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Cournot and Bertrand Competition in a Model of Spatial Price Discrimination with Differentiated Products

Chia-Hung Sun

Abstract

This study investigates spatial price discrimination with two types of market competition – price competition and quantity competition – and two kinds of cross-relations between goods – substitutes and complements – with endogenous location choices in a barbell model. The results herein present that the maximum differentiation (end point agglomeration) is the unique location equilibrium with substitutes (complements), irrespective of what type of competition. We demonstrate that if the unit transportation rate is sufficiently high, then consumer surplus, profits, and social welfare are higher under price competition than under quantity competition for both substitutes and complements. This means that introducing a spatial barrier to competition generated through transportation costs may solve the problem of inconsistency from the conflict interests between consumers, firms, and a social planner.

JEL Classification: D43; L22; R32

Appendix

Proof of Proposition 3

The derivations of the case of complements are omitted, because the results are directly implied by the non-spatial Singh and Vives (1984) model, in which each firm’s constant marginal cost is assumed to be the same. In the case of substitutes, direct calculations yield:

πjCπjB=2c322c2t+2ctct2(4c2)2(1c2),

where the function πjCπjB is easily shown to be concave in t, for j=A,B. Solving for πjCπjB=0 yields the following two roots:

t=1+c1c2c<0andt=1+c+1c2c>t2.

It follows that πjC>πjB for all t<t2. With a straightforward derivation, we yield:

πjCπ1L=(2+c+2t)8+8t+4c+2c22c2t+c3c4+c4tc2(4c2)2.

Here, the function πjCπ1L can be shown as a convex function with respect to t, for j=A,B. Solving for πjCπ1L=0 obtains the following two roots:

t=84c2c2c3+c482c2+c4t˜>t2andt=2c2=t1>t˜.

Thus, πjCπ1L for t2tt˜ and πjC<π1L for t˜<t<t1.

Regarding the social welfare and consumer surplus rankings, we solve the equilibrium prices and quantities as:

[17]p1AC=2c+ct(4c2),p2AC=2c+2tc2t(4c2),q1AC=2c+ct(4c2),q2AC=2c2t(4c2).
[18]p1AB=2c+ctc2(4c2),p2AB=2c+2tc2(4c2),q1AB=2c+ctc2(4c2)(1c2),q2AB=2c2tc2+c2t(4c2)(1c2).
[19]p1AL=1+c+tc,q1AL=1tc.

Here, pijC and qijC (pijB and qijB) are respectively firm i’s equilibrium price and quantity in market j under Cournot competition (under Bertrand competition), and pijL and qijL respectively denote firm i’s price and quantity in the limit-pricing equilibrium, for i=1,2 and j=A,B.9 After substituting eqs [17], [18], and [19] into the utility (total surplus) function, u(q1,q2)=q1+q2(q12+q22+2cq1q2)/2, and the consumer surplus function, u(q1,q2)+(Ip1q1p2q2)=(q1+q2(q12+q22+2cq1q2)/2)+(Ip1q1p2q2), straightforward calculations yield the following social welfare difference and consumer surplus difference in each market j, for j=A,B:

WjCWjB=c28+12c2c22c3+t4c+4c2+t243c22(4c2)2(1c2)<0,
WjCWjL=2+c+2t8+12c2c2c3+t86c2+2c42c2(4c2)2<0,
CSjCCSjB=c28+4c+6c22c3+t84c6c2+2c3+t24+3c22(4c2)2(1c2)<0,
[20]CSjCCSjL=2c2t2c484c+6c2+3c32c4+t86c2+2c42c2(4c2)2<0.

The four functions in eq. [20] are all quadratic functions with respect to t. A similar logic to the above analysis shows that the signs of the four functions are all negative for all the admissible values of the model’s parameters. Q.E.D.

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  1. 1

    The discrete linear-city model can be extended to consumers located at the nodes in a finite transportation network.

  2. 2

    Consider a firm that produces coffee. If its competitor produces coffee (cream, respectively), then the firm definitely wants its competitor to charge a high price for coffee (the firm definitely wants the quantity of cream to be high, respectively).

  3. 3

    More specifically, a limit-pricing strategy occurs in a situation of strategic entry deterrence, by setting the price at the highest level that is consistent with keeping the potential entrant out, and a firm works as a monopolist, but bears competitive pressure from its rival. Without loss of generality, it is assumed that the low transportation cost firm 1 (firm 2) obtains the whole share of market A (market B) in equilibrium.

  4. 4

    Analogous situations in the Hotelling’s (1929) mill-pricing model, where market demands touch each other at the margin when customers have finite reservation prices and the market is fully covered, are completely analyzed by Hinloopen and Van Marrewijk (1999).

  5. 5

    The detailed derivation of the sign of the function in eq. [14] is omitted to save space. It can be checked that if c<31, then the function in eq. [14] is concave in the transportation rate t and has two roots, with the large root greater than t1 and the small root smaller than t2. If c>31, then the function in eq. [14] is convex in the transportation rate t and has two roots, with the large root smaller than t2. In either case, the sign of the function in eq. [14] is positive for all t2t<t1.

  6. 6

    Empirical evidence proposed by Porter (2000) supports a similar idea that Manhattan hoteliers tend to locate new hotels sufficiently close to established hotels that are similar on one product dimension in order to benefit from agglomeration economies, but different on another product dimension in order to create complementary differences.

  7. 7

    In a non-spatial context, Zanchettin (2006) reverses the standard profit ranking between Bertrand equilibrium and Cournot equilibrium with substitutes, by relaxing Singh and Vives’ (1984) assumption of positive primary outputs.

  8. 8

    It is well known that a spatial model also has an explanation for product locations.

  9. 9

    Note that firm 2’s quantity supplied is zero in firm 1’s home market in the limit-pricing equilibrium.

Published Online: 2014-6-7
Published in Print: 2014-1-1

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