MONOPOLY LINEAR AND NONLINEAR PRICING

Transcription

1 MONOPOLY LINEAR AND NONLINEAR PRICING Babu Nahata Department of Economics University of Louisville Louisville, Kentucky January, 2005 ABSTRACT This pedagogical note explains how the same basic principle can be applied to explain the profit-maximizing behavior of a monopolist under both linear and nonlinear pricing by introducing an average price function. It is shown that optimal conditions under nonlinear pricing are similar to that of linear pricing. These conditions can be explained through a simple graphical exposition. The optimal conditions under monopolistic linear and nonlinear price discriminations are also similar.

2 MONOPOLY LINEAR AND NONLINEAR PRICING I. INTRODUCTION A monopolist determines the profit-maximizing output by equating marginal revenue to marginal cost. This well-known textbook result assumes that the monopolist uses a single uniform price, or in other words, the pricing is linear. In other words, the consumer s expenditure is a linear function of the quantity purchased. Since price and marginal revenue are related under linear pricing through price elasticity of demand, the profit-maximizing price can be expressed in terms of price elasticity of demand and marginal cost. However, when a monopolist is able to design and implement nonlinear pricing strategies, for example, a two-part tariff or package pricing, the textbooks offer no simple rule comparable to equating marginal revenue with marginal cost. The fundamental difference between linear and nonlinear pricing is that nonlinear pricing strategies are based on consumer s total willingness to pay (the area under the demand curve) while linear pricing is based on the willingness to pay for each unit (the height), or marginal willingness to pay. As a result, the revenue under nonlinear pricing is the total area under the demand as opposed to the area of the rectangle under linear pricing. Because the attainable revenue under nonlinear pricing is the area under the demand curve, marginal revenue function does not play any significant role in determining the profit-maximizing output and the outlay or the tariff paid by the consumers under nonlinear pricing. The purpose of this paper is to show that when a monopolist uses nonlinear pricing and consumers are homogeneous, (a prerequisite for nonlinear pricing) a relationship exists and it mimics the equation marginal revenue equals marginal cost. This relationship can be derived by introducing an average willingness to pay function 1

3 (AWP). For comparison between linear and nonlinear pricing three functions, namely the AWP, demand, and the MR should be considered. Because consumers do not pay the same price for each unit, the AWP, and demand functions become the two relevant functions in determining the profit-maximizing output and total tariff under nonlinear pricing. However, since the average and marginal prices are the same under linear pricing, the demand and the marginal revenue functions determine the optimal output and price. As shown below, the demand function plays a role similar to that of the marginal revenue under nonlinear pricing. It needs to be emphasized, however, that while linear pricing always remains a viable option, nonlinear pricing requires more information about consumers, in particular, information about homogeneity of consumers is critical for nonlinear pricing. II. PROFIT MAXIMIZATION UNDER NONLINEAR PRICING Consider a monopolistic producer of a good with (inverse) market demand or the MWP as MWP = p = f(q). (1) The total revenue R = pq = f(q)q and MR is given by MR = f(q)+qf 0 (q). (2) For any quantity q, themaximum attainable revenue is the total willingness to pay (TWP) or the area under the demand curve, and can be written as TWP = Z q 0 The average willingness to pay (AW P ) is given by pdx. (3) AW P = TWP q = p = R q 0 pdx. (4) q 2

4 Equation (4) or p represents the AWP or average price function. It is easy to see that MR < p < p. Graphically, the AW P function is above the demand function, and the demand function lies above the marginal revenue function. These three functions are shown in Figure 1. When consumers are homogeneous and the monopolist uses some form of nonlinear pricing, the total profits can be written as Maximize q Π = Z q 0 pdx C(q) =pq C(q). (5) The first-order condition yields p = MC,where MC is the marginal cost. Note that, under nonlinear pricing, p = MC represents the condition which is similar to marginal revenue equals marginal cost, except that now p replaces MR. The profit-maximizing quantity q is the solution of the equality p = MC. The optimal average price p can be obtained from the AW P function. After finding the optimal quantity and the average price, suitable pricing strategies such as two-part tariff, and package pricing, can be designed taking into considerations the type of product and ease of implementation. Essentially, designs of different nonlinear pricing strategies amount to partitioning the area under the demand curve for the optimal quantity in different ways. For example, the total willingness to pay for the optimal quantity q is partitioned in two pays, namely entry fee and usage fee, under two-part tariff; while a single partition (one payment) is used in package pricing or pure bundling. The optimal profits Π = p q q AC. Let η = dq p dp q AW P. The relationship between p and p can be given by 1 p =, which is the price elasticity of η p. (6) η 1 Note that the relationship between p and p,given above, is similar to the familiar relationship between p and MR (p = η η 1MR). Since at the profit-maximizing 1 d p = qp pdx dq q 2 = p p q or, d p q = p p = 1 η. dq p p 3

5 equilibrium p equals MC,wehavetherelationshipbetweenp and MC in terms of elasticity as follows p = η MC. (7) η 1 Note again that at the profit-maximizing equilibrium under nonlinear pricing, the relationship between p and MC also mimics the relationship that exists under linear pricing except that η is replaced by η, the elasticity of demand of average price. III. GRAPHICAL ILLUSTRATION For pedagogical purposes both linear and nonlinear pricing are illustrated using linear demand and linear total cost (constant marginal cost) functions. Figure 1 shows AW P, D, MR, and MC functions. The profit-maximizing quantity q under linear pricing is determined by equating MR to MC and the quantity q under nonlinear pricing is determined by equating p to MC. Further, when MC is constant, the profit-maximizing price p and p are equal, but q =2q. Figure 1 clearly shows that under nonlinear pricing there is no deadweight loss and consumer surplus is also zero. The output produced is the same as in perfect competition but without any consumer surplus. Figure 1 can be used to stress that the fact that monopoly, per se, is not the reason for the deadweight loss. The cause for deadweight loss is linear pricing and the reason why a monopolist uses linear pricing is because consumers are heterogeneous and they cannot be sorted into groups of homogeneous consumers. IV. PRICE DISCRIMINATION When buyers can be separated using some identifiable characteristics such as gender, age, profession etc., and, at the same time, arbitrage can be prevented, then a 4

6 monopolist can use price discrimination. If the pricing is still linear for each segmented groups of consumers the price discrimination is called third-degree price discrimination. Because heterogeneity exists within each segmented group in a thirddegree price discrimination, the pricing remains linear. The textbook result namely MR 1 = MR 2 =... = MR n = MC is the profit-maximizing condition to determine the output in each segmented market. When all conditions of third degree price discrimination are met, and additionally, when the identified consumers in each segmented market or group are homogeneous; then price discrimination with nonlinear pricing can be used, and the profit-maximizing condition resembles the one under third-degree price condition. This condition under nonlinear price discrimination becomes p 1 = p 2 =... = p n = MC. Note that the condition under nonlinear price discrimination also resembles the condition under third-degree price discrimination. 2 The difference is that under third-degree price discrimination there exists deadweight loss in each market but the output under price discrimination with nonlinear pricing is efficient. In addition to the two usual requirements of no arbitrage and market segmentation for price discrimination, group homogeneity is the added requirement for efficiency. Under third-degree price discrimination the inverse elasticity rule determines which group pays the higher discriminatory linear price the higher the price elasticity of demand, the lower the price. The same rule also applies under price discrimination with nonlinear pricing the higher the elasticity of AW P, the lower the 2 It should be noted that when consumers cannot be identified nonlinear price discrimination can still be used. When monopolist cannot identify the consumers, i.e., who-is-who is unknown, a monopolist offers a menu of options and consumers self-select. This type of price discrimination, generally known as second-degree price discrimination can take many different forms and the profit-maximizing condition becomes a particular pricing strategy specific. Optimal conditions under different types of pricing strategies is beyond the scope of this note. 5

7 price. In other words, consumers with higher elasticity get a quantity discount. V. CONCLUDING REMARKS The main contribution of this pedagogical note is to demonstrate that the economic principle governing linear and nonlinear pricing can be explained by considering average price, demand and marginal revenue functions on one diagram. Although it is true that the set of nonlinear pricing strategies, in general, may be quite large, the average price the consumers pay plays the central role in the design of any nonlinear pricing strategy. Many goods are sold using packages on a take-it-or-leave-it basis. The total outlay for each such package is based on the consumers total willingness to pay. However, it is a common observation that in most supermarkets, in addition to the outlay for a pre-determined quantity by the seller, or the total payment for a particular package, the average price is generally postedontheshelves. Generally, a bigger package is sold at a lower average price. On the other hand, when pricing is linear the price is quoted in dollars per unit and consumers get no quantity discount as price remains the same for each unit. 6

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