﻿ An Economic Welfare Analysis of Price Discrimination
San José State University
Department of Economics

applet-magic.com
Thayer Watkins
Silicon Valley
USA

 An Economic Welfare Analysis of Price Discrimination

This is an attempt to find the social welfare impact of the arbitrary setting of different prices for different categories of users. The case in mind is where a government agency sets one price for water going to cities and a lower price going to agriculture.

Suppose the market is initially in equilibrium with both groups paying the price P0 as shown in the diagram below.

To simplify the analysis suppose the government sets the prices of water for two market such that the increase in quantitiy demanded for the group getting the lower price is exactly the same as the decrease in quantity demanded for the group experiencing the higher price. This assures the market is exactly cleared. After the price is lowered to P1 for the first group and raised to P2 for the second group the situation is as shown below.

An economic welfare analysis then compares the gain in benefits for Market Group 1 with the loss for Market Group 2. The changes in benefits are the areas under the demand curves for the two groups. These are as shown.

Clearly there is a net social loss because the product is being transferred from one group where its marginal value is greater than P0 to the other group where its value is less than P0. The net social loss is the area of the quadrilateral whose width is ΔQ and the average of the heights of its two sides is

#### ½[(P2-P0)+(P0-P1)] = ½[P2-P1] = ½ΔP

Thus the social welfare loss of the price discrimination is

#### ½(ΔP)(ΔQ).

where ΔP is the price differential between the two groups and ΔQ is the quantity transferred between the two groups by the price differential.

To reduce matters to their simplest consider the transfer of the last unit from the low price consumers to the high price consumer. With these two prices being P1 and P2, as in the above diagrams, the loss to the low price consumers is P1 and the gain to the high consumers is P2. If the price the low price consumers sell the unit to the high price consumers is anything above P1 there is a gain for them and if the sale price is below P2 then there is a gain for the high price consumers. Together the two groups of consumers will have a combined gain of ΔP=P2−P1 for the transfer of that last unit. In other words, the beneficiaries of lower prices under price discrimination will benefit from that discrimination but they would benefit more if they were allowed to sell some of their consumption to the high price consumers.

It might be thought that the welfare conclusion of the preceding argument are limited because it assumed a fixed level of production. This is not the case. The preceding used that assumption only to simplify the presentation. Consider the level Q to be the output chosen under discrimination and then carry out the analysis of the transfer of output to the point of market equilibrium. The net gain for a transition to equilibrium is the same as the net loss of a transition from equilibrium to price discrimination.

## The Difficulties in Getting the Correct Result Using Consumer Surplus

A commonly used but flawed method of performing economic welfare analysis is to look at the changes in consumer surplus. For the above case the results are as shown below.

Clearly in this case the increase in consumer surplus for the group which gets the discriminatory lower price is greater than the decrease in consumer surplus for the group paying the higher price. This makes it look as though the welfare change depends upon the relative sizes of the two demand functions and their elasticities. This is not correct.

What was left out of the welfare analysis was the changes in revenue to the selling agency. These are shown below.

The revenue at the price P0 and the quanities Q1 and Q2 are common to the revenues after the price changes and cancel out. That leaves the quantities shown for gains and losses.

When these gains and losses in revenues are combined with the gain and loss in consumer surplus the results are as shown below.

This brings the results into agreement with the previous analysis which showed unambiguously that there is a net social loss from price discrimination.

The labels for the triangles in the above diagram are a bit inaccurate. They represent the differences between the gain or loss in consumer surplus and the gain or loss of revenue from the price changes. Likewise what is labeled a gain or loss in revenue is not a net gain or loss but the change in revenue from the transfer of ΔQ from one group to the other.

There is always a danger in evaluating social welfare changes in terms of changes in consumer and producer surpluses in leaving out of the analysis some essential component.

There are cases of the failure of analysis in terms of consumer and producer surpluses to give the correct result. Two other examples are given at