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Minimum wages and information
Journal of Economics and Business 53 (2001) 153–170
Minimum wages and information Charles F. Mason* Department of Economics & Finance, University of Wyoming, Laramie, WY 82071-3985, USA Received 20 August 1999; received in revised form 26 July 2000; accepted 31 August 2000
Abstract The paper investigates the impact of minimum wage legislation on the level of employment, both for skilled and unskilled labor, when employers are imperfectly informed about a potential employee’s skill level prior to hiring. In this context, imposition of a minimum wage can facilitate the skilled employee’s provision of information about his skill. When laborers can take a certifying test, minimum wages facilitate the sorting effect of the test. If the supply of skilled labor is sufficiently elastic relative to the demand for unskilled labor, this enhancement of the sorting effect can increase total employment, and reduce prices. © 2001 Elsevier Science Inc. All rights reserved. JEL classification: D8; J2; L1 Keywords: Asymmetric information; Testing; Signaling
1. Introduction Minimum wages have been a politically contentious issue for several decades. Over the course of the last decade, U.S. legislation has dramatically raised the Federal minimum wage, from $3.35 per hour to $5.15 per hour, by September 1997; a further increase of $1 per hour has recently been proposed. These increases in minimum wages have spawned considerable interest in the likely effects on employment levels, as well as unemployment rates. Since minimum wages will commonly induce firms to shift employment from lesser- to greaterskilled workers, perhaps the increase in employment of skilled labor would offset the decrease in employment of unskilled labor. In the political debate over the first of the recent
* Tel.: ⫹1-307-766-5336; fax: ⫹1-307-766-5090. E-mail address: [email protected] (C.F. Mason). 0148-6195/01/$ – see front matter © 2001 Elsevier Science Inc. All rights reserved. PII: S 0 1 4 8 - 6 1 9 5 ( 0 0 ) 0 0 0 4 8 - 5
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increases, for example, it was claimed that the preceding increase in the minimum wage did not lower employment: “More than 11 million new jobs have been created since 1981 [when the minimum wages was increased to $3.35]. . . ” (Subcommittee on Labor Standards, Committee on Education and Labor, US, 1987, p. 9). To the extent that the level of total employment is a meaningful indicator of economic performance, the net effect of a minimum wage is likely to be of interest to policy makers. Economists have long been suspicious of minimum wages, because they alter market conditions. This leads to an inefficient allocation of resources, and underemployment of lower paid workers (Stigler, 1946; Rottenberg, 1981). Moreover, the reduction in employment of lesser-skilled labor commonly more than offsets the increased employment of greater-skilled labor, so that total employment falls. Such arguments appeal to elementary market analysis, and so rely on perfect information regarding potential employee’s abilities. The argument I wish to raise in this paper is: To what extent does imperfect information regarding potential employee skill levels affect these efficiency arguments against minimum wages? Put differently, if many economists got their way and the minimum wage were reduced or eliminated, would we necessarily see an improved allocation of resources and greater employment of lesser-skilled labor? My main finding is that uncertainty about skill levels can create a scenario under which the relaxation of a price floor, such as a minimum wage, may decrease total employment. Because of the imperfect information on skill levels, there is an incentive for higher-skilled labor to signal their talents to prospective employees. However, I assume that the standard signaling model (Spence, 1973, 1974) cannot be applied, for example because the cost of the signal does not differ for different types of agents. In this vein, I consider an environment where potential employees have only a noisy signal available. This signal could be education if some lesser-skilled workers obtain degrees. Alternatively, it could be a certifying test that must be passed to obtain a certain type of job. The crucial aspect is that workers of both types can probabilistically obtain certification. Further, the certification test is able to partially discriminate between different skill classes, and so provides useful information in its own right. An important consequence of probabilistic certification is that separating equilibria are generally unlikely. Instead, equilibria commonly entail some pooling, with more-skilled labor inclined to seek certification to a greater extent than lesser-skilled labor. In this context, a minimum wage raises the reward from not seeking certification, which is comparatively more attractive for lesser-skilled labor, and so increases the screening power of the certification device. As a result, taking the test more accurately signals talent. Correspondingly, the wage paid to certified laborers increases, inducing a great volume of skilled labor into the market. When the supply of skilled workers is more responsive to this higher wage than the demand for unskilled workers, this increased volume of skilled workers will exceed the reductions in employment of unskilled workers and total employment will rise. In this case the newly added workers, who are more skilled, have higher marginal product than the displaced workers, who are less skilled. It follows that each firm’s supply curve shifts out, so that market price will fall. Thus, minimum wages can benefit consumers as well as skilled workers: with fortuitous combinations of labor supply for skilled workers, demand for unskilled workers, and final good demand, minimum wages can increase net surplus.
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The notion that minimum wages may be welfare-enhancing when employers are imperfectly informed about potential employees’ skills levels has been considered before (Lang, 1987). His model includes a device, such as higher education, that allows workers to perfectly signal their skill levels to potential employers. In this line of thought, it is less costly for more-skilled workers to pursue higher education than it is for less-skilled workers, and so only more-skilled workers go to college. The extent that the signal is pursued, by expending resources on the signal, depends on the length to which more-skilled workers must go to ensure that less-skilled laborers will not seek to mimic their actions. With the impositions of a minimum wage, the opportunity cost of pursuing higher education is raised for less-skilled laborers, and so they are less inclined to attempt to pass for more-skilled labor. As a consequence, more skilled laborers can successfully signal their talents by expending fewer resources, and this benefits society. There is, however, some question as to the efficiency of higher education as a perfect screening device (Hersch & Lowenstein, 1990; Hungerford & Solon, 1987; Layard & Psacharopoulos, 1974). In particular, unless one accepts the argument that the signal is sufficiently more costly for less-skilled agents to obtain, no separating equilibrium ensues. If, on the other hand, the signal is obtained by all types of agents and yet is more likely to be obtained by higher-talented individuals, it can still be useful to employers in forming their hiring decisions. Such a signal is a fortiori noisy, in that it fails to provide perfect information to the market. My result is very different from the corresponding result in the context of a perfect signal. In the latter type of model, the signal itself provides no useful information: all relevant information is conveyed by the identity of agents who purchase the signal. Hence, any benefits associated with minimum wages arise because there is a reduction in wasteful expenditures. In my model, since some unskilled workers succeed in obtaining certification, and since minimum wages reduce the incentive for unskilled workers to seek certification, price floors may lead to better (i.e. more accurate) information. When the social value of this improved information exceeds its cost, which in this context is the foregone gains from trade associated with the resultant unemployment, then net surplus can increase. This model and the accompanying results extend beyond labor markets: They are relevant to any market setting with adverse selection. This could be a scenario where an input can come in varying grades, and downstream firms cannot identify quality ex ante. Alternatively, it could apply to a final goods market, where a firm’s product could be of varying quality. In this last context, my results would suggest that a price floor on the final good is likely to generate a higher average quality than a “lemons law.”
2. The certifying model Consider a typical firm in a competitive market that cannot observe the productivity, or quality, of its workers prior to hiring. Further, suppose that it is either too costly or impractical to infer any one employee’s productivity from output, and that no perfect signal of skill exists. However, an imperfect signal correlated with the worker’s productivity does exist. For pedagogical purposes, I will refer to the signaling device as a certification test.1
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The players in this game are of two types: firms and potential employees. Potential employees are of varying generic ability, and for pedagogical simplicity I limit the types to higher and lower ability, or “skilled” and “unskilled” labor.2 The timing in this game is: (a) Nature designates some workers as skilled and some as unskilled, with each worker observing their skill level; (b) potential employees choose whether or not to test; (c) employers make wage offers; (d) potential employees choose whether or not to accept the wage. Labor supply and demand curves are common knowledge, so that the total number of potential employees who are skilled or unskilled under any wage structure can be inferred. However, only the worker knows her true skill level. I denote the reservation wage of the jth unit of type i labor by wR i (j), where i ⫽ 1 for skilled workers and i ⫽ 0 for unskilled workers. All these parameters are assumed to be common knowledge. Unless otherwise noted, I will assume that the supply of labor is upward sloping, for both skills: wR i ⬘(j) ⬎ 0. I assume that the probability of a skilled laborer passing the test is 1, while the probability of an unskilled laborer passing the test is 0. The certification test provides imperfect information on a prospective employee’s talents, so that 1 ⱖ 1 ⬎ 0 ⬎ 0. The cost of the test is A ⬎ 0.3 I assume that prospective employers know if a prospective employee has taken and passed the test, but cannot distinguish those who have failed the test from those who chose not to take the test. Correspondingly, two wages emerge: Those who have passed the test are offered wc, while all other employees are offered wn.4 The sequence of decisions and the ultimate financial outcomes is illustrated in Fig. 1. This diagram presents the decision tree for each type of worker. Each branch of the tree represents either a choice (e.g., “test” vs. “don’t test”) or a random outcome (pass vs. fail). Probabilities associated with random events are in parentheses, below the random outcome. The choice problem confronting a laborer has two nodes. First, she must determine if she wishes to take the certification test. If not, she chooses between taking a job at the uncertified wage wn and exiting the market. If she takes the test, she again must choose between taking a job and exiting the market; however now the available wage depends partially on the test result: passed workers are offered wc, while those who do not pass are offered wn. Clearly, anyone who chose to pay the test cost would not exit upon passing the test. On the other had, someone who failed the test would take a job (respectively, exit) if the uncertified wage exceeded (respectively, was less than) her reservation wage. The expected gain from taking the test can be determined by combining the information on ultimate rewards with the probability of reaching the final node. For example, if the jth skilled worker passes the test she will earn a net gain of wc ⫺ wR 1 (j) ⫺ A. Similarly, the jth (j) ⫺ A if she passes the test. The net gain earned by the unskilled worker will earn wc ⫺ wR 0 (j) ⫺ A if the worker enters the market for jth type i worker following a failure is wn ⫺ wR i uncertified labor; if she chooses to exit, she earns ⫺A. Combining these observations, the expected gain to the jth type i worker from taking the test is Gi(j): Gi(j) ⫽ i 关wc ⫺ wIR(j)] ⫹ (1 ⫺1)max[wn ⫺ wR i (j),0] ⫺ A,
(1)
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Fig. 1. Extensive Form with No Minimum Wage
where again i ⫽ 0 or 1, and j indexes workers. This expected gain is compared against the return that can be had with certainty, max[wn ⫺ wR i (j),0]. If follows that the jth type i worker will (weakly) prefer to take the test if R wc ⫺ wR i (j) ⫺ max[w n ⫺ w i (j),0] ⱖ A/ i.
(2)
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If (2) is satisfied as an equality, then the jth type i agent is indifferent between testing and her best alternative. This best alternative is to exit if her reservation wage exceeds wn; otherwise her best alternative is to remain in the market and take the lower wage. The most plausible equilibrium in this market has both types of workers taking the test and some unskilled workers not testing, i.e. a partial-pooling equilibrium.5 Note that (2) implies that the difference between the expected net gain from testing and the certain net gain from not testing is independent of the worker’s reservation wage, for any unskilled worker in the market. Correspondingly, if some but not all unskilled workers test, it follows that these two net gains are equal, so that wc ⫽ wn ⫹ A/0.
(3)
The marginal unskilled worker, U*, determines the volume of unskilled laborers in the market. He is the last unskilled laborer who is willing to work at the uncertified wage: woR(U*) ⫽ wn.
(4)
With all unskilled workers indifferent between testing and not testing, some of the U* unskilled workers present choose to test. Let UT represent the number of unskilled workers who opt to test. With A and 0 positive, the certified wage exceeds the uncertified wage: wc ⬎ wn. Since 1 ⬎ 0, one deduces that all skilled workers test or exit.6 Moreover, the marginal skilled worker would exit if she failed the test. To see this, let S⬘ represent that skilled worker who would just be indifferent between taking the wage paid uncertified workers and exiting:7 w1R(S⬘) ⫽ wn.
(5)
Combined with Eq. (3), this implies that the expected gain to skilled worker S⬘ is A/0. As 1 ⬎ 0, it follows that (2) is satisfied as an inequality for this worker, and so she strictly prefers testing to exiting. Continuity of the supply curve wR 1 (S) then ensures the existence of agents with reservation prices between wn and wc. Hence some skilled labor takes the test, planning to exit if failed. One may then use (2) to determine the identity of the marginal unit of skilled labor that tests, S*: wR 1 (S*) ⫽ w c ⫺ A/ 1.
(6)
The volume of skilled labor in the market is then S*.8 Wages are determined by firms’ expectations regarding the mix of skills they are acquiring. In particular, competitive pressures will force wages to adjust until the marginal effect on expected profit from hiring an additional unit of labor is nil in both the certified and uncertified segments. At this point, wages equal expected marginal value product for both certified and uncertified labor. Write the expected marginal value product from the last unit of certified labor as MVc, and let the expected marginal value product from the last unit of uncertified labor be MVn. Suppose that employers believe the probability that a worker who has been certified as skilled equals c. Letting the value of marginal product from the last skilled (respectively, unskilled) worker be MVS (respectively, MVu), the effects MVc and MVn are then readily determined as9
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MVc ⫽ cMVs ⫹ (1 ⫺ c)MVu;
(7)
MVn ⫽ nMVs ⫹ (1 ⫺ n)MVu.
(8)
Because MVc ⫺ wc ⫽ 0 ⫽ MVn ⫺ wn in equilibrium, Eqs. (7) and (8) may be used to derive equilibrium wages as wc ⫽ cMVs ⫹ (1 ⫺ c)MVu;
(9)
wn ⫽ nMVs ⫹ (1 ⫺ n)MVu.
(10)
In a rational expectations equilibrium, the believed probabilities c and n are confirmed by workers’ actions. In particular, the value c matches the true probability Pc that a certified worker is skilled, given the numbers of skilled and unskilled workers who truly take the test. From Bayes’ rule the true probability a certified worker is skilled is given by Pc ⫽ 1S*/[1S* ⫹ 0UT].
(11)
Similarly, the value n matches the true probability that an uncertified worker is skilled, Pn. To infer the true probability an uncertified worker is skilled, recall that all skilled workers seek certification, and only S⬘ skilled workers would remain in the market if they failed. On the other hand, U* ⫺ UT unskilled workers choose not to seek certification, and all of those would remain in the market even if they failed the test. It then follows that Pn ⫽ (1 ⫺1)S⬘/[(1 ⫺ 1)S⬘ ⫹ U* ⫺ UT ⫹ (1 ⫺ 0)UT] ⫽ (1 ⫺1)S⬘/[(1 ⫺ 1)S⬘ ⫹ U* ⫺ 0UT].
(12)
An equilibrium is a vector (UT,U*,S⬘,S*,c,n,wc,wn) satisfying Eqs. (3)–(6), (9), and (10), subject to the constraint that the believed probabilities c and n match the true probabilities Pc and Pn as given by Eqs. (11) and (12).10,11 The discussion can be greatly simplified if one imposes two relatively innocuous assumptions: that no skilled workers are willing to take the uncertified wage, and that the supply curve for unskilled workers is perfectly inelastic. The first of these assumptions implies that S⬘ ⫽ 0 ⫽ Pn, so that all workers in the uncertified segment are marked as unskilled. The second of these assumptions implies that all U* unskilled workers will be hired (with 0UT in the certified market and U* ⫺ 0UT in the uncertified market, on average). Correspondingly, wn equals the value of marginal product from the U*th unskilled worker.12 An equilibrium in this context is a vector (UT, S*, c, wc) satisfying Eqs. (3), (5), (9), and the constraint that c matches Pc (as determined by Eq. (11)). With a perfectly inelastic supply curve for unskilled workers, the number of unskilled workers is fixed. Calculation of the true probability Pc depends on S* and UT, so that one may view wc and the equilibrium expectation of c as implicit functions of S* and UT, and reduce the system to two (nonlinear) equations in two unknowns.
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3. Equilibrium with minimum wages Now imagine that the lower wage is restricted to equal wm, where presumably wm exceeds the old equilibrium uncertified wage. This would effectively fix the uncertified wage; if unskilled workers were indifferent between testing and not testing, the certified wage would then also be fixed (at wcm ⫽ wm ⫹ A/0). With both wages fixed, the levels of workers seeking employment would be fixed by Eqs. (4), (5) and (6); call these new values U*m, S⬘m, and S*m. But competitive pressures force adjustments until marginal value product equals the wage rate in both the certified and uncertified segments. Only the fraction of unskilled labor that chooses to test remains as a variable, and this cannot adjust to simultaneously meet both marginal conditions. An additional variable must change: employment. By hiring fewer uncertified workers, firms can manipulate marginal product until the marginal hire yields zero expected profit. At the same time, with the increased wage at least as many workers will seek employment in the uncertified market, and so some unemployment occurs. This alters the structure of the decision tree for both types of workers, as illustrated in Fig. 2. The rewards from passing the test or exiting remain as in Fig. 1, but a worker entering the uncertified market now faces an additional risk. Such an action leads to employment with probability ⬍ 1. If hired, the worker receives the wage wm; if not hired, he receives no wage income. The probability equals the fraction of workers seeking employment in the uncertified market who are hired. The possibility that a job in the uncertified segment may not be forthcoming generally alters one of the equilibrium conditions. In the pre-minimum wage testing equilibrium, the difference between the expected return from testing and the certain return from not testing is independent of the workers’ reservation wage, and so UT adjusts until all unskilled workers are indifferent between testing and not testing. Further, all but the marginal unskilled worker strictly prefer entering the uncertified segment to exiting the market altogether. It then follows that unemployment risk would lower the appeal of the uncertified segment, with the reduction in appeal depending on the worker’s reservation wage. With the risk of unemployment, some unskilled workers would strictly prefer testing, while others would strictly prefer not to test. The fraction of unskilled workers that test must adjust until the marginal unskilled worker who tests is just indifferent between testing and not testing. Since is the probability that an uncertified worker will be hired, the expected gain to the jth unskilled worker from not testing when the uncertified wage is wm becomes [wm ⫺ wR 0 (j)]. In contrast with the analysis of section 2, the incentive for an unskilled laborer to test can depend on his reservation wage in this context. Specifically, the difference between the expected gain from testing and the expected gain from not testing for the jth unit of unskilled labor is now:
0wcm ⫹ (1 ⫺ 0)[wm ⫹ (1 ⫺ )w0R(j)] ⫺ w0R(j) ⫺ A ⫺ [wm ⫺ w0R(j)] ⫽ 0w cm ⫺ A ⫺ 0关 w m ⫹ 共1 ⫺ 兲w 0R共 j兲兴. It is clear that this difference is positive (respectively, negative) for small (respectively, large) reservation wages. The intuition is simple enough. Because the risk of unemployment is associated with the uncertified market, the bonus from passing the test becomes relatively
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Fig. 2. Extensive Form with Minimum Wage
more important. But this bonus is just the wedge between the certified wage and one’s reservation wage, wcm ⫺ wR 0 (j), and so is larger for workers with smaller reservation wages. The implication of these remarks is that some unskilled worker, whom I will label as UTm,
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is just indifferent between testing and not testing. Comparing the expected gains from testing and from not testing, this worker’s reservation wage may be inferred: T w0R(Um ) ⫽ [0wcm ⫺ 0wm ⫺ A]/[0(1 ⫺ )].
(13)
Equivalently, one may use this worker’s identity to determine the equilibrium certified wage: T wcm ⫽ A/0 ⫹ wm ⫹ (1 ⫺ )w0R(Um ).
(14)
As above, the marginal unskilled laborer seeking employment is just indifferent between not entering, and entering but not testing: w0R(U*m) ⫽ wm.
(15)
Also as above, all skilled labor that enters the market will seek certification. In the event a skilled laborer fails the certification test, she may elect to seek employment in the uncertified segment of the market. The marginal such skilled unit, S⬘m, is determined by inserting the minimum wage into Eq. (5): w1R(S⬘m) ⫽ w m.
(16)
The marginal unit of skilled labor entering the market, S*m, is determined by inserting wcm into Eq. (6): w1R(S*m) ⫽ wcm ⫺ A/1.
(17)
Finally, the true probability that a certified worker is skilled, Pcm, may be derived by substituting S*m and UTm into Eq. (11) to get T ]. Pcm ⫽ 1S*m/[1S*m ⫹ 0Um
(18)
Eq. (12) can be similarly adapted to determine the fraction of uncertified hires that are skilled, Pnm. In keeping with the notation above, I denote the value of marginal product from the last unit of labor hired as MVim for worker type i ⫽ s (for skilled) or u (for unskilled). Suppose firms predict the fraction of certified (respectively, uncertified) workers who are skilled to be cm (respectively, nm). Then expected profits from the marginal worker in each segment are zero if
cmMVsm ⫹ (1 ⫺ cm)MVum ⫽ wcm;
(19)
nmMVsm ⫹ (1 ⫺ nm)MVum ⫽ wm.
(20)
An equilibrium with a minimum wage is a vector (UTm, U*m, S⬘m, S*m, , cm, wcm) that satisfies Eqs. (13), (15), (16), (17), (19), and (20), subject to the constraint that cm matches the true probability that a certified worker is skilled, Pcm. It is evident that some unemployment must be anticipated in this context; some of the unemployed can be skilled laborers who fail the test. As above, considerable simplification can be had if one assumes the supply of unskilled labor is perfectly inelastic. In this case, U*m ⫽ U*, and all unskilled labor would accept any
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Fig. 3. Testing Equilibrium: The Effect of a Minimum Wage
wage. Consequently, in equilibrium all unskilled labor must be indifferent between testing and not testing, and the equilibrium certified wage is wcm ⫽ wm ⫹ A/0.
(14⬘)
As discussed above, because both wages are then pegged, S*m and S⬘m are also determined. Pcm and Pnm are then uniquely determined by UTm and ; in equilibrium they simultaneously satisfy Eqs. (19) and (20), where cm ⫽ Pcm and nm ⫽ Pnm are used to evaluate expected profits. The effects of a minimum wage are illustrated in Fig. 3. Using Eq. (5), one may determine the identity of the marginal skilled worker, S*, by comparing the certified wage against the curve labeled wR 1 ⫺ A/1. Similarly, comparing the uncertified wage against the supply curve for unskilled labor, which is labeled wR 0 , allows determination of the marginal unskilled worker, U*. The market demand curves for skilled and unskilled labor that would arise if skill were observable are labeled MVs and MVu, respectively. In equilibrium, the demand curve for certified labor, MVc, reflects the true expected marginal value product of a certified worker. Initially, this yields a wage of wc, with S* skilled workers seeking certification. If the wage paid uncertified workers were set at wm, employment of unskilled labor would fall to Um while U*m unskilled laborers would seek jobs. At the same time, the expected marginal value product of certified labor would shift out to MVc⬘, the certified wage would rise to wcm, and the number of skilled workers seeking certification would rise to Sm. The main point of this paper is that there are circumstances under which total (expected)
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employment can be larger with a minimum wage. This point is most readily demonstrated when the reservation wages of all skilled laborers exceed the uncertified wage, both before and after a minimum wage is imposed. Proposition 1: If no skilled labor would accept the minimum wage and the supply of skilled labor is sufficiently elastic relative to the production function, test cost, and pass rates 1 and 0, then the relaxation of a minimum wage can lower total employment. Proof: If the supply of unskilled labor is not perfectly inelastic, the effect on the equilibrium certified wage due to an increase in the minimum wage may be deduced from Eq. (14) as: ⭸wcm/⭸wm ⫽ ⫹ (⭸/⭸wm)[wm ⫺ w0R(U*m)] ⫹ (1 ⫺ )w0R⬘(U*m)(⭸U*m/⭸wm).
(21)
The first and third terms on the right-hand side of Eq. (21) have positive sign while the second term in negative, so that general statements of the effect of a minimum wage are hard to draw. However, when Eq. (21) is evaluated at wm ⫽ wn, the uncertified wage in the pre-minimum wage equilibrium, U*m ⫽ U* and ⫽ 1 (i.e., there is no unemployment). Thus the last two terms in Eq. (21) vanish, yielding (⭸wcm/⭸wm)兩wn ⫽ 1.
(22)
If the supply of unskilled labor is perfectly inelastic, the effect on the equilibrium certified wage due to an increase in the minimum wage may be deduced from Eq. (14⬘) as: ⭸wcm/⭸wm ⫽ ,
(21⬘)
which again implies Eq. (22). Irrespective of the elasticity of supply for unskilled labor, the certified wage initially rises as rapidly as the uncertified wage. Expected employment of skilled labor may be calculated as 1S*m in the equilibrium with a minimum wage. Using Eq. (17), then, the increase in expected employment of skilled labor due to a small increase in the uncertified wage above wn may be determined as 1/wR 1 ⬘(S*). Since the rate of decrease in employment of unskilled labor is the magnitude of the slope of the demand curve for unskilled labor, 兩dMVu/dwn兩, it follows that a small increase in the minimum wage above wn will raise total employment if
1/w1R⬘(S*) ⬎ 兩 dMVu/dwn兩. Q.E.D. Five remarks are germane. First, the proof demonstrates that a small increase in the uncertified wage can raise total employment; this effect will generally not persist when larger increases are considered. Indeed, the effects in Eq. (21) that are negligible at wm ⫽ wn become more pronounced as wm is raised, and these tend to impede the expansion of skilled employment. At the same time, layoffs of uncertified labor rise, so that total employment must fall at some point. My intention is not to argue that one case is more important. I wish only to point out that total employment does not necessarily decrease following the imposition of a minimum wage (or, conversely, that total employment need not increase following a reduction in the wage floor), when workers’ skill levels are private information.
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Second, the increase in total employment discussed in Proposition 1 can only occur when employers are imperfectly informed about their workers’ productivity, and only then if the resultant equilibrium is a pooling equilibrium. If employers know or can exactly infer each individual worker’s productivity, a minimum wage must lower total employment.13 In this vein, I note the implicit assumption that 0 ⬎ 0 used in determining the equilibrium certified wages. If instead 0 ⫽ 0, then the testing equilibrium must separate (Mason & Sterbenz, 1994), and hence imposition of a minimum wage could not increase total employment. Third, expected employment of skilled labor will generally rise following the imposition of a minimum wage, which lowers the expected marginal value product of skilled labor. For Eq. (19) to still be satisfied, the probability of a certified worker being skilled must rise. This is accomplished by a decrement in the fraction of unskilled workers who test: fewer unskilled laborers seek certification. Consequently, the distribution of the number of skilled workers in the certified segment is favorably altered; the mean rises, while the variance falls. Because profits are concave in labor, this reduction in variance raises expected profits.14 Fourth, it is evident that skilled workers benefit from the higher certified wage that emerges as a result of the minimum wage. This suggests skilled labor may collectively have an incentive to offer all unskilled workers some inducement to forego taking the certification test.15 This would yield a separating equilibrium, very similar to the signaling equilibrium in Lang (1987), where skilled labor pays some amount (here, the sum of some per-capita bribe and the test cost) to obtain certification, unskilled labor opts to not seek certification, and the certified (respectively, uncertified) wage equals the true marginal value product of a unit of skilled (respectively, unskilled) labor. The most optimistic case for existence of such a bribe is the simple variation of the model above, where the supply of unskilled labor is perfectly inelastic, and where no skilled labor would accept the uncertified wage. Writing these wages as ws and wu and the bribe paid to uncertified labor as B, and assuming that one can only receive the certified wage by passing the test, an unskilled worker seeking certification in this regime compares 0ws ⫹ (1 ⫺ 0)wu ⫺ A against wu ⫹ B.16 The bribe which just makes this worker indifferent between testing and not testing is thus B ⫽ 0(ws ⫺ wu) ⫺ A. The expected gain for a skilled laborer is the difference between the certified wage with and without bribes, multiplied by the probability of passing the test: 1(ws ⫺ wc). Using Eq. (3) and noting that the original uncertified wage equals the new uncertified wage (since no skilled labor accepts it in either case), this expected gain can be expressed as (1/0)B. The aggregate net gain to skilled labor expected to pass in the market needs to be large relative to the expected volume of unskilled labor which would pass if all unskilled labor tested. Finally, I note that each unit of increased skilled labor adds more to output than each displaced unit of unskilled labor forfeits. If the typical firm’s increment in skilled labor is sufficiently large relative to the decrement in unskilled labor, its supply must increase. In this case, each firm’s output increases, for any given final goods price. This increase in final goods supply translates into a reduction in the equilibrium price of the final good; it follows immediately that imposition of a minimum wages can increase net surplus: Proposition 2: For sufficiently elastic supply of skilled labor, minimum wages can lower final goods prices and increase net surplus.
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An immediate corollary to Proposition 2 is that the elimination of a wage floor might have deleterious consequences, raising final goods prices and lowering net surplus.
4. Concluding remarks To the extent that employers’ information about potential employees’ skill levels is limited a priori and that a noisy signal regarding skills such as a certification test is available, a minimum wage can facilitate the sorting role of a certifying test. This obtains because the increased wage paid uncertified workers raises the opportunity cost of testing for unskilled labor, since it increases the net gain associated with entertaining the uncertified segment of the labor market. This induces some unskilled workers to forego testing, raising the expected net gains from testing. In turn, this yields additional incentives for skilled workers to participate in the market, and it lowers the risk associated with hiring a certified worker. Both of these effects lead to increased employment of certified labor. When combined with the higher wage paid certified labor, this doubly benefits certified workers, who are disproportionately skilled. At the same time, some uncertified labor will become unemployed. With fortuitous combinations of labor supply and demand curves, net surplus can be increased. The artificial increase in the uncertified wage has enhanced the certifying test’s ability to provide an employer with information on a prospective employee’s skill. Ultimately the benefits of this improvement in information manifest themselves in terms of greater output and lower prices, while the costs manifest themselves in terms of unemployment. When these benefits exceed the costs, net surplus is larger with a minimum wage than without one. There are two caveats to these results. First, if a continuum of labor types exists, then a separating equilibrium generally exists even with a noisy signal. Here, a small increase in the minimum wage would only affect the marginal worker (who was initially just indifferent between testing and not testing). This fails to generate any significant increase in the certified wage. But notice that a discrete increase in minimum wage can lead to a measurable increase in the certified wage, yielding similar results to those discussed above. Second, if markets meet repeatedly, the prospect for retaking the test exists. The dynamics of such a market are far more complex than those I addressed above, though the essential insights seem relevant: A price floor provides incentives for less talented agents to forego testing, and hence improves the ex ante mix of tested agents. This must push up the certified wage, inducing more higher-talent agents to remain in the market. As a closing note, I observe that my model suggests that price floors may offer a more attractive alternative than laws that require merchandisers to guarantee that their product exceeds a certain quality, or to provide consumers with impartial information on quality. Such laws, which contain the essence of the “lemons law” suggested by the FTC in the 1980s, would seem unlikely to increase average product quality if some sellers initially opt to test (Mason & Sterbenz, 1994). Price floors, on the other hand, fundamentally alter the testing equilibrium, to the advantage of sellers of higher-quality products. In light of Proposition 1, this seems likely to yield an increase in average product quality.
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Notes 1. Although my terminology implies a focus of attention on tests of job related skills, my model could be applied in principle to any costly imperfect signal related to skill. Thus, to the extent that higher education does not perfectly signal true skills levels nor cost unskilled workers more than skilled workers, the “test” I speak of could be the ordeal of pursuing a college degree. 2. The crucial aspect of this assumption is that there are a finite number of types, and not the dichotomy between skilled and unskilled. With a larger number of skill levels, the key comparison would still be between the lowest and the next-lowest levels. 3. Because there are only two types of workers, only one test will be used. For if two tests existed, then either all workers would prefer one, or one group would be indifferent between the two tests, and the other group would strictly prefer one of the tests. In the latter case, the choice of test would signal skill. Correspondingly, any pooling equilibrium entails the use of only one test. The same kind of logic may be applied if more than two tests exist. 4. I am assuming here that the firm cannot conduct a screening test, and that the testing firm does not reveal the outcome of any tests taken by potential workers. These assumptions are largely for analytical convenience: allowing for both workers and firms to engage in testing would greatly complicate the model, while allowing for all test results to be revealed would add a third wage (for those who have tested but failed). Since the main point in my paper is that wage floors raise the payoff to not testing, and so create the possibility of expanded employment of those with more skill, relaxing either assumption would only complicate the analysis without altering the main conclusion. 5. In the context of the model presented here, a separating equilibrium would entail certified workers receiving ws upon passing the test. Any skilled individuals who fail the test, along with all unskilled individuals, are offered the uncertified wage w ˆ , which must be larger than ws ⫺ A/0 and smaller than ws ⫺ A/1. While such a combination can emerge under fortuitous parameter condition, the underlying expectations firms must hold would fail the universal divinity criterion (Banks & Sobel, 1987). A detailed discussion on necessary conditions for existence of these separating equilibria can be found in Mason and Sterbenz (1994). It is also conceivable that a pooling equilibrium exists (in which all workers take the test), though this strikes me as somewhat less plausible than the partial-pooling equilibrium I focus on. 6. In light of Eqs. (1) and (3), if the jth skilled laborer tests, her expected gain is at least
1[wn ⫹ A/0 ⫺ w1R(j)] ⫹ (1 ⫺ 1)[wn ⫺ w1R(j)] ⫺ A ⫽ wn ⫺ w1R(j) ⫹ (1 ⫺ 0)A/0 ⬎ wn ⫺ w1R(j). But the final expression in this chain is the certain return from not testing, and so it follows that this worker must prefer testing to not testing.
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7. If no skilled laborer would remain in the market after failing the test, then Eq. (5) fails to hold for any positive S⬘. 8. This implicitly assumes that one may ignore the integer problem. A more precise identification of the marginal skilled worker is that w1R(S*) ⱕ wc ⫺ A/1 ⬍ w1R(S* ⫹ 1). One could get around this complication by allowing the marginal skilled worker to adopt a mixed strategy, taking the test with probability and exiting with probability 1 ⫺ . In such an event, the number of skilled workers in the market is S* ⫹ ⫺ 1. Alternatively, one could assume the number of workers is arbitrarily large, so that the supply curve is (approximately) continuous. In the discussion that follows I will ignore these complications, assuming instead that Eq. (6) identifies the marginal skilled worker. 9. Note that, because each firm’s production function is concave in both types of labor, expected marginal value product differs from marginal value product evaluated at the expected levels of employment. For the representative firm, let p denote market price, let f(s,u) be the production function (based on s units of skilled and u units of unskilled labor), and let g(x) be the probability that exactly x units of the c* units of certified labor are skilled. This probability is binomial; with N units of certified labor hired, the probability that x of them will be skilled is g(x) ⬀ xc (1 ⫺ c ) 共N⫺x兲 . (This can be well approximated by a normal distribution with mean Nc and variance Nc(1 ⫺ c).) Then expected profit accruing from the employment of n* uncertified and c* certified workers is
冕
c*
pf(x,c* ⫹ n* ⫺ x)g(x)dx ⫺ wcc* ⫺ wnn*.
0
10. Recognizing the relatively minor complications discussed in footnote 8 and allowing for the possibility that UT ⫽ U*, an equilibrium is a fixed point of a multidimensional function; existence follows from standard fixed point arguments (see, e.g., Chapter 6 in Border, 1985). In general, uniqueness is not guaranteed; this allows for the implausible event that the initial equilibrium may consist of lesser volumes of employment than some alternative equilibrium. Then, an imposed regime shift, such as the imposition of a minimum wage, can cause the system to move towards the equilibrium with larger volume of employment. But this increase in employment is purely coincidental, and cannot form the basis for a serious argument in favor of price controls. 11. While the zero-profit conditions embodied in the equilibrium conditions hold on average ex ante for each firm, they need not hold ex post. Some firms will have fortuitous draws, and will earn above-average profits; others will have uncommonly poor draws and will earn below-average profits. The lucky firms would likely pay
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12.
13.
14.
15.
16.
169
higher wages to their certified employees, whose productivity has been demonstrated as above average, so as to retain their services. The unlucky firms, on the other hand, would either go out of business or replace their certified employees. Assuming a new crop of potential enters the market each period, while others retire, one might expect to see higher wages paid to certified workers with higher seniority, since their (longer-lived) employers would be among those with above-average draws. While the assumption of perfectly inelastic supply of unskilled labor is clearly overly restrictive, it greatly simplifies the analysis. Note also that it biases the analysis against price floors, in that it leads to an overestimate of the degree of unemployment. For an analysis of imperfect testing within the context of upwards sloping supply curves for both types, see Mason and Sterbenz (1994). If firms were perfectly informed about each worker’s skill type, firms would employ mixes of worker types such that MVs/MVu ⫽ ws/wu, where wi is the wage paid type i ⫽ s (skilled) or u (unskilled). With ws ⬎ wu, the marginal rate of technical substitution between skilled and unskilled labor exceeds unity. Increasing wu to wm, via a minimum wage, alters the expansion path. On any given iso-cost line this yields an increase in skilled labor used, and a decrease in unskilled labor. But, with quasi-concave production functions, isoquants are convex and so the ratio of decreased unskilled labor to increased skilled labor must exceed ws/wm ⬎ 1 in magnitude; i.e., total employment must fall. This is very similar to the effects discussed in Mason (1986), where additional information via a perfect test applied at the appropriate point in a sequence of transactions yields market expanding benefits from both extra information and reduced variance. Of course, there is the potential free-riding problem to be surmounted. If some policing organization exists, such as a trade union, it could be straightforward to get each skilled laborer to contribute her pro-rated share of the bribe. Alternative mechanisms for public good provision have been suggested which could resolve this issue [see Bagnoli and McKee (1991) for a survey and experimental evidence]. While this could be a rational expectations equilibrium outcome, there is a subtle problem with assuming that workers who fail the test are paid a lower wage: Since no unskilled labor tests in this proposed equilibrium, all failed labor must be skilled. Alternatively, if firms can condition their wage offers on the attempt to seek certification, then all workers compare ws ⫺ A against wu ⫹ B. The bribe which just makes workers indifferent between testing and not testing in this equilibrium becomes B ⫽ ws ⫺ wu ⫺ A, and aggregate net gains to skilled labor from offering this bribe are B(S* ⫺ U*) ⫺ S*(1 ⫺ 0)/0.
Acknowledgments An earlier version of this paper was presented at the WEA International conference in San Diego, California. I wish to acknowledge helpful discussions with Ted Bergstrom, Joni
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Hersch, Coleman Kendall, Jim Pierce, and Sherrill Shaffer. Of course, any remaining shortcomings are my responsibility.
References Bagnoli, M., & McKee, M. (1991). Voluntary contribution games. Economic Inquiry, 29, 351–366. Banks, J. S., & Sobel, J. (1987). Equilibrium selection in signaling games. Econometrica, 55, 647– 661. Border, K. (1985). Fixed point theorems with applications to economics and game theory. Cambridge, UK: Cambridge University Press. Hersch, J., & Lowenstein, M. (1990). Degree effects and the screening hypothesis. University of Wyoming mimeo. Hungerford, T., & Solon, G. (1987). Sheepskin effects in the returns to education. Review of Economics and Statistics, 69, 175–177. Lang, K. (1987). Pareto improving minimum wages. Economic Inquiry, 25, 145–158. Layard, R., & Psacharopoulos, G. (1974). The screening hypothesis and the returns to education. Journal of Political Economy, 82, 985–998. Mason, C. (1986). Cherries, lemons, and the FTC, revisited. Economic Inquiry, 24, 363–365. Mason, C., & Sterbenz, F. (1994). Imperfect product testing and market size. International Economic Review, 35, 61– 85. Rottenberg, S. (Ed.). (1981). The economics of legal minimum wages. Washington, DC: American Enterprise Institute for Public Policy Research. Spence, A. M. (1973). Job market signaling. Quarterly Journal of Economics, 87, 355–374. Spence, A. M. (1974). Market signaling: informational transfer in hiring and related processes. Cambridge, Mass: Harvard University Press. Stigler, G. (1946). The economics of minimum wage legislation. American Economic Review. Subcommittee on Labor Standards, Committee on Education and Labor, US. (1987). House of Representatives. Hearings on H.R. 1834, The Minimum Wage Restoration Act of 1987. Vol. 1, Washington DC: US GPO.
Minimum wages and information Charles F. Mason* Department of Economics & Finance, University of Wyoming, Laramie, WY 82071-3985, USA Received 20 August 1999; received in revised form 26 July 2000; accepted 31 August 2000
Abstract The paper investigates the impact of minimum wage legislation on the level of employment, both for skilled and unskilled labor, when employers are imperfectly informed about a potential employee’s skill level prior to hiring. In this context, imposition of a minimum wage can facilitate the skilled employee’s provision of information about his skill. When laborers can take a certifying test, minimum wages facilitate the sorting effect of the test. If the supply of skilled labor is sufficiently elastic relative to the demand for unskilled labor, this enhancement of the sorting effect can increase total employment, and reduce prices. © 2001 Elsevier Science Inc. All rights reserved. JEL classification: D8; J2; L1 Keywords: Asymmetric information; Testing; Signaling
1. Introduction Minimum wages have been a politically contentious issue for several decades. Over the course of the last decade, U.S. legislation has dramatically raised the Federal minimum wage, from $3.35 per hour to $5.15 per hour, by September 1997; a further increase of $1 per hour has recently been proposed. These increases in minimum wages have spawned considerable interest in the likely effects on employment levels, as well as unemployment rates. Since minimum wages will commonly induce firms to shift employment from lesser- to greaterskilled workers, perhaps the increase in employment of skilled labor would offset the decrease in employment of unskilled labor. In the political debate over the first of the recent
* Tel.: ⫹1-307-766-5336; fax: ⫹1-307-766-5090. E-mail address: [email protected] (C.F. Mason). 0148-6195/01/$ – see front matter © 2001 Elsevier Science Inc. All rights reserved. PII: S 0 1 4 8 - 6 1 9 5 ( 0 0 ) 0 0 0 4 8 - 5
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increases, for example, it was claimed that the preceding increase in the minimum wage did not lower employment: “More than 11 million new jobs have been created since 1981 [when the minimum wages was increased to $3.35]. . . ” (Subcommittee on Labor Standards, Committee on Education and Labor, US, 1987, p. 9). To the extent that the level of total employment is a meaningful indicator of economic performance, the net effect of a minimum wage is likely to be of interest to policy makers. Economists have long been suspicious of minimum wages, because they alter market conditions. This leads to an inefficient allocation of resources, and underemployment of lower paid workers (Stigler, 1946; Rottenberg, 1981). Moreover, the reduction in employment of lesser-skilled labor commonly more than offsets the increased employment of greater-skilled labor, so that total employment falls. Such arguments appeal to elementary market analysis, and so rely on perfect information regarding potential employee’s abilities. The argument I wish to raise in this paper is: To what extent does imperfect information regarding potential employee skill levels affect these efficiency arguments against minimum wages? Put differently, if many economists got their way and the minimum wage were reduced or eliminated, would we necessarily see an improved allocation of resources and greater employment of lesser-skilled labor? My main finding is that uncertainty about skill levels can create a scenario under which the relaxation of a price floor, such as a minimum wage, may decrease total employment. Because of the imperfect information on skill levels, there is an incentive for higher-skilled labor to signal their talents to prospective employees. However, I assume that the standard signaling model (Spence, 1973, 1974) cannot be applied, for example because the cost of the signal does not differ for different types of agents. In this vein, I consider an environment where potential employees have only a noisy signal available. This signal could be education if some lesser-skilled workers obtain degrees. Alternatively, it could be a certifying test that must be passed to obtain a certain type of job. The crucial aspect is that workers of both types can probabilistically obtain certification. Further, the certification test is able to partially discriminate between different skill classes, and so provides useful information in its own right. An important consequence of probabilistic certification is that separating equilibria are generally unlikely. Instead, equilibria commonly entail some pooling, with more-skilled labor inclined to seek certification to a greater extent than lesser-skilled labor. In this context, a minimum wage raises the reward from not seeking certification, which is comparatively more attractive for lesser-skilled labor, and so increases the screening power of the certification device. As a result, taking the test more accurately signals talent. Correspondingly, the wage paid to certified laborers increases, inducing a great volume of skilled labor into the market. When the supply of skilled workers is more responsive to this higher wage than the demand for unskilled workers, this increased volume of skilled workers will exceed the reductions in employment of unskilled workers and total employment will rise. In this case the newly added workers, who are more skilled, have higher marginal product than the displaced workers, who are less skilled. It follows that each firm’s supply curve shifts out, so that market price will fall. Thus, minimum wages can benefit consumers as well as skilled workers: with fortuitous combinations of labor supply for skilled workers, demand for unskilled workers, and final good demand, minimum wages can increase net surplus.
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The notion that minimum wages may be welfare-enhancing when employers are imperfectly informed about potential employees’ skills levels has been considered before (Lang, 1987). His model includes a device, such as higher education, that allows workers to perfectly signal their skill levels to potential employers. In this line of thought, it is less costly for more-skilled workers to pursue higher education than it is for less-skilled workers, and so only more-skilled workers go to college. The extent that the signal is pursued, by expending resources on the signal, depends on the length to which more-skilled workers must go to ensure that less-skilled laborers will not seek to mimic their actions. With the impositions of a minimum wage, the opportunity cost of pursuing higher education is raised for less-skilled laborers, and so they are less inclined to attempt to pass for more-skilled labor. As a consequence, more skilled laborers can successfully signal their talents by expending fewer resources, and this benefits society. There is, however, some question as to the efficiency of higher education as a perfect screening device (Hersch & Lowenstein, 1990; Hungerford & Solon, 1987; Layard & Psacharopoulos, 1974). In particular, unless one accepts the argument that the signal is sufficiently more costly for less-skilled agents to obtain, no separating equilibrium ensues. If, on the other hand, the signal is obtained by all types of agents and yet is more likely to be obtained by higher-talented individuals, it can still be useful to employers in forming their hiring decisions. Such a signal is a fortiori noisy, in that it fails to provide perfect information to the market. My result is very different from the corresponding result in the context of a perfect signal. In the latter type of model, the signal itself provides no useful information: all relevant information is conveyed by the identity of agents who purchase the signal. Hence, any benefits associated with minimum wages arise because there is a reduction in wasteful expenditures. In my model, since some unskilled workers succeed in obtaining certification, and since minimum wages reduce the incentive for unskilled workers to seek certification, price floors may lead to better (i.e. more accurate) information. When the social value of this improved information exceeds its cost, which in this context is the foregone gains from trade associated with the resultant unemployment, then net surplus can increase. This model and the accompanying results extend beyond labor markets: They are relevant to any market setting with adverse selection. This could be a scenario where an input can come in varying grades, and downstream firms cannot identify quality ex ante. Alternatively, it could apply to a final goods market, where a firm’s product could be of varying quality. In this last context, my results would suggest that a price floor on the final good is likely to generate a higher average quality than a “lemons law.”
2. The certifying model Consider a typical firm in a competitive market that cannot observe the productivity, or quality, of its workers prior to hiring. Further, suppose that it is either too costly or impractical to infer any one employee’s productivity from output, and that no perfect signal of skill exists. However, an imperfect signal correlated with the worker’s productivity does exist. For pedagogical purposes, I will refer to the signaling device as a certification test.1
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The players in this game are of two types: firms and potential employees. Potential employees are of varying generic ability, and for pedagogical simplicity I limit the types to higher and lower ability, or “skilled” and “unskilled” labor.2 The timing in this game is: (a) Nature designates some workers as skilled and some as unskilled, with each worker observing their skill level; (b) potential employees choose whether or not to test; (c) employers make wage offers; (d) potential employees choose whether or not to accept the wage. Labor supply and demand curves are common knowledge, so that the total number of potential employees who are skilled or unskilled under any wage structure can be inferred. However, only the worker knows her true skill level. I denote the reservation wage of the jth unit of type i labor by wR i (j), where i ⫽ 1 for skilled workers and i ⫽ 0 for unskilled workers. All these parameters are assumed to be common knowledge. Unless otherwise noted, I will assume that the supply of labor is upward sloping, for both skills: wR i ⬘(j) ⬎ 0. I assume that the probability of a skilled laborer passing the test is 1, while the probability of an unskilled laborer passing the test is 0. The certification test provides imperfect information on a prospective employee’s talents, so that 1 ⱖ 1 ⬎ 0 ⬎ 0. The cost of the test is A ⬎ 0.3 I assume that prospective employers know if a prospective employee has taken and passed the test, but cannot distinguish those who have failed the test from those who chose not to take the test. Correspondingly, two wages emerge: Those who have passed the test are offered wc, while all other employees are offered wn.4 The sequence of decisions and the ultimate financial outcomes is illustrated in Fig. 1. This diagram presents the decision tree for each type of worker. Each branch of the tree represents either a choice (e.g., “test” vs. “don’t test”) or a random outcome (pass vs. fail). Probabilities associated with random events are in parentheses, below the random outcome. The choice problem confronting a laborer has two nodes. First, she must determine if she wishes to take the certification test. If not, she chooses between taking a job at the uncertified wage wn and exiting the market. If she takes the test, she again must choose between taking a job and exiting the market; however now the available wage depends partially on the test result: passed workers are offered wc, while those who do not pass are offered wn. Clearly, anyone who chose to pay the test cost would not exit upon passing the test. On the other had, someone who failed the test would take a job (respectively, exit) if the uncertified wage exceeded (respectively, was less than) her reservation wage. The expected gain from taking the test can be determined by combining the information on ultimate rewards with the probability of reaching the final node. For example, if the jth skilled worker passes the test she will earn a net gain of wc ⫺ wR 1 (j) ⫺ A. Similarly, the jth (j) ⫺ A if she passes the test. The net gain earned by the unskilled worker will earn wc ⫺ wR 0 (j) ⫺ A if the worker enters the market for jth type i worker following a failure is wn ⫺ wR i uncertified labor; if she chooses to exit, she earns ⫺A. Combining these observations, the expected gain to the jth type i worker from taking the test is Gi(j): Gi(j) ⫽ i 关wc ⫺ wIR(j)] ⫹ (1 ⫺1)max[wn ⫺ wR i (j),0] ⫺ A,
(1)
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Fig. 1. Extensive Form with No Minimum Wage
where again i ⫽ 0 or 1, and j indexes workers. This expected gain is compared against the return that can be had with certainty, max[wn ⫺ wR i (j),0]. If follows that the jth type i worker will (weakly) prefer to take the test if R wc ⫺ wR i (j) ⫺ max[w n ⫺ w i (j),0] ⱖ A/ i.
(2)
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If (2) is satisfied as an equality, then the jth type i agent is indifferent between testing and her best alternative. This best alternative is to exit if her reservation wage exceeds wn; otherwise her best alternative is to remain in the market and take the lower wage. The most plausible equilibrium in this market has both types of workers taking the test and some unskilled workers not testing, i.e. a partial-pooling equilibrium.5 Note that (2) implies that the difference between the expected net gain from testing and the certain net gain from not testing is independent of the worker’s reservation wage, for any unskilled worker in the market. Correspondingly, if some but not all unskilled workers test, it follows that these two net gains are equal, so that wc ⫽ wn ⫹ A/0.
(3)
The marginal unskilled worker, U*, determines the volume of unskilled laborers in the market. He is the last unskilled laborer who is willing to work at the uncertified wage: woR(U*) ⫽ wn.
(4)
With all unskilled workers indifferent between testing and not testing, some of the U* unskilled workers present choose to test. Let UT represent the number of unskilled workers who opt to test. With A and 0 positive, the certified wage exceeds the uncertified wage: wc ⬎ wn. Since 1 ⬎ 0, one deduces that all skilled workers test or exit.6 Moreover, the marginal skilled worker would exit if she failed the test. To see this, let S⬘ represent that skilled worker who would just be indifferent between taking the wage paid uncertified workers and exiting:7 w1R(S⬘) ⫽ wn.
(5)
Combined with Eq. (3), this implies that the expected gain to skilled worker S⬘ is A/0. As 1 ⬎ 0, it follows that (2) is satisfied as an inequality for this worker, and so she strictly prefers testing to exiting. Continuity of the supply curve wR 1 (S) then ensures the existence of agents with reservation prices between wn and wc. Hence some skilled labor takes the test, planning to exit if failed. One may then use (2) to determine the identity of the marginal unit of skilled labor that tests, S*: wR 1 (S*) ⫽ w c ⫺ A/ 1.
(6)
The volume of skilled labor in the market is then S*.8 Wages are determined by firms’ expectations regarding the mix of skills they are acquiring. In particular, competitive pressures will force wages to adjust until the marginal effect on expected profit from hiring an additional unit of labor is nil in both the certified and uncertified segments. At this point, wages equal expected marginal value product for both certified and uncertified labor. Write the expected marginal value product from the last unit of certified labor as MVc, and let the expected marginal value product from the last unit of uncertified labor be MVn. Suppose that employers believe the probability that a worker who has been certified as skilled equals c. Letting the value of marginal product from the last skilled (respectively, unskilled) worker be MVS (respectively, MVu), the effects MVc and MVn are then readily determined as9
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MVc ⫽ cMVs ⫹ (1 ⫺ c)MVu;
(7)
MVn ⫽ nMVs ⫹ (1 ⫺ n)MVu.
(8)
Because MVc ⫺ wc ⫽ 0 ⫽ MVn ⫺ wn in equilibrium, Eqs. (7) and (8) may be used to derive equilibrium wages as wc ⫽ cMVs ⫹ (1 ⫺ c)MVu;
(9)
wn ⫽ nMVs ⫹ (1 ⫺ n)MVu.
(10)
In a rational expectations equilibrium, the believed probabilities c and n are confirmed by workers’ actions. In particular, the value c matches the true probability Pc that a certified worker is skilled, given the numbers of skilled and unskilled workers who truly take the test. From Bayes’ rule the true probability a certified worker is skilled is given by Pc ⫽ 1S*/[1S* ⫹ 0UT].
(11)
Similarly, the value n matches the true probability that an uncertified worker is skilled, Pn. To infer the true probability an uncertified worker is skilled, recall that all skilled workers seek certification, and only S⬘ skilled workers would remain in the market if they failed. On the other hand, U* ⫺ UT unskilled workers choose not to seek certification, and all of those would remain in the market even if they failed the test. It then follows that Pn ⫽ (1 ⫺1)S⬘/[(1 ⫺ 1)S⬘ ⫹ U* ⫺ UT ⫹ (1 ⫺ 0)UT] ⫽ (1 ⫺1)S⬘/[(1 ⫺ 1)S⬘ ⫹ U* ⫺ 0UT].
(12)
An equilibrium is a vector (UT,U*,S⬘,S*,c,n,wc,wn) satisfying Eqs. (3)–(6), (9), and (10), subject to the constraint that the believed probabilities c and n match the true probabilities Pc and Pn as given by Eqs. (11) and (12).10,11 The discussion can be greatly simplified if one imposes two relatively innocuous assumptions: that no skilled workers are willing to take the uncertified wage, and that the supply curve for unskilled workers is perfectly inelastic. The first of these assumptions implies that S⬘ ⫽ 0 ⫽ Pn, so that all workers in the uncertified segment are marked as unskilled. The second of these assumptions implies that all U* unskilled workers will be hired (with 0UT in the certified market and U* ⫺ 0UT in the uncertified market, on average). Correspondingly, wn equals the value of marginal product from the U*th unskilled worker.12 An equilibrium in this context is a vector (UT, S*, c, wc) satisfying Eqs. (3), (5), (9), and the constraint that c matches Pc (as determined by Eq. (11)). With a perfectly inelastic supply curve for unskilled workers, the number of unskilled workers is fixed. Calculation of the true probability Pc depends on S* and UT, so that one may view wc and the equilibrium expectation of c as implicit functions of S* and UT, and reduce the system to two (nonlinear) equations in two unknowns.
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3. Equilibrium with minimum wages Now imagine that the lower wage is restricted to equal wm, where presumably wm exceeds the old equilibrium uncertified wage. This would effectively fix the uncertified wage; if unskilled workers were indifferent between testing and not testing, the certified wage would then also be fixed (at wcm ⫽ wm ⫹ A/0). With both wages fixed, the levels of workers seeking employment would be fixed by Eqs. (4), (5) and (6); call these new values U*m, S⬘m, and S*m. But competitive pressures force adjustments until marginal value product equals the wage rate in both the certified and uncertified segments. Only the fraction of unskilled labor that chooses to test remains as a variable, and this cannot adjust to simultaneously meet both marginal conditions. An additional variable must change: employment. By hiring fewer uncertified workers, firms can manipulate marginal product until the marginal hire yields zero expected profit. At the same time, with the increased wage at least as many workers will seek employment in the uncertified market, and so some unemployment occurs. This alters the structure of the decision tree for both types of workers, as illustrated in Fig. 2. The rewards from passing the test or exiting remain as in Fig. 1, but a worker entering the uncertified market now faces an additional risk. Such an action leads to employment with probability ⬍ 1. If hired, the worker receives the wage wm; if not hired, he receives no wage income. The probability equals the fraction of workers seeking employment in the uncertified market who are hired. The possibility that a job in the uncertified segment may not be forthcoming generally alters one of the equilibrium conditions. In the pre-minimum wage testing equilibrium, the difference between the expected return from testing and the certain return from not testing is independent of the workers’ reservation wage, and so UT adjusts until all unskilled workers are indifferent between testing and not testing. Further, all but the marginal unskilled worker strictly prefer entering the uncertified segment to exiting the market altogether. It then follows that unemployment risk would lower the appeal of the uncertified segment, with the reduction in appeal depending on the worker’s reservation wage. With the risk of unemployment, some unskilled workers would strictly prefer testing, while others would strictly prefer not to test. The fraction of unskilled workers that test must adjust until the marginal unskilled worker who tests is just indifferent between testing and not testing. Since is the probability that an uncertified worker will be hired, the expected gain to the jth unskilled worker from not testing when the uncertified wage is wm becomes [wm ⫺ wR 0 (j)]. In contrast with the analysis of section 2, the incentive for an unskilled laborer to test can depend on his reservation wage in this context. Specifically, the difference between the expected gain from testing and the expected gain from not testing for the jth unit of unskilled labor is now:
0wcm ⫹ (1 ⫺ 0)[wm ⫹ (1 ⫺ )w0R(j)] ⫺ w0R(j) ⫺ A ⫺ [wm ⫺ w0R(j)] ⫽ 0w cm ⫺ A ⫺ 0关 w m ⫹ 共1 ⫺ 兲w 0R共 j兲兴. It is clear that this difference is positive (respectively, negative) for small (respectively, large) reservation wages. The intuition is simple enough. Because the risk of unemployment is associated with the uncertified market, the bonus from passing the test becomes relatively
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Fig. 2. Extensive Form with Minimum Wage
more important. But this bonus is just the wedge between the certified wage and one’s reservation wage, wcm ⫺ wR 0 (j), and so is larger for workers with smaller reservation wages. The implication of these remarks is that some unskilled worker, whom I will label as UTm,
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is just indifferent between testing and not testing. Comparing the expected gains from testing and from not testing, this worker’s reservation wage may be inferred: T w0R(Um ) ⫽ [0wcm ⫺ 0wm ⫺ A]/[0(1 ⫺ )].
(13)
Equivalently, one may use this worker’s identity to determine the equilibrium certified wage: T wcm ⫽ A/0 ⫹ wm ⫹ (1 ⫺ )w0R(Um ).
(14)
As above, the marginal unskilled laborer seeking employment is just indifferent between not entering, and entering but not testing: w0R(U*m) ⫽ wm.
(15)
Also as above, all skilled labor that enters the market will seek certification. In the event a skilled laborer fails the certification test, she may elect to seek employment in the uncertified segment of the market. The marginal such skilled unit, S⬘m, is determined by inserting the minimum wage into Eq. (5): w1R(S⬘m) ⫽ w m.
(16)
The marginal unit of skilled labor entering the market, S*m, is determined by inserting wcm into Eq. (6): w1R(S*m) ⫽ wcm ⫺ A/1.
(17)
Finally, the true probability that a certified worker is skilled, Pcm, may be derived by substituting S*m and UTm into Eq. (11) to get T ]. Pcm ⫽ 1S*m/[1S*m ⫹ 0Um
(18)
Eq. (12) can be similarly adapted to determine the fraction of uncertified hires that are skilled, Pnm. In keeping with the notation above, I denote the value of marginal product from the last unit of labor hired as MVim for worker type i ⫽ s (for skilled) or u (for unskilled). Suppose firms predict the fraction of certified (respectively, uncertified) workers who are skilled to be cm (respectively, nm). Then expected profits from the marginal worker in each segment are zero if
cmMVsm ⫹ (1 ⫺ cm)MVum ⫽ wcm;
(19)
nmMVsm ⫹ (1 ⫺ nm)MVum ⫽ wm.
(20)
An equilibrium with a minimum wage is a vector (UTm, U*m, S⬘m, S*m, , cm, wcm) that satisfies Eqs. (13), (15), (16), (17), (19), and (20), subject to the constraint that cm matches the true probability that a certified worker is skilled, Pcm. It is evident that some unemployment must be anticipated in this context; some of the unemployed can be skilled laborers who fail the test. As above, considerable simplification can be had if one assumes the supply of unskilled labor is perfectly inelastic. In this case, U*m ⫽ U*, and all unskilled labor would accept any
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Fig. 3. Testing Equilibrium: The Effect of a Minimum Wage
wage. Consequently, in equilibrium all unskilled labor must be indifferent between testing and not testing, and the equilibrium certified wage is wcm ⫽ wm ⫹ A/0.
(14⬘)
As discussed above, because both wages are then pegged, S*m and S⬘m are also determined. Pcm and Pnm are then uniquely determined by UTm and ; in equilibrium they simultaneously satisfy Eqs. (19) and (20), where cm ⫽ Pcm and nm ⫽ Pnm are used to evaluate expected profits. The effects of a minimum wage are illustrated in Fig. 3. Using Eq. (5), one may determine the identity of the marginal skilled worker, S*, by comparing the certified wage against the curve labeled wR 1 ⫺ A/1. Similarly, comparing the uncertified wage against the supply curve for unskilled labor, which is labeled wR 0 , allows determination of the marginal unskilled worker, U*. The market demand curves for skilled and unskilled labor that would arise if skill were observable are labeled MVs and MVu, respectively. In equilibrium, the demand curve for certified labor, MVc, reflects the true expected marginal value product of a certified worker. Initially, this yields a wage of wc, with S* skilled workers seeking certification. If the wage paid uncertified workers were set at wm, employment of unskilled labor would fall to Um while U*m unskilled laborers would seek jobs. At the same time, the expected marginal value product of certified labor would shift out to MVc⬘, the certified wage would rise to wcm, and the number of skilled workers seeking certification would rise to Sm. The main point of this paper is that there are circumstances under which total (expected)
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employment can be larger with a minimum wage. This point is most readily demonstrated when the reservation wages of all skilled laborers exceed the uncertified wage, both before and after a minimum wage is imposed. Proposition 1: If no skilled labor would accept the minimum wage and the supply of skilled labor is sufficiently elastic relative to the production function, test cost, and pass rates 1 and 0, then the relaxation of a minimum wage can lower total employment. Proof: If the supply of unskilled labor is not perfectly inelastic, the effect on the equilibrium certified wage due to an increase in the minimum wage may be deduced from Eq. (14) as: ⭸wcm/⭸wm ⫽ ⫹ (⭸/⭸wm)[wm ⫺ w0R(U*m)] ⫹ (1 ⫺ )w0R⬘(U*m)(⭸U*m/⭸wm).
(21)
The first and third terms on the right-hand side of Eq. (21) have positive sign while the second term in negative, so that general statements of the effect of a minimum wage are hard to draw. However, when Eq. (21) is evaluated at wm ⫽ wn, the uncertified wage in the pre-minimum wage equilibrium, U*m ⫽ U* and ⫽ 1 (i.e., there is no unemployment). Thus the last two terms in Eq. (21) vanish, yielding (⭸wcm/⭸wm)兩wn ⫽ 1.
(22)
If the supply of unskilled labor is perfectly inelastic, the effect on the equilibrium certified wage due to an increase in the minimum wage may be deduced from Eq. (14⬘) as: ⭸wcm/⭸wm ⫽ ,
(21⬘)
which again implies Eq. (22). Irrespective of the elasticity of supply for unskilled labor, the certified wage initially rises as rapidly as the uncertified wage. Expected employment of skilled labor may be calculated as 1S*m in the equilibrium with a minimum wage. Using Eq. (17), then, the increase in expected employment of skilled labor due to a small increase in the uncertified wage above wn may be determined as 1/wR 1 ⬘(S*). Since the rate of decrease in employment of unskilled labor is the magnitude of the slope of the demand curve for unskilled labor, 兩dMVu/dwn兩, it follows that a small increase in the minimum wage above wn will raise total employment if
1/w1R⬘(S*) ⬎ 兩 dMVu/dwn兩. Q.E.D. Five remarks are germane. First, the proof demonstrates that a small increase in the uncertified wage can raise total employment; this effect will generally not persist when larger increases are considered. Indeed, the effects in Eq. (21) that are negligible at wm ⫽ wn become more pronounced as wm is raised, and these tend to impede the expansion of skilled employment. At the same time, layoffs of uncertified labor rise, so that total employment must fall at some point. My intention is not to argue that one case is more important. I wish only to point out that total employment does not necessarily decrease following the imposition of a minimum wage (or, conversely, that total employment need not increase following a reduction in the wage floor), when workers’ skill levels are private information.
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Second, the increase in total employment discussed in Proposition 1 can only occur when employers are imperfectly informed about their workers’ productivity, and only then if the resultant equilibrium is a pooling equilibrium. If employers know or can exactly infer each individual worker’s productivity, a minimum wage must lower total employment.13 In this vein, I note the implicit assumption that 0 ⬎ 0 used in determining the equilibrium certified wages. If instead 0 ⫽ 0, then the testing equilibrium must separate (Mason & Sterbenz, 1994), and hence imposition of a minimum wage could not increase total employment. Third, expected employment of skilled labor will generally rise following the imposition of a minimum wage, which lowers the expected marginal value product of skilled labor. For Eq. (19) to still be satisfied, the probability of a certified worker being skilled must rise. This is accomplished by a decrement in the fraction of unskilled workers who test: fewer unskilled laborers seek certification. Consequently, the distribution of the number of skilled workers in the certified segment is favorably altered; the mean rises, while the variance falls. Because profits are concave in labor, this reduction in variance raises expected profits.14 Fourth, it is evident that skilled workers benefit from the higher certified wage that emerges as a result of the minimum wage. This suggests skilled labor may collectively have an incentive to offer all unskilled workers some inducement to forego taking the certification test.15 This would yield a separating equilibrium, very similar to the signaling equilibrium in Lang (1987), where skilled labor pays some amount (here, the sum of some per-capita bribe and the test cost) to obtain certification, unskilled labor opts to not seek certification, and the certified (respectively, uncertified) wage equals the true marginal value product of a unit of skilled (respectively, unskilled) labor. The most optimistic case for existence of such a bribe is the simple variation of the model above, where the supply of unskilled labor is perfectly inelastic, and where no skilled labor would accept the uncertified wage. Writing these wages as ws and wu and the bribe paid to uncertified labor as B, and assuming that one can only receive the certified wage by passing the test, an unskilled worker seeking certification in this regime compares 0ws ⫹ (1 ⫺ 0)wu ⫺ A against wu ⫹ B.16 The bribe which just makes this worker indifferent between testing and not testing is thus B ⫽ 0(ws ⫺ wu) ⫺ A. The expected gain for a skilled laborer is the difference between the certified wage with and without bribes, multiplied by the probability of passing the test: 1(ws ⫺ wc). Using Eq. (3) and noting that the original uncertified wage equals the new uncertified wage (since no skilled labor accepts it in either case), this expected gain can be expressed as (1/0)B. The aggregate net gain to skilled labor expected to pass in the market needs to be large relative to the expected volume of unskilled labor which would pass if all unskilled labor tested. Finally, I note that each unit of increased skilled labor adds more to output than each displaced unit of unskilled labor forfeits. If the typical firm’s increment in skilled labor is sufficiently large relative to the decrement in unskilled labor, its supply must increase. In this case, each firm’s output increases, for any given final goods price. This increase in final goods supply translates into a reduction in the equilibrium price of the final good; it follows immediately that imposition of a minimum wages can increase net surplus: Proposition 2: For sufficiently elastic supply of skilled labor, minimum wages can lower final goods prices and increase net surplus.
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An immediate corollary to Proposition 2 is that the elimination of a wage floor might have deleterious consequences, raising final goods prices and lowering net surplus.
4. Concluding remarks To the extent that employers’ information about potential employees’ skill levels is limited a priori and that a noisy signal regarding skills such as a certification test is available, a minimum wage can facilitate the sorting role of a certifying test. This obtains because the increased wage paid uncertified workers raises the opportunity cost of testing for unskilled labor, since it increases the net gain associated with entertaining the uncertified segment of the labor market. This induces some unskilled workers to forego testing, raising the expected net gains from testing. In turn, this yields additional incentives for skilled workers to participate in the market, and it lowers the risk associated with hiring a certified worker. Both of these effects lead to increased employment of certified labor. When combined with the higher wage paid certified labor, this doubly benefits certified workers, who are disproportionately skilled. At the same time, some uncertified labor will become unemployed. With fortuitous combinations of labor supply and demand curves, net surplus can be increased. The artificial increase in the uncertified wage has enhanced the certifying test’s ability to provide an employer with information on a prospective employee’s skill. Ultimately the benefits of this improvement in information manifest themselves in terms of greater output and lower prices, while the costs manifest themselves in terms of unemployment. When these benefits exceed the costs, net surplus is larger with a minimum wage than without one. There are two caveats to these results. First, if a continuum of labor types exists, then a separating equilibrium generally exists even with a noisy signal. Here, a small increase in the minimum wage would only affect the marginal worker (who was initially just indifferent between testing and not testing). This fails to generate any significant increase in the certified wage. But notice that a discrete increase in minimum wage can lead to a measurable increase in the certified wage, yielding similar results to those discussed above. Second, if markets meet repeatedly, the prospect for retaking the test exists. The dynamics of such a market are far more complex than those I addressed above, though the essential insights seem relevant: A price floor provides incentives for less talented agents to forego testing, and hence improves the ex ante mix of tested agents. This must push up the certified wage, inducing more higher-talent agents to remain in the market. As a closing note, I observe that my model suggests that price floors may offer a more attractive alternative than laws that require merchandisers to guarantee that their product exceeds a certain quality, or to provide consumers with impartial information on quality. Such laws, which contain the essence of the “lemons law” suggested by the FTC in the 1980s, would seem unlikely to increase average product quality if some sellers initially opt to test (Mason & Sterbenz, 1994). Price floors, on the other hand, fundamentally alter the testing equilibrium, to the advantage of sellers of higher-quality products. In light of Proposition 1, this seems likely to yield an increase in average product quality.
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Notes 1. Although my terminology implies a focus of attention on tests of job related skills, my model could be applied in principle to any costly imperfect signal related to skill. Thus, to the extent that higher education does not perfectly signal true skills levels nor cost unskilled workers more than skilled workers, the “test” I speak of could be the ordeal of pursuing a college degree. 2. The crucial aspect of this assumption is that there are a finite number of types, and not the dichotomy between skilled and unskilled. With a larger number of skill levels, the key comparison would still be between the lowest and the next-lowest levels. 3. Because there are only two types of workers, only one test will be used. For if two tests existed, then either all workers would prefer one, or one group would be indifferent between the two tests, and the other group would strictly prefer one of the tests. In the latter case, the choice of test would signal skill. Correspondingly, any pooling equilibrium entails the use of only one test. The same kind of logic may be applied if more than two tests exist. 4. I am assuming here that the firm cannot conduct a screening test, and that the testing firm does not reveal the outcome of any tests taken by potential workers. These assumptions are largely for analytical convenience: allowing for both workers and firms to engage in testing would greatly complicate the model, while allowing for all test results to be revealed would add a third wage (for those who have tested but failed). Since the main point in my paper is that wage floors raise the payoff to not testing, and so create the possibility of expanded employment of those with more skill, relaxing either assumption would only complicate the analysis without altering the main conclusion. 5. In the context of the model presented here, a separating equilibrium would entail certified workers receiving ws upon passing the test. Any skilled individuals who fail the test, along with all unskilled individuals, are offered the uncertified wage w ˆ , which must be larger than ws ⫺ A/0 and smaller than ws ⫺ A/1. While such a combination can emerge under fortuitous parameter condition, the underlying expectations firms must hold would fail the universal divinity criterion (Banks & Sobel, 1987). A detailed discussion on necessary conditions for existence of these separating equilibria can be found in Mason and Sterbenz (1994). It is also conceivable that a pooling equilibrium exists (in which all workers take the test), though this strikes me as somewhat less plausible than the partial-pooling equilibrium I focus on. 6. In light of Eqs. (1) and (3), if the jth skilled laborer tests, her expected gain is at least
1[wn ⫹ A/0 ⫺ w1R(j)] ⫹ (1 ⫺ 1)[wn ⫺ w1R(j)] ⫺ A ⫽ wn ⫺ w1R(j) ⫹ (1 ⫺ 0)A/0 ⬎ wn ⫺ w1R(j). But the final expression in this chain is the certain return from not testing, and so it follows that this worker must prefer testing to not testing.
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7. If no skilled laborer would remain in the market after failing the test, then Eq. (5) fails to hold for any positive S⬘. 8. This implicitly assumes that one may ignore the integer problem. A more precise identification of the marginal skilled worker is that w1R(S*) ⱕ wc ⫺ A/1 ⬍ w1R(S* ⫹ 1). One could get around this complication by allowing the marginal skilled worker to adopt a mixed strategy, taking the test with probability and exiting with probability 1 ⫺ . In such an event, the number of skilled workers in the market is S* ⫹ ⫺ 1. Alternatively, one could assume the number of workers is arbitrarily large, so that the supply curve is (approximately) continuous. In the discussion that follows I will ignore these complications, assuming instead that Eq. (6) identifies the marginal skilled worker. 9. Note that, because each firm’s production function is concave in both types of labor, expected marginal value product differs from marginal value product evaluated at the expected levels of employment. For the representative firm, let p denote market price, let f(s,u) be the production function (based on s units of skilled and u units of unskilled labor), and let g(x) be the probability that exactly x units of the c* units of certified labor are skilled. This probability is binomial; with N units of certified labor hired, the probability that x of them will be skilled is g(x) ⬀ xc (1 ⫺ c ) 共N⫺x兲 . (This can be well approximated by a normal distribution with mean Nc and variance Nc(1 ⫺ c).) Then expected profit accruing from the employment of n* uncertified and c* certified workers is
冕
c*
pf(x,c* ⫹ n* ⫺ x)g(x)dx ⫺ wcc* ⫺ wnn*.
0
10. Recognizing the relatively minor complications discussed in footnote 8 and allowing for the possibility that UT ⫽ U*, an equilibrium is a fixed point of a multidimensional function; existence follows from standard fixed point arguments (see, e.g., Chapter 6 in Border, 1985). In general, uniqueness is not guaranteed; this allows for the implausible event that the initial equilibrium may consist of lesser volumes of employment than some alternative equilibrium. Then, an imposed regime shift, such as the imposition of a minimum wage, can cause the system to move towards the equilibrium with larger volume of employment. But this increase in employment is purely coincidental, and cannot form the basis for a serious argument in favor of price controls. 11. While the zero-profit conditions embodied in the equilibrium conditions hold on average ex ante for each firm, they need not hold ex post. Some firms will have fortuitous draws, and will earn above-average profits; others will have uncommonly poor draws and will earn below-average profits. The lucky firms would likely pay
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12.
13.
14.
15.
16.
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higher wages to their certified employees, whose productivity has been demonstrated as above average, so as to retain their services. The unlucky firms, on the other hand, would either go out of business or replace their certified employees. Assuming a new crop of potential enters the market each period, while others retire, one might expect to see higher wages paid to certified workers with higher seniority, since their (longer-lived) employers would be among those with above-average draws. While the assumption of perfectly inelastic supply of unskilled labor is clearly overly restrictive, it greatly simplifies the analysis. Note also that it biases the analysis against price floors, in that it leads to an overestimate of the degree of unemployment. For an analysis of imperfect testing within the context of upwards sloping supply curves for both types, see Mason and Sterbenz (1994). If firms were perfectly informed about each worker’s skill type, firms would employ mixes of worker types such that MVs/MVu ⫽ ws/wu, where wi is the wage paid type i ⫽ s (skilled) or u (unskilled). With ws ⬎ wu, the marginal rate of technical substitution between skilled and unskilled labor exceeds unity. Increasing wu to wm, via a minimum wage, alters the expansion path. On any given iso-cost line this yields an increase in skilled labor used, and a decrease in unskilled labor. But, with quasi-concave production functions, isoquants are convex and so the ratio of decreased unskilled labor to increased skilled labor must exceed ws/wm ⬎ 1 in magnitude; i.e., total employment must fall. This is very similar to the effects discussed in Mason (1986), where additional information via a perfect test applied at the appropriate point in a sequence of transactions yields market expanding benefits from both extra information and reduced variance. Of course, there is the potential free-riding problem to be surmounted. If some policing organization exists, such as a trade union, it could be straightforward to get each skilled laborer to contribute her pro-rated share of the bribe. Alternative mechanisms for public good provision have been suggested which could resolve this issue [see Bagnoli and McKee (1991) for a survey and experimental evidence]. While this could be a rational expectations equilibrium outcome, there is a subtle problem with assuming that workers who fail the test are paid a lower wage: Since no unskilled labor tests in this proposed equilibrium, all failed labor must be skilled. Alternatively, if firms can condition their wage offers on the attempt to seek certification, then all workers compare ws ⫺ A against wu ⫹ B. The bribe which just makes workers indifferent between testing and not testing in this equilibrium becomes B ⫽ ws ⫺ wu ⫺ A, and aggregate net gains to skilled labor from offering this bribe are B(S* ⫺ U*) ⫺ S*(1 ⫺ 0)/0.
Acknowledgments An earlier version of this paper was presented at the WEA International conference in San Diego, California. I wish to acknowledge helpful discussions with Ted Bergstrom, Joni
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Hersch, Coleman Kendall, Jim Pierce, and Sherrill Shaffer. Of course, any remaining shortcomings are my responsibility.
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