (mean) export price 5 is unityu, as are wages. w, then i f W is t h e expected u t i l i t y we wish t o measure by L, from equations (25) a,~d(A2) - - - w= v(;, q, po, -L) = 2 log ( 1 -$I - s l o g q, q = 1 rv/p. + This can be z r i t t e n a s L = 2( 1 - 49) , JI z exp {W + 6 log ( 1 + rvl;) ). (A301 ?n t h i s form, L measures the cost of risk as a fraction of original average income. Alternatively, W can be measured by 1 L, and t h i s is the measure - adopted i n Table 1. References Akerlo:, G. and Miyapski, H. (1980), "The Implicit Contract Theory of Unemployment Meets the Wage Bill Argument", Review of Economic Studies, - 47, (2) ,321-338. Arnott, R., Hosios, A. and Stiglitz, J. 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