Minimum Wages and Hours of Work

Transcription

1 Minimum Wages and Hours of Work Preliminary and Incomplete Ross Doppelt August 10, 2017 Abstract I investigate, both empirically and theoretically, how minimum-wage laws aect the intensive margin of labor, or the number hours per employee. Using CPS data, I document the fact that minimum-wage employees work longer hours when the minimum wage is higher. To explain this pattern, I introduce a theoretical model of search and bargaining, subject to minimum-wage laws. Within a match, the number of hours is determined by an upward-sloping labor-supply curve, so people are willing to work more when the minimum wage goes up. As long as a worker's productivity exceeds the minimum wage, an employer is willing to accept the extra labor. However, higher wages diminish total prots, vacancy creation, and employment. I derive conditions under which a minimum wage can be welfare-improving, and I discuss empirical tests to determine whether those conditions are satised. After deriving these analytical results, I extend the model to facilitate a quantitative analysis with heterogeneity in wages and hours. Contact: rdoppelt@andrew.cmu.edu. Comments are welcome. This draft is preliminary and incomplete, so please do not cite nor circulate this version without permission. I thank Gaston Chaumont and Eunbi Ko for research assistance. 1

2 1 Introduction The majority of minimum-wage research focuses on the extensive margin, or the number of workers who are employed. However, understanding the intensive margin, or the number of hours per worker, is necessary to answer certain fundamental questions about how minimum wages aect labor markets. Does a statemandated increase in wages induce an increase in incomes? Does the aggregate quantity of labor, as measured in total person hours, go up or down? What are the costs and benets of a distortionary policy when there are two margins of labor being distorted? A higher minimum wage leads to an increase in hours, and I will advance this argument along three fronts. First, I document the association between minimum wages and hours of work using data from the Current Population Survey (CPS). Second, I introduce a theoretical model of search and bargaining, subject to minimum-wage laws; the model is simple enough to impart analytical results for both the decentralized equilibrium and the planner's problem. Third, as a framework for quantitative analysis, I develop a dynamic general-equilibrium model with heterogeneity in wages and hours. These quantitative results are preliminary: I derive the likelihood and discuss identication issues, but the current draft of the paper only contains a calibration exercise. When the minimum wage is higher, we observe minimum-wage employees working longer hours. This positive correlation appears in both cross-sectional and longitudinal data. The CPS's rotating-panel structure allows us to see two observations, 12 months apart, on a worker. If the minimum wage goes up during those 12 months, then we can see the change in hours amongst workers whose initial wage was above the old minimum, but below the new minimum. As a comparison, we can also see the change in hours amongst workers who are observationally similar, but whose initial wage was above the new minimum. A regression estimated with both groups suggests that aected employees increase their hours following a minimum-wage increase: If a worker gets a 10% real wage increase, then her expected hours of work increases 17.9%, conditional on remaining employed. That being said, minimum-wage employees are more likely to lose their jobs: If a worker gets a 10% real wage increase, then her probability of being employed one year later decreases by 4.5%. The positive relationship between minimum wages and hours of work is dicult to explain with competitive labor markets, but this pattern emerges naturally from search and bargaining. To illustrate the basic idea, I provide a one-shot model: Firms post vacancies, are matched with workers, and then bargain over the terms of employment. When the minimum wage binds, workers and rms only bargain over hours, taking the wage as given. Within a match, the number of hours is determined by an upward-sloping labor-supply curve. In particular, the bilaterally bargained labor supply is a convex combination between the labor sup- 2

3 ply that would would prevail under pure competition, and the labor supply that would prevail under pure monopsony. When the state increases wages, an employee is willing to work more. As long as the wage is less than the worker's productivity, the rm is willing to accept the extra hours. But a rm is only willing to hire more labor within a match. Higher wages push down prots, so rms post fewer vacancies, leading to higher unemployment. The fact that employment and hours move in opposite directions underscores the importance of studying the distinct forces that shape each margin of labor. When setting the minimum wage, policymakers face a tradeo between distorting the intensive and extensive margins. Generically, the equilibrium allocation is inecient, because of a congestion externality: By posting a vacancy, a rm makes it harder for all other rms to nd workers. Like Hosios [1990], I nd that the equilibrium will only be ecient when the worker's bargaining power coincides with the elasticity of the matching function. A minimum wage can improve welfare if the worker's bargaining power is too low, but this policy can never attain the unconstrained optimal allocation. Absent a minimum wage, workers and rms would each get a xed fraction of the joint match surplus; consequently, they would agree to set hours in a way that maximizes the surplus. By pushing hours above this laissez-faire level, the minimum wage shrinks the surplus within each match. In the context of the one-shot model, I characterize the welfare-maximizing minimum wage that balances the benet of reducing congestion against the cost of reducing surpluses. Besides deriving the theoretical optimum, I show that choosing the minimum wage to maximize welfare is equivalent to maximizing total payrolls. This result is useful, because governments (and econometricians) can observe payrolls, but not welfare. The dynamic model embellishes the one-shot model by incorporating heterogeneity across matches. The equilibrium features dispersion in wages and hours, with the minimum wage binding on some matches, but not others. Still, the policy aects everyone: Workers cycle in and out of unemployment, so the worker's threat point when bargaining is endogenous, and depends on the minimum wage. I construct the likelihood function by deriving the equilibrium distribution over wages, hours, unemployment durations, and employment statuses. Not all of the model's deep parameters can be identied, but it's still possible to make inferences by taking a Bayesian approach with an informative prior. The estimation is work in progress; in the current version of the paper, I perform a calibration exercise to generate some preliminary results. Like the one-shot model, the dynamic model suggests that a higher minimum wage causes both hours and unemployment to rise, although the magnitude of the increase is modest. Under the preliminary parameterization, the model's welfare implications are fairly sensitive to the size of the worker's bargaining power, relative to the Hosios condition. To the best of my knowledge, this paper provides the rst model with all three of the following ingredients: a minimum wage, both margins of labor, and search frictions. Clearly, the rst two components are necessary 3

4 to answer the question at hand, but search frictions are also important for understanding the problem. The most obvious risk of implementing a minimum wage is higher unemployment, and the most natural way of modeling unemployment is with search. There are numerous models that analyze the minimum wage using the tools of search theory, but abstracting from the intensive margin. Of these, the most closely related are due to Flinn [2006, 2011]. I generalize Flinn's framework by adding variable hours and incorporating a richer form of heterogeneity across matches. In the context of competitive labor markets, without search frictions, several authors have developed theories of how the minimum wage aects both margins of labor. Examples include Strobl and Walsh [2011] and Michl [2000], who assume that employment and hours enter the production function as separate arguments. 1 Michl nds that higher wages have a negative eect on hours, whereas Strobl and Walsh nd an ambiguous eect. These results are largely driven by assumptions about the curvature of the production function. I will adopt a simpler specication: Each person's output is linear in hours of work. Consequently, the relationship between wages and hours is primarily driven by the way workers and rms bargain over the terms of employment, not assumptions about technology. Several studies have explored the statistical relationship between minimum wages and hours of work. The results have been mixed. Zavodny [2000] nds a positive relationship; Connolly and Gregory [2002] and Allegretto et al. [2011] nd no signicant relationship; Stewart and Swaeld [2008] and Couch and Wittenburg [2001] nd negative relationships. The regressions that I perform are closest to Zavodny's. She initially proposed studying the hours response of individual workers to the minimum wage, comparing those who are directly aected by a policy change to those who are not. Although we both work with CPS data, we use almost entirely non-overlapping samples: She studies teenage workers between 1979 and 1993, whereas I study all workers between 1990 and The correlations that are evident in these data match the predictions of the theory: When the minimum wage goes up, aected workers are more likely to enter unemployment, but those who remain employed work longer hours. If one ascribes a causal interpretation to these regressions, then the economic theory changes the interpretation of the coecients. In a textbook competitive model, increasing the minimum wage traces out an upward movement along a market-wide labor-demand curve. In the search-and-bargaining model, increasing the minimum wage traces out an upward movement along a match-specic labor-supply curve. In that respect, my results echo an old line of argument about how minimum wages interact with monopsony power. Stigler [1946] pointed out that a monopsonist, who takes the market labor-supply curve as given, could choose to keep employment low in order to keep wages low. A minimum wage may therefore induce a monopsonist to hire more people. Bhaskar and To [1999] make a similar point with a modern model of 1 Lee and Saez [2012] study how the minimum wage can be combined with optimal non-linear taxes when labor markets are competitive, but subject to incomplete information. They consider variable hours of work as an extension to their baseline model. 4

5 monopsonistic competition, although those authors abstract from search frictions and the intensive margin. The literature has long recognized that search models have monopsony-like features, because a worker is only matched with one rm at a time. But unlike monopsony models, most search models (including mine) imply that minimum wages lead to higher unemployment. The monopsony analogy is much more applicable to the intensive margin than the extensive margin. Extensively, there are many rms that could hire a worker out of unemployment. Intensively, a worker's current employer will be the only rm in a position to purchase that worker's time. Indeed, the model predicts that minimum wages depress employment while inating hours. I will proceed as follows. Section 2 provides evidence from the regressions using CPS data. Section 3 introduces the one-shot model. Section 4 contains the dynamic model with heterogeneity. Section 5 discusses structural estimation of the dynamic model. Section 6 contains preliminary quantitative results from a calibration exercise. All proofs are in the appendix. 2 Regression Analyses Before expositing the theory, I will document the correlation between minimum wages and hours of work using CPS data. Section 2.1 presents cross-sectional evidence, and Section 2.2 presents longitudinal evidence. Section 2.2 also documents the relationship between changes in the minimum wage and the probability of exiting employment. The sample consists of workers from the CPS outgoing rotation group (ORG) samples between 1990 and The CPS has a rotating-panel design, and each household in the sample is tracked over 16 months. Respondents are surveyed for four months, ignored for the next eight months, and then surveyed again for four months. The CPS only asks about hourly wages and usual hours of work when a respondent is in months 4 and 16 (i.e., in an outgoing rotation). I will adopt the following notation. Let W i,t denote worker i's nominal wage at date t; let W m,i,t denote the nominal minimum wage in the state where worker i lives at date t. Let mwe i,t be an indicator variable that's equal to one if worker i is a minimum-wage earner at date t (i.e., if W i,t = W m,i,t ). Let w i,t denote worker i's real wage at date t; let w m,i,t denote the real minimum wage in the state where worker i resides at date t. 2 Let e i,t be an indicator variable that's equal to one if worker i is employed at date t. Let l i,t denote worker i's usual weekly hours of work at date t. Throughout, I will restrict the sample to workers who are paid on an hourly basis. The household data are downloaded from IPUMS CPS, which also provides detailed documentation. 3 2 Real wages are computed as nominal wages, divided by the CPI. 3 See: The data on state-level minimum wages is 5

6 Table 1: Cross-Sectional Regressions Included Coecient on Standard Coecient on Standard Covariates log (w m,i,t ) Error log (w m,i,t ) mwe i,t Error None CS CS CS compiled by Ben Zipperer, who makes the data available on his website Cross-Sectional Regressions I will regress log (l i,t ) on log (w m,i,t ), log (w m,i,t ) mwe i,t, mwe i,t, and covariates. I will consider the following congurations of covariates for the cross-sectional regressions: No covariates Covariates CS-1: age, age squared, race (black, white, other), Hispanic origin (yes or no), education (no high-school diploma, high-school graduate, college graduate) Covariates CS-2: Covariates CS-1, plus industry, occupation, and state-level unemployment rate Covariates CS-3: Covariates CS-2, plus state xed eects and year xed eects All regressions include a constant. Including workers who do not earn the minimum wage serves two functions. First, it gives us an idea of whether minimum wages are systematically related to the hours of non-minimumwage workers. Second, these people provide a comparison group. That is, for minimum-wage earners, the regressions capture the conditional correlation between minimum wages and hours of work, relative to people who don't earn the minimum wage, but who are otherwise observationally similar. The coecient on log (w m,i,t ) summarizes the association between minimum wages and hours of work for non-minimum-wage earners. For minimum-wage earners, the minimum wage is positively associated with hours of work if the coecients on log (w m,i,t ) and log (w m,i,t ) mwe i,t sum to a positive number. Table 1 shows results from the cross-sectional regressions. In each specication, the sum of the coecients on log (w m,i,t ) and log (w m,i,t ) mwe i,t is, in fact, positive. Although the coecient on log (w m,i,t ) comes out negative, the association between minimum wages and hours appears weak for non-minimum-wage earners. When the regression includes state and year xed eects (CS-3), the coecient on log (w m,i,t ) has only borderline signicance, with a p-value of 0.054; furthermore, the magnitude of the coecient is practically quite small. The positive correlation between wages and hours for minimum-wage earners is both stronger and 4 See: 6

7 more robust. Of course, the cross-sectional regressions tell us nothing about what's driving this correlation. One possibility is that individuals earning the minimum wage increase their hours when the minimum wage goes up. Another possibility is that, when the minimum wage is higher, the pool of minimum-wage earners consists of people who tend to work more. To demonstrate that the correlation is not just coming from composition eects, I will turn to longitudinal regressions to study the hours responses of individual workers. 2.2 Longitudinal Regressions Following Zavodny [2000], I will study the hours responses of workers who are caught up in a minimum wage increase, relative to those who are not. Because the CPS has a rotating panel structure, we can observe a worker at two points in time, 12 months apart. Suppose that we observe a worker at dates t and t 12. I will say that worker i is aected by a minimum-wage increase if: W m,i,t 12 W i,t 12 < W m,i,t. (2.1) That is, the worker's initial wage weakly exceeds the old minimum wage, but is less than the new minimum wage. 5 Notice that, by construction, workers are not considered aected unless there is a statutory increase in the minimum wage where they live. Beyond categorizing workers as aected and not aected, we can capture how strongly the policy change binds on an individual. To that end, dene the wage gap as: { } wm,i,t w max i,t 12 w i,t 12, 0 if worker iis aected gap i,t 0 if worker iis not aected. (2.2) In other words, gap i,t represents the real wage increase, in percent terms, an aected worker needs in order to comply with the new minimum wage. Minimum-wage statutes govern nominal wages, which is why equation (2.1) is in nominal terms. However, economic behavior is governed by real wages, which is why equation (2.2) is in real terms. I will study regressions in which the response variable is % l i,t li,t li,t 12 l i,t 12. The sample is restricted to workers with e i,t = e i,t 12 = 1. I will regress % l i,t on gap i,t and covariates. I will consider the following congurations of covariates for the longitudinal regressions: No covariates Covariates L-1: Age, age squared, race (black, white, other), Hispanic origin (yes or no), education (no 5 I am using the word aected as shorthand to mean that a worker satises the condition given in equation (2.1). In a broader sense, other workers could be subject to some kind of spillover eect, even if they earn more than the minimum wage. For instance, Lee [1999] argues that the minimum wage indirectly pushes up the wages of non-minimum-wage workers. 7

8 Table 2: Longitudinal Hours Regressions Included Covariates Coecient on gap i,t Standard Error None L L L high-school diploma, high-school graduate, college graduate) Covariates L-2: Covariates L-1, plus the change in the state-level unemployment rate for the state in which worker i is located between dates t 12 and t Covariates L-3: Covariates L-2, plus state xed eects and year xed eects All regressions include a constant. Again, including workers who are not aected by a minimum-wage increase provides a comparison group. Table 2 shows results from the longitudinal hours regressions. In each specication, the coecient on gap i,t is positive and signicant. To put the magnitudes in perspective, consider the coecient from the regression that includes year and state xed eects (L-3). If a worker gets a 10% wage increase following a change in the minimum wage, then we would expect that worker's hours to go up by about 17.9%, conditional on remaining on employed. These results are striking, but some qualications are necessary for interpreting the results. The numbers in Table 2 summarize the conditional correlation between changes in the minimum wage and changes in hours for people who are caught up in the policy change and this correlation is the opposite of what we would expect from a textbook model of supply and demand. There are two possible barriers to interpreting the regression coecients as causal eects. First, the CPS doesn't record a respondent's employer, so we cannot see whether someone changes jobs. It's possible that a higher minimum wage purges the economy of low-productivity, part-time jobs. Then, some aected workers might eventually nd new jobs that require longer hours. Second, the regressions only include workers employed at date t 12 who remain employed at date t, which raises the possibility of sample-selection eects. However, these would be dierent from the composition eects entering the cross-sectional regressions. In the cross-section, we have to consider the possibility that the people who retain their minimum-wage jobs are people who happen to work more. In the longitudinal regressions, selection eects would arise if unobserved factors that aect a worker's propensity to remain employed are correlated with unobserved factors that aect the growth rate of hours. The theory presented in Section 4 suggests that changes in the minimum wage will alter the composition of jobs, but within a job, hours are constant unless the minimum wage changes. The data also allow us to examine how changes in the minimum wage aect the probability of remaining employed. For the sample with e i,t 12 = 1, I will regress e i,t on gap i,t on covariates. The results are 8

9 Table 3: Longitudinal Employment Regressions Included Covariates Coecient on gap i,t Standard Error None L L L summarized in Table 3. Again, to put the magnitudes in perspective, consider the coecient from the regression that includes year and state xed eects (L-3). If a worker gets a 10% wage increase following a change in the minimum wage, then the worker's probability of remaining employed one year later goes down by about 4.5%. This result conforms to the convention wisdom; i.e., low-wage workers are more likely to lose their jobs following a minimum wage increase. 3 A One-Shot Model Section 3.1 introduces an environment where workers and rms must search for matches, and then bargain over wages and hours, subject to a minimum wage law. Section 3.2 characterizes the equilibrium to ush out the model's positive predictions. Sections 3.3 and 3.4 analyze the model's welfare properties. Section 3.5 links the positive and normative implications of the model, by proposing an empirical test for when it's optimal to have a binding minimum wage. 3.1 Environment Agents and Timing There is a unit measure of workers, all of whom are initially unemployed. The measure of rms is determined by free-entry. There is no heterogeneity across workers, nor across rms. First, rms post vacancies. Second, workers are matched to rms. Third, workers and rms bargain over wages and hours. Finally, workers supply labor and produce output. Production Technology Production takes place in worker-rm pairs. Let l denote the intensive labor supply, or the number of hours worked by a single employee. If a rm hires a worker for l hours, then it produces zl units of output. Search Technology A rm posting a vacancy pays cost κ. Let θ denote market tightness, or the ratio of vacancies to job seekers. 6 The probability of a rm meeting a worker is q (θ), and the probability of a worker meeting a rm is p (θ) θq (θ). Standard assumptions apply: q ( ) is strictly decreasing and convex, and 6 In the one-shot model, all workers are seeking jobs, so θ is just equal to the number of vacancies. In the dynamic model, θ will be the ratio of vacancies to unemployed workers. 9

10 p ( ) is strictly increasing and concave. It will be convenient to dene η (θ) θ q (θ) q(θ). Note that η (θ) must be bounded between zero and one, because q (θ) is decreasing with elasticity η (θ), and p (θ) is increasing with elasticity 1 η (θ). Preferences Workers seek to maximize utility, and rms seek to maximize prots. A worker who receives wage w and provides l hours of labor gets utility wl v (l), where v ( ) is a positive-valued, twice-dierentiable, strictly increasing, and strictly convex function that represents the disutility of labor. To ensure the existence of an equilibrium, I will maintain the assumption that v (0) = 0. The utility associated with unemployment is normalized to zero. 7 A rm that hires a worker for l hours at wage w earns prot (z w) l; a rm that does not match with a worker gets nothing. Bargaining When a match is formed, wages and hours are determined by Nash bargaining, and β (0, 1) is the worker's bargaining power. Wages and hours must solve: max w,l [wl v (l)]β [(z w) l] 1 β s.t. w w m and l 0, (3.1) where w m is the minimum wage. I will maintain the assumption that z > w m, so rms will actually be willing to hire workers. 3.2 Equilibrium Free entry requires that the cost of posting a vacancy equalizes with the expected prot: κ = q (θ) (z w) l. (3.2) Because the size of the labor force is normalized to one, the number of employed workers, denoted e, is given by the probability of an individual worker nding a job: e = p (θ). (3.3) We can now dene an equilibrium. Denition. An equilibrium consists of a wage w, an intensive labor supply l, an employment level e, and a value of market tightness θ such that (w, l) maximizes the constrained Nash product (3.1), the free-entry 7 This normalization comes without loss of generality: What matters is the dierence between the utility from employment and the utility from unemployment. Assuming that a worker gets utility b during unemployment is equivalent to replacing v (l) with v (l) + b. 10

11 condition holds (3.2), and the employment level equals the job-nding probability (3.3). Let (w n, l n, e n, θ n ) denote the equilibrium outcome when there is no minimum wage (or, equivalently, when the minimum wage is nonbinding). Let (w m, l m, e m, θ m ) denote the equilibrium outcome when the minimum wage is binding. First, I will characterize l, θ, and e as functions of w; these characterizations will be valid in any equilibrium, regardless of whether the minimum wage binds. In turn, I'll analyze an equilibrium without a binding minimum wage, and an equilibrium with a binding minimum wage. The rst-order condition of the Nash product (3.1) with respect to l establishes a link between wages and hours worked: w = βv (l) + (1 β) v (l). (3.4) l The above expression can be interpreted as a labor-supply curve. In particular, it's a convex combination between a competitive labor-supply condition and a monopsonistic labor-supply condition. If labor markets were frictionless and competitive, then a worker would take w as given and choose l to maximize wl v (l): w = v (l). (3.5) If labor markets were frictionless and monopsonistic, then a single rm would dictate wages and hours, taking into account workers' labor-supply decisions. To ensure worker participation, a monopsonist would have to make wl v (l), the utility from employment, equal to zero, the utility from unemployment: w = v (l). (3.6) l The bilaterally bargained labor supply (3.4) is a convex combination between equations (3.5) and (3.6): The worker's bargaining power β determines the weight on the competitive labor-supply function, and the rm's bargaining power 1 β determines the weight on the monopsony labor-supply function. Furthermore, to interpret equation (3.4) as a labor-supply curve, we need to establish that it's upwardsloping. To that end, dene the function S (l) as being equal to the right-hand side of equation (3.4). Proposition 1. If S (l) l v (l) 0, then S (l) is a strictly increasing function of l. The condition S (l) l v (l) 0 must hold in an equilibrium: It simply means that w and l satisfy equation (3.4) while simultaneously providing the worker with some weakly positive surplus. To analyze how changes in the minimum wage aect hours, it will be convenient to write l as a function of w, rather than w as a function of l. This is trivial, because S ( ) is monotone. Corollary. Suppose S (l) l v (l) 0 and w = S (l). Then, there exists an inverse function s ( ) S 1 ( ) 11

12 such that l = s (w). The function s (w) is strictly increasing. It remains to characterize market tightness and employment as functions of w. Evaluating the free entry-condition (3.4) at l = s (w) provides a relationship between market tightness and wages: κ = q (θ) (z w) s (w). (3.7) From the above, we can recover equilibrium market tightness, given the wage, as θ = Θ (w), where I have dened the function: ( ) Θ (w) q 1 κ. (3.8) (z w) s (w) It follows that equilibrium employment, as a function of the wage, is e = p (Θ (w)). Equilibrium without a Binding Minimum Wage Without a binding minimum wage, the value of w that maximizes the Nash product (3.1) must satisfy the rst-order condition: wl v (l) = β [zl v (l)]. (3.9) The total surplus generated by a match is zl v (l), the sum of the rm's prot and the worker's utility. The bargained wage ensures that a fraction β of the surplus goes to the worker, and a fraction 1 β goes to the rm. Combining equations (3.4) and (3.9) shows that l n maximizes the total match surplus: z = v (l n ). (3.10) Given l n, it's straightforward compute the remaining equilibrium variables: w n = S (l n ), θ n = Θ (w n ), and e n = p (θ n ). These outcomes provide a useful benchmark for analyzing the distortions caused by minimum wage laws. When wages and hours are left unrestricted, they can perform distinct functions. The number of hours is the instrument that determines the total value of the match. The wage is the instrument that determines how much of the match value goes to the worker, relative to the rm. When wages are constrained, bargaining over l must perform two functions simultaneously: It determines both the amount of surplus and the division of surplus. Equilibrium with a Binding Minimum Wage The minimum wage will bind if w m (w n, z). In that case, l m = s (w m ). Mathematically, s ( ) is strictly increasing, so it follows immediately that l m > l n. 12

13 Economically, s ( ) is an upward-sloping labor-supply function, so when wages increase from w n to w m, workers are willing to increase their hours from s (w n ) to s (w m ). For their part, employers are willing to accept more work from their employees: A rm's prot is (z w m ) l, so as long as w m is less than z, the rm can make more prot by hiring more hours of labor. To understand this result further, let's revisit the bargaining problem. If hours were xed, then increasing wages would be a means of transferring surplus from the rm to the worker. But if wages are xed, as they are when the minimum wage binds, then increasing hours is a means of transferring surplus from the worker to the rm. Observe that, given w m, raising l above l n lowers the worker's utility: d dl [w ml v (l)] l ln = [w m v (l)] < z v (l n ) = 0, (3.11) l ln where the inequality comes from the convexity of v ( ) and the assumption that w m < z. Conversely, the rm's prots (z w m ) l are increasing in l. By beneting rms at the expense of workers, the increase in hours counteracts the increase in wages when dividing the match surplus. However, adjusting the choice of hours decreases the amount of surplus to be divided, because zl v (l) is maximized at l = l n. In contrast to hours, employment declines. Furthermore, the magnitude of this decline depends on the elasticity of the intensive labor-supply function. Equilibrium market tightness is given by θ m = Θ (w m ), and equilibrium employment is e m = p (Θ (w m )). The elasticity of employment with respect to the minimum wage is: w m e m de m dw m = [1 η (Θ (w m ))] w m Θ (w m ) Θ (w m ), (3.12) and the elasticity of the market-tightness function (3.8) is: [ w Θ (w) Θ (w) = 1 w s (w) η (Θ (w)) s (w) w ]. (3.13) z w Equation (3.13) shows how the intensive margin inuences the extensive margin of labor. Because of the free-entry condition, the protability of hiring a worker dictates how many vacancies are posted and, by extension, the number of jobs created. In equation (3.13), the term in square brackets is the elasticity of prots (z w) s (w) with respect to w. When the wage goes up, the rm's prot per hour (z w) goes down, but the number of hours supplied by the worker s (w) goes up. Consequently, the sign of w Θ (w) Θ(w) depends on two terms: the elasticity of the intensive labor-supply function w s (w) s(w), and the payroll-to-prot ratio w z w. On one hand, if intensive labor supply is highly elastic, then rms can easily increase their revenues when the minimum wage goes up, because employees are willing to work more. On the other hand, if the payroll-to-prot ratio is high, then the rm's cost of labor is sensitive to changes in the minimum 13

14 wage. Ultimately, the increase in costs exceeds the increase in revenues, leading to a decline in prots: The following proposition establishes that the minimum wage does, in fact, depress vacancy creation. Proposition 2. An increase in the minimum wage leads to a decrease in market tightness: w Θ (w) Θ (w) < 0, w w n. (3.14) To summarize, when the minimum wage increases: The sign of the hours response is positive, the sign of the employment response is negative, and the magnitudes of these responses are linked via equation (3.13). Total personhours el and total payrolls elw respond ambiguously. Whether the increase in hours exceeds the decrease in employment depends on the elasticity of s ( ), the intensive labor-supply function. For interpreting empirical studies, these theoretical results provide an interesting counterpoint to the textbook supply-and-demand model. Both models predict a drop in employment. However, many regression analyses fail to nd strong negative associations (if any at all) between employment and the minimum wage. One conjecture in the literature, summarized by Schmitt [2013], is: Even within the competitive framework, employers might choose to respond to a minimum-wage increase by reducing workers' hours, rather [than] by reducing the total number of workers (p. 15). In other words, the competitive model suggests that a quantitatively small employment response can be explained by a quantitatively large reduction in hours. The search model suggests exactly the opposite: A quantitatively small employment response can be explained by a quantitatively large expansion in hours. This fact comes from equation (3.13), which connects the magnitudes of w Θ (w) Θ(w) and w s (w) s(w). 3.3 An Unconstrained Planner First, consider a planner who chooses vacancies and hours to maximize total output, minus the disutility of labor and vacancy-creation costs. The planner's objective function is: W (θ, l) = p (θ) [zl v (l)] θκ. (3.15) Let (θu, l u) denote the unconstrained maximizer of W (θ, l). In this context, the planner is unconstrained in the sense that (θu, l u) need not be consistent with a decentralized equilibrium. However, the following proposition establishes when the optimal allocation coincides with the equilibrium allocation. Proposition 3. The unconstrained planner's allocation coincides with an equilibrium if the minimum wage is not binding and β = η (θu). Regardless of whether β equals η (θu), l u = l n. 14

15 The rst part of Proposition 3 is the familiar Hosios [1990] condition: Eciency requires that the worker's share of the match surplus reect the congestion she creates by participating in the search process. Generically, the equilibrium is inecient, so there is at least some possibility that policy can improve welfare. However, the second part of Proposition 3 shows that a minimum wage can never implement the optimal allocation, because l m > l n = l u. In an equilibrium without a minimum wage, the number of hours worked within each match will be ecient, even if the number of matches is inecient. 3.4 A Ramsey Planner Now, consider a Ramsey planner who uses a single policy instrument, the minimum wage, to maximize welfare, taking as given the responses of workers and rms. The goal is still to maximize (3.15), but the Ramsey planner is constrained in the sense that (θ, l) needs to be consistent with an equilibrium. For (w, θ, l, e) to constitute an equilibrium with a binding minimum wage, it's necessary and sucient that w = w m > w n, θ = Θ (w), l = s (w), and e = p (Θ (w)). We can therefore consolidate these constraints into the planner's objective function by dening R (w) W (Θ (w), s (w)). The optimal minimum wage, denoted wm, solves: w m = arg max w [w n,z] R (w). (3.16) If the minimum wage does not improve welfare, then the planner can simply set wm = w n. Before solving equation (3.16), it's instructive to contrast the Ramsey criterion with individual payos. The planner is eectively maximizing the ex ante expected utility of unemployed workers, before matching takes place. Ex post, the minimum wage will always make some individuals better o, even if the policy reduces aggregate welfare. In particular, if w m > wm = w n, then job holders benet at the expense of job seekers. Proposition 4. Ex ante, the minimum wage increases the expected utility of unemployed workers if, and only if, the minimum wage increases aggregate welfare: p (θ m ) [w m l m v (l m )] p (θ n ) [w n l n v (l n )] R (w m ) R (w n ), w m > w n. (3.17) Ex post, the minimum wage always increases the realized utility of employed workers: w m l m v (l m ) > w n l n v (l n ), w m > w n. (3.18) A worker who nds a job benets from the minimum wage, regardless of whether the policy is optimal 15

16 from a social-welfare perspective. This fact would be trivial in a model without the intensive margin. When hours are determined endogenously, though, this result is less obvious: The minimum wage makes people work ineciently hard. According to equation (3.11), when a worker supplies an extra hour of labor at a xed wage w m, the value of leisure lost outweighs the value of consumption gained. However, as w m increases, the level of hours adjusts along the equilibrium labor-supply schedule, so an employee ultimately prefers the contract (w m, l m ) to the contract (w n, l n ), even if w m wm. As established by Proposition 3, a policy maker cannot attain the unconstrained optimum by adjusting the minimum wage. Instead, the best the Ramsey planner can do is negotiate a tradeo between distorting the intensive and extensive margins of labor. To see this more explicitly, we can decompose the marginal welfare eect of increasing the minimum wage as: R (w) = R E (w) + R I (w), (3.19) where I have dened the functions: R E (w) Θ W (θ, l) (w) θ (3.20) (θ,l)=(θ(w),s(w)) R I (w) s W (θ, l) (w) l. (3.21) (θ,l)=(θ(w),s(w)) We can interpret R E (w) as the welfare eect of adjusting extensive margin of labor: Incrementally increasing the wage causes vacancies to change by Θ (w), and the marginal vacancy causes welfare to change by W θ. Similarly, R I (w) is the welfare eect of adjusting the intensive margin of labor. For a binding minimum wage to be optimal, the welfare eects of adjusting the two margins of labor must balance each other out. That's because an interior solution to the Ramsey problem (3.16) requires R (w m) = 0, or equivalently, R E (w m) = R I (w m). The equilibrium number of hours is ecient when the minimum wage does not bind, and inecient when it does bind. Accordingly, the intensive-margin channel has a weakly negative eect on welfare. Proposition 5. R I (w n ) = 0, and R I (w) < 0 for any w > w n. Consequently, the Ramsey planner will only choose a binding minimum wage if the extensive-margin channel has a positive eect on welfare. In general, the sign of R E (w) is ambiguous, and it's dicult to characterize w m without making assumptions about the functional forms of v ( ) and q ( ). Nevertheless, we can say something about the sign of R E (w n ), which is useful because R E (w n ) = R (w n ). When R (w n ) is positive, it means that a marginally binding minimum wage leads to welfare gains, so R E (w n ) > 0 is a 16

17 sucient condition for w m > w n. Proposition 6. R E (w n ) 0 if, and only if, η (θ n ) β. Corollary. If η (θ n ) > β, then wm > w n. If the worker's bargaining power is too low, relative to the Hosios condition, then a binding minimum wage is optimal. The benet of the minimum wage operates exclusively through the extensive margin, by ameliorating the congestion externality. Firms post fewer vacancies, but each vacancy has a higher probability of being lled. However, Proposition 5 demonstrates that intensive-margin distortions will reduce welfare as the minimum wage increases, and this loss must be weighed against the possible gains from reducing congestion. 3.5 Sucient Statistics for Welfare Improvement It's possible to extend the above results by specializing to the case where v ( ) and q ( ) are isoelastic. There are two things I want to demonstrate. First, the Ramsey problem admits a closed-form solution, which makes it clear, in theory, when a binding minimum wage will be optimal. Second, there is a connection between the economy's positive behavior and the model's normative implications: I will show that choosing the minimum wage to maximize welfare is equivalent to maximizing total payrolls. Consequently, the elasticity of payrolls with respect to the minimum wage provides an indicator of whether the minimum wage is too high or too low. Suppose that v (l) = χ 1+1/φ l1+1/φ. This is the dominant specication of v ( ) in the applied literature. 8 In a standard frictionless model, the parameter φ would represent the Frisch elasticity of labor supply. In the present model, φ will play a similar role, as the elasticity of the intensive labor-supply function. Proposition 7. The intensive labor-supply function is isoelastic: w s (w) s(w) wage, the equilibrium wage is w n = β+φ 1+φ z. = φ. Without a binding minimum Suppose further that q (θ) = θ η, so η (θ) is just a constant, η. 9 Proposition 6 established that η > β is a sucient condition for a binding minimum wage to be optimal. Subject to the functional-form assumptions, η > β is both necessary and sucient, and the optimal minimum wage w m has a closed-form solution. Proposition 8. A binding minimum wage is optimal if, and only if, η > β. In that case, w m = η+φ 1+φ z. 8 See, e.g., Keane and Rogerson [2012] for a review. 9 This is equivalent to assuming that the total number of matches is given by a Cobb-Douglas function of vacancies and job-seekers, which is a popular choice in the literature. See, e.g., Petrongolo and Pissarides [2001]. None of the results that I'm about to present would change if q (θ) were scaled up by a positive constant, i.e., if q (θ) were equal to cθ η, where c > 0. 17

18 Ideally, a policymaker would want to know η and β. More practically, to discern whether a minimum wage is a good idea, it's useful to have a criterion that's based upon observable labor-market outcomes. Total payrolls elw provides such a criterion. Proposition 9. Solving the Ramsey problem (3.16) is equivalent to choosing the minimum wage that maximizes total payrolls elw. If the Ramsey problem has an interior solution, then only if, w m w m. If the Ramsey problem has a corner solution, then Much of the empirical literature asks, How can we use data to identify wm l m w m d(e ml mw m) e ml mw m dw m 0 if, and w m d(e ml mw m) e ml mw m dw m 0 and wm w m. dl m dw m and wm de m e m dw m? Popular identication strategies include state-level panel regressions and quasi-natural experiments. However, even if policymakers did vary the minimum wage as part of a genuine randomized experiment, we would only be able to observe the resulting changes in hours and employment not welfare. Suppose that an econometrician could estimate wm l m dl m dw m and wm e m de m dw m perfectly. The question remains, How do these quantities relate to optimal policy? Proposition 9 provides clear guidelines: If an econometrician estimates that w m d(e ml mw m) e ml mw m dw m is positive, then a policymaker should raise the minimum wage. Conversely, if an econometrician estimates that w m d(e ml mw m) e ml mw m dw m is negative, then a policymaker should lower the minimum wage. Labor economists have made dierent assumptions about how the measured changes in employment and hours translate into normative implications. Allegretto et al. [2011] use total payrolls as a proxy for welfare, but those authors do not invoke any specic, formal theory. 10 Nevertheless, Proposition 9 validates their interpretation. Alternatively, Neumark and Wascher [2008] state: total hours is the most relevant statistic for testing the validity of the competitive model of labor demand, although perhaps not necessarily the most important statistic from a policy perspective (p. 78). However, the response of total personhours is, in fact, informative of optimal policy: Because w m d(e ml m) w e ml m dw m equals m d(e ml mw m) e ml mw m dw m 1, the elasticity of total personhours can also be used as a sucient statistic for whether the minimum wage is too high or too low. 4 A Dynamic Model I will now extend the simple model to a dynamic setting with heterogeneous match quality. These features serve several purposes. Incorporating random match quality allows for two important sources of heterogeneity that are found in the data. There will be an endogenously-determined mass of workers earning the minimum wage, plus a non-degenerate distribution of wages above the minimum. Additionally, there will be heterogeneity in hours worked, even amongst workers earning identical wages. By including a time dimension, the model allows individual workers to cycle in and out of unemployment; consequently, the worker's 10 They assert: If the wage bill elasticity is negative, teens as a whole are worse o from the increase in minimum wage. If it is positive, teens as a whole are better o (p. 221). 18

19 threat point when bargaining is endogenous, because it reects the value of continuing to search. Unless stated otherwise, the notation in the dynamic model mirrors the notation from the static model. 4.1 Environment Time is continuous, with an innite horizon, and I will focus on steady states. All agents discount the future at rate ρ. I will assume that workers get ow utility b during unemployment. Instead of denoting probabilities, p (θ) and q (θ) now represent Poisson rates of arrival. Prospective rms pay κ as a ow cost for as long as a vacancy remains open. Matches dissolve exogenously at rate λ. When a worker meets a rm, they realize a draw from the distribution of idiosyncratic match quality; then, the worker and the rm engage in Nash bargaining over the terms of a contract. A contract consists of a pair (w, l), where w is the wage and l is the intensive labor supply. Match quality consists of a pair (z, ɛ), where z represents idiosyncratic productivity, and ɛ represents idiosyncratic utility. Concretely, in a match with contract (w, l) and quality (z, ɛ), the ow output is zl, the rm's ow prot is zl wl, and the worker's ow utility is wl v (l) + ɛ. Draws of (z, ɛ) are i.i.d. across matches with joint CDF F (z, ɛ). I will assume that F (z, ɛ) is smooth with density f (z, ɛ) and full support on R + R. The Value of Being Matched Let h (w, l, ɛ) denote the worker's value of being employed with contract (w, l) and match-specic utility ɛ. Let u denote the value associated with being unemployed. The worker's value of employment must satisfy the Bellman equation: ρh (w, l, ɛ) = wl + ɛ v (l) + λ [u h (w, l, ɛ)]. (4.1) I will defer providing an expression for u until after I detail the bargaining process. Let g (w, l, z) denote the rm's value of employing a worker for l hours at wage w with productivity z. The rm's value of operating must satisfy the Bellman equation: ρg (w, l, z) = zl wl λg (w, l, z). (4.2) Because of free entry, rms do not have any value if not matched with a worker. Notice that h (w, l, ɛ) and g (w, l, z) are the values that agents would get from an arbitrary contract (w, l), not necessarily an equilibrium contract. In an equilibrium, the contract will be a function of (z, ɛ). 19

20 Bargaining When a worker and a rm meet, a value of (z, ɛ) is realized, but the agents do not bargain over a contract unless it's possible to achieve some gains from trade. For a given (z, ɛ), a contract must fall within the bargaining set, which is dened as: B (z, ɛ) {(w, l) h (w, l, ɛ) u, z w w m, l 0}. (4.3) The constraint h (w, l, ɛ) u requires that the worker weakly prefers the contract (w, l) to unemployment. The constraints z w w m and l 0 imply that the contract provides the rm with some non-negative prots, while also comporting with the minimum-wage law. 11 Any contract (w, l) in the bargaining set B (z, ɛ) will make both the worker and the rm weakly better o, relative to remaining unmatched. Conversely, if B (z, ɛ) is empty, then there does not exist a contract that makes a match worthwhile. Dene: A {(z, ɛ) B (z, ɛ) }. (4.4) The set A consists of all match-quality draws that will induce a match to form. Provided that (z, ɛ) A, the specic contract selected by the worker and the rm is determined by Nash bargaining. Wages and hours satisfy: (w (z, ɛ), l (z, ɛ)) = argmax (w,l) B(z,ɛ) [h (w, l, ɛ) u] β g (w, l, z) 1 β. (4.5) Because wages and hours are determined as functions of (z, ɛ), we can also write the worker's and the rm's values of being matched in terms of (z, ɛ): h (z, ɛ) h (w (z, ɛ), l (z, ɛ), ɛ), g (z, ɛ) g (w (z, ɛ), l (z, ɛ), z). (4.6) 11 In other words, the constraints z w and l 0 are sucient for g (w, l, z) 0. Strictly speaking, the constraint z w is not necessary for g (w, l, z) 0. By imposing z w, I'm ruling out the possibility of a rm hiring a worker for zero hours (l = 0) at a wage that exceeds her productivity (w > z); in that case, g (w, l, z) would be zero, even though h (w, l, ɛ) could be positive for suciently high ɛ. Empirically, it's not helpful to talk about jobs in which employees do not supply any labor; theoretically, it's benign to assume that the rm does not accept contracts with w > z and l = 0, because the rm would only be indierent to forming a match. 20