22/06/2025
"Price Discrimination, Elasticity, and Monopolistic Profit Maximization
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A Modern Microeconomic Perspective"
If price discrimination is applied, only the additional units are sold at gradually lower prices, and hence the MR shifts to the position MR'. The new equilibrium position is defined by q', at which the quantity is 0X' > 0X. In the limiting case of perfect discrimination, the MR will coincide with the DD' curve, since each unit is sold at its own price, the highest price that the buyers are willing to pay on a 'take-it-or-leave-it' basis. The equilibrium will be at e', the output will be 0X' and the following condition will hold:
MC = MR = AR = P
and the seller will have achieved the maximum increase in his revenue, reaping all consumers' surplus.
Mathematical derivation of the equilibrium position of the price-discriminating monopolist
Given the total demand of the monopolist:
P = f(X)
Assume that the demand curves of the segmented markets are:
P1 = f1(X1) and P2 = f2(X2)
The cost of the firm is:
C = f(X) = f(X1 + X2)
Price Discrimination
The firm aims at the maximisation of its profit:
π = R1 + R2 – C
The first-order condition for profit maximisation requires:
∂π/∂X1 = 0 and ∂π/∂X2 = 0
That is:
(a)
∂π/∂X1 = ∂R1/∂X1 – ∂C/∂X1 = 0 and
∂π/∂X2 = ∂R2/∂X2 – ∂C/∂X2 = 0
or
(b)
MR1 = MC1 and MR2 = MC2
But:
MC1 = MC2 = MC
Therefore:
MR1 = MR2 = MC
The second-order condition for profit maximisation requires:
∂²R1/∂X1² < ∂²C/∂X1² and
∂²R2/∂X2² < ∂²C/∂X2²
That is, the MR in each market must be increasing less rapidly than the MC for the output as a whole.
A numerical example
We use the same basic equations as in the example of the simple monopolist (p.176) so as to be able to compare results.
Assume that the total demand is:
X = 50 – 0.5P or P = 100 – 2X
Assume further that the demand functions of segmented markets are:
X1 = 32 – 0.4P1 or P1 = 80 – 2.5X1
X2 = 18 – 0.1P2 or P2 = 180 – 10X2
(Clearly X1 + X2 = X)
Finally, assume that the cost function is:
C = 50 + 40X = 50 + 40(X1 + X2)
The firm aims at the maximisation of its profit:
π = R1 + R2 – C
(1)
R1 = X1 * P1 = X1(80 – 2.5X1) = 80X1 – 2.5X1²
MR1 = ∂R1/∂X1 = 80 – 5X1
(2)
R2 = X2 * P2 = X2(180 – 10X2) = 180X2 – 10X2²
MR2 = ∂R2/∂X2 = 180 – 20X2
(3)
MC = ∂C/∂X = ∂C/∂(X1 + X2) = 40
Setting MR in each market equal to the common MC we obtain:
80 – 5X1 = 40 → X1 = 8
180 – 20X2 = 40 → X2 = 7
The prices are:
P1 = 80 – 2.5X1 = 80 – 2.5(8) = 60
P2 = 180 – 10X2 = 180 – 10(7) = 110
The profit is:
π = R1 + R2 – C = 500
The elasticities are:
e1 = ∂X1/∂P1 * P1/X1 = (–0.4) * 60/8 = 3
e2 = ∂X2/∂P2 * P2/X2 = (–0.1) * 110/7 = 1.57
Thus e1 > e2 and P1 < P2.
Comparing the above results with those for the example of the simple monopolist, we observe that X is the same in both cases but the profit (π) of the discriminating monopolist is larger.
IV. PRICE DISCRIMINATION AND THE PRICE ELASTICITY OF DEMAND
We have established (p. 173) that:
MR = P (1 – 1/e)
In the case of price discrimination, we have:
MR1 = P1 (1 – 1/e1)
MR2 = P2 (1 – 1/e2)
And:
MR1 = MR2
Therefore:
P1 (1 – 1/e1) = P2 (1 – 1/e2)
Where:
e1 = elasticity of D1
e2 = elasticity of D2
If e1 = e2, the ratio of prices is equal to unity:
P1 = P2
That is, if elasticities are the same, price discrimination is not possible.
The monopolist will charge a uniform price for his product.
If price discrimination is possible, price will be higher in the market where demand is less elastic.
This is obvious from the equality of the MRs:
P1 (1 – 1/e1) = P2 (1 – 1/e2)
If e1 > e2, then:
(1 – 1/e1) < (1 – 1/e2)
Thus for the equality of MRs to be fulfilled:
P1 < P2
So, the market with the higher elasticity will have the lower price.
In modern microeconomics, understanding how firms with market power operate is essential to evaluating real-world pricing strategies. The concept of price discrimination—where a monopolist charges different prices to different consumer groups for the same product—provides a clear window into how firms extract consumer surplus and enhance profits without increasing output. This topic sits at the intersection of market structure analysis, welfare economics, and behavioral pricing theory.
The mathematical derivations and numerical examples explored above are not just academic exercises; they reflect actual strategies used by businesses today. From airline tickets and software licenses to education fees and utility pricing, price discrimination based on elasticity of demand is widespread. Modern economists analyze this behavior using marginal analysis, elasticity calculations, and consumer behavior insights—just as shown in the derivations of marginal revenue and cost alignment (MR = MC) across segmented markets.
Additionally, this topic aligns with modern tools in industrial organization, a field that investigates how firms compete and how market structure influences pricing and output decisions. Price discrimination is considered a strategy that can improve efficiency in some cases, while in others, it may raise ethical or regulatory concerns—particularly when it exploits consumers with fewer options.
For students of economics, studying price discrimination and elasticity not only builds a strong foundation in theory but also develops practical understanding of how monopolistic firms operate in reality. This topic teaches the importance of marginal thinking, shows how market segmentation can lead to different pricing outcomes, and introduces the role of elasticity in determining consumer behavior. Students learn that a firm does not need to increase output to boost profits—it can simply alter its pricing based on consumer responsiveness. Moreover, the link between mathematical rigor and economic intuition becomes clearer as students work through real-world equations and outcomes. By mastering this area, students gain valuable insights into how firms make strategic decisions, how consumers are affected, and how policy makers might respond to protect welfare and market fairness. It is a powerful illustration of how abstract economic models are deeply connected to the real economic world.
