Abstract
This paper studies amenities and wage premiums in a service economy where individuals with different skills choose cities with different amenities and choose occupations to produce different services, namely the highquality service or the lowquality service. Workers with higher skills have stronger preferences for amenity and choose the highamenity city. Within each city, workers with higher skills choose to produce the highquality service, and workers with lower skills choose to produce the other. Workers with higher skills are willing to sacrifice more wages to live in the highamenity city. As a result, the price of the highquality service, relative to the price of the lowquality service, is lower in the highamenity city, because the wage equals the price times skill or productivity. The wage of a worker with a given skill in the highquality service sector, relative to the wage of a worker in the lowquality service sector, or the wage premium, is thus lower in the highamenity city. A quantitative analysis shows that the wage premium is about 3% lower when amenity is 10% higher. However, the average wage of highquality service workers over that of lowquality service workers may be lower or higher in the highamenity city due to skill concentration in the highamenity city.
1 Introduction
A body of research has studied the effects of amenities on wages (Roback 1982; Brueckner etÂ al. 1999; Glaeser etÂ al. 2001; Brueckner and Neumark 2014; Diamond 2016; Lee and Lin 2017). With free mobility, individuals choose their locations to maximize their utilities, and wages must be lower in higheramenity locations. Another literature has considered how amenities affect the wage premium, namely the difference between the wage of highskilled workers and the wage of lowskilled workers, and has shown that the wage premium is lower in higheramenity cities (Black etÂ al. 2009; Lee 2010). This paper adds to this literature by studying the role played by the local provision and consumption of services in determining the effects of amenities on the wage premium.
This paper builds on two observations. First, services are nontradable and locally produced and consumed in that one has to live in an area to consume the services produced in the area (David and Dorn 2013; HortaÃ§su and Syverson 2015; Hsieh and RossiHansberg 2019).^{Footnote 1} The prices of services are then determined by the demand for and the supply of services in the area and vary across areas. Second, the preferences for amenities differ among individuals. In particular, workers with higher skills and higher incomes tend to choose higheramenity locations (Black etÂ al. 2009; Lee 2010; Gagliardi and SchlÃ¼ter 2015; Diamond 2016; Carlino and Saiz 2019).
In the model, there are two cities, the highamenity city and the lowamenity city, two sectors, the highquality service sector and the lowquality service sector, and a continuum of individuals with different skills. A worker produces in one sector but consumes both the highquality service and the lowquality service, and housing. A workerâ€™s productivity in the highquality service sector increases in her skill, but a workerâ€™s productivity in the lowquality service sector is independent of her skill. This fixedproductivity assumption in the lowquality service sector facilitates the comparison of the wage premium between two cities by allowing us to focus on the productivity of highskilled workers. The price of a service in a city is determined by the demand for and the supply of the service in the city. There are two services, and only the relative price (the price of the highquality service relative to the price of the lowquality service) can be determined in a city.
The comparison of the relative price between two cities turns out to be crucial to the comparison of the wage premium between two cities. While the relative price in a city is determined in the servicemarket equilibrium of the city, the relative price can be compared between two cities by considering individualsâ€™ choice of a city and an occupation. Workers choose a city and a sector in the city to maximize their utility. In each city, there are both highquality service workers and lowquality service workers, as both services are essential. Lowquality service workers have the same productivity, as noted earlier, and are indifferent between two cities. Highquality service workers sort into two cities, and there is a highquality service worker who is indifferent between two cities. The indifferent highquality service worker earns more and is willing to sacrifice her wage more to enjoy the highamenity city than lowquality service workers. Since lowquality service workers earn the same wage in both cities,^{Footnote 2} the wage of the indifferent highquality service worker should be lower in the highamenity city. The wage equals the marginal product revenue or the price times skill, and the (relative) price should be lower in the highamenity city. Since the price in a city applies to all workers in the city, the wage premium of a highquality service worker with any given skill over the wage of a lowquality service worker is lower in the highamenity city than the wage premium of a highquality service worker with the same skill in the lowamenity city.
The discussion above compares the wage premium of a highquality service worker with the same skill between two cities. However, as in the literature below, the wage premium can be more broadly defined as the average wage of highskilled workers over the average wage of lowskilled workers. Since highquality service workers with higher (lower) skills than the indifferent highquality service worker choose the highamenity (lowamenity) city, the average skill of highquality service workers is higher in the highamenity city. Considering highquality service workers as highskilled workers and lowquality service workers as lowskilled workers, this higher averageskill effect counteracts the lower price effect, and the broad wage premium may be higher or lower in the highamenity city than in the lowamenity city.
All individuals consume one unit of housing, and the housing cost increases in the city size. In equilibrium, housing is more expensive in the highamenity city, and the highamenity city is larger. The reason is that the utility from consumption of two services decreases in the price, and lowquality service workers enjoy a higher utility from consumption of two services in the highamenity city than in the lowamenity city, given that the price is lower in the highamenity city. In addition, they enjoy high amenity in the highamenity city. However, their productivity is fixed, and they earn the same wage between two cities. Thus, the housing cost must be higher in the highamenity city in equilibrium.
A quantitative analysis shows a few additional results. The highamenity city hosts a larger fraction of workers among its workforce in the lowquality service sector, amidst hosting the highest skill workers in its highquality service sector. For example, it appears that New York and London may have among their inhabitants the most productive workers in their highquality service sectors but they tend to have disproportionately larger lowquality service sectors in comparison to Newark NJ and Coventry in the UK. Relatedly, due to this mix of workers in the highamenity city, the mean consumption of a service, either the highquality or the lowquality service, among the residents of the highamenity city is not necessarily higher than that in the lowamenity city. As the amenity difference between the two cities widens, the mean consumption in the highamenity city falls whereas the mean consumption in the lowamenity city increases.
The assumption that services are nontradable plays an important role in the analysis. However, in recent years, technological advances have made some services tradable such as finance, insurance and retail. For example, Amazonâ€™s online sales account for a large fraction of retail services, and traditional retailers such as Walmart and Target have grown their businesses online (Thomas 2019). Indeed, according to the most recent report on ecommerce (US Census Bureau 2021a), the percentage of revenue from electronic sources out of the total revenue for service industries in the U.S. had more than doubled from 3.4% in 2012 to 7.6% in 2019. Nevertheless, the percentage of revenue from electronic sources is still below 10% of the total revenue for service industries. During the Covid19 pandemic, online businesses have increased substantially, and other types of services beyond retail services such as medicine have shifted to online services to some extent, as consumers and providers have tried to minimize close contacts. According to the Bureau (US Census Bureau 2021b), the third quarter 2020 online retail sales increased 36.1 percent from the third quarter of 2019, showing that consumers and providers prefer online transactions during the pandemic. The substantial increase in ecommerce during the pandemic may or may not continue after the pandemic, but ecommerce still does not appear to be large enough to invalidate the assumption that services are largely nontradable in this paper. Thus, the nontradableservice assumption will be maintained throughout, and a model with tradable services is left for future research.
The paper is organized as follows. The next section discusses the relation of this paper to the literature. SectionÂ 3 considers a simple model of the production and consumption of two services in a given city. SectionÂ 4 extends the model to a twocity setting to study the interplay between locational choices and local labor markets. The analysis enables the comparison of the wage premium between the two cities. SectionÂ 5 provides a quantitative analysis, and the last section concludes.
2 Relation to the literature
This paper is related to a number of strands of literature. A body of research has studied the relationship between city sizes and wages and has shown that wages are higher in larger cities (for example, Glaeser 2008; BaumSnow and Pavan 2013). Such benefits of density come from agglomeration economies (Glaeser and MarÃ© 2001; Rosenthal and Strange 2008; Tse 2010; Davis and Dingel 2019). For instance, denser labor markets facilitate the matching of workers and employers and the exchange of productive ideas. Another reason for higher wages in larger cities is that highskilled workers sort into larger cities (Carlsen and RattsÃ¸J 2016; De La Roca and Puga 2017). A related literature has studied the effects of city sizes on wage premiums (Glaeser and MarÃ© 2001; Di Addario and Patacchini 2008; Bacolod etÂ al. 2009; BaumSnow and Pavan 2012, 2013). The literature has shown that wage premiums tend to be larger in larger cities, but the opposite also occurs.^{Footnote 3}
In our model, by contrast, the wage premium is lower in the larger city with higher amenity than in the smaller city with lower amenity. The city size would be endogenous and depend on some underlying factors, but the literature does not consider them and takes city sizes as exogenous parameters when studying the relationship between city sizes and wages.^{Footnote 4} In this paper, the city size is endogenously determined in equilibrium, and the highamenity city is larger in equilibrium. Our result thus concerns the effects of amenity, not the city size, on the wage premium. What drives our result is that workers with higher levels of skill are more likely to choose the highamenity city and choose to produce the highquality service with the price determined endogenously.
The result that the wage premium is lower in the highamenity city comes in part from the narrow definition of the wage premium in our result. Defining the wage premium more broadly as the average wage of highquality service workers over the average wage of lowquality service workers in a city, the wage premium in the highamenity and large city may be higher or lower than in the lowamenity and small city. The reason is that workers with higher skills prefer the highamenity city, and the average skill or productivity of highquality service workers in the highamenity city is higher while the average productivity of lowquality service workers is the same between two cities. The higher average productivity of highquality service workers due to concentration of workers with higher skills in the highamenity city counteracts the lower relative price there, making ambiguous the comparison of the wage premium between two cities. If the higher skillconcentration effect outweighs the lower relativeprice effect in the highamenity city, the wage premium is higher in the highamenity city. If the opposite occurs, the wage premium is lower in the highamenity city.
Another line of research has investigated real wage inequality that takes into account the cost of living. Since the cost of living differs across cities, the pattern of the real wage premium across cities would differ from that of the nominal wage premium. Moretti (2013) demonstrates that highskilled workers have been concentrated in highcost cities since 1980, making the difference in the real wage premium between cities smaller than that in the nominal wage premium. Diamond and Moretti (2021) measure consumption of market goods that workers of different incomes enjoy across commuting zones. They find that highincome workersâ€™ consumption varies little across commuting zones, but lowincome workers enjoy consumption more in lowcost areas than in highcost areas. These papers provide important results, as we care about consumption and utility, not just wages. However, these papers are not related, at least directly, to this paper or the literature above, because they do not explain why the nominal wage premium exists or the cost of living differs between locations but take them as given and show that the real wage premium in a larger city over a smaller city is smaller than the nominal wage premium.
A strand of literature considers the role of amenity in the wage premium and is closely related to this paper. In Black etÂ al. (2009) and Lee (2010), the wage premium is lower in the highamenity city than in the lowamenity city, the same result as the one in this paper. However, the model and mechanism behind the same result in these two papers differs from that in this paper and is worth discussion. Both papers assume that highskilled workers value amenity more than lowskilled workers like this paper. In Black et al., lowskilled workers in the highamenity city value or enjoy the high amenity little, but the housing cost is higher in the highamenity city due to amenity being capitalized into housing prices. As a result, with free mobility, lowskilled workers in the highamenity city must be paid more by the difference in the housing cost than in the lowamenity city (by assuming that they do not value amenity at all for easy exposition). Highskilled workers in the highamenity city enjoy the high amenity, and their wages in the highamenity city do not increase as much as the difference in the housing cost in equilibrium. Thus, the wage premium is smaller in the highamenity city. From an analytical point of view, Black et al. do not model production or wages or occupational choices, unlike in this paper. Rather, they compare an exogenous triple (amenity, wage, housing cost) in the highamenity city with another exogenous triple (amenity, wage, housing cost) in the lowamenity city for a lowskilled worker and for a highskilled worker. Utility equalization for each type of worker in equilibrium enables them to infer the properties of equilibrium wages and hence to compare the wage premium between two cities. City sizes do not matter in their analysis, as the housing cost depends on amenity not the city size.
In Lee, workers value consumption variety, which is considered amenity and a luxury good. It is assumed that a larger city offers greater consumption variety, and rents are higher in a larger city. Individuals consume many tradable goods that give rise to consumption variety and a local nontradable service, namely health care service, but Lee models production of health care service only and workers mean health care workers. Firms in a city hire workers of different skills or occupations (say, doctors and nurses) and produce health care to maximize profits. Profit maximization determines the wage for each occupation in the health care sector as a function of the city size, as the number of workers in an occupation depends on the city size. Thus, all three variables that affect the utility of a worker in a city, namely wage, rent and variety, depend on the city size. Lee then conducts a comparative statics exercise on the effect of an increase in the city size on those three variables by holding the utility of a worker constant. The exercise shows that the wage premium is lower in a larger city, because highskilled workers are willing to accept lower wages to live in a larger city that offers greater consumption variety more than lowskilled workers are willing. Lee considers production and wages in one sector, namely the health care sector, although a health care worker consumes many goods and services. The reason for the onesector assumption is to ensure that production does not affect the price of health care service or the prices of other goods (the health care sector is too small relative to the economy). What matters in Lee is not the health care sector per se, but the assumption that the price level does not change. If all sectors were considered, or the price level changed like in this paper when workers move into or out of a city and production in the city changes, it would make more complicated the comparison of the wage premium between two cities. In this paper, there are two sectors, the highquality service and the lowquality service, and both occupations and prices are endogenously determined, and the lower wage premium in the highamenity city comes from the fact that highskilled workers prefer to live in the highamenity city and choose to produce the highquality service. Thus, the exogenous price level and one sector are crucial to Lee, but endogenous prices and occupational choices between two sectors are crucial to this paper. In addition, Lee considers one city and conducts a comparative statics exercise, but in this paper, city sizes and wages and prices are determined in a twocity equilibrium.
3 Model
3.1 Basics
The economy is populated by a continuum of a unit mass of consumerworkers, who differ in their skills h, distributed according to the (continuously differentiable) distribution function \(\Phi (h)\) with \(\phi (h)=\Phi '(h)>0\) over the support \(h\in [\underline{h},\overline{h}].\) A worker with skill h will be called an h sometimes for simplicity. There are two cities, indexed by superscripts \(i=1,2,\) and workers choose to reside in one of the two cities.
A consumer in a given city consumes a unit of housing and two types of services, H and L, produced in the same city. This feature of services is not entirely realistic, as an individual who lives in one city may travel to another to consume the services produced there. Traveling and other related expenses, however, would make such crosscity consumption of services a rare event.^{Footnote 5} The two services, H and L,Â are interpreted as different qualities of the same type of service as in the model of vertical product differentiation (Shaked and Sutton 1982).^{Footnote 6} For instance, they represent upscale restaurants and ordinary restaurants, or luxury theaters and regular movie theaters. Unlike Shaked and Sutton, we allow for a given consumer to consume both H and L.^{Footnote 7} Strictly speaking, our model is thus a hybrid model of vertical and horizontal product differentiation.
In reality, an individual consumes many services and manufactured goods. We choose to model one service for analytical tractability. The one service should then be interpreted as a composite service, and we abstract from the consumption and production of manufactured goods to focus on services.
3.2 Preferences and utility maximization
A resident of city i derives utility,
from the consumption of \(C_{H}\) units of H and \(C_{L}\) units of L, with \(\beta \in (0,1)\) denoting a share parameter in the utility function and \(\theta ^{i}\) the level of consumption amenity city i offers. Amenity is broadly interpreted as any natural conditions such as climate and geographical characteristics that affect the general welfare of the cityâ€™s inhabitants. We assume that \(\theta ^{1} >\theta ^{2},\) so that city 1 is a more desirable place to live than city 2, other things equal.
Let \(p_{L}\) and \(p_{H}\) denote, respectively, the price of L and H, and w denote a consumerâ€™s income. Assume that a unit of housing is provided by converting c(n) units of L, where n is the cityâ€™s population.^{Footnote 8} Housing cost c(n) is assumed to increase in n. A consumer chooses \(C_L\) and \(C_H\) to maximize the utility subject to the budget constraint, \(p_{L}C_{L}+p_{H}C_{H}+p_{L}c(n) = w,\) where \(p_{L}c(n)\) is the consumerâ€™s expenditure on housing. The prices and a consumerâ€™s income depend on where she resides, but this section considers a given city, and it does not matter whether the city is city 1 or city 2.
Assuming \(wp_{L}c(n) >0\), the consumerâ€™s utilitymaximizing choices of \(C_{H}\) and \(C_{L}\) are given by
respectively, yielding the indirect utility function,
where
3.3 Production and occupational choice
A worker supplies one unit of labor inelastically to produce either H or L in a given city. In particular, an h produces \(zh^{\tau }\) units of H and earns the wage
if the worker chooses to be employed in the H sector, where \(z>0\) and \(\tau >0\) are parameters of the production function. For instance, if \(\tau =1\), an h produces zh unit of H, but \(\tau\) can be any positive constant. The Appendix provides a foundation for the production technology, but all that matters is that output \(zh^{\tau }\) and the wage \(w_{H}(h)\) are increasing in skill h.
A worker of any h is assumed to produce the same x units of L.^{Footnote 9} The wage of any worker in the L sector is then
Given (4) and (5), worker h is better off and chooses to work in the H sector if and only if
3.4 Product and labor market equilibrium
Let \(C_{H,S}\), \(C_{L,S}\), and \(V_{S}\), respectively, denote the optimal consumption of H and L and the indirect utility of a sector S worker, \(S = L, H.\) An H worker has income \(w_{H}(h)=p_{H}zh^{\tau }\), so (2) and (3) for the worker are
Any L worker has the same income \(w_{L}=p_{L}x,\) so (2) and (3) for the worker are
In equilibrium, workers make optimal occupational choices between the sector H and the sector L, and the markets for H and L clear. Occupational choices are characterized by (6). With a slight abuse of notation, letting \(\Phi \left( h\right)\) denote for now the distribution of h in the given city, the market clearing conditions are
On the RHS of the second condition above, nc(n) is subtracted due to the production of housing by L workers. There are two services, and only one price can be determined and the two market clearing conditions reduce to
(14) can be solved for \(h_c\), which in turn determines the relative price, by (6),
The utility from housing consumption is omitted, as its inclusion would add a constant term to the utility function. The Appendix models a housing market, and it is briefly discussed here. Lowskilled workers with \(h \le h_c\) choose to produce L or housing. The wage from either occupation must be the same, and the wage from producing one unit of housing equals \(p_L c(n)\), which in turn equals the price of housing, as in (7)â€“(8) and (10)â€“(11). The demand for housing is n, as each resident needs one unit of housing. In equilibrium, the number of workers producing housing equals nc(n)/x,Â because a lowskilled worker provides one unit of labor and produces x units of L or x/c(n) units of housing. This number is thus subtracted from the RHS of the marketclearing condition for L in (13).
4 Twocity equilibrium, skill sorting and wage premiums
4.1 Twocity equilibrium
To model a twocity equilibrium, let \(n^{i}\) denote the population of city i and \(n^i_S\) denote the number of workers in sector \(S = L, H\) of city i. The adding up constraints are
The market clearing conditions are
where \(D^i_S\) and \(S^i_S\) denote the demand for service S and the supply of service S in city i, respectively. Unlike in the previous section with one city, the market clearing condition cannot be stated until the identities of workers in the H sector and the L sector are determined in each city. For future reference, let \(A^i_S\) denote the set of workers in sector S of city i.
A TwoCity equilibrium is \(\left\{n^{1},n^{2}, n^1_L, n^1_H, n^2_L, n^2_H, A_{H}^{1},A_{L}^{1}, A_{H}^{2}, A_{L}^{2}, p^1 \equiv \frac{p^1_H}{p^1_L}, p^2 \equiv \frac{p^2_H}{p^2_L}\right\}\) in which (15)â€“(18) are satisfied with \(n^{1},n^{2}\in \left( 0,1\right) ,\) and workers choose a city and choose an occupation in the city to maximize their utility. The subsequent analysis considers an equilibrium in which both H and L are produced in each city.^{Footnote 10}
Although the model is simple, it captures the interactions between the productlabor market equilibrium and the mobility of workers between two cities. Such interactions turn out to be crucial for our main results on the wage premium. To make the utility of a worker dependent explicitly on the choice of city, the indirect utility functions in (9) and (12) are rewritten as
where B is dropped for simplicity, and \(p^i \equiv p^i_H/p^i_L\) and \(c^i \equiv c(n^i), i = 1, 2.\)
4.2 Skill sorting of workers between two cities
This section discusses the sorting of workers between the two cities, and the following lemma helps in determining the pattern of skill sorting:
Lemma 1
In a TwoCity equilibrium,

(a)
\(V_{L}^{1}=V_{L}^{2}\) (the utility of an L worker is equalized between two cities).

(b)
\(V_{H}^{1}(\tilde{h})=V_{H}^{2}(\tilde{h})\) (there is a unique H worker, denoted by \(\tilde{h}\), who is indifferent between two cities).
The lemma (and PropositionÂ 1 below) is proved in the Appendix, and the intuition is discussed here. Part (a) has a simple intuition. All L workers earn the same wage in a given city and therefore are equally well off in the city. Then, if, for example, \(V_{L}^{1}>V_{L}^{2}\), there can be no L workers staying in city 2 to produce L. Thus, it must be that \(V_{L}^{1}=V_{L}^{2}\) in equilibrium, so that both H and L are produced in each city.
To see part (b), observe that there is at least one H worker who is indifferent between two cities because \(\phi (h) > 0\) for all \(h \in [\underline{h},\overline{h}]\) and each city must have both L workers and H workers. Suppose to the contrary that there are two different H workers, \(h'\) and \(h''\), who are indifferent between two cities. Since the term, \(\beta \; ln\;p^i + ln\;\theta ^i,\) in the indirect utility function of an H worker in city i is independent of skill h, the difference in log disposal incomes (incomes net of the housing cost) between two cities for \(h'\) should be the same as that for \(h'',\) so
The equality implies \(p^2/p^1 = c^2/c^1.\) The housing cost differential between two cities is then offset by the relative price differential, and any L worker would prefer to live in city 1 due to high amenity, a contradiction to (a). Therefore, there must be a unique \(\tilde{h}\). More simply, the housing cost differential and the relative price differential are the same for all workers, and there cannot be more than two workers with different skills who are indifferent between two cities. Given that any L worker is indifferent between two cities, only one H worker can be indifferent.
Using LemmaÂ 1, the following proposition states the properties of equilibrium:
Proposition 1
In a TwoCity equilibrium,

(a)
\(p^1 \equiv \frac{p_{H}^{1}}{p_{L}^{1}}< p^2 \equiv \frac{p_{H}^{2}}{p_{L}^{2}}\) and \(n^{1}>n^{2}\) (the price of H relative to that of L is lower in the highamenity city, and the highamenity city is larger).

(b)
\(A_{H}^{1} = [\tilde{h}, \overline{h}], A_{L}^{1} = [\underline{h}, h_c^2], A_{H}^{2} = [h^2_c, \tilde{h}]\), and \(A_{L}^{2} = [\underline{h}, h_c^2]\) (H workers in city 1 are those with \(h\in \left[ \widetilde{h},\overline{h}\right] ,\) H workers in city 2 are those with \(h\in [ h_{c}^{2}, \widetilde{h}]\), and L workers in the two cities are those with \(h\in [ \underline{h},h_{c}^{2}]\)).
To understand the proposition, observe that a higher \(p^i\) decreases the utility of any worker, as reflected in the term \( \beta \;ln\;p^i\) of \(V^i_H(h)\) or \(V^i_L\). The reason is that a higher price decreases disposable incomes when the same quantities of services are consumed. A higher \(p^i\) also increases the wages and the utility of any H worker, as reflected in the term \(ln\;(p^i z h^\tau  c^i)\) of \(V^i_H(h).\) The wage effect outweighs the price effect, and a higher \(p^i\) increases the utility of an H worker, as H workers are net suppliers of H.^{Footnote 11} Since a higher \(p^i\) has no effect on the wages of L workers, a higher \(p^i\) decreases the utility of an L worker.
To see the first part of (a), note that because of the incomeamenity complementarity in the utility function, higher income workers value the high amenity in city 1 more and are willing to pay a higher housing cost in city 1 than lower income workers.^{Footnote 12} An H worker then should sacrifice more income to live in city 1 to enjoy the high amenity than an L worker does. Both workers pay the same housing cost, so an H workerâ€™s wage in city 1 should be lower than in city 2, given that an L workerâ€™s wage is the same, x, between two cities. Since the wage of an H worker in city i equals \(w^i_H(h) = p^i z h^\tau\), it must be that \(p^1 < p^2\) and \(w^1_H(h) < w^2_H(h).\)
As for the second part of (a), L workers pay the lower service price due to \(p^1 < p^2\) and enjoy the high amenity in city 1. Thus, to satisfy LemmaÂ 1, \(V_L^1 = V_L^2\), the housing cost must be higher in city 1 and it must be that \(n^1 > n^2.\)
The result that the population is larger and the housing cost is higher in the highamenity city is the same as that in the literature (Black etÂ al. 2009; Lee 2010). However, the mechanism in this paper differs from that in those two papers. Black et al. assume that high amenity is capitalized into the housing price, and Lee assumes that the housing cost is higher and amenity in terms of consumption variety is also higher in a larger city, so the housing cost is higher in the high amenity city. In this paper, the choice of a city and that of an occupation between the H sector and the L sector, along with the endogenous service prices, determine the relationship between \(n^1\) and \(n^2\) and hence between \(c^1\) and \(c^2.\)
Part (b) directly follows from the observation that higher income workers value the high amenity in city 1 more than lower income workers. As \(\tilde{h}\) is the only H worker who is indifferent between two cities in LemmaÂ 1 (b) and the wage of an H worker increases in her skill, H workers with \(h > \tilde{h}\) choose city 1 while H workers with \(h < \tilde{h}\) choose city 2.^{Footnote 13} Thus, there is an \(h^2_c\) in city 2, so that workers with \(h \in [h_c^2, \tilde{h}]\) choose the H sector and workers with \(h \in [\underline{h}, h_2^c]\) choose the L sector in city 2, because city 2 has both H workers and L workers and \(\phi (h) > 0\) for all h in the economy. This explains \(A_{H}^{2} = [h^2_c, \tilde{h}]\) and \(A_{L}^{2} = [\underline{h}, h_c^2].\)
By LemmaÂ 1 (a), L workers in city 1 are those with \(h\in \left[ \underline{h},h_{c}^{2}\right]\), because L workers must enjoy the same utility between two cities, regardless of their skills. Since H workers in city 1 are those with \(h \ge \tilde{h}\), as noted above, \(A_{H}^{1} = [\tilde{h}, \overline{h}]\) and \(A_{L}^{1} = [\underline{h}, h_c^2].\)
FigureÂ 1 illustrates the skill sorting of workers between the two cities described in PropositionÂ 1 (b). Unlike in city 2, \(h^1_c\) does not exist in the sense that no worker in city 1 is indifferent between the H sector and the L sector. However, it is still true in city 1 that workers with higher skills \(h \in A_{H}^{1}\) choose the H sector, and workers with lower skills \(h \in A_{L}^{1}\) choose the L sector. That is, in city 2, \(\tilde{h} > h_c^2\), so \(V^2_H(\tilde{h}) > V^2_L\) and the \(\tilde{h}\) chooses the H sector. Since \(V_{L}^{1}=V_{L}^{2}\) and \(V_{H}^{1}(\tilde{h})=V_{H}^{2}(\tilde{h})\) by LemmaÂ 1, \(V^1_H(\tilde{h}) > V^1_L\) and the \(\tilde{h}\) chooses the H sector in city 1. As a result, any worker with \(h > \tilde{h}\) chooses the H sector in city 1. In an analogous manner, workers with \(h \in [\underline{h}, h_c^2]\) choose the L sector in city 1.
Although the paper focuses on amenities, amenities affect city sizes. The pattern of skill sorting in PropositionÂ 1(b) thus can be related to the literature on skills and city sizes. For the purpose of discussion, call H workers with \(h \ge (\le )\;\tilde{h}\) higherskilled (lowerskilled) H workers. According to PropositionÂ 1(b), higherskilled H workers choose city 1, and lowerskilled H workers choose city 2. Since city 1 is larger by PropositionÂ 1(a), the result in PropositionÂ 1(b) implies that higherskilled H workers choose a larger city. This does not necessarily mean that a larger city has a larger fraction of higherskilled workers in its H sector or in its economy, because it depends on the shape of \(\phi (h)\) for \(h \in [\tilde{h}, \overline{h}]\) and that of \(\phi (h)\) for \(h \in [h^2_c, \tilde{h}]\). Thus, the pattern of skill sorting in this paper may or may not be consistent with an empirical literature on skills and city sizes, discussed in Sect.Â 2. In particular, Moretti (2013) shows that highskilled workers have been concentrated in highcost cities. To the extent that highcost cities are large, a large city has a larger fraction of highskilled workers. Likewise, highskilled workers choose a larger city (Carlsen and RattsÃ¸J 2016; De La Roca and Puga 2017). However, the result in PropositionÂ 1(b) implies that higherskilled H workers are concentrated in city 1 in the sense that the average skill of H workers is higher in city 1. This difference in the average productivity of H workers between two cities plays an important role in the comparison of the broad wage premium between two cities in the next section.
The prices, \(p^1\) and \(p^2\), are determined by the market clearing conditions. Given the skill sorting in PropositionÂ 1 (b), the conditions are
The condition for city 2, namely (20), is identical to (14) in the onecity model, except that the upper limit of the integral \(\overline{h}\) is replaced by \(\tilde{h}\) due to \(A^2_H = [h^2_c, \tilde{h}]\) in PropositionÂ 1 (b), and \(\Phi (h_c)\) is replaced by \(n^2_L\) due to both being the number of L workers. The condition for city 1 in (19) includes z and the price \(p^1\), unlike in (20). The reason is that \(h^1_c\) does not exist in city 1. If it existed, \(h^1_c = (p_L^1 x/p^1_H z)^{1/\tau } = (x /p^1 z)^{1/\tau }\) and (19) would be written in a manner analogous to (20) without z or \(p^1\).
There are 10 variables to be determined: \(n^{1},n^{2}, n^1_L, n^1_H, n^2_L, n^2_H, p^1, p^2, h^2_c,\) and \(\tilde{h}.\) There are eight conditions: (6) for city 2 or \(h^2_c = (x/p^2 z)^{1/\tau }\), (15), (16), (17), \(V_{L}^{1}=V_{L}^{2},\) \(V_{H}^{1}(\tilde{h})=V_{H}^{2}(\tilde{h}),\) (19) and (20). The additional conditions are
These three conditions simply reflect the skillsorting outcome in PropositionÂ 1 (b) and in FigureÂ 1. However, (23) is not an independent condition, as (15), (16), (17), (21) and (22) imply (23). Thus, the 10 conditions determine the 10 variables in equilibrium. The identities of workers in each sector of each city, namely \((A_{H}^{1},A_{L}^{1}, A_{H}^{2}, A_{L}^{2}),\) will then be determined by \(h^2_c\) and \(\tilde{h}\), as in PropositionÂ 1(b).
Although an equilibrium consists of 10 variables, it can be characterized more intuitively by two variables \((n^1, \tilde{h})\) by eliminating a number of trivial conditions such as (15) through (17). Using \(n^2 = 1  n^1,\) two equalutility conditions in LemmaÂ 1 can be solved for \(p^1\) and \(p^2\) as functions of \((n^1, \tilde{h})\). Since \(h^2_c = (x/p^2 z)^{1/\tau }\), it is also a function of \((n^1, \tilde{h}).\) Substituting \(p^1(n^1, \tilde{h}), p^2(n^1, \tilde{h})\) and \(h^2_c(n^1, \tilde{h})\) into two marketclearing conditions, (19) and (20), \(n^1\) and \(\tilde{h}\) are determined. The remaining variables \((n^1_H, n^2_H, n^1_L, n^2_L)\) are then trivially determined by (21) through (23) along with \(n^1_L + n^1_H = n^1.\)^{Footnote 14}
4.3 Wage premiums
The wage premium has been definedÂ differently. First, the wage premium may refer to an extra wage paid to workers in a city with less desirable living conditions. In a similar vein, the wage premium may be defined as the difference in wages between a larger city and a smaller city due to stronger agglomeration economies in the former city. Second, the wage premium may refer to the difference between the wage of higherskilled/moreeducated workers and the wage of lowerskilled/lesseducated workers in a particular place. This usage is sometimes also referred to as the skill premium.^{Footnote 15} This section studies the implications of the model for the skill premium, the second notion of the wage premium above, as in Black etÂ al. (2009) and Lee (2010).
The wage of an L worker, regardless of the workerâ€™s skill h, in city i equals
The wage of an H worker with skill h in city i equals
A workerâ€™s wage thus depends on the workerâ€™s skill h and choice of city, as the price differs between cities. Then, an H worker of a higher skill in one city does not necessarily earn a higher wage than a worker of a lower skill in another city. This feature of wage highlights the difference from the literature, mentioned in Sect.Â 2, that focuses on tradables such as manufactured goods. That is, since the price of tradables is independent of locations, workers with higher skills in one city must earn higher wages than workers with lower skills in the same city or another city.
Let \(\omega ^{i}(h)\) denote the skill premium for an H worker of skill h in city i over the L workers in the same city. By (24) and (25),
The wage of an H worker in city i is necessarily higher than the wage of an L worker in the same city given optimal occupational choice. A more interesting question is if \(\omega ^{1}(h) > \omega ^{2}(h)\) or \(\omega ^{1}(h) < \omega ^{2}(h),\) i.e., if a given worker may enjoy a higher skill premium in one city than in another city. The given worker is equally productive in the two cities, and the difference in the skill premium comes from the difference in the (relative) price between the two cities. By PropositionÂ 1(a), \(p^{1} < p^{2},\) and the following result can be stated:
Proposition 2
\(\omega ^{1}(h)<\omega ^{2}(h)\) (the skill premium for a given H worker is lower in the highamenity city than in the lowamenity city).
Black etÂ al. (2009) and Lee (2010) also find that the skill premium is lower in cities with better amenities, but their model and mechanism behind the same result differs from that in this paper. The difference was discussed in detail in Sect.Â 2, and it will be briefly mentioned here. In Black et al., amenity is assumed to be capitalized into housing prices, and housing is more expensive in the highamenity city. Highskilled workers value amenities more and are willing to pay a higher housing price to live in the highamenity city than lowskilled workers. Since each worker should enjoy the same utility between the highamenity city and the lowamenity city with mobility, the wages of highskilled workers in the highamenity city decrease more than the wages of lowskilled workers. The wage premium is thus lower in the highamenity city. Black et al. do not model production or wages or occupational choices, unlike in this paper. Rather, they compare the wage premium based on two cities with exogenously different amenities, housing costs, and wages.
In Lee, a larger city offers greater consumption variety, but rents are higher in the larger city. Highskilled workers value consumption variety more than lowskilled workers. Since all workers pay the same rent, and a worker should enjoy the same utility between cities, highskilled workers are willing to accept lower wages to live in the larger city with greater consumption variety more than lowskilled workers. The wage premium is thus lower in the highamenity city. Lee models production and wages, but assumes that the prices are fixed. Like Black et al., Lee also compares two cities with exogenously different amenities, housing costs, and wages. However, in this paper, the comparison of the wage premium between two cities hinges on occupational choices and endogenous prices, along with city choices.
PropositionÂ 2 is based on a narrow definition of the skill premium. That is, the proposition compares the skill premium for a skill h in city 1 with that for the same skill h in city 2. However, there is a continuum of workers with different skills in each city, and the skill premium of a city can be more broadly defined as the average wage of highskilled workers over the average wage of lowskilled workers, called â€˜the broad skill premiumâ€™ and denoted by \(\hat{\omega }^i.\) In fact, this broad definition has been used in most of the literature on city sizes and wage premiums, mentioned in Sect.Â 2. Considering H workers as highskilled workers, their skills range from \(\tilde{h}\) to \(\overline{h}\) in city 1 by PropositionÂ 1 (b). Using (25), the average wage of highskilled workers in city 1 equals
where
denotes the average productivity of H workers in city 1. Considering L workers as lowskilled workers, their wage in city 1 is \(p_L^1 x\) regardless of their skills. Thus,
Likewise, the broad skill premium in city 2 equals
where
given that H workers in city 2 are those with skills \(h \in [h_c^2, \tilde{h}]\) by PropositionÂ 1 (b).
The difference in the broad skill premium between two cities reads as
To understand the sign of (27), it is necessary to compare \(Q_H^1\) with \(Q_H^2.\) Observe that \(h^\tau\) is increasing in h, so \(h^\tau \ge \tilde{h}^\tau\) for all \(h \in [\tilde{h}, \overline{h}].\) Thus, \(Q_H^1\) can be rewritten as
In a manner analogous to (28), \(Q_H^2\) can be rewritten as
Thus, the sign of (27) is ambiguous, as \(p^1 < p^2\) but \(Q_H^1 > Q_H^2.\) This result can be stated as:
Proposition 3
\(\hat{\omega }^{1} \le \;or\ge \; \hat{\omega }^{2}\) (the broad skill premium may be lower or higher in the highamenity city than in the lowamenity city).
The result in the proposition has a simple intuition. An Hworkerâ€™s wage in a city depends on the price of H, \(p_H^i\), and her skill, h. The price of H relative to the price of L is lower in city 1 than in city 2, which makes the broad skill premium lower in city 1 than in city 2. However, H workers with higher skills are concentrated in city 1, as highskilled workers prefer a city with high amenity. As productivity increases in skills in the H sector, the average productivity of H workers is higher in city 1 than in city 2. Thus, if the relative price effect outweighs the skillconcentration effect, the broad skill premium is lower in city 1. If the opposite occurs, the broad skill premium is higher in city 1. The relationship between \(\hat{\omega }^{1}\) and \(\hat{\omega }^{2}\) cannot be determined in general, and the next section compares them numerically.
PropositionÂ 3 can be related to the known result in the literature that the skill premium is higher in larger cities. As noted earlier, the literature does not consider endogenous prices. Thus, if the prices are constant and do not vary across cities, as is the case of manufactured goods, \(p^1 = p^2 = p.\) In this case, (27) reduces to
so the broad skill premium is higher in the high amenity and large city. This discussion underscores the importance of the role of endogenous prices in the determination of the skill premium in this paper.
5 Numerical analysis
This section presents a quantitative analysis of the model and illustrates a few implications of the analysis. The economyâ€™s distribution of h is assumed to follow a powerlaw distribution,
for some \(\alpha >0\), which generalizes the Pareto distribution.^{Footnote 16} The virtue of the powerlaw distribution of h is that the wage distribution similarly follows a powerlaw distribution, which resembles the wage distribution commonly found in empirical studies of the labor market (Bombardini etÂ al. 2012; David and Dorn 2013). The housing cost function is assumed to be \(c( n) =an^{b}\) for \(a>0\) and \(b\ge 1\). The parameters are set as follows: \(\overline{h}=1.1\), \(\underline{h}=0.1\), \(a=8,\) \(b=1.2,\) \(x=z=6\) , \(\beta =0.5\), \(\alpha =2.5\) and \(\tau =1\). TableÂ 1 presents the results.
The top row of the table depicts the assumed difference in amenities between city 1 and city 2 in terms of the \(\theta ^{1}/\theta ^{2}\) ratio, the key parameter of the model. To relate the ratio to empirical findings, consider Albouy (2016). He estimates quality of life across a large number of metro areas and nonmetro areas. Quality of life depends on a number of factors such as cooling degree days, heating degree days and distance to coast. The quality index varies widely, ranging from 0.01 in Phoenix, AZ to 0.05 in Boston, MA and to 0.21 in Honolulu, HI. Thus, the difference in amenity, ranging from 10% with \(\theta ^{1}/\theta ^{2}\) = 1.1 to 100% with \(\theta ^{1}/\theta ^{2}\) = 2, in the table does not seem unreasonable.
The first three panels show the relative price \(p^i = p^i_H/p^i_L\), the population \(n^i\), and the skill thresholds \(\tilde{h}\) and \(h^2_c\). As in PropositionÂ 1, \(p^1 < p^2, n^1 > n^2,\) and \(\tilde{h} > h_c^2\) for all \(\theta ^{1}/\theta ^{2}.\) In addition, as city 1 becomes more attractive or the ratio \(\theta ^{1}/\theta ^{2}\) increases, the differences in the population \(n^1  n^2\), the relative price \(p^2  p^1\), and the skill threshold \(\tilde{h}  h^2_c\) increase, decreasing the ratio \(p^1/p^2\) and increasing the ratios \(n^1/n^2\) and \(\tilde{h}/h^2_c\) in the table. These results can be understood intuitively. First, as city 1 becomes more attractive, an H worker is willing to sacrifice her income more than an L worker to live in city 1, so the H workerâ€™s income decreases more. The H workerâ€™s income is proportional to the relative price p, so \(p^1\) must decrease more and the ratio \(p^1/p^2\) must decrease. Second, as amenity increases and the price decreases more in city 1, the housing cost must increase more in city 1 for an L worker to enjoy the same utility between two cities. As a result, \(n^1\) must increase and the ratio \(n^1/n^2\) must increase, as \(\theta ^1/\theta ^2\) increases. Third, an increase in the amenity level of city 1 attracts workers with higher skills and incomes who are willing to pay for higher housing costs in city 1, increasing the skill threshold for the H sector in city 1, \(\tilde{h},\) and hence the ratio \(\tilde{h}/h^2_c.\)
The fourth panel presents the average productivity of H workers in each city, \(Q^i_H.\) As noted above, as \(\theta ^1/\theta ^2\) increases, more H workers with higher skills move to city 1, increasing the skill threshold \(\tilde{h}\) in city 1. As a result, the ratio \(Q^1_H/Q^2_H\) increases. The next panel shows the broad wage premium \(\hat{\omega }^i\). As in PropositionÂ 3, the broad wage premium in general cannot be compared between two cities, because the price is lower but the average productivity of H workers is higher in city 1. However, the panel shows \(\hat{\omega }^1 > \hat{\omega }^2\) for all \(\theta ^1/\theta ^2,\) because the higher averageproductivity effect outweighs the lower price effect in city 1. In addition, as city 1 becomes more attractive, it decreases the wage premium in city 1 but increases the wage premium in city 2, decreasing the ratio \(\hat{\omega }^1/\hat{\omega }^2.\)
The sixth panel considers the fraction of L workers in a city. Since L workers are indifferent between two cities regardless of their skills, it is in general not possible to determine which city has a larger fraction of L workers. However, under the parameter values, the fraction is larger in city 1 for all \(\theta ^1/\theta ^2.\) A more important result is that the fraction in city 1 increases but the fraction in city 2 decreases, as city 1 becomes more attractive and \(\theta ^1/\theta ^2\) rises. The reason is that as housing becomes more expensive in city 1, a smaller number of H workers with the highest skills move to city 1, increasing the fraction of L workers in city 1. Expensive housing in city 1 forces many H workers who cannot afford housing in city 1 to stay in city 2, increasing the fraction of H workers and decreasing the fraction of L workers in city 2.^{Footnote 17} As a result, the fraction of L workers in city 1, relative to that in city 2, increases as \(\theta ^1/\theta ^2\) rises.
The last two panels compare the mean consumption of the two services between two cities. Define, respectively,
as the mean consumption of H in the two cities and
as the mean consumption of L in the two cities. Notice that the numerators of \(\overline{C}_{L}^{i}\) denote the output of L in the city net of the amount used up for housing provision. The results in the table show that as \(\theta ^{1}/\theta ^{2}\) increases, \(\overline{C}_{H}^{1}\) falls while \(\overline{C}_{H}^{2}\) rises. This occurs due to the interaction of three factors. First, as \(\theta ^1/\theta ^2\) increases, the average productivity of H workers in city 1 \(Q^1_H\) increases, increasing the production of H and the average consumption of H. Second, \(\tilde{h}\) increases, decreasing the number of H workers in city 1, decreasing the production of H and the average consumption of H. Third, the population of city 1, \(n^1,\) increases, decreasing the average consumption of H. Thus, in general, as \(\theta ^1/\theta ^2\) increases, the average consumption of city 1 may increase or decrease. Under the parameter values, the last two consumptiondecreasing effects outweigh the first consumptionincreasing effect, so the average consumption of H in city 1 decreases. In an analogous manner, the average consumption of H in city 2 increases. As for the consumption of L, \(\overline{C}_{L}^{1}\) falls and \(\overline{C}_{L}^{2}\) rises, as \(\theta ^1/\theta ^2\) increases. The reason is that as \(n^1\) increases, more L workers produce housing for a larger population and fewer L workers produce L in city 1. This, along with the larger population itself, decreases the average consumption of L in city 1. By an analogous argument, the average consumption of L in city 2 increases, as \(\theta ^1/\theta ^2\) increases.
6 Conclusion
In this paper, we study the role played by service industries in determining the skill composition and the wage premium in a city. The analysis has shown that the patterns of skill distribution across cities hinge on the difference in the level of amenity between the two cities. Our main result is that the wage premium for a given skill is lower in the highamenity city but the broad wage premium may be higher or lower in the highamenity city.
Agglomeration economies have been found to be crucial to the spatial distribution of productivity and wages in urban and regional economics. The productivity of workers and the wages they earn tend to be higher in larger cities due to the various productivityenhancing factors inherent in a larger city, including but not restricted to the complementarity between skills, thick labor markets and the more frequent exchanges of ideas. This paper focuses on the role played by the local production and consumption of services and the choice of occupations in the spatial distribution of productivity and wage. In our model, both product prices and wages are determined locally, given that services are nontradable. Most importantly, the way the wages are determined in a city and across cities differs from the mechanisms emphasized in the agglomeration literature. While the agglomeration benefits in a large city have received a good deal of attention, the role of service industries has received little attention. This lack of attention to service industries from scholars is unfortunate, as services account for a large share of production and consumption activities in most countries, and the share keeps rising. While this paper studies one aspect of services in relation to skills and wages, more research on services in general would be useful to advance our understanding of the spatial aspect of economic activities.
Notes
It is possible for consumers in an area to enjoy services produced in another area by traveling. However, defining an area broadly, the transaction cost of consuming services in another area would be high enough to discourage such consumption.
The same wage can be understood easily by assuming for simplicity that the price of the lowquality service is one in each city.
For example, in a wage regression of Glaeser and MarÃ©, the coefficient of the interaction term, the metro dummy x the 16year education dummy, is positive but the coefficient of the metro dummy x the morethan16year education dummy is negative. In a wage regression of Di Addario and Patacchini, the coefficient of first degree or above x Pop is negative.
A city thus should be interpreted more broadly as a geographical area made up of multiple cities, as in a metropolitan area, in which an individual can consume services produced in any city of the area by incurring a nonprohibitive travel cost.
This interpretation is not necessary. What matters to the analysis is that highskilled workers produce highvalue services and lowskilled workers produce lowvalue services.
At a point in time, a consumer may only consume either H or L. Over a longer time horizon, the consumer would prefer to consume both varieties.
A housing market is considered below.
An earlier version assumes that an h produces \(xh^{\lambda }\) units of L,Â so that higherskilled workers are more productive. This assumption does not affect the result qualitatively as long as \(\lambda < \tau\) but makes the presentation complicated.
The proof of existence of an equilibrium is in the online Appendix.
Simple differentiation can show that \(dV^i_H(h)/d p^i = [(1\beta )p^i z h^\tau + \beta c^i]/[p^i(p^i z h^\tau  c^i)] > 0.\)
To see the relationship between income and the willingness to pay for amenity in the simplest manner, focus on income, denoted by y, and amenity \(\theta .\) The indirect utility is \(V = ln\; y + ln\; \theta .\) Differentiation gives \(\partial y/\partial \theta =  y/\theta\) at a given level of utility, so \(\partial [\partial y/\partial \theta ]/\partial y =  1/\theta < 0.\) Thus, workers are willing to accept lower income or to pay more for amenity, and the willingness to pay increases in income.
It is straightforward to verify that \(d[V^1_H(h)  V^2_H(h)]/dh > 0\) when evaluated at \(h = \tilde{h}\) due to \(c^1 > c^2\) and \(p^1 < p^2,\) so that workers with \(h > \tilde{h}\) choose city 1.
(21) determines \(n^1_H\), because it is a function of \(\tilde{h}\). The resulting \(n^1_H\) then determines \(n^1_L = n^1  n^1_H\). Likewise, (22) determines \(n^2_H\), as it is a function of \(\tilde{h}\) and \(h^2_c(n^1, \tilde{h}).\) Using \(n^1_L = n^1  n^1_H\), (23) determines \(n^2_L\), as it is a function of \(n^1, n^1_H(\tilde{h})\) and \(h^2_c(n^1, \tilde{h}).\)
This notion of the wage premium has received a good deal of attention in relation to wage inequality (Acemoglu 2002; Goldin and Katz 2007; Autor etÂ al. 2008). The wage premium in this literature is typically measured by the ratio of collegegraduate wages to highschoolgraduate wages and is called the college premium.
The powerlaw distribution above is a truncated Pareto distribution with an upper bound \(\overline{h}.\)
This also explains in the fourth panel above why the average productivity of H workers in city 2 increases as \(\theta ^1/\theta ^2\) increases.
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Appendix
Appendix
1.1 A model of housing market in SectionÂ 4
Letting R denote the price of a unit of housing, the budget constraint becomes \(p_{L}C_{L}+p_{H}C_{H}+R = w.\) Assuming \(wR>0\), the consumerâ€™s utilitymaximizing choices of \(C_{H}\) and \(C_{L}\) are then given by
respectively. Given \(w_H(h) = p_H z h^\tau\) and \(w_L = p_L x\),
Letting \(n_L\) and \(n_R\) denote the number of workers that produce service L and the number of workers that produce housing, respectively, in equilibrium
The first condition states that workers with \(h < h_c\) produce service L or housing. The second one shows that lowskilled workers are free to produce housing or service L, so the wage from producing one unit of housing R equals the wage from producing c(n) units of L, \(p_L c(n)\), given that a unit of housing is provided by converting \(c\left( n\right)\) units of L. The third one is the housing market equilibrium condition. The LHS shows the demand, as each resident consumes one unit of housing and there are n residents. The RHS is the supply, as \(n_R\) workers produce housing and each worker produces x units of service L or x/c(n) units of housing. The remaining two conditions are the equilibrium conditions for services L and H.
Substituting \(R = p_L c(n)\) into (32), the resulting expressions are the same as those in (7)â€“(8) and (10)â€“(11). In addition, substitution of \(n_L = n\Phi ( h_{c})  n_R = n\Phi ( h_{c})  n c(n)/x\) into the RHS of the fourth condition of (33) yields \(n_L x = n(\Phi ( h_{c})x  c(n)).\) Thus, the two marketclearing conditions in (13) continue to hold.
Foundation for (4)
A standard textbook firm chooses labor input (the number of workers) n to maximize its profits, \(\pi =pf(n)wn,\) where p and w denote the output price and the wage, respectively. An interior solution exists for where \(f' >0, f'' <0.\) In this paper, workersâ€™ skills differ, and a firm chooses the number of workers, in addition to the types of workers (skills) to hire. The firmâ€™s profit maximization becomes more involved where there are two separate but related decisions to make, especially with a continuum of skills. The production technology in Kremer (1993) provides one possible but not the only reference.
Following Kremer (1993), assume that the production of H requires \(\tau\) tasks to be completed, each of which is to be carried out by a given worker. Interpreting \(h_{i}\) as the probability of task \(i=1,...,\tau\) being successfully completed by the worker assigned to the task, the probability of one unit of H being produced successfully equals \(\Pi _{i=1}^{\tau }h_{i}.\) Defining z as output per worker if all tasks are completed successfully, a firm that hires \(\tau\) workers is expected to produce \((\Pi _{i=1}^{\tau }h_{i})z\tau\) units of H. The firmâ€™s profit is then,
where p is the price of H, and \(w(h_{i})\) is the wage for an \(h_{i}\) worker. The choices of \(h_{i},i=1,2,...,\tau\), satisfy the firstorder condition,
In a positive assortative matching equilibrium, the firm chooses the same h for all \(\tau\) tasks, as explained in Kremer (1993). As a result, the firmâ€™s output becomes \((\Pi _{i=1}^{\tau }h_{i})z\tau =h^{\tau }z\tau\), so each worker produces \(zh^{\tau }\) units of H on average as in the main text. The firstorder condition becomes \(ph^{\tau 1}z\tau =w'(h).\) Integrating the conditionÂ and setting the constant of integration to zero, \(w(h)=pzh^{\tau },\) which is (4). The firm earns zero profit.
In the above, \(\tau\) and \(h_{i}\) are interpreted, respectively, as the number of tasks and the probability of task i being completed successfully. However, such interpretations are not necessary, and \(\tau\) or \(h_{i}\) can be any positive number, not just a positive integer and a probability between 0 and 1, respectively, as discussed in Kremer (1993).
Proof of LemmaÂ 1

(a)
Suppose to the contrary that \(V^1_L > V^2_L.\) Since all L workers in city 2 enjoy the same utility \(V^2_L\) regardless of their skill h, all of them prefer to move to city 1. There will be no L worker in city 2, a contradiction to the notion of equilibrium that each city has both H workers and L workers. Thus, it must be that \(V^1_L = V^2_L\) in equilibrium. Likewise, if \(V^1_L < V^2_L,\) it leads to a contradiction to the notion of equilibrium, so it mus be that \(V^1_L = V^2_L\) in equilibrium.

(b)
Suppose to the contrary that there are \(h'\) and \(h''\) such that \(V_H^1(h') = V_H^2(h')\) and \(V_H^1(h'') = V_H^2(h'').\) These two equalities imply
$$\begin{aligned} \frac{p^1 z (h')^\tau  c^1}{p^2 z (h')^\tau  c^2} = \frac{p^1 z (h'')^\tau  c^1}{p^2 z (h'')^\tau  c^2}, \end{aligned}$$which in turn implies \(p^1/p^2 = c^1/c^2.\) Using the definition of \(V^i_L\),
$$\begin{aligned} V^1_L  V^2_L = ln\;\frac{x  c^1}{x  c^2}  \beta \;ln\;\frac{p^1}{p^2} + ln\;\frac{\theta ^1}{\theta ^2}. \end{aligned}$$If \(p^1 \le p^2\) and hence \(c^1 \le c^2, \frac{x  c^1}{x  c^2} > 1\) and \(\frac{p^1}{p^2} < 1\). Thus, given \(\frac{\theta ^1}{\theta ^2} > 1\), \(V^1_L  V^2_L > 0\), a contradiction to LemmaÂ 1. If \(p^1 > p^2\) and hence \(c^1 > c^2,\)
$$\begin{aligned} V^1_L  V^2_L = ln\;\frac{x  c^1}{x  c^2}  ln\;\frac{p^1 z (h')^\tau  c^1}{p^2 z (h')^\tau  c^2} < 0, \end{aligned}$$again a contradiction to LemmaÂ 1. The equality follows from \(V_H^1(h') = V_H^2(h')\) by hypothesis, and the inequality comes from \(p^1> p^2, c^1 > c^2,\) and \(p^i z (h')^\tau  c^i > 0\) and \(x  c^i > 0.\) Thus, there is a unique h that satisfies \(V_H^1(h) = V_H^2(h).\)
\(\square\)
Proof of PropositionÂ 1

(a)
\(p^1 < p^2\): Since \(V^1_L  V^2_L = 0\) and \(V^1_H(\tilde{h})  V^2_H(\tilde{h}) = 0\) by LemmaÂ 1,
$$\begin{aligned} \frac{p^1 z \tilde{h}^\tau  c^1}{p^2 z \tilde{h}^\tau  c^2} = \frac{x  c^1}{x  c^2}, \end{aligned}$$(36)which reduces to
$$\begin{aligned} xz\tilde{h}^\tau (p^2  p^1) + x (c^1  c^2) + z \tilde{h}^\tau (c^2p^1  c^1p^2) = 0. \end{aligned}$$(37)(37) can be rewritten as
$$\begin{aligned} p^2 z\tilde{h}^\tau (x  c^1)  p^1 z\tilde{h}^\tau (x  c^2) + x (c^1  c^2) = 0. \end{aligned}$$(38)Suppose to the contrary that \(p^1 \ge p^2.\) Then, \(c^1 > c^2,\) because otherwise any H worker wants city 1. That is, the utility of an H worker increases in the price, as discussed in the main text and in footnote 11, and any H worker pays a lower housing cost and enjoys higher amenity in city 1. Since \(x c^i > 0\) by assumption, (38) can be rewritten as
$$\begin{aligned} 0 \le p^2 z\tilde{h}^\tau (x  c^1)  p^2 z\tilde{h}^\tau (x  c^2) + x (c^1  c^2) = (p^2 z\tilde{h}^\tau  x) (c^2  c^1) \end{aligned}$$(39)due to \(p^1 \ge p^2\) by hypothesis. (39) is a contradiction, because \(c^2c^1 < 0\), as noted above, and \(p^2 z \tilde{h}^\tau  x > 0\) due to an H worker earning more than an L worker. Thus, it must be that \(p^1 < p^2\).
\(n^1 > n^2\): An L worker enjoys the lower price of services due to \(p^1 < p^2\) and the high amenity in city 1, so the housing cost must be higher in city 1 and \(n^1 > n^2,\) because otherwise \(V^1_L > V^2_L\) and LemmaÂ 1 would be violated.

(b)
The proof is the same as that in the main text.
\(\square\)
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