who benefits from pricing regulations when ... - Wiley Online Library

The Japanese Economic Review The Journal of the Japanese Economic Association

The Japanese Economic Review Vol. 61, No. 2, June 2010

doi: 10.1111/j.1468-5876.2009.00484.x

WHO BENEFITS FROM PRICING REGULATIONS WHEN ECONOMIC SPACE MATTERS?* HONG HWANG†‡, CHAO-CHENG MAI‡§ and HIROSHI OHTA¶ †National Taiwan University ‡Academia Sinica §Tamkang University ¶Aoyama Gakuin University Reinterpreting Hwang-Mai (AER, 1990) by both simplifying and generalizing their analysis in terms of two key demand parameters representing income and market size, we probe the welfare effects of spatial price discrimination to determine how robust the previous welfare findings in the literature are. Endogenous location matters when a monopolist chooses asymmetric location under different pricing schemes. If he/she remained at the same location, outcomes of fixed and endogenous location models must be analytically the same. Endogenous and different location changes outcomes radically. When non-discriminatory pricing regulations benefit the poor, different transportation burdens mean that the rich become poor, even worse off than the ex-poor, and the society becomes worse off accordingly. JEL Classification Number: L11, L12, R32. jere_484

1.

218..233

Introduction

Casual observation indicates that price discrimination is common in many industries. Baumol (2006) further observes that even in highly competitive markets, firms may have no choice but to practice discriminatory pricing. Robinson’s (1933) classic work launched the formal inquiry into the output effects of third-degree price discrimination.1 Building on the intuition presented by Pigou (1929), she showed that if a monopolist faces two independent linear demands, the adoption of price discrimination would not affect the industry output. Later on, Schmalensee (1981) demonstrated that price discrimination would reduce welfare when demands are linear. He extended his analysis to the general non-linear case and shows that an increase in total industry output is a necessary condition for price discrimination to be welfare improving. The key to Schmalensee’s result is that price differentials in third-degree price discrimination reduce the quantity purchased in some market segments while increasing purchases in other segments, thereby shifting output among buyers and creating a divergence in marginal valuations of a unit of output. This result may lead to income or welfare redistribution associated with changes in output. Note that all the above discussions are predicated on a non-spatial world, however. It is now well recognized that economic space has a significant impact upon the microeconomic theory of pricing. In a seminal paper, Greenhut and Ohta (1972; hereafter GO) constructed a spatial model with a linear demand and demonstrated that a spatial monopolist

*

We are grateful to two anonymous referees for their useful comments and critiques. The third author also wants to express his gratitude for invitation to Tamkang University on their Chair Lecture program, 2006, which has led to the present inquiry among others.

1

The subsequent work includes Greenhut and Ohta (1972, 1976), Formby et al. (1983), Mai and Shih (1984) and Shih et al. (1988).

218 Journal compilation © 2009 Japanese Economic Association No claim to original US government works

H. Hwang, C-C. Mai and H. Ohta: Who Benefits from Pricing Regulations

practicing discriminatory pricing leads to a higher level of total output than does a mill-pricing firm. Along the same line of thinking, Holahan (1975) further proved that a spatial discriminatory pricing generates greater social welfare. By sharp contrast, Beckmann (1976) found that non-discriminatory mill pricing would always provide higher consumer surplus and social welfare than spatially discriminatory pricing. The seeming contradiction between the results of GO (1972)–Holahan 1975) and those of Beckmann (1976) stems from the different treatment of market areas, the former being endogenous and the latter fixed. In any event, their analyses are based on the assumption that the location of the firm is exogenously given. This assumption being somewhat restrictive, however, called for relaxation in the subsequent literature on price discrimination; e.g. Hwang and Mai (1990; hereafter HM), Anderson et al. (1992), Cheung and Wang (1995), and Claycombe (1996). In particular, by endogenizing the firm’s location, HM set up a barbell model with consumers being located at the end-points of an interval to examine the effects of price discrimination on optimal location, output and social welfare. They found that the welfare effect of spatial price discrimination could be positive and the accompanying output effect negative. This negative output effect of discriminatory pricing a la HM should be striking in light of the contrary Schmalensee-Varian consensus that a positive output effect is a necessary condition for discriminatory pricing to be welfare enhancing. Unfortunately, however, HM’s analysis was incomplete inasmuch as their requisite simulation was not predicated upon parameters that would ensure all the assumed markets to be served regardless of either pricing to be adopted, discriminatory or nondiscriminatory. The objective of this paper is two-fold. It first reinterprets the HM findings by both simplifying and generalizing their analysis in terms of two key demand parameters in addition to a distance cost. Proper interpretation of each is crucial to the present inquiry. With proper interpretation and use of these parameters we then proceed to our secondfold objective to remedy HM’s surprising, but incomplete theorem. By using the same simple model of linear demands as in the literature, while remedying HM’s analysis we show that when economic space matters, banning discriminatory pricing can indeed diminish community welfare despite an accompanying positive effect it has on output. Moreover, we will also show a nonempty set of reservation price and distance cost parameters under which pricing regulations could mean that the rich become poor, even worse off than the poor, who are made better off. An intriguing feature of the HM model is that it is predicated upon the same number of separate markets with linear demands that a monopolist faces regardless of the two alternative pricing methods to be employed by the firm. Under such well-defined, specific conditions, the monopolistic discriminatory pricing yields no output effects over simple non-discriminatory pricing, but the counterpart welfare effects are necessarily negative regardless of distance costs according to the prior literature aforementioned. But HM have shown that if location of the firm is endogenous, then the welfare effect of discrimination may become positive, while the related output effect of discrimination negative. Their findings on both output and welfare effects are thus in sharp contrast, almost contrary, to the literature. In this connection endogenous location matters when the firm can choose relocation should non-discriminatory pricing regulation be imposed upon it. If it remained at the same location, outcomes of fixed and endogenous location models must be analytically the same. With these preliminary observations above we set forth in what follows our basic model in Section II, followed by Section III, which classifies four alternative combinations 219 Journal compilation © 2009 Japanese Economic Association No claim to original US government works

The Japanese Economic Review

of the monopolist’s pricing and location for the subsequent welfare analysis of pricing under endogenous location in Section IV. Section V probes deeper into the related questions of what would happen to individual as well as social welfare, or who, either the poor or the rich in particular, would benefit from pricing regulations under what conditions when economic space matters. Section VI concludes.

2.

The model

Our model of the paper is similar to the barbell model developed by HM.2 Consider a monopolist facing two separate markets, which are a fixed distance, say, one mile, apart. The market demands at the two ends of the barbell model, called hereafter nodes 1 and 2, are as follows:

p = 1 − q,

(1)

p = a (1 − q n) ,

(2)

and

where a, reservation price, is a measure for income and n for market size at node 2 relative to that at node 1. These demand specifications are derivable from a certain particular quasi-linear utility function for a representative consumer.

U ( y, q; y0, q0 ) = y + ay0 q − ( ay0 2q0 ) q 2, a > 0,

(3)

where y is the income remaining or saved after a given income y0 is spent on q (or anything else, albeit assumed away herein for simplicity) whereas q0 is the amount of q that will turn its marginal utility to zero. Maximizing U subject to the budget constraint y0 = y + pq yields the linear demand of the form p = ay0(1 - q). Normalizing y0 to unity yields (2) above for one person, i.e. n = 1.3 The demand in (1) becomes equivalent to that in (2) if we further assume a = 1 and n = 1. Note that Equations (1) and (2) above are not only simpler than HM’s counterpart equations: with fewer parameters without sacrificing any generality, the present model is also subtly different from HM’s intrinsically. Note in particular that there are only two parameters, a and n, in our linear demand specification. The parameter a is used to represent the income disparity between the two markets. Thus, if a is greater (smaller) than unity, it implies

2

The barbell model was subsequently employed by Gross and Holahan (2003) and Liang et al. (2006).

3

The present demand specifications are no more fully general than the constant elasticity demand derivable from the conventional constant elasticity of substitution utility function, but the former proves to be more realistic in the senses to follow. First, the demand for anything in general must vanish if price becomes high enough. On the other extreme of vanishing price, one would not increase consumption of anything endlessly even if it became a free good. Both of these observations are incorporated in the linear demand along with the variable price elasticity. Second, the linear demand specifications above require the income effect to be positive for strictly positive prices only. When price approaches zero the income effect must vanish accordingly. The present linear demand model is also subtly different from Tabuchi-Peng’s (2007), which assumes consumers with no effective budget constraint.



that the consumer at node 2 is richer (poorer) than at node 1. Regarding the other parameter n, note that we can measure the consumer population or market size at node 2 as relative to node 1 by n in equation (2) insofar as consumers are identical in preference. If this value is equal to unity, then there is only one consumer (or the same number of consumers normalized to unity) in each market node and the market size in both markets must be the same accordingly. If this value exceeds unity so that n > 1, then population at node 2 must be larger than at node 1, and vice versa.4 We henceforth define the demand at market node 2 to be poor (rich), and small (large) in comparison to that at node 1 according as a ) 1 and n ) 1.5 As proved by HM, the barbell model yields a corner solution for location under either mill pricing or discriminatory pricing. In any case, consumers are assumed to either travel to the firm’s plant or stay home to receive delivery. However, in case of mill pricing all consumers pay the same free on board (FOB) mill price.6 In case of discriminatory pricing, by comparison, consumers at different locations pay different mill prices.7 Hence, we can analyze a monopolist’s optimal pricing policy while assuming its location at market node i. While also assuming cost of production to be nil for simplicity, we can readily calculate the sales proceeds (= revenue = profit, net of freight costs tq, where t is freight rate, and assumed constant) from each market and aggregate profits under alternative, either simple monopoly or discriminatory monopoly pricing policies, given that the firm locates either at market node 1 or 2. Hereafter simple monopoly and discriminatory monopoly will be called SM and DM, respectively.

3.

Four alternative cases of pricing and location

Four alternative combinations of these pricing and location policies yield respective profit outcomes as follows. We initially consider two cases of discriminatory monopoly, DM, locating at either node 1 or node 2 as Case (1) and Case (2).

4

If the number of consumers at node 1 is n1 and that at node 2 is n2, then node 1 will have the inverse market demand of the form p = 1 - q/n1 and node 2 will have p = 1 - q/n2. Accordingly, for any given price, the quantity demanded of the demand at node 1 is greater (or smaller) than that at node 2 where n1 > (or or < 1.

5

We define n here as a continuous and non-negative real number including a non-integer such as 0.9. Note in a related vein that the market size defined here is subtly different from HM’s. For example, according to HM, demand is small if the reservation price is small, but not necessarily here. Even if the parameter a is very small, the poor market can be large if the demand slope is flat enough or a/n is small enough.

6

FOB mill pricing requires buyers to pay a unique mill price plus other relevant costs including freight and insurance. This total cost, called cost, insurance and freight (CIF) price, increases directly with the costs of distance under uniform FOB mill pricing. Two related notes are in order: (i) The fact that buyers pay freight under mill pricing does not require them to visit the mill. Home delivery is acceptable and also prevalent under mill pricing. (ii) Although buyers pay full freight under FOB mill pricing, the firm absorbs part of it by reducing mill price depending on its demand elasticity. This implies both the firm and consumers to bear part of freight even under FOB mill pricing. See Greenhut (1956), and Ohta (1988) for more details on spatial pricing techniques and related analysis.

7

In a related vein uniform CIF pricing, under which mill price varies inversely with distance, is of particular importance, empirically as well as theoretically (a la Greenhut, 1956, 1981; Kats-Thisse, 1993; Ohta-Lin-Naito, 2005, etc.), albeit-non-sequitur to the present inquiry.



Case 1: DM at Node 1 The maximum profit p11 from node 1, when the firm locates at node 1, is:8

π11 ≡ max ( p1q1 − cq1 ) = 1 4 , c = 0. The maximum profit p21 from node 2, given the firm’s location at node 1, is:

π 21 ≡ max ( p2 q2 − cq2 − tq2 ) = n ( a − t ) 4a , c = 0, t > 0, 2

where freight rate t is required to be less than the reservation price a for DM to make any (non-negative) sales to the node 2 market. Hence the aggregate profit of DM locating at node 1 is: 2 π1 ≡ π11 + π 21 = ⎡⎣a + n ( a − t ) ⎤⎦ 4a , a > t.

(3)

Case 2: DM at Node 2 The aggregate profit given the firm locating at node 2, by comparison, is: 2 2 π 2 ≡ π 22 + π12 = na 4 + (1 − t ) 4 = ⎡⎣na + (1 − t ) ⎤⎦ 4 ,

(4)

where t < 1 is required (hence assumed) for obvious reason. To see which location is better for DM, we may calculate the difference between p1 and p2. Thus,

Δπ ≡ π1 − π 2 = t [( n − a ) t + 2a (1 − n)] 4a 2−t , a > t > 0. ∴ Δπ > 0 ⇔ n < n ( a; t ) = 2−t a

(5)

The above inequalities reveal the following relations. If a = 1, then the inequality (5) is satisfied if and only if n < 1. This implies that when the two markets are equally rich, the DM firm must locate at the large market. (With n < 1, node 1 is the large market.) If n = 1, then similarly from (5) a < 1. Given equal sizes, a < 1 requires the DM firm to locate at the rich market. (With a < 1, node 1 is the rich market.) More generally, derived from above are the trade-off combinations of a and n that yield Dp = 0, or n = (2 - t)/(2 - t/a). This trade-off relationship can be represented by the solid hyperbola of Figure 1. Note that while the hyperbola shows the combinations of (n, a) such that the DM firm is indifferent to its location at either node, the shaded area points to the set of a and n that yields Dp > 0. The profitable location is node 1, if the combination of (a, n) falls within the shaded area and is node 2, if otherwise. Given this area under the hyperbola, no matter how large the income, a, at node 2, there exists a small enough n that will keep the DM firm from locating there.

8

Also assumed here are that fixed costs of locating at either node are high enough to keep the firm from locating also at the other node. Such fixed costs, though not explicitly included herein for analytical simplicity, are required to exceed nt(a - t)/2a so as to exclude double location.



F IGURE 1.

Combinations of (a, n) yielding discriminatory monopoly’s profitable location at node 1

This implies that the price intercept (or income disparity) matters, in the sense that the DM should locate at a richer market provided the market sizes are equal, i.e. the number of consumers at node 2 being n = 1. This is not to say, however, that the market size does not matter. It in fact does. But even if market size at node 2 is strictly larger than at node 1 such that n > 1, the DM will not locate at node 2, if that market is poor enough with a small enough a to approach t, unless n is large enough to exceed 2 - t.9 Another interesting relationship that can readily be seen from Figure 1 is that, an increase in freight rate t tends to twist the hyperbola around the point (a = 1, n = 1), as shown by a dotted curve in the Figure. This implies that for a poor (rich) market at node 2 (node 1), the larger the freight rate t, the larger (smaller) the minimum required size of the market, n, that would entice the DM there. Thus, for example, if a < 1, then the minimum market size required for the DM to be enticed therein should increase with t. But if a > 1, then the minimum required size decreases as t increases. Impacts that economic space (freight cost) has on the poor market and the rich market may appear to be asymmetric, but they are not really.10 We now turn to the analysis of simple monopoly, SM. In the case of SM, the market size does not matter in location optimization; the reservation price or the richness of the market does not matter, either. The SM firm locates at a market node having a flatter demand provided that both markets are served.11 While this is HM’s most intriguing discovery, we attempt to provide an alternative interpretation that is simpler with a stronger intuitive appeal. We begin with Case (3), followed by Case (4).

9

If it happens to exceed this critical value, then the DM will locate at node 2, the (a, n) combinations lying outside the shaded area of Figure 1.

10

The poor market, at node 1, requires a greater population to entice the DM there when t increases. With a > 1, it is the node 2 that is the rich market. Then it requires lesser population to entice the DM when t increases.

11

As shown by Thisse (1993) this result of “end-point” location, may hold only under some restricted conditions such as on demand convexity in price, freight in distance.



F IGURE 2.

Simple monopoly’s aggregate average revenue when local demands are parallel

Case 3: SM’s Location when Demands are Parallel in Both Markets Consider for a departure point analysis SM’s aggregate market demand, or more precisely, average revenue (AR) when demands at both nodes 1 and 2 are parallel. These parallel demands, denoted AR1 and AR2, respectively, are now defined in light of Equations (1) and (2) and in consideration of freight rate t as follows: AR1: p = 1 - t - q1, t > 0 if located at node 2, t = 0 if at node 1. AR2: p = a - t - q2, t > 0 if located at node 1, t = 0 if at node 2.

Summating these ARis horizontally yields the aggregate average revenue AR defined over the proper domains of Q (= q1 + q2) as follows: AR: p = 1 - t - Q, 0 < Q < 1 - (a - t), if 0 < Q < 1 - (a + t), if = (1 + a)/2 - t - Q/2, 1 - (a - t) < Q < 1, if 1 - (a + t) < Q < 1, if

located located located located

at at at at

node node node node

1; 2. 1; 2.

Figure 2 illustrates the AR under consideration. It depicts two parallel local average revenues, ARis, before and after freight rate t is subtracted, and the aggregate AR as the horizontal sum of the net ARis, net of freight t. In the Figure, note ki stands for a “kink” that will appear on an aggregate AR at an intersection with local demand, either AR1 or its net counterpart, net of freight rate t. Specifically a “kink” k1 appears on AR1 by adding net AR2, net of freight t, horizontally. The horizontal sum of AR1 and net AR2 thus obtained represents aggregate AR that the SM can obtain by locating at node 1, while generating a kink at k1. Likewise the horizontal sum of net AR1 and AR2 thus obtained represents aggregate AR that the SM can obtain by locating at node 2, thereby generating a kink at k2. It should be obvious then that SM’s aggregate ARs will be identical over the relevant overlapping domain, despite their separate kinks, insofar as the ARis are parallel. Therefore profits are the same regardless of its location at either node 1 or node 2 provided that both markets have parallel local ARs and are served under SM. Of these two provisos, the latter one that both markets are served can be shown to hold under the following inequality condition:12 12

1, 2 1, 2 1 > π SM This follows from the requirement of π SM , where π SM is the total profits under SM from both market 1 nodes 1 and 2, and π SM the local profit from node 1 alone that is the larger (as well as richer) market.



F IGURE 3.

Simple monopoly’s aggregate average revenue when demand at node 2 is flatter

a > a ( n; t ) =

n . ( n + 1 − t )2 − 1

(6)

The former proviso, by comparison, requires the slope of the demand at node 2 to be equal to that at node 1, i.e. n/a = 1. Combining these two provisos requires: n > 2 − 1 + t . What this requirement means is: when the two (linear) demand functions under consideration are parallel, the SM firm is indifferent in choosing either the large market or the small market node and makes sales to both markets, subject, however, to this size requirement. Otherwise, i.e. if the small market is small enough, then that market must be abandoned.13 Case 4: SM’s Location when Demand at One Node is Flatter We now relax one of the two provisos above and suppose that the demand at node 2 is flatter than that at node 1. This condition yields a dotted AR in Figure 3, and a related aggregate AR in a dotted line accordingly. Note, however, that the dotted AR passes a “kink” at k2, not k1. This is because, by HM, SM locates at node 2, i.e. with flatter demand. What if it located at node 1 instead, having steeper demand? Then the aggregate AR will pass a “kink” at k1, yielding a dotted line parallel to the one passing k2. As the slope of SM’s aggregate AR must remain the same regardless of its location, it immediately follows that the AR passing through k2 must stay strictly above that passing through k1. Alternatively put, the former AR’s reservation price must be strictly higher than the latter’s. The conclusion above holds generally regardless of which node may be flatter, richer, or larger in size. The SM will be better off by locating at a market having flatter demand provided that both markets are served. Remember that this proviso requires n > (1 + n/a)1/2 - (1 - t) as shown in (6), where n/a is the reciprocal of the demand slope at node 2. Within the confines of this proviso suppose, for example, the demand at node 2 is as flat as with

13

When node 2 is the richer market with a > 1, the counterpart requirement becomes π 2SM > π 22SM , which, combined with n = a > 1, yields n = a < (1 - t) + (2 - 4t + t2)1/2. This implies that unless the large market is small enough the SM will locate at the rich node 2 and serve that node alone, abandoning node 1.



F IGURE 4.

Income a, market size n, and demand slope a/n parameters for simple monopoly (SM) and discriminatory monopoly (DM) to locate at different nodes

a/n = 1/3, then n is required to be larger than 1, or a > 1/3, ignoring t. That is, a at node 2 needs to be at least as large as 1/3 the reservation price at market node 1. If smaller, then SM will locate at node 1, and serves node 1 alone. These demand-parameter conditions for SM location being combined with those for DM location given by (5) supra, in turn, yield the Figure 4 representation of the general relations between a and n that yield SM and DM location at either node 1 or 2. In the Figure an area represented by (SMi, DMj) indicates SM location at node i, and DM location at node j. Thus, for example, the shaded area surrounded by (5), (6) and the 45° (n/a = 1) line show the parameter combinations that would entice SM to locate at node 2, and DM at node 1.14 An asterisk in the Figure refers to SM making sales only to the single market node where it is located.15 We now summarize the main features of our analysis above as Proposition 1 by focusing on a well-defined subset of parameters, normalizing the reservation price at node 1 as unity while assuming node 2 to be either the poor market or the rich market. 14

Figure 4 assumes away the case for a < t, which case requires SM located at node 1 to make no sales to node 2, but it does not require either DM or SM to locate there at node 2. If n is large enough, then not only DM but also SM may locate indeed at node 2 and make sales to both nodes. The parametric conditions under which DM will locate at such an extremely poor market is given by a rectangular hyperbola n > (2 - t)t/a, which is not shown in Figure 4, however, to avoid optical bruise. If both a and n are small enough, then even DM would make no sales to such an unattractive market node.

15

Under the parameter conditions a/n < 1 the demand at node 2 is flatter than at node 1, but it won’t entice SM to locate there unless its market size is large enough. If not, i.e. if n < (1 + (n/a))1/2 - (1 - t), then SM should rather locate at node 1, and abandon node 2. Figure 4 also assumes away the case in which a and/or n are large enough to yield location at node 2 with no sales at node 1. The parameter combinations (n, a) that would lead to SM2* is given by a > (1 - t)n + 2(1 - t)2 + (1 - t)3/n, which could be superimposed upon Figure 4, but is omitted for brevity.



Proposition 1: There exists a nonempty set of parameters (a, n; t) that would entice DM and SM to locate at different nodes while making sales to both nodes. The parameters set under consideration is given by the relations n > a, (5) and (6), and illustrated by Figure 4 as areas (SMi, DMj), i ⫽ j.

4.

Welfare comparisons, given SM or DM firm’s optimal location

We are now in a position to examine the welfare impact of SM and DM. Henceforth we assume, albeit not needed, that the market at node 2 is poorer than at node 1, i.e. a < 1. We further assume that node 2 is not large enough as to entice DM to locate there. This implies that even if a/n < 1, the DM will locate at node 1, but SM will locate at node 2. Note we are now interested in the shaded area in Figure 4 that yields the outcome of asymmetric location with (SM2, DM1) rather than (SM1, DM2). With this in mind we now calculate the profits and consumer surplus (CS) from the two markets and the social welfare under DM locating at node 1, as follows:

π11DM = 1 4 , CS11DM = 1 8 π12DM = ( a − t ) 4b , CS12DM = n ( a − t ) 8a 2

2

3 ⎡ n (a − t ) ⎤ 1+ ⎥ 8 ⎢⎣ a ⎦ 2

∴W1DM ≡ π11DM + π12DM + CS11DM + CS12DM =

where the subscripted numbers refer to the plant location chosen and the superscripted numbers the market sites; and a > t again is a necessary condition for DM to serve both markets while locating at node 1. Similarly, the profit and consumer surplus and welfare under SM, by comparison, locating at node 2, assuming a/n < 1, are given by:

π 2 SM =

[ n + (1 − t )]2 4 (1 + n a )

CS21SM =

[(1 + 2n a ) (1 − t ) − n]2 2 8 (1 + n a )

CS22SM =

n [ n + 2a − (1 − t )] 2 8a (1 + n a )

2

∴W2 SM = π 2 SM + CS21SM + CS22SM [ n + (1 − t )] [(1 + 2n a ) (1 − t ) − n]2 n [ n + 2a − (1 − t )]2 = + + 2 2 4 (1 + n a ) 8 (1 + n a ) 8a (1 + n a ) 2



The ranking of the two pricing schemes in terms of social welfare is therefore given by DW ≡ W1DM - W2SM:

3 ⎡n a + ( n − tn a ) ⎤⎦ ΔW = ⎣ 8n a 2 2 2 ⎡ [ n + (1 − t )] [(1 + 2n a ) (1 − t ) − n] n [ n + 2a − (1 − t )] ⎤ + + −⎢ ⎥. 2 2 8 (1 + n a ) 8a (1 + n a ) ⎣ 4 (1 + n a ) ⎦ 2

By assuming that both markets are equal in size, i.e. n = 1, while also assuming a < 1, the welfare differential above is reduced to:

ΔW =

{

}

1 2 2 ⎡(1 + a ) 3a + 3 ( a − t ) − 4a2 − a {(1 − t ) ( 2 + 3a ) + a} 2 ⎣ 8a (1 + a ) {(1 − t ) ( 2 + a ) − a} + a2( 2 − t )2 ⎤⎦ .

Letting DW ⱖ 0 yields the following inequality:

(a − t ≥ τ ( a ), τ ( a ) ≡

)

3a − 3a3 (1 − a ) 3a + a − 3 2

, a < 1.

(7)

Note this is a necessary condition for the aggregate welfare under DM to exceed that under SM. Also note that the requirement (6) that both markets are served by SM or DM now reappears as (8) below when n = 1:

t ≤ T (a) , T (a) ≡ 2 − 1 + 1 a .

(8)

Combining (7) and (8), while assuming a < 1, thus yields the following proposition and a related corollary as a special case for n = 1 and a < 1, i.e. node 2, the poor market, has a flatter demand, but the same population as node 1 does. Proposition 2: There exists a nonempty set of parameters (a, t; n = 1) that would entice DM and SM to yield strictly higher welfare under DM than under SM. Corollary to Propositions 1 and 2: There exists a nonempty set of parameters (a, t; n = 1) that would entice DM and SM to locate at different nodes and yield strictly higher welfare under DM than under SM. The parameters set for the corollary above is given by (7) and (8), and illustrated by the shaded area of Figure 5. Figure 5 shows that welfare is necessarily lower under DM than SM if the transport rate is nil. This confirms the findings of Beckmann (1976), Schmalensee (1981) and Varian (1985). However, the ranking could be reversed if the transport rate becomes positive. In general, the welfare under DM is higher than that under SM if the value of a is close to 228 Journal compilation © 2009 Japanese Economic Association No claim to original US government works


F IGURE 5.

The parameters set (a, t; n = 1) yielding strictly higher welfare under discriminatory monopoly than under simple monopoly

1 (i.e. the income of node 2 is close to that of node 1) and this critical value of a increases along the t = T(a) curve in Figure 5 as t increases.16 This confirms the HM simulation results, and moreover shows more generally and analytically (and diagrammatically as well) the parameter conditions (a, t) under which DM yields greater aggregate welfare than SM does. For example, when a is large enough and close to unity, even a very low freight rate is sufficient to make DM superior to SM in aggregate welfare. It is in this sense that endogenous location matters when a monopolist chooses different locations depending upon different pricing policies either freely chosen or compulsory. If they choose to remain at the same location, outcomes of fixed and endogenous location models must be analytically the same. The robustness of the GO–Holahan welfare findings is thus confirmed under endogenous location. So much for aggregate welfare comparisons based on the above fairly general formula, albeit not fully general. We now ask under what conditions the poor consumers may gain from non-discriminatory pricing. Remember that in the absence of economic distance costs, the poor are invariably better off under DM than under SM provided that demand from the poor is more elastic than that from the rich so that a < 1. With this proviso the poor can never benefit from simple monopoly.

5. Would the poor benefit from non-discriminatory mill pricing when economic space matters? We now ask if the poor may benefit from non-discriminatory mill pricing rather than discriminatory pricing when economic space matters. We find that the poor can indeed 16

However, as long as t remains small enough, the critical value of a increases along the t = t(a) curve of Figure 5 as t decreases. When t becomes low enough to be less than 0.2, the peak value of t(a), even a very small income differential between the two markets or a very high a would suffice for DM to yield higher welfare than SM does. Conversely if a is small enough, then t could be either too low or too high to yield this welfare outcome. When t is low enough to fall below t(a) both markets may be served by both DM and SM, but with strictly less welfare under DM than SM. When t is high enough to exceed ⌻(a), then it becomes impossible for both markets to be served by SM. 229 Journal compilation © 2009 Japanese Economic Association No claim to original US government works


F IGURE 6.

Mill price under simple monopoly may be lower than full price under discriminatory monopoly, i.e. a*/2 < (a + t)/2, when freight t is large enough

benefit from non-discriminatory pricing if the distance cost t is large enough to turn the imaginary price intercept value of the SM firm’s perceived aggregate demand a* low enough relative to the poor market’s reservation price a. It follows that if SM locates its plant at the poor market, given its demand is flatter than that at the rich market with a/n < 1, then the SM mill price there at node 2 can be strictly lower than the counterpart price charged by the DM firm locating at node 1. This is a full price that the poor located t distance away from the firm incur under DM, and is defined as a sum of DM’s optimal mill price for the poor market, i.e. (a - t)/2, and full freight t borne by the poor. The full price the poor bear under DM becomes (a + t)/2 accordingly. Figure 6 illustrates how mill price a*/2 under SM may become lower than this full price under DM at node 1 when t is large enough relative to small a, viz., t > (1 - a)/ (2 + n/a). Note here that while the optimal SM mill price is half the imaginary reservation price a* of the aggregate demand, the optimal DM mill price applied to the poor market is half the net reservation price, net of freight rate t, i.e. (a - t)/2, but the full price that the poor bear under DM is (a - t)/2 + t = (a + t)/2. Thus, the poor can benefit from non-discriminatory mill pricing regulations if only t is high enough so that (a + t)/2 exceeds a*/2, which relation is readily reduced to:

t > t (a) , t (a) ≡

1− a , 2 +1 a

(9)

where t(a) is a concave function of a similar to, but strictly less than, t(a) of Figure 5. Superimposing t(a) upon Figure 5 yields Figure 7, where the dark gray area under t(a) is the complement of the set given by (9) above, i.e. t ⱕ (1 - a)/(2 + 1/a). Note that this dark gray area is strictly smaller than the area under t(a) that yields a higher aggregate welfare under SM than DM. This implies that as long as t remains small enough below t(a), the poor will benefit from DM, or lose from SM even though the rich and the society as a whole benefit from SM. But within a range between t(a) and t(a) there exist combinations of a and t under which the poor benefits from SM, not DM, at the same time that society also benefits from SM. 230 Journal compilation © 2009 Japanese Economic Association No claim to original US government works


F IGURE 7.

Welfare under simple monopoly: the poor versus the rich

An interesting question is if this implies that given (9), the poor become strictly better off under non-discriminatory SM mill pricing than under DM, then what causes the aggregate surplus to decrease under SM as compared to DM? Remember that given the linear demand functions and in the absence of economic distance costs, it is the rich who are the sole winners of the SM regulation while both the poor and the firm are the losers. The rich indeed are big winners under SM whose gains outweigh the loss of both losers so that the aggregate community surplus increases under SM. The big winners under SM, however, turn into big losers when transportation costs induce the firm upon SM regulations to move away from the rich market to the poor market. If t is large enough to turn the net reservation price of the rich lower than the poor market reservation price a, i.e. (1 - t) < a, then the rich effectively become poorer than the poor! Thus, the biggest losers of non-discriminatory pricing regulations, when transportation costs are significant, are the rich. This is because when freight rates are high enough to entice SM to locate at the poor market, the SM price would be pushed up higher than it would be if the firm remained at the rich market. The rich have to pay the freight rates that would have been nil should the SM have located at their site. It is thus at the cost of the rich who would have benefited from SM either in the absence of distance costs or if SM located at their site but actually lose so big that the aggregate welfare subject to SM regulations shrinks, strictly below welfare obtainable under DM as implied by the HM simulation. Given the firm’s location at the poor site, the rich become less rich, and can become even worse off than the poor, provided that the transportation costs are large enough. The firm’s location or relocation so as to save transportation can mean that the rich become poor, even poorer than the poor under conditions of SM regulation. Thus while the poor do benefit from SM pricing when transportation costs are large enough, they do so at a huge cost to the rich and also to the society as a whole.

6.

Conclusion

Reinterpreting Hwang-Mai (1990) by both simplifying and generalizing their analysis in terms of two key demand parameters representing income and market size, in addition to a 231 Journal compilation © 2009 Japanese Economic Association No claim to original US government works


distance cost, this paper probed the welfare effects of spatial price discrimination to determine how robust the Greenhut-Ohta (1972)–Holahan (1975) welfare findings are. Endogenous location matters when a monopolist chooses asymmetric location. If he/she remained at the same location regardless of pricing techniques to be chosen, outcomes of fixed and endogenous location models must be analytically the same as the findings by Beckmann (1976) that non-discriminatory mill pricing provides strictly higher social welfare than does spatial discriminatory pricing. Beckmann’s spatial results are analytically the same as the non-spatial counterpart results by Schmalensee (1981) and Varian (1985). In sharp contrast the present paper shows that the Hwang-Mai findings (1990) on positive welfare effects of spatial price discrimination under endogenous and asymmetric location are fairly robust. When a monopolist is allowed to freely choose the location of his plant the optimal location will depend on the pricing policy to be chosen and the demand parameters assumed. If the demand parameters are such that the monopolist is enticed to locate at the same market node regardless of the pricing policy chosen, then the welfare effect of discriminatory pricing must be negative, his market size being fixed. This confirms the spatial as well as non-spatial results shown by Beckmann, Schmalensee and Varian. However, if the monopolist were regulated to practice non-discriminatory mill pricing, and if he/she would relocate moving away from the location he/she would choose if allowed to practice discrimination, then such regulation is most likely to reduce social surplus. Indeed we find a fairly general parameter set under which the welfare effect of discrimination proves positive, and that of mill pricing negative. In a related vein, we also pondered what would happen to individual welfare, or who would benefit from pricing regulations under what conditions when economic space matters. We found in particular that the poor rather than the rich tend to benefit from SM pricing when transportation costs are large enough. But they do so at a huge cost to the rich upon SM relocation from their market to the poor’s market. If the policy objective is to protect the poor, then SM pricing should never be adopted unless transportation costs are large enough. In contrast, if transportation costs are large enough, then mill pricing may be enforced to protect the poor but at a cost of the loss of community welfare, in addition to a huge loss the rich incur. In sum the present paper has presented an analysis of the spatially separate markets for the monopolist in terms of a simple, yet general model with linear demand functions. Our findings, following and extending Hwang-Mai (1990), challenge not only the Pigou (1929)–Robinson (1933) findings on output effects of discrimination, but also the Beckmann (1976)–Schmalensee (1981) findings on welfare effects of price discrimination, all based on linear demand functions with different reservation prices. Final version accepted 8 February 2009.

REFERENCES Anderson, S., P. De Palma and J. F. Thisse (1992) “Social Surplus and Profitability under Different Spatial Pricing Policies”, Southern Economic Journal, Vol. 58, No. 4, pp. 934–949. Baumol, W. (2006) Regulation Misled by Misread Theory. Perfect Competition and Competition-Imposed Price Discrimination, 2005 Distinguished Lecture AEI-Brookings Joint Center for Regulatory Studies, Washington DC: The AEI Press. Beckmann, M. J. (1976) “Spatial Price Policies Revisited”, Bell Journal of Economics, Vol. 7, pp. 619–630.


H. Hwang, C-C. Mai and H. Ohta: Who Benefits from Pricing Regulations Cheung, F. K. and X. Wang (1995) “Spatial Pricing Discrimination and Location Choice with Non-uniform Demands, ”Regional Science and Urban Economics, Vol. 25, pp. 59–73. Claycombe, R. J. (1996) “Mill Pricing and Spatial Price Discrimination: Monopoly Performance and Location with Spatial Retail Markets”, Journal of Regional Science, Vol. 36, No. 1, pp. 111–127. Formby, J. P., S. K. Layson and W. J. Smith (1983) “Price Discrimination, Adjusted Concavity, and Output Changes under Conditions of Constant Elasticity”, Economic Journal, Vol. 93, pp. 892–899. Greenhut, M. L. (1956) Plant Location in Theory and in Practice, Chapel Hill: University of North Carolina Press. Greenhut, M. L. (1981) “Spatial Pricing in the USA, West Germany and Japan”, Economica, Vol. 48, pp. 79–86. —— and H. Ohta (1972) “Output under Alternative Spatial Pricing Technique”, American Economic Review, Vol. 62, pp. 705–713. —— and —— (1976) “Joan Robinson’s Criterion for Deciding Whether Market Discrimination Reduces Output”, Economic Journal, Vol. 82, pp. 96–94. Gross, J. and W. L. Holahan (2003) “Credible Collusion in Spatially Separated Markets”, International Economic Review, Vol. 44, pp. 299–312. Holahan, W. L. (1975) “The Welfare Effects of Spatial Price Discrimination”, American Economic Review, Vol. 65, pp. 498–503. Hwang, H. and C. C. Mai (1990) “Effects of Spatial Price Discrimination on Output, Welfare, and Location”, American Economic Review, Vol. 80, pp. 567–575. Kats, A. and J. F. Thisse (1993) “Spatial Oligopolies with Uniform Delivered Pricing”, in H. Ohta and J. F. Thisse (eds), Does Economic Space Matter? Festschrift to Honor Melvin L. Greenhut, London: McMillan Press and New York: St. Martin’s Press. Liang, W. L., H. Hwang and C. C. Mai (2006) “Spatial Discrimination: Bertrand vs. Cournot with Asymmetric Demands”, Regional Science and Urban Economics, Vol. 36, pp. 790–802. Mai, C. C. and J. J. Shih (1984) “Output Effect of the Labour-Managed Firm Under Price Discrimination”, Economic Journal, Vol. 94, pp. 931–935. Ohta, H. (1988) Spatial Price Theory of Imperfect Competition, C. S. Texas: Texas A. & M. University Press. Ohta, H., Y. S. Lin and M. K. Naito (2005) “Spatial Perfect Competition: A Uniform Delivered Pricing Model”, Pacific Economic Review, Vol. 10, pp. 539–556. Pigou, A. C. (1929) The Economics of Welfare, 3rd edn. London: Macmillan. Robinson, J. (1933) The Economics of Imperfect Competition, London: Macmillan. Schmalensee, R. (1981) “Output and Welfare Implications of Monopolistic Third Degree Price Discrimination”, American Economic Review, Vol. 71, pp. 242–247. Shih, J. J., C. C. Mai and J. C. Liu (1988) “A General Analysis of the Output Effect Under Third-Degree Price Discrimination”, Economic Journal, Vol. 98, pp. 149–158. Tabuchi, T. and S-K. Peng (2007) “Spatial Competition in Variety and Number of Stores”, Journal of Economics and Management Strategy, Vol. 16, pp. 227–250. Thisse, J.-F. (1993) “The Optimal Location of a Firm: Mill Versus Uniform Delivered Pricing”, in H. Ohta and J. F. Thisse (eds), Does Economic Space Matter?, Festschrift to Honor Melvin L. Greenhut, London: McMillan Press and New York: St. Martin’s Press. Varian, H. R. (1985) “Price Discrimination and Social Welfare”, American Economic Review, Vol. 75, pp. 870–875.
