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The Ramsey problem, or Ramsey pricing, or Ramsey–Boiteux pricing, is a second-best policy problem concerning what prices a public
monopoly A monopoly (from Greek language, Greek el, μόνος, mónos, single, alone, label=none and el, πωλεῖν, pōleîn, to sell, label=none), as described by Irving Fisher, is a market with the "absence of competition", creating a situati ...
should charge for the various products it sells in order to maximize
social welfare Welfare, or commonly social welfare, is a type of government support intended to ensure that members of a society can meet basic human needs such as food and shelter. Social security may either be synonymous with welfare, or refer specifical ...
(the sum of producer and consumer surplus) while earning enough revenue to cover its fixed costs. Under Ramsey pricing, the price markup over marginal cost is inverse to the price elasticity of demand: the more elastic the product's demand, the smaller the markup.
Frank P. Ramsey Frank Plumpton Ramsey (; 22 February 1903 – 19 January 1930) was a British philosopher, mathematician, and economist who made major contributions to all three fields before his death at the age of 26. He was a close friend of Ludwig Wittgens ...
found this 1927 in the context of
Optimal tax Optimal tax theory or the theory of optimal taxation is the study of designing and implementing a tax that maximises a social welfare function subject to economic constraints. The social welfare function used is typically a function of individuals ...
ation: the more elastic the demand, the smaller the optimal tax. The rule was later applied by
Marcel Boiteux Marcel Boiteux (9 May 1922 – 6 September 2023) was a French economist, mathematician, and senior civil service member. He was the "architect of the French nuclear program" that created 61 nuclear reactors and kept the French electricity secto ...
(1956) to natural monopolies (industries with decreasing average cost). A
natural monopoly A natural monopoly is a monopoly in an industry in which high infrastructural costs and other barriers to entry relative to the size of the market give the largest supplier in an industry, often the first supplier in a market, an overwhelming ad ...
earns negative profits if it sets price equals to marginal cost, so it must set prices for some or all of the products it sells to above marginal cost if it is to be viable without government subsidies. Ramsey pricing says to mark up most the goods with the least elastic (that is, least price-sensitive) demand.


Description

In a first-best world, without the need to earn enough revenue to cover fixed costs, the optimal solution would be to set the price for each product equal to its marginal cost. If the average cost curve is declining where the demand curve crosses it however, as happens when the fixed cost is large, this would result in a price less than average cost, and the firm could not survive without subsidy. The Ramsey problem is to decide exactly how much to raise each product's price above its marginal cost so the firm's revenue equals its total cost. If there is just one product, the problem is simple: raise the price to where it equals average cost. If there are two products, there is leeway to raise one product's price more and the other's less, so long as the firm can break even overall. The principle is applicable to pricing of goods that the government is the sole supplier of (public utilities) or regulation of natural monopolies, such as
telecommunication Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that ...
s firms, where it is efficient for only one firm to operate but the government regulates its prices so it does not earn above-market profits. In practice, government regulators are concerned with more than maximizing the sum of producer and consumer surplus. They may wish to put more weight on the surplus of politically powerful consumers, or they may wish to help the poor by putting more weight on their surplus. Moreover, many people will see Ramsey pricing as unfair, especially if they do not understand why it maximizes total surplus. In some contexts, Ramsey pricing is a form of
price discrimination Price discrimination is a microeconomic pricing strategy where identical or largely similar goods or services are sold at different prices by the same provider in different markets. Price discrimination is distinguished from product different ...
because the two products with different elasticities of demand are one physically identical product sold to two different groups of customers, e.g., electricity to residential customers and to commercial customers. Ramsey pricing says to charge whichever group has less elastic demand a higher price in order to maximize overall social welfare. Customers sometimes object to it on that basis, since they care about their own individual welfare, not social welfare. Customers who are charged more may consider unfair, especially they, with less elastic demand, would say they "need" the good more. In such situations regulators may further limit an operator’s ability to adopt Ramsey prices.Body of Knowledge on Infrastructure Regulation
“Tariff Design: Economics of Tariff Design – Deviations from Marginal Cost Pricing: Ramsey Pricing”


Formal presentation and solution

Consider the problem of a regulator seeking to set prices \left(p_1,\ldots,p_N\right) for a multiproduct monopolist with costs C(q_1,q_2,\ldots,q_N) =C( \mathbf), where q_ is the output of good ''i'' and p_ is the price. Suppose that the products are sold in separate markets so demands are independent, and demand for good ''i'' is q_\left( p_\right) , with inverse demand function p_i(q). Total revenue is R\left( \mathbf\right) =\sum_i p_i q_i (p_i). Total welfare is given by :W\left( \mathbf\right) =\sum_i \left( \int\limits_0^p_i( q) dq\right) -C\left( \mathbf\right). The problem is to maximize W\left( \mathbf\right) by choice of the subject to the requirement that profit \Pi = R-C equal some fixed value \Pi^* . Typically, the fixed value is zero, which is to say that the regulator wants to maximize welfare subject to the constraint that the firm not lose money. The constraint can be stated generally as: :R( \mathbf) -C( \mathbf) \geq \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 \mathbf are :\begin p_i - C_i \left(\mathbf\right) &= -\lambda \left( \frac - C_\left( \mathbf\right) \right) \\ &= -\lambda \left( p_i \left( 1 - \frac\right) - C_i \left(\mathbf\right) \right) \end where \lambda is a Lagrange multiplier, ''C''''i''(q) is the partial derivative of ''C''(q) with respect to ''q''''i'', evaluated at q, and Elasticity_i= -\frac\frac is the elasticity of demand for good i. Dividing by p_i and rearranging yields :\frac=\frac where k=\frac< 1. . That is, the price margin compared to marginal cost for good i is again inversely proportional to the elasticity of demand. Note that the Ramsey mark-up is smaller than the ordinary monopoly markup of the Lerner Rule which has k=1 , since \lambda=1 (the fixed-profit requirement, \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 loss In economics, deadweight loss is the difference in production and consumption of any given product or service including government tax. The presence of deadweight loss is most commonly identified when the quantity produced ''relative'' to the amou ...
es, 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).


See also

* Amoroso–Robinson relation *
Lerner Index The Lerner index, formalized in 1934 by British economist of Russian origin Abba Lerner, is a measure of a firm's market power. Definition The Lerner index is defined by: L=\frac where P is the market price set by the firm and MC is the firm's ...


References

{{reflist Economic policy Monopoly (economics) Mathematical economics Tax