《宏观经济学 Macroeconomics》课外读物：Wages and Employment under Uncertain Demand

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Wages and Employment under Uncertain Demand MARTIN NEIL BAILY This paper examines some implications of two postulates for firms wage and emple policies. The first is that firms, or stockholders, have easier access to capital lower costs or higher returns than do small investors, such as workers. Seco are important mobility and turnover costs incurred when a worker moves from to another The existence of mobility costs means that the labour market is not a perfect market. Each firm is not restricted to taking as given some exogenous market wage, period by period, but has some amount of freedom about the wage strategy it sets. The firm annot choose any wage-employment path it wishes, however, and I shall assume that in the long-run the firm must offer the same(expected) utility as that available elsewhere In the short-run there is a constraint that the wage offer must never be so bad that all the firms workers will quit and incur the mobility cost There are solid grounds for believing that great differences exist between stockholders and workers with regard to capital markets. The majority of stocks are held by very wealthy persons indeed, who also hold almost all the state and local bonds and large pro- portions of the property and other assets. In addition, stockholders are frequently company executives or professionals with greater financial expertise and salaries many times that of the average industrial worker. The worker typically has a rather small net worth. his assets are durable goods and a rather small holding of money. He frequently has consumer redit liabilities outstanding He also has much less knowledge of financial assets and institutions bo, a principal function of capital markets is to allow wealth-holders to diversify their Idings and so reduce the risk of their total portfolios. Stockholders, through their greater wealth and expertise, are much better able to bear risks than are workers. The difference in ability to bear risk between the two groups immediately suggests an opportunity to trade. In deciding what wage-employment strategy to set, the firm will be willing to reduce worker risk. By doing so, the firm is offering a joint product, employment plus an insurance or financial intermediation service. The firm does not do this simply because workers prefer it. Risk-reducing policies are the cheapest and hence most profitable way of attracting any given work-force The choice of a risk-reducing policy by the firm will have an important impact on both the wage set and on employment variations-and hence the probability of unemploy ment. The firm will, in general, wish to reduce the uncertainty of the workers'incomes An important feature of the model presented here is that the tendency of the firm to reduce risk has an asymmetrical effect on the wage strategy and on the employment strategy. 1 First version received February 1973 ersion received May 1973(Eds ond the members of the theory omments and criticisms. i reta In addition, one might feel that stockholders as a class are more willing to bear risk simply because of differences in aversion to risk

REVIEW OF ECONOMIC STUDIES Subject to the( somewhat restrictive) assumptions of the model, one can show that a pre-announced non-stochastic wage strategy will be set by the firm. This is true even though the future path of employment is stochastic and hence there is a positive probability of being laid off. An equivalent result is not true for the employment strategy. The firm will wish to vary the size of its work-force Workers dislike income uncertainty and dislike being laid off. The asymmetry etween the wage and employment strategies arises because when a worker is laid off he receives a non-zero income. To attract workers, the firm must pay a higher wage if there is some positive probability of unemployment than it would if employment were guaranteed. As against this, the firm can save on its wage costs by cutting its work-force during periods of slack product demand. Provided workers have some alternative sources of income when they are laid off, the savings from employment variations will outweigh the higher wage necessary, even though workers are risk averse. The alternative sources of income are from unemployment compensation, from working outside the sector considered nd from the income equivalent of avoiding the disutility of work. Workers dislike lay-offs the question is how much do they dislike them relative to how much the firm saves on its One way of thinking about the alternatives facing the firm is to compare two possible policies, one which implies more employment variation and another which implies more yage variation. Such comparisons are an intuitively appealing approach, but it is not the one followed here, nor is it necessary to make comparisons of this type to prove the desired result. This is discussed further in Section IV(including footnote 2, p. 44) The result, that a pre-announced non-stochastic wage is set by firms, is intended to provide an explanation of the phenomenon of sticky wages. In particular, it suggests an explanation for the fact that the real wage does not seem to adjust in the short-run to clear the labour market There is a question of what is meant by sticky wages. In the formal model presented there are a number of rather strong assumptions which allow a clear result. The wage rate is strictly non-stochastic, pre-announced. It does not respond at all to fluctuations in demand. To reach this conclusion firms are assumed to be risk neutral, maximizing the present value of expected profits. Workers are risk averse, and do not operate in capital markets. These assumptions are intended to reflect the asymmetry between workers and capitalists that i described, but clearly they are strong. In Sections VI and VII the consequences of changing these assumptions are discussed . A SINGLE FIRM AND UNCERTAIN DEMAND I shall discuss first the case of a single firm which has a stochastic demand for its product. The simplest assumption to make is that the firm is a perfect competitor in the product market. It is a price-taker and sets quantity as the decision variable. I shall consider in this model a finite time-horizon of T periods Tle assumption 1. The price of the firm's product in period t, P, is a random variable P are jointly distributed and are bounded above and below. pEQ,(t=l,., T) If the joint density function of the prices is F(Pi,.,Pr), then there is a prior or un- conditional expectation of p, defined by E(p)=,… F(p1,…,pr)dlp (1) examining what determines price. One could assume a fluctuating world price. Also ption is not essential a fluctuating demand curve would serve as well

BAILY WAGES AND EMPLOYMENT The unsubscripted expectations operator will be used to denote this expected value, i.e. when the state variables from 1, ., T(the prices Pi,..., Pr in the above case)are unknown The operator Er will denote the expected value of a variable when state variables up to period t are known. It is the conditional expectation defined in the usual way It is convenient to take the technology and capital equipment to be constant. It is doubtful if any essential features of the results depend upon this, but a number of awkward Assumption 2. The firms output in period t is x, and is given by x1=9(L)g′>0,g"0, U"<O and p is a constant The assumption implies all workers are alike in their preferences. This makes life much easier, as I shall comment later, but obviously is pretty strong. The firm and industry are assumed small relative to the rest of the economy. The overall price level is constant ects in the rest of the are given exogenously. There may, in fact, be many choices open to a worker who is not employed with the firm we are analysing. But just as one assumes an equilibrium wage prevailing in the economy in the standard textbook theory, I shall assume a given(possibly stochastic) path of income available elsewhere Assumption 4. The path of income available to a worker elsewhere in the economy is yi,..., yT. The y2 are stochastic and may be jointly distributed Consider a worker who does join the firm we are analysing. If he is employed in criod t his income is simply the firm wage wr. If there were no mobility or information costs in this economy the firm would have no real choice. It would pay w,= yo in each riod as a perfect competitor in the labour market. more realistically, however, although a firm may face a long-run horizontal supply of labour, this is not true period by period because of information and mobility costs The fact that mobility costs exist has been well recognized in the literature. - In ctual practice the cost of changing jobs may be quite different from one worker to another. Some workers may have many jobs open to them in the same location and have a general skill needed in many industries. There would seem to be many workers, however, for whom changing jobs would involve considerable costs. These might be moving expenses search expenses, income foregone and possible retraining. For almost all firms there are significant costs involved when a new worker joins the company. He may have to be fROTh us. E ( p) is t te the ite ral r Feo eh conditional density of Pr, defined as the ratio of the integral 3 The length of the period can be taken as the order of er Is as an analysis of very short-run fluctuations and overtime

REVIEW OF ECONOMIC STUDIES given some on-the-job training or equivalently his productivity may be lower during the first few periods after he is hired, as he learns by doing It is hard to do justice to the full complexity of the factors described, particularly differences between workers at different skill levels and different geographical locations Instead the following simple parametrization is used Assumption 5. If a worker leaves the firm where he has been seeking work and moves to another firm in period t he suffers a mobility cost in periodτ given by C≥0τ≥t The mobility cost experienced by the worker he pays directly. The firm's turnover cost is assumed to take the form of a reduced wage for the first few periods after the worker ns the firm I L FIRM WAGE AND EMPLOYMENT STRATEGIES Once we include mobility and information costs the single firm has a measure of freedom about the wage and employment policies it can set. To model this the distributions of the state variables are assumed known and the firm announces at time zero a strategy with respect to wages and employment. The strategy will consist of two decision rules con- ditional on the values of the state variables, which are the prices Pi,., Pr and the incomes available elsewhere yi, .. yT2 A particular(and not in fact very likely) strategy would be to choose a constant level of employment and let the wage always equal the marginal product times price in each period. The announced strategy can be defined by two sets of mappings from the state variables into employment and wages Assumption 6. The firm announces at time zero a strategy s defined by two sets of mappings(a1,…,ar)and(b1,…,br) such that L=a1(P1,…,2y3,…,y9) (4) y,…,y) These mappings could be analytic functions or perhaps integral equations where L, and w,depend on some function of the expected values. This assumption is very weak in the ense that the class of possible strategies is very general. The announced strategy together with the known distribution of the variables mean that the worker can evaluate the expected utility of seeking work at the firm The knowledge requirements of this formulation are quite considerable but the stylized model makes the framework seem more unrealistic than it actually Workers learly do not make complex calculations upon announced joint probability distributions They are, however, concerned with the past behaviour of firms, how frequently lay-offs occur and what is the likely path of wages. Firms are concerned about their reputations as employers, suggesting that short-sighted decisions do not necessarily imply long-run profit maximization. In terms of the model, they stick to an implicit strategy since it is in their long run- interests to do so. In addition, the conclusions of this paper suggest that the wage strategy will be one of setting a non-stochastic pre-announced wage path; this is the strategy which reduces the knowledge requirements and simplifies the calculation of This section deals with an easy case. Employment variations are excluded so that the wage strategy is considered given a constant employment level. This keeps everything revious periods in prefcrence toha ntew thaknawnworkeh. hence the ost expaerehced iven dene far as the firm is concerned, the product price and incomes available elsewhere have exogenously

BAILY WAGES AND EMPLOYMENT very simple and gives the favour of the more complex case where employment varies, which is handled subsequently. Consider the properties then of two specific strategies Strategy S,: the firm pays a constant wage w and employs a constant number of workers Any worker taken on is guaranteed employment up to time T. He may, of course quit if he wishes. Strategy S2: the firm employs a constant number of workers L but the path of wages is unknown at time zero, 1. e. w,= b, (Pi,., Pr, yi,.. . yi) so that w, is a stochastic In order for these strategies to be meaningful the level of wages set must be such that the firm actually can employ workers Definition. Strategies S, and S2 are said to be feasible if the firm has at least L worker To model the assumption of a long-run horizontal labour supply, assume that at time zero all workers search for firms to find the best expected utility offer A single firm can then ensure L workers at the beginning of period one by offering the same expected utility as that available elsewhere Assumption 7. If the labour supply condition(5)is satisfied by strategies S, and S2 then the firm will have L workers available at the beginning of period one v=E∑Uo1+p) =∑U()(1+p)forS1 U(w)(1+p) The simple form of Assumption 7 depends upon the fact that workers were to have identical preferences. If this assumption were changed the analysis would assumed a lot more complicated. It seems intuitively likely that if workers, on the average, are risk averse, then the risk -reducing strategy S, is going to turn out to be the cheapest way of attracting any given size of work-force. It might be tricky to prove, however. Once workers have chosen to come to this firm they can re-evaluate their positions at any time. If they decide to move they incur the mobility costs Cr. For S1, consider the 200+)+(-2E-1亿202-cx+p)+ If this inequality is satisfied for t= 2,..., T then a worker who leaves the firm operating S, will always be worse off. The firm will retain its workers and S will be feasible. To nterpret this condition consider a strategy S which satisfies the labour supply condition (5. It is clear that if the variations of yo are large relative to mobility costs then the constant wage strategy will not be feasible. A large sustained increase in yo will force the firm to adjust its own wage upwards The force of the inequality( 6) is therefore that the constant wage strategy, that satisfies the labour supply condition, will be feasible provided mobility costs are sufficiently large relative to the short-run fluctuations in yo (the wage income available elsewhere in the economy)

REVIEW OF ECONOMIC STUDIES costs relative to fluctuations in the wage income available elsewhere in the economy are large enough so that a strategy s, which satisfies(5) will satisfy the inequality(6) Strategy S2 is really the class of strategies with stochastic wage paths which the firm may wish to choose. The set of these which is feasible must also satisfy a feasibility con- straint as well as the labour supply condition. This is given by Er-12 U(w (1+p)+(e-112 the RHS of (6) The formulation has a slight musical chairs air to it, since everybody searches at and then joins a firm when period one starts. This feature results from the synchronization of everyone's actions, rather as the exact consumption loan model does. Let me try and relate the model a little more closely to reality as follows Workers enter the labour force or retire at random. Some workers quit to look for better conditions and some others come from other firms. What I am trying to capture in the labour supply condition is that, provided the firm offers an expected utility over a period equal to that available elsewhere in the economy, it will find that it can quits and retirements with new entrants and hiring. The feasibility condition is a of the amount of period by period freedom open to the firm resulting from the costs of the labour market The strategy S, is defined in terms of a constant wage. Constancy is stronger than certainty but it is the latter that is really the key feature of S,. This model has ignored such factors as capital accumulation and technical change. In a more general model or in thinking about the relevance of the results, one might plausibly generalize by allowing ing fluctuations around the trend. I have commented(and will comment) on uncertain or stochastic strategies compared with pre-announced, non-stochastic wage strategies in the course of the general discussion since it is the wage certainty not the wage constancy that is important. IIL EXPECTED PROFITS AND ALTERNATIVE STRATEGIES The firm will make a profit in period t given by: m+=P29(L4)-w:L The present value of expected profits evaluated at the beginning of period one is given by E(m)=∑(1+r)E{Pg(L1)-w2L4}, where r is the discount rate and the mappings (4)define the distributions of w, and Lt There is a minor complication introduced by the parameter p in the utility function and the discount rate r. i will set r= p to keep things simple. with the framework developed the following proposition can be shown come Proposition 1. Provided the feasibility constraint is satisfied, the strategy s, with tant wage and employment yields larger expected profits than S2 with a stochastic wage Notice that the worker's expected utility and the firm,s expected profits are evaluated at the beginning of period one, when future values of the state variables are unknown The expected values are, therefore, the prior or unconditional values. Developing the formal proof of the proposition for this case does not seem worth while result is constant capital nly for the long-run shutdown decision ssary if S1 was not deir t effects familiar isher interest theory but not central to the issues here

BAILY WAGES AND EMPLOYMENT intuitively clear. Both strategies S, and s, have the same constant employment path The difference in expected profits between the two is therefore given by E(TI|,)-E(I|S2)=2(1+r)"(E(w )-W)L, (10) which is the difference in the present value of wage costs. The two strategies must both yield the same expected utility for workers from condition(5). Since workers are risk averse, the non-stochastic wage w can be less than the expected value of the stochastic wage E(w). The firms costs are, therefore, lower and its expected profits higher. In the next section simultaneous employment and wage variations are dealt with, but in a somewhat different context. Instead of a single relative price changing, overal fuctuations in an economy are considered. IV, WAGES UNDER UNCERTAIN AGGREGATE DEMAND We now consider an economy where the price level and aggregate output fluctuate. The economy consists of M firms producing a single(composite) good with the same tech nology. Even though the general equilibrium framework (in which wages and profits feed back into aggregate demand)is not allowed for, the force of the result, it will be argued does not depend upon this Workers and producers are assumed to be uncertain about the level of aggregate demand over the future period t= 1,..., T. There is uncertainty about the actions of consumers or investors or the government or the foreign trade sector or some combination of these Producers, in turn, will react to changes in aggregate demand-leading to price and output movements. There is no very satisfactory theory of price and output dynamics in response to overall fluctuations. As long as prices do not respond fully and instantaneously then output will certainly fluctuate. seems to occur, at least in the short-run. Each producer has to guess what demand will be and how other producers will respond. He then forms expectations about the movements of output and the price level. Based upon these expectations each producer, as before, sets a strategy for wages and employment. This will in general, be conditional on the values of output and price that actually occur. The strategy then defines the distributions of wages(Wi1,.,Wir) and employment(Lil, . LIT)fc All firms operate under the same conditions with the same technology. For the purposes here, they differ only in scale. Consider first the properties of equilibria such that u firms adopt the same wage, employment and output strategy. 2 The results do not necessarily mean that this economy would actually reach or remain at such a point under competitive conditions. This question is examined subsequently The labour force consists of N workers who seek work over the periods t=l,..., T. Each firm sets the same wage and employment strategy over the period so that workers distribute themselves between firms to equalize the probability of employment at each firm ssumption 9. The probability of finding employment in period t is the same in each and is, hence, equal to the overall probability of employment q, given by L (11) fence we are dropping the assumption that each firm can sell all it wants at the going price. 3 Since the str ot matter here whether there are mobility costs and workers search only at time zero or whether there are no mobility costs and they can move freely in each

REVIEW OF ECONOMIC STUDIES There is no money illusion in the economy. Workers are concerned about their real wage and producers about the real value of profits. The real wage in t is ur. In any period when a worker is laid off he receives unemployment compensation and avoids the disutility of work. Realistically one might also allow him to do temporary work within the household sector, but this is not specifically modelled here. The worker's real income in any period when he is unemployed is D, a constant. Consider any strategy, S, set by all M firms which involves some uncertainty of future wages. The wage under S may respond to fluctuations in demand and employment. The strategy may also involve employment variations and hence a positive probability of a lay-off for each worker. I shall now show the following proposition Proposition 2. There exists a strategy s with a non-stochastic wage, and the same path of employment and output as S, which has the following properties. (a) Each worker's expected utility under s is the same as under S.(b)The present value of expected real profits of the firms is higher under S th We are comparing two economies, as it were, one where all firms adopt S and the other where all adopt S. notice that the proposition states that the profit-maximizing strategy (for all firms) involves a non-stochastic wage, whatever the path of employment. If you consider any strategy S with a stochastic wage path, there is always another that yield higher profits, for a given expected utility. 2 For those who find it natural to make com- parisons between more employment uncertainty with less wage uncertainty versus less employment uncertainty with more wage uncertainty, the form of Proposition 2 may seem incomplete, since it compares strategies with the same employment path. This is not so Compare two strategies: (i)Sa with a large degree of wage uncertainty but with a smal or zero probability of unemployment and (ii) Sp with little or no wage uncertainty but a large probability of unemployment. It is not possible(without much more information) to say which of the two is preferred or yields larger profits. Proposition 2, however, tells us that there exists a strategy S, which gives higher expected profits than S,. This means that a trategy like Sa cannot be profit-maximizing even though it might possibly be better than SA There is, however a much more fundamental, and much harder, question of the extent to which the existence of a market equilibrium involving a wage that does not adjust to fluctuations in demand is inefficient, and may exacerbate unemployment in the economy or sector as a whole. Proposition 2 says nothing about the social efficiency of alternative strategies. It is quite possible for a strategy like S to maximize expected profits and for an alternative strategy, with less unemployment to be more socially efficient. There are transactions(or mobility) costs and other market imperfections involved in the model A formal proof of Proposition 2 is desirable for this case, where employment can vary, since the result is less obvious. When employment varies, a worker's income is still some- what uncertain even under a constant wage strategy. The proposition goes through because variations in L, affect only the probability of being in each of the two states-employed or unemployed. The setting of a constant, non-stochastic, wage represents a partial reduction of risk for the worker. Income, in the event of being employed, is non-stochastic. The firms expected profits under strategy S are given by: E(|s)=(+12x)- and under strategy S by E(I|S)=∑(1+r)E 1 Two arguments of the utility function---income and leisure luce into an income equivalent 2 In particular, if we knew the guarantee, then the profit-maximizing wage strategy would be to offer a non-stochastic wage in addi

BAILY WAGES AND EMPLOYMENT where d is the non-stochastic wage under S and g, (Lit is the output of the ith firm.The difference between the two is given by: E(IS)-E(m|S=∑(1+)E{2L4-L1} (14) In the economy under S the worker has an income v, with probability q, and an income u with probability(1-qt). His expected utility when he joins the firm is given by: V=2(1+p)"Eq, U(u,+(1-q U(Ow) s we have p E{qU()+(1-q)U(n) If we now choose b so that(15)and (16) are equal-as specified in the proposition-we have that I (1+p)E{LU(n)-LU()}=0 (17) U(v, can be expanded in a Taylor series to give U(v2)=U(的)+U(0)(v2-创)+U"(中)(v2-0) (18) where lies between v, and 6. Multiply through by L(1+p), sum overt and take expected values. Substitution of the condition(17) then gives (1+p)-1EL2-bL}=2(+p)1E[-UL(-6) (19 We know that U0, L,20 and (U,-0)->0 so that the left hand side of (19) is positive(strictly positive unless v, =0 for all Li>O). Since r= p the left hand side of (19)is equal to(14), the difference in expected profits between the two strategies, so the proposition is proved. If we ignore Umand higher order terms the expression in(19)for the profit difference can be simplified to E(I|S)-E(|S)=R4∑(1+n)E{L42-0 hereR is the degree of absolute risk-aversion of workers. The morerisk averse the workers, the greater is the return from the non-stochastic wage. 2 Proposition 2 has shown that it is profitable for all firms together to set a constant wage. This does not show that it will be profit-maximizing for each firm taken singly. To show this requires introducing mobility costs once again We have not really considered mobility in this economy so far. Let us now apply essentially the same framework as that used in the previous section. Suppose that all firms in the economy follow a strategy S with a stochastic wage. Consider a single firm deciding on its wage strategy over the period. Let this firm decide to offer S instead. This implies the same employment path and the same expected utility for a worker joining the firm, so that the firm will expect the same number of workers to join it at time zero as it would have had under S. The firm's expected profits will be higher, provided it can ensure that it still has the same share of the work force over the time-period t=l,, T(or at least whenever 2 Wherea seems tip be hy terribly tan ina ineg am thptatioh of the term in parenthesis. Clearly the greater the variation of vr aror

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