# Ramsey problem

The Ramsey problem, or Ramsey–Boiteux pricing, is a Second best policy problem concerning what price a public monopolist or a firm faced with an irremovable revenue constraint should set, in order to maximize social welfare. A closely related problem arises in relation to optimal taxation of commodities.

In a First best world, the optimal solution would be to use prices equal to marginal cost and charge an optimal lump-sum charge that would cover the fixed cost or revenue requirement. Nevertheless, this is usually impossible to implement, thus price distortion is inevitable.

This principle is applicable to pricing of goods that the government is the sole supplier of (public utilities) or regulation of natural monopolies, such as telecommunications firms. It is also applicable to situations where there is perfect competition in the private sector, but the government needs to distort the prices of the goods it provides in order to break even, or to earn a profit. In this case the "constraint" is that the revenue requirement cannot be covered by a lump-sum tax, so prices must be distorted.

## Description

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. Frank P. Ramsey found such a result in 1927 in the context of taxation. The rule was later applied by Marcel Boiteux [fr] (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, subject to Economies of Scale being exhausted. 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.

Ramsey pricing is sometimes consistent with a government’s objectives because Ramsey pricing is economically efficient in the sense that it can maximize welfare under certain circumstances. There are, however, problems with Ramsey pricing. A profit-maximizing operator will choose Ramsey prices only if all markets are equally monopolistic or equally competitive. If markets are not equally monopolistic or competitive, then the regulator has an interest in ensuring that the extent to which the operator can use Ramsey pricing is limited to groups of services that are subject to similar degrees of competition. Regulators typically do this by forming groups of services that are subject to similar degrees of competition and allowing the operator price flexibility within each service group.[citation needed]

Even though Ramsey pricing can be economically efficient, it may not be consistent with the government’s goal of providing affordable service to the poor and the rate by which prices change to achieve Ramsey-efficient prices may not be consistent with political sustainability. As a result of these two concerns, the regulator sometimes limits the operator’s ability to pursue Ramsey pricing within a service group. In the case of services to the poor, the regulator may place upper limits on the prices. In the case of services where traditional prices were different from Ramsey prices, there are equity issues in changing from the traditional pricing structure to a new structure, even if the new structure would be more efficient in an aggregate sense. In such situations, the regulator may impose pricing restrictions that prevent Ramsey pricing or that impose a slower transition to Ramsey pricing than the operator would choose left to its own devices.

Lastly, regulators often note that Ramsey pricing is a form of price discrimination—although not necessarily a bad form of price discrimination—and customers sometimes object to it on that basis. The public sometimes believes that it is unfair to cause one type of customer to pay a greater mark-up above marginal cost than another type of customer. In such situations regulators may further limit an operator’s ability to adopt Ramsey prices.[1]

Practical issues exist with attempts to use Ramsey pricing for setting utility prices. It may be difficult to obtain data on different price elasticities for different customer groups. Also, some customers with relatively inelastic demands may acquire a strong incentive to seek alternatives if charged higher markups, thus undermining the method. Politically, customers with relatively inelastic demands may also be considered as those for whom the service is more necessary or vital; charging them greater markups can be challenged as unfair. Crucially, many economists deny this, considering less vital services as unnecessary depending on its price elasticity of demand.[citation needed]

## Formal presentation and solution

A formal presentation was given by Ramsey in a journal article titled: "A Contribution to the Theory of Taxation".[2] The mathematical derivation follows:[3]

Consider the problem of a regulator seeking to set prices ${\displaystyle \left(p_{1},\ldots ,p_{N}\right)}$  for a multi-product monopolist with costs ${\displaystyle C(q_{1},q_{2},\ldots ,q_{N})=C(\mathbf {q} )}$  where ${\displaystyle q_{n}}$  is the output of good n and ${\displaystyle p_{n}}$  is 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 ${\displaystyle q_{n}\left(p_{n}\right),}$  with inverse demand function ${\displaystyle p_{n}(q).}$  Total revenue is ${\displaystyle R\left(\mathbf {p,q} \right)=\sum _{n}p_{n}q_{n}(p_{n}).}$

Total welfare is given by

${\displaystyle W\left(\mathbf {p,q} \right)=\sum _{n}\left(\int \limits _{0}^{q_{n}(p_{n})}p_{n}(q)dq\right)-C\left(\mathbf {q} \right).}$

The problem is to maximize ${\displaystyle W\left(\mathbf {p,q} \right)}$  subject to the requirement that profit ${\displaystyle \Pi =R-C}$  should be equal to some fixed value ${\displaystyle \Pi ^{*}}$ . Typically, the fixed value is zero to guarantee that the profit losses are eliminated.

${\displaystyle R(\mathbf {p,q} )-C(\mathbf {q} )=\Pi ^{*}}$

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

{\displaystyle {\begin{aligned}p_{n}-C_{n}\left(\mathbf {q} \right)&=-\lambda \left({\frac {\partial R}{\partial q_{n}}}-C_{n}\left(\mathbf {q} \right)\right)\\&=-\lambda \left(p_{n}\left(1-{\frac {1}{\varepsilon _{n}}}\right)-C_{n}\left(\mathbf {q} \right)\right)\end{aligned}}}

where ${\displaystyle \lambda }$  is a Lagrange multiplier, Cn(q) is the partial derivative of C(q) with respect to qn, evaluated at q, and ${\displaystyle \varepsilon _{n}=-{\frac {\partial q_{n}}{\partial p_{n}}}{\frac {p_{n}}{q_{n}}}}$  is the elasticity of demand for good ${\displaystyle n.}$

Dividing by ${\displaystyle p_{n}}$  and rearranging yields

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

where ${\displaystyle k={\frac {\lambda }{1+\lambda }}<1.}$ . That is, the price margin compared to marginal cost for good ${\displaystyle n}$  is again inversely proportional to the elasticity of demand. Note, the Ramsey mark-up is smaller than the ordinary monopoly where ${\displaystyle k=1}$ , since ${\displaystyle \lambda =1}$  (the fixed profit requirement, ${\displaystyle \Pi ^{*}=R-C}$  is non-binding). The Ramsey-price setting monopoly is in a second-best equilibrium, between ordinary monopoly and perfect competition.

## Ramsey condition

An easier way to solve this problem in a two-output context is the Ramsey condition. According to Ramsey, as to minimize deadweight losses, one must increase prices to rigid and elastic demands in the same proportion, in relation to the prices that would be charged at the first-best solution (price equal to marginal cost).