The effect of minimum-wage laws on the distribution of employment: theory and evidence

Journal of Public Economics 69 (1998) 67–82

The effect of minimum-wage laws on the distribution ofemployment: theory and evidence

*Kevin Lang , Shulamit KahnDepartment of Economics, Boston University, 270 Bay State Road, Boston, MA 02215, USA

Received 31 August 1995; received in revised form 30 June 1997; accepted 18 August 1997

Abstract

Recent research casts doubt on the view that minimum-wage laws reduce employment.We show that in a simple model of bilateral search with heterogeneous workers, aminimum-wage law increases employment. However, the increased competition fromhigher productivity workers makes lower productivity workers worse off without makinghigher productivity workers better off. We provide evidence that minimum-wage laws shiftemployment from adults to teenagers and students. This raises concerns about thedistributional consequences of minimum wages even when they increase employment. 1998 Elsevier Science S.A.

Keywords: Minimum wages; Search; Unemployment

JEL classification: J30

1. Introduction

In the textbook analysis, a minimum-wage law reduces employment. Whilerecent research challenging this prediction remains controversial, it is safe to saythat the evidence that minimum-wage laws significantly reduce teenage and

1fast-food employment is very weak.It is natural to respond to these findings by concluding that minimum-wage laws

*Corresponding author. Tel.: (1-617) 353-5694; fax: (1-617) 353-4449; e-mail: [email protected] A summary of the evidence that concludes that there are zero or even positive employment effects

can be found in Card and Krueger (1995). Neumark and Wascher (1996) present a summary thatreaches the opposing conclusion.

0047-2727/98/$19.00 1998 Elsevier Science S.A. All rights reserved.PI I : S0047-2727( 97 )00085-6

68 K. Lang, S. Kahn / Journal of Public Economics 69 (1998) 67 –82

must be desirable. Indeed, Sylvia Nasar writing in the New York Times (Nasar,1992, p. D1) concludes that these studies make ‘‘the case for modest increases inthe minimum wage as an ingredient in future anti-poverty measures’’ (1992, p.D4).

At the same time, Nasar quotes one employer as saying: ‘‘Paying that extra 50cents helped us recruit. We’re getting better kids.’’ This suggests that minimum-wage laws not only affect the number of low-wage jobs but also who gets them. Ifthe experience of the employer just quoted is typical, low-wage employers may besubstituting workers they prefer for more disadvantaged workers. Yet in thetextbook model of minimum-wage laws, all workers are identical and, therefore,equally likely to share in the minimum-wage generated employment losses.

Textbook models also tend to be based on market-clearing models. In a worldincorporating search by employers and unemployed workers, the fact that labormarkets do not clear can have major implications on the impact of minimumwages.

In Section 2, we turn to a simple model incorporating both bilateral search andheterogenous workers developed in Lang and Dickens (1992). We show that, inthat model, a minimum-wage law increases employment by attracting more andbetter applicants to low-wage jobs with high vacancy rates. However, theequilibrium with a minimum wage is strictly pareto inferior to the equilibriumwithout a minimum wage. The increased competition from higher productivityworkers makes lower productivity workers worse off without making higherproductivity workers better off. In Section 3, we provide some evidence thatsuggests that minimum-wage laws do shift the composition of low-wage employ-ment. In Section 4, we compare our model to other models in which a minimum-wage law increases employment.

2. A model of bilateral search

For many jobs, firms set a single wage regardless of who fills it. The wage maydepend on seniority and, in some cases, on education and experience, but not onthe outcome of individual negotiations or on the quality of the worker. Of course,there are many jobs, particularly in professional labor markets, where suchnegotiation does take place, but individualized payments are atypical in low-wagelabor markets. Instead, there is a pre-set wage and, conditional on this wage, thefirm has a maximum number of vacancies it is willing to fill.

On the other side of the market, workers do not apply for all jobs. They mustweigh the cost of applying for a given job against the potential benefits.Everything else equal, they will prefer to apply where their prospects ofemployment are highest. This, in turn, implies that they will care about thequantity and quality of the applicants with whom they are likely to compete.

The following model, albeit stylized, is designed to capture these features. We

K. Lang, S. Kahn / Journal of Public Economics 69 (1998) 67 –82 69

model the labor market as a one-shot game with multiple stages in which firms setwages in order to attract applicants. Modelling the labor market as a one-shotgame is somewhat unnatural. However, it is fairly easy to transform the one-shotgame into a dynamic game provided that workers must be unemployed in order tosearch and firms must have a vacancy in order to recruit workers.

The structure of the game is as follows.

(A1) There are N workers and mN potential firms where both N and m are large.

This assumption ensures that we can use asymptotic approximations to thebinomial distribution and that there are sufficient potential firms so that expectedprofits can be zero in equilibrium.

(A2) There are two types of workers denoted type 1 and type 2. The number ofworkers of each type is large. Firms have lexicographical preferences: theymaximize profits; however, given equal profits, they prefer type 1 workers to type2 workers.

It might be more natural to model type 1 workers as more productive than type2 workers rather than simply preferred. Assuming lexicographic preferencessimplifies the presentation. However, the same results obtain if we assume insteadthat one group of workers is (slightly) more productive than the other.

(A3) The firms are indexed 1, . . . ,mN. They decide sequentially whether or not toenter.

This assumption allows us to ensure that the zero-profit equilibrium is reachedwhile relying on game-theoretic concepts such as sub-game perfection.

(A4) Firms that enter the market pay a capital rental fee of d.

This assumption implies that firms’ decisions regarding whether to openvacancies will be influenced by the probability of filling the vacancy.

(A5) After all firms have entered, firms simultaneously announce the wage theywill offer to any worker to whom they make an offer. The wage offer cannot bedependent on the type of worker hired.

There are two reasons firms may not announce wages conditional on workertype. The first is various anti-discrimination laws. The second is that although typeis observable it may not be verifiable. If so, the firm would always claim that anytype 1 worker it hires was actually a type 2.


(A6) After observing all the wage offers, workers may apply to at most one firm.

This assumption is designed to ensure that workers care about the probability ofgetting a job and thus about the number and type of other applicants.

(A7) Firms hire at most one worker. This worker is chosen at random from theapplicants belonging to the most preferred type from which the firm receivesapplicants.

This assumption rules out the possibility that the firm could credibly commit tohiring a type 2 worker even in the presence of type 1 applicants. It also rules outstrategies of the form ‘always hire Jane Doe if she applies’.

(A8) Workers who are hired produce v and receive the contracted wage.

This rules out various incentive and contracting problems.Together, these assumptions capture the idea that workers want to apply where

they are likely to obtain employment and firms open vacancies when they arelikely to fill them. Moreover, firms use wages to influence the probability of fillingtheir vacancies. When assessing the probability of getting a job, workers mustconsider what other types of workers are likely to apply. As a consequence,workers may prefer to apply to a low-wage job if low-wage jobs are less likely toattract more preferred applicants.

In equilibrium, workers use the following strategies. Type 1’s randomize amongjobs so that if all other type 1 workers followed the same mixed strategy, theexpected wage would be the same at all jobs among which they randomize with

2non-zero probability; the wage at jobs to which they apply with zero probabilitywould be no higher than the expected wage at jobs to which they apply withpositive probability.

Type 2’s randomize among jobs so that if type 1’s followed the strategy justdescribed and if all type 2’s followed the same random strategy, the expected wagefor type 2’s would be the same at all jobs among which they randomize withnon-zero probability; the expected wage at jobs to which they apply with zeroprobability would be no higher than the expected wages at jobs to which theyapply with positive probability.

Although it is self-evident that the strategies just described are an equilibrium ofthe worker sub-game, it is important to note that the strategies are complete forany set of wage offers including out-of-equilibrium wage distributions.

The model described above has been used by Lang and Dickens (1992) tomodel labor market segmentation. Before extending this model to study the impact

2 This is always equal to the expected wage since if a type 1 chose to apply, he would receive the jobwith certainty.


of minimum wages, we briefly discuss some of the aspects of this modeldeveloped in that earlier work. In equilibrium, type 1 workers apply to high-wagejobs while type 2 workers apply to low-wage jobs. The wages in the low-wagejobs are just sufficiently low to deter the preferred workers from applying.Preferred workers have both higher wages and higher expected wages than their

3less preferred counterparts.The expected number of applicants to high-wage jobs, z , is given by1

z 5 2 log(k /v) (1)1 1

where v is the value of the worker’s product. k is the expected wage received by1

type 1 workers and can be shown to equal

2z1k 5 w [(1 2 exp ) /z ] (2)1 1 1

where w is the level of the high-wage job and the term in square brackets equals14the probability of getting a job.

The zero-profit condition is

(1 2 exp(2z ))(v 2 w ) 2 d 5 0 (3)1 1

which says that the probability that the job is filled multiplied by the profitconditional on filling the job must equal the entry cost. Eqs. (1)–(3) can be solvedfor w , k and z .1 1 1

The low-wage is determined by the condition that type 1 workers are indifferentbetween applying to a high-wage firm and getting a low-wage job with probability1 (which, of course, is only possible if type 1 workers apply to low-wage firmswith probability zero) so that

w 5 w (1 2 exp(2z )) /z 5 k (4)2 1 1 1 1

which says that the low-wage is equal to the high-wage multiplied by theprobability of employment for a type 1 worker applying to a high-wage firm.

Finally, the number of applicants to low-wage firms can be derived from thezero-profit condition:

(1 2 exp(2z ))(v 2 w ) 2 d 5 0. (5)2 2

Not surprisingly, the three conditions Eqs. (1)–(3) that describe the equilibrium

3 The key point in the Lang and Dickens (1992) proof of a separating equilibrium is that if a firm’soffer attracts both types, it can increase the number of applicants by lowering the wage. (This isderived from Eq. (2) and an equivalent condition for type 2 workers.) Thus whenever an offer attractsboth types, the firm can always increase profits by lowering the wage. The equilibrium must thereforeinvolve separation of the two types.

4 Using the Poisson approximation to the binomial generates the result that the probability that thevacancy is filled is (12exp(2x)) where x is the expected number of applicants. The same approachyields (12exp(2x)) /x as the probability of employment when all workers are treated equally.


for type 1 workers are identical to the equilibrium conditions when workers arehomogeneous (as in Montgomery, 1991). The addition of a second type which isinferior to the first type has no effect on the first type. Since firms cannot committo hiring lower quality workers in preference to higher quality workers, theyensure that low quality applicants will not be in competition with high qualityapplicants by setting a wage that deters such applicants.

We now consider the effect of a minimum wage on this equilibrium. Aminimum wage set somewhat above the equilibrium low wage changes theequilibrium in the following way. It is no longer possible to set the low wage sothat high-wage workers do not wish to apply as in condition Eq. (4). Therefore,the low wage is the wage set by the minimum. Both type 1 and type 2 workersapply for the low-wage jobs. This increases employment rates for type 1 workersand reduces employment rates for type 2 workers. Indeed, the reduction inemployment for type 2 workers more than offsets the increased wage so that theirexpected wage falls. A sufficiently high minimum wage would actually inducesome type 2 workers to apply for high-wage jobs. However, provided that theminimum is not too high, the equilibrium will entail all type 2 workers applyingfor minimum wage jobs and type 1 workers randomizing between high andlow-wage jobs.

It is not obvious which of the two effects, increased employment for type 1’s orreduced employment for type 2’s, should dominate. The following theorem statesthat a sufficiently small minimum wage must increase total employment in themodel.

Theorem. Under assumptions (A1)–(A8), a law which slightly increases theminimum wage above w must increase employment.2

Proof. See Appendix A.

Despite the increase in employment, welfare is adversely affected. Expectedwages do not change for type 1 workers. Type 1 workers apply to jobs offering theminimum wage until the expected wage in the minimum wage jobs equals theexpected wage in the high-wage jobs. Therefore, they continue to obtain anexpected wage of k . However, the expected wage is lower for type 2 workers1

because of the increased competition from type 1 workers.It is not easy to move from the highly stylized model with only two types of

workers presented in this section to a more complex model. Nevertheless, themodel makes two useful points. First, higher wages attract more applicants tolow-wage jobs, causing low-wage employers to open more vacancies which islikely to increase employment in low-wage jobs and may increase overallemployment, as it does in this model. Secondly, minimum-wage laws may have asignificant impact on the distribution of employment regardless of any effect onthe level of employment.


In addition, at a somewhat more informal level, the model points to the potentialimportance of incentive effects for increased labor supply. One interpretation ofthe empirical work that follows is that the increased minimum wage attractedrelatively productive students into the labor force. Because, in the model, jobopenings respond to the number and quality of job applicants, the increased laborsupply, especially among better workers, increases the availability of jobs and,thus, overall employment. Principles students often argue that a minimum wagewill increase employment by giving people a greater incentive to work. In thestandard model, this argument is clearly wrong, because, in that model, employ-ment is determined by labor demand, and the labor demand curve is downward-sloping. In the model developed here, that intuition is correct.

3. The effect of the federal minimum wage increases: Evidence

At the beginning of 1988, the minimum wage was $3.35 almost everywhere inthe United States. Beginning with the increase in the minimum wage in Californiawhich survived legal challenges to become effective some time in late 1988 (Card,1992), a series of amendments to state and federal laws raised minimums until on

51st April 1991, the minimum was $4.25 in almost all states. In this section weexamine the impact of these changes on patterns of employment in low-wageoccupations in the food-service industry.

The data we use are drawn from the NBER Current Population Survey extractsof out-going rotation groups. Our sample consists of hourly workers employed inindustry 641 (eating and drinking establishments) and working in occupations433–444 (food service occupations). In 1987, these individuals earned an averageof $3.87 per hour. If wages of hourly workers in the food-service occupations (ineating and drinking establishments) had increased only at the average rate of wageincrease for hourly workers in all occupations in the economy between 1987 and1992, we would have expected, based on the 1987 distribution of earnings, that52% of food-service workers would be earning less than $4.25 per hour in 1992.In fact, only 24% earned less than $4.25 in 1992. Thus, these occupations seem tohave been significantly impacted by the minimum wage increase.

We follow a common practice of using prevailing wage levels (average hourlyearnings and percent earning less than $3.80/hour in 1987) to proxy theimportance of the minimum wage increase. The higher the prevailing wage in astate, the less impact on employment we would expect to find among workers in

5 Both Alaska and the District of Columbia had minimums exceeding the federal minimum by morethan 110% at the beginning of the period. At the end, the Alaskan minimum remained significantlyabove the federal minimum.


that state. The dependent variables are the change in various employmentcomposition variables between 1988 and the year immediately following thefederal minimum wage increase to $4.25 on 1st April 1991. The 1988 startingpoint was chosen because it predates the increases in nine states’ minimum wages.We use wage data from 1987 to eliminate potential bias. Had we used 1988 wages,if the CPS randomly over-sampled a low-wage group in 1988 in some state, wewould tend to both underestimate the prevailing wage in that state and see adecline in that group between 1988 and 1992, thus generating a spuriouscorrelation between low prevailing wages and a shift away from the types ofworkers who have low wages. We report results using a bounded influenceregression technique because some states, particularly Alaska with its very highminimum wage, were occasionally outliers. The results in this section do not

6change qualitatively if the estimates are based on OLS. Finally, we also re-estimated results by feasible GLS (to incorporate differing sample sizes) withsimilar qualitative results.

Studying food service occupations is complicated because federal changes inminimum wage abatements for tip income reduced the real minimum wage to be

7paid by employers for some workers between 1988 and 1992. However, only 5%of workers in the sample received the minimum wage for workers receiving

8substantial tip income. We do not expect the small decline in the employer-paidminimum wage for those workers to have had a measurable effect on employment.

Another way to account for tip income is to base the analysis on average hourlyearnings including tips. Even if the person receives some tip income, standardminimum wage rates [e.g. $3.35] applies to their total average hourly earningsincluding tips.

The total average hourly earnings is in principle derivable from usual weeklyearnings, which respondents are instructed to calculate including tips. By dividingthis number by usual weekly hours, we obtained ‘usual hourly earnings’. Weconcentrate on results using this usual hourly earnings measure. However, sincethere is likely to be significant measurement error in this variable, we alsore-estimated our equations using reported hourly wage and report these results.

In the first row of Table 1, the dependent variable is the change in a state’s ratio

6 Thus, although standard errors tend to be slightly larger in OLS, all parameters that are statisticallysignificant in the robust regression results are also significant in OLS results. The technique used is therreg routine in Stata (see Stata Corporation, 1993).

7 When the minimum wage was $3.35, tips could be used to replace up to 40% of the minimumwage which meant an effective employer-paid minimum wage of $2.01 for people earning at least$1.34 per hour in tips. When the minimum wage rose to $4.25, tips could be used to replace up to 50%of the minimum wage so that people earning more than $2.12 hourly in tips had an effectiveemployer-paid minimum wage of $2.13, implying a decline in real terms of 12.4% between 1988 and1992.

8 In 1988, substantial tip income was at least $1.34 /hour.


Table 1The effect of minimum wage laws on employment in food-service occupations

aDependent variable Based on earnings Based on hourly wage

Mean state %,3.80 in Mean state %,3.80 inearnings state earnings state(1987) (1987)

% employment in 0.04 (0.04) 20.12 (0.16) 20.01 (0.06) 0.16 (0.16)food service

Adult (.20 years) 0.30 (0.07) 21.26 (0.29) 0.47 (0.10) 21.56 (0.29)Student 20.24 (0.09) 0.86 (0.37) 20.41 (0.13) 1.17 (0.38)Teenage student 20.32 (0.11) 1.13 (0.43) 20.58 (0.14) 1.78 (0.43)High-school graduate 0.10 (0.06) 20.39 (0.27) 0.09 (0.08) 20.33 (0.25)Female 20.06 (0.10) 0.37 (0.39) 0.08 (0.14) 20.07 (0.42)Full-time worker 0.10 (0.08) 20.41 (0.31) 0.09 (0.10) 20.36 (0.33)Married 0.07 (0.09) 20.32 (0.37) 0.13 (0.13) 20.49 (0.40)a The dependent variable is the change in log(x /(1002x)) from 1988 to the year beginning 1st April1991. For the first row, x is the percentage of total state employment in hourly-paid food-serviceoccupations (in eating and drinking establishments). For all other rows, x is the percentage of hourlyfood-service workers in the state with that characteristic.Estimated with a bounded-influence regression technique. All coefficients have been multiplied by 100.

9of hourly workers in food-service occupations to total state employment. In thefirst and third column, the minimum-wage effect is proxied by the statewideaverage hourly earnings in these occupations in 1987; in the second and fourthcolumn, the minimum-wage effect is proxied by the proportion of hourly foodservice workers receiving less than $3.80 per hour in 1987. We cannot reject thehypothesis that employment growth across states is unaffected by varyingsusceptibility to the minimum wage.

However, there is a strong relation between these proxies for susceptibility tominimum-wage effects (in these occupations) and the proportion of food serviceworkers in several demographic sub-groups (see the remaining rows in Table 1). Instates with more susceptibility to minimum wages (as proxied by lower meanearnings or a higher proportion earning less than $3.80 /hour), the demographicmakeup of food-services occupations shifted toward teenagers and away fromadults. Similarly, in states with more sensitivity to minimum wage changes, therewere increasing proportions of students, particularly teenage students, working in

10the food services occupations. Thus, the results suggest a pattern of jobs shiftingaway from adults towards students including high school students. It is plausible

9 All dependent variables in Table 1 are based on changes in the log-odds transformation of therelevant variable.

10 The marginally significant result that higher minimum wages increase employment of non-highschool graduates may reflect this same shift towards current students.


that the higher minimum wage attracted more student workers to food-serviceoccupations. Many of these may have been talented workers who compared

11favorably with unskilled adult workers. Consistent with this finding, theparameter indicates a shift towards part-time work although it is not significant atconventional levels. Table 1 does not indicate that minimum wages shifted eitherthe gender or marital-status composition of these occupations.

In the light of these findings, it appears that minimum-wage laws have littleeffect on employment levels but may have important effects on who getslow-wage jobs.

No sizable industry or occupation other than food-service occupations in thefood service industry has a high proportion of minimum wage employees. Thus,we cannot easily replicate our approach for another occupation / industry combina-tion. The strongest candidate is sales workers in retail trade. Even after excluding3-digit industries and occupations that tend to have few low-wage workers (e.g.fuel dealers, hardware and building supplies sales workers), only 14% of the

12remaining workers earn $3.35 or less.Nevertheless, we investigated whether any significant compositional effects

could be detected, using the approach we applied to food service. Not surprisingly,we obtained very few statistically significant and consistent results. We cautiouslyidentify two possible effects. The minimum wage increase may have shiftedemployment in the retail trades towards part-timers and women. We obtainconsistent signs for these variables across a variety of specifications and estimationtechniques. In some cases, but not all, the effects are marginally significant but

13never more than at the 0.1 level. Results for the remaining variables were notconsistent across specifications and always fell well short of significance.

Thus we see some weak evidence that, as in the food service industry, theminimum wage change may have impacted the demographic composition oflow-wage jobs in retail sales. However, the results are not consistent acrossindustries regarding which groups are affected. In retail, higher minimum wagesseem to have shifted employment towards part-timers and women, while in food

11 One possible explanation for the shift towards students is that student employment may allowemployers to take advantage of the training sub-minimum wage. However, other studies have shownthat very little use is made of the sub-minimum wage (Katz and Krueger, 1992; Card et al., 1994). Inour data on food-service workers, there is absolutely no evidence of a spike in the data at thesub-minimum. For the year subsequent to the 1st April 1991 minimum wage increase, the sub-minimum wage was $3.62. Not a single member of the sample earned $3.62. Therefore, thesub-minimum wage is unlikely to account for this shift.

12 The sample of sales workers in the remaining retail trades is approximately 6500 per year.13 As with the food service industry, we estimated these relations using a variety of methods—robust

regression, OLS and GLS—and using both the hourly wage variable and the wage variable calculatedfrom usual weekly earnings and hours. For both female and full-time/part-time dependent variables,the sign of the effects was consistent across estimation techniques and choice of explanatory variablebut significance levels varied.


services higher minimum wages shifted employment most clearly towards youthand towards students, although a shift towards part-timers may have occurred inthis industry as well. These industry differences may be spurious. Our inability toisolate a sub-sector of retail trade with a high fraction of workers receiving aminimum wage may have induced us to look for effects that simply are notmeasurable. Alternatively, there may be genuine differences in the industry /occupation groups in the types of workers employers prefer and/or in thedemographics of the marginal workers attracted by a higher minimum wage. Forexample, the fact that retail workers tend to be much more highly female than foodservice jobs may partially reflect employer preferences. In sum, while the retailresults are weak enough to be only suggestive, it may be that there are importantbut somewhat different demographic impacts of minimum wages in differentoccupations.

We can also discern some indication from Card and Krueger (1994) that theminimum-wage law shifts food service employment from adults to teenageworkers. They find that, following the minimum wage increase in New Jersey,relative to Pennsylvania, teenage employment rose by 2 percentage points andadult employment fell by 2.9 percentage points (pp. 789–90 and Footnote 31).

In a sample not limited to a specific industry, Neumark and Wascher (1996) findcompositional effects of increasing minimum wages within the teenage population,an issue that our data are not sufficient to address. They find that increasing theminimum wage induces teenagers who had been simultaneously employed and inschool to leave school (and seemingly work longer hours) while teenagers who hadbeen employed but not in school were displaced.

4. Discussion and conclusions

There are two important criteria for assessing a stylized model such as the onedeveloped in Section 2. The first is whether the theory improves upon existingtheoretical explanations. Here we believe the answer is ‘yes’. A natural reaction tothe recent empirical findings of no or positive employment effects of minimum-wage laws has been to develop arguments for the presence of monopsony power inthe labor market. Thus, in search (Mortensen and Burdett, 1989) and certainefficiency wage models (Rebitzer and Taylor, 1995), firms increase wages whenthey want to employ more workers and thus exhibit monopsony-like behavior.However, although individual monopsonistic firms increase employment inresponse to a small minimum wage, aggregate employment need not increase in amonopsonistic labor market, because the increased output lowers price and inducesexit (Bhaskar and To, 1996).

Moreover, the monopsony-like models offer little insight into compositionalissues. The only predictions that monopsonistic models generate about which


group’s employment will be most impacted relate to the groups’ elasticity of labor14supply, and even here the predictions seem weak.

The second important criterion for evaluating the model is the power to predictan empirically verified result. The results presented in Section 3 at least suggestthat minimum-wage laws shift employment towards teenagers and students.Together with newspaper accounts that minimum-wage laws bring low-wageemployers ‘better kids’, this indicates to us that the issues raised by the model maybe relevant.

For the empirical work to be directly applicable to the theoretical model, theteenagers would have to be type 1 workers who gain no real benefit from theirenhanced opportunities. Of course, this model is clearly stylized and may notapply in its details. The higher minimum wage might help teenagers and studentssubsidize their education, allowing them to acquire additional skills that willimprove their future labor market opportunities. On the other hand, these teenagedstudents might be middle-class students who would find other work and would, inany event, be changing their economic circumstance within a few years with orwithout this job. It is even possible that the positive employment effect on thisgroup has a perverse effect, giving relatively productive teenagers less incentive topursue their education.

The complexity of the possibilities suggests that a full examination of theredistributive effects must be conducted by the policy and intellectual community.The examination requires not just identification of impacted groups, but also atheoretical context within which to evaluate these impacts. This paper represents acontribution to this process.

Acknowledgements

This paper grew out of work Lang coauthored with William Dickens. We aregrateful to him for helpful conversations, to Mukesh Chawla for his researchassistance and to an anonymous referee, Tony Atkinson, Eli Berman, Larry Katz,Alan Krueger, Steve Pischke, Jim Rebitzer, and participants in the Harvard /MITLabor Workshop for their comments and criticisms. This project was funded inpart by NSF grants SES-9223349 and SBR-9515052. A preliminary version of theempirical work described at the end of this paper was funded by the EmploymentPolicies Institute. The usual caveat applies.

14 Under monopsony, workers with more elastic labor supply will tend to receive higher wages, andtherefore be less likely to be affected by an increase in the minimum wage. On the other hand, amongthose who are affected by the minimum wage, the labor supply response will be greater for those typesof workers whose labor supply is more elastic.


Appendix A

Lemma.2z 2z 2z 2z(1 2 e 1 ze ) /(1 2 e ) # ln z 2 ln(1 2 e ) z $ 0. (L1)

Proof.2z 2z 2z 2z

≠(ln z 2 ln(1 2 e )) /≠z 5 (1 2 e 2 ze ) / [z(1 2 e )]. (L2)

2zSince ln z2ln(12e ) is concave in z

2z 2z≠(ln z 2 ln(1 2 e )) /≠z , [ln z 2 ln(1 2 e )] /z. (L3)

Combining Eqs. (L2) and (L3) and multiplying by z, proves the lemma.

Proof (that a sufficiently small minimum wage raises employment).Let: N 5number of high-wage firms; N 5number of low-wage firms; z 51 2 1

expected number of type 1 applicants to high-wage firms; z 5expected number of2

type 2 applicants to low-wage firms; z*5expected number of type 1 applicants tolow-wage firms; c 5number of type 1 workers (exogenous); c 5number of type 21 2

workers (exogenous).Then

z N 1 z*N 5 c (A1)1 1 2 1

z N 5 c . (A2)2 2 2

Fully differentiating,

z dN 1 z*dN 1 N dz* 5 0, (A3)1 1 2 2

or, evaluated at no minimum wage,

dN 5 2 (N /z )dz* (A4)1 2 1

and

dN 5 2 (N /z )dz . (A5)2 2 2 2

Total employment is given by the probability that a vacancy is filled in each sectormultiplied by the number of firms (vacancies) in that sector.

2z 2z 2z*1 2E 5 (1 2 e )N 1 (1 2 e )N . (A6)1 2

Fully differentiating the employment level yields

2z 2z 2z* 2(z 1z*)1 2 2dE 5 (1 2 e )dN 1 (1 2 e )dN 1 e N d(z 1 z*) (A7)1 2 2 2

which, evaluated at no minimum wage, equals


2z 2z 2z1 2 2dE 5 (1 2 e )dN 1 (1 2 e )dN 2 e N d(z 1 z*). (A8)1 2 2 2

Substituting for dN and dN gives1 2

2z 2z 2z 2z2 1 2 2dE 5 N h[e 2 (1 2 e ) /z ]dz* 1 [e 2 (1 2 e ) /z ]dz j (A9)2 1 2 2

To find dz*, note that equilibrium for type 1s requires that2z*w (1 2 e ) /z* 5 k (A10)2

so that2z* 2z* 2z* 2*dw (1 2 e ) /z* 1 [(z*e 2 (1 2 e )w ) /z ]dz* 5 0. (A11)2 2

Evaluating Eq. (A11) at z*50 and applying l’Hopital’s rule yields2z1dz* 5 (2 /w )dw 5 2/(ve )dw (A12)2 2 2

where the second equality uses Eqs. (1) and (4).To find d(z*1z ) use the zero-profit condition for low-wage firms2

2(z*1z )2(v 2 w )(1 2 e ) 2 d 5 0. (A13)2

Taking derivatives and evaluating at z*50 gives

2z z 2z2 2 1d(z* 1 z ) 5 (1 2 e ) /(ve (1 2 e )) (A14)2

where we have again used the fact that w 5k5v exp(2z ).2 1

Substituting Eqs. (A12) and (A14) into Eq. (A9) gives

2z 2z 2z 2z 22 1 2 21 2 e 2(1 2 e ) 2(1 2 e ) (1 2 e )]]] ]]]] ]]]] ]]]]]dE 5 (N /v) 2 1 2 .H J2z 2z 2z 2z 2z2 1 1 1 2 11 2 e z e z e z e (1 2 e )1 2 2

(A15)

To find z as a function of z , combine the two zero-profit constraints to get2 1

2z 2z1 2(1 2 e )(v 2 w ) 5 (1 2 e )(v 2 w ). (A16)1 2

or2z 2z 2z1 2 1(1 2 e) (v 2 w ) 5 (1 2 e )v(1 2 e ) (A17)1

which simplifies to2z2w 5 ve . (A18)1

Substituting for w in Eq. (4) gives1

2z 2z1 2w 5 v(1 2 e )e /z . (A19)2 1

Substituting for w and w in Eq. (A16) and rearranging terms1 2

2z1z 5 2 [ln z 2 z 2 ln(1 2 e )]. (A20)2 1 1


2Multiplying dE by v(1 2 exp(2z )) z z and substituting for z gives the1 1 2 2

following condition for dE to be positive:

2z 2z 2z 31 1 1A ; z hz (1 2 e 2 z e ) 2 2(1 2 e ) j1 1 1

2z 2z 2z 2z 2z 31 1 1 1 11 ln[(1 2 e ) /z ]hz (1 2 e 2 z e )e 2 2(1 2 e ) j1 1 1

2z 2z 2z 2z 2z 21 1 1 1 11 2z (1 2 e 2 z e )(1 2 e ) 2 (1 2 e 2 z e ) . (A21)1 1 1

Using the lemma, we have

2 2z 2z 2z 2z 31 1 1 1A . z (1 2 e 2 z e )e 2 2z (1 2 e )1 1 1

2z 2z 2 2z 2z 2z 2z1 1 1 1 1 12 (1 2 e 2 z e ) z e /(1 2 e ) 1 2(1 2 e 2 z e )(11 1 1

2z 2 2z 2z 2z 2z 2z 21 1 1 1 1 12 e ) 1 2z (1 2 e 2 z e )(1 2 e ) 2 (1 2 e 2 z e )1 1 1

2z1; A9 /(1 2 e ). (A22)

Rearranging terms and letting x5exp(2z )1

2 3 4 2 2A9 5 1 2 5x 1 9x 2 7x 1 2x 1 (ln x) x(21 1 3x 2 2x )2 2

2 ln x(1 2 2x 1 x 2 (ln x) x)x. (A23)

A sufficient condition for dE to be positive is that A9 be positive. A9 equals 0when x equals 1 (z 50). A sufficient condition for A9.0 (0,x,1) is that1

dA9 /dx,0. Now2 2dA9 /dx 5 2(x 2 1) (4x 2 3) 2 (ln x)(x 2 1)(7x 2 3) 1 (ln x) (3x 2 1)

31 2x(ln x) . (A24)

Since the last term is negative, a sufficient condition for the derivative to benegative is that the last three terms (;B) be negative. Note that B is continuousand equal to 2` when x50. If B is positive, it must therefore equal 0 for somevalue. If B50, then we can apply the quadratic formula to find ln x as a functionof x which gives

2 0.5(ln x) /(x 2 1) 5 [(7x 2 3)6(247x 1 62x 2 15) ] /(6x 2 2). (A25)

If there are no values of x such that ln x is the appropriate function of x, then Bcannot equal zero. For x,30/94, applying the quadratic formula implies that ln xis imaginary; so B cannot equal 0 for x,30/94.

We now show that dB /dx.0 for x.30/94 which is sufficient to show that B isnegative everywhere. Taking the derivative and rearranging terms we get

2dB /dx 5 [(x 2 1) /x][3(x 2 1)(8x 2 1) 2 2(ln x)(7x 2 1)] 1 3(ln x) . (A26)

A sufficient condition for this to be positive is that the second square-bracketedterm be negative. Note that this term is more negative than 3(x21)22(ln x) which


is negative for x.0.42. Note also that for x.30/94, this term is more negativethan 3(x21)(8x21)23.58(x21)(7x21) (since ln x.1.79(x21)) which is nega-tive for x,0.54. Hence, the square-bracketed term is negative for x.30/94 and,thus, dB /dx is positive for x.30/94, and B is everywhere negative.

References

Bhaskar, V., To, T., 1996. Minimum Wages for Ronald McDonald Monopsonies: A Theory ofMonopsonistic Competition, University of St. Andrews, unpubl.

Card, D., 1992. Using regional variation in wages to measure the effects of the federal minimum wage.Industrial and Labor Relations Review 46, 22–37.

Card, D., Katz, L.F., Krueger, A.B., 1994. Employment effects of minimum and subminimum wages:Panel data on state minimum wage laws: Comment. Industrial and Labor Relations Review 48,487–496.

Card, D., Krueger, A.B., 1994. Minimum wages and employment: A case study of the fast foodindustry in New Jersey and Pennsylvania. American Economic Review 84, 772–793.

Card, D., Krueger, A.B., 1995. Myth and Measurement, Princeton University Press, Princeton, NJ.Katz, L.F., Krueger, A.B., 1992. The effect of the minimum wage on the fast-food industry. Industrial

and Labor Relations Review 46, 6–21.Lang, K., Dickens, W.T., 1992. Labor Market Segmentation, Wage Dispersion and Unemployment,

NBER Working Paper No. 4073.Montgomery, J.D., 1991. Equilibrium wage dispersion and interindustry wage differentials. Quarterly

Journal of Economics 106, 163–180.Mortensen, D.T., Burdett, K., 1989. Equilibrium wage differentials and employer size, Northwestern

Center for Mathematical Studies in Economics and Management Science, Working Paper No. 860.Nasar, S., 1992. Forging new insight on minimum wages and jobs. New York Times (June 29) D1, D4.Neumark, D., Wascher, W., 1996. Reconciling the evidence on employment effects of minimum

wages—A review of our research findings, in: The Effects of the Minimum Wage on Employment,AEI Press, Washington, DC.

Rebitzer, J.B., Taylor, L.J., 1995. The consequences of minimum wage laws: Some new theoreticalideas. Journal of Public Economics 56, 245–255.

Stata Corporation, 1993, Stata Users Guide, Stata Press, College Station, Texas.

The effect of minimum-wage laws on the distribution of employment: theory and evidence

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