A Hotelling model with production

Abstract

This paper extends the Hotelling model of spatial competition by incorporating the production technology and labor inputs. A duopolistic game is constructed in which firms choose their locations simultaneously in the first stage, and decide the prices of the product and wages of labor in the second stage. We find that the equilibrium locations depend on the production technology. Specifically, when productivity increases, the two firms change from dispersion to agglomeration and then to dispersion again. We then analyze the case with a minimum wage requirement and show the robustness of the equilibrium locations. Furthermore, the socially optimal locations do not depend on the production technology and the minimum wage requirement. However, a higher minimum wage increases unemployment and prices, which may reduce the total welfare level.

Full text

A Hotelling Model with Production
Wen-Chung Guoa, Fu-Chuan Laib,c, Dao-Zhi Zengd,∗
aDepartment of Economics, National Taipei University, 151, University Rd., San-Shia, Taipei, 23741
Taiwan.
bResearch Center for Humanities and Social Sciences, Academia Sinica, Nankang, Taipei 11529,
Taiwan.
cDepartment of Public Finance, National Chengchi University, Taipei, Taiwan
dGraduate School of Information Sciences, Tohoku University, Aoba 6-3-09, Aramaki, Aoba-ku, Sendai,
Miyagi 980-8579, Japan.
Abstract
This paper extends the Hotelling model of spatial competition by incorporating the pro-
duction technology and labor inputs. A duopolistic game is constructed in which firms
choose their locations simultaneously in the first stage, and decide the prices of the prod-
uct and wages of labor in the second stage. We find that the equilibrium locations depend
on the production technology. Specifically, when productivity increases, two firms change
from dispersion to agglomeration and then to dispersion again. We then analyze the case
with a minimum wage requirement and show the robustness of the equilibrium locations.
Furthermore, the socially optimal locations do not depend on the production technol-
ogy and the minimum wage requirement. However, a higher minimum wage increases
unemployment and prices, which may reduce the total welfare level.
JEL classification: R3, L1
Keywords: Hotelling spatial model; Production; Minimum wage; Returns to scale
1. Introduction
This paper aims to investigate how the production technology and labor inputs impact
the locations of firms. Since Hotelling (1929), spatial oligopoly competition has been in-
vestigated by many scholars such as d’Aspremont et al. (1979), Economides (1986, 1989),
Neven (1986), Tabuchi (1994), Lambertini (1997), Irman and Thisse (1998), Anderson
(1988), and Fleckinger and Lafay (2010). These models examine firms’ locations when
∗Corresponding author. Tel./Fax: ++81 22 795 4380.
Email addresses: guowc@ntu.edu.tw (Wen-Chung Guo), uiuclai@gate.sinica.edu.tw
(Fu-Chuan Lai), zeng@se.is.tohoku.ac.jp (Dao-Zhi Zeng)
Preprint submitted to Elsevier June 6, 2014
*Manuscript
Click here to view linked References

they compete to sell a product to consumers, ignoring how the product is produced. How-
ever, some recent papers show that the production side of the Hotelling model deserves
more attention. For example, Lai and Tabuchi (2012) consider how the input locations
affect the location choices of firms. Brekke and Straume (2004) discuss the upstream-
downstream bilateral monopoly framework and the equilibrium locations through Nash
bargaining. Mai and Peng (1999) consider a framework in which the production costs are
reduced as the locations of firms become nearer to each other. Nevertheless, the major
roles of production, labor behavior and returns of scale have not been formally embedded
in the Hotelling model within the standard location then price framework. This paper
will fill this gap in the literature.
We also note that minimum wage laws have been enacted in most countries. Such
a minimum wage is desirable because it ensures the ability to pay for necessary living
costs, and reduces poverty. It is, therefore, important to explore the mechanisms through
which a minimum wage impacts various economic activities. The spatial competition in
the labor market with minimum wage legislation is an interesting topic in this area. Some
models consider spatial oligopsony without competition in the output market. For exam-
ple, Bhaskar and To (1999) consider a circular space of workers and uniformly distributed
oligopsonistic firms that offer jobs to nearby workers under a technology of constant re-
turns to scale. They find that a rise in the minimum wage increases social welfare. Bhaskar
and To (2003) examine the wage distribution arising from oligopsonistic competition in the
labor market, assuming a uniform distribution of firms. Whereas the two aforementioned
papers treat locations as exogenous (with equal distance), Kaas and Madden (2010) solve
the location-wage game of a duopsonic labor market. They also conclude that imposing
a minimum wage always improves welfare. The above studies attribute the result to the
effect of minimum wage on non-wage job characteristics like locations. However, all above
articles focus on a full-employment labor market, without discussing the production part.
It is well known that a minimum wage creates unemployment (McConnel and Brue, 1999,
p.594), which may affect the total welfare of workers. Accordingly, we need to carefully
examine the welfare issue incorporating production and the possibility of unemployment.
This paper considers both the demand (product) and the supply (labor and produc-
tion technology) sides to see how they interact with each other, which is carried out by
including production behavior in the familiar Hotelling duopoly model of d’Aspremont et
al. (1979). We examine how the firm locations depend on the production technology. The
production function is simple, but general enough to describe either increasing, constant
and/or decreasing returns to scale technology. While Anderson and Engers (1994) exam-
ined the role of general production costs earlier, we go deeper by letting labor be the only
2

input of production, so that the production costs are endogenously determined. Workers
are assumed to be uniformly distributed along a line with unit length. Each individual
not only acts as a consumer, purchasing the homogeneous good from a firm, but also
provides his/her labor to a firm. The commuting/transport costs for a worker to buy
from and work at a firm are quadratic functions of the distance between the residential
place and the location of the firm.
The analytical setting is a two-stage location-price (wage) game in which two firms
determine first their locations and then their prices of output and wages. We find that
the equilibrium locations crucially depend on the production technology. The equilibrium
locations (−1/4,5/4) of firms in Tabuchi and Thisse (1995) remain true only when the
production technology exhibits constant returns to scale (the benchmark case). In general,
two firms may be located either closer to or farther away from each other when the
production technology does not display constant returns to scale. More specifically, both
firms are closer to (farther from) each other when production function is characterized by
moderately increasing (decreasing or highly increasing) returns to scale. Intuitively, the
competition in both the product market and the labor market is important for location
choices of firms. Two main effects are associated with a higher degree of returns to scale.
On the one hand, the firms become more productive, resulting in a market effect by
which firms move toward the market center in order to attract more consumers. This is
because firms need a larger market share to digest their products. On the other hand,
this increased productivity creates a competition effect by which firms move farther from
each other in order to reduce price competition and gain a higher profit. For a low level
of returns, the market effect is weak due to the low productivity. As a result, firms locate
very far away from each other. When the degree of returns to scale increases from those
low levels, the market effect prevails initially over the competitive effect, because the
level of competition is low. However, as the degree of returns becomes large enough, the
competitive effect prevails over the market effect because of two reasons: First, a large
degree of returns implies stiffer competition for any location. Second, once the degree
of returns has become large enough, the firms are very close, which further amplifies the
intensity of competition.1
The setup herein is then easily extended to investigate the impact of a minimum wage
on location equilibrium and welfare. We find that our previous equilibrium results are
robust even when a minimum wage is imposed. Unlike the equilibrium, the optimal loca-
tions depend neither on the production technology nor on the minimum wage requirement.
1The authors are grateful to an anonymous referee for forming this intuitive explanation.
3

This is because the residents are symmetric, and firms have the same technology. In the
minimum wage model of Kaas and Madden (2010), social welfare is always improved
when minimum wage is imposed. In contrast, our model shows that social welfare may
decrease under minimum wage regulation. The different conclusions of the two models
suggest that it is important to incorporate the labor market when we analyze the welfare.
A minimum wage higher than the equilibrium wage causes unemployment and increases
prices, lowering the welfare of unemployed workers.
The rest of this paper is organized as follows. Section 2 establishes the model, and
Section 3 presents the equilibrium analysis. Section 4 examines the impact of a minimum
wage; and finally, Section 5 presents some concluding remarks. All proofs are provided in
the Appendix.
2. The model
Consider a Hotelling-type market in which residents are uniformly distributed in x∈
[0,1]. Two firms compete to sell their products to the residents. At the same time, two
firms use the labor of residents as their only input in production.
We formulate a two-stage game, and only pure strategies on locations, prices, and
wages are discussed. In the first stage, both firms decide their locations simultaneously.
Denote their locations by x1∈R1and x2∈R1, respectively. Without loss of generality,
let x1≤x2. In the second stage, two firms decide their product prices (p1and p2) and
wages (w1and w2) simultaneously. Backward induction will be employed to solve the
equilibrium.
As a consumer, each resident buys one unit of product (for instance, one car) from one
of the firms. As a worker, he/she provides labor to a firm to produce the product. Workers
are all identical except in their residential locations. We first assume full employment in
Sections 2 and 3, and then consider the case with unemployment in Section 4 where a
minimum wage is imposed. Workers determine their optimal effort based on the wage
rate. Following d’Aspremont et al. (1979) and Tabuchi and Thisse (1995), we assume a
quadratic function to describe both the transport cost and the commuting cost.
Consumers are assumed to have perfectly inelastic demand. The utility of a represen-
tative consumer at xwho is hired by firm iand buys one unit of product from firm jis
described by
uij(x) = u−αe2
i
2+mij (1)
4

where eiis the labor effort made by the worker when he works for firm i,2which is
associated with a degree of suffering described by a disutility parameter α > 0, while mij
represents the utility from a composite good chosen as the num´eraire. Consumption of
the product yields a positive constant surplus, ¯u, which is assumed to be sufficiently large
to ensure the participation of all consumers.
To keep the model as simple as possible, we assume that the commuting costs are the
same as the transportation costs in this paper. We further borrow the idea of commuting
shopping in Claycombe (1991), Calycombe and Mahan (1993), and Raith (1996) that
consumers can save on travel costs by shopping from the same firm that they work for.
We then have the following budget constraint
mij =(I+wiei−pj−k(x−xi)2−k(x−xj)2,if i6=j
I+wiei−pi−k(x−xi)2,if i=j(2)
where krepresents the commuting/transport costs, which are assumed to be quadratic in
distance to avoid the nonexistence of equilibrium in a Hotelling model (see d’Aspremont
et al., 1979).
The individual labor supply functions ei=wi/α,i= 1,2 are derived by maximizing
(1) subjected to constraint (2). If residents choose different firms to work at and buy
product from, Appendix A shows that there are no equilibrium wage rates and prices.
Therefore we focus on the case in which residents choose the same firm to work at and
buy product from.
Given w1,p1of firm 1 and w2,p2of firm 2, the indifferent worker is
ˆx=x1+x2
2+p2−p1
2k(x2−x1)+w2
1−w2
2
4αk(x2−x1).(3)
Residents of x < ˆxwork at firm 1 and buy product from firm 1, while residents of x > ˆx
work at firm 2 and buy product from firm 2.
The profit functions of firms 1 and 2 are calculated as:
π1=p1ˆx−w1Zˆx
0
e1dx =p1−w2
1
αˆx,
π2=p2(1 −ˆx)−w2Z1
ˆx
e2=p2−w2
2
α(1 −ˆx).
2We assume that both the labor effort and the utility function are observable information of workers
and firms.
5

For simplicity, we assume the production function as
Y=1
2(ηL)β,(4)
where Ldenotes the labor input, ηis the labor efficiency, and β > 0 characterizes the
production technology. Specifically, the production function exhibits increasing (constant
or decreasing) returns to scale if β > 1 (β= 1 or β < 1). We can choose units of labor
and product to obtain the coefficient 1/2 in (4), which can simplify some notations later.
The output of a firm is required to cover the demand. That is,
1
2ηZˆx
0
e1dxβ=1
2ηˆxw1
αβ
≥ˆx, (5)
1
2ηZ1
ˆx
e2dxβ=1
2hη(1 −ˆx)w2
αiβ
≥1−ˆx. (6)
These two equations imply that each firm hires enough labor (efforts) to meet its market
demand.
In the second stage of the game, the optimization problems for the firms are as follows:
max
p1, w1
π1=p1−w2
1
αˆxs.t. ˆx≤1
2ηˆxw1
αβ,(7)
max
p2, w2
π2=p2−w2
2
α(1 −ˆx) s.t. 1 −ˆx≤1
2hη(1 −ˆx)w2
αiβ.(8)
It is noteworthy that the constraint of (7) is binding. This is because the first-order
conditions of the objective function, which are written as
∂π1
∂p1
= ˆx−p1−w2
1
α1
2k(x2−x1)= 0,
∂π1
∂w1
=−2w1
αˆx+p1−w2
1
αw1
2αk(x2−x1)= 0,
do not hold simultaneously whenever ˆx6= 0, w16= 0. A similar argument applies to the
constraint of (8).
Predicting the prices and wages solved above in the second stage, both firms determine
their locations in the first stage. The following lemma further shows that the inequalities
in the constraints of (7) and (8) will be become equalities. This implies that each firm
does not have an incentive to dispose of excess production.
6

3. Equilibrium analysis
It is well known that equilibrium prices depend on commuting/transportation costs.
On one hand, when kis small, firms will engage in a Bertrand competition, and there
exists no subgame perfect equilibrium. On the other hand, if transport costs are high,
firms do not worry about losing their customers, and thus they charge higher prices to
earn higher profits. In the case of increasing returns to scale technology, the commut-
ing/transportation costs have a stronger impact on competition. This section imposes
the following technical assumption:
k≥4α
η21−1
β.(9)
As we will see later, this condition ensures nonnegative profits for firms. It also ensures
the existence of a pure strategy equilibrium. This assumption is not a restrictive one.
Indeed, (9) is automatically satisfied in a traditional Hotelling model with a technology
of constant return to scale (β= 1).
To simplify the notation, let
γ∗=α
2kη21−1
β1−2
β.(10)
Although our model contains various parameters, the following result shows that those
parameters are well summarized in γ∗. Figure 1 plots function (1 −1/β)(1 −2/β). It is
43
1
2
3
4
Β
0.2
0.4
0.6
0.8
1.0
1.2
H1-1ΒLH1-2ΒL
Figure 1: γ∗depends on β
observable that γ∗decreases when β < 4/3 and increases when β > 4/3. Furthermore,
γ∗is negative if β∈(1,2) and positive when β > 2 or β∈(0,1). It is noteworthy that
7

γ∗>−1/8 holds under (9).3
We are able to solve the two-stage location/price game as shown in Appendix B. Our
equilibrium result is summarized as follows.
Proposition 1. In the Hotelling model with both product and labor markets, we have a
Nash equilibrium with locations (x∗
1,1−x∗
1), where
x∗
1=−1
1 + s1 + 1
1
8+γ∗
,(11)
and prices and wages
p∗
1=p∗
2=4α
η2β+k(1 −2x∗
1), w∗
1=w∗
2=2α
η.(12)
The following result is immediately derived from Proposition 1 and Figure 1.
Corollary 1. Two firms come closer to each other when βincreases until β= 4/3and
they move farther apart from each other when βfurther increases.
Proposition 1 reveals how product prices, labor wages, and locations of firms depend on
the interaction of demand and supply behavior of firms at equilibrium. Because γ∗>−1/8
holds under (9), we know that (11) is well defined, x∗
1∈(−1/2,0), and x∗
1decreases in γ∗.
Although the consumer demand is perfectly inelastic, the two firms do not agglomerate at
the center. As explained by Tabuchi and Thisse (1995), the negativeness of x∗
1is the result
of avoiding price competition. Matsumura and Matsushima (2012) surprisingly find that
locating outside the linear city can improve the welfare of consumers with some strategic
reward contracts. This result is distinct from that of Anderson and Engers (1994), in
which price-taking duopolists agglomerate at the market center with the assumption of
exogenously given production costs. In our setup, being away from the rival, a firm may
gain from a reduction in competition in both the product market and the labor market.
The prices and wages of (12) give
p∗
i−(w∗
i)2
α=4α
η21
β−1+k(1 −2x1) (13)
3In fact, if β≤1 or β≥2, then γ∗≥0>−1/8. If 1 < β < 2, then (9) implies γ∗≥ −(1/8)·(2/β −1) >
−1/8.
8

≥4α
η21−1
β(−2x∗
1) (14)
for i= 1,2, where the inequality is from (9). Accordingly, p∗
i−(w∗
i)2/α ≥0 holds
from (13) when β∈(0,1] and from (14) when β > 1, ensuring nonnegative firm profits
according to (7) and (8).
In the typical Hotelling model in which the labor market is ignored, Tabuchi and Thisse
(1995) and Lambertini (1997) give the equilibrium locations (x∗
1, x∗
2) = (−1/4,5/4). We
reproduce the same result when β= 1 or 2. To understand the general result here, we
notice two effects associated with a higher degree of returns to scale (higher β). First,
the firms become more productive, so they need to secure a larger market share. This
market effect induces firms to move towards the market center to attract more consumers.
Meanwhile, this increased productivity increases the intensity of competition, for any
location, which creates an incentive to locate farther apart from each other. This is the
competition effect, which induces firms to stay apart to avoid price competition. Firm
locations are determined by the interaction of these two effects. More specifically, when β
is small, the productivity is low, so firms do not have incentive to locate close to the center.
As a result they locate very far away from each other. When βincreases, the market
effect prevails initially over the competitive effect, because the level of competition is low.
However, when βis large enough, the competitive effect prevails over the productive effect
for two reasons: On the one hand, a large βimplies higher competition for any location.
On the other hand, once βhas become large enough, the firms are very close, which
amplifies the intensity of competition. As a whole, we can see the process from dispersion
to agglomeration and from agglomeration to dispersion again when productivity increases,
as indicated in Corollary 1. Meanwhile, firms are closer than the case without production
when β∈(1,2), and farther when β∈(0,1) or β > 2.
It is interesting that equilibrium wages w∗
iand, therefore, labor efforts ei=w∗
i/α are
very simple in our model, in the sense that they are independent of parameters k,t, and
β. This can be attributed to the simple production function of (4).
In addition to the productivity parameter β, equilibrium locations x∗
iand wages w∗
iare
also dependent on other parameters. We have the following comparative statics results.
∂x∗
1
∂k >0,∂x∗
1
∂α <0 and ∂x∗
1
∂η >0 iff β < 1 or β > 2,
∂w∗
1
∂η <0,∂w∗
1
∂α >0.
Table 1 provides three examples of equilibrium locations and the prices when the
9

production technology is of decreasing returns, moderate increasing returns and significant
returns, respectively. Other parameters are k=α=η= 1. Compared with the locations
Table 1: Examples of locations and prices
x∗
1x∗
2p∗
1=p∗
2
β= 1/2−0.406 1.406 11.624
β= 3/2−0.229 1.229 5.584
β= 5/2−0.268 1.268 4.672
of (−0.25,1.25) without production, we can see more distant locations of firms in the case
of decreasing return to scale, but closer locations of firms and lower product prices in
the case of moderate increasing returns, and further-apart locations again in the case of
significant returns.
Finally, it is noteworthy that the socially optimal locations of firms are well known to
be (xo
1, xo
2) = (1/4,3/4) (see Lambertini (1997), Mai and Peng (1999) and Braid (1996)).
In the present model, the social welfare maximization is identical to the minimization of
total transportation costs and total production costs. Under symmetric locations, each
half of the market is served by one firm. Production suffers from no deadweight loss, so
the optimal location is still (xo
1, xo
2) = (1/4,3/4). Comparing (xo
1, xo
2) with (x∗
1, x∗
2), we
know that the firms are over-differentiated at the equilibrium locations.
4. Location analysis with a minimum wage
The first minimum wage law (the Industrial Conciliation and Arbitration Act, 1894)
was enacted by the government of New Zealand with compulsory arbitration between
employers and labor unions. Now minimum wage regulation is common in most coun-
tries, based on either legislation or binding collective bargaining agreements. This section
therefore derives the equilibrium and optimal locations under the constraint of a minimum
wage.
Consider an effective minimum wage wmin (> w∗
i, i = 1,2) which is exogenously im-
posed by the government. Other assumptions are the same as in Section 2. When a
minimum wage is imposed, full employment is not ensured. To consider the effect of
unemployment, we now let firms hire nearby workers, and the remotest workers are un-
employed. This assumption is reasonable, because nearby workers would be those most
motivated to take the job, and this motivation would decrease with distance. Such an
idea can be seen in some American urban policies to create new businesses and job op-
portunities to help high-unemployment inner city areas (Porter, 1995). Note that those
10

unemployed workers are assumed to still have a positive endowment, so they continue
to consume the good. The labor effort of each worker is now dependent on the mini-
mum wage. The minimum wage regulation is clearly binding because employers have no
incentive to set a wage higher than the minimum. Namely, w1=w2=wmin holds in
this section. Since the minimum wage is higher than the market equilibrium wage, the
individual labor supply increases to wmin/α, which is greater than the labor demand. The
excess supply of labor creates unemployment of the workers in the middle area, because
we have assumed that firms tend to hire nearby workers. Those workers are involuntarily
unemployed, even though the wage is higher than that in the previous sections.
In this new framework, the indifferent consumer is given by
ˆx=x1+x2
2+p2−p1
2k(x2−x1).(15)
Note that (3) and (15) are identical when w1=w2. This is because the indifferent
consumer is the farthest from both firms, so he/she is unemployed due to the assumption
of w1=w2=wmin > w∗
i. Since each resident consumes one unit of product, given
production function (4), the labor demands for firms 1 and 2 are (2ˆx)1
β/η and [2(1−ˆx)] 1
β/η,
respectively. Accordingly, the profits of the firms are written as:
π1=p1ˆx−wmin
(2ˆx)1
β
η,(16)
π2=p2(1 −ˆx)−wmin
[2(1 −ˆx)] 1
β
η.(17)
Both firms determine their prices to maximize the above profits.
Similar to (9), this section assumes
k≥2wmin
ηβ 1−1
β,(18)
which ensures nonnegative profits in equilibrium. We further let
γ∗∗ =(1 −β)wmin
4kηβ2.(19)
The following properties are useful later:
(i) γ∗∗ (decreases
increases )in βif β(∈(0,2)
>2),(20)
11

(ii) ∂γ∗∗
∂wmin
=1−β
4kηβ2R0 if β⋚1.(21)
Furthermore, it holds that γ∗∗ >−1/8 under (18). As shown in Appendix B, the two-
stage game is solvable again. Equilibrium locations and prices under the minimum wage
regulation are as follows:
Proposition 2. With a minimum wage, wmin , symmetric equilibrium locations are (x∗∗
1,1−
x∗∗
1), where
x∗∗
1=−1
1 + s1 + 1
1
8+γ∗∗
,(22)
and the equilibrium prices are
p∗∗
1=p∗∗
2=k(1 −2x∗∗
1) + 2wmin
βη .(23)
The following result is immediately derived from Proposition 2 and (20).
Corollary 2. Two firms come closer to each other when βincreases until β= 2 and they
move farther apart from each other when βfurther increases.
Proposition 2 resembles Proposition 1 in its form. Function form (22) is exactly the
same as (11), which is decreasing in its variable. Therefore, the qualitative results of
Proposition 1 and Corollary 1 are robust even when a minimum wage is imposed.
It holds that (x∗∗
1, x∗∗
2) = (−1/4,5/4) when β= 1. For a general β, we have x∗∗
1∈
(−1/2,0) and p∗∗
i>2wmin/η. The indifferent consumer is ˆx= 1/2 and the profits of two
symmetric firms are nonnegative.
Comparing γ∗∗ and γ∗yields
x∗∗
1< x∗
1iff γ∗∗ > γ∗iff (1 −β)[ηwmin + 2α(β−2)] >0.
Three cases are possible. First, when β < 1, we have x∗∗
1< x∗
1if wmin >2α(2 −β)/η.
Namely, if the production technology exhibits decreasing returns, firms are more differ-
entiated in their locations when wmin is high and less differentiated when wmin is low.
Second, when 1 < β < 2, we have x∗∗
1< x∗
1if wmin <2α(2 −β)/η. Namely, if the pro-
duction technology displays moderately increasing returns, firms are more differentiated
when wmin is low. Finally, when β > 2, we have x∗∗
1> x∗
1for any minimum wage.
12

According to (21), the impact of a minimum wage on equilibrium locations depends on
the production technology. Specifically, a higher wmin induces closer equilibrium locations
of firms if the technology is increasing returns to scale (β > 1) and separates the firms
when the technology is decreasing returns to scale (β < 1). Intuitively, the minimum
wage makes the market effect more important when productivity is high but reduces the
importance when productivity is low.
Moreover, from (23), we know that equilibrium prices are dependent on the minimum
wage in two ways. The first term is the indirect effect through x∗∗
1, whose relation with
wmin is discussed in the previous paragraph. The second term is the direct effect: a higher
minimum wage leads to a higher equilibrium price.
Subsequently we investigate the optimal location under a minimum wage. With sym-
metry, x2= 1 −x1holds. As in Section 3, each employed worker contributes e=wmin/α
units of labor input to the employer. The production of each firm is 1/2, requiring 1/η
units of labor. Then the number of workers each firm hires is
N1=
1
η
wmin
α
=w∗
1
2wmin
<1
2,
where the inequality is from the assumption of wmin > w∗
1. The total number of unem-
ployed residents is, therefore, 1 −w∗
1/wmin.
We keep the assumption of commuting shopping as well as the assumption that the
commuting costs are the same as the transportation costs. Since all consumers buy the
good from one of the firms, the socially optimal location is obtained by minimizing the
following total social costs (denoted by TSC):
TSC = 2 Z1
2
0
k(x−x1)2dx + 2 ZN1
0
α
2e2
1dx =k
12 −kx1
2+kx2
1
|{z }
indirect
+wmin
η
|{z}
direct
,
The impact of a minimum wage is also divided into direct and indirect effects. Since
∂TSC/∂wmin = 1/η > 0, the direct effect of wmin on the welfare is always negative. In
fact, a larger wmin increases both the number of unemployed residents and prices p∗∗
1=p∗∗
2,
reducing the total welfare level.
The socially optimal locations and their dependence on the minimum wage are sum-
marized as follows.
Proposition 3. (i) Imposing a minimum wage keeps the socially optimal locations xo
1=
1/4and xo
2= 3/4;(ii) The social welfare is decreasing in the minimum wage when βis
13

close to 1.
Together with the statement at the end of Section 3, Proposition 3 (i) tells us that
the optimal locations depend neither on the production technology nor on the minimum
wage regulation. Basically, this is because the residents are symmetric and firms have the
same technology. Comparing (xo
1, xo
2) with (x∗∗
1, x∗∗
2), we know that the firms are again
over-differentiated at the equilibrium locations.
Proposition 3 (ii) is in contrast to Kaas and Madden (2010) in terms of both short-run
welfare and long-run welfare.4Specifically, Kaas and Madden (2010, Theorem 5) find that
the short-run welfare is independent of the minimum wage requirement (when two firms
have the same productivity), but the imposition of a minimum wage improves the long-
run welfare over laissez-faire because the firm locations are more socially desirable. To the
contrary, in our setup, an increase in the minimum wage will decrease the social welfare
in the short-run (direct effect). Intuitively, a change in the minimum wage will change
the distribution of employed and unemployed workers. The production is reallocated to a
smaller group of workers who suffer increasing disutility, while the increased unemployed
workers still have to pay shopping costs, and thus the total commuting and shopping costs
remain unchanged. The indirect effect depends on the change of locations. Precisely,
closer (farther) locations of firms increase (decrease) the social welfare. Meanwhile, the
impact of the minimum wage on firm locations is small enough when βis close to 1, so the
indirect effect is negligible when the production technology is close to constant returns to
scale.
The different results can be attributed to the different assumptions of labor supply in
these two frameworks. First, Kaas and Madden (2010) assume perfectly elastic demand,
while this paper assumes constant (inelastic) demand for each resident `a la Hotelling
(1929). Second, Kaas and Madden (2010) assume that the productivity of each worker is
constant, while our study further incorporates the competition in the labor market. The
assumption that each worker produces just one unit of product for any wage rate leads to
the full employment of residents. In contrast, we assume endogenous productivity of each
worker, causing involuntary unemployment when a minimum wage is imposed. Finally,
they allow asymmetric productivities, while we assume identical setting and symmetric
solutions.
4Their analysis of short-run social welfare refers to the influences on the welfare through prices and
wages when the firm locations are fixed at the endpoints. In contrast, the term of long-run welfare is
used when firm locations are not fixed.
14

5. Conclusions
This paper generalizes the Hotelling model to include labor input in which two firms
compete in product prices and wages, and consumers are also the labor providers. This
study shows that the production technology influences equilibrium locations. When the
production function exhibits constant returns to scale, the duopolistic firms are located
at x∗
1=−1/4 and x∗
2= 5/4, which are identical to those obtained by Tabuchi and Thisse
(1995). When the production technology does not exhibit constant returns to scale,
the firms change their locations. The result is compared to the case when a minimum
wage is imposed. We find that locations are similarly affected regarding productivity,
showing the robustness of our results. We also analyze the social optimum in two cases.
Unlike the equilibrium, the socially optimal locations are independent of the production
technology and the minimum wage requirement. Meanwhile, an increase in the minimum
wage decreases the social welfare when the production technology is close to constant
returns to scale. Therefore, the assumption of full employment is crucial to the result of
Kaas and Madden (2010) that the imposition of a minimum wage always improves the
welfare.
Appendix A
If residents may choose different firms to work at and buy product from, let xwand
xcbe the indifferent worker and indifferent consumer, respectively.
If xc> xw, then the resident at xwbuys product from firm 1, but is indifferent toward
working at firm 1 or firm 2. Thus, we obtain
¯u−α
2w2
α2
+I+w2
w2
α−p1−k(xc−x2)2−t(xc−x1)2
= ¯u−α
2w2
α2
+I+w2
w2
α−p2−k(xc−x2)2,
which leads to p2≥p1and xc=x1+p(p2−p1)/t. Similarly, the resident at xcworks at
firm 2 but he/she gives equal consideration to buying the product from firm 1 and firm
2. Thus,
¯u−α
2w1
α2
+I+w1
w1
α−p1−k(x−x1)2
= ¯u−α
2w2
α2
+I+w2
w2
α−p1−k(x−x1)2−k(x−x2)2,
15

so that w2> w1and
xw=x2−rw2
2−w2
1
2kα .
Accordingly, xc> xwcan be rewritten as
x2−x1<rp2−p1
k+rw2
2−w2
1
2kα .(A.1)
On the other hand, the profit of firm 2 is calculated as
π2=p2(1 −xc)−w2Z1
xw
e2dx =p2(1 −xc)−w2
2
α(1 −xw)
=p21−x1−rp2−p1
k−w2
2
α1−x2rw2
2−w2
1
2kα .
As an equilibrium, the first-order conditions give
∂π2
∂p2
= 1 −x1−3p2−2p1
2pk(p2−p1)= 0,
∂π2
∂w2
=−w3
2
αp2kα(w2
2−w2
1)−w22(1 −x2)
α+r2(w2
2−w2
1)
kα3= 0.
They imply that
x2−x1=rp2−p1
k+rw2
2−w2
1
2kα +p2
2pk(p2−p1)+w2
2
2p2kα(w2
2−w2
1)
>rp2−p1
k+rw2
2−w2
1
2kα ,
which contradicts (A.1). Therefore, there is no equilibrium of wage rates and price when
xc> xw.
Similarly, we can prove that there is no equilibrium of wage rates and price when
xc< xw.
Appendix B
Proof of Proposition 1.
The case of β= 1:
In this case, (5) and (6) imply
w1=w2=2α
η,
16

so (7) and (8) are simply
max
p1p1−η2
4αˆxand max
p2p2−η2
4α(1 −ˆx),
respectively, where ˆxis given by (3). Solving these two maximization problems, we obtain
product prices:
p1=4α
η2−k(x1−x2)(2 + x1+x2)
3,
p2=4α
η2−k(x1−x2)(4 −x1−x2)
3.
Substituting p1and p2into π1and π2yields
π1=k(x2−x1)(2 + x1+x2)2
18 , π2=k(x2−x1)(x1+x2−4)2
18 .
Then back to the first stage. Solving ∂π1/∂x1= 0 and ∂π2/∂x2= 0 simultaneously yields
x1=−1/4 and x2= 5/4.
The case of β6= 1:
In this case, (5) and (6) imply
ˆx=ηβ
2αβ1
1−βw
β
1−β
1,1−ˆx=ηβ
2αβ1
1−βw
β
1−β
2.(B.1)
According to (3) and the first part of (B.1),
p1=w2
1−w2
2
2α+p2+kη
2β
1−β(x1−x2)w1
αβ
1−β−k(x2
1−x2
2).
Substituting the above equality into (7) yields
π1=2 1
β−1ηβ
1−βw1
αβ
1−βhp2+ 2k(x1−x2)η
2β
1−βw1
αβ
1−β
−k(x2
1−x2
2)−w2
1+w2
2
2αi.
The first-order conditions of optimality yield the following equation for the equilibrium
wages
(2 −β)w2
1+βw2
2−4αβk(x1−x2)η
2β
1−βw1
αβ
1−β
17

= 2αβp2−2αβk(x2
1−x2
2).
Similarly, from (3) and the second part of (B.1),
p2=w2
2−w2
1
2α+p1+k(x1−x2)(x1+x2−2)
+k(x1−x2)η
2β
1−βw2
αβ
1−β.
Then,
π2=2 1
β−1ηβ
1−βw2
αβ
1−βhp1+k(x1−x2)η
2β
1−βw2
αβ
1−β
+k(x1−x2)(x1+x2−2) −w2
2+w2
1
2αi,
and its first-order condition
(2 −β)w2
2+βw2
1−4αβk(x1−x2)η
2β
1−βw2
αβ
1−β
= 2αβp1−2αβk(x1−x2)(2 −x1−x2).
These equations imply that w1is determined (implicitly) by
(2 −β)
2αβ h2αβ
ηβ1
1−β−w
β
1−β
1i2(1−β)
β+ 6k(x1−x2)ηβ
2αβ1
1−βw
β
1−β
1−2−β
2αβ w2
1
=k(x1−x2)(x1+x2+ 2),(B.2)
while others variables are then given by
w2=h2αβ
ηβ1
1−β−w
β
1−β
1i1−β
β,(B.3)
p1=w2
1
αβ −2k(x1−x2)ηβ
2αβ1
1−βw
β
1−β
1,(B.4)
p2=w2
2
αβ −2k(x1−x2)ηβ
2αβ1
1−βw
β
1−β
2,(B.5)
ˆx=x1+x2
2+ηβ
2αβ1
1−β(w
1
1−β
2−w
1
1−β
1) + (2 −β)(w2
2−w2
1)
4αβk(x2−x1).(B.6)
The partial derivatives are calculated from (B.2) using the implicit function theorem
18

for w1and other variables in (B.3), (B.4), (B.5) and (B.6). They are listed as follows.
∂w1
∂x1
=
2αβkw1h3ηβ
2αβ1
1−βw
β
1−β
1−1−x1i
(2 −β)w
β
1−β
1w
2−3β
1−β
2+ (2 −β)w2
1+6αβ2
1−βηβ
2αβ1
1−βk(x2−x1)w
β
1−β
1
,
∂w1
∂x2
=
2αβkw1h1 + x2−3ηβ
2αβ1
1−βw
β
1−β
1i
(2 −β)w
β
1−β
1w
2−3β
1−β
2+ (2 −β)w2
1+6αβ2
1−βηβ
2αβ1
1−βk(x2−x1)w
β
1−β
1
,
∂w2
∂x1
=−w1
w22β−1
1−β∂w1
∂x1
,
∂w2
∂x2
=−w1
w22β−1
1−β∂w1
∂x2
,
∂p1
∂x1
=2w1
αβ
∂w1
∂x1
−2kηβ
2αβ1
1−βw
β
1−β
1
−2βk
1−βηβ
2αβ1
1−β(x1−x2)w
2β−1
1−β
1
∂w1
∂x1
,
∂p1
∂x2
=2w1
αβ
∂w1
∂x2
+ 2kηβ
2αβ1
1−βw
β
1−β
1
−2βk
1−βηβ
2αβ1
1−β(x1−x2)w
2β−1
1−β
1
∂w1
∂x2
,
∂p2
∂x1
=2w2
αβ
∂w2
∂x1
−2kηβ
2αβ1
1−βw
β
1−β
2
−2βk
1−βηβ
2αβ1
1−β(x1−x2)w
2β−1
1−β
2
∂w2
∂x1
,
∂p2
∂x2
=2w2
αβ
∂w2
∂x2
+ 2kηβ
2αβ1
1−βw
β
1−β
2
−2βk
1−βηβ
2αβ1
1−β(x1−x2)w
2β−1
1−β
2
∂w2
∂x2
,
∂ˆx
∂x1
=1
2−2βw
2β−1
1−β
1
1−βηβ
2αβ1
1−β∂w1
∂x1
+(2 −β)
4αβk
2w2∂w2
∂x1−w1∂ w1
∂x1(x2−x1) + w2
2−w2
1
(x2−x1)2,
∂ˆx
∂x2
=1
2−2βw
2β−1
1−β
1
1−βηβ
2αβ1
1−β∂w1
∂x2
+(2 −β)
4αβk
2w2∂w2
∂x2−w1∂ w1
∂x2(x2−x1) + w2
1−w2
2
(x2−x1)2.
19

Now, consider the symmetric equilibrium at which x∗
2= 1 −x∗
1. Then
p∗
1=p∗
2=4α
η2β+k(1 −2x1), w∗
1=w∗
2=2α
η.
The derivatives are simplified as
∂w1
∂x1
=αβkw1(1 −2x1)
2(2 −β)w2
1+3αβ2
1−βk(1 −2x1),
∂w1
∂x2
=∂w1
∂x1
,
∂w2
∂x1
=∂w2
∂x2
=−∂w1
∂x1
,
∂p1
∂x1
=h2
αβ w1−kβ
w1(1 −β)(2x1−1)i∂w1
∂x1
−k,
∂ˆx
∂x1
=1
2−hβ
1−β
1
w1
+2−β
αβk
w1
1−2x1i∂w1
∂x1
.
Then we have
∂π1
∂x1
=∂ˆx
∂x1p1−w2
1
α+ ˆx∂p1
∂x1
−2w1
α
∂w1
∂x1
=β
1−β
k
2
2x1−1
w∗
1
−(1 + β)
αβ w∗
1+(1 −β)(β−2)
α2β2(1 −2x1)kw∗
1
3∂w1
∂x1
+3(1 −β)
β
w∗
1
2
4α−k
2x1
at the symmetric equilibrium. Therefore, it holds that
h2(2 −β)w∗
1
2+3αβ2
1−βk(1 −2x1)i∂π1
∂x1
=k+t
2(1 −β)[αβ2k(4x1+ 1)(2x1−1) −(1 −β)(2 −β)(2x1+ 1)w∗
1
2].
The above relation can be rewritten as
∂π
∂x1
=k
4β2(x1+1
4)(x1−1
2)−γ∗(2x1+ 1)
γ∗+3
16 β2(1 −2x1),
which should be zero, obtaining the following first-order condition of x1for maximizing
π1:
(x1+1
4)(x1−1
2)−γ∗(1 + 2x1) = 0,(B.7)
where γ∗is given by (10). If (9) holds, then γ∗≥ −1/8, so that (B.7) has a solution (the
20

other possible solution of (B.7) minimizes π):
x∗
1=1
8+γ∗−r(1
8+γ∗)(9
8+γ∗) = −1
1 + q1 + 1
1
8+γ∗
,
so that
x∗
2= 1 −x∗
1=7
8−γ∗+r(1
8+γ∗)(9
8+γ∗) = 1 + 1
1 + q1 + 1
1
8+γ∗
.
Proof of Proposition 2
¿From (15),
∂ˆx
∂p1
=1
2k(x1−x2),∂ˆx
∂p2
=−1
2k(x1−x2),
∂ˆx
∂x1
=p2−p1
2k(x1−x2)2+1
2,∂ˆx
∂x2
=p1−p2
2k(x1−x2)2+1
2.
In optimal problems (16) and (17), the first-order conditions are
∂π1
∂p1
= ˆx+1
2k(x1−x2)p1−21
βwmin
βη ˆx1−β
β= 0,(B.8)
∂π2
∂p2
= 1 −ˆx−1
2k(x1−x2)−p2+21
βwmin
βη 1−ˆx1−β
β= 0.(B.9)
Subtracting (B.8) from (B.9) obtains
p1−p2
2k(x1−x2)= 1 −2ˆx+21−β
βwmin
kβη(x1−x2)ˆx1−β
β−1−ˆx1−β
β.(B.10)
Let
Φ≡3ˆx−1−21−β
βwmin
kβη(x1−x2)hˆx1−β
β−1−ˆx1−β
βi−x1+x2
2.
By use of (15), (B.10) is rewritten as Φ = 0. On the other hand, (B.8) implies
p1=21
βwmin
βη ˆx1−β
β−2k(x1−x2)ˆx, (B.11)
so that
∂p1
∂x1
=−2kˆx+21
β(1 −β)wmin
β2ηˆx1
β−2−2k(x1−x2)∂ˆx
∂x1
.(B.12)
21

Symmetry between x1and x2yields ˆx= 1/2 and x2= 1 −x1. Then Φ = 0 gives
∂ˆx
∂x1
=−∂Φ/∂x1
∂Φ/∂ˆx=k(2x1−1)β
6βk(2x1−1) −8(1 −β)wmin/(βη)=−∂ˆx
∂x2
,
while (B.11) and (B.12) are simplified as
p1=−k(2x1−1) + 2wmin
βη ,
∂p1
∂x1
=−k+41
β−1wmin
βη −2k(2x1−1)∂ˆx
∂x1
.
Then the first-order condition for firm 1 in selecting the optimal x1is
∂π1
∂x1
=p1
∂ˆx
∂x1
+∂p1
∂x1
ˆx−2wmin
βη
∂ˆx
∂x1
=−k
2+1
β−12wmin
βη −2k(2x1−1)∂ˆx
∂x1
=0,
whose solution is
x∗∗
1=−1
1 + s1 + 1
1
8+γ∗∗
,
where r∗∗ is defined by (19). Substituting x∗∗
1and x∗∗
2= 1 −x∗∗
1into ∂π1/∂p1= 0 and
∂π2/∂p2= 0 yields
p∗∗
1=p∗∗
2=k(1 −2x∗∗
1) + 2wmin
βη .
Proof of Proposition 3
(i) The first-order condition ∂TSC/∂x1= 0 leads to the socially optimal location
xo
1= 1/4 directly. The symmetric location of firm 2 is xo
2= 3/4. (ii) Totally differentiating
TSC with respect to wmin at x1=x∗∗
1yields
dTSC
dwmin
=∂TSC
∂wmin
+∂TSC
∂x1
·∂x∗∗
1
∂γ∗∗ ·∂γ∗∗
∂wmin
=1
η+1
4−x∗∗
1(1 + 2x∗∗)2
(−2x∗∗)(1 + x∗∗ )
1−β
ηβ2(B.13)
Since limβ→1x∗∗ =−1/4, (B.13) is positive when βis close to 1.
22

Acknowledgements
We wish to thank two anonymous referees for their highly valuable and detailed com-
ments. The financial support from JSPS KAKENHI of Japan (Grant Numbers 26380282,
24330072 and 24243036) for the third author is acknowledged.
References
Anderson, S.P., 1988. Equilibrium existence in the linear model of spatial competition.
Economica 55, 479-491.
Anderson, S.P., Engers, M., 1994. Spatial competition with price-taking firms. Economica
61, 125-136.
d’Aspremont, C., Gabszewicz, J.J., Thisse, J.F., 1979. On Hotelling’s “stability in com-
petition”. Econometrica 47, 1145-1150.
Bhaskar, V., To, T., 1999. Minimum wages for Ronald McDonald monopsonies: a theory
of monopsonistic competition. The Economic Journal 109, 190-203.
Bhaskar, V., To, T., 2003. Oligopsony and the distribution of wages. European Economic
Review 47, 371-399.
Braid, R.M., 1996. The optimal locations of branch facilities and main facilities with
consumer search. Journal of Regional Science 36, 217-234.
Brekke, K.R., Straume, O.R., 2004. Bilateral monopolies and location choice. Regional
Science and Urban Economics 34, 275-288.
Brueckner, J.K., Thisse, J.F., Zenou, Y., 2002. Local labor markets, job matching, and
urban location. International Economic Review 43, 155-171.
Claycombe, R.J., 1991. Spatial retail markets. International Journal of Industrial Orga-
nization 9, 303-313.
Claycombe, R.J., Mahan, T.E., 1993. Spatial aspects of retail market structure beef pric-
ing revisited. International Journal of Industrial Organization 11, 283-291.
Economides, N., 1986. Minimal and maximal product differentiation in Hotelling’s
duopoly. Economics Letters 21, 67-71.
23

Economides, N., 1989. Quality variations and maximal variety differentiation. Regional
Science and Urban Economics 9, 21-29.
Fleckinger, P., Lafay, T., 2010. Product flexibility and price competition in Hotelling’s
duopoly. Mathematical Social Sciences 60, 61-68.
Hotelling, H., 1929. Stability in competition. Economic Journal 39, 41-57.
Irmen, A., Thisse, J.F., 1998. Competition in multi-characteristics spaces: Hotelling was
almost right. Journal of Economic Theory 78, 76-102.
Kaas, L., Madden, P., 2010. Minimum wages and welfare in a Hotelling duopsony. Eco-
nomic Theory 43, 167-188.
Lai, F.C., Tabuchi, T., 2012. Hotelling meets Weber. Regional Science and Urban Eco-
nomics, 42, 1017-1022.
Lambertini, L., 1997. Optimal fiscal regime in a spatial duopoly. Journal of Urban Eco-
nomics 41, 407-420.
Mai, C.C., Peng, S.K.,1999. Cooperation vs. competition in a spatial model. Regional
Science and Urban Economics 29, 463-472.
Matsumura, T., Matsushima, N., 2012. Locating outside a linear city can benefit con-
sumers. Journal of Regional Science 52, 420-432.
McConnell, C.R., Brue, S.L., 1999. Economics, Irwin-McGraw Hill (14th ed.).
Neven, D., 1986. On Hotelling’s competition with non-uniform customer distributions.
Economics Letters 21, 121-126.
Porter, M., 1995. The competitive advantage of the inner city. Harvard Business Review
73, 55-71.
Raith, M., 1996. Spatial retail markets with commuting consumers. International Journal
of Industrial Organization 14, 447-463.
Tabuchi, T., 1994. Two-stage two-dimensional spatial competition between two firms.
Regional Science and Urban Economics 24, 207-227.
Tabuchi, T., Thisse, J.F., 1995. Asymmetric equilibria in spatial competition. Interna-
tional Journal of Industrial Organization 13, 213-227.
24