Ann Reg Sci 40:531–544 (2006)
DOI 10.1007/s00168-006-0075-5
ORIGIN AL PAPER
Song-ken Hsu
Simple monopoly price theory in a spatial
market
Received: 24 July 2005 / Accepted: 16 January 2006 / Published online: 24 May 2006
© Springer-Verlag 2006
Abstract One of main conclusions drawn by prior studies is that in a spatial
market, the shape of the demand function can fully determine whether one simple
pricing policy is superior to another either on the basis of the firm’s preference or
social desirability while there is one assumption that is generally stipulated,
namely, that the fixed market area assumption under which the market area is
exogenously determined and remains the same under alternative pricing policies.
In order to fully understand the impact of a demand function, this paper attempts to
reexamine the relative economic advantages between two simple spatial pricing
policies in a world with variable market area, that is, the market area is
endogenously determined the price charged. We show that the fixed market area
assumption is valid only where demand is linear, but no longer holds where
demand is nonlinear. Moreover, and more importantly, we show that in a world
with variable market area, some conclusions drawn by prior studies on the relative
economic benefits of two pricing policies cannot remain valid. The main
conclusion of this paper is that even the relative economic benefits of two simple
spatial pricing policies is concerned, the impact of economic space is significant.
JEL Classification D42 . R32
S.-k. Hsu
Industrial Economics Department, Tamkung University, Tamsui,
Taipei, Taiwan, Republic of China
S.-k. Hsu (*)
130, Section 2, Academia Road, Taipei, Taiwan, Republic of China
E-mail: [email protected]
532
S.-k. Hsu
1 Introduction
In his pioneering contribution to spatial price theory, Smithies (1941) among others
shows that, when demand is linear, the firm is indifferent between two simple
spatial pricing policies1, whereas profit is greater (smaller) under mill pricing than
under uniform pricing when demand is concave (convex). A recent paper by
Cheung and Wang (1996) shows further that the above-mentioned finding with
regard to the firm’s preference can be extended to include other economic benefits
such as monopoly output and welfare. One possible interpretation naturally arises
from these results, namely, that the shape of demand alone can completely
determine the matter of whether one simple pricing policy is superior to another,
and, thus, the economic space plays a non-significant role in the comparison of
economic benefits between the two simple pricing policies.
The work of Smithies (1941) and Cheung and Wang (1996) has assumed, among
other things, that under both mill and uniform pricing, the firm serves a fixed and
same market area (see also Beckmann 1976). Nevertheless, some studies on spatial
price theory in the literature have instead considered the case in which the market area
of a firm is endogenously determined by the price charged (see, for example,
Greenhut and Ohta 1972; Holahan 1975; Heffley 1980; Hsu 1979, 1983a,b). It shows
that the price charged varies with the function form of a consumer demand, and so
does the size of the market area a firm serves. In other words, the function form of a
consumer demand may have some impact on the size of the market area in a world
with variable market area, namely that the market area is endogenously determined.
In order to fully understand the complete impact of a consumer demand, the purpose
of this paper is thus to reexamine the relative economic advantages between two
simple spatial pricing policies in a world with variable market area We will show that
the fixed market area assumption is valid only for linear demand, but no longer holds
for nonlinear demand. This result confirms the expectation of Cheung and Wang
(1996, p.142). Moreover and more importantly, we will show that in a world with
variable market area, some findings on the relative economic benefit of two simple
spatial pricing policies drawn by prior studies no longer hold.
The organization of this paper is as follows: in Section 2, we present a model
that is the same as that postulated by prior studies except that the market area is
variable. In Section 3, we first derive the optimal prices and market areas under the
two simple pricing policies, and relate those to the shape of the demand function.
Section 4 is devoted to a comparative study of the relative economic benefits of the
two simple pricing policies that include monopoly output, profits, the aggregate
benefits of consumers, and social welfare. Section 5 contains some concluding
remarks.
1 They
are mill pricing and uniform pricing. Under mill or constant F.O.B. pricing, a consumer
pays for transportation costs incurred, while the firm is responsible for such expenses under
uniform or constant C.I.F. pricing. Both belong to simple pricing since the firm will charge a
constant mill or delivered price while different prices will be charged under another category of
pricing policy, namely, spatial price discrimination.
Simple monopoly price theory in a spatial market
533
2 Basic model
Consider a linear spatial market over which consumers are continuously and
uniformly distributed. A monopolist sells a standardized product subject to a
strictly positive and constant freight cost, say t. The demand is given by
qð xÞ ¼ f ð pÞ ¼ ða bpÞð1=vÞ
(1)
where x denotes the distance from the seller’s mill, q(x)=the quantity demanded at
the market site x, p=the delivered price, i.e. the amount a consumer is to pay for a
unit of commodity, and a,b as well as v are positive parameters of the demand
function. Equation 1 is employed by Greenhut et al. (1975) to explore in detail the
properties of spatial (total) demand under competition since it is completely general
for the purpose of spatial price theory (see fn. 10, p. 673 for the argument), and by
Greenhut (1977) to examine the output effect of a movement from spatial price
discrimination to mill pricing. Moreover and more importantly, Eq. 1 not only
includes the linear demand generally postulated by prior studies as a special case,
but also allows the shape of the demand curve to be concave, or else convex. This is
because it follows from Eq. (1) that
00
(2)
f ¼ @2 f ð pÞ @p2 ¼ ð1 vÞ b2 v2 ða bpÞð1=v2Þ
and thus, v<1, v=1, and v>1, respectively, infers that f ″ is greater than, equal to, or
less than zero and, therefore, the demand curves are convex, linear, or concave
based on the above. Finally, we assume throughout this paper that v>0 in order to
ensure that the slope of the demand function is negative since f 0 ¼ ðb=vÞ
ða bpÞð1=v1Þ ; and refer the increase in the value of v to the situation where the
demand function becomes less convex, or more concave.
The firm’s profit, π, can generally be written as
Z B
π¼
½ðm cÞqð xÞdx F
(3)
0
where B=the extent of the market area over which the seller actually serves, m=the
mill price=p−tx =the amount the seller can receive by selling one unit of product,
c=the constant marginal cost of production, and F=total fixed costs. The profit
under mill pricing becomes
Z B
πf ¼
½ðm cÞqð xÞdx F ¼ ðm cÞQf F;
(4)
RB
0
RB
where Qf ¼ 0 qð xÞdx ¼ 0 ða bpÞð1=vÞ dx ¼ the monopoly output under mill
pricing. On the other hand, the profit under uniform pricing is given by
Z B
Z B
½ðp c txÞqð xÞdx F ¼ f ð pÞ
ðp c txÞdx F
(5)
πu ¼
0
0
in which the second equality is based on the fact that, under uniform pricing, the
quantity demanded is independent of the distance.
534
S.-k. Hsu
Finally, we assume in accordance with Holahan (1975); Heffley (1980); Hsu
(1983a,b) that the market area is endogenously determined by the price charged.
Thus, the following relation is used to determine the market area under mill 2
pricing
qð BÞ ¼ f ðpmax Þ ¼ 0;
(6)
where Pmax = the reservation price, i.e. the maximum price a consumer is willing to
pay for a unit of commodity. Based on the belief that a consumer will refuse to buy
any quantity if the selling price is more than the maximum he is willing to pay, the
above relation states that, under mill pricing, the market area will equal the distance
from his site to the point at which the delivered price equals the consumer’s
reservation price. Moreover, when demand is given by Eq. 1, Pmax=a/b.
Instead, the profit-maximizing firm under uniform pricing will refuse to sell
beyond the market point at which the amount he receives for a unit of commodity
equals the average variable cost (see Beckmann 1968; Heffley 1980; Hsu 1983a).
Thus, the following relation can be used to determine the boundary of the market
area under uniform pricing3
ðp c tBÞ ¼ 0:
(7)
3 Optimal prices and market areas
The marginal profit under mill pricing is given by
dπf dm ¼ Qf þ ðm cÞdQf dm:
(8)
2 This relation can also be derived from the profit-maximization with respect to the market area To
RB
see this, note
first that
since under mill pricing, f ¼ ðm cÞQ F, where Q ¼ 0 qð xÞdx.
Thus, @f @B ¼ @Q @B ¼ qðBÞ, from which we see that as long as the quantity demanded is
positive at the boundary B, the marginal profit increases with the market area, and equals zero
while the relation (6) holds. Moreover, the marginal profit is always negative for those sites
beyond the boundary of the market area B. It follows that the optimal size of the market area must
satisfy Eq. 6 since beyond that site B, the delivered price is higher than the reservation price.
3 Similar to mill pricing, this relation can be derived from the profit-maximization with respect to
the market area rather than by
R Bimposing it as an assumption. To see this, note first that under
uniform pricing,
u ¼ f ðpÞ 0 ðp c txÞdx, and thus, @u =@B ¼ ðp c tBÞf ð pÞ since
u ¼ f ð pÞ ðp cÞB tB2 2 . Here we see that the marginal profit under uniform pricing equals
zero at the market site satisfying Eq. 7. Moreover, the marginal profit is always negative for those
sites beyond the boundary of the market area B, and it increases with the market area as long as
ðp c txÞ > 0.
Simple monopoly price theory in a spatial market
535
If demand is given by Eq. 1, then the monopoly output under mill pricing is
Z Bf
Z Bf
Qf ¼
qðxÞdx ¼
ða bm þ btxÞð1=vÞ dx
0
0
(9)
¼ ð1=bt Þ½v=ð1 þ vÞða bmÞð1=vþ1Þ
in which the third equality is based on Eq. 6. The marginal profit under mill pricing
can thus be rewritten as
dπf dm ¼ v ð1 þ vÞbt ða bmÞð1=vþ1Þ þ ðm cÞð1=t Þða bmÞð1=vÞ ; (8’)
(8)
since dQf dm ¼ ð1=t Þða bmÞð1=vÞ : Accordingly, the optimal mill price, mf, is4
mf ¼ ða=bÞ v ð1 þ 2vÞ þ c ð1 þ vÞ ð1 þ 2vÞ :
(10)
The boundary of the market area under mill pricing, Bf, is then determined by
substituting Eq. 10 in Eq. 6, and is given by
(11)
Bf ¼ ða bcÞ bt ð1 þ vÞ ð1 þ 2vÞ :
The marginal profit under uniform pricing is
@πu @p ¼ f ð pÞðp tB cÞ dB dp
Z B
Z
ðp tx cÞdx þ
þf 0 ð pÞ
0
B
f ð pÞdx:
(12)
0
Note first that in view of Eq. 7, the first term on the right hand side (hereafter RHS)
of Eq. 12 disappears. Moreover, the second term on the RHS of Eq. 12 equals
f 0 ð pÞðp cÞ2 ð1=2t Þ; and the last term can be written as f ð pÞðp cÞ t since
RB
0 f ð pÞdx ¼ f ð pÞB and from Eq. 7, B ¼ ðp cÞ t: Thus, the marginal profit
under uniform pricing becomes
dπu dp ¼ f 0 ð pÞðp cÞ2 ð1=2t Þ þ f ð pÞðp cÞ t:
(13)
Therefore, the optimal uniform delivered pricing, Pu, is
pu ¼ ½ð2va=b þ cÞ ð1 þ 2vÞ
(14)
4 In an earlier version of this paper, one of readers suggested that the examination of the secondorder optimal conditions for profit-maximization maybe helpful for the relevant analysis. The
formal examination is presented in Appendix. The finding hereupon is that our discussion in
Section 2 shows that the value of v must be greater than zero for the slope of a demand being
negative, and therefore, the second-order optimal conditions for two optimization problems of
this paper are always satisfied.
536
S.-k. Hsu
since Eq. 13 together with Eq. 1 can be written as
h
i
dπu dp ¼ ðb=vÞða bpÞð1=v1Þ ðp cÞ þ 2ða bpÞ1=v ðp cÞ 2t
h
i
¼ ½ðb=vÞðp cÞ þ 2ða bpÞ ða bpÞð1=v1Þ ðp cÞ 2t
(15)
Moreover, the boundary of the market area under uniform pricing, Bu, is
determined by substituting Eq. 15 in Eq. 7, and is given by
(16)
Bu ¼ ða bcÞ bt 2v ð1 þ 2vÞ :
We shall first compare two market areas under alternative pricing policies.
Subtracting Eq. 16 from Eq. 11 yields
(17)
Bf Bu ¼ ða bcÞ bt ð1 vÞ ð1 þ 2vÞ :
The term (a−bc) in Eq. 17 is positive since both mf −c>0 and pu−c>0 must hold.
Thus, it follows from Eq. 17 that when v>1(v <1), Bf <Bu (Bf >Bu), and Bf =Bu if and
only if v =1. Formally, we have
Proposition 1 The market area is larger (smaller) under mill pricing than it would
be under uniform pricing when demand is convex (concave), and the firm serves
the same market area under the two simple pricing policies if and only if demand is
linear.
Proposition 1 indicates that when the market area is variable, the fixed market
area assumption made in prior studies applies only when demand is linear, but
cannot be valid otherwise. In other words, the conclusions drawn by Beckmann
(1976) based on linear demand with regard to the relative economic benefits of the
two simple pricing policies remain valid when the market area is variable.
Nevertheless, the findings of Smithies (1941); Cheung and Wang (1996) in relation
to nonlinear demands need to be re-examined since in this case their common
assumption, namely, that the firm serves the same market area under the two price
policies, cannot be true.
Figures 1a,b depicts two optimal delivered prices and optimal market areas
when demand is concave (convex), that is, v>1(v <1)5. In both figures, the boundary
of the market area under mill pricing is determined by the intersection of Pmax and
Pf, but that under uniform pricing, by the intersection of Pu-c-tx and the x-axis. Note
also that the delivered price at the firm’s site under mill pricing is always smaller
than that under uniform pricing for any value of v.6 Thus, the two optimal delivered
price patterns under alternative pricing policies are depicted as those in Figs. 1a and b.
Finally, we shall examine the effects
of the shape.of the demand function on.prices
.
and market areas. First, since @pf @v ¼ ða=b cÞ ð1 þ 2vÞ2 > 0 , and @pu @v ¼
5 Figures
1a and b are drawn using Microsoft Excel with a=b=1, c=0.5, t=0.1.
see this, note first that the difference
between the
two delivered prices at the firm’s site is
given by pf ð0Þ pu ¼ ða=b cÞ ðvÞ ð1 þ 2vÞ , where pf ð xÞ ¼ mf þ tx = the optimal
delivered price under mill pricing. It follows that pf (0)<pu since v>0.
6 To
Simple monopoly price theory in a spatial market
537
Pf
Pmax
Pu
Pu - c - tx
Bu
DISTANCE
Bf
Fig. 1a Optimal prices and market areas: concave local demand
Pf
Pmax
Pu
Pu - c - tx
Bf
Bu
DISTANCE
Fig. 1b Optimal prices and market areas: convex local demand
.
2ða=b cÞ ð1 þ 2vÞ2 > 0, both delivered prices increase with an increase in the
value of v. Moreover, an increase in v enlarges the market area under uniform pricing
increases, but
mill pricing
since @Bf i=@v ¼
reduces both
the market .area under
h .
2
ða bcÞ= bt ð1 þ 2vÞ < 0; and @Bu @v ¼ ða bcÞ bt 2 ð1 þ 2vÞ2 > 0.
538
S.-k. Hsu
In short, the shape of the demand function plays a significant role in determining the
optimal prices and the market areas under simple monopoly pricing in the spatial
market. Formally,
Proposition 2 As the demand curve becomes less convex or more concave, that is,
the value of v increases, the two delivered prices of simple pricing policies will
become higher, and the market area under uniform pricing will become larger, but
the market area under mill pricing will become smaller.
4 Economic benefits
In this section, the analysis of the economic benefits of the two pricing policies
consists of two parts, the first that in relation to the monopoly output, and, then, that
in relation to the profits, the consumers’ surplus as well as the social welfare. We
shall first examine the output effect of a move from mill pricing to uniform pricing.
Monopoly output under mill pricing is obtained by substituting Eq. 10 into
Eq. 9, and is
ð1=vþ1Þ
:
(18)
Qf ¼ ðv=bt Þð1 þ vÞ1=v ða bcÞ ð1 þ 2vÞ
Similarly, monopoly output under uniform pricing is
1=v
Qu ¼ ð1 bcÞ ð1 þ 2vÞ Bu
(19)
RB
since Qu ¼ 0 u f ðpu Þdx ¼ f ðpu ÞBu and a bpu ¼ ða bcÞ ð1 þ 2vÞ: By combining Eq. 19 with Eq. 16, we obtains
1=v Qu ¼ ða bcÞ ð1 þ 2vÞ
ða bcÞ bt 2v ð1 þ 2vÞ
(20)
ð1=vþ1Þ
ð2v=bt Þ
¼ ða bcÞ ð1 þ 2vÞ
2
CSf /CSu
1
Q f / Q u = π f /π u
SWf / SWu
v
0
0. 2
1
1.8
Fig. 2 Economic benefits and the shape of demand function
2.6
Simple monopoly price theory in a spatial market
539
Thus, dividing Eq. 18 by Eq. 20 yields
.
Qf Qu ¼ ð1 þ vÞ1=v 2
(21)
It is easy to see from Eq. 21 that whether or not mill pricing yields a larger
monopoly output than delivered pricing is a matter that is completely dependent
upon the shape of the demand function, that is, the value of v. Moreover, although it
is easy to show from Eq. 21 that Qf =Qu when demand is linear, i.e. v=1, a
numerical simulation is needed to examine the value of Qf /Qu with a change in v
other than one. Figure 2 depicts the level of Qf /Qu with an increase in v, the value
ranging from 0.2 to 3. It shows that Qf /Qu is greater (smaller) than one if the
demand function is convex (concave), that is, v<1 (v>1). Formally,
Proposition 3 In a world with variable market area, the total output sold under mill
pricing is the same as it would be under uniform pricing if and only if demand is
linear, and the monopoly output must be larger (smaller) under mill pricing than
uniform pricing when demand is convex (concave).
Proposition 3 shows that in the variable market area case, the shape of the
demand curve alone can fully determine whether the output effect of one pricing
policy can occur. In other words, the conclusions drawn by Cheung and Wang
(1996, p. 136, Proposition 3) with the fixed market area assumption in this regard
remains valid in a world with variable market area.
The relative economic benefits of the two pricing policies, in addition to
monopoly output, are evaluated, respectively, by the firm’s profits, consumers’
surplus and social welfare since there are two groups of participants involved in the
market, namely, one firm and a group of consumers. Three measures under mill
pricing are given, respectively, by
πf ¼ ð1 þ vÞð1=vÞ AB
(22)
h
.
i
CSf ¼ ð1 þ vÞð1=vþ1Þ ð1 þ 2vÞ AB
(23)
SWf ¼ ð1 þ vÞð1=vÞ ð2 þ 3vÞ ð1 þ 2vÞ AB
(24)
and
ð1=vþ2Þ
; and B ¼ v2 ðb2 t Þ : The firm’s profit
where A ¼ ða bcÞ ð1 þ 2vÞ
under mill pricing is obtained by substituting Eqs. 10 and 11 in Eq. 4. Equation 23
can be obtained
by substituting
Eqs. 10 and 11 in the consumers’ surplus, namely,
i
R B hR pmax
CS ¼ 0 p f ðτ Þdτ dx; while the level of social welfare is the sum of the
540
S.-k. Hsu
firm’s profits plus the consumer’ surplus, that is, SW ¼ π þ CS: Similarly, these
three measures under uniform pricing are as follows
πu ¼ 2AB
(25)
CSu ¼ 2 ð1 þ vÞ AB
(26)
SWu ¼ 2ð2 þ vÞ ð1 þ vÞ AB
(27)
and
Consider first the profit effect of a move from mill pricing to uniform pricing.
Dividing Eq. 25 by Eq. 22 yields
.
(28)
πf πu ¼ ð1 þ vÞð1=vÞ 2
The RHS of Eq. 28 is exactly the same as that of Eq. 21, and therefore the firm’s
preference between the two simple pricing policies depends completely on the
shape of the demand function, that is, the value of v. Moreover, Fig. 2 can be used
to answer the issue of how the concavity of the demand function determines the
difference in the seller’s profit between two simple spatial pricing policies7. It is
clear from Fig. 2 that πf /πu is greater (smaller) than one when the demand function
is convex (concave), that is, v<1 (v>1), and πf =πu if and only if demand is linear.
Formally,
Proposition 4 In a world of variable market area, the firm’s profits is greater
(smaller) under uniform pricing than under mill pricing when demand is convex
(concave), and both pricing policies have the same profits if and only if demand is
linear.
Proposition 4 shows that these results that are obtained based on the fixed
market area assumption by Smithies (1941); Beckmann (1976); Cheung and Wang
(1996) with regard to the firm’s preference between the two pricing policies remain
valid in a world with variable market area.
In order to make more comparison of the economic benefits between the two
pricing policies, we first divide Eq. 23 by Eqs. 26 and 24 by Eq. 27 to obtain two
ratios for the consumers’ surplus and social welfare, respectively, as follows
.
(29)
CSf CSu ¼ ð1 þ vÞð1=vþ2Þ ½2ð1 þ 2vÞ
7 The data set for Fig. 2 is computed using Microsoft Excel with the value of v ranging from 0.2 to
3, and is available upon request. Moreover, since the values of πf /πu and Qf /Qu are the same, only
one curve is drawn for them.
Simple monopoly price theory in a spatial market
541
and
h
i.
SWf SWu ¼ ð1 þ vÞð1=vþ1Þ ð2 þ 3vÞ ½2ð1 þ 2vÞð2 þ vÞ
(30)
It is clear from Eqs. 29 and 30 that, in terms of consumer surplus or social welfare,
the relative economic benefit of a move from mill pricing to uniform pricing is a
matter that is completely dependent on the shape of the demand function.
Figure 2 depicts the curves relating πf /πu, CSf /CSu, and SWf /SWu, respectively,
to the value of v. It is clear from Fig. 2 that the value of CSf /CSu is greater than one
for all v, that is, CSf >CSu. Formally,
Proposition 5 Despite concavity of the demand function, the aggregate consumer
benefit is always greater under mill pricing than it would be under uniform pricing
in a world of variable market area.
Proposition 5 ensures that in a world with variable market area, the conclusion
made by Beckmann (1976, p. 628) with regard to the consumers’ surplus remains
valid despite the shape of the demand function. In other words, the shape of the
demand curve has no significant impact on the relative aggregate consumer
benefits of the two pricing policies.
The shape of the demand curve nevertheless matters when evaluating the
welfare effect of a move from mill pricing to uniform pricing. The value of SWf /
SWu, as shown in Fig. 2, starts with the value being greater than one when v=0.2,
the value always decreases with an increase in v, and eventually, the value is
smaller than one when v>1.3. In other words, mill pricing yields greater social
welfare than uniform pricing when demand is convex, linear, or less concave, but
uniform pricing is socially superior to mill pricing when the concavity of the
demand function exceeds a certain degree. Formally,
Proposition 6 In a world of variable market area, social welfare is greater under
mill pricing than it would be under uniform pricing when demand is convex, linear,
or less concave, but the opposite result holds for more concave demand functions.
The main reasoning underlying Proposition 6 is that the more concave the
demand function is, the more profits that the monopoly can earn (see Fig. 2), and
therefore, the level of social welfare under uniform pricing is greater than it would
be under mill pricing despite the fact that the latter pricing policy is always socially
superior as far as the aggregate consumers’ surplus is concerned, and the level of
the consumers’ surplus increases with the concavity of the demand function.
Proposition 6 shows that in a world with variable market area, the finding by
Beckmann (1976) with linear demand can be generalized to all convex demands
and some less concave ones, but is no longer true otherwise since uniform pricing
yields a larger social welfare than mill pricing when demand is more concave. As
far as the paper by Cheung and Wang (1996) is concerned, their finding with regard
to the comparison of welfare between the two simple pricing policies are presented
as Propositions 4 (p. 139) and 5 (p. 140), which can be summarized as follows: if
demand is convex (concave), then welfare is greater (smaller) under mill pricing
than welfare under uniform pricing. Thus, Proposition 6 shows that the conclusion
drawn by Cheung and Wang (1996) in this instance for convex demand is not true
for less convex demand in a world with variable market area.
542
S.-k. Hsu
5 Concluding remarks
In this paper, we have provided a detailed exploration of the pricing behaviors and
resultant economic benefits of a simple monopoly under mill and uniform pricing
in a world with variable market area. Proposition 1 confirms the expectation of
Cheung and Wang (1996, p. 142), namely, that the fixed market area assumption is
invalid in a world with variable market area and nonlinear demand. Moreover,
Propositions 3 and 4 ensure those conclusions drawn by Smithies (1941) and
Cheung and Wang (1996) with regard to monopoly output and the firm’s
preference based on the fixed market area assumption can be generalized to the
variable market area case.
This paper has instead shown that the finding presented by Cheung and Wang
(1996) with regard to the social superiority between the two simple pricing policies
as measured by welfare is not true in a world with variable market area (Proposition
6). In other words, the last conclusion reached by Cheung and Wang (1996),
namely, that “the price policy chosen by the firm could be consistent with the
welfare maximization objective” (p. 142) no longer remains valid without the fixed
market area assumption. In addition, this paper has shown that the aggregate
consumers’ surplus is always greater under mill pricing than it would be under
uniform pricing (Proposition 5). Thus, the above-mentioned conclusion reached by
Cheung and Wang (1996) is not true even when the social desirability only takes
consumer benefits into account in simple pricing. Therefore, there is a departure in
terms of the social superiority of simple pricing from the firm’s preference in a
spatial market with variable market area.
This paper has shown that in a world with variable market area, the shape of the
demand function cannot fully determine the social superiority between mill and
uniform pricing. This is in sharp contrast to the finding provided by Cheung and
Wang (1996) with the fixed market area assumption. In other words, the economic
space does matter in determining whether mill pricing is socially superior to its
uniform counterpart in a spatial market.
Acknowledgements The author would like to thank Professor Kim as well as three anonymous
readers of this journal, and Professor Kan of the Institute of Economics at Academia Sinica,
Taiwan for helpful comments; however, he alone is responsible for any remaining errors or
deficiencies in this paper. Financial aids from National Science Council (Taiwan) are gratefully
acknowledged.
Simple monopoly price theory in a spatial market
543
Appendix
1 Appendix
This appendix is to show that the second-order optimal conditions for two
optimization problems of this paper are always satisfied since for the slope of a
demand being negative, the value of v in Eq. 1 is greater than zero.
In the case of mill pricing, the second derivative of the firms’ profit with respect
to the mill price is
@2 π @m2 ¼ 2 @Q @m þ ðm cÞ @2 Q @m2
¼ 2 @Q @m Q @Q @m @2 Q @m2
h (A1)
i 2
¼ 2 @Q @m Q @2 Q @m2 @Q @m
¼ K1 @Q @m
2
where K1 ¼ 2 @Q @m Q @2 Q @m2 ; and the second equality is based on that
the first order condition for profit maximization is @π=@m ¼ Q þ ðm cÞ
R
@Q @m ¼ 0; and thus, m c ¼ Q @Q @m : Since @Q @m ¼ ðb=vÞ
ða bpÞ1=v1 dx < 0; the fulfillment of the second-order
condition,
optimal
in
R
turn, requires that K1>0. Moreover, since @2 Q @m2 ¼ b2 ð1 vÞ v2
ða bpÞð12vÞ=v dx; we have
R
R K1 ¼ 2 ðb=vÞ2 ða bpÞ2=v2 dx b2 v ð1=v 1Þða bpÞð22vÞ=v dx
i2
Rh
¼ ð1 þ vÞ ðb=vÞða bpÞ1=v1 dx
(A2)
Here we see that while v>0, K1>0, and therefore, @2 π=@m2 < 0:
pricing, the firm’s profit can be written as πu ¼ f ð pÞ
In the case 2of delivered
ðp cÞB tB 2 ; and thus, the marginal profit with respect to the delivered
price is
@π @p ¼ f 0 ð pÞ ðp cÞB tB2 2 þ f ð pÞ B þ ðp cÞ t tB=t
(A3)
¼ ½f 0 ð pÞðp c tB=2Þ þ f ð pÞB
¼ ½f 0 ð pÞðtB=2Þ þ f ð pÞB
where the third equality is based on that B ¼ ðp cÞ t: Thus, we have from the
first order optimal condition that
tB=2 ¼ f ð pÞ f 0 ð pÞ
(A4)
544
S.-k. Hsu
The second derivative of the firms’ profit with respect to the delivered price
under delivered pricing is
d 2 πu =dp2 ¼ ½f 00 ðpÞðtB=2Þ þ f 0 ðpÞ=2 þ f 0 ðpÞB þ ½f 0 ðpÞðtB=2Þ þ f ðpÞ=t
¼ f½f 00 ðpÞf ðpÞ=f 0 ðpÞ þ f 0 ðpÞ=2 þ f 0 ðpÞgB þ ½f ðpÞ þ f ðpÞt=t
¼ K2 ½B=f 0 ðpÞ
.
(A5)
where K2 ¼ f 00 ð pÞf ð pÞ þ 3½f 0 ð pÞ2 2; and the second equality is based on Eq.
A4. Note in addition that
K2 ¼ ð3=2Þðb=vÞ2 ða bpÞ2=v2 ð1 vÞðb=vÞ2 ða bpÞð12vÞ=v ða bpÞ1=v
h
i2
¼ ðv þ 1=2Þ ðb=vÞða bpÞ1=v1
(A6)
follows that while v>0, K1>0, and therefore, @2 πu =@p2 < 0 since the term
It B f 0 ð pÞ in Eq. A5 is always negative.
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