Discussion Paper No. 761

REGIONAL RESTRICTION,STRATEGIC DELEGATION, AND WELFARE

Toshihiro Matsumura
Noriaki Matsushima

November 2009

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Regional restriction, strategic delegation, and welfare
Toshihiro Matsumura
Institute of Social Science, University of Tokyo
Noriaki Matsushima∗
Institute of Social and Economic Research, Osaka University
November 14, 2009

Abstract
We investigate the effects of restricting locations of firms into Hotelling duopoly models. In
the standard location-price models, the equilibrium distance between firms is too large from theviewpoint of consumer welfare. Thus, restricting locations of firms and reducing the distance
between firms improve consumer welfare, through lower prices and smaller transportation costs
for consumers. We introduce strategic reward contracts into the location-price models. We findthat in contrast to the above existing result, restriction of the locations of firms reduces consumer
welfare. Restricting locations of the firms reduces transportation costs but increases the prices
through the change of strategic commitments by the firms, and it yields a counterintuitive
result.

JEL classification: R52, R32, L13
Key words: product selection, delegation, Hotelling, locational restriction∗ Corresponding author: Noriaki Matsushima, Institute of Social and Economic Research, Osaka University,
Mihogaoka 6-1, Ibaraki, Osaka 567-0047, Japan. Phone: +81-6-6879-8571, Fax: +81-6-6878-2766, E-mail:nmatsush@iser.osaka-u.ac.jp1

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Introduction

In this paper, we consider the effects of regional restrictions on consumer welfare. We use twostandard location-price models on the Hotelling line—one that restricts the locations of firms within
the city (d’Aspremont et al. (1979)) and the that other does not impose any such restriction(Tabuchi and Thisse (1995) and Lambertini (1997)). The latter two papers show that the firms
locate outside the city if they are allowed to do so. The outside locations yield higher equilibrium
prices and larger transportation costs of consumers. Thus, it is obvious that restricting the locationsof firms improves consumer welfare. In our paper, however, we show that this welfare property does
not hold under a plausible strategic environment.
We introduce strategic reward contracts that are quite popular in the literature on management
science and industrial organization. We find that the symmetric equilibrium price in which thefirms are not allowed to locate outside the city is higher than that in which the firms are allowed tolocate outside if the firms can make strategic commitments through reward contracts. On the other
hand, the restriction on the firms’ locations reduces the transportation costs of consumers. The
higher price and the lower transportation costs represent a trade-off. In this setting, the negative
effect dominates the positive one—that is, restricting the locations of firms reduces consumer welfare
when firms can make strategic commitments through reward contracts. Moreover, we show that thesymmetric equilibrium price without strategic reward contracts is higher (smaller) than that withthem when the firms are (not) allowed to locate outside the city.
The matter discussed here is related to the following fact. In many countries, large stores often
locate in suburban areas where the population densities are quite small. Regulations governing
stores’ locations in suburban areas have been repeatedly discussed to realize a “compact city” thatwill yield reduction in the environmental damage arising from commute to and from stores, and will
also help consumers who do not drive and/or own cars. For example, in Japan, the so called “Three
Laws of Community Building” have been in force since 2000 and were further strengthened in 2006;
even then, there are discussions on further regulations with regard to the centralization of the store

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locations.1 Given such discussions, it is important to discuss whether or not regional restrictions on
suburban areas are beneficial.
There are two closely related articles that discuss optimal zoning. Using a standard Hotellingmodel a la d’Aspremont et al. (1979), Lai and Tsai (2004) show that restricting the locations offirms reduces both transport costs and prices, resulting in an improvement in consumer surplus aswell as total social surplus. Chen and Lai (2008) discuss optimal zoning in a spatial Cournot model.
There are at least two differences between these two papers and our paper. First, we consider the
relation between the strategic commitments of firms and equilibrium outcomes whereas these two
papers do not. Second, we mainly consider whether or not the firms should be allowed to locate
outside the linear city whereas the above two papers discuss zoning policies within the city. Our
setting is closely related to the problem of urban sprawl whereas the two papers are related to the
problem of zoning within a city.
We now briefly mention the importance of studying the case where ownership and management are separated. Traditional economic theories assume that the single aim of the firms is profit
maximization. In large companies, however, ownership and management are separated, and managerial decision processes are rather complex. In his classical paper, Baumol (1958) mentioned thatmanagers may be guided by objectives other than pure profit-maximization, and he suggested asales-maximization model as a more realistic alternative. Later, economists noticed that the ownersof firms have strategic incentives to link sales to managers’ rewards. Fershtman (1985), Vickers(1985), Fershtman and Judd (1987), and Sklivas (1987) made path-breaking contributions in thisregard. They considered the separation of owners and managers, and examined the following twostage game: in the first stage, the owner makes the manager’s contract and announces it publicly; inthe second stage, after observing the contract, the manager maximizes the payoff given the reward
contract. In these papers, the managers’ rewards are proportional to a linear combination of profits
and outputs or sales. Since the model formulation of the abovementioned works is quite realistic and1 Before 2000, there existed a Law on the Locations of Large Stores, which restricted, rather than promoted, theplacement of large stores in the city’s central district. For the emergence of “Big Box” retail stores in the U.S., seeDenning and Lary (2005), Haltiwanger et al. (in press), and the references therein.

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explains many of the actual behaviors of the firms, their model has become one of the most popularmodels in the fields of industrial organization and management science (Jansen et al., 2007).
Subsequent works on this separation of ownership and management showed that the reward
contract is crucially dependent on model formulation such as price or quantity competition (strategic
complement or substitute). In the context of price competition, the managers’ rewards are usually
decreasing in the firms’ sales; further, this also holds in the location-price model where the firms’
locations are restricted within the city. Our result suggests, however, that this does not hold in thelocation-price model where the firms are allowed to locate outside the city. As mentioned earlier, themanagers’ rewards are increasing in the sales of the firms and this yields aggressive price competition
and improves consumer welfare.
The remainder of this paper is organized as follows. Section 2 formulates the model. Section 3
presents the equilibrium outcomes in the two models. We compare the outcomes and the results
derived by the standard location models. Section 4 shows the main result. Section 5 concludes thepaper.

2

Model

Consider a linear city along the unit interval [0, 1], where consumers are uniformly distributed along
the interval. Firm i (i = 1, 2) is allowed to locate at xi ∈ [α, β] where α ≤ 0 and β ≥ 1. If α = 0and β = 1, the model corresponds to that in d’Aspremont et al. (1979), i.e., the firms are allowed
to locate within the city. If α is sufficiently small and β is sufficiently large, the model corresponds
to that in Tabuchi and Thisse (1995) and Lambertini (1997). Without loss of generality, we assume
that x1 ≤ x2 .
Each consumer buys exactly one unit of the good, which can be produced by either firm 1 or 2.
Let pi denote the price of firm i (i = 1, 2). The utility of the consumer located at x is given by½ux =v − t(x1 − x)2 − p1
v − t(x2 − x)2 − p2

if bought from firm 1,
if bought from firm 2,(1)

where v is a positive constant and sufficiently large, and t represents the exogenous parameter of4the transport cost incurred by the consumer. For a consumer living at x(p1 , p2 , x1 , x2 ), where−t(x1 − x(p1 , p2 , x1 , x2 ))2 − p1 = −t(x2 − x(p1 , p2 , x1 , x2 ))2 − p2 ,(2)

the utility is the same regardless of the firm chosen. We can rewrite (2) as follows:x(p1 , p2 , x1 , x2 ) =x1 + x2
p 2 − p1+.
22t(x2 − x1 )Thus, the demand facing firm 1, D1 , and that facing firm 2, D2 , are given by
D1 (p1 , p2 , x1 , x2 ) = min{max(x(p1 , p2 , x1 , x2 ), 0), 1},D2 (p1 , p2 , x1 , x2 ) = 1 − D1 (p1 , p2 , x1 , x2 ).

(3)

We assume that the owners of both the firms hire a manager to delegate the price and thelocation decisions. The manager is offered a contract that is publicly observable. In this contract,
the manager receives a fixed salary and a bonus related to the firm’s profits and market share. In
particular, if profits πi are positive, the manager of firm i receives a bonus that is proportional to
the linear combination
Ui = πi + λi Di ,where the weight λi is a real number chosen by owner i in order to maximize his profits, and profits
πi (the owner i’s profits) are represented by
π1 = (p1 − c)D1 (p1 , p2 , x1 , x2 ), π2 = (p2 − c)D2 (p1 , p2 , x1 , x2 ),where c is the marginal cost of both the firms. Note that, in this model, Di is not only the quantity
supplied by firm i but also the market share of firm i because the total demand is normalized to 1.
Thus, we can say that the model setting in this paper follows that in Jansen et al. (2007).
Consumer surplus and social welfare are given by
Z D1
CS =

Z 1

2

(v − p1 − t(x − x1 ) )dx +
0

Z D1
SW =

(v − c − t(x − x1 )2 )dx +0(v − p2 − t(x2 − x)2 )dx,

D1
Z 1

(v − c − t(x2 − x)2 )dx.

D1

The game runs as follows. In the first stage, each owner seeks to maximize his profits by properly
choosing the weight in the manager’s contract λi . In the second stage, manager i chooses its location5

xi , and in the third stage, manager i chooses its price pi ∈ [c − ci , ∞). Note that, we consider twocases concerning the location choices of the firms in the second stage: (i) the firms are not permitted
to locate outside the line and (ii) the firms are permitted to locate outside the line.

3

Equilibrium

In this section, we derive the equilibrium locations, the managers’ contracts, profits, consumer
surplus, and total social surplus in the two models. One is the model wherein α = 0 and β = 1.
The other is the model wherein α is sufficiently small and β is sufficiently large that the restriction
α ≤ xi ≤ β is not binding.
First, we investigate the price competition stage. Given the locations of the firms, x1 and x2 ,the managers face Bertrand competition. The manner of competition is common between the twomodels. Let the superscript “B” denote the equilibrium outcome at the price-competition stage
given x1 and x2 . The equilibrium prices are
pB
1

=

pB
2

=

3c − 2λ1 − λ2 + t(2 + x1 + x2 )(x2 − x1 ),33c − 2λ2 − λ1 + t(4 − x1 − x2 )(x2 − x1 ).
3Then, the managers’ rewards are given by

3.1

U1B

=

U2B

=

(λ1 − λ2 + t(2 + x1 + x2 )(x2 − x1 ))2,18t(x2 − x1 )(λ2 − λ1 + t(4 − x1 − x2 )(x2 − x1 ))2.
18t(x2 − x1 )Locations are restricted

As a benchmark, we consider the case in which the locations of the firms are restricted within thelinear city, that is, α = 0 and β = 1.
We consider the location choices by the managers. If the difference determined in the first stage(|λ1 − λ2 |) is not so large,2 the maximum differentiation appears in equilibrium, that is,x1 = 1 − x2 = 0.
2 This difference is related to the discussion on the large cost difference in Matsumura and Matsushima (2009).

6

Then the profits of the firms, the managers’ rewards, the consumer surplus, and social welfare are
given by(3t − 2λi − λj )(3t + λi − λj )(3t + λi − λj )2 R
3(c + t) − 2λi − λj, UiR =
, pi =,
18t
18t
336tc − 18t(λi + λj ) − (λi − λj )2 + 39t2
, (i = 1, 2, i 6= j),
CS R = −
36t

πiR =

where the superscript “R” denotes the equilibrium outcome in the case where the locations are
restricted.
We now obtain the equilibrium value of λi (i = 1, 2). In the first stage, each owner i independently
chooses λi so as to maximize its profit πi = πiR . The first-order condition is−

3t + 4λi − λj
= 0, i = 1, 2, i 6= j.
18t

(4)

From (4), we have the following lemma:
Lemma 1 When the firms’ locations are restricted, each owner i chooses λi at
λER
= λER
= −t.
1
2

(5)

The profits and the consumer surplus are as follows:xER= 0, xER= 1, pER= c + 2t, πiER = t, CS ER = v −12i

12c + 25t
, (i = 1, 2).
12(6)

If each owner is unable to set a negative value of λi , the result is
xER= 0, xER= 1, λER= 0, pER= c + t, piER
=
1
2
i
i
i

12c + 13tt
, CS ER = v −, (i = 1, 2).
2
12

(7)

In the standard location-price model in d’Aspremont et al. (1979), the equilibrium outcome isas follows:
Result 1 (d’Aspremont et al., 1979): Suppose that each owner directly manages its own firm.
When the firms are not allowed to locate outside the linear city, the equilibrium outcome isx1 = 0, x2 = 1, p1 = p2 = c + t, pi1 = π2 =t
12c + 13t
, CS = v −
.
2
12

(8)

Henceforth, we refer to this case as “the non-delegation case.” From (6), (7), and (8), we have thefollowing proposition:
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Proposition 1 Suppose that the firms are not allowed to locate outside the linear city. If eachowner can set a negative value of λi , the following three properties hold. The equilibrium prices
in the delegation case are larger than those in the non-delegation case. The consumer surplus in
the delegation case is smaller than that in the non-delegation case. The equilibrium profits in thedelegation case are larger than those in the non-delegation case. If each owner is unable to set a
negative value of λi , the equilibrium outcomes in the delegation case and in the non-delegation caseare the same.
We now mention the intuition behind the result. In the two cases, the firms fully maximize thedegree of product differentiation within the Hotelling line. The incentive contracts λi (i = 1, 2) only
affect the price strategies. A higher value of λi leads to a lower price pi because the manager i tends
to put more stress on the sales volume. Because of the strategic complementarity, the competition
between the firms is intensified. Anticipating this, each owner sets a lower λi so as to mitigate thesubsequent price competition.

3.2

Locations are not restricted

We now consider the case in which the locations of the firms are not restricted. In other words, α
is sufficiently small and β is sufficiently large that the constraint α ≤ xi ≤ β is not binding.
The result in the third stage (price competition) has already been derived. We consider thelocation choices by the managers. The first-order conditions lead tox1 =4(λ1 − λ2 ) − 3t
15t − 4(λ2 − λ1 ), x2 =.
12t
12t

The resulting profits of the firms, the managers’ rewards, the consumer surplus, and social welfare
are given by(9t + 4(λi − λj ))(9t − 2(λi + 2λj ))(9t + 4(λi − λj ))2 U
6c + 9t − 2(λi + 2λj ), UiU =
, pi =
,
108t
108t
6
432tc + 765t2 − 216(λi + λj )t + 16(λi − λj )2
CS U = −, (i = 1, 2, i 6= j),
432t
πiU =

where the superscript “U” denotes the equilibrium outcome in the case where the locations areunrestricted.
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We now obtain the equilibrium value of λi (i = 1, 2). In the first stage, each owner i independently
chooses λi so as to maximize its profit πi = πiU . The first-order condition is
9t − 8λi − 4λj= 0, i = 1, 2, i 6= j.
54t

(9)

From (9), we have the following lemma:
Lemma 2 When the firms’ locations are not restricted, each owner i chooses λi at
λEU
= λEU
=
1
2

3t
.
4

(10)

The profits and the consumer surplus are as follows:
53t
3t
48c + 49t
1
= , pEU= c + , πiEU = , CS EU = v −, (i = 1, 2).
xEU= − , xEU
2
i
1
4
4
4
8
48

(11)

In the standard location-price model in Tabuchi and Thisse (1995), the equilibrium outcome isas follows:
Result 2 (Tabuchi and Thisse, 1995): Suppose that each owner directly manages its own firm.
When the firms are allowed to locate outside the linear city, the equilibrium outcome is
53t
3t
38c + 85t
1.
x1 = − , x2 = , p1 = p2 = c + , pi1 = π2 = , CS = v −
4
4
2
4
38

(12)

From (11) and (12), we have the following proposition:
Proposition 2 Suppose that the firms are not allowed to locate outside the linear city. The equilibrium prices in the delegation case are smaller than those in the non-delegation case. The consumer
surplus in the delegation case is larger than that in the non-delegation case. The equilibrium profits
in the delegation case are smaller than those in the non-delegation case.
Proposition 2 is quite different from Proposition 1. When the firms are allowed to locate outside
the linear city, price competition in the delegation case is tougher than that in the non-delegation
case. We now mention the intuition behind this result. The incentive contracts λi (i = 1, 2) affectnot only the price strategies but also the location strategies. A higher value of λi leads to a lower
price pi because the manager i tends to put more stress on the sales volume. In the case where the9firms are allowed to locate outside, a lower price pi caused by a higher value of λi also affects thelocation choices of the firms. The rival firm j (j 6= i) moves far away from the center to escape thetougher competition resulting from the higher value of λi . As a result, the quantity supplied by firmi substantially increases. This demand-enhancing effect dominates the competition-enhancing effectbecause of the strategic complementarity of the price strategies. Anticipating this, each owner setsa higher λi ; this accelerate competition.

4

Main result

We compare the equilibrium outcomes in the two models discussed above and mention welfare
implications. Comparing (6) and (7) with (11), we have the following proposition.
Proposition 3 Suppose that α is sufficiently small and β is sufficiently large—that is, the firms
are allowed to locate outside the linear city. In this case, the profits of the firms are lower than thosein the case where the locations are restricted (α = 0, β = 1). The consumer surplus is higher thanin the case where the locations are restricted (α = 0, β = 1).
In our delegation case, the restriction on the locations of firms that reduces the distance between thefirms mitigates competition and then reduces consumer welfare. At first glance, the relation between
the location restriction and consumer welfare is counterintuitive. As explained in the previous
section, no restriction on the locations accelerates the competition between the firms because of thedemand-enhancing effect resulting from the high-powered incentive contracts. Note that, since thisresult holds with and without the non-negative constraint of λi (i = 1, 2), Proposition 3 is highlyrobust.
Proposition 3 is in sharp contrast to the relation between the results obtained in d’Aspremont etal. (1979) and Tabuchi and Thisse (1995). These two papers show that the profits are higher when
the locations are unrestricted. Moreover, they show that when the locations are unrestricted, theconsumer surplus is lower. Proposition 3 indicates that incorporating the separation of ownershipand management and the reward contracts of the managers into the models of d’Aspremont et al.
(1979) and Tabuchi and Thisse (1995) drastically changes the results.
105Concluding Remarks

In this paper, we investigate the effects of the location restriction on consumer welfare. We use twostandard location-price models on the Hotelling line—one restricts the firms’ locations within the
Hotelling line and the other does not impose any such restriction. We incorporate strategic reward
contracts into the models. We find that the former model with the location restrictions yields
smaller consumer welfare than the latter. This means that the restrictions on locations reduce
consumer welfare. This result is quite different from the two results derived by the standard models(d’Aspremont et al. (1979) and Tabuchi and Thisse (1995)). We think that our result provides a
caveat to the recent debate on urban sprawl in the U.S. and Japan.
Our paper also sheds light on the value of strategic commitment in Hotelling models. In the model
without location restrictions, the firms have strong incentives for exhibiting aggressive behavior in
the subsequent stages; in contrast, in the model with location restrictions, the firms have strong
incentives for exhibiting less aggressive behavior. Thus, our result indicates that the strategic valueof commitment crucially depends on the model formulation of the location choices.
Our result depends on the assumption that the managers choose both the locations and theprices. If the owners choose the locations and the managers choose the prices, our results do nothold. Here, less aggressive behavior is exhibited in both cases—with and without the restriction oflocations—because the owners have already chosen the locations. However, no restrictions on thelocations always reduces consumer surplus.
Our results may be consistent with the retail prices set by the supermarkets located in suburban
areas, such as Wal-Mart in the U.S. and ÆON in Japan. These retailers set their prices at lower
levels—this is not consistent with previous researches (d’Aspremont et al. (1979) and Tabuchi andThisse (1995)) but is consistent with our main result. We believe that our result also has implicationsin the context of retailer strategies.

11References
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