Ramsey problem

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The Ramsey problem is a policy rule by Frank Ramsey concerning what price a monopolist should set, in order to maximize social welfare, subject to a constraint on profit. A closely related problem arises in relation to optimal taxation of commodities.

The rule states that the price markup should be inverse to the price elasticity of demand: The more elastic demand for the product, the smaller the price markup. It is applicable to public utilities or regulation of natural monopolies, such as telecom firms.

[edit] Formal presentation and solution

Consider the problem of a regulator seeking to set prices \left( p_{1},...p_{N}\right) for a multi-product monopolist with costs C\left( z_{1},z_{2}....,z_{N}\right) =C\left( \mathbf{z}\right) where zn is the output of good n and pnis the price. Suppose that the products are sold in separate markets (this is commonly the case) so demands are independent, and demand for good n is z_{n}\left( p_{n}\right) , with inverse demand function p_{n}\left( z\right) .Total revenue is

R\left( \mathbf{p,z}\right) =\sum_{n}p_{n}z_{n}\left( p_{n}\right)

Consumer surplus is given by

W\left( \mathbf{p,z}\right) =\sum_{n}\left( \int\limits_{0}^{z_{n}\left( p_{n}\right) }p_{n}\left( z\right) dz\right) -C\left( \mathbf{z}\right)

The problem is to maximize W\left( \mathbf{p,z}\right) subject to the requirement that profit Π = R − C should be equal to some fixed value Π * %

R\left( \mathbf{p,z}\right) -C\left( \mathbf{z}\right) =\Pi ^*

This problem may be solved using the Langrange multiplier technique to yield the optimal output values, and backing out the optimal prices. The first order conditions on \mathbf{z} are

p_{n}-C_{n}\left( \mathbf{z}\right) =\lambda \left( \frac{\partial R}{\partial z_{n}}-C_{n}\left( \mathbf{z}\right) \right)

= \lambda \left( p_{n}\left( 1+z_{n}\frac{\partial p_{n}}{\partial z_{n}}\right)-C_{n}\left( \mathbf{z}\right) \right)

where λ is a Lagrange multiplier.

Dividing by pn and rearranging yields

\frac{p_{n}-C_{n}\left( \mathbf{z}\right) }{p_{n}}=-\frac{k}{\varepsilon _{n}}

where k=\frac{\lambda }{1+\lambda } and \varepsilon _{n}=\frac{\partial z_{n}}{\partial p_{n}}\frac{p_{n}}{z_{n}} is the elasticity of demand for good n. That is, the price markup over marginal cost for good n is inversely proportional to the elasticity of demand.

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