Ramsey problem
Encyclopedia
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 tax
ation 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.
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.
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.
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.
for a multi-product monopolist with costs 
where 
is the output of good n and 
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 
with inverse demand function 
Total revenue is 
Total surplus is given by
The problem is to maximize
subject to the requirement that profit 
should be equal to some fixed value 
. Typically, the fixed value is zero to guarantee that the profit losses are eliminated.
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
are
where
is a Lagrange multiplier and Cn(z) is the partial derivative of C(z) with respect to zn, evaluated at z.
Dividing by
and rearranging yields
where
is lower than 1 and 
is the elasticity of demand for good 
That is, the price markup over marginal cost for good 
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.
Monopoly
A monopoly exists when a specific person or enterprise is the only supplier of a particular commodity...
should set, in order to maximize social welfare
Welfare economics
Welfare economics is a branch of economics that uses microeconomic techniques to evaluate economic well-being, especially relative to competitive general equilibrium within an economy as to economic efficiency and the resulting income distribution associated with it...
, subject to a constraint on profit
Profit (economics)
In economics, the term profit has two related but distinct meanings. Normal profit represents the total opportunity costs of a venture to an entrepreneur or investor, whilst economic profit In economics, the term profit has two related but distinct meanings. Normal profit represents the total...
. A closely related problem arises in relation to optimal tax
Optimal tax
Optimal tax theory is the study of how best to design a tax to minimize distortion and inefficiency subject to increasing set revenues through distortionary taxation. A neutral tax is a theoretical tax which avoids distortion and inefficiency completely....
ation of commodities
Commodity
In economics, a commodity is the generic term for any marketable item produced to satisfy wants or needs. Economic commodities comprise goods and services....
.
For any monopoly, the price markup should be inverse to the price elasticity of demand
Price elasticity of demand
Price elasticity of demand is a measure used in economics to show the responsiveness, or elasticity, of the quantity demanded of a good or service to a change in its price. More precisely, it gives the percentage change in quantity demanded in response to a one percent change in price...
: 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
Frank P. Ramsey
Frank Plumpton Ramsey was a British mathematician who, in addition to mathematics, made significant and precocious contributions in philosophy and economics before his death at the age of 26...
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
Natural monopoly
A monopoly describes a situation where all sales in a market are undertaken by a single firm. A natural monopoly by contrast is a condition on the cost-technology of an industry whereby it is most efficient for production to be concentrated in a single form...
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
Telecommunication
Telecommunication is the transmission of information over significant distances to communicate. In earlier times, telecommunications involved the use of visual signals, such as beacons, smoke signals, semaphore telegraphs, signal flags, and optical heliographs, or audio messages via coded...
firms.
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.
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
Price discrimination
Price discrimination or price differentiation exists when sales of identical goods or services are transacted at different prices from the same provider...
—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.
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.
Formal presentation and solution
Consider the problem of a regulator seeking to set prices for a multi-product monopolist with costs where is the output of good n and 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 with inverse demand function Total revenue is Total surplus is given by
The problem is to maximize
subject to the requirement that profit should be equal to some fixed value . Typically, the fixed value is zero to guarantee that the profit losses are eliminated.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
arewhere
is a Lagrange multiplier and Cn(z) is the partial derivative of C(z) with respect to zn, evaluated at z.Dividing by
and rearranging yieldswhere
is lower than 1 and is the elasticity of demand for good That is, the price markup over marginal cost for good 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.