Question

A natural gas LDC in a certain region has two types of costumers: residential (R) and commercial (C). Their demands are given by Qr = 2 − Pr and Qc = 1 − 1/ 4*Pc, where bot Qr and Qc are in millions and P is in dollars. The firm’s only cost is a fixed cost of 1 million dollars.

Given the above demands, derive Ramsey Pricing.

Answer 1

Qr = 2 − Pr and Qc = 1 − 1/ 4*Pc

Total cost = $ 1 million

Marginal cost(c) = 0

If price is set is equal to MC, the producer would incur losses.

Ramsey pricing formula:

(p − c) / p = − k / ε

where ε is the price elasticity of demand and k is a positive constant.

This formula tells us that if a regulator wants to meet a break-even constraint to encourage a firm to produce for both types of consumers, prices should be increased in proportion to the elasticities.

Calculating price elasticity of demand for both Qr and Qc

ε =

Hence,

ε(r) = -1*Pr /(2-Pr)

ε(c) = -1/4 * Pc / (1-1/4*Pc)

For two groups, set the ratio of the mark-ups equal to the ratio of demand elasticities (divide Ramsey pricing formula for r by the Ramsey pricing for c).

((pr − c) / p1)/ ((pc − c) / p2) = ε(c)/ ε(r)

As c = 0, LHS equals to 1, we have

ε(c) = ε(r)

-1*Pr /(2-Pr) = -1/4 * Pc / (1-1/4*Pc)

Solving we get,

Pc = 2 Pr

Thus prices will be raised above MC in this proportion such that the total cost is covered.