Probation, layoffs, and wage-tenure profiles: A sorting explanation

Labour Economics 5 Ž1998. 359–383

Probation, layoffs, and wage-tenure profiles: A sorting explanation Ruqu Wang a

a,),1

, Andrew Weiss

b,1

Economics Department, Queen’s UniÕersity, Kingston, Ont., K7L 3N6, Canada b Economics Department, Boston UniÕersity, Boston, MA 02215, USA

Abstract In this paper, we demonstrate that sorting considerations alone generate steep wage-tenure profiles, high turnover rates of newly hired workers, an increase in the variance of wages with seniority, and mandatory retirement rules. We show that ‘excessive monitoring’ is sometimes necessary to deter applications from low productivity workers. We derive conditions under which firms randomly test workers, as well as conditions under which firms retain some workers that fail its test. Finally, we show that competition for workers can lower total output. q 1998 Elsevier Science B.V. All rights reserved. JEL classification: D82; J41; J63 Keywords: Probation; Monitoring; Sorting; Hiring

1. Introduction It is a commonplace observation that the ‘best’ workers on a job are generally more than twice as productive as the ‘worst’ workers. Naturally, firms are concerned about the quality of their work force and try to attract workers with higher productivity. Workers typically possess private information relevant to their )

C orresponding author. T el.:q 1-613-5452272; fax:q 1-613-5456668; e-m ail: [email protected]. 1 An earlier version of this paper was circulated under the title: ‘‘A sorting explanation of layoffs, retentions, and wage-tenure profiles.’’ We would like to thank Robert Rosenthal, Kevin Lang, Michael Riordan, Glenn MacDonald, and especially Kenneth Burdett and two anonymous referees for their very helpful comments. Of course, the usual caveats apply. 0927-5371r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 5 3 7 1 Ž 9 7 . 0 0 0 1 9 - 5

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own ability and productivity which are vital to firms. By monitoring the performance of a worker during an initial probation period, a firm can gain valuable information on his future productivity. Wages that are contingent upon such information can be used to improve a firm’s profitability. Since a high ability worker usually performs better than a low ability worker during the probation period, a firm can reward good performance while penalize bad performance to attract high ability workers and deter low ability workers. Papers analyzing this kind of sorting effect include Salop and Salop Ž1976.; Guasch and Weiss Ž1982.; Nalebuff and Scharfstein Ž1987.. In Salop and Salop, steep wage tenure profiles are used to deter applications from workers with high probabilities of quitting. In Guasch and Weiss, and Nalebuff and Scharfstein, tests, application fees and contingency wages are used to deter applications from workers with low expected productivity. Each of these papers offers an explanation for the variance of wages to increase with tenure. Questions remained such as why inexperienced workers are more likely to be fired, what determines the length of the probation period, and why low performance workers are sometimes retained by a firm. In this paper, we analyze a model in which firms evaluate their newly hired workers during a probationary period. This test is valuable because it provides information which can be used to improve job assignments. By letting wages and retentions depend on test results, firms can design contracts which are attractive to high ability workers but not appealing to low ability workers. This self-selection effect can increase firms’ productivity and profitability. We show that sorting considerations alone can explain the inverse relation between tenure and dismissal rates, the length of the probation period, retention of some Žbut not all. of low performance new hires as well as upward sloping wage-tenure profiles, the increase in the variance of wages with experience, and the use of mandatory retirement provisions in labor contracts. Each of these phenomena are generated by the desire of firms to reward successful workers, and to effectively assign workers to jobs. 2 Finally, we show that competition among firms can sometimes decrease aggregate output. This last result comes about because competition among firms can lead to increased dead-weight losses from expensive testing. In our model, every job starts with a probation period; however, not all workers are necessarily monitored. An employment contract includes an application fee, a probability that a worker will be tested, the length of the probation period, a wage

2

Topel Ž1990. and Kletzer Ž1989. provide convincing evidence that the return to seniority is large and significantly above what is found by Altonji and Shakotko Ž1987.; Abraham and Faber Ž1987.; Marshall and Zarkin Ž1987.; Topel Ž1986.. Lang Ž1987. had shown that, except for Marshall and Zarkin Ž1987., the latter estimates were estimates of the lower bound of the returns to tenure.

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during probation, wages contingent on performance measured during the probation period and a probability that a worker whose performance was low is fired. 3 We focus on the sorting features of contracts and ignore both learning on the job and incentive effects of contracts. These omitted factors are likely to have important effects on the design of contracts. The model we present here, therefore, should be viewed as an attempt to see the extent to which labor market phenomena can be explained solely by sorting considerations. We are not arguing that other factors are unimportant. We are especially interested in seeing whether a sorting model can explain empirical regularities, such as the importance of probationary periods in wage determination, which are most difficult to explain using a purely human capital theoretic approach.

2. The preliminaries of the model We shall investigate the equilibrium employment contracts for two different market structures: Ža. a competitive market with many identical firms so that test results are equally relevant to all firms; versus Žb. a monopsonistic labor market in which the test results for one firm are irrelevant to other firms. In our model, every newly hired worker starts his work life with a probationary period. During this period the worker’s performance may be Žsecretly. monitored. The monitoring process generates a passrfail result at the end of the probationary period. Those workers who were monitored are informed whether they pass or fail at the end of this period; only at that time, do the other workers learn that they were not monitored. We have chosen to emphasize this apprenticeship-type monitoring approach in which testing takes place over a nontrivial length of time for several reasons, even though our model is immediately applicable to instantaneous testing. First, there is strong evidence that much testing takes the form of observing the performance of workers on the job. 4 Second, models in which all testing takes place before hiring seem at odds with evidence that layoffs decline rapidly with tenure, that at least some promotions and dismissals are based on job performance, and that current wages are positively correlated with the length of the probationary period. That correlation ranges from 0.120 in the National Center for Research in Vocational Education survey to 0.245 in the survey of employers conducted by the Federal 3 Previous research in this area has treated all these decisions, except the length of the probation period, the choice of contingency wages and, in a restricted sense, the accuracy of the test, as exogenous. Exceptions include Nalebuff and Scharfstein Ž1987. who allow the test to approach perfect accuracy and Cho Ž1991. who endogenizes the accuracy of testing. 4 72% of the jobs surveyed by the National Center for Research in Vocational Education ŽNCRVE. had probationary periods, and 90% of the union contracts on file with Prentice-Hall in a 1980 survey contain a probation clause. See Groshen and Loh Ž1993.; Loh Ž1994a..

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Reserve Board of Cleveland. ŽSee Loh Ž1995.; Groshen and Loh Ž1993... 5 Third, previous sorting models with instantaneous tests assume that workers incur significant costs to be tested. Workers rarely have enough assets or access to credit to pay those costs. We maintain the following assumptions throughout: There are only two types of workers in the population: types 1 and 2; the productivity of a type 2 worker is a large negative number so that a firm only wishes to hire type 1 workers; all workers have the same work-life L, and the same savings S that can be used to pay for any application fee. ŽWe allow for S s 0, which is equivalent to not allowing application fees.. It takes m periods for a firm to obtain all productivityrelated information from a worker; if a probationary period is shorter than m periods no information is provided; a probationary period longer than m periods provides no additional information. The probability of a type i worker passing the test is denoted by pi , where p 1 ) p 2 . A worker knows his own type. A type i worker has a reservation wage of wi , where w1 G w 2 . Firms incur a cost H G 0 for each worker they hire. Workers are risk neutral. There is no discounting. The per period productivity of an untested type 1 worker is Q1 during and after the test. Those type 1 workers who are being tested have a per period productivity net of monitoring costs of Q1) , where Q1) - Q1. We assume that the total cost of monitoring is proportional to the length of the testing period. 6 ŽWe use the terms probationary period and testing period interchangeably.. The productivities of tested type 1 workers that pass or fail the test are Q1p and Q1f respectively, where Q1 - p 1 Q1p q Ž 1 y p 1 . Q1f ,

Ž 1.

and Q1f - Q1 - Q1p . Inequality ŽEq. Ž1.. captures the efficiency gain from training and from correctly assigning workers to jobs. We assume that Q1 ) w 1 , but do not restrict the sign of Q1f y w 1. A firm can randomly test workers. A worker knows his probability of being 5 See Gibbons and Katz Ž1991. for evidence that workers whose performance is low are more likely to be laid off for ‘economic’ reasons. The Longitudinal Employer–Employee Data ŽLEED. Žcf. Topel and Ward Ž1983.; MacDonald Ž1988., Table II. shows that most turnover occurs during the first quarter of employment. 6 Most previous work on labor contracts with tests have assumed that tests are arbitrarily short, and that there is a fixed cost of testing a worker. See for example, Burdett and Mortensen Ž1981.; Nalebuff and Scharfstein Ž1987.; Guasch and Weiss Ž1980.; Guasch and Weiss Ž1981.; Guasch and Weiss Ž1982.. The instantaneous test analyzed in those models can be considered as special cases of our more general testing technology. To analyze a short but costly test we can fix the length of the testing period, m, at an arbitrarily small number and choose Q1) such that mŽ Q1 y Q1) . represents the fixed cost of testing.

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tested, but not whether he has actually been chosen for the test. ŽA worker usually cannot distinguish whether or not he is being monitored.. Firms cannot fire untested workers. 7 However, workers who failed the test may be retained. The contract a firm offers can be described by Ž m, t, f, w, w p , w f ., where

m: t: f: w: w p: wf:

the actual length of the testing period; m G m, where m is the minimum length of time to reveal the productivity of a worker; the probability that a newly hired worker will be tested; the probability that a worker who fails the test will be fired; the post-testing-period wage of an untested worker; the post-testing-period wage of a worker that passes the test; the post-testing-period wage of a worker who failed the test but is retained.

We omit the wage during the probationary period from the contract space. This is because to deter type 2 workers from applying, it is always optimal for a firm to pay all workers the legal minimum wage, which we set equal to zero, during the testing period. 8 For the same reason, we do not include the application fees in the contract space: it is always optimal for a firm to charge a worker an application fee that is equal to the worker’s savings S. We assume that a firm can commit to its contract. However, workers can quit their job at any time. Specifically, workers can quit at the end of the probation period; they can then either be hired by other firms or get their reservation wages. How much they will get when they quit may not be included in the optimal contract; but it does affect the design of the optimal contract. Firms know all the population parameters, such as pi , wi , S, but not the type of any particular worker. If a worker is fired, he is fired at the end of the testing period. The rest of the paper is organized as follows. In Section 3, we analyze the case when there are many identical firms competing for type 1 workers. In Section 4, we characterize the optimal contract when test results are useful only to the testing firm. In Section 5, we compare the amount of testing and output in the above two cases. Section 6 contains a summary. The proofs of the lemma and the theorems are in Appendix A.

7 This restriction is consistent with U.S. court decisions on unjust dismissal Žsee Krueger Ž1989... Weiss and Wang Ž1990. analyzed the case in which firms could fire untested workers. 8 The proof is simple. In order to deter the type 2 workers from applying, it is better for the firm to pay every worker zero before he knows his status and to compensate him by increasing w p , the wage the firm pays to workers who pass the test. Since p1 ) p 2 , an increase in w p which exactly compensates type 1 workers for their lower wages during the testing period would make type 2 workers worse off. This would help the firms to separate type 1 workers from the type 2.

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3. Many identical firms — test results are public knowledge In this section, we assume that two or more identical firms are competing for type 1 workers. Each firm’s output is a concave function of the quantity of efficiency units of labor it employs. An initial employer chooses the most profitable contract which attracts type 1 workers while deterring type 2 workers from applying for jobs. It is obvious that a firm would not pay untested workers or failures more than their productivity during the post-testing period. Because other firms can observe the test results and thus know which workers were not tested and which workers failed the test, competition for those workers drives their wages up to their respective productivities. Firms are indifferent between keeping these workers or firing them. Without loss of generality, we assume that all failures are fired. Suppose that only type 1 workers apply for the jobs. Each worker must pay an application fee equal to the worker’s savings S. With probability 1 y t, a worker is untested; he receives zero in the test period and for the remaining L y m periods he receives Q1. With probability t, the worker is tested; again, he receives zero during the test and with probability p 1 he passes the test and receives w p for the remaining L y m periods; with probability 1 y p 1 , he fails the test and is fired. Since a firm has to incur a hiring cost of H, the total expected cost of hiring a type 1 worker is given by Ž1 y t .Ž L y m. Q1 q t Ž L y m. p 1w p y S q H. An untested type 1 worker produces Q1 every period. A tested type 1 worker produces Q1) Žnet of monitoring costs. during the probationary period and Q1p during the remaining L y m periods if he passes the test Žrecall that he is fired if he fails the test.. The total productivity of a type 1 worker is given by Ž1 y t . LQ1 q tmQ1) q t Ž L y m. p 1 Q1p. Therefore, the cost per efficiency unit of labor for a firm is: es

Ž 1 y t . Ž L y m . Q1 q t Ž L y m . p 1 w p y S q H . Ž 1 y t . LQ1 q tmQ1) q t Ž L y m . p1 Q1p

Ž 2.

The expected utility of a type 1 who applies to work for the firm is: U1 s Ž 1 y t . Ž L y m . Q1 q t Ž L y m . p 1w p q Ž 1 y p 1 . w1 y S,

Ž 3.

wi s max Q1f ,

where wi 4 is the post-testing-period earnings of type i workers when they are identified as type 1 failures Žand competition will drive up their wages to Q1f .. Similarly, the utility of a type 2 applicant is: U2 s Ž 1 y t . Ž L y m . Q1 q t Ž L y m . p 2 w p q Ž 1 y p 2 . w 2 y S.

Ž 4.

Note that both type 1 and type 2 workers who failed the test get an outside wage of Q1f , since types are not distinguishable and in equilibrium only type 1 workers apply. Of course, they can always quit and get their reservation wage w i . In order to attract type 1 workers and deter type 2 workers from applying for the job, a firm offers a contract that maximizes type 1’s utility subject to the firm’s

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zero-profit constraint and type 2’s non-participation constraint and the minimal test-length constraint: 9 max

U1

s.t.

e s 1,

Ž 5. Ž 6. Ž 7.

U2 F w 2 L, m G m.

We assume that the type 2 workers will not choose the contract if U2 s w 2 L. We shall refer to the resulting contract as the optimal contract. ŽIt is, of course, not necessarily socially optimal.. Solving for w p from Eqs. Ž2. and Ž5., we have wps

tmQ1) q Ž 1 y t . mQ1 q t Ž L y m . p 1 Q1p q S y H t Ž L y m . p1

.

Ž 8.

It is easy to see that w p is increasing in m. This is consistent with the evidence in Groshen and Loh that for workers who have jobs with probationary periods their current wages are positively correlated with the length of the probation period. For obvious reasons, w p is also increasing in Q1p and S, but decreasing in H. Substituting for w p in Eqs. Ž3. and Ž4., U1 s Ž 1 y t . LQ1 q tmQ1) q t Ž L y m . p 1 Q1p q Ž 1 y p 1 . w 1 y H , U2 s Ž 1 y t . Ž L y m . Q1 q t Ž L y m . Ž 1 y p 2 . w 2 y S p2 q tmQ1) q Ž 1 y t . mQ1 q t Ž L y m . p 1 Q1p q S y H . p1

Ž 9.

Ž 10 .

From Eq. Ž1.,

E U1 Em

s yt p 1 Q1p q Ž 1 y p 1 . w 1 y Q1) - 0,

and

E U2 Em

ž

s yŽ 1 y t . 1 y

p2 p1

/

Q1 y t

p2 p1

p 1 Q1p q

Ž 1 y p 2 . p1 p2

w 2 y Q1) - 0.

Ž 11 . 9 While we are restricting our analysis to cases where firms earn zero profits, these particular results hold even if firms earn positive profits. Under some conditions, firms could earn positive profits in equilibrium. This occurs when contracts are at boundaries so that any wage changes that increase the utilities of type 1 workers also increase the utilities of type 2 workers. Once type 2’s participation constraint is binding, firms cannot offer more utility to type 1 workers without attracting type 2 workers. Therefore, a positive profit equilibrium may result. Weiss and Wang Ž1990. present conditions under which competitive firms make positive profits.

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Eq. Ž11. holds because Ž1 y p 2 . p 1rp 2 ) 1 y p 1 and w 2 G Q1f ; therefore, the term in square brackets in Eq. Ž11. is positive: given a firm’s zero-profit constraint Eq. Ž5., an increase in m hurts both type 1 and type 2 workers. Therefore, given any t, a firm chooses the smallest m that deters type 2 workers from applying. Let R Ž m . s mQ1) q Ž L y m . p 1 Q1p q Ž 1 y p 1 . w 1 y LQ1 .

Ž 12 .

Thus RŽ m. is the difference in the expected productivity between tested and untested type 1 workers when the test is of length m. Using RŽ m., Eq. Ž9. can be rewritten as: U1 s LQ1 q tRŽ m. y H. Therefore,

E U1 Et

s R Ž m. .

Ž 13 .

Depending on m, Eq. Ž13. could be positive or negative. From Eq. Ž10., we have

E U2 Et

s Ž L y m . p 2 Q1p q Ž 1 y p 2 . w 2 y Q1 y

p2 p1

Ž Q1 y Q1) . m.

Ž 14 .

Given E U2rE m - 0, the sign of E U2rE t helps determine both the slope of U2 s w 2 L in Ž m, t . space and the trade-off between m and t. The following lemma provides a necessary and sufficient condition for the slope to be positive: Lemma: Given a firm’s zero-profit constraint Eq. Ž5., the slope of U2 s w 2 L is upward sloping in Ž t, m. space if and only if p 2 Q1p q Ž 1 y p 2 . w 2 y Q1 Q1 y Q1) G

p2 p1

Ž w2 L q S . y

LQ1 y Ž w 2 L q S . q

p2 p1

p2 p1

Ž SyH . .

Ž Q1 L q S y H . Ž 15 .

Proof: See Appendix A. The intuition behind the conditions determining the sign of E trE m along U2 s w 2 L stems from whether, at the margin, type 2 workers benefit from a higher probability of being tested. They will benefit if the wage of a type 2 worker who has been tested is large, relative to that of an untested worker. They also benefit if the per period cost of testing is small. These effects are captured by the first term in Eq. Ž15.. Note that in the special case in which p 2 Q1p q Ž1 y p 2 . w 2 y Q1 - 0 and E trE m < U2sw 2 L ) 0, U2 lies outside the feasible region of Ž t, m. space of w0, 1x = w0, L x. Thus in that case the sign of E trE m < U2sw 2 L is irrelevant since the participation constraint of type 2 workers is not binding.

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The marginal benefit from additional testing will also be large relative to its cost if the length of the testing period is short, since then the increased income to a tested type 2 worker occurs over more periods and the loss of output over fewer periods. The easier it is to deter applications from type 2 workers Žat any level of t . the shorter will be the testing period m. The factors that make it relatively easy to deter type 2 workers include large outside wealth which enables the firm to charge a large application fee; high outside income Ž w 2 L. which raises the per period opportunity cost of being tested; and low values of p 2rp1 which make it relatively unlikely that type 2 workers will succeed on the test. All these terms enter Eq. Ž15. with the predicted signs. Also when untested workers are relatively productive Ž Q1 large., type 2 workers, at the margin, are less likely to benefit from more testing, since replacing an untested worker with a tested one will have less effect on increasing the total surplus to be distributed via wages. Again, Q1 enters Eq. Ž15. with the predicted sign. Classifying situations by whether U2 s w 2 L is upward sloping or downward sloping simplifies the characterization of the optimal contract. How much testing is enough to deter type 2 workers from applying is the central issue. It is relatively easy to see that if testing increases type 1’s productivity, a firm wants more testing; if testing decreases type 1’s productivity, it wants less testing. Let t be the minimum t that satisfies U2 F w 2 L given m s m. Let m ˜ and m ) be the infimum of m that satisfy U2 F w 2 L when t s 1 and t ™ 0, respectively. Note that m must be at least m. Therefore, max  m ) , m4 is the minimum length for the test to deter type 2 workers when t ™ 0 while max  m, ˜ m4 is the minimum length when t s 1. In Theorem 1 below we show that when there is a dead-weight loss from testing, i.e. RŽ m. F 0, and U2 s w 2 L is upward sloping, firms will minimize the

Fig. 1. Feasible deterrence regions and optimal contracts.

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probability of testing a worker. Conversely, in the opposite case in which testing has productivity enhancing effects, i.e. RŽ m. ) 0, and U2 s w 2 L is downward sloping, then all workers will be tested. In the other cases where testing is productive and U2 s w 2 L is upward sloping, or testing is unproductive and U2 s w 2 L is downward sloping, the proportion of tested workers will generally be between zero and one. ŽSee Fig. 1.. Theorem 1: Suppose that U2 s w 2 L is upward sloping Ži.e Eq. Ž15. holds.. Ž1. If RŽmax  m ) , m4. F 0, then m s max  m ) , m4 , t ™ 0; Ž2. if RŽmax  m ) , m4. ) 0, then m g wmax  m ) , m4 , max  m, ˜ m4x, t g w t, 1x. Suppose that U2 s w 2 L is downward sloping Ži.e Eq. Ž15. does not hold.. Ž3. If RŽmax  m, ˜ m4. ) 0, then m s max  m, ˜ m4, t s 1; Ž4. if RŽmax  m, ˜ m4. F 0, then m g wmax m, m˜ 4, max  m, m ) 4x, t g Ž0, t x. Proof: See Appendix A. The following example shows that for plausible parameter values, it is optimal for the firm to set the length of the testing period longer than m — the length that is required to become informed about the productivity of a worker. Example 1. Let m ™ 0 and H s 0; i.e., it is an instantaneous test and there is no hiring cost. Let Q1 s w 2 and S s 0, so that as m ™ 0 and t ™ 0 type 2 workers are deterred from applying. Let Q1p be sufficiently high that Eq. Ž15. is satisfied. It is easy to see that U2 s w 2 L is upward sloping. Since U1 s LQ1 q tRŽ m., as t ™ 0, U1 ™ LQ1. ŽFrom Eq. Ž12., RŽ0. ) 0.. A small increase in t requires a small increase in m to deter type 2 workers, but RŽ m. remains positive for small m. Therefore, U1 is increased. Thus, in the optimal contract, m ) m. This result is quite intuitive. Testing in our example increases the utility of both types. To deter type 2 workers from applying, increasing the probability of workers being tested makes it necessary to increase the length of the testing period. The efficiency loss from a longer test at m s m is outweighed by the gain from testing a higher proportion of workers, so m will be set greater than m.

4. Idiosyncratic firms r monopsony — test results are private information In this section, we consider the optimization problem of a representative firm offering an idiosyncratic test: the firm’s test results are uninformative to other firms. Alternatively, we could describe the firm in this section as a monopsonist in the labor market. Both are equally valid interpretations of the model. We assume that there is an excess supply of type 1 workers when the firm offers a contract that maximizes its profits subject to attracting type 1 workers and not attracting type 2. Since there is excessive supply of type 1 workers, those who are not tested

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or who failed the test, but are retained by the firm, are paid a post-testing-period wage w1 , a type 1’s reservation wage. This is because paying them more is a waste of money, and paying less is equivalent to firing them. Note that in this model a firm can make positive profits. Our objectives in this section are to characterize the optimal contract, to derive conditions under which some workers who fail the test are retained by the firm, and to show that the profit maximizing contract may entail probationary periods that exceed m. We also show that jobs with longer probationary periods will pay higher wages. The result that some failures may be retained by the firm is in contrast to the usual assumption in the sorting literature that workers who fail the test are fired. The reason that workers who fail may be retained is that if testing is costly it may not be cost effective to replace a failure with a new hire. Similarly to Section 3, the firm’s cost per efficiency unit of labor and the workers’ utility function can be written as: Ž Ly m . w tp1w p q t Ž 1y p1 . Ž 1y f . w1 q Ž 1y t . w1 x y Sq H ) m w tQ1 q Ž 1y t . Q1 x q Ž Ly m . tp1 Q1p q t Ž 1y p1 . Ž 1y f . Q1f q Ž 1y t . Q1

,

Ž 16 .

U1 s t Ž L y m . p 1w p q Ž 1 y p 1 . w 1 q Ž 1 y t . Ž L y m . w 1 y S,

Ž 17 .

es

w

x

U2 s t Ž L y m . p 2 w p q Ž 1 y p 2 . fw 2 q Ž 1 y p 2 . Ž 1 y f . w1 q Ž 1 y t . Ž L y m . w 1 y S.

Ž 18 .

We maintain the previous assumption that an untested worker can never be fired. We again assume that the firm’s output is a concave function of the quantity of efficiency units of labor it employs. It is a standard result that maximizing profit is equivalent to minimizing Eq. Ž16. subject to workers’ incentive constraints if the labor supply constraint is not binding and if the firm’s production function is concave Žsee Weiss Ž1980... That is, min s.t.

e U1 G w 1 L,

Ž 19 .

U2 F w 2 L,

Ž 20 .

m G m.

Ž 21 .

Again, we assume that the type 2 workers choose the contract if and only if U2 ) w 2 L, and we refer to the resulting contract as the optimal contract. In the optimal contract, Eq. Ž19. is binding. 10 From U1 s w 1 L, we have wps

10

w1 L q S y Ž 1 y t . Ž L y m . w1 y t Ž L y m . Ž 1 y p1 . w1 t Ž L y m . p1

.

Ž 22 .

If a firm offers a contract that satisfies constraints Eqs. Ž19. and Ž20., and Eq. Ž19. is not binding, then the firm can lower w p without violating the constraints and thereby lower e.

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Note that w p is increasing in m and decreasing in t. Substituting Eq. Ž22. into Eqs. Ž16. and Ž18., we have es

H q w1 Ly tf Ž 1y p1 . w1Ž Ly m . tmQ1) q Ž 1y t . mQ1 q Ž Ly m . w tp1 Q1p q t Ž 1y p1 . Ž 1y f . Q1f q Ž 1y t . Q1 x

U2 s Ž L y m . w1 q

p2 p1

,

Ž 23 .

Ž w1 m q S . y tf Ž w1 y w 2 . Ž L y m . Ž 1 y p 2 . y S. Ž 24 .

Note that in contrast to the case with identical firms E U2rE t - 0 and E U2rE f 0. The firm chooses m, f, t to minimize Eq. Ž23., given that the above U2 is no greater than w 2 L. From Eq. Ž23., if p2 Ž L y m . w1 q Ž w1 m q S . y S F w 2 L Ž 25 . p1 holds for a given m G m, then any positive f and t deter the type 2 workers from applying. When Eq. Ž25. holds, a type 2 worker is very easy to deter: Eq. Ž25. holds if S or w 2 is large, or p 2 is small. If Eq. Ž25. does not hold, then t and f must be large enough to deter the type 2 workers. In this case, df

f < U2sw 2 L s y . dt t

Any Ž f, t . combination in the shaded area in Fig. 2 deters type 2 workers from applying: for some parameter values there are no values of f, t and m that deter type 2 from applying. We now characterize the firm’s optimal contracts. Define g Ž m . s Ž L y m . p 1 Q1p q Ž 1 y p 1 . Q1f y Q1 q m Ž Q1) y Q1 . .

Fig. 2. Feasible region for deterring type 2 workers given m.

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Thus g Ž m. is the productivity enhancing effect on the labor input of the firm from testing a previously untested type 1 worker, when workers that fail the test are retained and the test is of length m. We first assume that the length of the probationary period is fixed at m s m. Define the following ratios as the expected cost per additional efficiency unit of labor from: Ži. retaining one more worker who fails the firm’s test; Žii. hiring one more worker who is not going to be tested, and Žiii. hiring one more worker who is only retained if he passes the test:

d fs

w1 Q1f

,

d us

H q w1 L Q1 L

,

d ps

H q Ž L y m . p1 q m w1

Ž L y m . p1 Q1p q mQ1)

.

We shall refer Žloosely. to d f , d u and d p respectively as the cost per efficiency unit of labor from increasing the number of ‘failures’, ‘untested’, and ‘successful’ workers employed by the firm. The values of t and f in the optimal contract depend on the ordering of these numbers and the sign of g Ž m.. In particular, all workers who fail the test are fired Ž f s 1., unless keeping the workers who fail the test is ‘cheaper’ than hiring a new worker; i.e., unless d f - min  d u , d p 4 . But even if d f - min  d u , d p 4 , it is possible for f s 1, if this is needed to deter type 2. We also find that not all workers are tested Ž t - 1. if either Ža. the untested have the lowest cost per effective unit of labor, or Žb. the failures have the lowest cost per effective unit of labor and testing is wasteful, g Ž m. - 0. Let f ) be the minimum f needed to deter type 2 when t s 1 and let t ) be the minimum t needed to deter type 2 when f s 1. Theorem 2 characterizes f and t in the optimal contract given a fixed pre-determined length of the apprenticeship program. Because Ž3c. describes the important case where some workers who fail the test are retained, that case is presented in boldface. Theorem 2: 1. If d p F min  d f , d u4 , then f s 1, t s 1; 2. If d u F min  d p , d f 4 , then f s 1, t s t ) ; 3. If d f F min { d p , d u }, and Ž3a. g Ž m. F 0, and Eq. Ž25. does not hold, then f s 1, t s t ) ; Ž3b. g Ž m. F 0, and Eq. Ž25. holds, then f s 0, t ™ 0; (3c) g ( m ) ) 0, then f s f ) , t s 1. In all cases, w p is determined by Eq. Ž22.. Proof: See Appendix A. The intuition for these results is as follows. In Ž1., the firm’s cost per efficiency unit of labor is lowest if it tests all workers and only retains those who pass the test. Since this policy also deters type 2 the firm will do it.

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One way of interpreting Ž2., Ž3a. and Ž3b. is to note that in each of those cases either g Ž m. F 0 or d p G d u, so the firm tries to minimize the proportion of workers that it tests. If the cost per efficiency unit of labor from retaining workers who fail the test is greater than that of hiring a worker who will not be tested, or if increasing f helps minimize t, then the firm will fire all the workers who fail the test; i.e., set f s 1 as in cases Ž2. and Ž3a.. Otherwise the firm chooses f s 0 as in Ž3b.. Note that Eq. Ž25. implies that type 2 workers are always deterred along U1 s w 1 L. The most interesting of these results is Ž3c., since in that case all workers are tested but some workers who fail the test are retained while others are fired. In Ž3c. testing is efficient: g Ž m. ) 0, and retaining workers that fail the test lowers the firm’s cost per efficiency unit of labor. Consequently the firm tests all its workers and minimizes the proportion of workers that fail its test who are fired, while still deterring applications from type 2; i.e., sets t s 1 and f s f ) . This conclusion is somewhat different from those in the existing literature where firing all failures is usually assumed. Here, some of the failures are retained, because keeping them is cheaper than testing a new worker and incurring the hiring costs. 11 When the hiring cost is zero, however, this can never happen: Corollary: When the hiring cost is zero, all workers that fail the test are fired. Proof: H s 0 implies that d u - d f . The only possible situations are Ž1. and Ž2. in Theorem 2, which implies that f s 1. I Because hiring is costless and untested workers are paid the same wage as workers that fail the test Ž w1 ., firing failures and replacing them with untested workers costlessly improves the average productivity of the firm and helps the firm deter applications from type 2 workers. The previous analysis is based on the assumption that m is fixed. We should now characterize the optimal contract when the firm can choose the length of the test m Žwhich should not be shorter than m.. Unfortunately, we were not successful in doing so because there are too many choice variables. Nevertheless, when H s 0, f s 1 must be true in the optimal contract, because it is true for any fixed value of m. This reduces one choice variable and helps us analyze the optimal contract where m is a choice variable of the firm. Similar to Section 3, we find that holding U1 s w 1 L fixed, U2 s w 2 L is downward sloping in Ž t, m. space if and only if Ž w 2 L q S .rŽ w 1 L q S . ) p 2rp1. 11 There could be other reasons for a firm to keep its failure workers. For example, unsuccessful workers can be reassigned to other positions in the firm, as in Demougin and Siow Ž1991a.; Demougin and Siow Ž1991b..

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In that case, type 2 workers are easy to deter because of their relatively high reservation wage and relatively low probability of passing the test. As in the previous section, this classification is helpful in characterizing the optimal contract. Let m ˜ and m ) be respectively the minimum length of test to deter a type 2 worker when t s 1 and t ™ 0. Of course, the actual length of the test must not be shorter than m. Define G Ž m . s Ž L y m . p 1 Ž Q1p y Q1 . q m Ž Q1) y Q1 . . Thus GŽ m. is the productivity enhancing effect on the labor input of the firm from testing a previously untested type 1 worker, if workers that fail the test are fired and replaced by untested workers for the rest of their work lives, and when the test is of length m. Note that the previously defined g Ž m. is the productivity enhancing effect of increasing the proportion of tested workers when the firm keeps the failures, and is therefore less than GŽ m.. The following theorem characterizes the firm’s optimal contract: Theorem 3: Suppose that H s 0. If (w2 L q S) r (w1 L q S) F p2 r p1 , then m s m and

ts

½

1,

if G Ž m . G 0,

t,

if G Ž m . - 0

If (w2 L q S) r (w1 L q S) ) p2 r p1 , then: (1) if G(max {m, m ˜ }) G 0, m s max {m, } ( ) ( { }) m ˜ and t s 1; 2 if G max m, m˜ - 0, t ™ 0 and m s max {m, m ) }. Proof: See Appendix A. To understand these results let us first look at the determination of t. If testing is productive Žat the relevant level of m. everyone is tested. If testing is unproductive then the minimum level of testing is done that will still deter type 2 workers. The length of the testing period is determined by the ease with which type 2 workers can be deterred from applying. Turning now to the determination of m. If Ž w 2 L q S .rŽ w 1 L q S . - p 2rp1 , it is relatively easy to deter applications from type 2 workers and m is set at its technological minimum m. If Ž w 2 L q S .rŽ w 1 L q S . ) p 2rp1 , it may be necessary to choose m ) m in order to deter type 2 workers. As in Section 3, the testing period may be longer than is necessary to screen applicants, i.e., the testing period may be chosen for its selection effects Žcf. Ž2. in Theorem 3.. Since all workers receive w ) s 0 during the testing period, lengthening the testing period increases the cost of applying to the firm and thus deters applications from the type 2 workers.

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5. Potential loss in output from competition The total output in our models could be lower than the socially optimal level due to the information asymmetry between firms and workers. Consequently, firms may have to design socially inefficient schemes to deter applications from low productivity workers. Furthermore, the total output can be lower in the competitive setting than with a monopsonist. We illustrate this possibility through a simple example; the result holds for a range of fairly plausible parameters. The intuition is that in a competitive environment firms may have little scope for punishing failures since even if those workers are fired they are paid Q1f by other firms. Thus if Q1f is large relative to w 2 firms in a competitive environment will generally need to impose a long testing period in order to deter applications from type 2 workers. If testing is costly the long probationary period will impose large dead-weight losses that lower total output. In a monopsonistic setting these extra costs do not occur since a fired worker is paid his reservation wage irrespective of Q1f . If there are no hiring costs we know from the corollary that all failures are fired. Consequently, if w 2 is low relative to Q1f , it is much easier to deter applications from type 2 workers in the monopsonistic setting and the dead-weight loss from either excessive testing or an excessively long monitoring period will be significantly smaller in the monopsonistic setting. These points generate the result in example 2. In that example, the length of the monitoring period is strictly longer in the competitive setting than in the monopsonistic setting. This longer monitoring period causes output to be strictly lower in the competitive labor market. Example 2 Let Q1 s 8, Q1p s 20, Q1f s 4, Q1) s 0, p 1 s 0.5, p 2 s 0, L s 1, w 1 s 2, w 2 s 1, S s 0, H s 0, and m s e , where e is arbitrarily small. Because there is no hiring cost, all failures are fired in both settings. A type 1 worker’s output in the monopsony case can be expressed as

Ž 1 y t . LQ1 q tmQ1) q t Ž L y m . p1 Q1p q Ž 1 y p1 . w1 . A type 1 worker’s output in the competitive case can be expressed as

Ž 1 y t . LQ1 q tmQ1) q t Ž L y m . p1 Q1p q Ž 1 y p1 . w1 . Output under monopsony is Žalmost. 11 since instantaneous testing of workers deters type 2 workers. ŽSee Ž1. in Theorem 3, where m ˜ s 0 in this example.. The monopsonist tests all workers, but the testing period is arbitrarily short. Output in the competitive environment is approximately 8, since almost no workers are tested, but the monitoring period lasts 78 of the worker’s total length of employment. ŽSee Ž4. in Theorem 1, where m ) s 78 and m ˜ s 34 . Calculations show that ) . m s m , t f 0 is optimal. The robustness of this example can be seen by letting Q1 vary between 8 and 10 while holding all other parameter values fixed. In all

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those cases the output per type 1 worker in the competitive environment is approximately Q1 while output per type 1 worker under monopsony stays fixed at 11, which is a strictly larger number. Of course, if we increase Q1 to above 11 in this example, the opposite holds; that is, the output in the competitive environment is higher than in the monopsony environment.

6. Summary We have characterized the optimal labor contracts offered by firms when workers have information about their own productivity that is not available to firms, and firms can administer passrfail tests to workers. These tests occur during a probationary period and may be interpreted as performance evaluations. In order to discourage applications from workers with bad private information about their own productivity, firms initially pay workers less than their expected productivity and later pay workers that pass their tests more than their expected productivity. These results are consistent with steep wage tenure profiles, and the use of mandatory retirement rules. The reason for mandatory retirement in this model is similar to that in Lazear Ž1979. and Akerlof and Katz Ž1989.: because senior workers are paid more than the value of their output, they would not quit when the value of their output falls below the opportunity cost of their time. Mandatory retirement or wage cuts around the time of retirement are efficient, and both can increase the total return from the employment relationship. Another implication from our analysis is that an increase in the length of the test increases the wages for the successful workers, and thus also increases the growth rates of their wages. Thus our analysis can explain findings by Barron et al. Ž1989. who estimated that a 10% increase in on-the-job training Žmeasured by its length. raised wage growth by 1.5%, and by Loh Ž1994a. who finds that expected returns to on-the-job training only occur in firms with probationary periods. 12 Another result of our analysis is that because monitoring reduces productivity during the probation period, a firm may randomly monitor workers to save costs. Firms are still able to select good workers by the judicious choice of wage-tenure profile and the length of the monitoring period. When firms are idiosyncratic, there are conditions under which some low performance workers are retained after being tested. For workers that fail the test to be retained two conditions must be satisfied: Ži. the firm’s cost per efficiency unit of labor from retaining those workers must be less than its cost per efficiency 12 Loh and Groshen Ž1992. find that the effect of a probationary period on wage growth is more than 6 times the total effect of training on wage growth when both variables are included as determinants of wage growth.

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unit of labor from replacing them with new hires and Žii. the productivity enhancing effects of testing a type 1 worker must be positive. It seems likely that these conditions would be satisfied in a significant portion of jobs. Finally, the analysis in this paper is consistent with the widespread use of probationary periods, and with the low starting wages and rapid wage growth associated with jobs that have probationary periods. Even when researchers control for all human capital variables including on-the-job training, previous experience and education, jobs with probationary periods have exceptionally low starting wages and exceptionally high increases in wages. The finding of high wage growth persists even after controlling for productivity growth. 13 These empirical regularities seem easiest to explain if employment contracts are being used to induce self selection: results in Loh Ž1994b. provide convincing evidence for this explanation.

Appendix A. Proofs of the lemma and theorems 1–3 in the text Proof of the Lemma: From Eq. Ž10., U2 s w 2 L can be expressed as a function of t: ts

N0 q mN1 D 0 q mD1

,

Ž A.1 .

where

ž

N0 s L Ž Q1 y w 2 . y 1 y

p2 p1

/

Sy

p2 p1

ž

H,

N1 s y 1 y

p2 p1

/

Q1 ,

D 0 s yL Ž p 2 Q1p q Ž 1 y p 2 . w 2 y Q1 . , D 1 s p 2 Q1p q Ž 1 y p 2 . w 2 y

p2 p1

ž

Q1) y 1 y

p2 p1

/

Q1 .

t is always a monotone Žincreasing or decreasing. function in Ž m, t . space, It is increasing in m if and only if D 0 N1 G D 1 N0 , which can be re-arranged as p 2 Q1p q Ž 1 y p 2 . w 2 y Q1 G LQ1 y Ž w 2 L q S . q

Ž w2 L q S . y p2 p1

Ž SyH .

p2 p1

p2 p1

I

13

The ‘t’-statistic is equal to 3.16 in Loh and Groshen Ž1992..

Ž Q1 L q S y H .

Ž Q1 y Q1) . .

Ž A.2 .

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Proof of Theorem 1: When U2 s w 2 L is upward sloping, that is, max  m ) , m4 is the minimum value of m that is feasible in deterring applications from type 2 workers. If for this m, RŽ m. F 0, then t ™ 0 is optimal. If for this m, RŽ m. ) 0, then Ž m, t . lies on U2 between max  m ) , m4 and max m, m ˜ 4, and the corresponding t lies between t and 1. When U2 s w 2 L is downward sloping, max  m, ˜ m4 is the minimum m that is feasible in deterring applications from type 2 workers. If for this m, RŽ m. G 0, then t s 1 at this m will be optimal. If for this m, RŽ m. - 0, then since RŽ m. is decreasing in m, RŽ m. - 0 for all m along U2 s w 2 L. The optimal m will lie on the boundary between max  m, m ˜ 4 and L and the corresponding t is between w t ) , ) t x, where t is the value t approaches along U2 s w 2 L as m ™ L. I Proof of Theorem 2: Denoting the denominator in Eq. Ž23. as D, we have

Ee Ef

1 sy

Ee Et

D2

1 s

D2

t Ž L y m . Ž 1 y p 1 .  tw1 g Ž m . y Q1f Ž H q w 1 L . y w 1 LQ1

4,

 f Ž L y m . Ž 1 y p1 .

= Q1f Ž H q w 1 L . y w 1 LQ1 y g Ž m . w H q w 1 L x 4 .

Ž A.3 .

If Eq. Ž25. holds, the feasible set of contracts that deter applications from type 2 workers is the whole set w0, 1x = w0, 1x in Ž t, f . space. However, if Eq. Ž25. does not hold, the restriction imposed by U2 F w 2 L is binding. Therefore, we need to evaluate

Ee

de s dt

U 2 sw 2 L

Ee d f q

Et 1

s

D2



E f dt

U 2 sw 2 L

Q1f Ž H q w 1 L . y w 1 LQ1 Ž L y m . Ž 1 y p 1 . f

yg Ž m . w H q w 1 L x 4 y

1 D2

t Ž L y m . Ž 1 y p 1 .  tw1 g Ž m .

y Q1f Ž H q w 1 L . y w 1 LQ1 1 sy

D2

4 Ž yfrt .

g Ž m .  Ž H q w 1 L . y Ž L y m . Ž 1 y p 1 . tfw1 4 .

Ž A.4 .

The term in braces in Eq. ŽA.4. is always positive, so the sign of d erdt along the participation constraint of type 2 workers depends on the sign of g Ž m. only.

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There are two cases: Case I w1 g Ž m . G Q1f Ž H q w1 L . y w1 LQ1

Ž i.e. d p F d f .

Ž A.5 .

In this case, two situations can occur: Case I.1

Ž L y m . Ž 1 y p1 . Q1f Ž H q w1 L . y w1 LQ1 y g Ž m . w H q w1 L x F0 = Ž i.e. d p F d u . f

Ž A.6 .

u

Under these conditions, if d F d , that is, Q1f Ž H q w 1 L . y w 1 Q1 L G 0,

Ž A.7 .

then from Eqs. ŽA.5. and ŽA.7., we have g Ž m. G 0, which with Eq. ŽA.6. implies ; f g w0, 1x, f Ž L y m . Ž 1 y p 1 . Q1f Ž H q w1 L . y w 1 LQ1 y g Ž m . w H q w1 L x F 0; i.e., E erE t F 0. Therefore, t s 1 in the optimal contract. But at t s 1, Eq. ŽA.5. implies E erE f F 0. Thus f s 1 must be true in the optimal contract. If d f G d u , that is, Q1f Ž H q w 1 L . y w 1 Q1 L F 0, then from Eq. ŽA.5., no matter whether g Ž m. is positive or negative, tw1 g Ž m . G Q1f Ž H q w 1 L . y w 1 LQ1 , ; t g w0, 1x. Therefore, E erE f F 0, which implies that f s 1 in the optimal contract. But at f s 1, from Eq. ŽA.6. E erE t F 0, which implies that t s 1. Thus, in Case I.1, in which d p F min  d f , d u 4 , f s 1, t s 1 must be true in the optimal contract, regardless of whether Eq. ŽA.7. holds. Case I.2

Ž L y m . Ž 1 y p1 . Q1f Ž H q w1 L . y w1 LQ1 y g Ž m . w H q w1 L x G0 Ž i.e. d p G d u .

Ž A.8 .

In this case, g Ž m. must be non-positive and thus from Eq. ŽA.5. we have Q1f Ž H q w 1 L . y w 1 Q1 L F 0,

Ž i.e. d f G d u .

From this and Eq. ŽA.8., we have ; f g w0, 1x, f Ž L y m . Ž 1 y p 1 . Q1f Ž H q w1 L . y w 1 LQ1 y g Ž m . w H q w1 L x G 0 which implies that E erE t G 0. In the case when type 2 are deterred for all 0 - f, t - 1, Ži.e. Eq. Ž25. holds., t ™ 0 in the optimal contract. In the case when Eq. Ž25.

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does not hold, Ž t, f . in the optimal contract must be on the boundary of U2 s w 2 L. Since g Ž m. F 0 in this case, d erdt along U2 s w 2 L must be nonnegative, which implies that t s t ) , f s 1 in the optimal contract. Thus, if we define t ) s 0 in the case when Eq. Ž25. holds, we have the following result: if d u F d p F d f , then t s t ) , f s 1 in the optimal contract. Case II w1 g Ž m . - Q1f Ž H q w1 L . y w1 LQ1 ,

Ž i.e. d p ) d f .

Ž A.9 .

Again, in this case, two situations can occur: Case II.1

Ž L y m . Ž 1 y p1 . Q1f Ž H q w1 L . y w1 LQ1 y g Ž m . w H q w1 L x F0 = Ž i.e. d p F d u .

Ž A.10 .

Eqs. ŽA.9. and ŽA.10. imply Q1f Ž H q w 1 L . y w 1 Q1 L G 0,

Ž i.e. d f F d u .

and g Ž m. G 0, which together with Eq. ŽA.9. imply that tw1 g Ž m . F Q1f Ž H q w 1 L . y w 1 LQ1 . Therefore, E erE f G 0, and thus f s 0 in the case when Eq. Ž25. holds and Ž t, f . on the boundary of U2 when Eq. Ž25. does not hold. Notice that from Eq. ŽA.3. at f s 0, E erE t has the opposite sign of g Ž m.; if Ž t, f . is on the boundary, from Eq. ŽA.4., d erdt has the opposite sign of g Ž m. also. From this, we can conclude that t s 1, f s f ) must be true in the optimal contract. Therefore, we obtain the following result: if d f F d p F d u , then t s 1, f s f ) hold in the optimal contract. 14 Case II.2

Ž L y m . Ž 1 y p1 . Q1f Ž H q w1 L . y w1 LQ1 y g Ž m . w H q w1 L x )0 = Ž i.e. d p ) d u .

Ž A.11 .

If d f F d u , Q1f Ž H q w 1 L . y w 1 Q1 L G 0,

Ž A.12 .

then ; t g w 0, 1 x ,

tw1 g Ž m . F Q1f Ž H q w 1 L . y w 1 LQ1 .

14 Note that d f - d p - d u implies g Ž m. ) 0, so that case Ž3c. subsumes d f - d p - d u . The proof is simple. Denote the denominators of d f , d p and d u as a, c and e respectively, and the numerators as b, d and f respectively. These terms are all positive. For a ) 0, ar b) cr d) er f implies that Ž a aq c .rŽ a bq d . ) cr d) er f. Let a be Ž Ly m.Ž1y p1 .. Thus, a bq ds f. This implies that a aq c) e, therefore g Ž m. ) 0.

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Therefore, E erE f G 0, and thus f s 0 in the case when Eq. Ž25. holds and Ž t, f . on the boundary of U2 when Eq. Ž25. does not hold. From Eq. ŽA.3. at f s 0, E erE t has the opposite sign of g Ž m.; if Ž t, f . is on the boundary, from Eq. ŽA.4., d erdt has the opposite sign of g Ž m. also. Consequently if g Ž m. G 0, then t s 1; if g Ž m. - 0, then t s t ) , where in the case when Eq. Ž25. holds, t ) s 0. It follows that if d f F d u F d p , then t s t ) , f s 1 if g Ž m. F 0 and Eq. Ž25. does not hold; t s t ) , f s 0 if g Ž m. F 0 and Eq. Ž25. holds; t s 1, f s f ) if g Ž m. G 0. If d u F d f , that is, Q1f Ž H q w 1 L . y w 1 Q1 L F 0,

Ž A.13 .

we can conclude that g Ž m. - 0 from Eq. ŽA.9., which also implies that ; f g w0, 1x, f Ž L y m . Ž 1 y p 1 . Q1f Ž H q w1 L . y w 1 LQ1 y g Ž m . w H q w1 L x ) 0. Ž A.14 . Therefore, E erE t ) 0. In the case when Eq. Ž25. holds, t s 0, which implies that E erE f s 0. So without loss of generality, we assume that f s 1. In the case when Eq. Ž25. does not hold, Ž t, f . is on the boundary of U2 s w 2 L and from g Ž m. - 0, we know that t s t ) , f s 1 holds in the optimal contract. So whether Eq. Ž25. holds or not, t s t ) , f s 1 holds in the optimal contract if d u F d f - d p. I Proof of Theorem 3: Substituting f s 1 into Eqs. Ž23. and Ž24., we have: w1 L y t L y m 1 y p1 w1 ) tmQ1 q 1 y t LQ1 q t L y m p 1 Q1p

Ž

min e s

Ž

s.t. U2 s Ž L y m . w 1 q

.Ž

.

.

Ž

p2 p1

Ž A.15 .

.

Ž w1 m q S . y t Ž w1 y w 2 . Ž L y m . Ž 1 y p 2 . y S

F w 2 L.

Ž A.16 .

Substituting f s 1 in Eq. ŽA.3., we have

Ee Et

1 s D

2

w 1 L  m Ž Q1 y Q1) . y Ž L y m . p 1 Ž Q1p y Q1 . 4 s y

w1 L D2

G Ž m. .

Ž A.17 . From Eq. ŽA.15.,

Ee Et

1 s

D2

tw1 L  p 1 Ž Q1p y Q1) . q Ž 1 y p 1 . Ž 1 y t . Ž Q1 y Q1) . 4 ) 0.

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Fig. 3. Feasible region and optimal contract in case 1

From Eq. Ž24., at f s 1,

Ee Et

s y Ž w1 y w 2 . Ž L y m . Ž 1 y p 2 . - 0.

Therefore if U2 were drawn in Ž m, t . space Žsee Fig. 3 ., any Ž m, t . above U2 s w 2 L can deter applications from the type 2 workers. Let m i s Ž wi L q S .rpi . From Eq. ŽA.16., U2 s w 2 L can be expressed as ts

w 1 Ž 1 y p 2rp1 .

Ž 1 y p 2 . Ž w1 y w 2 .

q

p2 Ž m1 y m 2 .

Ž L y m . Ž w1 y w 2 . Ž 1 y p 2 .

.

The slope of U2 s w 2 L depends Žsee Fig. 4. on the sign of Ž m 1 y m 2 .. Moreover, it is easy to show that it is concave in Ž m, t . space when it is

Fig. 4. Feasible region and optimal contract in case 2

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downward sloping and convex when it is upward sloping. Thus, we have two cases: Case 1; m 1 G m 2 In this case, U2 s w 2 L is upward sloping in Ž m, t . space. Since E U2rE t - 0 along boundary U2 s w 2 L, any Ž m, t . above the boundary is feasible in deterring applications from type 2 workers. From E erE m ) 0, we know that m s m. From E erE t s yw1 LGŽ m.rD 2 , we know that ts

½

1,

if G Ž m . G 0;

t,

if G Ž m . - 0.

Case 2; m 1 - m 2 In this case, U2 s w 2 L is downward sloping in Ž m, t . space. Recalling that m ˜ is the minimum m that satisfies U2 F w 2 L given f s 1 and t s 1, max  m, m ˜ 4 is the smallest m that is feasible in deterring type 2 workers. If GŽmax  m, m ˜ 4. G 0, then m s max  m, m˜ 4, t s 1. If GŽmax  m, m ˜ 4. - 0, then m s m ) , t ™ 0. This is because the contract in this case must be on the boundary of U2 s w 2 L which is concave. It is easy to see that the firm’s iso-profit Žiso-efficiency–wage. curves are convex, 15 therefore Ž m,t . in the optimal contract must be at one of the two extreme ends of U2 s w 2 L. As t ™ 0, m ™ m ) , e approaches its minimum for any m that satisfies GŽ m. F 0. I

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15

From Eq. ŽA.15., es constant can be expressed as t s crŽ aq bm., which is convex in m as long as t is positive.

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