Oligopolistic nonlinear tariffs

transcript

International Journal of Industrial Organization 6 (1988) 199-221. North*Holland

O L I G O P O L I S T I C N O N L I N E A R TARIFFS*

Esther GAL-OR University of Pittsburgh, Pittsburgh, PA 15260, USA

Final version received June 1987

We consider the welfare implication of increased variety in industries with differentiated products, where nonlinear price schedules are feasible. We use an oligopoly model and demonstrate that increased variety (1) allows more consumers of low valuation for the product to participate in the market, (2) lowers the marginal price charged from consumers thus inducing increased consumption. With infinite brands we get convergence to the Pareto optimal marginal cost pricing and the whole market is served.

1, Introduction

Our objective in this paper is to extend the analysis of nonlinear pricing to study the welfare effects of such pricing in an oligopoly. We construct a model of price behavior in a differentiated product oligopoly, and investigate the impact of increased competition upon the terms of sale and upon welfare. In analogy to the traditional models of linear pricing, we also investigate whether the Pareto optimal outcome is obtained with infinite firms in the market.

The insurance or computer industries may serve as examples for industries where nonlinear pricing methods replace the more traditional linear pricing practice. The insurance companies offer different contracts depending upon the premium paid by the insured, and IBM charges a lump sum monthly rental for the right to buy machine time. Since the above markets are oligopolistic, it is natural to ask about the effects of nonlinear pricing in oligopolies.

Pigou (1932) was the first to explain the advantages of employing nonlinear pricing. As he suggests, using a multi-part tariff may serve two purposes: one is to sort between large and small demanders and the second is to extract surplus from the consumers. Rent extraction using quantity discounts, is advantageous to the firm even if the population of consumers is homogenous. For sorting purposes using quantity discounts may be optimal

*The insightful comments of Robert Masson and two anonymous referees greatly improved both the exposition and the interpretation of the results. This work was supported by NSF grant SES-8420492.

200 E. Gal-Or, Oligopolistic nonlinear tariffs

when in the relevant region large demanders also have more elastic demand given price. Otherwise, sorting noninear tariffs may be rising in quantity, However, tariffs rising in quantity are only relevant when small quantities cannot be added together.

The modern theory of nonlinear pricing was initiated by Oi (1971) who analyzed a uniform two part tariff pricing arrangement. Murphy (1977) generalized Oi's results to the case of multi-part tariffs and Leland and Meyer (1976) considered the two block tariff price'schedule (a sequence of marginal prices for different demand 'blocks').

Feldstein (1972), Mayston (1973), Ng and Weisser (1974) and Auerback and Pellecio (1978) demonstrated the benefit of applying a two part tariff to decreasing cost industries. Littlechild (1975) further generalized the model to include externalities in consumption. Spence (1977) extended these models to consider a general form nonlinear price not restricted to the class of n part or n block price schedule, and Willig (1978) and Roberts (1979) followed him. Rothschild and Stiglitz (1976) and Stiglitz (1977) derived nonlinear price schedules for insurance contracts when the market is a monopoly and when it is competitive. None of the above papers relates to oligopolistic markets.

Two recent papers examine nonlinear pricing in oligopolistic markets. One is a paper by Oren, Smith and Wilson (1983) and the second is by Spulber (1980). While in the first paper firms produce a homogenous product, in the second a differentiated duopoly is considered. The welfare implication of increased competition are not explicitly considered in either of the above two papers. In the present paper, a differentiated oligopoly is considered. Our emphasis is different from the earlier papers in that we conduct a comparative statics analysis to investigate the impact of entry upon marginal prices and welfare. We also investigate the limiting properties of the equilibrium with infinite brands in a similar way to the investigation of Oren, Smith and Wilson. However, while we consider the Bertrand equilibria where each firm chooses an outlay schedule they consider the Cournot equilibria where each firm chooses a distribution of quantities arrayed by purchase size, purchase price or buyer's type.

Each consumer in our model is represented by a vector of taste parameters which determines his willingness to pay for the different products in the market. This representation of tastes was previously employed by Perloff and Salop (1985) and by Sattinger (1984) (although not in the price discrimination context). In a similar way to earlier papers on nonlinear pricing, we assume the absence of an ex-ante mechanism for identifying consumers with different demands. The selected tariff structure, however, serves as a monitor- ing device that allows some degree of ex-post discrimination.

Since all the firms in our model face the same technology and since each brand is preferred the most by an equal number of consumers, we only consider symmetric equilibria. This does not preclude the possibility of

E. Gal-Or, Oligopolistic nonlinear tariffs 201

asymmetric equilibria. Our measure of the degree of competition is the number of brands that are available in the market, and a comparative statics analysis is conducted to find how prices at the equilibrium are affected by the degree of competition.

We consider a class of equilibria for which each producer may find it optimal to neglect the lower tail of the distribution of consumers. More explicitly, in order to extract more surplus from consumers of high valuation each firm charges prices which are prohibitive to consumers of low valuation, so such consumers are eliminated from the market. With increased competition the segment of the market covered by each brand increases, so that more consumers participate in the market. Since each producer lowers the 'expenditure function', that relates the expenditure of the consumer to the volume purchased by him, new consumers with lower taste parameters are attracted to participate in the market. We also derive conditions under which increasing the number of firms lowers the marginal price paid by the consumer and induces increased consumption. With infinite brands, the oligopolistic outcome converges to the Pareto optimal outcome, where the sum of consumer and producer surpluses is maximized. A similar asymptotic optimality result is obtained also by Perloff and Salop in their model of product differentiation with linear pricing. Hence employing more sophisti- cated pricing schemes does not violate the convergence to the Pareto optimum. The outline of our paper is as follows: in section 2 we present our model, in section 3 we consider the impact of increased competition and the Pareto optimal outlay schedule, and in section 4 we conclude.

2. The model

We assume a world of two commodities X and Z. The X industry consists of N oligopolistic firms each offering a different brand of product X. Each firm charges its customers an amount related to the quantity bought, the relationship however need not be linear. It is assumed that there exists a common unit of measurement for all the brands offered, and that there is no secondary resale market for X. Product Z is sold in a perfectly competitive market and its price is equal to unity (Z can be interpreted as income not spent on X). The marginal utility of product Z (for some particular consumer consuming a particular brand) is assumed to be a constant, independent of the amount consumed of X. Hence the area under the demand function of product X serves as an index for the consumer's willingness to pay for the product [Willig (1976)].

There are an unlimited number of distinct possible brands indexed by i=1,2 ..... Each consumer is characterized by the vector X = ( x l , x z . . . . ) of taste parameters that determine his willingness to pay for the different brands. The market consists of a large number of (small) consumers and a

202 E, Gal-Or, Oligopolistic nonlinear tariffs

finite number of brands that are actually produced. We denote by Y the vector of taste parameters for the brands that are actually offered in the market.

A consumer of type Y is represented by N inverse demand functions D(qt;y~) i= 1 . . . . . N, where ql is the amount bought of i and y~ is the taste parameter that determines the willingness to pay for brand i. The demand function for brand i is conditional on this being the only brand purchased. It does not depend upon the willingness to pay parameter of any other brand j. Each of the conditional demands of a consumer are independent of N. We assume that Dq,<O and D r , > 0 , namely each demand curve is downwards sloping, and shifts outwards with the taste parameter. We consider Yt to be a shift parameter of the demand, that can be rescaled so that Dr,r,=0,11 The generic functional form of the inverse demand is identical for all brands,

In what follows we assume that each consumer buys only one brand, thus the only possible substitution in consumption arises from brand switching (in the case of a switch, the consumption level may vary). It is either impossible or undesirable to partially substitute brands. This assumption is reasonable in at least two cases: if each producer offers discounts for high volume purchases so that the tariff schedule is concave, or if there are high fixed transition costs from one producer to the other. As it was mentioned already, there may be two purposes to offer nonlinear tariffs: one is for the purpose of surplus extraction and the second is for the purpose of sorting among different consumers. For the purpose of surplus extraction the producer would always want to offer discounts for high volume purchases, and for the purpose of sorting it might want to do the same if the demand of consumers of high taste parameters is more elastic given price. In appendix 1 we derive a necessary condition for this increased elasticity. Generally speaking, if rent extraction is the dominating factor in determining the tariff structure, nonlinear schedules are likely to be concave. As far as the second (transition cost) explanation for nondiversification in the consumption, the transition cost may either arise since producers are geographically located at different places, or because of any other queuing or search costs. For instance, even if several fast food restaurants are located together, a consumer will usually have his whole meal consumed at only one of the restaurants.

The distribution of consumers is represented by an N dimensional random variable with its cumulative distribution function denoted by G(Y). That is, G(y'I . . . . . Y'N) is the joint probability that yi'<y~ for all i. This N-dimensional distribution function can be interpreted either as the actual frequency of consumers with different vectors of parameters, or as the probability that one individual consumer has a given vector of parameters. We assume a simple symmetric preference distribution. By symmetry, we mean, that the prefer-

1This assumption implies either a linear additive shift parameter, or a linear multiplicative shift parameter.

E, Gal-Or, Oligopolistic nonlinear tariffs 203

ences for each particular brand are independently and identically distributed, or G(Y)=]-[~I F(yt). The distribution function F(.) is defined over the support [y, y-]. Under the above specification, each brand is most preferred by an equal share of consumers in the market. The independence assumption is made for expositional simplicity. In an earlier version of this paper, we characterized a much wider family of multivariate distribution functions G(.) for which the main theorems of the paper hold. The independence assumption implies that entering firms into the industry may choose the attributes of their brands independently of the attributes alreasy available in the market. If there was a negative correlation among the yf's then entering firms would choose their attributes to be as different as possible from those already available. 2 If there was a positive correlation entrants would be expected to closely imitate existing brands.

The symmetry assumption, that we make, is similar to the assumption made in previous models of horizontal differentiation, where brands are assumed to be symmetrically differentiated [see for instance, Chamberlin (1950), Dixit and Stiglitz (1977), Lancaster (1975), and Salop (1979)]. An identical preference structure is also assumed by Perloff and Salop in their model of product differentiation.

The strategy of firm i is an outlay schedule Ri(q), that specifies the expenditure of any potential consumer if he purchases the quantity q from firm i. Given any set of outlay schedules (R~(q), q__> 0, i = 1, N} for the firms, a consumer of type Y chooses which brand to purchase and how much to purchase of his preferred brand so as to maximize his net utility (or he may refrain from purchasing). The firms know the optimization problem for the consumers and the distribution of tastes in the population, and choose their pricing strategy to maximize profits for given strategies chosen by the other firms. Our goal is to derive the symmetric Nash equilibrium of the above game when outlay schedules are the strategies chosen. It is noteworthy that if product X were homogeneous and outlay schedules were the strategies chosen by the firms, the only Nash equilibrium of the game would be a price equal to marginal cost independent of the quantity purchased by the consumer or the number of firms in the market (as long as N>2) . This happens, since in the absence of capacity constraints on the firms, a each will undercut the price of the other as long as costs are covered. If there are, however, several differentiated brands of X, marginal cost pricing is generally not an equilibrium strategy.

The technology that we assume is of constant returns to scale where the

2For instance, if the degree of sweetness is the dominating attribute that determines how much consumers value the different brands of cereal, then with negative correlation among the y[s, the consumers that highly value the sweetened brands are those that value the unsweetened brands less,

3With capacity constraints mixed strategy equilibria with prices exceeding marginal cost are possible.

unit cost is denoted by c. We assume that the intercept of the inverse demand function of the consumer of the lowest taste parameter exceeds the unit cost c (i.e., limq~oD(q;y)>c).

2.1. Formulation of the problem

We first consider the maximization problem facing each consumer. The kth consumer, that is represented by the taste parameter vector yk, chooses from which firm i k to buy and what quantity qk to buy from this firm (or chooses to buy none of the products) so as to

Maximize V k = {~b(q k, y~) - R i ( q k ) } , where qk, ik

qk C~(qk, yk) = ~ D(Q; y~) dQ.

That is, consumer surplus, V* equals gross consumer value for product i (the integral of D) net of consumer payments to the producer (Ri).

If q*>O for some j, then for that j eq. (1) must satisfy the first and second order conditions

D(qk; yk) _ R)(qk) = O, and (2)

Dq(qk; y~) -- R'j (qk) < 0, where (3)

j = argmax { vk}.

The symbol argmax is used to designate the argument that maximizes the objective function V k.

We require that second order condition (3) holds as a strict inequality so as to guarantee the existence of a unique solution to the maximization of the consumer. If the maximal V k for each brand, i, is negative, this means that consumer k purchases none of the products.

To develop the firm's maximization problem in its most convenient form, we define a consumer net surplus function, conditional on the consumer purchasing good i. Suppressing the consumer superscript k, this is

w(yi) = max {qS(q, y , ) - R,(q)}. (4) q

Since our focus is on the derivation of symmetric equilibria, suppose that all remaining firms except i employ an ,identical pricing strategy R(q) (= Rj(q)

E. Gal-Or, Oligopolistic nonlinear tariffs 205 r

j 4: i). Hence if the consumer is restricted not to buy from i, he will clearly select from the remaining ( N - I ) brands the one that he prefers the most. If Y-i designates the taste parameter vector of the consumer excluding its ith entry, the consumer will select the brand that corresponds to the maximal element of Y-i, because each R(q) function is the same. Let this maximal element be denoted by y~_]", and define the net surplus function of the consumer, who is excluded from the purchase of i as follows:

v(ym] ~) = m a x {0, max {q~(q, ym_]x)-R(q)} t ' (5)

If by purchasing from the most preferred rival of i the consumer can derive only a negative net surplus, v(') is defined to be equal to zero. Otherwise, the net surplus of a consumer (that is excluded from the purchase of i) is equal to the gross consumer value for the most preferred brand except i net of the consumer payments for this brand. From firm i's point of view the function v(.) is viewed as the reservation utility level of a consumer with tastes described by Y. The consumer of type Y will purchase brand i rather than any other competitive brand if w(yi)> v(y~]~), or stated differently, if

yr2_~x < V- l(w(yl) ). (6)

That is, the right-hand side of (6) describes what parameter level would be required to make a potential buyer of i indifferent to buying some j, i f j were priced by the uniform competitive price function Rj, j¢ i . Firm i cannot observe the value of y~]*. It can, however, use the joint distribution function of Y to compute the probability of the event stated in (6). The random variable on the LHS of (6) is equal to the maximum of ( N - - l ) identically and independently distributed random variables. Hence the cumulative distribution function of this random variable is given by IF(.)] N-l , and the probability of the event stated in (6) is given as follows:

Prob {y~] x < v- l(w(y~))} = IF(v- l(w(yO))]U- l. (7)

Given the above explanation the maximal expected value of the profits of firm i is as follows:

Max S [Ri(q(Yi)) - cq(y,)] [F(v - l(w(yi)))]N- l f(yi) dyi, (8) R~(q) y

where q(Yl) = argmax {~b(q, Yi)- R,(q)}, w(yi) = max {q~(q, y , ) - R,(q)}.

Notice that in formulating its maximization problem firm i takes as given t h e reservation utility level v(.), that is determined by the pricing strategies of its rivals. The integrand of (8) corresponds to the expected profits, that firm i can derive from a consumer of taste parameter vector whose ith entry is equal to y~. The definitions describe the behavior of a consumer with that specific value of y~. The expected profits in the integrand is the product of three terms: (1) the gross revenues net of the production costs of selling to such a consumer, (2) the probability that such a consumer selects to purchase from i rather than any other alternative brand, and (3) the relative frequency of the population of such consumers. Notice that this applies to profits from consumer k and, due to constant returns to scale, this applies to total profits. Accordingly, we interchangeably use the distribution F(.) to describe both the probability of drawing a consumer of specific characteristics, and to describe the actual number of different types of consumers.

The method to solve the constrained maximization in (8), will not be to directly derive Rg(q), but to indirectly derive it after the derivation of the functions q(y~) and w(y~). Notice that the revenue from selling to a consumer of type y~ (ith entry of y) can be expressed as r(y~)=R~(q(y~)), where

r(yi) = (D( q(yt), y~) -- w(yi).

That is, revenue is the difference between gross and net consumer surplus. Then, if q(Yi) is invertible, we can derive Rl(q) as

Ri(q) = r(q - l(q)).

The producer's problem reduces, therefore, to a simple optimal control problem with q(y~) the control variable and w(y~) the state variable. The problem is stated in full and then explained (we omit the index i in the statement of the problem).

Problem 1

Max S [~b(q(y), y) - w(y) - cq(y)] [F(v - l(w(y)))]N- i f ( y ) dy (9) q(y) r

subject to q(y)

w'(y)= S D , ( Q ; y ) d Q , (10) 0

w"(y)>0, (11)

q(y)~_ O, (12)

w(y)>O. (13)

The first term of the integrand appearing in (9) is the profit to firm i, conditional upon a sale to a consumer of type y. This profit is ryy)-cq(y) , where r(.) is separated into its consumer surplus terms from its definition. The conditional profit is multiplied by the probability that the consumer purchases from i, and by the probability that the consumer has a parameter of value y, i.e., the density function f( .) . When the integrand is integrated over the interval [y,37], we get the expected profits of firm i. Constraints (10) and (11) are self selection constraints. From the definition of w(y), w'(.)= [D(q; y) -R ' (q)]q '+S~Dyd Q. Recall from the consumer's value maximization first order condition, D - R ' = O . So, by the envelope theorem, w'(.) is defined as in condition (10). Given that we rescaled the variable y so that Dry=0, it follows that w"(.)=Dyq' which should be positive to guarantee the invertabi- lity of q(.).4 Constraints (12) and (13) guarantee that consumers can never sell the product and that the firm cannot force a negative payoff upon a consumer, respectively.

2.2. The solution of Problem 1

The Hamiltonian of Problem 1 may be written as (the argument y is suppressed in q(y), w(y), 2(y) and #(y)):

H(q, w, y) = [qS(q, y) - w - cq] IF(v- l(w))]s- i f(y) + 2 1 D,(Q; y) dQ + uq, 0

where 2 is the multiplier attached to the differential equation (10), that determines the rate of change of the state variable w. At every value of y the multiplier 2(y) measures the 'shadow price' of the payoff w(y), of a consumer with taste parameter y. More explicitly, 2 determines the rate of change in the expected profits of the firm as a result of sightly raising the payoff of the consumer. The variable /~ is the Lagrange multiplier attached to the non- negativity constraint of the control variable q ['eq. (12)]. The first order conditions that the optimal pair (q, w) must satisfy (14)--(17) below

Hq = 0 = [O(q; y) - c] IF(v- l(w))Js- i f(y) + 2Dy(q; y) +/~, (14)

'~Notice that from (2) and (3) q'= -DJ(Dq-R")> 0. This inequality follows from the demand restriction Dy>0, for all q>0.

- - H w = 2' = I F ( v - I ( W ) ) ] N - l f (y)

[q~(q, y) -- w-- cq](N - 1)[f(v- l(w))]N- 2 f ( v - l(w)) f ( y )

q/.t=0, q>0, /~>0,

2(y)w(y)= 2(~)w(y) =0, w(y) >= o.

v'(v- l(w)) (15)

The second order sufficient condition is

nqq = Dq(q; y) [ F (v - ~(w))] N- i f ( y ) + ,~,Dqr(q; y) < O. (18)

Conditions (14) and (18) guarantee the existence of a unique control variable q that maximizes the Hamiltonian for given 2 and w, Condition (15) determines the rate of change of the shadow price (2). Condition (16) is a Kuhn-Tucker condition derived from the non-negativity constraint (12). Condition (17) is the transversality condition implied by (13). Since the state variable is not restricted to prespecified levels at the boundaries, the product of the state variable by its shadow price at the boundaries should vanish l'see Kamien and Schwartz (1981, p. 147)].

Several observations follow immediately from (14)-(18). The first thing to note from (14) is that 2(y)<0 for all y, since the optimal schedule must satisfy the property that marginal price is never below the unit cost of production ( D ( q ; y ) - c > O for all y). Otherwise the producer can raise his profits by excluding the consumers for whom (D(q; y ) - c ) < 0 . Secondly, from the assumption that D r > 0 for all q, it follows that q is strictly increasing in y. Hence if q is conditional upon buying i or nothing, from (12) the solution is characterized by at most two regions as follows:

q(y)=O for y=y~,

q(y)>0 for y > y * . (19)

Consumers with a taste parameter lower than y* are asked to pay for i a price above their willingness to pay. Hence they never purchase this brand. The consumers with a taste parameter bigger than y* can expect a positive payoff by purchasing from i,

Notice that the second order condition (18) is certainly satisfied if Dqy>0, since 2<0.

We use (19) to derive the symmetric equilibrium from (10), (14)--(17). At the symmetric equilibrium s v- l(w(y)) = y and y* = y* = y*.

Since from (16) /z=0 if q>0 , (14) reduces to

D(q;y)-c=-2D~,(q;y)/{[F(y)] 'v-l f(y)} if q>O. (20)

Integrating the RHS of (15), requiring the continuity of the shadow price 2, and noticing that from ( 1 7 ) 2 ( ~ ) = 0 (since w '>0 , w07)>0), yields the following solution for 2(y):

). = -- [1 -- [F(y)]N/N

y + I [q~(q' s) - w - cq] (N - 1) [F(s)] N- Elf(s)] 2 ds y* W f

= - [1 - [F(y)]N]/N

+~ [ fb(q ' s ) -w-cq](N-1)[F(s)]N-2[ f (s )]2ds y > y * . (21) y W e

Notice that the solution for 2 at 37 satisfies the transversality condition (17). Since w(37)>0 207)=0. In addition, in stating the expression for 2 we use symmetry to replace v' by w'.

Solving the differential equation (10) and noticing that w(y*)=0, since q(y*)=0, yields the solution for w(y)

w=0, y< y*

y q(s)

=~ ~ Dr(Q;s)dQds, y>y*. (22) y* 0

Define the function M as the maximized value of a producer's objective function (9) when evaluated at the equilibrium levels of q and w (given by (20)--(22)), then by substituting (22) into (9) M is equal to

M = c~(q(y), y) -- S Dr(Q, s) dQ d s - cq(y) * y* 0

xE (v l(!q! o. o s, ~At the symmetric eqmllbnum R~.)=Rj(.)=R(') for all i,j. Hence from the definition of v(.)

in (5) v-X(w(~))=~, where ~ still varies across individuals.

The cutoff point y* that determines the segment of the market covered is optimally selected by each firm to maximize M. Differentiating M with respect to y* yields minus the value of the 'maximized Hamiltonian' evaluated at the cutoff point, y*, as follows:

dM - - = - H(q(y*), w(y*), y*). (23) dy*

Differentiating it again with respect to y* while using the first order conditions (14)-(17) yields by the envelope theorem that

d2M (dy.) 2 = - H r ( q ( y * ), w(y*), y*). (24)

Notice that for every y > = y * ( d H / d y ) = H q q ' + H , , w ' + H x 2 ' + H r From (14) Hq=0, from (15) H w = - - 2 ' and from the definition of the Hamiltonian Hz=w' . Hence the first three terms of d H / d y vanish and for every y > y * d H / d y = H y , thus (24) is implied. If H y > 0 for every y the function M is a strictly concave function of y* and a unique cutoff point maximizes M. Imposing the restriction that Hy> 0 for all y is stronger than is necessary to guarantee the existence of an interior maximum of the function M. We impose this restriction in order to guarantee the globality of the interior maximum. Given the definition of the Hamiltonian

H y = I ! Dr(Q;Y)dQ][F(v - I (w) ) ]N- l f (Y )

+ [~b(q, y) - w - cq] [F(v - l(w))]N- l f , (y). (25)

Notice, that in deriving the expression for H r, we partially differentiate only those terms of the Hamiltonian, that are direct functions of y. In particular, we do not differentiate with respect to the state and control variables, which are indirect functions of y.6 Since Dy>0 for all y and q, a sufficient condition for H r > 0 is that f ' ( y ) ~ O for every y, namely the density is a nondecreasing function of y.7

We will assume that M is indeed a strictly concave function of y* to guarantee the uniqueness of an optimal cutoff point. If at the equilibrium each firm chooses to neglect a segment of the market (i.e., y* >y), the cutoff

6This indirect differentiation is eliminated by the envelope theorem (i.e., Hqq'+ H~,w'+ H~2'= 0 for all y>y*.)

VThis condition is stronger than is required to guarantee the existence of a local maximum. For an interior maximum it is sufficient that f ' > 0 in the neighborhood of y*.

point satisfies the requirement that H(q(y*), w(y*),y*)=O. 8 All consumers are served (i.e., y* = y_) if H(q(y), w(y), y) > 0 for all y including y. In this case dM/ dy* in (23) is negative for all y* and each firm chooses to sell to all the consumers. The condition that the maximized Hamiltonian vanishes at a cutoff point is quite standard in optimal control problems defined over time, when the planning horizon is unspecified [see Kamien and Schwartz (1981, p. 56)]. Our optimal control problem is defined over the taste parameter of the consumer. The producer can optimally select what segment of the potential population to serve. This is similar to dynamic optimal control problems where firms are not constrained to produce over periods of prespecified length. In the absence of such a constraint firms optimally select how long to operate in the market.

3. The impact of increased competition

Since inequality (18) is assumed strict there exists a unique q that maximizes the Hamiltonian for a given 2, and satisfies condition (14) (for #=0). Substituting this solution for q in terms of 2 into conditions (15) and (10) yields a system of two first order differential equations in 2 and w with two initial conditions specified by: 207)=0 and w(y*)=0, where y* is selected so that the maximized Hamiltonian is zero at y*. The existence and uniqueness theorem of differential equations guarantees the existence of a unique solution for 2 and w, which implies the existence of a unique symmetric equilibrium. This obviously does not preclude the possible existence of asymmetric equilibria. In the present section, we exploit the uniqueness result to conduct a comparative statics analysis.

We investigate the impact of increased competition upon the terms of sale offered to each consumer. Increased competition is measured by an increase in the number of brands offered in the market, or put differently, by an increase in the variety of products offered in the market. Increased competition might have two effects; (a) it might change the segment of the market chosen to be covered by each producer, (b) it might change the quantity offered and marginal price charged from a consumer that is considered as a potential buyer by some producer. While Theorem 1 considers the first effect, Theorem 2 considers the second.

Theorem 1. The critical value y* is a nonincreasing function of N if f '(y)>= 0 for all y.

Proof See appendix 2.

According to Theorem I when more firms participate in the market each SH(q(y* + e), w(y* + 8), y* + e)) > 0 for e > 0 since dH/dy > O.

212 E. Gal.Or, Oligopolistic nonlinear tariffs

firm tends to reduce its tariffs so that additional consumers having lower valuations for the product are attracted to the market. Those consumers purchase none of the products if competition is less intense. It is important to point out the role of the assumption f'(y)>_O in the proof. This assumption is used so as to guarantee the monotonicity of the maximized Hamiltonian with respect to the taste parameter. More explicitly, this assumption guarantees that the expected profits derived from consumers having high valuations for the brand, exceed the expected profits derived from consumers with low valuations. This monotonicity, guarantees the uniqueness of the cutoff point y* that determines the size of the potential population considering the purchase of a given brand. Notice that a similar sufficient condition had to be imposed even if only a single producer existed in the market. In order to guarantee, that a monopoly may eliminate the lower tail of the distribution of consumers, it is sufficient to impose the condition that i f(y)>0. This is, however, a sufficient rather than a necessary condition 9 both for a monopoly or an oligopoly. Hence it is a stronger requirement than is necessary to guarantee the monotonicity of the maximized Hamiltonian. The uniform distribution function satisfies this condition. Hence if consumers are uniformly distributed according to types, the size of the potential population considering each brand expands with more intense competition.

The next question we address is the impact of increased competition on the quantity sold to each consumer that participates in the market (a consumer of type y>y*). To answer this question, we derive the expression for dq(y)/dN in (26).

dq(y)/dN = [ 1/Hqq] Dy( q; y) . L(y) [F(y)]N- i f(y) , (26)

[-q(Y*) 1 L(y) =- - B(y)-- S I ~ Dr(Q; y*) dQ (dy*/dN) ykO

x (([F(s)] N - z[ f (s )]2(N- 1))/w'(s)) ds

and B(y) is defined in the proof of Theorem 1.1° It is difficult to sign the right-hand side of (26). Several conclusions can

however be drawn. The first conclusion is that a consumer of type y buys the same quantity of X independent of the number of firms operating in the

9The necessary condition is stated in footnote 7. 1°The expression for dq(y)/dN is derived by totally differentiating first order condition (14)

with respect to N. More explicitly, dq/dN-~-HqN/Hqq.

industry, since dq(~)/dN =0. The marginal price he pays D(q(37), p) is always equal to marginal cost of production, since the shadow price is zero for a consumer that values the product the most (2(37)=0), 11 Notice that this does not imply that such a consumer does not benefit from increased competition. Even though he pays the same marginal price for the last unit that he purchases, he may pay lower prices for previous units. The second conclusion is summarized in Theorem 2.

Theorem 2. I f f '(y)> 0 for all y then

(a) dq(y)/dNlN=l >O all 37>y>y* (b) if y*=y_for N* then dq(y)/dN>O all N>=N*.

Proof From (26)

-- E1/Hqq]Dr(q; y)B(y) _ I~N dq(y)/dNlu=l = ~ = , ,q,Y),

In addition, if y * = y for N* then y * = y for N > N * since dy*/dN<O from Theorem 1 and y*>y . Hence the second term of L(y) vanishes for N>=N* and dq(y)/dN=l(N,q,y) for every N>N*. When f'(y)>O, B(y)>0 for all y > y* and since Hqq < O, I(N, q, y) > O. Q.E.D.

According to Theorem 2 the marginal price paid by each consumer declines when the number of firms increases and the whole market is covered or if the number of firms locally increases beyond the monopolistic market, A sufficient condition for the decline in marginal prices is that f'(y)>=O. This condition, once again, guarantees that expected profits from consumers of high valuation exceed expected profits from consumers of low valuation. By expected profits, we mean, the product of the profit per consumer by the relative frequency of this type of consumers.

We cannot establish, in general, the impact of increased variety upon the marginal prices paid by consumers. The ambiguity arises since the expression derived for dq(y)/dN includes two terms that contradict in sign. More explicitly

sign[dq(y)/dN]=+signIB(Y)+![F(s)]N-2[f(s)]2(N-1)ds l w ' ( s )

xlqti')Dr(Q;y*)dQ][(dy*/dN)].

:1Monopoly extraction, first degree discrimination leads to a marginal price equal to MC. This result is stronger than what we obtain. It pertains to a price structure applicable to all buyers. In our model only .9 pays p=MC, for y<)7, p>MC. In addition, the result that p=MC for )7 is more general than the result that dq(~)/dN=O. The latter relies upon zero income effects from lowering inframarginal unit prices.

While the sign of B(y) is non-negative, the sign of the second term is nonpositive by Theorem 1.

At the symmetric equilibrium, the consumers that buy brand i are those that value it most. Firm i faces, therefore, a population of potential buyers characterized by the density IF(y)] N- if(y) over the support [y~, y-]. When the number of firms rises, the density function declines at every point and the support of the density is larger. As an outcome, increased variety has two contradicting effects on the degree of homogeneity of the consumers that buy one particular brand. (a) Each consumer is expected t o find a product that matches closer his tastes and as an outcome each producer may expect a more homogeneous population consisting of consumers of higher taste parameters. (b) Each producer lowers his tariffs in order to maintain his market. The lower tariffs attract additional consumers with lower taste parameters. Hence a more heterogeneous population consisting of additional consumers with lower taste parameters also enter into the potential market of each producer.

When the population of customers buying from some producer becomes more homogeneous and consists of those that value the brand more highly, the producer has a tendency to lower marginal prices (offer discounts for high volume purchases). The discounts offered induce a significant increase in consumption for those consumers who value the product more highly. When the population becomes more heterogeneous, the opposite effect may be expected. In the expression derived for dq(y)/dN, the two contradictory effects discussed above are demonstrated.

In fig. 1 we further demonstrate how competition affects the degree of homogeneity of the consumers, that purchase one particular brand.

D C I I I I B / I I / / i I / ~ " I I J + - - Y A

'~'E 'l I I I I I I I I

YM Y~ Fig. 1

In the figure y~t 1 denotes the cutoff point selected by firm I if it is the only firm in the market, y~l and y~2 designate the cutoff points selected by firm 1 and firm 2, respectively, as a result of the entry of firm 2 into the market. At the symmetric equilibrium y~l =y~; and from Theorem 1 y~t <y~ta. Prior to the entry of firm 2, the population of consumers that bought from firm 1 consisted of all consumers for which Yl > y , t . Upon the entry of firm 2 all the consumers with preferences in CAB switched to the consumption of brand 2. This switch has a homogenizing effect on the population of consumers that purchase brand 1, However, new consumers with lower taste parameters in the region AEy*ly *~ are now purchasing brand 1, as well. This addition to the population of consumers has an heterogenizing effect on the population of consumers, that buy brand 1.

In order to compare the equilibria of an oligopoly and a monopoly we consider an example. Let D(q;y)=a-bq+y, c = 0 and let the population of consumers be uniformly distributed (the distribution F(,)) over the support [0, 27] where y > a . Solving (20)--(23) for N = 1 yields the following solution:

q, =[y_ 27-a12 2 _]b'

w - [ -37-CI~i "-L y -2-J ~'

y.=y a,

where subscript m designates a monopoly. Solving the same conditions for N > 1 yields the following second order differential equation for the payoff function w(.):

,, W - - b w ' +bw y=2y+-~(N-1),

with initial conditions w(y*)=0, w'(y*)=0, and w'(y)=(a+27)/b. The first two initial conditions guarantee that the critical consumer of type y* gets zero surplus and buys zero units of the product. The last initial condition guarantees that 2(37) = 0.

It is difficult to solve the second order differential equation specified above for all N. However, if .~ =(237/a)- 1 we obtain the following solution:

[-• + a7

( y + a ) 2

E. Gal-Or, Oligopolistic nonlinear tariffs

Y~7-O,

R ~ ( q ) = ( y + ~ q l y + a - b-~ 1,

where subscript /~/ designates the equilibrium with (2)5/a)-1 firms. For instance, when y = 2a, N = 3. Comparing the equilibrium under the monopoly and oligopoly yields, that while under the oligopoly all the consumers participate in the market, under the monopoly the lower tail of the distribution is neglected, The tariff structure under the oligopoly is lower at all levels of q. As a result each consumer purchases more of the product (i.e., Y~7 < Y,,, q~7 > q~ all y).

It is interesting to investigate what happens as N tends to infinity. With infinite brands, we can derive the following expression for ~.(y):

l im2(y)=0 if y=)5

[qS(q(33), )5) -- w(y) -- cq()5)-] = f ( ; ) if y<)5.

Since the marginal price is never below the marginal cost of production 2(y)<0 all y; hence c~(q()5),y)-w(~)-cq(~f)=O implying that with infinite brands ;t(y)=0 for all y. We get, therefore, convergence to linear pricing of the form R(q)=cq. With such pricing, the whole market is covered (y*=_y) since 12

w(y) = 4)( q(£), y_) - cq(y) > 0 where D( q(y); y) = c.

The oligopolistic outcome with infinite brands coincides with the Pareto optimal outcome. By the last we mean an outcome that maximizes the sum of consumer and producer surpluses. The consumer surplus in our case is

~2This follows from our assumption that limq.o D(q; y)> c, namely even the lowest intercept of the inverse demand exceeds unit cost.

equal to

and producer surplus is given by N times the profit of each firm. The sum of both is equal, therefore, to

S [qS(q(y), y) -- cq(y)]N[F(y)] N- ~f(y) dy, (28) y*

that is maximized by marginal cost pricing and by the whole market being covered, namely y* = y and D(q(y), y ) = c all y. Even though we allow firms to use nonlinear pricing methods, we still get convergence to the Pareto optimal outcome, in the same way we get the convergence result in a model where firms employ linear pricing.

From the discussion held earlier, it should be clear that the index used for measuring social welfare rises when the number of firms increases locally beyond a monopolistic market or if the number of firms increases when the market is completely covered. Such a monotonicity result cannot be estab- lished in general, because of the ambiguous effect of increased variety upon the marginal price charged by producers. However, as N tends to infinity, the oligopolistic outcome coincides with the Pareto optimal outcome.

4. Concluding remarks

We have developed an oligopoly model of nonlinear pricing and investi- gated the implications of increased competition. If the population of consumers is characterized by increased frequency of the consumers that value the product more highly, producers charge lower marginal prices, and serve a larger segment of the market, when competition is more intense.

The assumption on f ' ( . ) is designed to eliminate low demanders, and to give a larger weight to high demanders in the objective function of each producer. It is also related to the existence of quantity discounts. That is, if low demanders were a vast proportion of the population producers might have selected to induce increased consumption of such consumers rather than consumption of high demanders. As it was already mentioned, for sorting purposes quantity discounts may not be optimal unless large demanders also have more elastic demand given price. When large demanders are fewer in number providing quantity discounts becomes even less advantageous to the producer.

Appendix A

Increased elasticity of high demanders Using the implicit function theorem it is possible to solve for q in terms of

p and y from the inverse demand D(q, y) as follows:

q = F(p, y), (A.1)

where aq/ap= 1/Dq and Oq/t?y=-Dy/Dq. For a given price the elasticity of the demand is

E - aq. p _ l p (A.2) c3p q Dq(F, y) F'

where F is the implicit function stated in (A.1). Totally differentiating (A.2) with respect to y, while holding price constant yields

P{[DqqOa-~y+Dqr]F+d~-~yDq}_ P{Dq, F-D,[ I+D~qFI~ o e _

Oy [Dq(F, y)]ZF2 [D~(F, y)2F2 '

where the last equation follows upon substitution of aq/Oy. For a given price elasticity increases (in absolute value) when y increases if

DqyF > Dy I1 DqqF1 +T J In particular, if the inverse demand is concave or linear in q (Dqq<0), a necessary condition for (A.3) is that Dqy > 0. The latter condition implies that the slope of the inverse demand is flatter (less negative) for a consumer with a higher taste parameter.

Appendix B

Proof of Theorem 1 The cutoff point y* satisfies the equation

dM - - = - H(q(y*), w(y*), y*) =0. dy*

Totally differentiating the above with respect to N yields

dy* M:N H2v

dN M : : M : : '

where the last equality follows from (23). Hence the sign of dy*/dN is opposite to the sign of HN since M : : < 0 by second order condition (24),

HN = [q~(q(y*), y*) -- cq(y*)] [-F(y*)] N- if(y,) log F(y*)

V~(P') ] dA(y*) (B.2) +L ! Dr(Q;Y*)dQ dN

From - H ( ' ) = 0 we get

H N=[~!') Dr( Q; y*) dQ ] [ - 2(y*) Iog F(y*) + ~ I

=El!')Dr(Q;y.)dQ][l(logF(y,)_t 1 - ~ ( y * ) )

( g - 1) IF(y)] N- z [ f (y)] 2 [q~(q(y), y) _ w(y) - cq(y)] , x ['log F(y) - log F(y*)] dy. +I

,. w'(y)

+ : F(Y)N-2['f(Y)]2['dP(q(Y)'Y) dYl. (B.3) ~ [ w ~ -- w(y) -- cq(y) ] _1

In the derivation of HN in (B.3) we partially differentiated only those terms of the Hamiltonian that are direct functions of N. t3

We define the function B(y)

(N - 1 ) [F(s)] ~ - 2 [ f (s)] 2 9 + S [qb(q(s), s) -- w(s) -- cq(s)]]A(s) ds , w'(s)

+ ). [F(s)]" - 2If(s)] 2 [¢(q(s), s) - w(s) - cq(s)] ds

, w'(s) ' (B.4)

~3From (21)), is directly related to N.

J,I.O,-- C

220 E. Gal.Or, Oligopolistic nonlinear tariffs

where A(s) = [log F(s) - log F(y)]

B,, , f (y )[ -1-[F(y)] u-]

f (y) ~ (N - 1) IF(s)] s - 2 I f ( s ) ] 2 [q~(q(s), s) - w(s) - cq(s)] F(y) Jy w'(s)

f(Y) [F(y)]N-l f (y)[dp(q(y) ,y)-w(y)-cq(y)] F(y) w'(y)

f (y) 1 F(y) w'(y) H(q(y), w(y), y). (B.5)

The function H(q(y), w(y), y) is positive for all y > y*, since H(q(y*), w(y*) ,y*)=0 and dH/dy>O for all y>=y* by assumption. (As noted above a sufficient condi t ion for dH/dy>O is f'(y)=>O). Since sgn[B'(y)]=sgn[-H(q(y) ,w(y) ,y)] , B ' ( y ) < 0 for all y>y*. In addi t ion B(#) = 0, hence B(y) > 0 for all )7 > y > y*, in particular, B(y*) > O.

rq(y*) 1 HN = L ! D,(Q; y*) dQ B(y*) > O. (B.6)

Hence dy*/dN =< 0. Q.E.D.

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Oligopolistic nonlinear tariffs

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