Adjusted concavity and the output effect under monopolistic price discrimination.

Xinghe Wang

I. Introduction

The motive of engaging in third degree monopolistic price discrimination is rather simple, but its welfare implication is not. A necessary condition for price discrimination to improve welfare, as demonstrated by Schmalensee |5~ and Varian |8~, is that output increases under price discrimination. Because of this well-known result, a focal point of analysis of the social impact of price discrimination has been on the direction of change in output. The few general rules advanced to determine output change under price discrimination can be applied to three different situations: (i) all demand curves are linear, (ii) demand curves fall into two groups with opposing general curvatures, and (iii) all demand curves have similar general curvature.

Definite answers have been obtained for the first two cases.(1) Regarding the third case, less complete results have been developed. Robinson |4~ introduces the "adjusted concavity" criterion that the direction of output change is determined by the relative size of the adjusted concavity of the two demand curves valued at the simple monopoly output levels. This criterion, as noted by Robinson herself and further elaborated by Edwards |1~, depends critically upon the local assumption that the demand elasticities in the two markets are not very different.(2) Greenhut and Ohta |3~ confirm that, without the local assumption, the adjusted concavity criterion could lead to a wrong prediction of output change by constructing an example with two constant elasticity demand functions. Formby, Layson, and Smith |2~ further show that the adjusted concavity criterion always leads to a wrong prediction for the class of constant elasticity demand functions. Schmalensee |5~ then claims that there is apparently no simple criteria to determine the output effect of price discrimination for the general multiple markets situation.(3) However, Shih, Mai, and Liu |6~ demonstrate that a general criterion can be obtained.

The general criterion set forth by Proposition 2 in Shih, Mai, and Liu requires information about the slopes of the marginal revenue functions at some output levels which are between simple monopoly and discriminating output levels. A more applicable version--their Corollary 2.3--is obtained by imposing conditions on the rate of change of the slope of the marginal revenue functions, which are conditions on the third derivatives of the demand functions. Furthermore, their analysis leads them to conclude that "Robinson's adjusted concavity ... alone can not determine the output effect" |6, 157~.

In this paper, it is demonstrated that Robinson's adjusted concavity can lead to definite conclusions on the output effect of price discrimination. Specifically, if the maximum (minimum) value of adjusted concavity over the range of output levels between the simple monopoly and discriminatory outputs in each of the weak markets is less (greater) than or equal to the minimum (maximum) value of adjusted concavity over the corresponding ranges of output in all of the strong markets, then total output under discrimination will be greater (less) than that under simple monopoly when all demand curves are strictly convex (concave). For the cases where the adjusted concavity condition is retained and the general curvature of the demand curves is reversed, examples are provided to show that definite answers can not be obtained.

II. Analysis

Consider a monopolist producing a homogeneous good at a constant unit cost c and selling it in n separate markets with (inverse) demand function |p.sub i~(|q.sub.i~) for market i (i = 1,....,n). If third degree price discrimination is allowed, the firm chooses an output |q.sub.i~ to maximize its profit ||Pi~.sub.i~(|q.sub.i~) = ||p.sub.i~(|q.sub.i~) - c~|q.sub.i~ in market i. Assume that all profit functions are strictly concave (i.e., ||Pi~|double prime~.sub.i~(|q.sub.i~) |is less than~ 0). The first order conditions are

|p|prime~.sub.i~(|q.sub.i~)|q.sub.i~ + |p.sub.i~(|q.sub.i~) - c = 0. (1)

Dividing (1) by |p|prime~.sub.i~(|q.sub.i~) and summing over i give the total output

|Mathematical Expression Omitted~,

where hat ( ) of a variable denotes its optimal value under price discrimination.

Under simple monopoly, the firm chooses a total output q to maximize its total profit |p(q) - c~q, where p(q) is the (inverse) aggregate demand function. Assume that all markets are served under simple monopoly. The first order condition gives the total output

|Mathematical Expression Omitted~,

where bar (???) of a variable denotes its optimal value under simple monopoly.

The effect of price discrimination on output can be obtained by comparing (3) with (2). To begin with, decompose the slope of the aggregate demand function in (3) in terms of the slopes of individual demand functions as follows. At the equilibrium price |Mathematical Expression Omitted~, the output of each market |Mathematical Expression Omitted~ is determined by |Mathematical Expression Omitted~. Let |d.sub.i~(p) be the direct demand function for market i and |Mathematical Expression Omitted~ be the direct aggregate demand function. Since |Mathematical Expression Omitted~, (3) can be written as

|Mathematical Expression Omitted~.

Subtracting (4) from (2) gives

|Mathematical Expression Omitted~.

Thus, the change in total output due to price discrimination is

|Mathematical Expression Omitted~,

where |L.sub.i~(|q.sub.i~) |is equivalent to~ ||p.sub.i~(|q.sub.i~) - c~/|p.sub.i~(|q.sub.i~) is the Lerner index of market i, ||Epsilon~.sub.i~(|q.sub.i~) |is equivalent to~ -|p.sub.i~(|q.sub.i~)/||q.sub.i~|p|prime~.sub.i~(|q.sub.i~)~ is the demand elasticity for market i, and |R.sub.i~(|q.sub.i~) |is equivalent to~ |q.sub.i~|p|double prime~.sub.i~(|q.sub.i~)/|p|prime~.sub.i~(|q.sub.i~) is the adjusted concavity of market i. Thus, the output effect of price discrimination is determined by the weighed sum of the change in output in each market, where the weights are given by the product of the Lerner index, the demand elasticity and the adjusted concavity in each market. This indicates the complexity of determining the output effect of price discrimination. In order to obtain some general rules, one will need to place conditions on these elements.

As usual, weak markets are defined to be those where price decreases under discrimination and strong markets those where price increases under discrimination. Let W be the collection of weak markets and S the collection of strong markets. Note that, in (5), for all weak markets the upper limit of the corresponding integration exceeds the lower limit and for all strong markets the lower limit exceeds the upper limit.

It is obvious that if all demand functions are linear (i.e., |p|double prime~.sub.i~ = 0, or equivalently, |R.sub.i~(|q.sub.i~) = 0) then the integrands in (5) all vanish and thus total output remains unchanged under price discrimination. Since |L.sub.i~(|q.sub.i~) and ||Epsilon~.sub.i~(|q.sub.i~) are both positive, it is also easy to see that if for all strong markets i, |R.sub.i~(|q.sub.i~) |is greater than or equal to~ 0 (i.e., |p|double prime~.sub.i~ |is less than or equal to~ 0), and for all weak markets i, |R.sub.i~(|q.sub.i~) |is less than~ 0 (i.e., |p|double prime~.sub.i~ |is greater than~ 0), then every integral in (5) is negative, implying that total output expands under price discrimination. Thus, the following well-known proposition is obtained.

PROPOSITION 1. (Robinson |4~, Schmalensee |5~, Shih, Mai, and Liu |6~.) Suppose all markets are served under simple monopoly. If all demand curves are linear then total output remains unchanged under price discrimination. However, if the demand curves are concave in all strong markets and strictly convex in all weak markets then total output increases under price discrimination, and vice verse.

Proposition 1 can be used to predict the direction of output change under price discrimination in the cases of linear demands or two groups of demands with opposing general curvatures. However, the more concerned case in the literature has been where all demands have the same general curvature. To analyze this latter case, we first examine the (product) function |L.sub.i~(|q.sub.i~)||Epsilon~.sub.i~(|q.sub.i~). A refreshment of the inverse elasticity rule suggests that this function is related to the marginal profit function and thus some of its properties can be obtained from the concavity of the profit function. In fact, the marginal profit function can be written as

||Pi~|prime~.sub.i~(|q.sub.i~) = |

q.sub.i~) - 1~. (6)

Since ||Pi~|prime~.sub.i~(|q.sub.i~) is a decreasing function in |q.sub.i~, by (1) and the fact that |-|p|prime~.sub.i~(|q.sub.i~)|q.sub.i~~ |is greater than~ 0, (6) implies that ||L.sub.i~(|q.sub.i~)||Epsilon~.sub.i~(|q.sub.i~) - 1~ |is greater than~ (=, |is less than~) 0 if and only if |Mathematical Expression Omitted~. Thus, |L.sub.i~(|q.sub.i~)||Epsilon~.sub.i~(|q.sub.i~) |is greater than~ 1 if i |is an element of~ W and |Mathematical Expression Omitted~, and |L.sub.i~(|q.sub.i~)||Epsilon~.sub.i~(|q.sub.i~) |is less than~ 1 if i |is an element of~ S and |Mathematical Expression Omitted~.

When all demand curves are strictly convex (i.e., |R.sub.i~(|q.sub.i~) |is less than~ 0 for all i), replacing |L.sub.i~(|q.sub.i~)||Epsilon~.sub.i~(|q.sub.i~) in (5) by 1 leads to

|Mathematical Expression Omitted~.

Let |Mathematical Expression Omitted~; |Mathematical Expression Omitted~, and |Mathematical Expression Omitted~; |Mathematical Expression Omitted~. If |Mathematical Expression Omitted~, then replacing |R.sub.i~(|q.sub.i~) in (7) by |Mathematical Expression Omitted~ implies

|Mathematical Expression Omitted~.

Since ||Pi~|double prime~.sub.i~(|q.sub.i~) = 2|p|prime~.sub.i~(|q.sub.i~) + |q.sub.i~|p|double prime~.sub.i~(|q.sub.i~) |is less than~ 0 implies that |R.sub.i~(|q.sub.i~) |is greater than~ -2, it follows that |Mathematical Expression Omitted~. Therefore, (8) leads to |Mathematical Expression Omitted~.

When all demand curves are strictly concave (i.e., |R.sub.i~(|q.sub.i~) |is greater than~ 0 for all i), similar derivation as above shows that if |Mathematical Expression Omitted~, where |Mathematical Expression Omitted~; |Mathematical Expression Omitted~, and |Mathematical Expression Omitted~; |Mathematical Expression Omitted~, then (5) implies

|Mathematical Expression Omitted~.

Since |Mathematical Expression Omitted~, it follows from (9) that |Mathematical Expression Omitted~. The following proposition summarizes these results.

PROPOSITION 2. Suppose all markets are served under simple monopoly. If all demand curves are strictly convex (concave) and the maximum (minimum) value of adjusted concavity in each of the weak markets is less (greater) than or equal to the minimum (maximum) value of adjusted concavity in all of the strong markets, i.e., |Mathematical Expression Omitted~, then total output under discrimination will be greater (less) than that under simple monopoly.

Proposition 2 can be applied, for example, to the class of constant adjusted concavity demand curves, introduced by Shih, Mai, and Liu |6~, |Mathematical Expression Omitted~(i = 1,..., m, |a.sub.i~, |b.sub.i~ |is greater than~ 0, R |is greater than~ -1), where R is the constant adjusted concavity for all markets. Since |Mathematical Expression Omitted~ and all demand curves are strictly convex (concave) if R |is less than~ 0 (R |is greater than~ 0), Proposition 2 implies that total output under discrimination will be greater (less) than that under simple monopoly if R |is less than~ 0 (R |is greater than~ 0).

Proposition 2 indicates that Robinson's adjusted concavity criterion, which applies only to situations where the demand elasticities in different markets are nearly equal, can be partially generalized to cover situations where the local assumption is relaxed. Specifically, consider the case where the minimum value of adjusted concavity in each of the weak markets is greater than or equal to the maximum value of adjusted concavity in all of the strong markets (i.e., |Mathematical Expression Omitted~). When all demand curves are strictly concave, Proposition 2 implies that total output decreases under price discrimination. When all demand curves are strictly convex, however, total output does not necessarily decrease under price discrimination (which differs from what Robinson's criterion would predict if it were fully generalized), as shown by the examples in the next section. Similar statements can be made for the case where |Mathematical Expression Omitted~.

III. Examples

Examples 1 and 2 show that if |Mathematical Expression Omitted~ then total output can either increase or decrease under price discrimination when all demand curves are strictly convex; examples 3 and 4 show that if |Mathematical Expression Omitted~ then total output can either increase or decrease under price discrimination when all demand curves are strictly concave.

Example 1

Consider the class of strictly convex demand functions |p.sub.i~(|q.sub.i~) = |a.sub.i~|e.sup.-|b.sub.i~|q.sub.i~~, |a.sub.i~, |b.sub.i~ |is greater than~ 0 (i = 1,...,n). Assume that all markets are served under simple monopoly (this is satisfied if, for example, |a.sub.i~ = a |is greater than~ c for all i). Market i's demand elasticity is ||Epsilon~.sub.i~(|q.sub.i~) = 1/||b.sub.i~|q.sub.i~~ which is decreasing in |q.sub.i~; its adjusted concavity is |R.sub.i~(|q.sub.i~) = -|b.sub.i~|q.sub.i~. Since weak markets have relatively higher elasticity under price discrimination, it follows that |Mathematical Expression Omitted~ for all i |is an element of~ W and for all j |is an element of~ S. Thus, |b.sub.i~|q.sub.i~ |is less than~ |b.sub.j~|q.sub.j~ for all i |is an element of~ W, |Mathematical Expression Omitted~, and for all j |is an element of~ S, |Mathematical Expression Omitted~; it follows that |Mathematical Expression Omitted~. Since for any i, ||Epsilon~.sub.i~(|q.sub.i~)|R.sub.i~(|q.sub.i~) = -1, |L.sub.i~(|q.sub.i~) is decreasing in |q.sub.i~, and |Mathematical Expression Omitted~ (here |Mathematical Expression Omitted~ is the price under simple monopoly), (5) implies

|Mathematical Expression Omitted~.

Therefore, |Mathematical Expression Omitted~. This example shows that if |Mathematical Expression Omitted~, total output can decrease under discrimination when all demand curves are strictly convex.

Example 2

Consider the class of strictly convex (constant elasticity) demand functions |Mathematical Expression Omitted~, ||Epsilon~.sub.i~ |is greater than~ 1, |b.sub.i~ |is greater than~ 0 (i = 1,...,n) where ||Epsilon~.sub.i~ is the demand elasticity of market i. It is easy to see that all markets are served under simple monopoly. Assume, without loss of generality, that ||Epsilon~.sub.1~ |is less than~ ||Epsilon~.sub.2~ |is less than~ |center dot~|center dot~|center dot~ |is less than~ ||Epsilon~.sub.n~ and that the number of strong markets is s(|is less than~ n). This implies that markets 1 to s are the strong markets and markets s + 1 to n are the weak markets. Since |R.sub.i~(|q.sub.i~) = -(1 + 1/||Epsilon~.sub.i~), it is obvious that |Mathematical Expression Omitted~. Formby, Layson, and Smith |2~ show that, for a large variety of elasticities, total output increases under discrimination. This example shows that if |Mathematical Expression Omitted~, total output can increase under discrimination when all demand curves are strictly convex.

Example 3

Consider the class of strictly concave demand functions |p.sub.i~(|q.sub.i~) = |(|a.sub.i~ - |b.sub.i~|q.sub.i~).sup.1/2~, |a.sub.i~, |b.sub.i~ |is greater than~ 0 (i = 1,...,n). Assume that all markets are served under simple monopoly (this is satisfied if, for example, |a.sub.i~ = a |is greater than~ |c.sup.2~ for all i). Market i's demand elasticity is ||Epsilon~.sub.i~(|q.sub.i~) = 2(|a.sub.i~ - |b.sub.i~|q.sub.i~)/(|b.sub.i~|q.sub.i~), which is decreasing in |q.sub.i~; its adjusted concavity is |R.sub.i~(|q.sub.i~) = |b.sub.i~|q.sub.i~/|2(|a.sub.i~ - |b.sub.i~|q.sub.i~)~, which is increasing in |q.sub.i~. Note that ||Epsilon~.sub.i~(|q.sub.i~)|R.sub.i~(|q.sub.i~) = 1 for all i. Since weak markets have relatively higher elasticity under price discrimination, i.e., |Mathematical Expression Omitted~ for all i |is an element of~ W and for all j |is an element of~ S, it follows that ||Epsilon~.sub.i~(|q.sub.i~) |is greater than~ ||Epsilon~.sub.j~(|q.sub.j~) for all i |is an element of~ W, |Mathematical Expression Omitted~, and for all j |is an element of~ S, |Mathematical Expression Omitted~; this implies that |Mathematical Expression Omitted~. Since for any i, ||Epsilon~.sub.i~(|q.sub.i~)|R.sub.i~(|q.sub.i~) = 1, |L.sub.i~(|q.sub.i~) is decreasing in |q.sub.i~, and |Mathematical Expression Omitted~, (5) implies

|Mathematical Expression Omitted~.

Therefore, |Mathematical Expression Omitted~. This example shows that if |Mathematical Expression Omitted~, total output can increase under discrimination when all demand curves are strictly concave.

Example 4

Consider two strictly concave demand functions |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~. The adjusted concavities of markets 1 and 2 are |R.sub.1~(|q.sub.1~) = 1 and |R.sub.2~(|q.sub.2~) = 2. Simple calculation yields:

|Mathematical Expression Omitted~

Therefore, both markets are served and |Mathematical Expression Omitted~. Since market 1 is the weak market, market 2 is the strong market, and |R.sub.1~(|q.sub.1~) |is less than~ |R.sub.2~(|q.sub.2~) for all |q.sub.1~ and |q.sub.2~, this example shows that if |Mathematical Expression Omitted~, total output can decrease under discrimination when all demand curves are strictly concave.

IV. Concluding Remarks

This paper, using straightforward calculus, analyzed the effect of third degree price discrimination on output. Several elements were identified to determine this effect; these were the Lerner index, the demand elasticity and the adjusted concavity. A general criterion was then developed to determine the output effect of discrimination based on the adjusted concavity concept suggested by Robinson sixty years ago.

This result adds to the existing rule advanced by Shih, Mai, and Liu |6~ in providing general conclusions on the output effect of price discrimination when all demand curves have the same general curvature. In particular, the present analysis, for the first time, has shown that convexity and concavity of the demand curves may have different implications on the output effect of price discrimination. Furthermore, Proposition 2 underlines the true reason why Greenhut and Ohta |3~ and Formby, Layson, and James |2~ found counter-examples to Robinson's adjusted concavity criterion, as illustrated in the argument developed in Example 2.

1. The results, as summarized by Shih, Mai and Liu |6~, are reproduced in Proposition 1 below.

2. As observed by Robinson |4, 193~, this local assumption amounts to assuming linear marginal revenue curves. Shih, Mai and Liu |6, 154~ further elaborate on this point.

3. In the case of two markets, Edwards |1~ obtains several sufficient global conditions on the rates of change of marginal revenue functions to determine the direction of output change under discrimination; Smith and Formby |7~ demonstrate the generality of the SMR-DMR criterion in deciding the output effect of discrimination.

References

1. Edwards, Edgar O., "The Analysis of Output under Discrimination." Econometrica, April 1950, 161-72.

2. Formby, John P., Stephen K. Layson and W. James Smith, "Price Discrimination, 'Adjusted Concavity', and Output Changes under Conditions of Constant Elasticity." Economic Journal, December 1983, 892-99.

3. Greenhut, Melvin L. and Hiroshi Ohta, "Joan Robinson's Criterion for Deciding Whether Market Discrimination Reduces Output." Economic Journal, March 1976, 96-97.

4. Robinson, Joan. The Economics of Imperfect Competition. London: Macmillan, 1933.

5. Schmalensee, Richard, "Output and Welfare Implications of Monopolistic Third Degree Price Discrimination." American Economic Review, March 1981, 242-47.

6. Shih, Jun-ji, Chao-cheng Mai and Jang-chao Liu, "A General Analysis of the Output Effect under Third Degree Price Discrimination." Economic Journal, March 1988, 149-58.

7. Smith, James W. and John P. Formby, "Output Changes under Third Degree Price Discrimination: A Reexamination." Southern Economic Journal, 1981, 164-71.

8. Varian, Hal R., "Price Discrimination and Social Welfare." American Economic Review, September 1985, 870-75.