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Temporarily asymmetric information and labour contracts
Labour
Economics
1 (1994) 269 -287. North-Holland
Temporarily asymmetric and labour contracts AsaRosh* Received
May 1992; final version
received
September
information
1993
Abstract
Under the assumption that workers are more heavily credit rationed than firms, the standard model of testing and self-selectton in the labour market IS extended. The two main findings are that ex post inefficient termination may be used as a self-selection device and that when workers can be of more than two different producitivities, only the best worker should be overpaid. Kr), ~,ords: Self-selection; Credit rationing
Wage-productivity
gaps;
Inefficient
termination;
Up-or-out
contracts:
JEL c~/as.sifrc~a~ion: D82: 53 1; 541
1. Introduction
In the literature on adverse selection in the labour market it has been common to assume either (i) that the firm can never learn anything about the abilities of individual workers, or (ii) that learning is instantaneous. The first assumption is made by e.g., Weiss (1980), whereas the second is associated with the work of Guasch and Weiss (1980, 1981, 1982). Arguably, the intermediate case is more realistic; when a worker first applies for a job, the firm’s estimate of the worker’s productivity is imperfect and less precise than that held by the worker himself. However, as time goes by, the employer’s estimate improves possibly converging to the true productivity. This paper is concerned with the optimal labour contracts in such an environment. In a world with no capital market imperfections, this intermediate case does not differ in any important respect from the situation where the employer learns Corrrspondencr fo: .6sa Rest-n. FIEF, Trade Union Institute for Economic Research, Wallingatan 38, 111 24. Stockholm,
Sweden.
*This is a revised version of chapter two of my Ph.D.-thesis at London School of Economics. Valuable comments and suggestions have been received from Mike Burkart, Tore Ellingsen, Eric Hansen. Alison Hole, Alan Manning, Dilip Mookherjee, John Moore, Tore Nilssen, Chris Pissarides. Bengt Rostn and an anonymous referee. The work was financially supported by Gcteborgs Handelshiigskolefonder, Helge Ax: son Johnsons Stiftelse, Torsten och Ragnar SGderbergs stiftelse, Tore Browaldhs stiftelse and Stiftelsen Blanceflor Boncompagni-Ludovisi fiidd Bildt.
0927-5371 94:X07.00 C; 1994 Elsevier Science B.V. All rights reserved SSDI 0927.5371(93)EOO27-P
productivity instantaneously, because payment can just be deferred until productivity is revealed. However, capital markets are empirically known to be imperfect, with a substantial number of workers being severely credit rationed.’ The starting point of this paper is the work by Guasch and Weiss (1980, 1981 and 1982). In their papers, applicants for a job can be of two types. The type is known to the potential worker, but not to the employer. However, if the worker can take a test which gives scores related to their expected productivity, then by requiring that each worker pays an application fee and offering contracts contingent on a test score, the firm can induce workers of different types to choose different contracts, i.e. to self-select. Here the models of Guasch and Weiss are modified in three dimensions. 1 have already indicated the first two: Credit rationing and slow revelation of information. In addition I allow for a continuous number of productivity levels and test results. This formulation leads to important departures from the results of Guasch and Weiss. When workers are credit rationed, a labour contract which specifies a very low wage early on will impose costs on the worker. I show that firing some productive workers might be a cheaper way to induce self-selection than deferring payment until productivity is revealed. The contract specifies that after the first period, when the firm learns the worker’s productivity, the worker will then either be retained or fired. Examples of such up-or-out contracts are common in academic institutions (getting tenure) and law firms (becoming a partner). While other papers have dealt with up-or-out contracts, they do not view them as a self selection device.2 The longer it takes to learn the workers productivity, the more important is the capital market imperfection, and the more likely it will be that the workers prefer the use of termination threats as a self-selection device. I also show that only the very best workers will receive a payment in excess of their marginal product, and that the result of Guasch and Weiss that a substantial number of workers will be overpaid is a misleading artefact of their assumption that there are only two test levels. The paper is organized as follows. Section 2 describes the basic model. The optimal productivity dependent wage contracts are derived in section 3. In section 4, I discuss the robustness of the contracts. Finally, a discussion of the results and the relation to the literature is in section 5.
’ Jappelli (1990) estimates that 20% of the U.S. families are credit rationed. The important variables in his study are: low age. no established credit history. and IOU income. It should also be emphasized that Jappelli define5 a person as credit rationed only if he IS unable to obtain any crcdlt at all. In another recent paper, Mayfield (1990) finds that 43% of households under the age of 30 arc credit constralned and that 37% of households under the age of SO face rignificant constraints. ‘See Harris and Weiss (1984). Kahn and Huberman Siou (1992).
(19X8), Waldman
(1990) and O’Flahert)
and
2. The model Employees live for two periods and are of two types, SIor /3. Type c( has higher expected productivity in the industry than type /?. A worker’s type is private information to that worker. Productivity, s, for an employee is an independent random variable, and is equal in both periods. The distribution of x differs between the two types. For each type it is described by a density function,f;(x) together with a point mass in m (I = X,/J’). The point mass in t?i is assumed to avoid some future technical problems. I use the somewhat non-standard notation withj;(_x) also denoting the point mass in m. The corresponding distribution function is denoted by F,(u), and it has support [I, m]. I assume that,f,(.u):fil(x) is increasing in x over [I/, m], thus the Monotone Likelihood Ratio Property (MLRP) ho1ds.3 E, denotes expected value with respect to F,(_u). Productivity in period 1 is observed (by the worker and the firm) only ex post, just before period 2. The wage in period I is denoted by it’s. and i~,(u)denotes wage in period 2 given that the productivity in period 1 was .x. Firms and employees are risk neutral. Firms have a zero discount rate, but employees are assumed to have a positive discount rate r. This is a simple way to capture that workers have less access to credit than firms.4 Expected utility for a worker of type I joining the industry is u, = \I’,, + ~ Ifr
1
n, \v,,(.u)dF,(x) -xc I
,
+ F,(u,,)w
I = (x,/l) ,
i
where \V is the best wage available in another industry and s,, is the minimum productivity level required to stay in the firm in period 2. That is, the expected utility is equal to the wage in period 1 plus the expected discounted wage payments in period 2. This specification implicitly excludes ex-post fines for low performance levels. Assuming that productivity levels are not verifiable justifies this exclusion. The criterion for an r-worker to join the industry (the industry participation constraint) is
I assume ‘For
that (PC) is satisfied
a thorough
diacuwon
for both z-types
of the MLRP
see Milgrom
and @-types. (1981).
‘A more rigorous approach would be to define workers’ utility over consumption in the two periods. and let preferences be convex. Since workers are assumed to be risk neutral, there IS no room for an Insurance market. Let interest rates be zero. A risk neutral worker would wish to equate marginal utility of consumption in period I with the marginal utility of expected consumption in period 2. Of course. If the wage in period I 1s lower than the wage I” period 2, this would require borrowing. If the worker faced an absolute credit constraint, or would have to pay a larger &rest than the employer. a higher wage III period I. ceteris paribus. yields a greater benefit to the worker than it ia costly to the firm
All the firms in the industry have an identical constant returns to scale technology. Each firm maximizes profit and takes the product price, the workers reservation utility and the employment contracts of other firms as given. Without loss of generality the product price is normalized to 1. Consider the expected profit to firm j from employing a z-type worker:
s ,,I
Ilj,
= E,[.x] -
~‘1,
+
(x -
\t’z,(x-))dF,(.x)
(2)
XC!
Profit is equal to expected productivity in period one, minus period one payment plus expected profit in the second period. Firms live forever. There is free entry into the industry, so that in equilibrium, it is not possible for a firm to enter and earn positive profits. Since we have a constant returns to scale technology, the equilibrium number of firms is indeterminate. I assume that firms can observe each other’s contracts, but not a particular worker’s productivity. However, workers are able to communicate their wage to an outside firm. Employees are equally productive in each firm. and productivity is invariant to the wage contract.’ It is assumed that workers cannot commit themselves to stay in the firm in period 2. This implies that the feasible period 2 wage structure will crucially depend on the current employers ability to match an outside firm’s wage offer. The two polar cases are (if the current employer can always match the offer, (ii) the current employer can never match the offer.6 Arguably some intermediate case is most plausible. Under assumption (i) it will be possible in period 2 for firms to pay all workers a wage that is as low as the productivity of the least productive worker, due to the winners curse problem faced by the outside firms.’ Under assumption (ii) each wage group has to be paid at least their expected productivity. (Remember that wage contracts are observable). I will here make the second assumption, thus allowing workers to make binding agreements to accept an outside offer.*
‘Assuming that workers’ ively as long as productivity ‘Assumption e.g. Waldman
productivities differ across firms does not change the results qualitatin one firm is positively correlated with productivity in another firm.
(i) is made in e.g. Waldman
(1990) and Milgrom
and Oster (1987). assumption
(ii) in
(1984).
‘The winners curse problem in this context is that since a current employer has more information about a particular worker, the only time he will not match an offer is when it is above the worker’s productivity. Thus, the outside firm succeeds in attracting workers only when it offers more than their productivity. *An alternatlve specification where information about 2 across firms. but the former employer has more information end of period one. would give similar results.
productivity is symmetric m period about the workers who got tired at the
The sequence
of events and the information
structure
is summarised
below.
of’ period 1 Firms offer two-period contracts. - Firms observe each other’s contracts. - Workers know their type. Beginning
-
Beyinniny
cfperiod
2
Workers and the current employer observe an individual worker’s productivity. - The worker is fired or offered continuing employment at the period 2 wage specified in the contract. __Other firms observe hiring and firing decisions and the wage offered. - Other firms can give competing wage offers.
-
Since workers are equally productive in all firms and contracts (and wages) are observable, the wage has to be greater than or equal to the average expected productivity, for each group of workers who earn the same wage in period 2. Otherwise they would leave the firm since another firm could profitably offer them a higher wage. Formally, r~~,(.x’) 2 E,[xlwz,(s’)]
for all u’EX,
I = [a,/I)
That only contracts are observable implies that any two workers earning the same wage look the same to an outside firm, as do all workers who are fired. This assumption provides a link between this article and the literature on visible and invisible workers, as developed by Waldman (1984) and Milgrom and Oster (1987) and the work by e.g. Gibbons and Katz (1991). There too, promotion, wage rises and firing decisions are signals to the market about a specific worker’s quality, and outside firms have less information than the present employer. Gibbons and Katz find that wage losses for those workers that were fired on employer’s discretion were larger compared to those who lost their jobs due to plant closing. This is consistent with the assumptions that there is asymmetric information between firms and that outside firms take firing as a signal of low quality. I will assume that the expected productivity given that a worker is fired is less than the alternative wage, i.e. E,[s 1.x5 ,I.,,] < C. This implies that fired workers will move to another industry. This is only a simplification and does not affect any of the main results. The cut-off level, x,, will be determined endogenously. Ex-post efficiency implies that .x,( = W and ex-post inefficient terminations occur iff s,, > W. Note that if outside firms can discriminate between the workers who are fired, s,, cannot be above C. Workers with productivity above \C would then immediately be hired by another firm and competition would ensure that their wage
would be equal to their productivity. By implication, firing would not be an effective threat in the first place. It can be seen that for inefficient terminations to occur it is necessary that those fired do not all have the same performance level. The assumption in this paper of a continuum of performance levels ensures that this holds.’ Finally, I make the assumption that E,[x] < W. This assumption ensures that my set-up where workers prefer wage payments in period 1 to wage payments in period 2 is consistent.”
3. Equilibrium reward schemes Since workers have information about their productivity that is not available to the firm, we will run into the standard problem that bad types @-types) want to mimic good types (z-types) unless the contracts take the asymmetric information into account. As in most papers on self-selection, I assume that once a worker has signed a contract, reputation effects allow the firm to credibly commit itself not to renegotiate due to reputation effects. Also, the worker cannot leave the firm immediately after he has signed the contract. For simplicity, assume that the worker cannot leave until the beginning of period 2. A separating equilibrium is defined by three conditions. (i) There exists no other contract that workers prefer and which gives a non-negative profit to the firm, given that the other firms pursue the equilibrium strategy. (ii) For each group of workers earning the same period 2 wage, the wage has to be greater than or equal to their average expected productivity. (iii) Finally, a P-worker does not prefer an r-contract to a /I-contract, and vice versa. We are now ready to define the maximization problem that solves out the best separating contracts.’ ’
st.
n,, =
E, [s]
-
11’,
I
+
t*t I
(s - w,,(.x))dF,(.x)
2 0,
I = Y, /r,
(3)
A‘!
vvl,(.Y’) 2 E,[xlwz,(s’)]
for all x’EX,
I = c~,p,
(4)
“To be precise. what is needed is more than one performance level ofthoae who arc tired. and that the former employer has more information about the workers than a potential employer. ‘“If I had assumed that the firm had to weaker assumption would have been Ex[r]
pay a
training cost, k. to make use of a new worker the ~ k -c 12.
’ I A pooling equilibrium where r- and /i-types opt for the same contract dots not exist, see Roskn (IYY I) for proof. This is a standard result in this kind of model. see e.g. Rothschild and Stightz (1976).
Constraint (3) is the non-negative profit condition. Constraint (4) is the wage condition. Constraints (IC-/I) and (IC-r) are the incentive compatibility constraints. ciO(~) is the utility a ,!j’worker gets from an y-contract and U,(p) the utility an cc-type gets from a p-contract and U: is the maximum utility a r-type can get given a separating contract. The optimal separating contracts are described in the following proposition. Proposition
I.
Part A: The equilibrium
Part B: The equilibrium
r-contract
j?-contract is
is Xcr
(1 + r)EplI-xl - E,[I.yl(.f#4/L(m)) + Wla =
w2Jm) = m +
I
(x - w)dF#
E
1 + r - Chk4M4)
LCXI -
Wlsl
L(m) Part C: The cut-off level, x,, . tions @
The r-contract
entails rx-post
ine$ficient termina-
(5)
and to
Proof
of’ Proposition
1.
See Appendix
A.
Since the proof of Proposition 1 is somewhat involved, I here discuss the intuition underlying the proposition. Consider the best /3-contract. This is found by assuming that (IC-r) is non binding and later checking that this assumption is correct. Given that (IC-x) is non-binding, the result given in part A of Proposition 1 is intuitive. The firm offers the highest possible period 1 wage subject to the requirement that workers in each wage category in period 2 are paid at least their average expected marginal product. Note that because workers are risk neutral, the wage schedule in period 2 is indeterminate. One example is where the wage is equal to productivity. i.e. MIX, = x. Another is when all workers earn a wage equal to the average productivity of those workers that stay in the firm, i.e. rc,,(.u) = E,[s(s 2 xc,] for all s 2 .Y~,.Since (IC-x) is non-binding, inefficient terminations will serve no purpose, so xcB = \?. We now turn to part B of Proposition 1. It will be the case that (IC-p) is binding, i.e. the constraint that b-types should not mimic x-types will bind. Also the non-negative profit constraint [constraint (3)] will be binding for the r-contract. Given this we can solve for the contract that minimizes U,(r). for a given profit and U,. When we do this we find that there will be no pooling over different productivity levels i.e. \v~~(.Y’)# \Q~(.Y”) if s’ # Y”. This is because x-workers’ utilities are unaffected by pooling over productivity levels while for a given z-contract, MLRP implies that pooling increases b-workers utilities. There are two potential ways to deter low productivity workers from choosing a contract intended for the high productivity workers. Either the %-contract can specify a high cut-off, xcZ, or it can specify a low initial wage with the worker being compensated by a higher expected wage in period 2. Consider first the latter device. When the a-contract involves period 2 payments above the expected productivity this will be done by rewarding the performance level m with a ‘prize’ and paying a wage equal to productivity at all other productivities The reason only performance level I?? is overpaid, is that by MLRP, ,\;(s),i/~~(s) is maximized at m, i.e. the probability that a worker is an r-type if both x- and p-types opt for the same contract is at its highest level at productivity
k. Ros3n. A.symmrtric
information
and lahour
contrac’ts
217
level m. Note that since the workers are risk neutral they care only about their expected payments. The feature that only the very highest performance level is paid a wage exceeding its value should be emphasised. In previous models where there have only been two test levels (pass or fail), all those who pass are paid a wage in excess of the value of their performance. Hence the assumption of two test levels gives the misleading impression that a large proportion of retained workers should be overpaid. As mentioned earlier, technical complications would have arisen if I had not assumed a point mass at m.” It would still have been the case that only the highest performance levels would have been overpaid, because .f; (.x)/“~ (x) is increasing in X, but for any ‘over-payment’ interval there exists another smaller interval, which yields a better contract. On the other hand it is not possible to overpay only the performance level m, since the probability of performing WI(without the point mass property) is zero. A similar problem was identified by Mirrlees (1974). 1 are found by The expressions for nlIZ and MJ~~(VI)in part B of Proposition requiring that in equilibrium, profits are zero, and /I-workers are indifferent between the a-contract and the /I-contract. That is, we require that both (3) and (IC-p) hold with equality. I now turn to the issue of the optimal cut-off level, .xcZ. By raising the cut-off level s,.,, the wage-schedule can be flattened while still deterring /?-types from opting for the x-contract. The condition for inefficient terminations [condition (5)] is obtained by differentiating U, with respect to x,, around E, given that w,;, and w~Jx) are as in part B of Proposition I. Similarly, the formula for the optimal cut-off level, x,,, [eqs. (6) and (7)] are obtained by maximizing Ui,, given that w,* and Wan are as in part B of Proposition 1, As can be seen from Proposition 1, eq. (5) a necessary condition for ex-post inefficient terminations is that fb(W).!f;(\V) > 1. Ex-post inefficient terminations are more likely the higher is r, ,fb(G)/fa(G) or ,&(m)[f;(m). These results can be understood as follows. A high r reflects a long period 1 duration. Since firing allows flatter wage schedules, this option becomes more attractive when the revelation of productivities is slow. The more likely P-workers are to have a productivity just above Wcompared to the cc-workers [the higher,fi(*),‘f2(w) is] the more likely are ex-post inefficient terminations. This is because ex-post inefficient terminations then lower an cr-worker’s ex ante utility less than a potential b-worker’s. Finally, the more likely a [I-worker is to have a productivity of m, compared to the likelihood for an x-worker [the highmji(m)/f;(m) is], the more likely it is that the optimal self-selecting contracts will involve ex-post inefficient firing rather than giving m-performers a high premium. This is “Another way around Holmstriim (1979).
this
problem,
is to
put an upper
limit
on the wage payment. see
because for given x,, the higher.fb(m)/fz(m) is, the steeper does the wage schedule have to be to keep the p-worker from applying, and thus ex-post inefficient firing becomes more attractive as the self-selection device. When both steepening of the wage schedule and ex-post inefficient terminations are used as self-selection devices, the optimal x,, is defined by (6) of Proposition 1. If self-selection is obtained solely by a high cut-off level, then optimal x,., is defined by (7). In this case u’, 1 = E,[x] and VL.~~(_X) = .Y for all Y above s,,. If parameter values are such that self-selection is achieved mainly through involuntary terminations, the market will offer r-contracts with a high initial wage but high requirements for being allowed to stay. The &contracts will have a lower initial wage but less demanding requirements for staying. Examples of such requirements could be passing an intermediate exam, or gaining promotion, such as tenure in an university, or becoming a partner in a law firm. It is also worth noting that the contracts are derived under the assumption that reservation wages are equal for the two types of worker. Under the assumption of different alternative wages, with W, > \cII, as e.g., in Weiss (1980) the condition for ex-post inefficient terminations becomes
(8) and there will ~[KYJ~Sbe some involuntary terminations since, as x,, goes to @,. the denominator in the first term goes to zero. Having established the best separating contract, I discuss the problem of its credibility in the next section.
4. Robustness of contracts There is a credibility problem with the ex-ante optimal contracts. Consider the contract for r-workers. The contract specifies that productive workers should be fired, and/or that workers of productivity m should be paid more than their productivity. The common way to defend such deviations from sequential rationality is to say that the firm lives for many periods and can establish a reputation for sticking to the ex-ante optimal contract. However, this explanation does not look entirely convincing when it comes to the premium paid to the m-performer. If,f@(rn) is low, the very best production level will almost never be attained by any worker, and the premium would have to be enormous. Since the productivity levels are (realistically) assumed not to be not perfectly observable, it would be difficult to keep the firm from cheating if the case were to arise.13 “The assumptton of not perfectly observable possibility of ex-post fines for low performance.
productivity
levels was made earlier to exclude the
A more robust contract, avoiding this problem, would be a rank-order contract, where the firm pays workers a wage which depends on their relative performance levels. This makes the firm’s payment in period 2 invariant to the actual productivity of the workers in period 1.14 In Rosen (1991) I showed that in a rank-order contract where self-selection is achieved at least partially through steepening the wage schedule, the wage will be equal to average marginal productivity in period 2 except for the first rank, which earns a prize [as we would expect from Malcomson (1986)]. This wage schedule is similar to the one obtained in Rosen’s (1986) study of elimination career ladders. In his model, the large first prize is necessary to induce contestants to maintain a high effort level late in the game. This wage schedule, which has been used as an explanation of top-executives’ high wages does thus not necessarily result from a moral hazard problem but could result from adverse selection as well.
5. Discussion
of results and relation to the literature
The two central results of this paper are firstly that termination of productive relationships may be used as a self-selection device and secondly that only the very best worker (if any) should be overpaid. Let me now discuss the first result together with the related issue of the contracts up-or-out feature, and its relation to the literature, in more detail. Following the work of Stiglitz and Weiss (1983) there have been a number of attempts trying to explain why productive labour market relationships are sometimes terminated. The mechanism, presented by Stiglitz and Weiss (1983), is that the threat of termination can induce higher effort by workers in a model where the employer can observe each worker’s production but not their effort. Variations of this latter idea have recently been presented by Kahn and Huberman (1988) and Waldman (1990). In these models, it is crucial that productivity is not verifiable. It is exactly the provision that workers with a sub-standard productivity are not retained which ensures that the employer does not falsely claim that a worker has a low productivity. However, the robustness of this type of argument has been seriously questioned by Mookherjee (1986). The simple StiglitzzWeiss model creates involuntary retention rather than involuntary layoffs for a range of plausible parameter values, and the more sophisticated model with two-sided uncertainty hinges on the assumption that the firm cannot use rank-order tournaments. If the firm could commit itself to promote a certain proportion of the work force, according
‘“Such
rank-order
tournaments
have been suggested
by CarmIchael (1983) and Malcomson of productivity levels. Malcomson a separating equilibrium achieved
( 1984, 1986) as a way to mitigate the problem of non-verifiability (1986)showed that, for a large enough number of participants, with piece rates can be achteved
by a tournament.
to relative productivity, this would efficiently solve the incentive problems on both sides.’ 5, I6 Malcomson (1984) derived the optimal contract in such a setting. If effort inducement is the reason for an up-or-out contract, then it is hard to understand why the alternative to being fired is lifetime employment, as is common in the academic market and in law firms. This suggests that sorting on labour quality provides a more convincing explanation for this type of contract. Models with the up-or-out feature that are based on ability and sorting are presented by Harris and Weiss (1984), O’Flaherty and Sow (1992) and Gibbons and Katz (1991). Of these only in Gibbons and Katz does there exist equilibria with ex-post inefficient firing. ” They make the same assumption as made here that outside firms can only observe contracts and hiring and firing decisions and not a particular worker’s productivity. However, inefficient firing does not arise due to a self-selection device, as in the present model, but as a Bayesian equilibrium. It is crucial in their model that firms cannot write two-period contracts. The mode1 presented here shows that if workers are credit rationed and there is asymmetric information about productivity at the hiring stage, then inefficient terminations can also arise where commitment and reputation are possible. Although Guasch and Weiss (198 1 p. 278) hinted at the idea that termination threats may supplement wage profiles in sorting the work force, they never substantiated the argument, and as it stands it is incorrect. In their 1981 paper, Guasch and Weiss analyse self-selection contracts both under the assumption of risk-neutral workers and under the assumption of risk-averse workers. They come to the conclusion that self-selection will always be optima1 when workers are risk neutral, but not necessarily when they are risk averse. Throughout their paper, however, they ussum~ that self-selection implies that the firm only hires those who pass the test. In Rosen (1991) I show that in their model, if the workers are risk neutral, all workers should be hired ex-post. Only above a certain level of risk aversion can it be optima1 to engage in inefficient firing. Again, of course, the intuition is that the steep wage schedule is costly ~ now due to workers’ risk aversion - and it may be profitable to introduce a second selfselection device. But even with risk aversion, it is necessary to have different
“This critique can be circumvented by assuming that the firm must incur a small cost in order to observe the workers’ productivity, and that the workers cannot know whether or not the firm promotes randomly. Personally, I am not convinced by this counter argument: m most cases tirmr learn something about workers’ abilities whether they want to or not. Furthermore, a firm with a random promotion policy would quickly earn a bad reputation. “‘Ifapplication fees are allowed in the Stiglltz Weiss analysis the Involuntary terminations would disappear. The reason why the tiring of productive workers can be rational is that workers are earning rents. due to eficiency wages. Thus. the threat to termmate induces higher effort. However. effort depends only on the difference between the wage if successful and the wage if not. With risk-neutral workers an application fee makes it possible for the firm to widen this difference without decreasing profits. Ii In Gibbons and Katz inefficlent firing is, however, not the unique eqmhbrium such equilibria are not robust even to ‘small’ reputation effects.
outcome, Also,
reservation wages to produce the termination results in Guasch and Weiss’ set-up. The results presented here do not need this difference. Another important difference between the present model and Guasch and Weiss (1981) is that I allow for competition in the labour and product markets. In Guasch and Weiss (1980, 1982), there is also competition in the labour and product markets, but ex-post inefficient terminations are not optimal (even with different reservation wages). The reason terminations may be optimal in the present model is that there is asymmetric information between the firm that fires a worker and a firm that hires the same worker. Since a fired worker who is productive cannot be distinguished from other less productive workers, there is a net loss associated with having to leave the firm. To better assess the relative merits of various theories, micro level data on inefficient terminations would be very welcome. The only indisputable example I am aware of where terminations of productive relationships have been used as part of a conscious strategy to increase overall efficiency, is from the Mongolian army under Ghengis Khan. There, a number of Mongolian soldiers would be executed if one of their leaders was lost in fight. The more prominent the leader, the higher was the number of soldiers punished. Although this practice was obviously ex-post inefficient, it seems to have been ex-ante efficient. The empirical and theoretical issues of whether the wage ~ productivity ratios increase over tenure, and with this, the related question of whether workers are paid more than their marginal productivity late in tenure, have been addressed in many papers.‘” The model in this paper predicts that the wageeproductivity ratio will increase over tenure. due to sorting and the premium paid to the top performer. However, it does not predict that a large group should be paid more than their productivity late in tenure. Empirical evidence on wageeproductivity gaps are inconclusive and we should remember that the above contract is only in a setting where matching ability and learning productivity are important. Thus it will not apply to the whole labour market. If strong evidence is found that large groups are overpaid late in tenure, then this paper suggests that we have to resort to explanations other than asymmetric information of productivity at the hiring stage.
Appendix A: Proof of Proposition
1
The proof is in the five following
steps:
(i) Assume (IC-c() is non-binding and solve for the best /&contract. (ii) Show that given nj,, U, and (4), minimizing U,(g) implies (a) u.~~(x’) # M.~~(x”) if x’ # x” and (b) Nail = Y if x < m and u’,,(m) 2 M. ‘“See Hutchens
(1989) for an overview
282
(iii) Show that nj~ = 0 and that (IC-fi) is binding. (iv) Find the optimal x,,. (v) Show that (IC-r) is non binding. Constraint (3) will hold with equality since (i) Assume (IC-x) is nonbinding. a worker’s utility can always be increased by lowering profits. Constraint (4) will hold with equality since workers prefer wage payments in period 1 to wage payments in period 2. From this part A of Proposition 1 follows. (ii) (a) We want to show that minimizing the utility of B-workers implies that the second-period wage for workers of different productivities will be different. Consider two wage contracts A and B, that cost the firm the same amount and give the same level of expected utility to the x-workers. The contracts are the same except for the second-period wage that they specify for workers whose productivities lie in an interval [x”,.Y*]. Contract A gives the same wage to all workers in the interval. \t’2a(X) = U’O iff
X0 < .x2,
.x E [x0, s’],
where M” 2 E,[xls E [x0, x2]]. Contract B splits the interval vals, paying workers in the two intervals as follows: M.*&) = u.’
if
xE[x~,.Y’],
u.2Z(_Y)= 1.(:2 if
xE[x~,,Y~],
where wi = E, [X 1x E [.x0,x’]] [F&2)
X0 < xi < x2,
and u.~ is defined
- F,(xO)]wO = [F,(x’)
into two subinter-
by
- F,(xO)]w’
+ [FJ?)
- F,(X’)]W2. (A.11
This implies that 1~’ > M2. We want to show that a potential wage contract A to B. Formally,
IFs(x2)- Fp(xo)]wo >
[F&x')
- F,j(xo)]wl
p-worker
prefers
+ [FB(x2) - Fa(.~‘)]w2
(A.21 Inserting
eq. (A.l) into (A.2) and rewriting
-
FAX’) - F,(.Y')~,, + F,(.x2)
, _
-
, _ F,(x') - F&x0)
F,(u')
F,(x')
W2) - F&‘) F&Y’)
we get
- Fp(xo)
w2.
F,(_Y')
F,,(x')
w2 ’ Fo(x2)
--
F&U') F&Y')
“‘* (A.3)
283
sinceMJ < w2 and MLRP
F&i)
*
- F,(xO)
Fp(x’) - Fp(XO)
F,(x2) - F,(xO) < FB(XZ) - F&O)
(A.2) holds. Thus if a contract specifies the same second-period wage for workers with productivities in some interval, then there exists another contract which costs the firm the same amount, gives z-workers the same expected utility, and gives a potential P-worker a lower utility. Hence the original contract cannot have minimized U,(U), given nj31 and U,. (b) Here we will show that, for given levels of profit for the firm, and a given level of utility for cc-workers, the contract which minimizes the utility of bworkers is one in which workers get paid their productivity if it is less than m, and at least their productivity if it is equal to m. This is done by supposing the contrary, and showing that for any contract A’ not satisfying this condition, there is another contract, B’, which cost the firm the same amount, gives cc-workers the same level of utility, and which gives a lower utility to a potential P-worker. Contract A’ and B’ specify the same first-period wage. (I will use [] for closed interval and () for open interval.) Let contract A’ specify lvzz(x) = x + y(x),
x
E
[x,,, m),
where J(X) 2 0 and
s
y(xU&)dx > 0
LX<.. ml
w2,(m)
Let contract
=
m+
y(m),
y(m)
B’ specify
w2z(x) = x,
xE [x,,,m)
w2z(m) = m + y(m) + y’,
where
2 0.
, y(m) 2 0,
y’ > 0:
The firm’s expected payments and r-workers’ expected utilities are the same under wage contracts A’ and B’. I will now show that b-types prefer wage contract A’ to wage contract B’, i.e., that lrXcz,,,~(.u),f~(x)dx >f;,(m)y’:
> (by MLRP) Thus if \v~~(.x) > x for some x < I?I, then there exists another contract that costs the firm the same, gives the x-workers the same utility and gives a potential b-worker a lower utility. (iii) Here we will show that Hj, = 0 and that (IC-0) is binding. Firstly, we show that nj, = 0. Suppose not, then the utility of r-workers (c/J, can be increased without affecting the incentive compatibility constraint of the /3-workers, (ICY-/3), or violating the wage condition, (4) with, for example, the following procedure. Increase am, by z (2 > 0) and lower W, by z(&(m)/(l + r)). This will lower profits by z(.f,(m) -.f;(m)l(l + r)), increase U, by zC(.f;(m) -.f+i))/(l + 41, leave U,(r) unchanged and it will not violate (4). That together (ii) implies that 1~‘2,(1r7) = m nj,=O with +
(J%C-~l - ~~l.)!f;lO4
Next we show that (IC-fi) is binding. Suppose not, in this case we can establish in the same way as in (i) that the best x-contract is
w*z(x’) = E,[xl wz,(x’)]
)
For a given profit and wage payment to x-types in period 2, we did establish in (ii) that p-types get the lowest utility from a wage contract where w(.u’) # w(x”) if s’ # x” thus, not taking (IC-/Y) into account give us
=“p[x]+&
“I
xdF,(x) r(, xdF&)
+ Fp(\+
+ F,(K)%
I U,(x)
k. Rasin.
A.symmeiric
mfbrmufmn
und lahour
confruct.s
285
Thus, without considering (IC-p), the p-types would prefer the optimal Zcontract to the optimal a-contract. (iv) To find xca, use the properties established above that wZe(x) = x if x < m, that w2&n) 2 m, that Ulp*= Ep[x] + [l/(1 + r)](~~xdF,(x) + F,,(G)%), that (3) and (IC-8) are binding. We can write the utility of the x-types as
u, =
WI1
+
xdF,(x)
~
+ E,[x]
- wlZ + F,(x,,)G
(A.4)
where w la
(1 + r)Elc[x] -
=
E,[x$fi
+ j-(x ii
01m
- ,,,,.,(x)]
‘[I I’
+ r -$E]. .zm
The optimal x,, is found by maximizing (A.4) s.t (i) w,,(m) 2 m, and (ii) x,, 2 W. That w,,(m) 2 m * E,[x] = w,%. Taking the first derivative of U, with respect to x,, we get r;lJ ac7 x,,
-
r,f&J
(xc* - 4
-LCd
[
(
1+ r -
%)]/(I
+r)(l
+r-2)
(A.3 From (A.5) it follows that
Since~~(x,,)/~;(x,,) is decreasing in x,, andfp(nr)/l;(m) we know that the optimal cc-contract entails ex-post
Mu’)
r fo(4 ‘--_I
[
and that the optimal
1>&!9
h(m) ’
< 1 (because of MLRP), inefficient terminations iff
(A.61
x,,, if above @, is found where
if the requirement that E,[x] - wIZ 2 0 holds at this level of x,,. Otherwise x,, is given by setting E,[x] - wlZ = 0 * j!Jx - w)dFl,(x) = (1 + r)(E,[x] -
J% [xl).
(v) Finally, we will show that (IC-c() is nonbinding. In part (ii) of the proof of Proposition 1 we established that, given l7, and U,, minimizing U,(a) implied that wZZ(x’) # wZI(x”) if x’ # x”. In a similar way it can be established that given n,, U, and w,~(x) = E[xlwzp(x)] then U,(b) is maximized when wZO(x) = x. Thus (IC-a) is non-binding if
lJ;=
Mill+-
The established
xdF,(x)
+ E,[x]
xdF,(x)
+ F,(G)%
fact that (lC-fi) is binding
- M;,%+ F,(x,,)G
(A.7)
implies
&3(m) xdFp (x) + ,f;o + (Ez[x]
- wla) + F, (x,,)*
xdF,, (x) + F, (W)W
Inserting
(A.8) into (A.7) and rewriting
we get that (IC-x) is non-binding
if
m
“1
xdF,(x)
- ~.t’~% + Fa(x,,)\i: +
+ E,[x]
-b3:
m
>
(A.81
xdF P (x) +‘!(m) fowl 2
Xcr
- w1.) +
xd F, (x) + F, (C)C
w
F, (.x,,)G
-m
+
!
xdF, (x) + F,(W)M:
E
o
>
- w)dFO(x) +
(x - G)dF,(x).
(J%Cxl - ~11)
(A.9)
By (5) in Proposition 1 and MLRP it follows thatfp(x)lfS(x) > 1 for all x I xcl, when x,, > W. Thus ~~‘(x - S)dFp(x) 2 j:‘(x - G)dF,(x). From MLRP we also know that ,fg(m)
287
References Carmichael, Lorne, 1983, Firm specific human capital and promotion ladders, Rand Journal of Economics 14, Spring, 251-258. Gibbons, Robert and Lawrence F. Katz, 1991, Layoffs and lemons, Journal of Labour Economics 9. Oct., 351-380. Guasch, J. Luis and Andrew Weiss. 1980, Wages as a sorting mechanism in competitive markets with asymmetric information: A theory of testing, Review of Economic Studies 47, July, 6533664. Guasch, J. Luis and Andrew Weiss 1981, Self-selection in the labor market. American Economic Review 7 I. June, 2755284. Guasch, J. Luis and Andrew Weiss 1982, An equilibrium analysis ofwage productivtty gaps, Review of Economic Studies 49, Oct., 4855497. Harris, Milton and Yoram Weiss, 1984, Job matching with finite horizon and risk aversion. Journal of Political Economy 92, Aug., 758-779. Holmstrom, Bengt 1979, Moral hazard and observability, Bell Journal of Economics IO, Spring. 74-91. Hutchens, Robert M.. 1989 Seniority, wages and productivity: A turbulent decade, Journal of Economic Perspectives 3. Fall, 49964. Jappelli, Tullio, 1990 Who is credit constrained in the U.S. Economy?, Quarterly Journal of Economics 105, Feb., 219 234. Kahn, Charles and Gur Huberman, 1988, Two-sided uncertainty and ‘Up-or-out’ contracts, Journal of Labour Economics 6, Oct., 423 444. Malcomson, James M., 1984, Work incentives. hierarchy, and internal labor markets, Journal of Political Economy 92, June, 4866507. Malcomson, James M., 1986, Rank-order contracts for a principal with many agents, Revtew of Economic Studies 53, Oct.. 807 817. Mayfield, Scott E., 1990, Borrowing constraints under secured loan contracts, Working paper no. 189 (Department of Economics, Boston College, Boston, MA). Milgrom, Paul, 1981, Good news and bad news: Representation theorems and applications, Bell Journal of Economics 12, Autumn, 380-391. Milgrom, Paul and Sharon Oster. 1987 Job dtscrimination, market forces and the invisibility hypothesis, Quarterly Journal of Economics 102, Aug.. 4533476. Mirrlees, James A., 1974, Notes on welfare economics, information and uncertainty, in: Balch, McFadden and Wu. eds.. Essays on economtc behavtor under uncertainty (North-Holland, Amsterdam). Mookherjee, Dilip, 1986, Involuntary unemployment and worker moral hazard. Review of Economic Studies 53, Oct. 7399754. O’Flaherty, Brendan and Aloysius Siow. 1992. On the job screening, up or out rules, and firm growth, Canadian Journal of Economics 25, May, 346-368. Rosen, Shervin, 1986. Prizes and incentives in elimination tournaments, American Economic Review 76. Sep.. 701-715. Rosen, Asa, 1991, Contributions to the theory of labour contracts, Ph. D.-thesis (London School of Economics and Political Science, London). Rotschild, Michael and Joseph E. Stightz, 1976, Equilibrium in competitive insurance markets: An essay in the economics of imperfect information, Quarterly Journal of Economics 90, Nov., 629-649. Stiglitz, Joseph E. and Andrew Weiss 1983, Incentive effects of terminations: Application to the credit and labor markets, American Economic Review 73, Dec., 912- 927. Waldman, Michael, 1984. Job assignments. stgnalling and efficiency. Rand Journal of Economics 15, Summer, 255- 267. Waldman. Michael, 1990. Up-or-out contracts: A signaling perspective, Journal of Labour Economics 8, April, 230-250. Weiss, Andrew, 1980. Job queues and layoffs in labor markets with flexible wages, Journal of Political Economy 88, June. 526- 538.
Economics
1 (1994) 269 -287. North-Holland
Temporarily asymmetric and labour contracts AsaRosh* Received
May 1992; final version
received
September
information
1993
Abstract
Under the assumption that workers are more heavily credit rationed than firms, the standard model of testing and self-selectton in the labour market IS extended. The two main findings are that ex post inefficient termination may be used as a self-selection device and that when workers can be of more than two different producitivities, only the best worker should be overpaid. Kr), ~,ords: Self-selection; Credit rationing
Wage-productivity
gaps;
Inefficient
termination;
Up-or-out
contracts:
JEL c~/as.sifrc~a~ion: D82: 53 1; 541
1. Introduction
In the literature on adverse selection in the labour market it has been common to assume either (i) that the firm can never learn anything about the abilities of individual workers, or (ii) that learning is instantaneous. The first assumption is made by e.g., Weiss (1980), whereas the second is associated with the work of Guasch and Weiss (1980, 1981, 1982). Arguably, the intermediate case is more realistic; when a worker first applies for a job, the firm’s estimate of the worker’s productivity is imperfect and less precise than that held by the worker himself. However, as time goes by, the employer’s estimate improves possibly converging to the true productivity. This paper is concerned with the optimal labour contracts in such an environment. In a world with no capital market imperfections, this intermediate case does not differ in any important respect from the situation where the employer learns Corrrspondencr fo: .6sa Rest-n. FIEF, Trade Union Institute for Economic Research, Wallingatan 38, 111 24. Stockholm,
Sweden.
*This is a revised version of chapter two of my Ph.D.-thesis at London School of Economics. Valuable comments and suggestions have been received from Mike Burkart, Tore Ellingsen, Eric Hansen. Alison Hole, Alan Manning, Dilip Mookherjee, John Moore, Tore Nilssen, Chris Pissarides. Bengt Rostn and an anonymous referee. The work was financially supported by Gcteborgs Handelshiigskolefonder, Helge Ax: son Johnsons Stiftelse, Torsten och Ragnar SGderbergs stiftelse, Tore Browaldhs stiftelse and Stiftelsen Blanceflor Boncompagni-Ludovisi fiidd Bildt.
0927-5371 94:X07.00 C; 1994 Elsevier Science B.V. All rights reserved SSDI 0927.5371(93)EOO27-P
productivity instantaneously, because payment can just be deferred until productivity is revealed. However, capital markets are empirically known to be imperfect, with a substantial number of workers being severely credit rationed.’ The starting point of this paper is the work by Guasch and Weiss (1980, 1981 and 1982). In their papers, applicants for a job can be of two types. The type is known to the potential worker, but not to the employer. However, if the worker can take a test which gives scores related to their expected productivity, then by requiring that each worker pays an application fee and offering contracts contingent on a test score, the firm can induce workers of different types to choose different contracts, i.e. to self-select. Here the models of Guasch and Weiss are modified in three dimensions. 1 have already indicated the first two: Credit rationing and slow revelation of information. In addition I allow for a continuous number of productivity levels and test results. This formulation leads to important departures from the results of Guasch and Weiss. When workers are credit rationed, a labour contract which specifies a very low wage early on will impose costs on the worker. I show that firing some productive workers might be a cheaper way to induce self-selection than deferring payment until productivity is revealed. The contract specifies that after the first period, when the firm learns the worker’s productivity, the worker will then either be retained or fired. Examples of such up-or-out contracts are common in academic institutions (getting tenure) and law firms (becoming a partner). While other papers have dealt with up-or-out contracts, they do not view them as a self selection device.2 The longer it takes to learn the workers productivity, the more important is the capital market imperfection, and the more likely it will be that the workers prefer the use of termination threats as a self-selection device. I also show that only the very best workers will receive a payment in excess of their marginal product, and that the result of Guasch and Weiss that a substantial number of workers will be overpaid is a misleading artefact of their assumption that there are only two test levels. The paper is organized as follows. Section 2 describes the basic model. The optimal productivity dependent wage contracts are derived in section 3. In section 4, I discuss the robustness of the contracts. Finally, a discussion of the results and the relation to the literature is in section 5.
’ Jappelli (1990) estimates that 20% of the U.S. families are credit rationed. The important variables in his study are: low age. no established credit history. and IOU income. It should also be emphasized that Jappelli define5 a person as credit rationed only if he IS unable to obtain any crcdlt at all. In another recent paper, Mayfield (1990) finds that 43% of households under the age of 30 arc credit constralned and that 37% of households under the age of SO face rignificant constraints. ‘See Harris and Weiss (1984). Kahn and Huberman Siou (1992).
(19X8), Waldman
(1990) and O’Flahert)
and
2. The model Employees live for two periods and are of two types, SIor /3. Type c( has higher expected productivity in the industry than type /?. A worker’s type is private information to that worker. Productivity, s, for an employee is an independent random variable, and is equal in both periods. The distribution of x differs between the two types. For each type it is described by a density function,f;(x) together with a point mass in m (I = X,/J’). The point mass in t?i is assumed to avoid some future technical problems. I use the somewhat non-standard notation withj;(_x) also denoting the point mass in m. The corresponding distribution function is denoted by F,(u), and it has support [I, m]. I assume that,f,(.u):fil(x) is increasing in x over [I/, m], thus the Monotone Likelihood Ratio Property (MLRP) ho1ds.3 E, denotes expected value with respect to F,(_u). Productivity in period 1 is observed (by the worker and the firm) only ex post, just before period 2. The wage in period I is denoted by it’s. and i~,(u)denotes wage in period 2 given that the productivity in period 1 was .x. Firms and employees are risk neutral. Firms have a zero discount rate, but employees are assumed to have a positive discount rate r. This is a simple way to capture that workers have less access to credit than firms.4 Expected utility for a worker of type I joining the industry is u, = \I’,, + ~ Ifr
1
n, \v,,(.u)dF,(x) -xc I
,
+ F,(u,,)w
I = (x,/l) ,
i
where \V is the best wage available in another industry and s,, is the minimum productivity level required to stay in the firm in period 2. That is, the expected utility is equal to the wage in period 1 plus the expected discounted wage payments in period 2. This specification implicitly excludes ex-post fines for low performance levels. Assuming that productivity levels are not verifiable justifies this exclusion. The criterion for an r-worker to join the industry (the industry participation constraint) is
I assume ‘For
that (PC) is satisfied
a thorough
diacuwon
for both z-types
of the MLRP
see Milgrom
and @-types. (1981).
‘A more rigorous approach would be to define workers’ utility over consumption in the two periods. and let preferences be convex. Since workers are assumed to be risk neutral, there IS no room for an Insurance market. Let interest rates be zero. A risk neutral worker would wish to equate marginal utility of consumption in period I with the marginal utility of expected consumption in period 2. Of course. If the wage in period I 1s lower than the wage I” period 2, this would require borrowing. If the worker faced an absolute credit constraint, or would have to pay a larger &rest than the employer. a higher wage III period I. ceteris paribus. yields a greater benefit to the worker than it ia costly to the firm
All the firms in the industry have an identical constant returns to scale technology. Each firm maximizes profit and takes the product price, the workers reservation utility and the employment contracts of other firms as given. Without loss of generality the product price is normalized to 1. Consider the expected profit to firm j from employing a z-type worker:
s ,,I
Ilj,
= E,[.x] -
~‘1,
+
(x -
\t’z,(x-))dF,(.x)
(2)
XC!
Profit is equal to expected productivity in period one, minus period one payment plus expected profit in the second period. Firms live forever. There is free entry into the industry, so that in equilibrium, it is not possible for a firm to enter and earn positive profits. Since we have a constant returns to scale technology, the equilibrium number of firms is indeterminate. I assume that firms can observe each other’s contracts, but not a particular worker’s productivity. However, workers are able to communicate their wage to an outside firm. Employees are equally productive in each firm. and productivity is invariant to the wage contract.’ It is assumed that workers cannot commit themselves to stay in the firm in period 2. This implies that the feasible period 2 wage structure will crucially depend on the current employers ability to match an outside firm’s wage offer. The two polar cases are (if the current employer can always match the offer, (ii) the current employer can never match the offer.6 Arguably some intermediate case is most plausible. Under assumption (i) it will be possible in period 2 for firms to pay all workers a wage that is as low as the productivity of the least productive worker, due to the winners curse problem faced by the outside firms.’ Under assumption (ii) each wage group has to be paid at least their expected productivity. (Remember that wage contracts are observable). I will here make the second assumption, thus allowing workers to make binding agreements to accept an outside offer.*
‘Assuming that workers’ ively as long as productivity ‘Assumption e.g. Waldman
productivities differ across firms does not change the results qualitatin one firm is positively correlated with productivity in another firm.
(i) is made in e.g. Waldman
(1990) and Milgrom
and Oster (1987). assumption
(ii) in
(1984).
‘The winners curse problem in this context is that since a current employer has more information about a particular worker, the only time he will not match an offer is when it is above the worker’s productivity. Thus, the outside firm succeeds in attracting workers only when it offers more than their productivity. *An alternatlve specification where information about 2 across firms. but the former employer has more information end of period one. would give similar results.
productivity is symmetric m period about the workers who got tired at the
The sequence
of events and the information
structure
is summarised
below.
of’ period 1 Firms offer two-period contracts. - Firms observe each other’s contracts. - Workers know their type. Beginning
-
Beyinniny
cfperiod
2
Workers and the current employer observe an individual worker’s productivity. - The worker is fired or offered continuing employment at the period 2 wage specified in the contract. __Other firms observe hiring and firing decisions and the wage offered. - Other firms can give competing wage offers.
-
Since workers are equally productive in all firms and contracts (and wages) are observable, the wage has to be greater than or equal to the average expected productivity, for each group of workers who earn the same wage in period 2. Otherwise they would leave the firm since another firm could profitably offer them a higher wage. Formally, r~~,(.x’) 2 E,[xlwz,(s’)]
for all u’EX,
I = [a,/I)
That only contracts are observable implies that any two workers earning the same wage look the same to an outside firm, as do all workers who are fired. This assumption provides a link between this article and the literature on visible and invisible workers, as developed by Waldman (1984) and Milgrom and Oster (1987) and the work by e.g. Gibbons and Katz (1991). There too, promotion, wage rises and firing decisions are signals to the market about a specific worker’s quality, and outside firms have less information than the present employer. Gibbons and Katz find that wage losses for those workers that were fired on employer’s discretion were larger compared to those who lost their jobs due to plant closing. This is consistent with the assumptions that there is asymmetric information between firms and that outside firms take firing as a signal of low quality. I will assume that the expected productivity given that a worker is fired is less than the alternative wage, i.e. E,[s 1.x5 ,I.,,] < C. This implies that fired workers will move to another industry. This is only a simplification and does not affect any of the main results. The cut-off level, x,, will be determined endogenously. Ex-post efficiency implies that .x,( = W and ex-post inefficient terminations occur iff s,, > W. Note that if outside firms can discriminate between the workers who are fired, s,, cannot be above C. Workers with productivity above \C would then immediately be hired by another firm and competition would ensure that their wage
would be equal to their productivity. By implication, firing would not be an effective threat in the first place. It can be seen that for inefficient terminations to occur it is necessary that those fired do not all have the same performance level. The assumption in this paper of a continuum of performance levels ensures that this holds.’ Finally, I make the assumption that E,[x] < W. This assumption ensures that my set-up where workers prefer wage payments in period 1 to wage payments in period 2 is consistent.”
3. Equilibrium reward schemes Since workers have information about their productivity that is not available to the firm, we will run into the standard problem that bad types @-types) want to mimic good types (z-types) unless the contracts take the asymmetric information into account. As in most papers on self-selection, I assume that once a worker has signed a contract, reputation effects allow the firm to credibly commit itself not to renegotiate due to reputation effects. Also, the worker cannot leave the firm immediately after he has signed the contract. For simplicity, assume that the worker cannot leave until the beginning of period 2. A separating equilibrium is defined by three conditions. (i) There exists no other contract that workers prefer and which gives a non-negative profit to the firm, given that the other firms pursue the equilibrium strategy. (ii) For each group of workers earning the same period 2 wage, the wage has to be greater than or equal to their average expected productivity. (iii) Finally, a P-worker does not prefer an r-contract to a /I-contract, and vice versa. We are now ready to define the maximization problem that solves out the best separating contracts.’ ’
st.
n,, =
E, [s]
-
11’,
I
+
t*t I
(s - w,,(.x))dF,(.x)
2 0,
I = Y, /r,
(3)
A‘!
vvl,(.Y’) 2 E,[xlwz,(s’)]
for all x’EX,
I = c~,p,
(4)
“To be precise. what is needed is more than one performance level ofthoae who arc tired. and that the former employer has more information about the workers than a potential employer. ‘“If I had assumed that the firm had to weaker assumption would have been Ex[r]
pay a
training cost, k. to make use of a new worker the ~ k -c 12.
’ I A pooling equilibrium where r- and /i-types opt for the same contract dots not exist, see Roskn (IYY I) for proof. This is a standard result in this kind of model. see e.g. Rothschild and Stightz (1976).
Constraint (3) is the non-negative profit condition. Constraint (4) is the wage condition. Constraints (IC-/I) and (IC-r) are the incentive compatibility constraints. ciO(~) is the utility a ,!j’worker gets from an y-contract and U,(p) the utility an cc-type gets from a p-contract and U: is the maximum utility a r-type can get given a separating contract. The optimal separating contracts are described in the following proposition. Proposition
I.
Part A: The equilibrium
Part B: The equilibrium
r-contract
j?-contract is
is Xcr
(1 + r)EplI-xl - E,[I.yl(.f#4/L(m)) + Wla =
w2Jm) = m +
I
(x - w)dF#
E
1 + r - Chk4M4)
LCXI -
Wlsl
L(m) Part C: The cut-off level, x,, . tions @
The r-contract
entails rx-post
ine$ficient termina-
(5)
and to
Proof
of’ Proposition
1.
See Appendix
A.
Since the proof of Proposition 1 is somewhat involved, I here discuss the intuition underlying the proposition. Consider the best /3-contract. This is found by assuming that (IC-r) is non binding and later checking that this assumption is correct. Given that (IC-x) is non-binding, the result given in part A of Proposition 1 is intuitive. The firm offers the highest possible period 1 wage subject to the requirement that workers in each wage category in period 2 are paid at least their average expected marginal product. Note that because workers are risk neutral, the wage schedule in period 2 is indeterminate. One example is where the wage is equal to productivity. i.e. MIX, = x. Another is when all workers earn a wage equal to the average productivity of those workers that stay in the firm, i.e. rc,,(.u) = E,[s(s 2 xc,] for all s 2 .Y~,.Since (IC-x) is non-binding, inefficient terminations will serve no purpose, so xcB = \?. We now turn to part B of Proposition 1. It will be the case that (IC-p) is binding, i.e. the constraint that b-types should not mimic x-types will bind. Also the non-negative profit constraint [constraint (3)] will be binding for the r-contract. Given this we can solve for the contract that minimizes U,(r). for a given profit and U,. When we do this we find that there will be no pooling over different productivity levels i.e. \v~~(.Y’)# \Q~(.Y”) if s’ # Y”. This is because x-workers’ utilities are unaffected by pooling over productivity levels while for a given z-contract, MLRP implies that pooling increases b-workers utilities. There are two potential ways to deter low productivity workers from choosing a contract intended for the high productivity workers. Either the %-contract can specify a high cut-off, xcZ, or it can specify a low initial wage with the worker being compensated by a higher expected wage in period 2. Consider first the latter device. When the a-contract involves period 2 payments above the expected productivity this will be done by rewarding the performance level m with a ‘prize’ and paying a wage equal to productivity at all other productivities The reason only performance level I?? is overpaid, is that by MLRP, ,\;(s),i/~~(s) is maximized at m, i.e. the probability that a worker is an r-type if both x- and p-types opt for the same contract is at its highest level at productivity
k. Ros3n. A.symmrtric
information
and lahour
contrac’ts
217
level m. Note that since the workers are risk neutral they care only about their expected payments. The feature that only the very highest performance level is paid a wage exceeding its value should be emphasised. In previous models where there have only been two test levels (pass or fail), all those who pass are paid a wage in excess of the value of their performance. Hence the assumption of two test levels gives the misleading impression that a large proportion of retained workers should be overpaid. As mentioned earlier, technical complications would have arisen if I had not assumed a point mass at m.” It would still have been the case that only the highest performance levels would have been overpaid, because .f; (.x)/“~ (x) is increasing in X, but for any ‘over-payment’ interval there exists another smaller interval, which yields a better contract. On the other hand it is not possible to overpay only the performance level m, since the probability of performing WI(without the point mass property) is zero. A similar problem was identified by Mirrlees (1974). 1 are found by The expressions for nlIZ and MJ~~(VI)in part B of Proposition requiring that in equilibrium, profits are zero, and /I-workers are indifferent between the a-contract and the /I-contract. That is, we require that both (3) and (IC-p) hold with equality. I now turn to the issue of the optimal cut-off level, .xcZ. By raising the cut-off level s,.,, the wage-schedule can be flattened while still deterring /?-types from opting for the x-contract. The condition for inefficient terminations [condition (5)] is obtained by differentiating U, with respect to x,, around E, given that w,;, and w~Jx) are as in part B of Proposition I. Similarly, the formula for the optimal cut-off level, x,,, [eqs. (6) and (7)] are obtained by maximizing Ui,, given that w,* and Wan are as in part B of Proposition 1, As can be seen from Proposition 1, eq. (5) a necessary condition for ex-post inefficient terminations is that fb(W).!f;(\V) > 1. Ex-post inefficient terminations are more likely the higher is r, ,fb(G)/fa(G) or ,&(m)[f;(m). These results can be understood as follows. A high r reflects a long period 1 duration. Since firing allows flatter wage schedules, this option becomes more attractive when the revelation of productivities is slow. The more likely P-workers are to have a productivity just above Wcompared to the cc-workers [the higher,fi(*),‘f2(w) is] the more likely are ex-post inefficient terminations. This is because ex-post inefficient terminations then lower an cr-worker’s ex ante utility less than a potential b-worker’s. Finally, the more likely a [I-worker is to have a productivity of m, compared to the likelihood for an x-worker [the highmji(m)/f;(m) is], the more likely it is that the optimal self-selecting contracts will involve ex-post inefficient firing rather than giving m-performers a high premium. This is “Another way around Holmstriim (1979).
this
problem,
is to
put an upper
limit
on the wage payment. see
because for given x,, the higher.fb(m)/fz(m) is, the steeper does the wage schedule have to be to keep the p-worker from applying, and thus ex-post inefficient firing becomes more attractive as the self-selection device. When both steepening of the wage schedule and ex-post inefficient terminations are used as self-selection devices, the optimal x,, is defined by (6) of Proposition 1. If self-selection is obtained solely by a high cut-off level, then optimal x,., is defined by (7). In this case u’, 1 = E,[x] and VL.~~(_X) = .Y for all Y above s,,. If parameter values are such that self-selection is achieved mainly through involuntary terminations, the market will offer r-contracts with a high initial wage but high requirements for being allowed to stay. The &contracts will have a lower initial wage but less demanding requirements for staying. Examples of such requirements could be passing an intermediate exam, or gaining promotion, such as tenure in an university, or becoming a partner in a law firm. It is also worth noting that the contracts are derived under the assumption that reservation wages are equal for the two types of worker. Under the assumption of different alternative wages, with W, > \cII, as e.g., in Weiss (1980) the condition for ex-post inefficient terminations becomes
(8) and there will ~[KYJ~Sbe some involuntary terminations since, as x,, goes to @,. the denominator in the first term goes to zero. Having established the best separating contract, I discuss the problem of its credibility in the next section.
4. Robustness of contracts There is a credibility problem with the ex-ante optimal contracts. Consider the contract for r-workers. The contract specifies that productive workers should be fired, and/or that workers of productivity m should be paid more than their productivity. The common way to defend such deviations from sequential rationality is to say that the firm lives for many periods and can establish a reputation for sticking to the ex-ante optimal contract. However, this explanation does not look entirely convincing when it comes to the premium paid to the m-performer. If,f@(rn) is low, the very best production level will almost never be attained by any worker, and the premium would have to be enormous. Since the productivity levels are (realistically) assumed not to be not perfectly observable, it would be difficult to keep the firm from cheating if the case were to arise.13 “The assumptton of not perfectly observable possibility of ex-post fines for low performance.
productivity
levels was made earlier to exclude the
A more robust contract, avoiding this problem, would be a rank-order contract, where the firm pays workers a wage which depends on their relative performance levels. This makes the firm’s payment in period 2 invariant to the actual productivity of the workers in period 1.14 In Rosen (1991) I showed that in a rank-order contract where self-selection is achieved at least partially through steepening the wage schedule, the wage will be equal to average marginal productivity in period 2 except for the first rank, which earns a prize [as we would expect from Malcomson (1986)]. This wage schedule is similar to the one obtained in Rosen’s (1986) study of elimination career ladders. In his model, the large first prize is necessary to induce contestants to maintain a high effort level late in the game. This wage schedule, which has been used as an explanation of top-executives’ high wages does thus not necessarily result from a moral hazard problem but could result from adverse selection as well.
5. Discussion
of results and relation to the literature
The two central results of this paper are firstly that termination of productive relationships may be used as a self-selection device and secondly that only the very best worker (if any) should be overpaid. Let me now discuss the first result together with the related issue of the contracts up-or-out feature, and its relation to the literature, in more detail. Following the work of Stiglitz and Weiss (1983) there have been a number of attempts trying to explain why productive labour market relationships are sometimes terminated. The mechanism, presented by Stiglitz and Weiss (1983), is that the threat of termination can induce higher effort by workers in a model where the employer can observe each worker’s production but not their effort. Variations of this latter idea have recently been presented by Kahn and Huberman (1988) and Waldman (1990). In these models, it is crucial that productivity is not verifiable. It is exactly the provision that workers with a sub-standard productivity are not retained which ensures that the employer does not falsely claim that a worker has a low productivity. However, the robustness of this type of argument has been seriously questioned by Mookherjee (1986). The simple StiglitzzWeiss model creates involuntary retention rather than involuntary layoffs for a range of plausible parameter values, and the more sophisticated model with two-sided uncertainty hinges on the assumption that the firm cannot use rank-order tournaments. If the firm could commit itself to promote a certain proportion of the work force, according
‘“Such
rank-order
tournaments
have been suggested
by CarmIchael (1983) and Malcomson of productivity levels. Malcomson a separating equilibrium achieved
( 1984, 1986) as a way to mitigate the problem of non-verifiability (1986)showed that, for a large enough number of participants, with piece rates can be achteved
by a tournament.
to relative productivity, this would efficiently solve the incentive problems on both sides.’ 5, I6 Malcomson (1984) derived the optimal contract in such a setting. If effort inducement is the reason for an up-or-out contract, then it is hard to understand why the alternative to being fired is lifetime employment, as is common in the academic market and in law firms. This suggests that sorting on labour quality provides a more convincing explanation for this type of contract. Models with the up-or-out feature that are based on ability and sorting are presented by Harris and Weiss (1984), O’Flaherty and Sow (1992) and Gibbons and Katz (1991). Of these only in Gibbons and Katz does there exist equilibria with ex-post inefficient firing. ” They make the same assumption as made here that outside firms can only observe contracts and hiring and firing decisions and not a particular worker’s productivity. However, inefficient firing does not arise due to a self-selection device, as in the present model, but as a Bayesian equilibrium. It is crucial in their model that firms cannot write two-period contracts. The mode1 presented here shows that if workers are credit rationed and there is asymmetric information about productivity at the hiring stage, then inefficient terminations can also arise where commitment and reputation are possible. Although Guasch and Weiss (198 1 p. 278) hinted at the idea that termination threats may supplement wage profiles in sorting the work force, they never substantiated the argument, and as it stands it is incorrect. In their 1981 paper, Guasch and Weiss analyse self-selection contracts both under the assumption of risk-neutral workers and under the assumption of risk-averse workers. They come to the conclusion that self-selection will always be optima1 when workers are risk neutral, but not necessarily when they are risk averse. Throughout their paper, however, they ussum~ that self-selection implies that the firm only hires those who pass the test. In Rosen (1991) I show that in their model, if the workers are risk neutral, all workers should be hired ex-post. Only above a certain level of risk aversion can it be optima1 to engage in inefficient firing. Again, of course, the intuition is that the steep wage schedule is costly ~ now due to workers’ risk aversion - and it may be profitable to introduce a second selfselection device. But even with risk aversion, it is necessary to have different
“This critique can be circumvented by assuming that the firm must incur a small cost in order to observe the workers’ productivity, and that the workers cannot know whether or not the firm promotes randomly. Personally, I am not convinced by this counter argument: m most cases tirmr learn something about workers’ abilities whether they want to or not. Furthermore, a firm with a random promotion policy would quickly earn a bad reputation. “‘Ifapplication fees are allowed in the Stiglltz Weiss analysis the Involuntary terminations would disappear. The reason why the tiring of productive workers can be rational is that workers are earning rents. due to eficiency wages. Thus. the threat to termmate induces higher effort. However. effort depends only on the difference between the wage if successful and the wage if not. With risk-neutral workers an application fee makes it possible for the firm to widen this difference without decreasing profits. Ii In Gibbons and Katz inefficlent firing is, however, not the unique eqmhbrium such equilibria are not robust even to ‘small’ reputation effects.
outcome, Also,
reservation wages to produce the termination results in Guasch and Weiss’ set-up. The results presented here do not need this difference. Another important difference between the present model and Guasch and Weiss (1981) is that I allow for competition in the labour and product markets. In Guasch and Weiss (1980, 1982), there is also competition in the labour and product markets, but ex-post inefficient terminations are not optimal (even with different reservation wages). The reason terminations may be optimal in the present model is that there is asymmetric information between the firm that fires a worker and a firm that hires the same worker. Since a fired worker who is productive cannot be distinguished from other less productive workers, there is a net loss associated with having to leave the firm. To better assess the relative merits of various theories, micro level data on inefficient terminations would be very welcome. The only indisputable example I am aware of where terminations of productive relationships have been used as part of a conscious strategy to increase overall efficiency, is from the Mongolian army under Ghengis Khan. There, a number of Mongolian soldiers would be executed if one of their leaders was lost in fight. The more prominent the leader, the higher was the number of soldiers punished. Although this practice was obviously ex-post inefficient, it seems to have been ex-ante efficient. The empirical and theoretical issues of whether the wage ~ productivity ratios increase over tenure, and with this, the related question of whether workers are paid more than their marginal productivity late in tenure, have been addressed in many papers.‘” The model in this paper predicts that the wageeproductivity ratio will increase over tenure. due to sorting and the premium paid to the top performer. However, it does not predict that a large group should be paid more than their productivity late in tenure. Empirical evidence on wageeproductivity gaps are inconclusive and we should remember that the above contract is only in a setting where matching ability and learning productivity are important. Thus it will not apply to the whole labour market. If strong evidence is found that large groups are overpaid late in tenure, then this paper suggests that we have to resort to explanations other than asymmetric information of productivity at the hiring stage.
Appendix A: Proof of Proposition
1
The proof is in the five following
steps:
(i) Assume (IC-c() is non-binding and solve for the best /&contract. (ii) Show that given nj,, U, and (4), minimizing U,(g) implies (a) u.~~(x’) # M.~~(x”) if x’ # x” and (b) Nail = Y if x < m and u’,,(m) 2 M. ‘“See Hutchens
(1989) for an overview
282
(iii) Show that nj~ = 0 and that (IC-fi) is binding. (iv) Find the optimal x,,. (v) Show that (IC-r) is non binding. Constraint (3) will hold with equality since (i) Assume (IC-x) is nonbinding. a worker’s utility can always be increased by lowering profits. Constraint (4) will hold with equality since workers prefer wage payments in period 1 to wage payments in period 2. From this part A of Proposition 1 follows. (ii) (a) We want to show that minimizing the utility of B-workers implies that the second-period wage for workers of different productivities will be different. Consider two wage contracts A and B, that cost the firm the same amount and give the same level of expected utility to the x-workers. The contracts are the same except for the second-period wage that they specify for workers whose productivities lie in an interval [x”,.Y*]. Contract A gives the same wage to all workers in the interval. \t’2a(X) = U’O iff
X0 < .x2,
.x E [x0, s’],
where M” 2 E,[xls E [x0, x2]]. Contract B splits the interval vals, paying workers in the two intervals as follows: M.*&) = u.’
if
xE[x~,.Y’],
u.2Z(_Y)= 1.(:2 if
xE[x~,,Y~],
where wi = E, [X 1x E [.x0,x’]] [F&2)
X0 < xi < x2,
and u.~ is defined
- F,(xO)]wO = [F,(x’)
into two subinter-
by
- F,(xO)]w’
+ [FJ?)
- F,(X’)]W2. (A.11
This implies that 1~’ > M2. We want to show that a potential wage contract A to B. Formally,
IFs(x2)- Fp(xo)]wo >
[F&x')
- F,j(xo)]wl
p-worker
prefers
+ [FB(x2) - Fa(.~‘)]w2
(A.21 Inserting
eq. (A.l) into (A.2) and rewriting
-
FAX’) - F,(.Y')~,, + F,(.x2)
, _
-
, _ F,(x') - F&x0)
F,(u')
F,(x')
W2) - F&‘) F&Y’)
we get
- Fp(xo)
w2.
F,(_Y')
F,,(x')
w2 ’ Fo(x2)
--
F&U') F&Y')
“‘* (A.3)
283
sinceMJ < w2 and MLRP
F&i)
*
- F,(xO)
Fp(x’) - Fp(XO)
F,(x2) - F,(xO) < FB(XZ) - F&O)
(A.2) holds. Thus if a contract specifies the same second-period wage for workers with productivities in some interval, then there exists another contract which costs the firm the same amount, gives z-workers the same expected utility, and gives a potential P-worker a lower utility. Hence the original contract cannot have minimized U,(U), given nj31 and U,. (b) Here we will show that, for given levels of profit for the firm, and a given level of utility for cc-workers, the contract which minimizes the utility of bworkers is one in which workers get paid their productivity if it is less than m, and at least their productivity if it is equal to m. This is done by supposing the contrary, and showing that for any contract A’ not satisfying this condition, there is another contract, B’, which cost the firm the same amount, gives cc-workers the same level of utility, and which gives a lower utility to a potential P-worker. Contract A’ and B’ specify the same first-period wage. (I will use [] for closed interval and () for open interval.) Let contract A’ specify lvzz(x) = x + y(x),
x
E
[x,,, m),
where J(X) 2 0 and
s
y(xU&)dx > 0
LX<.. ml
w2,(m)
Let contract
=
m+
y(m),
y(m)
B’ specify
w2z(x) = x,
xE [x,,,m)
w2z(m) = m + y(m) + y’,
where
2 0.
, y(m) 2 0,
y’ > 0:
The firm’s expected payments and r-workers’ expected utilities are the same under wage contracts A’ and B’. I will now show that b-types prefer wage contract A’ to wage contract B’, i.e., that lrXcz,,,~(.u),f~(x)dx >f;,(m)y’:
> (by MLRP) Thus if \v~~(.x) > x for some x < I?I, then there exists another contract that costs the firm the same, gives the x-workers the same utility and gives a potential b-worker a lower utility. (iii) Here we will show that Hj, = 0 and that (IC-0) is binding. Firstly, we show that nj, = 0. Suppose not, then the utility of r-workers (c/J, can be increased without affecting the incentive compatibility constraint of the /3-workers, (ICY-/3), or violating the wage condition, (4) with, for example, the following procedure. Increase am, by z (2 > 0) and lower W, by z(&(m)/(l + r)). This will lower profits by z(.f,(m) -.f;(m)l(l + r)), increase U, by zC(.f;(m) -.f+i))/(l + 41, leave U,(r) unchanged and it will not violate (4). That together (ii) implies that 1~‘2,(1r7) = m nj,=O with +
(J%C-~l - ~~l.)!f;lO4
Next we show that (IC-fi) is binding. Suppose not, in this case we can establish in the same way as in (i) that the best x-contract is
w*z(x’) = E,[xl wz,(x’)]
)
For a given profit and wage payment to x-types in period 2, we did establish in (ii) that p-types get the lowest utility from a wage contract where w(.u’) # w(x”) if s’ # x” thus, not taking (IC-/Y) into account give us
=“p[x]+&
“I
xdF,(x) r(, xdF&)
+ Fp(\+
+ F,(K)%
I U,(x)
k. Rasin.
A.symmeiric
mfbrmufmn
und lahour
confruct.s
285
Thus, without considering (IC-p), the p-types would prefer the optimal Zcontract to the optimal a-contract. (iv) To find xca, use the properties established above that wZe(x) = x if x < m, that w2&n) 2 m, that Ulp*= Ep[x] + [l/(1 + r)](~~xdF,(x) + F,,(G)%), that (3) and (IC-8) are binding. We can write the utility of the x-types as
u, =
WI1
+
xdF,(x)
~
+ E,[x]
- wlZ + F,(x,,)G
(A.4)
where w la
(1 + r)Elc[x] -
=
E,[x$fi
+ j-(x ii
01m
- ,,,,.,(x)]
‘[I I’
+ r -$E]. .zm
The optimal x,, is found by maximizing (A.4) s.t (i) w,,(m) 2 m, and (ii) x,, 2 W. That w,,(m) 2 m * E,[x] = w,%. Taking the first derivative of U, with respect to x,, we get r;lJ ac7 x,,
-
r,f&J
(xc* - 4
-LCd
[
(
1+ r -
%)]/(I
+r)(l
+r-2)
(A.3 From (A.5) it follows that
Since~~(x,,)/~;(x,,) is decreasing in x,, andfp(nr)/l;(m) we know that the optimal cc-contract entails ex-post
Mu’)
r fo(4 ‘--_I
[
and that the optimal
1>&!9
h(m) ’
< 1 (because of MLRP), inefficient terminations iff
(A.61
x,,, if above @, is found where
if the requirement that E,[x] - wIZ 2 0 holds at this level of x,,. Otherwise x,, is given by setting E,[x] - wlZ = 0 * j!Jx - w)dFl,(x) = (1 + r)(E,[x] -
J% [xl).
(v) Finally, we will show that (IC-c() is nonbinding. In part (ii) of the proof of Proposition 1 we established that, given l7, and U,, minimizing U,(a) implied that wZZ(x’) # wZI(x”) if x’ # x”. In a similar way it can be established that given n,, U, and w,~(x) = E[xlwzp(x)] then U,(b) is maximized when wZO(x) = x. Thus (IC-a) is non-binding if
lJ;=
Mill+-
The established
xdF,(x)
+ E,[x]
xdF,(x)
+ F,(G)%
fact that (lC-fi) is binding
- M;,%+ F,(x,,)G
(A.7)
implies
&3(m) xdFp (x) + ,f;o + (Ez[x]
- wla) + F, (x,,)*
xdF,, (x) + F, (W)W
Inserting
(A.8) into (A.7) and rewriting
we get that (IC-x) is non-binding
if
m
“1
xdF,(x)
- ~.t’~% + Fa(x,,)\i: +
+ E,[x]
-b3:
m
>
(A.81
xdF P (x) +‘!(m) fowl 2
Xcr
- w1.) +
xd F, (x) + F, (C)C
w
F, (.x,,)G
-m
+
!
xdF, (x) + F,(W)M:
E
o
>
- w)dFO(x) +
(x - G)dF,(x).
(J%Cxl - ~11)
(A.9)
By (5) in Proposition 1 and MLRP it follows thatfp(x)lfS(x) > 1 for all x I xcl, when x,, > W. Thus ~~‘(x - S)dFp(x) 2 j:‘(x - G)dF,(x). From MLRP we also know that ,fg(m)
287
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