Ramsey problem

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The Ramsey problem or Ramsey-Boiteux pricing, is a policy rule 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.
For any monopoly, the price markup should be inverse to the price elasticity of demand: the more elastic demand for the product, the smaller the price markup. This was stated by J. Robinson (1933) but it has been recognized later that Frank Ramsey has found the result before (1927) in another context (taxation). The rule was later applied by Marcel Boiteux (1956) to natural monopolies (decreasing mean cost): a natural monopoly experiences profit losses if it is forced to fix its output price at the marginal cost. Hence the Ramsey-Boiteux pricing consists into maximizing the total welfare under the condition of non-negative profit, that is, zero profit. In the Ramsey-Boiteux pricing, the markup of each commodity is also inversely proportional to the elasticities of demand but it is smaller as the inverse elasticity of demand is multiplied by a constant lower than 1.
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)

Total 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 * . Typically, the fix value is zero to guarantee that the profit losses are eliminated.

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

where λ is a Lagrange multiplier and Cn(z) is the partial derivative of C(z) with respect to zn, evaluated at z.

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 } is lower than 1 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 again inversely proportional to the elasticity of demand but it is smaller. The monopoly is in a second-best equilibrium, between ordinary monopoly and perfect competition.

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