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Concentration and the growth of market demand: a comment on Gaskins limit pricing model
JOURNAL
OF ECONOMIC
THEORY
Concentration a Comment
5, 303-305 (1972)
and the Growth of Market on Gaskins Limit Pricing
Demand: Model
N. J. IRELAND Department of Economics, University of Warwick, Coventry, England Received May 15, 1972
Gaskins [l] considers a model where market demand grows autonomously (i.e., the market demand curve shifts to the right autonomously) in a situation where sales of other than the dominant firm respond positively to that firm’s pricing above the limit price. The steady-state solution or long-run equilibrium is characterised by the optimal price for the dominant firm being positive and above the limit price, and the dominant firm’s share of the market being constant. Gaskins also argues that if the curvature of the demand curve at the equilibrium price is not too great an increase in the growth rate of market demand will increase the equilibrium market share. As Gaskins himself says [l, p. 3171 this is a disturbing result because “growth of the product market not only raises the long-run price level but it also allows dominant firms with insignificant cost advantages to maintain a constant market share over the long haul”.l He goes on to conclude [ 1, p. 3201: “By-products of faster economic growth are increased concentration levels and higher prices in dominated industries.” The purpose of this note is to point out that Gaskins’ conclusions come from a questionable specification and that a slightly different specification of the reaction function will permit a long-run solution to the market growth model which is analogous to his solution for his initial model with fixed market demand curve. Iff(P(t)) evt is the market demand at time t, with dominant firm price P(t) then the dominant firm’s sales are q(P(t), t) =f(P(t)) evt - x(t) where x(t) is the level of rivals’ sales. The profit per unit sale for the dominant firm is P(t) - c where c is constant. In Gaskins’ model the reaction of other firms’ sales is given by the reaction function k(t) = koeyt(P(t) - P), (0 1 Jacquemin and l’hisse [2] wonder if “this . . . result does not depend heavily upon the peculiar hypotheses of an exponential growth in demand.” In the present note a different explanation is advanced.
303 Copytight All rights
Q 1972 by Academic Press, Inc. of reproduction in any form reserved.
304
IRELAND
where P is the limit price and k,, is constant, and y is the rate of growth of market demand. Thus the rate of response of other sales grows at the same rate as market demand. But suppose the dominant firm prices at the limit price is. Then 9(t) is zero, and all of the increase in market demand is in terms of the dominant firm’s product. Other firms’ sales (x) do not respond at all to the growth in demand. This implies very strong assumptions about the iimit-price. In particular, it implies that fringe firms that have taken a share of the market face the same barriers in trying to keep that share of the market in the face of market growth, as they do in attempting to increase their share of the market. Also the limit price is invariant to the size of market demand, and to the allocation of the market between the dominant firm and other firms. The result of this specification is that the steady-state solution consists of a constant price greater than P such that x grows at the rate of growth of market demand-thus allowing the dominant firm an optimal constant share of the market. In particular, even if there is no cost advantage (F = c), the same result holds.2 If the assumption that increases in market demand are monopolized by the dominant firm is not made, but instead these increases are shared out according to the current shares in market sales then the reaction function (1) is replaced by (2): $(f) = p(t)
+ lcoeYt(P(t) - P).
(2)
Here is is the price which is the lower bound of the dominant firm’s prices which would cause the dominant firm to have its share of the market reduced. If (2) is inserted into Gaskins’ model, then a long-run steadystate solution is found such that price is equal to P, and the dominant firm’s market share is constant, and zero if it has no cost advantage-a result analogous to Gaskins’ static demand model.3 The purpose of this note has been to point out that it is the implicit assumptions concerning the role and existence of the constant limit price * See Gaskins [l, Table 11. 3 The analogy is immediate. The problem is to maximize co (P - c)Lf(P)&‘” - x] e-” dt subject to i = yx + eyt k,(P - P). I0 Let y = xe-yr and then j = (-yx + 2) e- yt = k,(P - p) and the problem to maximizing m (P - c)v(P) s0
reduces
- y] ept dr
subject to jt = k,(P - p) where /I = 7 - r -C 0. This is formally identical to the static demand model in Gaskins’ paper.
A COMMENT
ON
GASKINS
LIMIT
PRICING
MODEL
305
which creates the disturbing result which Gaskins found. The fact that all market growth was specified to be in terms of the dominant firm’s demand curve even though x > 0, allowed Gaskins’ dominant firm to continually charge above “limit-price” thus permitting some “entry” but maintaining its market share. A change in specification of the model by replacing (1) by (2) would not allow this long-run solution to exist. Price equal to P would then imply a constant market share,4 and if i’ = c (no cost advantage) this equilibrium share would be zero. REFERENCES
1. D. W. GASKINS, Dynamic limit pricing: Optimal pricing under threat of new entry. J. Econ. Theory, 3 (1971), 306-322. 2. A. P. JACQU~M~NAND J. THISSE, Recent application of optimal control theory to industrial organisation. Working Paper No. 7205. Institut des Sciences I%onomiques, Universite Catholique de Louvain, Louvain, Belgium.
’ In Gaskins’ notation, the constant long-run market share is
which is positive if (p - c) > r and f’(p) is negative. Notice this assumes that P is not a this constitutes a considerable
0 and zero if (p - c) = 0 as y is assumed to be less than that the derivative with respect to -y is positive, although function of y. Even with specification (2) rather than (1). assumption.
OF ECONOMIC
THEORY
Concentration a Comment
5, 303-305 (1972)
and the Growth of Market on Gaskins Limit Pricing
Demand: Model
N. J. IRELAND Department of Economics, University of Warwick, Coventry, England Received May 15, 1972
Gaskins [l] considers a model where market demand grows autonomously (i.e., the market demand curve shifts to the right autonomously) in a situation where sales of other than the dominant firm respond positively to that firm’s pricing above the limit price. The steady-state solution or long-run equilibrium is characterised by the optimal price for the dominant firm being positive and above the limit price, and the dominant firm’s share of the market being constant. Gaskins also argues that if the curvature of the demand curve at the equilibrium price is not too great an increase in the growth rate of market demand will increase the equilibrium market share. As Gaskins himself says [l, p. 3171 this is a disturbing result because “growth of the product market not only raises the long-run price level but it also allows dominant firms with insignificant cost advantages to maintain a constant market share over the long haul”.l He goes on to conclude [ 1, p. 3201: “By-products of faster economic growth are increased concentration levels and higher prices in dominated industries.” The purpose of this note is to point out that Gaskins’ conclusions come from a questionable specification and that a slightly different specification of the reaction function will permit a long-run solution to the market growth model which is analogous to his solution for his initial model with fixed market demand curve. Iff(P(t)) evt is the market demand at time t, with dominant firm price P(t) then the dominant firm’s sales are q(P(t), t) =f(P(t)) evt - x(t) where x(t) is the level of rivals’ sales. The profit per unit sale for the dominant firm is P(t) - c where c is constant. In Gaskins’ model the reaction of other firms’ sales is given by the reaction function k(t) = koeyt(P(t) - P), (0 1 Jacquemin and l’hisse [2] wonder if “this . . . result does not depend heavily upon the peculiar hypotheses of an exponential growth in demand.” In the present note a different explanation is advanced.
303 Copytight All rights
Q 1972 by Academic Press, Inc. of reproduction in any form reserved.
304
IRELAND
where P is the limit price and k,, is constant, and y is the rate of growth of market demand. Thus the rate of response of other sales grows at the same rate as market demand. But suppose the dominant firm prices at the limit price is. Then 9(t) is zero, and all of the increase in market demand is in terms of the dominant firm’s product. Other firms’ sales (x) do not respond at all to the growth in demand. This implies very strong assumptions about the iimit-price. In particular, it implies that fringe firms that have taken a share of the market face the same barriers in trying to keep that share of the market in the face of market growth, as they do in attempting to increase their share of the market. Also the limit price is invariant to the size of market demand, and to the allocation of the market between the dominant firm and other firms. The result of this specification is that the steady-state solution consists of a constant price greater than P such that x grows at the rate of growth of market demand-thus allowing the dominant firm an optimal constant share of the market. In particular, even if there is no cost advantage (F = c), the same result holds.2 If the assumption that increases in market demand are monopolized by the dominant firm is not made, but instead these increases are shared out according to the current shares in market sales then the reaction function (1) is replaced by (2): $(f) = p(t)
+ lcoeYt(P(t) - P).
(2)
Here is is the price which is the lower bound of the dominant firm’s prices which would cause the dominant firm to have its share of the market reduced. If (2) is inserted into Gaskins’ model, then a long-run steadystate solution is found such that price is equal to P, and the dominant firm’s market share is constant, and zero if it has no cost advantage-a result analogous to Gaskins’ static demand model.3 The purpose of this note has been to point out that it is the implicit assumptions concerning the role and existence of the constant limit price * See Gaskins [l, Table 11. 3 The analogy is immediate. The problem is to maximize co (P - c)Lf(P)&‘” - x] e-” dt subject to i = yx + eyt k,(P - P). I0 Let y = xe-yr and then j = (-yx + 2) e- yt = k,(P - p) and the problem to maximizing m (P - c)v(P) s0
reduces
- y] ept dr
subject to jt = k,(P - p) where /I = 7 - r -C 0. This is formally identical to the static demand model in Gaskins’ paper.
A COMMENT
ON
GASKINS
LIMIT
PRICING
MODEL
305
which creates the disturbing result which Gaskins found. The fact that all market growth was specified to be in terms of the dominant firm’s demand curve even though x > 0, allowed Gaskins’ dominant firm to continually charge above “limit-price” thus permitting some “entry” but maintaining its market share. A change in specification of the model by replacing (1) by (2) would not allow this long-run solution to exist. Price equal to P would then imply a constant market share,4 and if i’ = c (no cost advantage) this equilibrium share would be zero. REFERENCES
1. D. W. GASKINS, Dynamic limit pricing: Optimal pricing under threat of new entry. J. Econ. Theory, 3 (1971), 306-322. 2. A. P. JACQU~M~NAND J. THISSE, Recent application of optimal control theory to industrial organisation. Working Paper No. 7205. Institut des Sciences I%onomiques, Universite Catholique de Louvain, Louvain, Belgium.
’ In Gaskins’ notation, the constant long-run market share is
which is positive if (p - c) > r and f’(p) is negative. Notice this assumes that P is not a this constitutes a considerable
0 and zero if (p - c) = 0 as y is assumed to be less than that the derivative with respect to -y is positive, although function of y. Even with specification (2) rather than (1). assumption.