- Email: [email protected]
Contingent contracts and educational screening
Economics
of Education
Review,
PrintedinGreatBritain.
Vol. 9.
Contingent
No. 2, pp. 149-1.56.
1990.
0?7?-7757Ml
s3.00
@ 1990Pergamon
Contracts and Educational
+ o.oll
Press
plc
Screening
TIMOTHY J. PERRI
Department
of Economics, Appalachian State University, Boone, NC 28608, U.S.A.
Abstract -This paper is concerned with a world where firms initially do not know workers’ productivity and workers do not know the level of match specific job satisfaction at any firm. We find that, even if firms will not default on contingent contracts, there is generally a positive private and social return to educational screening. Contingent contracts do not prevent some screening. In general, some screen efficiently and others screen when it is inefficient to do so. However, if screening cost is either very small or very large there may be too little screening.
I. INTRODUCTION THE PIONEERINGwork
of Arrow (1973) and Spence (1974) demonstrated that individuals might obtain socially excessive levels of schooling if education is used as a signal or screen.’ Arrow treated education as a test: at some cost, identical to all, individuals’ productivity would be revealed (with some error). Spence modeled education as a pure signal: the level of education reveals nothing directly about an individual’s productivity, but in equilibrium there is a positive relation between productivity (ability) and the level of education chosen.’ Most of the screening literature considers models with pure signals, and almost all of this work has focused on the problem of the existence of equilibrium.3 In contrast, this paper focuses on a test where the alternative to individuals revealing their productivity by screening prior to employment (taking the test) is a contingent contract in which individuals accept a relatively low wage until their productivity is discovered on the job. There are two reasons why we assume that screening involves a test and not a pure signal. First, we avoid the existence questions4 which arise in models with a pure signal. Thus we can focus on the important, but recently neglected, issue of screening efficiency. Second, one might think that education would directly reveal something about one’s ability. Grade completed, major field, grade point average, etc. would seem to be indices of ability from which prospective em-
ployers could infer productivity. With a pure signal, these indices only indirectly reveal productivity when, for example, the more able are induced to stay in school longer than the less able. Researchers recently have ignored the question of efficiency of screening, which was the main focus of the early screening literature. The reason apparently involves the difficulty of determining the extent to which firms will renege on promises to workers. If firms are unreliable and will cheat workers, then it appears that individuals may invest in screening which is of private but not social value. If firms are reliable, then Barzel (1977) and Spence (1981) both demonstrate that educational screening can be avoided by contingent contracts which pay relatively low wages until productivity is discovered and then compensate workers for being underpaid initially. This assumes that observing ability on the job is cheaper than educational screening.5 Previous work on educational screening suggests that: (a) contingent contracts will completely replace screening if firms do not cheat; and (b) if contingent contracts do not replace screening, then the more able (higher productivity) individuals will screen, and this screening is a social waste. What distinguishes this paper from the rest of the screening literature is that we assume that individuals are initially uncertain about some aspect of the job, call it satisfaction, at any employer. The introduction of uncertain job satisfaction leads to the possibility that some individuals will want to quit if they realize a low level of
[Manuscript received 20 January 1989; revision accepted for publication 19 October 1989.1
149
150
Economics
of Education Review
satisfaction. The possibility of turnover was not considered in the analysis of contingent contracts in Barzel (1977) and Spence (1981). If individuals are compensated for being underpaid initially, independent of whether they remain with the firm, then contingent contracts will induce individuals to bypass educational screening unless the problem of firms cheating or default is significant. Such compensation effectively constitutes a completely vested pension. What we will demonstrate is: (a) with uncertain job satisfaction and the possibility of turnover, contingent Contracts will not completely replace educational screening unless pensions are completely vested; and (b) assuming that pensions are not vested, we will generally find that the least productive accept the contingent contract, the most productive screen when it is efficient to do so, and those in between screen when it is inefficient to do so. The reason that some screening is efficient is that the existence of uncertain job satisfaction implies a social return to screening which does not exist if compensation consists solely of pecuniary earnings. Clearly, two key assumptions in this paper involve the nature of job satisfaction and the extent of pension vesting. Previous analysis of screening and contingent contracts ignores the possibility that part of one’s compensation may be unknown ex ante. However, such an assumption is common in the labor economics literature. For example, Hall and Lazear (1984) explain the apparent sensitivity of quits and layoffs to demand by assuming that individuals and firms are initially uncertain about some aspect of the job. This assumption is used in the recent specific human capital literature6 to explain the sharing of investment costs and the existence of promotion ladders. Also, if firms take time to learn individual productivity, presumably there are aspects of the job about which individuals require time to learn, The assumption that pensions are not completely vested is also not unusual.’ In general, there would appear to be little incentive for firms to compensate those who quit. We assume that firms will compensate those who do not quit, but that they do not promise to compensate quitters. Presumably firms find it too costly to cheat continuing employees. However, these employees have no interest in those who have quit. In the absence of general screening, no one outside of one’s place of employment knows an individual’s productivity. This implies that any cost to a firm from cheating must involve its ability
to maintain and derive effort from its current workforce. A more complete analysis would consider the costs and benefits of pension vesting. However, this wouid lead us away from the main focus herein, which is that of efficiency when contingent contracts do not completely replace We could have assumed complete screening. pension vesting with some possibility of firm cheating. Either incomplete vesting or some possibility of default by firms would lead to our results. The rest of the paper proceeds as follows. In the next section we develop a two-period model with contingent contracts. In the third section we summarize our results and consider extensions of our analysis.
II. A MODEL WITH CONTINGENT
CONTRACTS
(A} Assumptions
We make the following assumptions of contingent contracts.
in our model
Assumption 1. The productivity of an individual is initially known only to that individual unless he has invested in a general productivity screen. The employer learns the individual’s productivity costlessly after one period of employment. This information is private and cannot be learned by other firms. One’s productivity is the same at all firms. Individuals work two periods and then retire. Assl&mpt~on2. A screening mechanism exists that is costly but that allows all potential employers to iearn an individual’s productivity. This screening occurs prior to employment and is a perfectly accurate test which costs the same for all individuals. The screen may be education, but the model does not require this to go through. Assumption 3. The firm is reliable in that it wili not defauit on wage promises to individuals. Assumption 4. A productivity-contingent claim is paid to an individual only if the individuai remains with the firm after one period. Thus the claim acts as a nonvested pension. For simplicity we assume that a retiree is treated like a current employee. Thus one who quits after one period ultimately receives x from his new employer - xL initially, and then x xl_ at retirement when productivity is learned.
Contingent
Contructs and Educational
Assumption 5. The job satisfaction of the worker is match specific and is unknown to the individual or the firm until the end of the first period of employment. At the beginning of one’s employment, the distribution of satisfaction is independent of both the individual and the firm.8 Job satisfaction is denoted by s, s - f(s), SE [sL. SH], and the expected value of s is zero, where braces enclose the arguments of a function. We assume that the distribution of s is continuous and symmetric so that:
\59r(s}ds = Flf(s}d.s = l/2.
(1)
6. Productivity is denoted by x and is continuously in the population.
Assumption
distributed
Assumption 7. Following Barzel (1977) we focus on the simplest kind of contingent contract. In the first period, an individual is paid a wage equal to the marginal value product of the lowest productivity individual in the labor force, xL. If the individual remains with the firm, he receives a wage in the second period equal to his productivity plus the difference between this productivity level and the wage in the first period.’ An alternative contingent contract allows an individual to announce his productivity, with the wage in both periods dependent on the individual’s announcement. In a model with no turnover, Spence (1981) demonstrates that such a contract can induce individuals to accurately reveal their productivity. Individuals with the lowest productivity receive a wage in the first period equal to their productivity. The higher one’s actual productivity, the lower is the first period wage. With this contract all but the least productive individuals receive a lower first period wage than in the contract considered herein, and they would have more to lose if they quit after one period. Hence, they would have more to gain from investing in screening. Also, with the first period wage declining with productivity, negative wages for the most productive in the first period might result. For these reasons, contingent contracts in which individuals announce their productivity do not appear to dominate the contract which we consider.“’
(B) The Individual’s Utility Without Screening Consider a model in which individuals work for two periods. Utility in each period is assumed to be equal to the sum of one’s wage and job satisfaction.
151
Screening
Without investing in screening, the individual’s (conditional expected) first period utility is xL. Discounting is ignored. After one period, the individual learns job satisfaction at this firm, so second-period utility at this firm is 2x - XL + s. The wage in the second-period equals productivity plus compensation for being “underpaid” in the first period. If the individual quits after one period, expected utility elsewhere equals n. The individual will receive a wage of XL initially, and then a payment of x - xL at “retirement”. Only one who quits with one period of work remaining is not compensated for being underpaid initially (at his original employer). A worker will quit only if utility will not be reduced, or if 2r-xL+s5x s 5 XL - x = s*.
(2)
Thus, with the cumulative density function of s denoted by F(s), the probability that an individual who has not invested in screening will quit after one period is F{s*}. Note that F{s*} is inversely related to x, so that the maximum value of s* is obtained when x = x1_. or s* = 0. The maximum value of F(s*} is l/2. At a high enough x, F(s*} = 0. This occurs when s* = SL. or -
XL x
=
XL
x -
-
SL,
SL
=
x,.
(3)
Note that lim x, = 00 SL-* - M. If sL is a large enough (in absolute value) negative number, then F(s*) > 0 for all x. We will analyze screening allowing for the possibility that .rC < 0~. Those with x z x, have F{s*} = 0: they have zero probability of quitting if they have not screened. If these individuals accept a contingent contract and do not screen, the wage loss from quitting after one period is large enough so that there is no possible level of job satisfaction low enough to induce them to quit. With a probability of 1 - F{s*} the individual will remain with his original employer with utility in the second period of 2r - xL + [l/(1 - F(s*})]if%f{sds}.
(4)
152
Economics
of Education
With a probability of F{s*} the individual will quit after period one and receive x. The expected utility from not screening, f_I,,,is first period utility plus the utility in either state (quit or stay) in the second period times the probability of being in that state. We have: u, = (2 - F{s*}).r f F{S*}XL -t J$%f(s}d.s.
1.3
Note that the firm earns profit on all those who quit after the first period except those with x = xL. Later in this section we will consider the effect on our results of imposing a zero profit constraint on firms. (C) The Individual’s Utility with Screening Let the cost of screening to any individual equal K. If an individual invests in screening, utility in the first period will be x - K. After one period, job satisfaction is discovered. The individual wilt quit only if: XfS”X, s 5 0.
(6)
Thus the probability of quitting is 112, and utility elsewhere is x. Utility of a stayer is x + 2!dH = sf(s}dr. The expected utility of one who does invest in screening, U,, is then iJd = 2x c psf{s}ds
- K.
(7)
D. The Returns to Screening and Effkiency An individual wifl invest in screening if Ud 2 ff,,, or if R, 2 K, where R, = Ud + K - U,, = the private return to screening. We have, with “E” the expected value operator, RP = 1112 - F@*y f- E{s\s’ 5 s 5 O}] + F(P) x - XL] z 0. (8) When x = xL, F{s*} = l/2, so = 0, and R, = 0. When x > xL, F{s*} < l/2, s* < 0, E{s(s* 5 s 5 0} < 0, and Rp > 0. When x z x,, F{s*} = 0, s* = sL, and Rp = 112 (E{sjsL 5 s I 0)) > 0. For x 2 xc, Rp is independent of X. With screening, an individual will quit when s 5 0; without screening, the individual will quit when s r s*. A quitter has conditional expected job satisfaction of zero. By screening, a stayer increases
Review
job satisfaction by E{sls* 5 s 5 0). The first term on the right-hand side of Equation (8) is the increased likelihood of quitting as the result of screening” times the net gain in job satisfaction from quitting. The second term on the right-hand side of Equation (8) represents the income gain from screening. Without screening there is a probability of F{s*} that an individual would quit after one period. By quitting the individua1 loses by not receiving the “pension” of x - xf_ at the original firm. Thus, by screening, the individual’s income is increased by the amount F{s*}[x - XL]. With multiple periods, each time an individual who has not invested in screening leaves an empioyer after one period he loses x - xr_. The social return to screening R,, is the difference between expected job satisfaction with and without screening. This is the first term on the right-hand side of Equation (8): R, = (112 - F{s*})
(-
E{sls*
1s s I 0)).
(9)
When x =r x,, R, = Rp. In the Appendix. we show that, for xL < x < x,, (aR,,/ax) > 0, (~*R,.~&~*) < 0, (8R,ldx) > 0, and (a*R,,iax*) >I< 0. Thus R, and R, will look as shown in Fig. 1. Now Rp - R, = F{s*}(x - xL) z 0. For those with minimum productivity. x,_. the private and social returns to screening are zero. For individuals with higher productivity, we find that the private return to screening exceeds the social return as long as F{s*} is positive. There is an externality only if there is a positive probability that an individual who does not screen would quit after one period. With no possibility of quitting if one has not invested in screening, the factors which affect the screening decision are job satisfaction and screening costs, which are social returns and costs of screening respectively. When F(s*} is positive. then the expected income gain from screening affects the screening decision, and this is a private return only. Thus the externality results. In the Spencian model of screening, there is no social return to screening so any screening is inefficient. Only individuals with relatively high productivity screen. Assuming that it pays someone to screen efficiently - K 5 12 [- E{SiSL c: s I O}] -we obtain different results. as illustrated in Fig. 1. Those with productivity XE[.X~. x’] do not screen since R, % K. Individuals with productivity XE[X’,x”] screen, and for them screening is inefficient since R,,
Contingent
Contracts and Educational
153
Screening
a
S
X (productivity) zer
Figure 1.
X (productivity)
/ < K. Those with x 2 x” screen efficiently since R, 2 K. Thus, even if screening has no effect on output, all screening is not inefficient. Some screening is efficient because there is a social gain from screening due to the increase in expected job satisfaction. In general, the least productive do not screen, the most productive screen efficiently, and those in between screen inefficiently. Again if F{s*} is positive for all x, which occurs if sL is small enough, all who screen do so inefficiently. (E) A Zero Profit Constraint So far we have not imposed a zero profit constraint on firms. Since some workers who do not screen will quit and were underpaid in the first period, then one’s original employer earns profit of x - xL on these individuals. Expected profit from an individual of productivity x is F{s*}(x - xL). Let the maximum value (with respect to x) of F{s*)(x - xL) be denoted by M. If competition for workers forces expected profit to zero, then workers who accept a contingent contract will be paid a signup bonus, B, equal to the expected value of F{s*}(x - x~). Thus R,, will be reduced by B. Since without the bonus Rp - R, = F{s*}(x - xL), and B < M, introducing the zero profit constraint shifts R,, down by B, but there remains a range of x for which Rp > R,. In fact B should equal the expected value of F{s*)(x - xL) over the range of x who choose the contingent contract in equilibrium. Thus B should be an increasing function of the average productivity of those who choose the contingent contract. Taking account of the dependence of B on the productivity
Figure 2.
of those who choose the contingent contract has no effect on the qualitative results illustrated in Fig. 2. Thus we treat B as if it were independent of the productivity of those who actually choose the contingent contract. The zero profit constraint affects our qualitative results only if K is particuiarly low or high as seen in Fig. 2. If K = K2, then we have the same result as we did without the zero profit constraint. However, if K = K[, then those from XL to a do not screen (and this is efficient), but those from a to b should screen (R, 2 K,) but do not since Rp < K1. If K = K3, no one screens, but those from e to the right would screen efficiently. Thus, with the zero profit constraint, if screening cost is low (high), there is too little screening by low (high) productivity individuals. Otherwise the zero profit constraint has no qualitative effect on our results. III. SUMMARY In the simple screening models, when output is independent of one’s initial job assignment and the screen does not directly affect output, then there is no social return to screening. By introducing the possibility of turnover via uncertain job satisfaction, we find a positive social return to screening for all but the least productive individuals. In general, there may be too much screening by some, those
154
Economics
of Education
neither at the top nor bottom of the productivity distribution, while the most productive screen efficiently. However, if screening cost is either particularly high or low, then there may be none who screen when it is inefficient to do so, but some may not screen when screening is efficient. Thus our results differ from the standard conclusions that the most productive screen and all screening is inefficient. We assumed that quitters are not compensated for being underpaid initially with contingent contracts. Such compensation would amount to a completely vested pension. Clearly, in our model, a state with completely vested pensions would dominate one with incomplete vesting. Since welfare would be increased with more complete vesting,12 a natural extension of our model would also consider the benefits of incomplete pension vesting. Such benefits include lower costs to employers from reduced turnover (Viscusi, 1985), and reduced shirking by workers (Lazear, 1979). Another extension of this paper would be a model in which there is the possibility of both individual and firm investment in screening. In this paper, firms costlessly learn productivity post-employment.
Review
In contrast, Waldman (1985) develops a model in which firms have greater access to capital markets, and, hence, have lower costs of investing in screening. In his model, with no worker turnover, efficient investment in screening occurs only when information is private, so the firm can reap the return from investment. Waldman suggests that with turnover there is a tradeoff: the more public information is, the lower investment in information will be, and the greater the utilization of this information (e.g. better matching of workers to firms). In this paper we illustrate the benefit of more public information in that there is an increased likelihood of individuals quitting and receiving higher job satisfaction. For this reason, there will be more investment in information by individuals as information is more public. A complete analysis of the efficiency of information investment, with investment possible by both individuals and firms, is left for the future. Acknowledgements - I am indebted to Donald Parsons, Dan Black, participants in Economics Department workshops at Appalachian State University and the University of Kentucky, and an anonymous referee for comments on earlier drafts of this paper. Remaining errors are my responsibility.
NOTES 1. I will use the words signalling and screening interchangeably throughout this paper. 2. In most screening models, when education or any screening device is a pure signal, it is necessary that the more able have a lower cost of obtaining the signal. (For an exception see Weiss (1983)). Such an assumption is not necessary if screening involves a test, since then the less able cannot mimic the more able by obtaining the same level of education. 3. In screening models which assume a pure signal, it was initially argued (e.g. Spence (1974)). that multiple equilibria would exist. Later, Riley (1975) questioned whether any equilibrium would survive experimentation by buyers. Subsequently-at least ihree possible solutions to the existence problem have been suggested. First Riley (1979) and Engers and Fernandez (1987) argue that if potential price-
searching- agents anticipate possible reactions by other agents. then otherwise profitable actions are _ . . not profitable and will not be undertaken. Thisbehaviorcan result in a unique-reactive equilibrium. Second, Riley (1985) shows that a Nash equilibrium is possible if an alternative sector exists in which individuals can find employment at a fixed wage, and if the marginal cost of screening rises sufficiently rapidly. Third, Weiss (1983) demonstrates that a Nash equilibrium exists if screening involves both years of schooling and a pass/fail test, and if the more able are more likely to pass the test. Other papers which consider the existence of equilibrium in signalling models are Cho and Kreps (1987) and Stiglitz and Weiss (1983). 4. See the discussion in note 3. 5. As Becker (1975, p. 6) has argued, ‘<. . surely a year on the job or a systematic interview applicant-testing program must be a cheaper and more effective way to screen”. 6. See Carmichael (1983), Hashimoto (1979, 1981), and Arnott and Stiglitz (1985). 7. Reasons for nonvested pensions include the reduction of turnover when it is costly to firms (Viscusi, 1985) and the reduction of worker shirking (Lazear, 1979). As will be demonstrated below, with complete vesting no one would screen and individuals would quit only when they obtained satisfaction less than the expected level. There is no way incomplete pension vesting can improve welfare in this model. The reason is that we do not consider any benefits of incomplete vesting. We examine the
Contingent
Contracts and Educational
Screening
efficiency of screening given that vesting is incomplete. An extension of this paper would consider the possible benefits of reduced vesting (mentioned above) along with the cost of less complete vesting (more resources devoted to screening). 8. For a similar assumption see Amott (1982) and Arnott and Stiglitz (1985). See Carmichael (1984) for an interesting paper which assumes that satisfaction differs at firms, and that incumbent workers can communicate their perceived satisfaction to potential employees. 9. Bane1 (1577) differs from us in that he assumes that the second period wage would be what we assumed minus the cost of screening by the individual. In his approach the gain from avoiding screening accrues to the employer. In our approach the gain accrues to the individual. We make this assumption in order to give individuals the greatest incentive not to screen. Also, competition for workers implies that they would receive any gain from avoiding screening. 10. Nalebuff and Scharfstein (1987) consider the case of two types of labor with firms offering two contracts, one with testing, and one without testing and paying a wage equal to the output of the low productivity types. Workers can be induced to identify themselves by the contract they choose. However, this result holds only if the wage paid to those who “fail” the test approaches negative infinity. 11. Our model implies that those who invest in screening are more likely to quit than those who do not screen. A referee has pointed out that job turnover appears to be negatively related to education. This would contradict the prediction of our model if education were undertaken only for screening purposes. Because education may serve both to screen and to increase individual productivity, a negative relation between education and turnover is not inconsistent with our theoretical results. 12. In case the reader is not clear why a state with complete vesting dominates any state with incomplete vesting in this model, consider the following. In the first period expected satisfaction is zero everywhere, so we focus on satisfaction in the second period. Let an individual who quits after one period receive a payment of a(x - x~) from his original employer. Thus a reflects the extent of vesting. In this paper, we assumed that o = 0. Complete vesting implies that cc = 1. Adding the term a(x - xL) to the right side of Equation (2) in the text, we have S* = (1 - a)@= - x). An individual who does not screen has a probability of 1 - F{s’) of staying with his original employer and receiving satisfaction of E(s(s* 5 s}. A quitter has a probability of F{s*} of receiving satisfaction of zero. Expected satisfaction is then /$sf{s}&= A;. When h = 1: si = 0, F(s*)= l/2, and expected satisfaction is \iHsf{s}ds = A*. Now A, - Al = -r.sf{s}ds 2 0. Thus complete vesting yields the highest level of expected job satisfaction and the lowest level of screening (zero).
REFERENCES ARNOTT,R. (1982) The structure of multi-period employment contracts with incomplete insurance markets. Can. /. Econ. 15, 51-76. ARNOT~,R. and Srm~rrz, J. (1985) Labor turnover, wage structures, and moral hazard: the inefficiency of competitive markets. /. Labor Econ. 3, 434-462. ARROW,K.J. (1973) Higher education as a filter. J. Public Econ. 2, 193-216. BARZEL.Y. 0977) Some fallacies in the internretation of information costs. J. Law Econ. 20.291-307. BECKER;G.S. (19%) Human Cupituf. 2nd edn. New York: Columbia University Press. CARMICHAEL, H.L. (1983) Firm-specific human capital and promotion ladders. Bell Journal 14.2.51-258. CARMICHAEL, H. (1584) Golden handcuffs: wage profiles with word or mouth reputations. Working paper, Hoover Institution. C&o; I.-K. and KREPS, D.M. Signaling games and stable equilibria. Q. J. Econ. 102, 179-221. ENGERS.M. and FERNANDEZ,L. (1987) Market equilibrium with hidden knowledge and self-selection. Econbmetrica 55, 425-439: ~ ’ HALL, R.E. and LAZEAR,E.P. (1984) The excess sensitivity of layoffs and quits to demand. 1. Labor Econ. 2, 233-257. HASHIMOTO,M. (1979) Bonus payments, on-the-job training and lifetime employment in Japan. 1. polit. Econ. 87, 1086-1104. HASHIMOTO, M. (1981) Firm-specific human capital as a shared investment. Am. Econ. Rev. 71, 475-482. LAZEAR,E.P. (1975) Why is there mandatory retirement? J. polit. Econ. 87, 1261-1284. NALEBUFF,B. and SCHARFSTEIN, D. (1987) Testing in models of asymmetric information. Rev. Econ. Studies 54, 265-277. RILEY,J.G. (1975) Competitive signalling. J. Econ. Theory 10, 174-186. RILEY,J.G. (1979) Informational equilibrium. Economefricu 47, 331-359. RILEY,J.G. (1985) Competition with hidden knowledge. 1. polif. Econ. 93, 958-976.
155
Economics
156
of Education Review
SPENCE,A.M. (1974) Marker Signaling. Cambridge: Harvard University Press. SPENCE,A.M. (1981) Signaling, screening and information. In Srudies in Lobor Markets (Edited by ROSEN, S.). Chicago: University of Chicago Press. Snc~trz, J.E. and WEISS, A. (1983) Sorting out the differences between screening and signaling models. Columbia University Department of Economics Discussion Paper No. 224. VISCUSI, W.K. (1985) The s&u&e of uncertainty and the use of nontransferable pensions as a mobility-reduction device. In Pensions, Labor, and Individual Choice (Edited by WISE, D.A.). Chicago: University of Chicago Press. WALDMAN,M. (1985) Information on worker ability: an analysis of investment within the firm. Department of Economics Working Paper No. 375, UCLA. WEISS, A. (1983) A sorting-cum-learning model of education. 1. polit. Econ. 91, 420-442.
APPENDIX In this Appendix we consider the shape of the Rp and R, functions with contingent contracts. Let z = E{s\s* 5 s 5 0) = ~~.sfls}dsl(l/2 - F(s*}). Differentiating z with respect to x g = f{s*)(s* Using Equation (Al), differentiate
Rp with respect to x (from Equation
2 = -f{s’}(s’ Since S* = xL - x, (JRdJ,) -f{s*) < 0 for x < xr. Similarly we have
- 2)/(1/2 - F(s*}).
- 2) - zf{P}
-f{s.}(x
(Al) (8)).
- XL) + F(s’}).
= F{s’} 2 0, where the inequality holds if x < x,. Thus (d2RddxZ) =
aRg_ - -s’f{s’) ax
2
0,
,
,< c$ =f{s*} + s* -ah*1 JS”
o
(A4)
0, and (J’RdJxa) > 0. As x increases, s* < 0, and, ifs is distributed normally, since S* is Forx=xL.s’= on the left side of the distribution of s, (Jf(s*}/Js*) > 0. Hence, at a large enough x, (a2RJ3x’) < 0. Thus R, will tend to look as drawn in Figs 1 and 2.
of Education
Review,
PrintedinGreatBritain.
Vol. 9.
Contingent
No. 2, pp. 149-1.56.
1990.
0?7?-7757Ml
s3.00
@ 1990Pergamon
Contracts and Educational
+ o.oll
Press
plc
Screening
TIMOTHY J. PERRI
Department
of Economics, Appalachian State University, Boone, NC 28608, U.S.A.
Abstract -This paper is concerned with a world where firms initially do not know workers’ productivity and workers do not know the level of match specific job satisfaction at any firm. We find that, even if firms will not default on contingent contracts, there is generally a positive private and social return to educational screening. Contingent contracts do not prevent some screening. In general, some screen efficiently and others screen when it is inefficient to do so. However, if screening cost is either very small or very large there may be too little screening.
I. INTRODUCTION THE PIONEERINGwork
of Arrow (1973) and Spence (1974) demonstrated that individuals might obtain socially excessive levels of schooling if education is used as a signal or screen.’ Arrow treated education as a test: at some cost, identical to all, individuals’ productivity would be revealed (with some error). Spence modeled education as a pure signal: the level of education reveals nothing directly about an individual’s productivity, but in equilibrium there is a positive relation between productivity (ability) and the level of education chosen.’ Most of the screening literature considers models with pure signals, and almost all of this work has focused on the problem of the existence of equilibrium.3 In contrast, this paper focuses on a test where the alternative to individuals revealing their productivity by screening prior to employment (taking the test) is a contingent contract in which individuals accept a relatively low wage until their productivity is discovered on the job. There are two reasons why we assume that screening involves a test and not a pure signal. First, we avoid the existence questions4 which arise in models with a pure signal. Thus we can focus on the important, but recently neglected, issue of screening efficiency. Second, one might think that education would directly reveal something about one’s ability. Grade completed, major field, grade point average, etc. would seem to be indices of ability from which prospective em-
ployers could infer productivity. With a pure signal, these indices only indirectly reveal productivity when, for example, the more able are induced to stay in school longer than the less able. Researchers recently have ignored the question of efficiency of screening, which was the main focus of the early screening literature. The reason apparently involves the difficulty of determining the extent to which firms will renege on promises to workers. If firms are unreliable and will cheat workers, then it appears that individuals may invest in screening which is of private but not social value. If firms are reliable, then Barzel (1977) and Spence (1981) both demonstrate that educational screening can be avoided by contingent contracts which pay relatively low wages until productivity is discovered and then compensate workers for being underpaid initially. This assumes that observing ability on the job is cheaper than educational screening.5 Previous work on educational screening suggests that: (a) contingent contracts will completely replace screening if firms do not cheat; and (b) if contingent contracts do not replace screening, then the more able (higher productivity) individuals will screen, and this screening is a social waste. What distinguishes this paper from the rest of the screening literature is that we assume that individuals are initially uncertain about some aspect of the job, call it satisfaction, at any employer. The introduction of uncertain job satisfaction leads to the possibility that some individuals will want to quit if they realize a low level of
[Manuscript received 20 January 1989; revision accepted for publication 19 October 1989.1
149
150
Economics
of Education Review
satisfaction. The possibility of turnover was not considered in the analysis of contingent contracts in Barzel (1977) and Spence (1981). If individuals are compensated for being underpaid initially, independent of whether they remain with the firm, then contingent contracts will induce individuals to bypass educational screening unless the problem of firms cheating or default is significant. Such compensation effectively constitutes a completely vested pension. What we will demonstrate is: (a) with uncertain job satisfaction and the possibility of turnover, contingent Contracts will not completely replace educational screening unless pensions are completely vested; and (b) assuming that pensions are not vested, we will generally find that the least productive accept the contingent contract, the most productive screen when it is efficient to do so, and those in between screen when it is inefficient to do so. The reason that some screening is efficient is that the existence of uncertain job satisfaction implies a social return to screening which does not exist if compensation consists solely of pecuniary earnings. Clearly, two key assumptions in this paper involve the nature of job satisfaction and the extent of pension vesting. Previous analysis of screening and contingent contracts ignores the possibility that part of one’s compensation may be unknown ex ante. However, such an assumption is common in the labor economics literature. For example, Hall and Lazear (1984) explain the apparent sensitivity of quits and layoffs to demand by assuming that individuals and firms are initially uncertain about some aspect of the job. This assumption is used in the recent specific human capital literature6 to explain the sharing of investment costs and the existence of promotion ladders. Also, if firms take time to learn individual productivity, presumably there are aspects of the job about which individuals require time to learn, The assumption that pensions are not completely vested is also not unusual.’ In general, there would appear to be little incentive for firms to compensate those who quit. We assume that firms will compensate those who do not quit, but that they do not promise to compensate quitters. Presumably firms find it too costly to cheat continuing employees. However, these employees have no interest in those who have quit. In the absence of general screening, no one outside of one’s place of employment knows an individual’s productivity. This implies that any cost to a firm from cheating must involve its ability
to maintain and derive effort from its current workforce. A more complete analysis would consider the costs and benefits of pension vesting. However, this wouid lead us away from the main focus herein, which is that of efficiency when contingent contracts do not completely replace We could have assumed complete screening. pension vesting with some possibility of firm cheating. Either incomplete vesting or some possibility of default by firms would lead to our results. The rest of the paper proceeds as follows. In the next section we develop a two-period model with contingent contracts. In the third section we summarize our results and consider extensions of our analysis.
II. A MODEL WITH CONTINGENT
CONTRACTS
(A} Assumptions
We make the following assumptions of contingent contracts.
in our model
Assumption 1. The productivity of an individual is initially known only to that individual unless he has invested in a general productivity screen. The employer learns the individual’s productivity costlessly after one period of employment. This information is private and cannot be learned by other firms. One’s productivity is the same at all firms. Individuals work two periods and then retire. Assl&mpt~on2. A screening mechanism exists that is costly but that allows all potential employers to iearn an individual’s productivity. This screening occurs prior to employment and is a perfectly accurate test which costs the same for all individuals. The screen may be education, but the model does not require this to go through. Assumption 3. The firm is reliable in that it wili not defauit on wage promises to individuals. Assumption 4. A productivity-contingent claim is paid to an individual only if the individuai remains with the firm after one period. Thus the claim acts as a nonvested pension. For simplicity we assume that a retiree is treated like a current employee. Thus one who quits after one period ultimately receives x from his new employer - xL initially, and then x xl_ at retirement when productivity is learned.
Contingent
Contructs and Educational
Assumption 5. The job satisfaction of the worker is match specific and is unknown to the individual or the firm until the end of the first period of employment. At the beginning of one’s employment, the distribution of satisfaction is independent of both the individual and the firm.8 Job satisfaction is denoted by s, s - f(s), SE [sL. SH], and the expected value of s is zero, where braces enclose the arguments of a function. We assume that the distribution of s is continuous and symmetric so that:
\59r(s}ds = Flf(s}d.s = l/2.
(1)
6. Productivity is denoted by x and is continuously in the population.
Assumption
distributed
Assumption 7. Following Barzel (1977) we focus on the simplest kind of contingent contract. In the first period, an individual is paid a wage equal to the marginal value product of the lowest productivity individual in the labor force, xL. If the individual remains with the firm, he receives a wage in the second period equal to his productivity plus the difference between this productivity level and the wage in the first period.’ An alternative contingent contract allows an individual to announce his productivity, with the wage in both periods dependent on the individual’s announcement. In a model with no turnover, Spence (1981) demonstrates that such a contract can induce individuals to accurately reveal their productivity. Individuals with the lowest productivity receive a wage in the first period equal to their productivity. The higher one’s actual productivity, the lower is the first period wage. With this contract all but the least productive individuals receive a lower first period wage than in the contract considered herein, and they would have more to lose if they quit after one period. Hence, they would have more to gain from investing in screening. Also, with the first period wage declining with productivity, negative wages for the most productive in the first period might result. For these reasons, contingent contracts in which individuals announce their productivity do not appear to dominate the contract which we consider.“’
(B) The Individual’s Utility Without Screening Consider a model in which individuals work for two periods. Utility in each period is assumed to be equal to the sum of one’s wage and job satisfaction.
151
Screening
Without investing in screening, the individual’s (conditional expected) first period utility is xL. Discounting is ignored. After one period, the individual learns job satisfaction at this firm, so second-period utility at this firm is 2x - XL + s. The wage in the second-period equals productivity plus compensation for being “underpaid” in the first period. If the individual quits after one period, expected utility elsewhere equals n. The individual will receive a wage of XL initially, and then a payment of x - xL at “retirement”. Only one who quits with one period of work remaining is not compensated for being underpaid initially (at his original employer). A worker will quit only if utility will not be reduced, or if 2r-xL+s5x s 5 XL - x = s*.
(2)
Thus, with the cumulative density function of s denoted by F(s), the probability that an individual who has not invested in screening will quit after one period is F{s*}. Note that F{s*} is inversely related to x, so that the maximum value of s* is obtained when x = x1_. or s* = 0. The maximum value of F(s*} is l/2. At a high enough x, F(s*} = 0. This occurs when s* = SL. or -
XL x
=
XL
x -
-
SL,
SL
=
x,.
(3)
Note that lim x, = 00 SL-* - M. If sL is a large enough (in absolute value) negative number, then F(s*) > 0 for all x. We will analyze screening allowing for the possibility that .rC < 0~. Those with x z x, have F{s*} = 0: they have zero probability of quitting if they have not screened. If these individuals accept a contingent contract and do not screen, the wage loss from quitting after one period is large enough so that there is no possible level of job satisfaction low enough to induce them to quit. With a probability of 1 - F{s*} the individual will remain with his original employer with utility in the second period of 2r - xL + [l/(1 - F(s*})]if%f{sds}.
(4)
152
Economics
of Education
With a probability of F{s*} the individual will quit after period one and receive x. The expected utility from not screening, f_I,,,is first period utility plus the utility in either state (quit or stay) in the second period times the probability of being in that state. We have: u, = (2 - F{s*}).r f F{S*}XL -t J$%f(s}d.s.
1.3
Note that the firm earns profit on all those who quit after the first period except those with x = xL. Later in this section we will consider the effect on our results of imposing a zero profit constraint on firms. (C) The Individual’s Utility with Screening Let the cost of screening to any individual equal K. If an individual invests in screening, utility in the first period will be x - K. After one period, job satisfaction is discovered. The individual wilt quit only if: XfS”X, s 5 0.
(6)
Thus the probability of quitting is 112, and utility elsewhere is x. Utility of a stayer is x + 2!dH = sf(s}dr. The expected utility of one who does invest in screening, U,, is then iJd = 2x c psf{s}ds
- K.
(7)
D. The Returns to Screening and Effkiency An individual wifl invest in screening if Ud 2 ff,,, or if R, 2 K, where R, = Ud + K - U,, = the private return to screening. We have, with “E” the expected value operator, RP = 1112 - F@*y f- E{s\s’ 5 s 5 O}] + F(P) x - XL] z 0. (8) When x = xL, F{s*} = l/2, so = 0, and R, = 0. When x > xL, F{s*} < l/2, s* < 0, E{s(s* 5 s 5 0} < 0, and Rp > 0. When x z x,, F{s*} = 0, s* = sL, and Rp = 112 (E{sjsL 5 s I 0)) > 0. For x 2 xc, Rp is independent of X. With screening, an individual will quit when s 5 0; without screening, the individual will quit when s r s*. A quitter has conditional expected job satisfaction of zero. By screening, a stayer increases
Review
job satisfaction by E{sls* 5 s 5 0). The first term on the right-hand side of Equation (8) is the increased likelihood of quitting as the result of screening” times the net gain in job satisfaction from quitting. The second term on the right-hand side of Equation (8) represents the income gain from screening. Without screening there is a probability of F{s*} that an individual would quit after one period. By quitting the individua1 loses by not receiving the “pension” of x - xf_ at the original firm. Thus, by screening, the individual’s income is increased by the amount F{s*}[x - XL]. With multiple periods, each time an individual who has not invested in screening leaves an empioyer after one period he loses x - xr_. The social return to screening R,, is the difference between expected job satisfaction with and without screening. This is the first term on the right-hand side of Equation (8): R, = (112 - F{s*})
(-
E{sls*
1s s I 0)).
(9)
When x =r x,, R, = Rp. In the Appendix. we show that, for xL < x < x,, (aR,,/ax) > 0, (~*R,.~&~*) < 0, (8R,ldx) > 0, and (a*R,,iax*) >I< 0. Thus R, and R, will look as shown in Fig. 1. Now Rp - R, = F{s*}(x - xL) z 0. For those with minimum productivity. x,_. the private and social returns to screening are zero. For individuals with higher productivity, we find that the private return to screening exceeds the social return as long as F{s*} is positive. There is an externality only if there is a positive probability that an individual who does not screen would quit after one period. With no possibility of quitting if one has not invested in screening, the factors which affect the screening decision are job satisfaction and screening costs, which are social returns and costs of screening respectively. When F(s*} is positive. then the expected income gain from screening affects the screening decision, and this is a private return only. Thus the externality results. In the Spencian model of screening, there is no social return to screening so any screening is inefficient. Only individuals with relatively high productivity screen. Assuming that it pays someone to screen efficiently - K 5 12 [- E{SiSL c: s I O}] -we obtain different results. as illustrated in Fig. 1. Those with productivity XE[.X~. x’] do not screen since R, % K. Individuals with productivity XE[X’,x”] screen, and for them screening is inefficient since R,,
Contingent
Contracts and Educational
153
Screening
a
S
X (productivity) zer
Figure 1.
X (productivity)
/ < K. Those with x 2 x” screen efficiently since R, 2 K. Thus, even if screening has no effect on output, all screening is not inefficient. Some screening is efficient because there is a social gain from screening due to the increase in expected job satisfaction. In general, the least productive do not screen, the most productive screen efficiently, and those in between screen inefficiently. Again if F{s*} is positive for all x, which occurs if sL is small enough, all who screen do so inefficiently. (E) A Zero Profit Constraint So far we have not imposed a zero profit constraint on firms. Since some workers who do not screen will quit and were underpaid in the first period, then one’s original employer earns profit of x - xL on these individuals. Expected profit from an individual of productivity x is F{s*}(x - xL). Let the maximum value (with respect to x) of F{s*)(x - xL) be denoted by M. If competition for workers forces expected profit to zero, then workers who accept a contingent contract will be paid a signup bonus, B, equal to the expected value of F{s*}(x - x~). Thus R,, will be reduced by B. Since without the bonus Rp - R, = F{s*}(x - xL), and B < M, introducing the zero profit constraint shifts R,, down by B, but there remains a range of x for which Rp > R,. In fact B should equal the expected value of F{s*)(x - xL) over the range of x who choose the contingent contract in equilibrium. Thus B should be an increasing function of the average productivity of those who choose the contingent contract. Taking account of the dependence of B on the productivity
Figure 2.
of those who choose the contingent contract has no effect on the qualitative results illustrated in Fig. 2. Thus we treat B as if it were independent of the productivity of those who actually choose the contingent contract. The zero profit constraint affects our qualitative results only if K is particuiarly low or high as seen in Fig. 2. If K = K2, then we have the same result as we did without the zero profit constraint. However, if K = K[, then those from XL to a do not screen (and this is efficient), but those from a to b should screen (R, 2 K,) but do not since Rp < K1. If K = K3, no one screens, but those from e to the right would screen efficiently. Thus, with the zero profit constraint, if screening cost is low (high), there is too little screening by low (high) productivity individuals. Otherwise the zero profit constraint has no qualitative effect on our results. III. SUMMARY In the simple screening models, when output is independent of one’s initial job assignment and the screen does not directly affect output, then there is no social return to screening. By introducing the possibility of turnover via uncertain job satisfaction, we find a positive social return to screening for all but the least productive individuals. In general, there may be too much screening by some, those
154
Economics
of Education
neither at the top nor bottom of the productivity distribution, while the most productive screen efficiently. However, if screening cost is either particularly high or low, then there may be none who screen when it is inefficient to do so, but some may not screen when screening is efficient. Thus our results differ from the standard conclusions that the most productive screen and all screening is inefficient. We assumed that quitters are not compensated for being underpaid initially with contingent contracts. Such compensation would amount to a completely vested pension. Clearly, in our model, a state with completely vested pensions would dominate one with incomplete vesting. Since welfare would be increased with more complete vesting,12 a natural extension of our model would also consider the benefits of incomplete pension vesting. Such benefits include lower costs to employers from reduced turnover (Viscusi, 1985), and reduced shirking by workers (Lazear, 1979). Another extension of this paper would be a model in which there is the possibility of both individual and firm investment in screening. In this paper, firms costlessly learn productivity post-employment.
Review
In contrast, Waldman (1985) develops a model in which firms have greater access to capital markets, and, hence, have lower costs of investing in screening. In his model, with no worker turnover, efficient investment in screening occurs only when information is private, so the firm can reap the return from investment. Waldman suggests that with turnover there is a tradeoff: the more public information is, the lower investment in information will be, and the greater the utilization of this information (e.g. better matching of workers to firms). In this paper we illustrate the benefit of more public information in that there is an increased likelihood of individuals quitting and receiving higher job satisfaction. For this reason, there will be more investment in information by individuals as information is more public. A complete analysis of the efficiency of information investment, with investment possible by both individuals and firms, is left for the future. Acknowledgements - I am indebted to Donald Parsons, Dan Black, participants in Economics Department workshops at Appalachian State University and the University of Kentucky, and an anonymous referee for comments on earlier drafts of this paper. Remaining errors are my responsibility.
NOTES 1. I will use the words signalling and screening interchangeably throughout this paper. 2. In most screening models, when education or any screening device is a pure signal, it is necessary that the more able have a lower cost of obtaining the signal. (For an exception see Weiss (1983)). Such an assumption is not necessary if screening involves a test, since then the less able cannot mimic the more able by obtaining the same level of education. 3. In screening models which assume a pure signal, it was initially argued (e.g. Spence (1974)). that multiple equilibria would exist. Later, Riley (1975) questioned whether any equilibrium would survive experimentation by buyers. Subsequently-at least ihree possible solutions to the existence problem have been suggested. First Riley (1979) and Engers and Fernandez (1987) argue that if potential price-
searching- agents anticipate possible reactions by other agents. then otherwise profitable actions are _ . . not profitable and will not be undertaken. Thisbehaviorcan result in a unique-reactive equilibrium. Second, Riley (1985) shows that a Nash equilibrium is possible if an alternative sector exists in which individuals can find employment at a fixed wage, and if the marginal cost of screening rises sufficiently rapidly. Third, Weiss (1983) demonstrates that a Nash equilibrium exists if screening involves both years of schooling and a pass/fail test, and if the more able are more likely to pass the test. Other papers which consider the existence of equilibrium in signalling models are Cho and Kreps (1987) and Stiglitz and Weiss (1983). 4. See the discussion in note 3. 5. As Becker (1975, p. 6) has argued, ‘<. . surely a year on the job or a systematic interview applicant-testing program must be a cheaper and more effective way to screen”. 6. See Carmichael (1983), Hashimoto (1979, 1981), and Arnott and Stiglitz (1985). 7. Reasons for nonvested pensions include the reduction of turnover when it is costly to firms (Viscusi, 1985) and the reduction of worker shirking (Lazear, 1979). As will be demonstrated below, with complete vesting no one would screen and individuals would quit only when they obtained satisfaction less than the expected level. There is no way incomplete pension vesting can improve welfare in this model. The reason is that we do not consider any benefits of incomplete vesting. We examine the
Contingent
Contracts and Educational
Screening
efficiency of screening given that vesting is incomplete. An extension of this paper would consider the possible benefits of reduced vesting (mentioned above) along with the cost of less complete vesting (more resources devoted to screening). 8. For a similar assumption see Amott (1982) and Arnott and Stiglitz (1985). See Carmichael (1984) for an interesting paper which assumes that satisfaction differs at firms, and that incumbent workers can communicate their perceived satisfaction to potential employees. 9. Bane1 (1577) differs from us in that he assumes that the second period wage would be what we assumed minus the cost of screening by the individual. In his approach the gain from avoiding screening accrues to the employer. In our approach the gain accrues to the individual. We make this assumption in order to give individuals the greatest incentive not to screen. Also, competition for workers implies that they would receive any gain from avoiding screening. 10. Nalebuff and Scharfstein (1987) consider the case of two types of labor with firms offering two contracts, one with testing, and one without testing and paying a wage equal to the output of the low productivity types. Workers can be induced to identify themselves by the contract they choose. However, this result holds only if the wage paid to those who “fail” the test approaches negative infinity. 11. Our model implies that those who invest in screening are more likely to quit than those who do not screen. A referee has pointed out that job turnover appears to be negatively related to education. This would contradict the prediction of our model if education were undertaken only for screening purposes. Because education may serve both to screen and to increase individual productivity, a negative relation between education and turnover is not inconsistent with our theoretical results. 12. In case the reader is not clear why a state with complete vesting dominates any state with incomplete vesting in this model, consider the following. In the first period expected satisfaction is zero everywhere, so we focus on satisfaction in the second period. Let an individual who quits after one period receive a payment of a(x - x~) from his original employer. Thus a reflects the extent of vesting. In this paper, we assumed that o = 0. Complete vesting implies that cc = 1. Adding the term a(x - xL) to the right side of Equation (2) in the text, we have S* = (1 - a)@= - x). An individual who does not screen has a probability of 1 - F{s’) of staying with his original employer and receiving satisfaction of E(s(s* 5 s}. A quitter has a probability of F{s*} of receiving satisfaction of zero. Expected satisfaction is then /$sf{s}&= A;. When h = 1: si = 0, F(s*)= l/2, and expected satisfaction is \iHsf{s}ds = A*. Now A, - Al = -r.sf{s}ds 2 0. Thus complete vesting yields the highest level of expected job satisfaction and the lowest level of screening (zero).
REFERENCES ARNOTT,R. (1982) The structure of multi-period employment contracts with incomplete insurance markets. Can. /. Econ. 15, 51-76. ARNOT~,R. and Srm~rrz, J. (1985) Labor turnover, wage structures, and moral hazard: the inefficiency of competitive markets. /. Labor Econ. 3, 434-462. ARROW,K.J. (1973) Higher education as a filter. J. Public Econ. 2, 193-216. BARZEL.Y. 0977) Some fallacies in the internretation of information costs. J. Law Econ. 20.291-307. BECKER;G.S. (19%) Human Cupituf. 2nd edn. New York: Columbia University Press. CARMICHAEL, H.L. (1983) Firm-specific human capital and promotion ladders. Bell Journal 14.2.51-258. CARMICHAEL, H. (1584) Golden handcuffs: wage profiles with word or mouth reputations. Working paper, Hoover Institution. C&o; I.-K. and KREPS, D.M. Signaling games and stable equilibria. Q. J. Econ. 102, 179-221. ENGERS.M. and FERNANDEZ,L. (1987) Market equilibrium with hidden knowledge and self-selection. Econbmetrica 55, 425-439: ~ ’ HALL, R.E. and LAZEAR,E.P. (1984) The excess sensitivity of layoffs and quits to demand. 1. Labor Econ. 2, 233-257. HASHIMOTO,M. (1979) Bonus payments, on-the-job training and lifetime employment in Japan. 1. polit. Econ. 87, 1086-1104. HASHIMOTO, M. (1981) Firm-specific human capital as a shared investment. Am. Econ. Rev. 71, 475-482. LAZEAR,E.P. (1975) Why is there mandatory retirement? J. polit. Econ. 87, 1261-1284. NALEBUFF,B. and SCHARFSTEIN, D. (1987) Testing in models of asymmetric information. Rev. Econ. Studies 54, 265-277. RILEY,J.G. (1975) Competitive signalling. J. Econ. Theory 10, 174-186. RILEY,J.G. (1979) Informational equilibrium. Economefricu 47, 331-359. RILEY,J.G. (1985) Competition with hidden knowledge. 1. polif. Econ. 93, 958-976.
155
Economics
156
of Education Review
SPENCE,A.M. (1974) Marker Signaling. Cambridge: Harvard University Press. SPENCE,A.M. (1981) Signaling, screening and information. In Srudies in Lobor Markets (Edited by ROSEN, S.). Chicago: University of Chicago Press. Snc~trz, J.E. and WEISS, A. (1983) Sorting out the differences between screening and signaling models. Columbia University Department of Economics Discussion Paper No. 224. VISCUSI, W.K. (1985) The s&u&e of uncertainty and the use of nontransferable pensions as a mobility-reduction device. In Pensions, Labor, and Individual Choice (Edited by WISE, D.A.). Chicago: University of Chicago Press. WALDMAN,M. (1985) Information on worker ability: an analysis of investment within the firm. Department of Economics Working Paper No. 375, UCLA. WEISS, A. (1983) A sorting-cum-learning model of education. 1. polit. Econ. 91, 420-442.
APPENDIX In this Appendix we consider the shape of the Rp and R, functions with contingent contracts. Let z = E{s\s* 5 s 5 0) = ~~.sfls}dsl(l/2 - F(s*}). Differentiating z with respect to x g = f{s*)(s* Using Equation (Al), differentiate
Rp with respect to x (from Equation
2 = -f{s’}(s’ Since S* = xL - x, (JRdJ,) -f{s*) < 0 for x < xr. Similarly we have
- 2)/(1/2 - F(s*}).
- 2) - zf{P}
-f{s.}(x
(Al) (8)).
- XL) + F(s’}).
= F{s’} 2 0, where the inequality holds if x < x,. Thus (d2RddxZ) =
aRg_ - -s’f{s’) ax
2
0,
,
,< c$ =f{s*} + s* -ah*1 JS”
o
(A4)
0, and (J’RdJxa) > 0. As x increases, s* < 0, and, ifs is distributed normally, since S* is Forx=xL.s’= on the left side of the distribution of s, (Jf(s*}/Js*) > 0. Hence, at a large enough x, (a2RJ3x’) < 0. Thus R, will tend to look as drawn in Figs 1 and 2.