Industry Dynamics and the Minimum Wage: A Putty-Clay Approach

We document three new findings about the industry-level response to minimum wage hikes. First, restaurant exit and entry both rise following a hike. Second, the rise in entry and exit is concentrated in chains. Third, there is no change in employment among continuing restaurants. We develop a model of industry dynamics based on putty-clay technology and show that it is consistent with these findings. In the model, continuing restaurants cannot change employment, and thus industry-level adjustment occurs through exit of labor-intensive restaurants and entry of capital-intensive ones. We show these three findings are inconsistent with other models of industry dynamics.


Introduction
This paper presents new evidence on how the restaurant industry, the largest U.S. employer of low-wage labor, responds to minimum wage hikes. We document three new empirical findings. First, exit and entry among limited service (i.e. fast food) restaurants rise after a minimum wage hike. Second, the rise in entry and exit is higher among chains. Third, there is no change in employment among continuing limited service restaurants. Together, these results imply an economically small impact on employment two years after a minimum wage hike. We show that an augmented putty-clay model explains these responses but other models of the labor market cannot. To the best of our knowledge, we are the first to provide micro-level evidence supportive of the importance of putty-clay relative to competing models of firm dynamics.
Our empirical findings are derived from the Census of Employment and Wages (QCEW), a database used to compile unemployment insurance payroll records collected by each state's employment office. The QCEW provides detailed information on each establishment's name, location, and employment level at a monthly frequency. We follow Card and Krueger (1994), Addison, Blackburn, and Cotti (2009), and Dube, Lester, and Reich (2010), among others, and compare restaurants that reside in counties near state borders where the minimum wage has risen on one side of the border but not the other. Our results suggest that exit and entry, particularly among chains, increases in the year following a minimum wage hike. By contrast, we find no comparable exit or entry effect among full service restaurants and mixed evidence among other accommodation and food service industries, both of which make less use of low-wage labor.
To interpret these findings, we describe a model of industry dynamics that extends the putty-clay framework of Sorkin (2015) and Gourio (2011) to incorporate endogenous exit as in Campbell (1998). In the model, new entrants can choose from a menu of capital-labor intensities but, once the establishment is built, output is Leontief between capital and labor.
In this environment, adjusting the capital-labor mix in response to higher wages requires shutting down labor-intensive establishments and opening capital-intensive establishments. Hence, the model predicts that, given reasonable parameters, both entry and exit rise in response to a minimum wage hike.
1 Not only does the putty-clay model match our new empirical findings on exit and entry, but it generates two other predictions that appear consistent with the minimum wage literature. First, the model implies that the cost of higher minimum wages are fully passed onto consumers in the form of higher prices (Aaronson (2001), Aaronson, French, and MacDonald (2008)). Second, because continuing restaurants cannot adjust their employment levels, the putty-clay model generates a small short-run employment response, consistent with much of the literature. Importantly, however, under putty-clay technology the disemployment effect of the minimum wage grows over time, as labor intensive incumbent restaurants are slowly replaced with more capital intensive entrants. This paper is organized as follows. In section 2, we argue that increases in exit and entry resulting from minimum wage hikes are inconsistent with other models of industry dynamics, as well as models incorporating imperfect competition in labor markets. Sections 3 to 5 describe the QCEW data, the estimation strategy, and the results. In section 6, we present the putty-clay model, which is used in section 7 to show how a minimum wage hike impacts exit and entry. A calibration of the model is presented in section 8. Section 9 concludes.

Literature Review
Putty-clay models have been effective at matching aggregate business cycle (Gilchrist and Williams (2000) and Atkeson and Kehoe (1999)) and financial market (Gourio (2011)) facts in a number of settings. 1 Our results complement earlier research by documenting that firm entry and exit decisions are consistent with the predictions of putty-clay models. As such, we believe we are the first to provide firm-level empirical evidence supportive of the relevance of putty-clay technology.
The key feature of the putty-clay model -that potential entrants are able to pick a capitallabor ratio that is well-suited to the minimum wage while incumbents are not -is not present in several benchmark models typically used to describe industry dynamics or the impact of minimum wage hikes. For example, Hopenhayn (1992) assumes that factor proportions can freely change. Thus his model predicts an increase in exit and a fall in entry after a minimum wage hike.
Search models contain a mechanism that can potentially match our entry and exit results.
In Flinn (2006), a minimum wage hike causes the lowest productivity matches to break up, generating a spike in firm exit. Additional exit increases the number of job searchers, potentially raising the return to posting a vacancy and thus potentially causing a spike in firm entry. That said, because Flinn (2006) is solved in steady-state, as is standard in the literature, his model does not distinguish between entry and exit. Furthermore, Flinn's model does not speak to our continuing firm results because it is a model of a firm vacancy and a single potential worker. Models with multiple worker firms, as in Elsby and Michaels (2013) or Acemoglu and Hawkins (2014), could likely match our exit results but not the lack of employment change among continuing firms. 2 Thus we believe that putty-clay is a key part of any explanation of the industry dynamics that we empirically document. 3 Our paper also adds to the voluminous literature on the employment effects of the minimum wage, surveyed by Neumark and Wascher (2008). In particular, we believe we are among the first (see also Rohlin (2011)) to estimate the firm entry and exit responses to minimum wage hikes. 4 Estimation of these responses provides clearer tests of models of low-wage labor market structure, which is critical for evaluating labor market policies to help the poor.
Moreover, we show that the putty-clay model is consistent with other market responses to minimum wage hikes that have been studied in the literature, including higher price levels (e.g. Aaronson (2001), Aaronson, French, and MacDonald (2008), and Basker and Khan (2013)), lower profits (Draca, Machin, and Reenen (2011)), and larger disemployment in the long-run than the short-run (Baker, Benjamin, and Stanger (1999), Meer and West (2015), 2 In a multi-worker firm model, the minimum wage hike would have heterogeneous effects among firms. High-paying firms benefit from the increased ease of finding workers and therefore might expand. Low-paying firms for which the minimum wage hike is binding might contract.
3 Like us, Jovanovich and Tse (2010) document evidence of a simultaneous spike in entry and exit in response to industry-level technology shocks. They develop a vintage capital model to describe these facts. However, their model still allows firms to freely adjust their capital-labor ratio, and thus would not predict a simultaneous spike in entry and exit after a minimum wage, or other factor price, change.
An important aspect of the putty-clay model is the decision of when to scrap. In this sense we also contribute to the optimal scrapping and replacement literature (Adda and Cooper (2000)) by aggregating to the industry level. and Sorkin (2015)). Our findings also complement recent work that finds a reduction in hiring and separations after a minimum wage hike (Brochu and Green (2013), Dube, Lester, and Reich (2013), and Gittings and Schmutte (2014)). Our results imply minimum wage hikes increase firm turnover, while their results suggest worker turnover declines among firms that neither enter nor exit following a minimum wage hike. Nevertheless, we see these results as potentially complementary to ours in that each suggest important dynamic dimensions, either within or across establishments, in which there are responses to a labor cost shock. 5 The only paper that simultaneously studies exit and entry in response to a minimum wage increase to which we are aware is Rohlin (2011). Using detailed firm locations derived from the Dun and Bradstreet Marketplace data files, he finds that state minimum wages hikes instituted between 2003 and 2006 discouraged firm entry but had little impact on the exit and employment of establishments in existence at least 4 years prior. Rohlin identifies exit, entry, and employment effects within miles of state borders, rather than at the coarser county level that we use. However, his main results are reported at the 1 digit (6 industries) SIC level, far too aggregated to distinguish heavy minimum wage users. Strikingly, the largest negative entry appears in manufacturing, where only 10 percent of its workforce is paid within 150 percent of the minimum wage (and only 3% within 110% of the minimum wage). Dube, Lester, and Reich (2010) find no earnings or employment effects of minimum wage hikes in manufacturing. In contrast, just over half of restaurant industry workers are paid within 150 percent of the minimum. Given Rohlin's detailed geographic precision, sample sizes get quite small when results are reported at the more detailed 2 digit industry level.

Data
We study the restaurant industry (NAICS 722) because it is the largest employer of workers at or near the minimum wage, accounting for roughly 16 percent of such employees between 2003 and 2006 according to the Current Population Survey's Outgoing Rotation Groups. 6 Moreover, the intensity of use of minimum wage workers in the restaurant industry is amongst 5 Similarly, Brochu et al. (2015) emphasize different responses among continuing, beginning and ending employment matches, which is analogous to our distinctions among continuing, entering, and exiting firms.
6 The next largest employer, retail grocery stores, employs just under 5 percent of minimum or near minimum wage workers. 4 the highest of the industrial sectors (Aaronson and French (2007)). Like many studies before this one (e.g. Katz and Krueger (1992), Card and Krueger (1995), Card and Krueger (2000), Neumark and Wascher (2000), Aaronson and French (2007) and Aaronson, French, and MacDonald (2008)), we concentrate specifically on limited service establishments, which are especially strong users of minimum wage labor. 7 Our source of restaurant employment data is the Quarterly Census of Employment and Wages (QCEW). Under an agreement with the Bureau of Labor Statistics (BLS), we were granted access to the establishment-level employment data provided in the QCEW data file. 8 The QCEW program compiles unemployment insurance payroll records collected by each state's employment office. The records contain the number of UI-covered employees on the 12 th of each month.
The main advantages of the QCEW are that it covers virtually all firms, and has very little measurement error. But as is typical of administrative datasets, information about establishments is sparse. In particular, the key variables are establishment ID, employment, location, and trade and legal name. The former three are used to measure exit, entry, and employment changes by geographic location. The trade/legal name allows us to identify establishments that are part of large chains. 9 Our results are derived from five state-level minimum wage hikes -a 17 percent increase  Table 1). 10 While a number of other states 7 In "limited service" (LS) outlets, meals are served for on or off premises consumption and patrons typically place orders and pay at the counter before they eat. In "full service" (FS) outlets, wait-service is provided, food is sold primarily for on-premises consumption, orders are taken while patrons are seated at a table, booth or counter, and patrons typically pay after eating. Unfortunately, prior to 2001, industry codes were unable to differentiate limited service and full service outlets. This is one reason why we concentrate on minimum wage changes in the 2000s. Another reason is that there is significant concern about the accuracy of single establishment reporting prior to 2001. We describe this problem below. 8 We also use data prior to 2003 when the QCEW was referred to as the ES-202. 9 However, the BLS' confidentiality restrictions do not allow us to disclose the chain names nor how we developed our list.
10 Other than Oregon, the hikes are of comparable size; our results are robust to dropping the Oregon hike. passed minimum wage changes during the 2000s, we exclude them because either a) the state QCEW data was not accessible (e.g. Pennsylvania, Massachusetts, New York), b) the change is small (e.g. CPI adjustments), or c) bordering states also raised their minimum wage. 11 Nevertheless, these five states and their neighbors contain significant numbers of restaurants along the borders.
We face three measurement issues with regard to creating a consistent panel of QCEW restaurant employment, entry, and exit.
First, many small restaurants appear to exit and then re-enter within a year. These look like seasonal businesses that are open for example, only in the summer or the winter.
To address this concern, we define an entrant after the hike as an establishment without employment in the year before the hike but with average monthly employment above 15 in each of the two six-month periods starting a year after the hike. 12 Likewise, we define an exit after the hike as an establishment having average employment above 15 employees in each of two six month periods prior to the minimum wage hike and no employment starting a year after the hike. We document how our results vary when we alter the size requirement between 1 and 20 employees.
Second, the BLS did not collect industry NAICS codes until 2001. Therefore, we must use BLS imputations of industry for establishments that exited prior to 2001. This problem is only relevant for one of the five state-level minimum wage hikes that we exploit (the 2001-02 California hike); the other four state-level minimum wage hikes that we study take place well after 2000, and thus imputed data is not needed.
Third, firms sometimes group establishments together for reporting purposes. In a multiestablishment firm, an individual establishment's birth or death may look instead like growth or contraction of a larger continuing firm. Moreover, reporting arrangements can switch between multi-unit and individual establishment reporting over time. Switches from multiunit to individual establishment reporting ("breakouts") will appear in our data as multiple births with the possibility of a death. Switches from individual to multi-unit reporting ("consolidations") will appear as multiple deaths with the possibility of a birth. Fortunately, using the QCEW Breakout and Consolidations Link (BCL) file, we can identify and drop establishments that were ever involved in a breakout or consolidation. Furthermore, we consider the robustness of our results to imposing an upper bound of 100 employees on establishment size. Like Card and Krueger (2000), we find that our results are robust to changes in this upper-bound.
Appendix tables A4 and A5 provides more details on sample construction and summary statistics.

Empirical Strategy
States might be more likely to raise the minimum wage in good times. Thus, standard statelevel difference-in-difference regressions may confound the impact of the minimum wage with the economic conditions that allowed minimum wage legislation to move forward.
To circumvent this problem, we focus on restaurants near state borders. A key advantage of the QCEW is that it contains the establishment's precise location. Geographically nearby restaurants in different states with different minimum wages likely face similar economic envi-ronments (other than having a different minimum wage). Our approach compares restaurants near a state border to the nearby restaurants on the other side of the state border. 13 In particular, we use the Dube, Lester, and Reich (2010) approach and estimate where Y ispt is the outcome of interest, w ist is the minimum wage faced by restaurant i in state s at time t, a pt is a full set of border segment-time dummies (e.g., northern California-southern Oregon in Jan. 2013), α s is a state dummy, ispt is a residual that is assumed uncorrelated with the minimum wage, and β is the impact of the minimum wage. We concentrate on three measures of Y ispt : the entry rate, the exit rate, and the percent change in employment among continuously-operating establishments. Entry is an indicator variable of whether establishment i existed at time t and not at time t − 1 and exit is an indicator for whether establishment i existed at time t − 1 and not time t. For each minimum wage hike, we use two time periods -before and after the hike. We use only restaurants that are in counties on a state border or adjacent to a county on the state border.
When we study entry and exit, equation (1) is a regression in levels at the restaurant level.
β is the estimated percentage point increase in the entry or exit probability of a restaurant in the treated counties relative to the control counties in response to a 100% minimum wage hike. To facilitate interpretation, we convert our estimates to elasticities using the baseline entry and exit rates.
Equation (1) is a generalization of the usual difference-in-difference approach. To see this comparison, define t np as the time of the first minimum wage hike for border segment p.
Differencing equation (1) yields: Next define the state that raised the minimum wage hike as state s and the comparison state that borders state s but did not have the hike as state ς. Differencing equation (2) across states s and ς yields all of the dummy variables related to time and geography vanish. However, there are two key differences between equation (3) and the usual difference-in-differences specification. First, we use the level of the minimum wage rather than a dummy for whether the minimum wage increased. Second, instead of comparing just two states, equation (2) allows us to pool multiple state level minimum wage hikes. Third, our approach does not compare changes across states, but across border segments, so, for example, the change in entry in Northern California is compared to the change in entry in southern Oregon. Table 2 reports the impact of a minimum wage increase on the likelihood of exit (row A), entry (row B), and change in employment of continuing firms (row C). Results for limited service restaurants are presented in columns (1) to (3), full service restaurants in column (4), and establishments that are not restaurants but in the NAICS 72 hospitality and food services industry in columns (5) and (6). Bootstrapped standard errors are in parentheses.

Results
We find that exit of limited service establishments unambiguously rises in the year after a minimum wage increase. A 1 percent increase in the minimum wage causes exit rates in limited service establishments to go up 2.40 percent (standard error of 0.86 percent). Given these estimates, a 10 percent increase in the minimum wage would increase the limited service annual exit rate from its sample mean of 5.7 percent prior to the minimum wage increase to approximately 7.1 percent after the hike. Our exit estimates are statistically significant at conventional levels. 14 Exits rise faster among chains than non-chains.
By sharp contrast, there is no impact of a minimum wage increase on the exit of full service restaurants (column (4), row A), nor on other NAICS72 establishments other than restaurants and hotels and motels (column (6), row A), where minimum wage labor share 14 Results are also statistically significant at the 5 percent level for all establishments when standard errors are clustered at the state-border or state-border-segment level. is lower (Aaronson and French (2007)). We do find a large impact on hotels and motels (column (5), row A). Hotels are fairly intensive users of minimum wage labor, although the magnitude of the estimated effect is still surprising. 15 Row B reports results on entry rates. Similar to exit, entry also increases in the year after a minimum wage hike. We find a one percent increase in the minimum wage leads to 1.37 percent (standard error of 0.61 percentage points) increase in the entry rate in the year after the hike relative to the two years prior. 16 Given these estimates, a 10 percent increase in the minimum wage would increase the limited service annual entry rate from its sample mean of 8.7 percent prior to the minimum wage increase to approximately 9.9 percent after the hike. The estimated entry effect is larger in establishments affiliated with a chain; entry rises by 2.64 (1.02) percent among chains but 0.78 (0.74) percent among non-chains. Notably, there is again no impact on the entry of full service restaurants, hotel and motels, or other non-restaurant NAICS72 establishments.
Row C reports results on employment changes among continuing firms. We find little evidence of a significant change in employment among any NAICS72 industries, including limited service restaurants, after a minimum wage increase. Table 3 provides a number of robustness checks of our benchmark specification (shown again for convenience in row A). Rows (B) to (D) vary the minimum employee size required to be in our sample from 1 to 20 employees. Of particular note, the aggregate limited service exit and entry elasticities are economically small and statistically indistinguishable from zero when the smallest establishments are included (row B). Yet even within this sample, we find economically meaningful, albeit not always statistically significant, differences between chains and non-chains. That is especially the case for entry, where the elasticity for chains is 2.09 (0.76) and is -0.18 (0.37) for non-chains. Once we drop the smallest restaurants (rows C and D), the exit and entry results become larger, although entry remains concentrated among chains regardless of establishment size. This pattern by size may indicate the difficulty of measuring exit and entry among the smallest establishments or, plausibly, that the minimum wage shocks apply in particular to establishments with a sizable workforce, which typically are chains.
Our benchmark specification allows for state border segment dummies. This implies that, for example, the Illinois-Indiana border is part of one labor market. To allow for more flexibility, we also split each border into four equal-length segments (what we call state-bordersegments) based on air distance from the southern or eastern-most point of that border and include the state-border segment a pt as a control. Although these results, reported in row (E), are a bit weaker overall, we view their general tenor as again supportive of the benchmark results -exit is fairly broad-based but entry is concentrated among chains.
Other reasonable perturbations, including excluding 100+ employee establishments to avoid concern that there are multi-establishments in the sample (row F) and excluding LA, Orange, and San Diego counties (row G), 17 have little impact on our inferences. 18 Indeed, as a whole, entry, and in some cases exit, differences between chains and non-chains are, if anything, more apparent in many of these cases.
Together, the exit and entry results have roughly offsetting effects on net employment.
Combined with the economically small impact on the employment of continuing restaurants, we estimate a disemployment elasticity of -0.1, suggesting that a 10 percent increase to the minimum wage reduces employment about 1 percent, although that estimate is highly imprecise. Precision aside, the point estimate is squarely in the range of previous estimates in the literature, especially those that use a border discontinuity design (Dube, Lester, and Reich (2010); Addison, Blackburn, and Cotti (2009)).
Overall, we read the results as suggesting that restaurant exit and entry rise in response to a minimum wage hike. 19 Employment barely changes among establishments that remain open throughout the period.
For the putty-clay model, the result that entry and exit is strongest among the chains is interesting for three reasons. First, on average, chains appear to be more capital intensive than non-chains. According to a survey conducted by the National Restaurant Association (NRA), 20 the compensation to sales ratio per full-time equivalent employee is 0.26 for chains and 0.30 for non-chains (Table D-6). Per seat, the difference is more dramatic: 0.21 for chains versus 0.33 for non-chains (Table D-12). Using 10-K reports for 22 large limited service restaurant chains, we find a similar payroll to revenues ratio of 24 percent. 21 Second, according to the NRA survey and 10-K reports, the interquartile range of payroll to sales ratio is 8-9 percentage points for the chains, compared to 6 percentage points for the non-chains.
17 Los Angeles, Orange, and San Diego counties are next to counties bordering Nevada or Arizona, although the vast majority of population in these three counties is far from the border. 18 We also experimented with using an indicator for a minimum wage hike (i.e. define wst−τ as 1 if there was a minimum change and 0 otherwise), rather than the magnitude of the increase and found similar results. 19 We have also used the Census' Statistics of U.S. Businesses (SUSB), which collects industry-state-year level information on exit, entry, and employment changes among continuing firms. We find qualitatively similar although quantitatively smaller effects on exit and entry in the SUSB. In particular, we find that restaurant entry and exit both rise within two years of a minimum wage change. No such effect is observed among non-restaurant NAICS72 establishments.
20 See the 2013-2014 NRA Restaurant Operations Report at http://www.restaurant.org/News-Research/Research/Operations-Report. 21 We found these companies, which do not correspond to the chains we identify in the QCEW, by using a list of the largest restaurant companies published by Nation's Restaurant News (www.nrn.com). We excluded full service restaurant companies, as well as those that did not file a 10-K in 2014.
In the context of the model, this extra dispersion provides scope for exit as well as entry to be concentrated in the chains. Third, because they have national operating manuals, in line with the model chains are likely to be relatively inflexible in response to location-specific cost shocks such as a minimum wage hike. 22

The Putty-Clay Model
The previous section of the paper showed that restaurant entry and exit both rise, and employment at existing restaurants changes very little following a minimum wage hike. Furthermore, there is some evidence that exiting restaurants are relatively labor-intensive, whereas entering restaurants are less labor-intensive.
As we argued in section 2, these findings are inconsistent with benchmark models of industry dynamics. To reconcile these findings, this section develops a model of industry dynamics based on putty-clay technology. When a restaurant enters, it can freely choose its input mix, so its technology is flexible like putty. The novel feature of the putty-clay model is that, after entry, the technology hardens to clay and the input level and mix is fixed for the life of the restaurant.
The putty-clay model is consistent with these findings because, following a minimum wage hike, putty-clay technology introduces an asymmetry between incumbents and potential entrants. Incumbents have a fixed input mix optimized for the old minimum wage but potential entrants choose an input mix optimized for the new minimum wage. This puts incumbent restaurants at a cost disadvantage following a minimum wage hike. Indeed, some exiting incumbents would remain open if they could adjust their input mix. This displacement of incumbents by more capital-intensive entrants generates a spike in entry following the hike.
This section sketches the key features and results of the model. Further details and proofs are in the appendix.

Production
Restaurants produce food using four inputs: capital, high-skill labor, low-skill labor, and materials. Capital includes land, structures, and machinery. Low-skilled labor is paid the minimum wage. A restaurant bundles inputs to produce initial output y 0 .
Ex-ante, restaurants can flexibly substitute between inputs. Restaurants face a CES production function so that production at time 0 (the birth of the restaurant) is where α = α k +α m +α h implies constant returns to scale, A j is the productivity of a restaurant aged j, σ is the elasticity of substitution, k is capital, m is materials, h is high-skill labor and l is low-skill labor.
Ex-post, the production function is Leontief and restaurants cannot substitute between inputs. Let k , m , h and l denote the initial input choices. In subsequent periods, restaurant optimization and constant returns to scale imply that the restaurant either operates with its original proportions at full capacity, or does not operate: This equation emphasizes two features of the restaurant's technology. First, once a restaurant has entered, it gradually becomes less productive. A restaurant that is age j has total factor productivity (TFP) A j = A 0 e −δj , where δ is the deterministic TFP depreciation term.
Second, a continuing restaurant faces the decision only whether or not to produce in a given period and so it does not adjust its input mix or the levels of its input (below we discuss the exit decision). Given the Leontief assumption, an optimizing restaurant either produces with its original factor mix ({k , m , h , l }), or does not operate. The notation allows the restaurant to use more of a factor than it did in its initial input mix, but this is never optimal. Combining the rate of technology depreciation and the Leontief assumption, the output of the restaurant aged j while still alive is y j = y 0 e −δj .

Factor Demands
A restaurant makes two decisions at entry.
First, it decides its input mix which is then fixed once capital is installed. This is a forward-looking decision that therefore considers the effective factor prices over the life of the restaurant. In the model, restaurants assume all prices remain constant over the life of the restaurant, with P the price of the output good and p m , w h , and w as the rental prices of materials, high-skill labor, and low-skill labor (i.e., minimum wage), respectively.
The restaurant purchases capital at price p k and can resell the capital at price ηp k , where η < 1. This resale discount means that immediately after spending p k k on capital, an amount (1−η)p k k is sunk the instant a restaurant enters. Thus restaurants do not immediately replace their capital once it depreciates.
Second, it decides what exit rule to follow. For the moment take J, the date of restaurant exit, as given. We endogenize J in section 6.3.
Assuming an interest rate r, discounted payments over the life of a new restaurant for materials, high-skill labor, and low-skill labor are q m m = ( J 0 e −rj p m dj)m, q h h = ( J 0 e −rj w h dj)h, and q w = ( J 0 e −rj wdj)l, respectively. Recall that capital can be purchased at price p k and re-sold at price ηp k . Therefore, discounted payments to capital are q k k = p k (1 − e −rJ η)k.
Because a restaurant that initially produces y 0 at time 0 will produce y j = y 0 e −δj at time j, total revenue over the life of the restaurant is q p y 0 = ( J 0 e −(r+δ)j P dj)y 0 . Thus a restaurant's profit over its lifetime is: Consequently, an entering restaurant solves the following maximization problem: subject to equation (4), which implies the conditional factor demands, given the exit age J: (7)

Market Price Determination
In steady state, free entry pins down the market price. Let f denote the steady state mass of entrants each period. The free entry condition indicates that any profit opportunities will be bid away by new entrants, implying either expected profits or entry is zero: In steady state, there is entry so profits are zero. Plugging the conditional factor demands from equations (7) and setting π = 0 in equation (5), and also using equation (8) yields two equations with two unknowns (P and J). Although the analytic expressions for P and J are complicated, their solution is straightforward.

Market-Level Equilibrium
Having determined the restaurant's problem, we can solve for the total number of restaurants in a market. The industry faces an isoelastic product demand curve with elasticity γ: Product market clearing implies that quantity demanded equals quantity supplied, where the quantity supplied is: Market supply is comes from restaurants of vintage j supplying quantity e −δj y 0 , the density of each vintage of restaurant (and the mass of entrants each period) f , and the mass of different vintages of incumbent restaurants J. Integrating (11) and rearranging provides an explicit solution to the steady state mass of restaurants that enter in every time period: where Q is a function of P as in equation (10), and P and J are solved as in section 6.4.

Steady State Equilibrium
A steady state equilibrium is given by endogenous objects {k, h, m, l, Q, P, J, f } taking factor prices {p k , p m , w h , w, r} and the environment {δ, η, θ, γ, σ, α k , α m , α h , y 0 } as given such that: • Restaurants maximize profits, where profits are defined in equation (5); • Free entry holds (equation 9); and • The product market clears (equation 11).

A Minimum Wage Hike
In this section, we consider a permanent but unexpected minimum wage increase from w o to w n at time t n . 23 Such a hike affects employment through both a scale and substitution effect, sometimes referred to as the Hicks-Marshall channels.
When there is free entry and expected profits are zero, restaurants pass the higher labor costs to consumers in the form of higher prices. As a result, consumers purchase fewer meals and restaurants require fewer inputs. This reduction in sales causes net exit of restaurants immediately following a minimum wage hike and consequently an immediate fall in 23 Note t denotes calendar time whereas j denotes the age of a restaurant.
the employment of minimum wage workers (Aaronson and French (2007)). This channel is sometimes known as the "scale effect." A hike in the minimum wage also makes low-skilled workers more expensive, causing restaurants to substitute to cheaper factors of production. However, in a putty-clay model, all substitution occurs through entry and exit. Because remaining incumbents maintain their input mix, the substitution effect occurs gradually as the incumbents exit and are replaced by new restaurants that are free to choose the optimal input mix given the higher price of minimum wage labor.

Exit Dynamics
Since incumbent restaurants are committed to their input mix, the only margin on which they can respond to higher labor costs is by exiting earlier (or later). In this section, we endogenize exit.
Let J(w o , w n ) be the exit age of a restaurant that entered when the minimum wage was w o but is deciding to exit when the minimum wage is w n . We rewrite the restaurant exit decision equation (8) as where the left hand side is the marginal benefit of continuing to operate at exit age J(w o , w n ), and the right hand side is the marginal cost of continuing to operate. Note that we assume the product price jumps to its new steady state P n immediately. In the next section, we show when this assumption is satisfied.
If σ ≤ 1 and restaurants become more capital-intensive after a minimum wage hike (i.e.,  Following the spike in exit, the density of restaurants exiting in a given period remains the same as before the minimum wage hike until all of the incumbent restaurants have exited.
Appendix E provides a detailed discussion of exit dynamics. rises by less than for incumbents. This cost disadvantage causes incumbents to exit.

Product price and entry response
Next we show why entry rises and why product prices jump to their new steady state level immediately, as in equation (13), and consistent with the empirical findings of Aaronson (2001) and Aaronson, French, and MacDonald (2008), among others.
Let P n be the new steady state product price and Q n = θP −γ n be the new steady state market output. Equation (A29) in Appendix D shows that, to a first-order approximation, market output drops instantly from Q o to Q n when: The left hand side is the percent of incumbent restaurants that exit, and the right hand side   The moment that identifies the depreciation rate, δ, is the exit elasticity. All else equal, δ determines the slope of the marginal benefit curve (see figure 1). If δ is small, then the firms' productivity, and consequently marginal benefit of producing, declines slowly over time. In that case, productivity levels are similar for many incumbents, including those that are close to exiting. Thus a small hike causes many restaurants to exit. In contrast, a steep marginal benefit curve (high δ) means that few restaurants are close to exiting and the exit elasticity is small.
The moment that identifies the resale price of capital (η) is the steady state exit age J.
All else equal, η determines the relative level of the marginal cost curve shown in figure 1.
When the resale price is low, the opportunity cost of re-selling capital is low and thus the marginal cost of operation is low as well. When marginal cost is low, the restaurant remains open longer. In contrast, a high marginal cost curve (high η) signifies that the opportunity cost of operating is high and restaurants exit at a younger age.
Finally, the moment that identifies the elasticity of demand for restaurant output (γ) is the entry elasticity. Because price pass-through immediately follows a minimum wage hike, all else equal, γ determines the change in market quantity and hence output. A low γ indicates that output is unresponsive to a minimum wage hike and most exiting output is replaced by entry. A high γ means that output is very responsive to a minimum wage hike and therefore a spike in entry is unlikely.
The model interprets the estimated spike in exit and entry rates after the minimum wage as a small δ and a high η (table 4, panel B). Indeed, our calibrated value of δ is lower and our calibrated value of η is higher than most estimates in the literature. This is to be expected since we are studying an industry where a large part of the capital stock is land, which has a high resale value and a low depreciation rate. Because both the entry and exit elasticities are of similar size, the total disemployment effect is small. Our product demand elasticity is similar to Aaronson and French's (2007) preferred value of 0.5. This is lower than the estimate in Harasztosi and Lindner (2015). More detail on the calibration is in Appendix A. Figure 4 shows industry price, quantity and employment in the 17 years following a 10 percent one-time, unanticipated and permanent minimum wage increase from steady state. 25 Because of entrants, the product price jumps to the new steady state immediately, about 0.97 percent higher than before the hike. That price increase implies an elasticity of 0.097, in line with the evidence discussed in Aaronson and French (2007). Because the price jumps immediately, the industry quantity jumps to its new steady state level as well.

The effects of a minimum wage hike
Employment responds more slowly to the minimum wage hike. After one year, total Note: The minimum wage rises 10 percent immediately after time 0. We aggregate the data to an annual frequency. Panels depict the percent change in market prices for the output good, market quantity, and employment, relative to their levels before the hike.
26 employment (high-and low-skill) falls by 0.8 percent, implying an elasticity of -0.08. But the employment response grows over time such that in the steady state determined by the new minimum wage, the long-term elasticity is −0.40. This graph illustrates a key feature of the putty-clay model. Because restaurants turn over slowly following a minimum wage hike, the full employment effect of the minimum wage also unfolds slowly.
The employment results highlight two useful and related insights. First, putty-clay models imply larger long-run effects. While the evidence on long-run effects is scarce, what little exists is consistent with this prediction (Baker, Benjamin, and Stanger (1999)). Second, the minimum wage hike causes a trend in employment after the hike. Typical differencein-difference estimators measure short-run effects, assuming that the minimum wage only causes a one time jump down in employment. Both of these issues will cause a difference-indifferences estimator to understate the true disemployment effect of the minimum wage.
Why is the short-run employment effect so small? Recall the two Hicks-Marshall channels by which employment falls. The first is the scale effect. Free entry implies that expected profits are 0, so restaurants will pass the higher labor costs to consumers in the form of higher prices. As a result, consumers will purchase fewer meals and restaurants will require fewer inputs. The reduction in sales causes net exit of restaurants immediately following a minimum wage hike and consequently an immediate fall in the employment of minimum wage workers. Appendix E shows that product prices rise in proportion to minimum wage labor's share of costs and Aaronson and French (2007) compute that, even at fast food restaurants, low-skill labor is only about 10 percent of total costs. Since the scale effect is in proportion to minimum wage labor's share, the scale effect is small and only modestly reduces employment.
The second channel is the substitution effect. A hike in the minimum wage makes lowskilled workers more expensive, causing restaurants to substitute to cheaper factors of production. However, in a putty-clay model, all substitution occurs through entry and exit.
Because remaining incumbent restaurants maintain their input mix, the substitution effect occurs gradually only as the incumbent restaurants exit and are replaced by new restaurants that are free to choose the optimal input mix given the new price of minimum wage labor.
The net employment response masks an increase in both entry and exit following a minimum wage hike. The upper panel of Figure 5 shows the total amount of exit following the 27 Figure 5: Share of firms entering and exiting after a 10% minimum wage hike Note: The exit share is the share of firms in operation a year ago that are currently not in operation. The entry share is the share of firms currently in operation that were not in operation a year ago. In these calibrations, the minimum wage is boosted by 10 percent immediately after time 0.
minimum wage hike, and the lower panel shows the same for entry. So long as product demand is elastic, market quantity falls and thus the exit rate is higher than the entry rate.
However, given our parameter values, the spike in exit is so large that there is a spike in entry also.

8.2
The contribution of putty-clay to entry and exit behavior In the absence of putty-clay technology (where restaurants can re-optimize their factor mix after the hike), there is still an increase in exit after the minimum wage hike if σ < 1.
However, the increase is barely noticeable. Since the exit response generated by the model without putty-clay technology is smaller, the entry response will be smaller as well. In fact, entry drops. This decline is a robust qualitative feature of a model without putty-clay technology, highlighting that putty-clay is central to understanding the rise in entry.

Conclusion
We present new evidence on the effect of minimum wage hikes on establishment entry, exit, and employment among employers of low-wage labor. We show that small net employment changes in the restaurant industry may hide a significant amount of establishment churning that arises in response to a minimum wage hike. To capture these dynamics, we develop a putty-clay model with endogenous entry and exit. The key feature of the putty-clay model is that, after entry, technology and input mix is fixed for the life of the restaurant. After minimum wage hikes, inflexible incumbents are replaced by potential entrants who can optimize on input mix. Thus, the model is capable of predicting both restaurant entry and exit in response to a minimum wage hike.
Furthermore, we show that the putty-clay model generates employment and output price responses to minimum wage hikes that are consistent with those reported in the literature.
In particular, putty-clay yields sluggish employment responses to minimum wage hikes, with a short-run disemployment effect of around -0.1 that grows to -0.4 in the long-run. Similarly, Figure 6: Share of firms entering and exiting after a 10% minimum wage hike, putty clay model versus standard model Note: The "putty-clay" line depicts the entry and exit behavior using our calibrated putty-clay model. The "standard" line depicts entry and exit behavior when restaurants can adjust their factor demands after a minimum wage hike. The exit share is the share of firms in operation a year ago that are not currently in operation. The entry share is the share of firms not in operation a year prior that are in operation. In these calibrations, the minimum wage is boosted by 10 percent immediately after time 0.
30 the model predicts that restaurant prices are immediately and fully passed onto consumers in the form of higher prices, again consistent with the literature.
Other models, such as those that incorporate adjustment costs, can reconcile some of these facts but not others, especially the simultaneous rise of exit and entry. As such, we believe putty-clay models could be potentially useful for understanding the response to other labor market policies, including taxes, hiring subsidies, and firing costs and we view our paper as a novel contribution in that we provide micro level evidence on the empirical relevance of putty-clay in an important policy setting.

Appendix A: Calibration
This appendix details the parameter values we use in the calibration exercise, and borrows heavily from Aaronson and French (2007).
Factor Shares, s l , s h , s m , s k There are a number of sources for labor share, all of which tend to report similar numbers for the food away from home industry. First, 10-K company reports contain payroll to total expense ratios. Of the 17 restaurant companies that appear in a search of 1995 reports using the SEC's Edgar database, the unconditional mean and median of this measure of labor share is 30 percent and it ranges from 21 to 41 percent. 26 These numbers are in-line with a sampling of 1995 corporate income tax forms from the Internal Revenue Service's Statistics on Income Bulletin. Because operating costs are broken down by category, it is possible to estimate labor's share. 27 According to these tax filings, labor cost as a share of operating costs for eating place partnerships is roughly 33 percent.
Consequently, we set s l + s h to 30 percent.
We are particularly interested in labor share in low wage firms. We use the 1997 Economic Census for Accommodations and Food Services, which reports payroll for full service (FS) and limited service (LS) restaurants. LS includes fast-food stores and any restaurant without sit-down service and where customers pay at the counter prior to receiving their meals.
Therefore, they tend to be the primary employer of minimum wage labor. According to this 1997 census, labor share, as a fraction of sales, is slightly higher at FS (31 percent) than LS (25 percent) stores. 28 Therefore, there is little evidence of a significant difference in labor share across establishment type. Aaronson and French (2007) use Current Population Survey data to show that 1 3 of restaurant industry workers are paid less than 150% of the minimum wage, and are thus likely to be affected by the minimum wage. This group accounts for 17% of the wage bill in the restaurant industry. Since labor's share is 0.3, minimum wage workers have a 0.3×17% = 5.1% share in expenses overall. Given our calculations on the number of restaurants that are 26 The search uses five keywords: restaurant, steak, seafood, hamburger, and chicken. 27 The IRS claims that labor cost is notoriously difficult to decompose for corporations and therefore we restrict our analysis to partnerships, where there is less concern about reporting.
28 Several 10-K reports of individual restaurant companies show that wages account for 85 percent of compensation. Therefore, labor's share based on compensation is roughly 36 and 29 percent at full and limited service restaurants. limited service, and the fact that limited service restaurants use minimum wage labor much more intensively than the restaurant industry as a whole, we set s l = 0.1 and s h = 0.2.
Based on the same sample of company financial reports used to compute s l + s h , we assume that capital's share is 30 percent and material's share is 40 percent.
The Elasticity Parameter σ Aaronson and French (2007) could not find estimates of the elasticity of substitution σ for restaurants, but review the literature on estimates of σ, and find that σ = 0.8 is an average estimate in the literature.
Targets: age, exit and entry elasticity The exit age of a restaurant, J, is picked to match the average exit probability of 0.057 (see appendix This appendix first derives the explicit expression for the market price almost in terms of model fundamentals (the exit age, (J), is left implicit). The appendix then solves for the elasticity of product price with respect to the minimum wage.

The (effective) product price
Free entry implies that the maximand in (6) is equal to zero. Substituting in the equilibrium factor demands from equation (7) and the definition of y 0 from equation (5) to the maximand in equation (6) set equal to zero: We now want to simplify each term. For example, the term involving low-skill wages simplifies as follows: 36 Exploiting analogous simplifications on each term in (A1) gives the effective product price: To convert the effective product price to the product price, explicitly solve the expression relating these two prices given in the paragraph above equation (5): Combining equations (A4) and (A5) gives an explicit expression for the product price:

Response of product price to a minimum wage hike
We are interested in the effect of a change in the low-skill wage on the price level. The effective low-skill wage, q w = 1−e −rJ r w, depends on w directly and because J, the exit age, depends on w. We study the effect of w on J in appendix C. To see where J enters the expression, substitute in the definitions of the effective prices in the paragraph above equation (5) into (A7). To keep the expression somewhat more compact, define: Then the product price depends on the flow prices and exit age (J) as follows, where in this expression only we write J(w) to emphasize the dependence of J on w: 37 Take the derivative of the product price with respect to the low-skill wage: Convert to an elasticity (the expression for w P comes from rearranging (A8)): ∂P ∂w To simplify this expression further, derive expressions for some steady state factor shares.
For low-skill labor: For capital: Hence, substituting the factor shares ((A11) and (A13)) into (A10) gives the following expression for the elasticity of the product price with respect to the low-skill wage: The dependence of the exit age on the low-skill wage introduces two additional terms relative 38 to the standard result that the product price elasticity is s L .
Proof. For σ < 1, Result 1 gives that J(w n , w n ) < J(w o , w o ) and for σ = 1 J(w n , w n ) = The proof strategy is to analyze the exit condition. The difficulty arises because in steady state the relative prices that restaurants face when they enter differs from the relative prices they face when they exit because some of the cost of capital is sunk. A restaurant exits when MC=MB. So consider the exit condition, equation (8), for both the new entrants: e −δJ(wn,wn) y 0 P n = rηp k k n + w h h n + p m m n + w n l n , and the incumbents: Rearrange these expressions so that the left hand sides are equal: y 0 P n = rηp k k n + w h h n + p m m n + w n l n e δJ(wn,wn) (A17) Set them equal and rearrange: rηp k k n + w h h n + p m m n + w n l n (rηp k k o + w h h o + p m m o + w n l o ) = e δJ(wo,wn) e −δJ(wn,wn) .
Note that J(w n , w n ) ≤ J(w o , w o ) for σ ≤ 1 so that showing that the left hand side is less than 1 proves what we want. Hence, we would like to show: To do so, note that input bundles (k o , h o , m o , l o ) and (k n , h n , m n , l n ) are both on the y 0isoquant (both produce y 0 in a brand new restaurant). Cost minimization implies that: q k n k o + q h n h o + q m n m o + q w n l o > q k n k n + q h n h n + q m n m n + q w n l n .
When would a minimum wage increase lead to a decrease in the use of capital and our high-level sufficient condition to fail? This cannot happen when σ = 1, because in this case kn ko = wn wo 1−α and the sufficient condition is always satisfied. This might happen if the exit age is incredibly responsive to the minimum wage (i.e. if ∂1−e −rJ ∂w w 1−e −rJ > 1). Then it is possible that capital use declines. The central difficulty in ruling out this case is that we cannot solve for J in closed form so it is hard to bound its responsiveness to w.
Condition for the product price to jump immediately to its new steady state level Under the assumption that the product price immediately jumps to its new steady state level, the output of the exiting firms is: Under this assumption, the change in market quantity is Q o − Q n , where the market quantity is a function of the product price.
What has to happen for the exit spike to be large enough to accommodate the hypothe-sized decline in market quantity? The relevant inequality is: That is, the exit spike has to be (weakly) larger than the change in market quantity. This leaves room for there to be an entry spike as well (if the inequality is strict).
Now we manipulate (A26) to ask what has to be true for the inequality to be satisfied.
The condition for the product price to jump immediately to the new steady state level is: e δJ(wo,wo)−δJ(wo,wn) − 1 e δJ(wo,wo) − 1 Or, dividing by the right hand side, multiplying through by the denominator on the left hand side, adding one to both sides, and taking logs, the condition can be rewritten as: ln e δJ(wo,wo)−δJ(wo,wn) − 1 Qo−Qn Qo  [t n + J(w o , w n ), t n + J(w n , w n )] 0 0 Note: This table summarizes results in equations (A31)-(A36). A ∆ indicates that the pdf is bounded so that instantaneously there is no entry/exit. The ∆ is a time interval.

Entry Dynamics
In the old steady state the number of entrants is: Over the time interval (t n , t n + J(w o , w n )), Q n remains constant, and the amount of exiting A40, A39 (t n , t n + J(w o , w n )) ∆ {e −δJ(wo ,wn ) foy0+δQn} y0 ∆ e −δJ(wo,wn) f o y 0 + δQ n A42, A41 [t n + J(w o , w n ), ∆ δQn y0 ∆δQ n A44, A43 t n + J(w n , w n )] Note: This table summarizes results in equations (A37)-(A44). A ∆ indicates that the pdf is bounded so that instantaneously there is no entry/exit. The ∆ is a time interval. and depreciating output is given by ∆ e −δJ(wo,wn) f o y 0 + δQ n (A41) so that the entering number is ∆ e −δJ(wo,wn) f o y 0 + δQ n y 0 .
Finally, in (t n + J(w o , w n ), t n + J(w n , w n )), there is no exit and thus entry just replaces the depreciation. Output is then: and the resulting number of new entrants is ∆ δQ n y 0 .  Note: This table reports how three of our key samples were constructed. In the first row of each panel (labeled "all establishments in counties..."), we report the total number of establishments in our exit and entry samples.
To appear in our sample, BLS confidentiality requires that counties ultimately have a minimum number of establishments. The difference in samples between panels A and B reflect the counties that meet this minimum number of establishments with at least 1 employee but not with at least 15 employees. Recall to be in the exit sample, an establishment must meet minimum employment requirements at time t − 1 but may or may not remain open at time t. Analogously, to be included in the entry sample, an establishment must meet minimum employment requirements at time t but may or may not be open at time t − 1. The next row in each panel deletes establishments that do not meet the minimum size threshold (of 1 in panel A and 15 in panels B and C). Finally, the third row (labeled "and delete breakouts/consolidations") is our final sample after additionally removing establishments that are part of a QCEW breakout or consolidation. Note: This table reports the exit and entry rates, as well as the average employment size, for limited and full service restaurants with a minimum employment threshold of 15. The average size of limited service restaurants in the exit sample with at least 1 employee is 16.7 (all), 25.7 (chains), and 13.9 (non-chains).