Ramsey problem

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 telecommunications 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 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.[2]

Consider the problem of a regulator seeking to set prices ${\displaystyle \left(p_{1},\ldots ,p_{N}\right)}$ for a multiproduct monopolist with costs ${\displaystyle C(q_{1},q_{2},\ldots ,q_{N})=C(\mathbf {q} ),}$ where ${\displaystyle q_{i}}$ is the output of good i and ${\displaystyle p_{i}}$ is the price.[3] Suppose that the products are sold in separate markets so demands are independent, and demand for good i is ${\displaystyle q_{i}\left(p_{i}\right),}$ with inverse demand function ${\displaystyle p_{i}(q).}$ Total revenue is ${\displaystyle R\left(\mathbf {p,q} \right)=\sum _{i}p_{i}q_{i}(p_{i}).}$

Total welfare is given by

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

The problem is to maximize ${\displaystyle W\left(\mathbf {p,q} \right)}$ by choice of the subject to the requirement that profit ${\displaystyle \Pi =R-C}$ equal some fixed value ${\displaystyle \Pi ^{*}}$. 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:

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

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 ${\displaystyle \mathbf {q} }$ are

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

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

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

${\displaystyle {\frac {p_{i}-C_{i}\left(\mathbf {q} \right)}{p_{i}}}={\frac {k}{Elasticity_{i}}}}$

where ${\displaystyle k={\frac {\lambda }{1+\lambda }}<1.}$. That is, the price margin compared to marginal cost for good ${\displaystyle i}$ 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 ${\displaystyle k=1}$, 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.

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).

• Amoroso–Robinson relation
• Lerner Index