Minimum Wage and Individual Worker Productivity: Evidence from a Large US Retailer

We study workers who are employed by a large US retailer, work in many store locations, and are paid based on performance. By means of a border-discontinuity analysis, we document that workers become more productive and are terminated less often after a minimum wage increase. These effects are stronger among workers whose pay is more often supported by the minimum wage. However, when workers are monitored less intensely, the minimum wage depresses productivity. We interpret these findings through an efficiency wage model. After a minimum wage increase, profits decrease, and a calibration exercise suggests that worker welfare increases.

I. Introduction Since the US minimum wage was introduced in 1938, its effect on employment has been hotly debated. Much less attention has been paid to the effect of the minimum wage on the productivity of employed workers. This paper examines this intensive margin and shows that increasing the minimum wage causes the productivity of employed workers to increase through an efficiency wage mechanism. Our evidence comes from salespeople who work at a large US retailer employing more than 10% of department store employees nationwide and operating more than 2,000 stores across all 50 states.1 Our workers’ pay is in part based on performance (sales per hour), and their productivity is recorded administratively.2 When our worker’s average hourly pay falls below the minimum wage, the employer is required to make up the difference. Our data cover 70 minimum wage increases at the state and local levels. Using a border-discontinuity research design, we compare the productivity of workers in treated versus control stores on the other side of the same border, in a specification that includes worker and county-pair × month fixed effects. We document that an increase in the minimum wage causes individual productivity to increase. This effect is stronger for workers whose pay is more often supported by the minimum wage (referred to herein as “low types”). These findings are not an artifact of selection and cross-border migration, and they are not explained by demand shifts, price changes, or other organizational changes. The effects persist in a “state panel” research design. Interestingly, the effect of the minimum wage on worker productivity flips (becomes negative) when workers are relatively less monitored, as measured by a low supervisor-to-worker ratio within a store. We interpret these findings through the lens of a model that features two sources of incentives. Workers are incentivized by both the threat of termination, which is based on the direct monitoring of effort, and the variable component of pay, which reflects individual performance and which is in turn a noisy signal of effort.3 We refer to these channels as “efficiency wage” and “pay for performance,” respectively. In this model, a minimum wage increase has two opposite effects on incentives: it demotivates effort provision because it flattens the pay schedule (pay-for-performance channel), but it motivates effort provision because of the fear of losing a now higher-paying job (efficiency wage channel). We infer that the efficiency wage channel dominates in our setting because, empirically, worker performance increases with the minimum wage. Consistent with the efficiency wage channel, we find that the workers whose performance increases the most with the minimum wage also have the largest drop in termination rates; according to efficiency wage theory, this decrease in terminations is the workers’ reward for exerting more effort. While the efficiency wage channel appears to dominate in the data, our theory predicts that this channel will vanish if effort is not monitored directly. In this scenario, the only source of incentives is the pay-for-performance channel, so the minimum wage is expected to demotivate workers. We find empirical support for this prediction: when a store’s workers are monitored less intensely (i.e., when the supervisor-to-worker ratio is low), a minimum wage increase is shown to decrease performance, contrary to that which we find on average. Turning to store-level outcomes, we document that termination, hiring, and turnover rates all decrease with the minimum wage and that the effect is increasing in the fraction of low types in a store, in line with the model’s predictions. Meanwhile, employment does not change, store-level output increases, and average profits across all stores go down. This last result indicates that the endogenous increase in output is not large enough to offset the wage growth caused by the minimum wage. Finally, we study the effect of the minimum wage on the welfare of employed and unemployed workers, under the assumption that the probability of exiting unemployment is a decreasing function of the minimum wage. In theory, welfare could be decreasing if unemployment duration sharply increases in the minimum wage. Nevertheless, our calibration suggests that the welfare of employed and unemployed workers increases with the minimum wage. Our paper contributes to different strands of the literature. First, it contributes to the literature on the individual productivity effects of the minimum wage. To our knowledge, this literature consists of Ku (2022) and Hill (2018), who study tomato and strawberry pickers, respectively, in a single farm around one or two minimum wage events. They reach apparently conflicting results: Ku (2022) finds that increasing the minimum wage increases individual productivity, while Hill (2018) finds the opposite. As they both use relatively more productive workers as the control group, their research designs allow only relative estimates (low vs. high types) of the productivity gain. In contrast, we observe workers in nearby establishments experiencing no minimum wage increase, permitting an estimation of the absolute productivity gains for low and high types. Moreover, our model and our empirical findings reconcile Ku’s (2022) and Hill’s (2018) apparently opposing findings by appealing to variation in monitoring intensity. Second, our paper contributes to the minimum wage literature on aggregate flows of low-paid workers and employment.4 This literature tends to find a reduction in worker turnover and, in the county border-discontinuity strand of this literature, a lack of disemployment effect after a minimum wage increase. We replicate these effects but, whereas the existing studies are based on county- and state-level aggregates, our study is based on individual worker data and links these effects to an endogenous increase in worker productivity. Third, our study is related to the recent and growing empirical literature studying the effects of the minimum wage on firm-level profits.5 These studies tend to find nonpositive effects of the minimum wage on profits, as do we. Our paper adds to this literature by showing that the rising labor costs associated with a minimum wage increase can be partially offset by an increase in worker productivity, which presumably attenuates the negative effect of the minimum wage on profits. Finally, we contribute to the empirical literature on efficiency wages, which has mostly interpreted efficiency wages as gift exchange or reciprocity.6 Within this literature, perhaps the paper closest to ours is Jayaraman, Ray, and de Vericourt (2016), which shows that an increase in the minimum daily wage payable to Indian plantation workers increases output. This effect is attributed to reciprocity because the plantation workers cannot be fired, so an efficiency wage channel à la Shapiro and Stiglitz (1984) or Rebitzer and Taylor (1995) cannot be invoked. In contrast, our paper provides evidence for the incentive effect provided by termination. Overall, this paper contributes to the minimum wage literature by documenting the endogenous effort response of low-paid workers. This is another channel through which the minimum wage may affect firm productivity and worker welfare, separately from the conventional channel that labor becomes more expensive, causing profits to shrink and workers to lose their job. In addition, the paper suggests that the efficiency wage model can be a useful organizing framework for understanding the workers’ response to the minimum wage. The paper proceeds as follows. Section II presents the model, while section III describes the data, the institutional context, and the identification strategy. Section IV discusses our core results: the effect of the minimum wage on worker productivity. Section V examines the heterogeneous effects of the minimum wage on worker productivity by monitoring intensity. Section VI presents the store-level results on turnover, employment, output, and profits. Section VII discusses two alternative channels: demand and organizational adjustments. Section VIII calibrates the effect of a minimum wage increase on worker welfare. Section IX discusses the external validity of our findings and concludes.

II. Model In this section, we model the effort choice of a worker (in our empirical setting, a salesperson whose job is to interact with a customer) who has two sources of incentives: the probability of being terminated and the wage. The probability of termination is decreasing in worker effort and depends on the firm’s monitoring intensity. The expected wage is based on individual performance (in our setting, sales per hour) and is increasing in effort. By law, the wage cannot fall below the minimum wage. The model generalizes the efficiency wage model of Rebitzer and Taylor (1995).7 The model is used to characterize the relationship between the minimum wage and optimal worker effort, how this relationship changes as a function of the firm’s monitoring intensity, and how firm-level turnover is affected by the minimum wage. Each worker has a firm- (or match-)specific type x ≥ 0 and chooses a continuous effort level e ∈ [ 0 , 1 ] . Type x’s cost of effort c(x, e) is strictly increasing in effort. We assume that the marginal cost of effort vanishes at zero and is infinite at one; these assumptions will help ensure that optimal effort is interior to [0, 1]. Worker performance (i.e., value of output—in our case, sales revenue per hour) is a nonnegative random variable Y(x, e) that is uniformly bounded from above across all (x, e). Its density f Y (y; x, e) has interval support, is twice continuously differentiable in all its arguments, and enjoys the strict monotone likelihood ratio property (MLRP) in x and e.8 Intuitively, the MLRP means that higher types and greater effort levels produce stochastically higher output. The MLRP implies first-order stochastic dominance. Consider any continuous nondecreasing compensation scheme w ¯ ( ⋅ ) that transforms performance into pay. For example, w ¯ ( Y ) = b + c Y , where b represents the base salary and c represents the commission rate. The expected wage is denoted by w ( x , e ) = ???? ( w ¯ ( Y ( x , e ) ) ) . The function w is nondecreasing in each of its arguments because of the MLRP. The functionis nondecreasing in each of its arguments because of the MLRP. Type x’s effort choice problem is (1) V E ( x ) = max ⁡ e w ( x , e ) − c ( x , e ) + 1 ( 1 + r ) [ π ( e ) V E ( x ) + ( 1 − π ( e ) ) V A ] . The function VE(x) represents type x’s discounted value of being employed. The numbers r > 0 and VA respectively represent the discount rate and the lifetime value of becoming unemployed. Note that VA is not a function of x, consistent with the assumption that types are firm (or match) specific. The function π(e) represents the probability of continued employment, which is assumed to be nondecreasing and continuously differentiable over [0, 1]. We interpret the magnitude of π′(e) as reflecting the firm’s monitoring intensity; the limit case where π ′ ( e ) ≡ 0 for all e is referred to as the “no-monitoring” case. The function) represents type’s discounted value of being employed. The numbersandrespectively represent the discount rate and the lifetime value of becoming unemployed. Note thatis not a function of, consistent with the assumption that types are firm (or match) specific. The function π() represents the probability of continued employment, which is assumed to be nondecreasing and continuously differentiable over [0, 1]. We interpret the magnitude of π′() as reflecting the firm’s monitoring intensity; the limit case wherefor allis referred to as the “no-monitoring” case. 9 To simplify the worker’s problem, subtract the equation [ r / ( 1 + r ) ] V A = u A from (1). We get (2) V ( x ) = max ⁡ e u ( x , e ) + 1 ( 1 + r ) π ( e ) V ( x ) , where V ( x ) = V E ( x ) − V A represents the additional discounted value of being employed relative to being unemployed and (3) u ( x , e ) = w ( x , e ) − c ( x , e ) − u A represents the flow value of employment, net of flow opportunity cost uA, of a type x who is currently employed and exerts effort e. whererepresents the additional discounted value of being employed relative to being unemployed andrepresents the flow value of employment, net of flow opportunity cost, of a typewho is currently employed and exerts effort Problem (2) shows that the worker maximizes the sum of two terms: the flow value of employment, which is the source of standard pay-for-performance incentives, and the value from continued employment, which is the efficiency wage incentive channel. If π(⋅) is a strictly increasing function, the efficiency wage channel motivates the worker to exert more effort than is justified solely by pay for performance. We assume that u is continuously differentiable over its domain and make the following intuitive assumption. Assumption 1. u x > 0 , u x e > 0 , and u e e < 0 . The first two properties signify that higher types have higher payoffs and higher marginal return on effort. The third property, concavity of u in e, helps identify the optimal effort level. The required properties may be imparted to u by either of its components, w and c. For example, assumption 1 holds if the wage w is identically equal to the minimum wage, provided that the cost function is strictly convex in e, and higher types have lower effort cost and lower marginal cost of effort, which are standard assumptions. To avoid trivialities, we assume that it is strictly optimal for all types to show up for work. Formally, we assume that the set of individually rational effort levels, defined as the set of effort levels e such that u ( x , e ) > 0 , is nonempty for all x. Then, expression (2) implies that V ( x ) > 0 for all x. Assumption 2. π(e) is weakly concave in e. This assumption helps ensure that problem (2) is strictly concave in e. Assumption 2 is trivially satisfied in the no-monitoring case, because then π ′ ( e ) ≡ 0 for all e. Under assumptions 1 and 2, the maximization problem in (2) is strictly concave in e, and so type x’s optimal effort, if positive, is the unique solution to the first-order condition (4) u e ( x , e ) + 1 ( 1 + r ) π ′ ( e ) V ( x ) = 0. The next lemma states that the model behaves “nicely.” Lemma 1. Suppose that assumptions 1 and 2 hold. 1. Fix x. The worker’s maximization problem (2) is concave in e and has a unique solution e*(x). 2. e*(x) is nondecreasing in x, and it is strictly increasing if e*(x) is interior to [0, 1]. 3. If π ′ ( e ) > 0 for all e, then e*(x) is interior to [0, 1] for all x. The next lemma states that the model behaves “nicely.” Proof. See appendix B.1. QED A. Linking the Model to Our Empirical Setting We study a single firm with many store locations across the United States, and the above model describes the problem of a worker operating in a specific store. Compensation scheme.—Since in our firm all workers nationwide are subject to the same compensation scheme, the compensation scheme cannot be optimally adapted to the local conditions of most stores. At best, it is optimal on average. Hence, in our model, we cannot assume that the compensation scheme w ¯ ( ⋅ ) is optimally adapted to the local parameters, including the minimum wage M. We instead assume that when a locality increases M, w ¯ does not change.10 Thus, in a store that is subject to a local minimum wage M, the expected wage is (5) w ( x , e ; M ) = ???? ( max ⁡ [ M , w ¯ ( Y ( x , e ) ) ] ) . The function w(x, e; M) is bounded below by M and is nondecreasing in all its arguments. The function) is bounded below byand is nondecreasing in all its arguments. 11 Henceforth, w(x, e; M) will replace w(x, e), and we assume that assumption 1 continues to hold after this replacement. The worker’s optimal effort e*(x; M) will henceforth be indexed by the minimum wage level. 1. Low Types Definition 1 (Motivated by minimum wage [MMW], or low type). Type x is MMW or a low type if w ( x , e ; M ) = M for all e ∈ [ 0 , 1 ] . MMW types cannot increase their wage by exerting more effort, so their only incentive to exert effort is the fear of losing their job. In this respect, MMW types behave exactly as the workers in the Rebitzer and Taylor (1995) model. The set of MMW types, if nonempty, is an interval including zero; this is because the function w(x, e; M) is nondecreasing in x. It is therefore appropriate to refer to MMW types as low types. Empirically, we define a low type as a worker whose pay is often determined by the minimum wage and therefore has incentives similar to the MMW types in the theory. 2. Three Cases Nested by the Model The model nests two polar cases and a hybrid one. Polar case: pure efficiency wages.—If w ( x , e ; M ) ≡ M , pay does not depend on performance and all incentives to exert effort in the worker’s problem (2) come from reducing the probability of being terminated. This is the pure efficiency wage model. Polar case: pure pay for performance.—If π ′ ( e ) ≡ 0 (no-monitoring case), the worker’s maximization problem (2) reduces to maximizing the per-period value of the worker’s utility from employment. In this case, exerting effort does not alter the probability of being fired, so all incentives to exert effort come from performance pay. Hybrid case (our preferred model).—When π ′ ( e ) > 0 and M is not too high, the model is a hybrid of pure efficiency wages and pure pay for performance, meaning that some types (MMW types) will be motivated by efficiency wages only and others (higher types) will in part be motivated by performance pay. The pure efficiency wage case may be disregarded for empirical purposes—the great majority of our workers receive a substantial amount of variable pay. Therefore, only two cases can possibly match our setting: pure pay for performance and the hybrid case. 3. Value of Outside Option The model can be extended to allow the lifetime value of a job in the local economy to depend on the minimum wage, so that V A = V A ( M ) . All the results go through if the function VA(⋅) is decreasing, as would be the case if the main effect of a minimum wage increase were to slow the movement out of unemployment. Conversely, if the function VA(⋅) rises too steeply, a minimum wage increase will be demotivating. This is not the case in our setting because, empirically, we find that increasing the minimum wage promotes worker effort (see sec. IV). B. Core Theoretical Results: Effect of Minimum Wage on Individual Productivity Assumption 3. w ¯ ( ⋅ ) is a strictly increasing function, w M ( x , 1 ; M ) > 0 , and | w e M ( x , e ; M ) | < ∞ for all x ≥ 0 , e ∈ [ 0 , 1 ] . The assumption that w ¯ ( ⋅ ) is strictly increasing is made for convenience of exposition. Note that it does not imply that w(x, e; M) is strictly increasing in e, and indeed this is not the case for MMW types. The assumption w M ( x , 1 ; M ) > 0 says that even a worker who exerts maximum effort ( e = 1 ) earns the minimum wage with a positive probability, however small. The assumption that | w e M ( x , e ; M ) | < ∞ is purely technical. Proposition 1 (Effect of the minimum wage on productivity). Suppose that assumptions 1–3 hold and also that π ′ ( e ) > 0 for all e. 1. Effort is strictly increasing in M for MMW types (low types). 2. The set of types whose effort increases with M grows with M. 3. For M large enough, all types’ effort increases with M. 4. Increasing M has a negligible effect on the effort of types whose wage is negligibly affected by the minimum wage. Proof. See appendix B.1. QED It is worth emphasizing that part 1 requires the assumption that π ′ ( e ) > 0 for all e. This assumption fails in the no-monitoring case, in which case increasing M does not increase the low types’ effort (proposition 2, pt. 2, below). Empirically, the low types in part 1 will correspond to the workers who, at a given point in time, most benefit from the minimum wage, while the negligibly affected in part 4 will correspond to high types—see section III.A.4. We will show that in the average store, these types show the response predicted by proposition 1. The medium types’ response will depend on the monitoring intensity, as discussed in the next section. C. Role of Monitoring in Effort Exertion Monitoring intensity must intuitively enter the model through the sensitivity to effort of the probability of being fired. We now make this idea precise. Definition 2 (Monitoring intensity). Monitoring is more intense under π ˜ ( e ) than under π(e) if, for every e, the elasticity of [ 1 + r − π ˜ ( e ) ] is larger in absolute value than that of [ 1 + r − π ( e ) ] . Definition 2 establishes a partial order on the functions π(⋅).12 In general, the constant function is the smallest element in this partial order—this is the previously mentioned no-monitoring case where π ′ ( e ) ≡ 0 . At the opposite end of the spectrum, π(⋅) may be chosen so that [ 1 + r − π ( e ) ] has arbitrarily large elasticity (in absolute value), provided that r is small enough (i.e., the worker is sufficiently patient).13 The next result describes how effort response to the minimum wage varies by monitoring intensity and by type. Proposition 2 (Role of monitoring in effort exertion). 1 (Effect of increasing monitoring) When monitoring becomes more intense, all types exert more effort. 2 (Effect of increasing M, no-monitoring case) Suppose that π ′ ( e ) ≡ 0 . Then MMW types (low types) exert zero effort. Increasing M does not increase their effort, and it decreases the effort of any type that exerts positive effort. 3 (Effect of increasing M, high-monitoring case) If assumption 3 holds, increasing M increases any type’s equilibrium effort if monitoring is sufficiently intense. Functions π(⋅) exist under which monitoring is arbitrarily intense if r is small enough. Proof. See appendix B.1. QED Part 1 is intuitive because it confirms that increasing monitoring raises equilibrium effort. Parts 2 and 3 are instructive: increasing the minimum wage promotes effort when monitoring is high, but it promotes shirking when monitoring is low. In addition, parts 2 and 3 yield testable predictions by type. Among the nonmonitored workers, the low types do not change their effort as the minimum wage increases (because nonmonitored MMW types shirk regardless of the minimum wage level), whereas higher types decrease their effort owing to the attenuated pay-for-performance incentive. Among the highly monitored workers, an increase in M causes all types to exert more effort. Taken together, these predictions are a strong empirical test of the dual nature (efficiency wage and pay for performance) of the model. Proposition 2 suggests that a store’s workers should respond differently to a minimum wage increase depending on whether monitoring is low or high in that store. To make this idea precise, we extend the model such that a fraction ( 1 − μ ) of workers in a store, chosen at random independently of their type, is not monitored. That is, a shirking worker is detected with probability π ¯ independent of effort. The remaining fraction μ of workers is highly monitored, meaning that shirking workers are detected with a highly elastic probability π(e), as described in proposition 2 (pt. 3).14 We think of μ as a continuous measure of monitoring coverage and, for now, take μ as given. It helps to think of the store as being partitioned into two divisions. In the nonmonitored division, workers effectively operate on a pure pay-for-performance basis; they are never terminated for lack of effort. Workers in the highly monitored division behave as in the previous sections. Proposition 2 characterizes how effort, and therefore individual performance, changes in either division. Empirically, we expect a store to behave as described in part 2 when monitoring coverage μ is low and to behave as in part 3 when μ is high. Both predictions are found to hold in the data (see sec. V). Empirically, we find that on average a store behaves as a high-monitoring store. Thus far, we have assumed that the monitoring coverage μ is not endogenous to M. In appendix B.3, we work out a theory where μ is endogenous and can be purchased by the firm at a cost K(μ). The theory predicts that if store profits are nondecreasing in M (as indeed is the case empirically in our main sample),15 then μ should increase with M. However, the increase could be small depending on the shape of the function K(μ). This prediction is tested empirically in section V. The coefficient on M has the expected sign, but its magnitude is small and statistical significance is lacking. Overall, we believe that the evidence is consistent with the theory of endogenous coverage μ but points to a degree of endogeneity small enough to be ignored for practical purposes. We therefore proceed under the assumption that μ is exogenous. D. Effect of Minimum Wage on Turnover in a Store’s Steady State In this section, we characterize the steady-state turnover rate in a store where a fraction μ of workers are highly monitored and the rest are not monitored. Steady state means that, given M and the termination policy given by π(⋅), replacement workers are randomly drawn from the pool of the unemployed such that the fraction of employees terminated and hired are equal, and in the next period the type distribution in the store is reproduced identically. Note that in this definition, the absolute size of the labor force in the store is left unspecified. Denote by H the cumulative distribution function of the type distribution that our firm can expect after hiring a random worker from the unemployment pool, and let h be its density. Since types are firm (or match) specific, unemployed workers are not negatively (or positively) selected from a hiring firm’s perspective; hence, H is not a function of any of the model’s parameters.16 Denote by gM(x) the density of the steady-state type distribution in a highly monitored division given a certain M. The density gM(x) must solve g M ( x ) = π ( e * ( x ; M ) ) g M ( x ) + λ ( M ) h ( x ) , where λ(M) denotes the per capita inflow of workers (which in steady state coincides with the outflow) in a highly monitored division. Isolating gM(x) yields (6) g M ( x ) = λ ( M ) 1 − π ( e * ( x ; M ) ) h ( x ) . Because gM must integrate to one, we get, for all M, 1 = λ ( M ) ∫ 0 ∞ 1 1 − π ( e * ( x ; M ) ) d H ( x ) . Since in a highly monitored store e*(x; M) is increasing in M for all x (proposition 2, pt. 3), λ(M) is decreasing in M. where λ() denotes the per capita inflow of workers (which in steady state coincides with the outflow) in a highly monitored division. Isolating) yieldsBecausemust integrate to one, we get, for allSince in a highly monitored store*() is increasing infor all(proposition 2, pt. 3), λ() is decreasing in Turning to the nonmonitored division, recall that π ¯ is the constant probability of retention under no monitoring. In a nonmonitored division, the turnover is ( 1 − π ¯ ) independent of type, and the steady-state type distribution in that division coincides with H(x). The steady-state turnover rate for the entire store, averaging across the highly monitored and nonmonitored divisions, is μ λ ( M ) + ( 1 − μ ) ( 1 − π ¯ ) . This expression is decreasing in M because λ(M) is decreasing in M. This proves the following result. Proposition 3 (Impact of minimum wage on steady-state turnover and tenure in a store). In steady state, the average turnover rate in a store is decreasing (and therefore average tenure is increasing) in the level of the minimum wage. Both effects are driven by increased effort. This expression is decreasing inbecause λ() is decreasing in. This proves the following result. Intuitively, the decrease in turnover results from the fraction of highly monitored workers who, after an increase in the minimum wage, exert more effort and thus are terminated less frequently. Among nonmonitored workers, effort decreases after a minimum wage increase (proposition 2, pt. 2), but their turnover remains unchanged, as their probability of termination is independent of effort. E. Effect of Minimum Wage on Store Output, Profits, and Employment The minimum wage has two opposite effects on store-level output: workers (at least, highly monitored workers) exert more effort, but gM changes in a way that may increase the representation of low types. Section VI.C calibrates the size of these two effects. We now turn to profits. In our empirical setting, the compensation scheme w ¯ is set uniformly for all workers nationally and is thus not adapted to local store conditions. Therefore, increasing the local minimum wage may possibly cause profits to increase in some stores but to decrease in others.17 However, the average effect across all stores could never be positive if w ¯ is set to maximize aggregate profits at the national level. The above discussion is summarized in the following result. Lemma 2 (Impact of minimum wage on store-level profits). If w ¯ is set to maximize nationwide profits, increasing M cannot increase nationwide profits. This lemma says that, on average across all stores, profits must decrease with the minimum wage. Section VI.C explores this prediction. Next, we address store size, which we denote by L. Given a certain M, the optimal store size solves (7) max L L ⋅ Π ( F M , M ) − κ ( L ) , where Π(F M , M) denotes gross store profits per worker,F M , M), with higher values yielding a larger optimal store size. However, if the function κ(⋅) is very convex around some L ˆ , such that it resembles a step function, optimal store size will approximately equal L ˆ irrespective of the value of Π(F M , M). where Π() denotes gross store profits per worker, 18 and the convex function κ(⋅) captures the amortization or capital cost of operating at a given size. The solution to problem (7) depends on the value of the term Π(), with higher values yielding a larger optimal store size. However, if the function κ(⋅) is very convex around some, such that it resembles a step function, optimal store size will approximately equalirrespective of the value of Π(). Section VI.B explores the empirical effect of the minimum wage on employment.

III. Data and Empirical Strategy A. Data and Institutional Background We match the firm’s biweekly worker-level payroll data with monthly personnel records from February 2012 to June 2015. Restricting our attention to salespeople who are paid based on their performance produces our total sample of more than 40,000 hourly salespeople. Further restricting the sample to border stores as per our research design (sec. III.B) leaves us with more than 200 stores with over 10,000 salespeople. Table 1 reports the summary statistics of this border sample. Table 1. Descriptive Statistics Variable Mean

(1) Standard Deviation

(2) p10

(3) p50

(4) p90

(5) N

(6) A. Worker-Level Variables Productivity: Sales/hour (shrouded units) 2.085 1.468 .781 1.872 3.522 217,822 Tenure and hours: Tenure (months) 48.92 65.01 4 24 126 217,822 Part-time (%) 60.25 48.94 0 100 100 217,822 Number of hours 106.5 44.12 46.47 107.6 162.3 217,822 Compensation: Base rate: regular pay/hour ($) 6.120 1.181 4.500 6 7 217,822 Commission rate: variable pay/sale (%) 3.462 3.188 1.057 2.343 7.531 213,726 Variable pay/hour ($) 5.947 4.936 1.740 4.610 11.78 217,822 Minimum wage adjustments/hour ($) .225 1.736 0 0 .771 217,822 Total pay ($) 1,361 831.2 494.6 1,218 2,343 217,822 Total pay/hour ($) 12.51 4.620 8.734 11.15 17.94 217,822 B. Store-Level Variables Termination, hiring, and turnover: Terminated (%) 4.755 7.692 0 0 12.50 12,359 Hired (%) 2.060 4.285 0 0 7.692 12,359 Turnover (%) 3.408 4.404 0 2.500 8.333 12,359 Employment and profits: Number of workers 16.64 6.855 8 16 26 12,359 Supervisor-to-worker ratio (%) 6.990 4.886 3.448 5.882 11.11 12,359 EBITDA/hour (shrouded units) 5.946 11.97 −8.010 5.630 19.97 12,359 1. Workers and Compensation Our workers are consultative sales associates. They answer walk-in customer questions and demonstrate product features. What we call “exerting effort” consists of meeting and greeting the customer, taking the time to explain and persuade, upselling (to higher-margin products), and cross-selling (warranties, loans, credit cards, etc.).19 They record a customer purchase as their own sale, and their pay consists of a base salary plus commissions on individual sales. For every salesperson, we aggregate the following at the monthly level: hours worked (average: 107 hours), sales (average: two per hour; units shrouded for confidentiality),20 and pay (average: $1,361 per month; base: $6.12 per hour, variable: $5.95 per hour). Variable pay is the sum of various commissions earned on the sale of different items. We compute the average commission rate (average: 3.5%) by dividing variable pay by the value of sales. We compute sales per hour—corresponding to performance in our model—as the value of sales divided by the number of hours worked. Tenure averages 49 months, as measured from the hiring date indicated in the human resources (HR) records. 2. Stores and Employment There are on average 16.64 consultative sales associates in a store. As is typical in retail, store-level turnover is high: 3.4% per month (the average of a 4.8% termination rate and a 2.1% hiring rate).21 Within a store there are several departments, across which conditions vary somewhat.22 We control for this heterogeneity by adding department fixed effects in all our specifications. Store-level profits are measured by EBITDA (units shrouded). Profits are positive on average. Each store has a manager and sometimes one or more assistant managers. They are excluded from our “workers” sample because they fall into the category of “supervisors.” These figures are responsible for personnel decisions (hiring and termination) in coordination with central HR, and they monitor workers.23 We use the ratio μ of supervisors to workers in a store as a proxy for monitoring coverage, with the caveat that such a ratio captures the extensive margin of monitoring but not the intensive margin (supervisor effort). The ratio of supervisors to workers is decided by the store manager in coordination with central HR and varies both across and within stores. Panel A in table A.3 shows that within a store, variation in the supervisor-to-worker ratio over time does not correlate with turnover, profits, or the fraction of low types as defined below. 3. Minimum Wage Variation Our sample contains minimum wage increases enacted by states, counties, and cities; the relevant constraint is the highest requirement. From February 2012 to June 2015, stores in our sample were affected by 70 minimum wage increases: 49 at the state level and 21 at the county or city level.24 The mean minimum wage is $7.87 per hour. The mean minimum wage increase is $0.54. If a worker’s average hourly pay in a week (base plus variable) falls below the minimum wage, the employer is required to make up the difference, as prescribed by the Fair Labor Standards Act.25 We create a variable called “minimum wage adjustment,” which equals the amount paid by the employer to comply with the minimum wage (this variable is often zero and averages $0.23 per hour). In an average month, 5% of our workers are paid no more than the minimum wage and 42% receive an adjustment in at least one of the 4 weeks. A $1 increase in the minimum wage raises the minimum wage adjustment by $0.25 per hour (Table 2, col. 3).26 In addition, variable pay per hour increases by $0.44 per hour (Table 2, col. 4), reflecting the endogenous increase in performance that is the subject of this paper. Overall, a $1 increase in the minimum wage raises average total pay per hour by $0.65 per hour (col. 5), which corresponds to a 5% increase and an elasticity of 0.38. Table 2. Effect of Minimum Wage on Compensation Scheme w ¯ and Overall Pay Inclusive of Minimum Wage Adjustment Compensation Scheme ( w ¯ ) Overall Pay Inclusive of Minimum Wage Adjustment Base Rate: Regular Pay/Hour ($)

(1) Commission Rate: Variable Pay/Sale (%)

(2) Minimum Wage Adjustments/Hour ($)

(3) Variable Pay/Hour ($)

(4) Total Pay/Hour ($)

(5) Total Pay ($ in Hundreds)

(6) Minimum wage −.059 .126 .250*** .439* .645*** .856** (.042) (.077) (.044) (.235) (.172) (.336) Observations 217,822 213,697 217,822 217,822 217,822 217,822 Units Workers Workers Workers Workers Workers Workers Mean dependent variable 6.120 3.462 .225 5.947 12.51 13.61 Effect of minimum wage increase (%) −.957 3.628 111.3 7.390 5.154 6.289 4. Definition of Worker Types We divide workers into three types. A worker is classified as a high, medium, or low type at time t based on her performance at time t − 1 relative to the minimum wage at t − 1 . In the spirit of Aaronson, Agarwal, and French (2012) and Clemens and Wither (2019), and following definition 1 in the theory, low types are those paid the minimum wage in t − 1 (about 4% of our observations). The remainder of the workers are either medium or high types, with the threshold between the two being the third quartile of the pay distribution.27 As expected, higher types sell more per hour, benefit from the minimum wage adjustment less often, and are terminated less frequently (see Table 3). A low type’s monthly earnings at t equals the minimum wage with a frequency of 20.5% and, moreover, is boosted by a minimum wage adjustment roughly every other week, thus suggesting that low types’ incentives are significantly affected by the minimum wage. In contrast, a high type’s monthly earnings at t equals the minimum wage with a frequency of only 0.7%, and they benefit from a minimum wage adjustment only once every 10 weeks, implying that they are negligibly affected by the minimum wage. Table 3. Descriptive Statistics for Low, Medium, and High Types Low Types

(1) Medium Types

(2) High Types

(3) % workers 3.9 72.4 23.7 % terminated 6.8 5.2 3.0 Sales/hour 1.08 1.94 2.73 % weeks with minimum wage adjustment 48.9 18.5 12.2 % months with minimum wage adjustment, all weeks 20.5 3.1 .7 5. Headquarters versus Store-Level Decisions Headquarters set the nationwide compensation scheme (base and commission rates, not adjusted for minimum wage) uniformly across stores and jurisdictions. Accordingly, when a local minimum wage changes, the base and commission rates earned by individual workers do not change systematically in that location. We show this in Table 2 (cols. 1, 2). Such wage stickiness makes sense in the presence of menu costs. Our theory reflects these institutional features in the assumption that the compensation scheme w ¯ does not vary with M and by avoiding the assumption that w ¯ is optimized at the local level.28 As mentioned, local managers have relative autonomy in deciding whether to terminate a worker or hire a new one, subject to maintaining the number of workers close to an agreed-on level with HR. In the model, the total number of workers L in a store is chosen to maximize expression (7), store by store. Pricing for our company is nationwide, as is the case for most national retail chains (Della Vigna and Gentzkow 2019). In section VII.B, we compute a store-level price index for our company and confirm that it does not vary with the local minimum wage. B. Identification Strategy 1. Sample Selection and Border-Discontinuity Design Our main empirical specification implements a border-discontinuity design in the spirit of Card and Krueger (2000) and closely follows Dube, Lester, and Reich (2010) and Allegretto, Dube, and Reich (2011). Specifically, workers on the side of the border where the minimum wage increased (treatment group) are compared with workers on the other side, where the minimum wage did not increase (control group). This research design aims to ensure that, apart from the minimum wage change, treated and control groups are similarly situated in terms of local economic conditions and demand shocks. The pretrend analysis in section IV.C supports this presumption. Appendix C.3 describes how the border sample is constructed. After restricting to stores located in counties whose centroids are less than 75 km apart, we are left with more than 200 stores and over 10,000 salespeople, approximately half of whom experienced variations in the minimum wage during our study period. An alternative research design consists of the traditional state panel approach, as employed by Neumark and Wascher (1992, 2007) among others and recently summarized by Neumark (2019). This strategy uses the entire sample of stores, regardless of their distance from the border. In section IV.C, we show that our core estimates are similar when applying this alternative research design. 2. Deriving Testable Implications from the Model Letting e*(x; M) denote type x’s optimal effort at minimum wage M, type x’s equilibrium performance is given by Y * ( x , M ) = Y ( x , e * ( x ; M ) ) . Linearizing around M yields the following estimating equation: (8) Y * ( x , M ′ ) = Y * ( x , M ) + ( M ′ − M ) ⋅ β . When β is estimated across all worker types, β ˆ = ???? [ Δ Y * ( x , M ) / Δ M ] represents the effect of the minimum wage on an average worker’s performance across all worker types. The analog of x is (9) Y * ( x , M ′ ) = Y * ( x , M ) + ( M ′ − M ) ⋅ [ β L 1 L ( x ) + β M 1 M ( x ) + β H 1 H ( x ) ] , where each β i represents the within-category performance effect of the minimum wage. Linearizing aroundyields the following estimating equation:When β is estimated across all worker types,represents the effect of the minimum wage on an average worker’s performance across all worker types. The analog of equation (8) by worker typeiswhere each βrepresents the within-category performance effect of the minimum wage. Our testable predictions are as follows. In the pure pay-for-performance case, proposition 2 (pt. 2) predicts that β L = 0 and β M , β H ≤ 0 . We will reject these predictions. In the hybrid case, proposition 1 (pt. 1) predicts that β L > 0 ; furthermore, in the high-monitoring subcase of the hybrid case, proposition 2 (pt. 3) predicts that β M , β H ≥ 0 . We will not reject these predictions. 3. Empirical Specification We translate equation (8) into the following regression specification: (10) Y i j p t = α + β M i n W j t + X i t ⋅ ζ + η Z j t + δ i + ϕ p t + ϵ i j p t , where Y ijpt represents the performance (sales per hour) of worker i in store j of county-pair p in month t, MinW jt represents the prevailing minimum wage in store j’s jurisdiction in month t, X it is a vector of time-varying worker characteristics that are likely to predict employee performance—specifically, the worker’s tenure and the department in which she works—and Z jt includes the monthly county-level unemployment rate to account for time-varying local economic conditions and local demand shocks (see Lemieux, MacLeod, and Parent i means that we leverage within-worker variation in the minimum wage. whererepresents the performance (sales per hour) of workerin storeof county-pairin month, MinWrepresents the prevailing minimum wage in store’s jurisdiction in monthis a vector of time-varying worker characteristics that are likely to predict employee performance—specifically, the worker’s tenure and the department in which she works—andincludes the monthly county-level unemployment rate to account for time-varying local economic conditions and local demand shocks (see Lemieux, MacLeod, and Parent 2012 ). Adding worker fixed effects δmeans that we leverage within-worker variation in the minimum wage. 29 Equation (10) includes county-pair × month fixed effects ϕ pt that restrict the comparison to treated and control stores/workers on either side of the same border. We estimate this equation by stacking our data as in Dube, Lester, and Reich (2010, 2016), meaning that stores/workers located in a county sharing a border with n other counties appear n times in the final sample. The standard errors are two-way clustered at the state level and at the border-segment level. Refer to appendix C.3 for more details on the specification. To study the heterogeneous effects of the minimum wage on worker performance by worker type, we translate equation (9) into the following regression specification: (11) Y i j p t = β 0 + β 1 M i n W j t + β 2 MediumType i j t + β 3 HighType i j t + β 4 M i n W j t ⋅ MediumType i j t + β 5 M i n W j t ⋅ HighType i j t + X i t ⋅ ζ + η Z j t + δ i + ϕ p t + ϵ i j p t , where MediumType ijt and HighType ijt are indicators for whether worker i is a medium or a high type. The effect of minimum wage on low, medium, and high types—that is, the coefficients β L , β M , and β H in 1 , β 1 + β 4 , and β 1 + β 5 , respectively. t − 1 relative to the minimum wage in t − 1 ; refer to the definition of types in where MediumTypeand HighTypeare indicators for whether workeris a medium or a high type. The effect of minimum wage on low, medium, and high types—that is, the coefficients β, β, and βin equation (9) —corresponds here to β, and, respectively. 30 The indicators for low, medium, and high types are predetermined because they are defined based on a worker’s pay inrelative to the minimum wage in; refer to the definition of types in section III.A.4

IV. Core Empirical Results: Effect of Minimum Wage on Worker Productivity This section tests the predictions from section III.B regarding the effect of the minimum wage on worker productivity. A. Core Findings Figure 1 displays the estimates of the β’s from equation (11)—that is, the effect of a $1 minimum wage increase on the percentage change in the performance of low, medium, and high types (for details, see Table 4, col. 2). We find that a $1 increase in the minimum wage increases performance (sales per hour) strongly among low types—that is, by 0.244 (shrouded units) or 22.6%. In the notation from section III.B, this means that β ˆ L > 0 , which rejects the pure pay-for-performance case and is consistent with the hybrid case. Fig. 1. Minimum wage (MinW) has a robust positive effect on the productivity of low types and no effect on high types. Shown is the effect of a $1 increase in the minimum wage on the percentage change in Y (sales per hour) for low-, medium-, and high-type workers. “Low type” refers to workers for whom the total pay per hour in month t − 1 is at minimum wage, “medium type” refers to workers for whom the total pay per hour in month t − 1 is between the minimum wage and 180% of the minimum wage, and “high type” refers to workers for whom the total pay per hour in month t − 1 is above 180% of the minimum wage. Vertical bars represent 95% confidence intervals computed using the estimated coefficients ( β 1 ˆ , β 1 ˆ + β 4 ˆ , and β 1 ˆ + β 5 ˆ ) from equation (11) and the associated standard errors. Table 4. Effect of Minimum Wage on Worker Productivity Dependent Variable: Sales/Hour (1) (2) Minimum wage .094** .244*** (.039) (.042) Medium type .354*** (.032) High type 1.169*** (.072) Minimum wage ⋅ medium type −.085*** (.025) Minimum wage ⋅ high type −.182*** (.032) Observations 217,822 209,513 Units Workers Workers Mean dependent variable 2.085 2.085 Effect of minimum wage increase(%) 4.485 Effect of minimum wage increase for low type (%) 22.56 p-value .009 Effect of minimum wage increase for medium type (%) 8.186 p-value .009 Effect of minimum wage increase for high type (%) 2.273 p-value .179 The effect is weaker but still positive for medium types ( β ˆ M = 0.156 , or 8.2%). Again, the pure pay-for-performance case is rejected, and the hybrid case is not. According to the theory, this effect obtains because our medium types occasionally earn minimum wage;31 thus, their response somewhat aligns with that of the low types. However, the effect vanishes for high types ( β ˆ H = 0.062 , or 2.3%, statistically indistinguishable from zero). These workers’ pay is least affected by the minimum wage, such that they barely respond to it. Next, we study the effect of the minimum wage on average worker performance. Because the effect is nonnegative for every type, we expect average worker performance to increase. Column 1 of Table 4 shows that a $1 increase in the minimum wage raises average individual performance by 0.094 (shrouded units), or 4.5%. This individual performance gain is economically sizable and statistically significant at the 5% level. The overall implied elasticity is 0.35.32 We conclude with a sanity check: as expected, worker pay increases with the minimum wage (table A.1). This increase is explained not by a change in the compensation scheme (recall that we do not find one) but rather by the mechanical effect of the minimum wage increase (more minimum wage adjustments) combined with the endogenous effort boost (more variable pay). Interestingly, the effect on pay is sizable for low and medium types, suggesting that both earn more because of larger and more frequent minimum wage adjustments and also from becoming more productive. B. Dynamic Effects We explore pretrends and the time pattern of the minimum wage effect by estimating the following distributed lag specification: (12) Y i j p t = α + ∑ m = −2 2 β 1 3 m M i n W j , t − 3 m + ∑ m = −2 2 β 2 3 m M i n W j , t − 3 m ⋅ MediumType i j t + ∑ m = −2 2 β 3 3 m M i n W j , t − 3 m ⋅ HighType i j t + γ 1 MediumType i j t + γ 2 HighType i j t + X i t ⋅ ζ + η Z j t + δ i + ϕ p t + ϵ i j p t , where β 1 ℓ captures the contemporaneous effect of the minimum wage on the productivity of low types if ℓ is zero, the ℓ-lagged (posttreatment) effect for low types if ℓ is positive, and the ℓ-lead (pretreatment) effect for low types if ℓ is negative; β x ℓ captures the difference in these productivity effects for type x = 2 , 3 (medium or high) relative to low types. wherecaptures the contemporaneous effect of the minimum wage on the productivity of low types if ℓ is zero, the ℓ-lagged (posttreatment) effect for low types if ℓ is positive, and the ℓ-lead (pretreatment) effect for low types if ℓ is negative;captures the difference in these productivity effects for type, 3 (medium or high) relative to low types. Figure 2 and the corresponding table A.2 present the cumulative response of the minimum wage on worker productivity, normalizing the estimates relative to the last preperiod (period −3).33 They show that the cumulative leading coefficients are not statistically significant for any worker type, confirming that there is no pretrend within worker type and no differential pretrend across types. They also show that low types display a 29% (statistically significant) cumulative increase in performance at or after the 6-month mark, suggesting that the performance effect is persistent. The effect sets in immediately after the minimum wage increase. High types, in contrast, do not experience a statistically significant response to the minimum wage. Fig. 2. Dynamic effect of the minimum wage (MinW) on worker productivity. Shown is the effect of a $1 increase in the minimum wage on the percentage change in Y (sales per hour) for low-, medium-, and high-type workers. “Low type” refers to workers for whom the total pay per hour in month t − 1 is at minimum wage, “medium type” refers to workers for whom the total pay per hour in month t − 1 is between the minimum wage and 180% of the minimum wage, and “high type” refers to workers for whom the total pay per hour in month t − 1 is above 180% of the minimum wage. Diamonds, squares, and circles show the cumulative response to the minimum wage relative to event date −3 for low, medium, and high types, respectively. The posttreatment effects for each type are computed by summing each type’s contemporaneous and lagged coefficients from equation (12) (e.g., for low types, β ˆ 1 0 , β ˆ 1 0 + β ˆ 1 3 , and β ˆ 1 0 + β ˆ 1 3 + β ˆ 1 6 ) and dividing by that type’s mean productivity. The pretreatment effects are computed by summing each type’s lead coefficients, multiplying the sum by −1 (e.g., for low types, − β ˆ 1 3 − β ˆ 1 6 and − β ˆ 1 3 ) and dividing it by that type’s mean productivity. Vertical bars represent 95% confidence intervals. C. Threats to Identification and Robustness Checks This section explores three potential threats to identification: violation of the common trends assumption, cross-border movements, and worker selection. We show that our core findings (productivity of low types increasing following a minimum wage hike) are robust across various alternative implementations of the research design. We briefly discuss each of these below; in-depth discussion and tables are provided in appendix E. 1. Pretrends Figure 2 displays the dynamic effects of the minimum wage. It shows no pretrends in performance by type for the sample of 102,000 observations generated by workers who remain employed over a window with a 6-month preperiod and 6-month postperiod around the minimum wage event. Figure A.1 confirms the lack of pretrends in the smaller sample of 89,000 observations generated by workers who are continuously employed over the wider window with a 9-month preperiod and 6-month postperiod, with the caveat that some of these workers experience more than one minimum wage change in the window because changes often happen at a yearly cadence. We also observe no pretrends for the larger sample of 144,000 observations generated by workers who are continuously employed for 6 months before the minimum wage event and for the sample of 107,000 observations generated by workers who are continuously employed for 12 months before the minimum wage event (see table E.1, cols. 1–3). The fact that the pretrends agree in these different samples is encouraging, as the difference in numerosity is nonnegligible and any difference in the estimates could indicate the presence of sample selection effects. 2. Cross-Border Worker Movements Border-discontinuity research designs are vulnerable to the concern that workers may move across borders (Neumark, Salas, and Wascher 2014). However, our core results on individual productivity should not be subject to this issue given that we include worker fixed effects, thus effectively comparing the “same worker” at two minimum wage levels. Further evidence against endogenous cross-border movements is provided by the absence of a correlation between the minimum wage increases and the home-to-work distance of new hires (table E.2, col. 1), which rules out changes in our workforce’s commuting patterns after a minimum wage increase. One might worry that cross-county migrants (rather than commuters) may confound store-level estimates. Zhang (2018), however, finds that after a minimum wage increase, migrants flow toward the same counties as commuters. The null effect in column 1 of table E.2 accordingly suggests that migration patterns also do not change among our workforce after a minimum wage increase. Furthermore, migration is likely more costly across state lines than across county or city lines, yet our estimates are the same in the sample including only county-city minimum wage increases as in the sample including only state-level increases (see sec. IV.C.6). Finally, table E.2 shows that the minimum wage does not affect the home-to-work distance proportionally more for low, medium, or high types (table E.2, col. 3). In sum, we believe it is unlikely that the cross-border movement of workers plays a significant role in our estimates. 3. Worker Selection The estimated productivity gains could potentially suffer from a selection bias owing to the change in the composition of retained workers following the minimum wage increase. We expect this worker selection confounder to have been largely controlled for by the inclusion of worker fixed effects in all our specifications. However, worker fixed effects may not necessarily eliminate the entirety of the selection bias.34 To alleviate this concern, we present two sets of results. First, we replicate our findings, restricting the sample to a balanced panel containing only workers who are employed throughout the sample period. When we do this, the sample size drops but its pretrends are the same, and the results are similar to the main sample (tables E.3, E.4). Second, reverting to the full sample, we obtain bounds for the selection bias in the estimates of interest. We do this by modeling the portion of the productivity change that is due to the change in worker composition and providing an upper bound for it (see app. E.3). We find that the bounds are small relative to the size of the baseline estimates: selection bias accounts for at most 4.6% of the baseline estimate of the average worker’s productivity change and at most 8.1% of the estimate for the low types. 4. Alternative Classifications of Low, Medium, and High Types Our baseline definition of type does not guarantee that types in the control county of a given county pair occupy the same quantiles in that county’s wage distribution as the quantiles occupied by the types in the treated county. To ensure a perfect quantile-quantile match across counties within a pair, we can change the type definition in the control county only and define these types using quantiles, so that the type distribution in the control county matches that in the treated county. When using this alternative approach, the results are nearly identical to our main findings (see table E.5, col. 1).35 In columns 2–5 of table E.5, we explore alternative ways of defining types—classifying them based on average pay in the previous 3 months (as opposed to the previous month) and constructing time-invariant types based on pay in their first month of employment or performance in their first quarter of employment. In table E.6, we change the threshold that separates medium and high types to 120%, 140%, or 160% of the minimum wage. Reassuringly, the findings paint the same picture regardless of the classification method: when minimum wage increases, low types become significantly more productive, while high types do not. 5. Alternative Research Designs Our border-discontinuity research design discards a large portion of the sample. We now explore a state-level design à la Neumark and Wascher (1992), which uses the entire sample of stores regardless of their distance from a border. The state-level design raises the question of what controls to include. Adding more controls is generally thought to produce closer estimates to the border-discontinuity design. Accordingly, with the aim of demonstrating the robustness of our results, we examine three minimally controlled specifications: with worker and month fixed effects, adding linear state trends, or adding census division × month fixed effects (see table E.1, cols. 2–4). The specification with division × month fixed effects is preferred because it is the only one that eliminates pretrends in worker performance. In this specification, the minimum wage once again increases the performance of low types and does not affect the performance of high types (see table E.7; fig. 3A).36 Fig. 3. Minimum wage (MinW) has a robust positive effect on the productivity of low types and no effect on high types. Shown is the effect of a $1 increase in the minimum wage on the percentage change in Y (sales per hour) for low-, medium-, and high-type workers. “Low type” refers to workers for whom the total pay per hour in month t − 1 is at minimum wage, “medium type” refers to workers for whom the total pay per hour in month t − 1 is between the minimum wage and 180% of the minimum wage, and “high type” refers to workers for whom the total pay per hour in month t − 1 is above 180% of the minimum wage. Panel A includes all stores, regardless of their distance from the border, in a specification with division × month fixed effects and standard errors clustered at the state level. Panel B (respectively, C) considers our main sample but only for state (respectively, within-state) variations in the minimum wage. Panel D (respectively, E) considers the sample of stores that are located less than 37.5 km (respectively, 18.75 km) from the border. Panel F considers our main sample but with nonstacked data and with border-segment-month fixed effects. Vertical bars represent 95% confidence intervals computed using the estimated coefficients ( β 1 ˆ , β 1 ˆ + β 4 ˆ , and β 1 ˆ + β 5 ˆ ) from equation (11) and the associated standard errors. 6. State versus Local Variation in the Minimum Wage Restricting the analysis to state-level minimum wage changes only or to county and city changes only (fig. 3B, 3C; table E.8) does not change our findings. This is reassuring, as one could worry that the cross-state variation is contaminated by other state-level policy changes. 7. Alternative Definitions of Bordering Stores We explore alternative definitions of bordering that are based on the exact location of the store rather than its county’s centroid. In addition, we set distance from the border to less than 37.5 km or less than 18.75 km, both shorter than in the main definition. Reassuringly, our results are consistent across these samples. See figure 3D, 3E and table E.9. 8. Robustness to Unstacking The results are also robust to using the same county-level border-discontinuity design as in our main estimates but without stacking the observations, with border segment × month fixed effects and clustering standard errors at the border-segment level (fig. 3F; table E.10). This specification is closer in spirit to an experimental-event design. 9. Alternative Controls The findings are similar if we control for department × store time trends and take into account potential differential trends across departments of a given store or if we run our specifications by department. Likewise, we obtain nearly identical results if we remove potentially “bad controls”—that is, variables that might be endogenous to the minimum wage level (worker tenure and county-level unemployment). See tables E.11 and E.12.

V. Heterogeneous Effect by Monitoring Illuminates Dual Nature of Model The theory makes two kinds of predictions. First, monitoring a worker more intensely results in weakly increasing her individual performance (proposition 2, pt. 1). Second and more interestingly, the effect of the minimum wage is heterogeneous by monitoring. Among the nonmonitored workers, the low types should not change their effort, while higher types should decrease their effort (proposition 2, pt. 2). Among highly monitored workers, all types should increase their effort (proposition 2, pt. 3), at least to some extent (proposition 1, pt. 4). This bifurcated response to the minimum wage reflects the dual nature of worker incentives. If highly monitored, the efficiency wage logic dominates, meaning that the increase in the wage level due to a rise in M motivates the worker. If not monitored, the pay-for-performance logic dominates, meaning that the worker is demotivated by a rise in M due to the decrease in the sensitivity of the wage to effort. This bifurcated prediction is a strong test of the dual nature of the theoretical model. We test these predictions using within-store variation in μ. In the model, μ represents the fraction of workers (independent of type) who are highly monitored. We proxy for μ using the supervisor-to-worker ratio in a store-month. A store is classified as either low coverage if it falls within the bottom quartile of the supervisor-to-worker ratio distribution or high coverage otherwise. Consistent with proposition 2 (pt. 1), we find that high coverage does positively correlate with average worker performance (Table 5, col. 1). While reassuring, this is not a very strong test of the dual nature of our model, as the presence of supervisors could also improve performance through channels other than monitoring. Next, we test the predictions that are most revealing of our model’s dual nature. Table 5. Effect of Minimum Wage on Worker Productivity with High or Low Mo