Ramsey problem

The Ramsey problem, or Ramsey pricing, or Ramsey–Boiteux pricing, is a second-best policy problem concerning what prices a public monopoly should charge for the various products it sells in order to maximize social welfare (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 inversely related to the price elasticity of demand and the price elasticity of supply: the more elastic the product's demand or supply, the smaller the markup. Frank P. Ramsey discovered this principle in 1927 in the context of optimal taxation: the more elastic the demand or supply, the smaller the optimal tax. The rule was later applied by Marcel Boiteux (1956) . to natural monopolies (industries with decreasing average cost). A natural monopoly earns negative profits if it sets prices equal to marginal cost, so it must set prices for some or all of the products it sells above marginal cost if it is to remain viable without government subsidies. Ramsey pricing indicates that goods with the least elastic (that is, least price-sensitive) demand or supply should receive the highest markup.

The Ramsey problem, or Ramsey pricing, or Ramsey–Boiteux pricing, is a second-best policy problem concerning what prices a public monopoly should charge for the various products it sells in order to maximize social welfare (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 inversely related to the price elasticity of demand and the price elasticity of supply: the more elastic the product's demand or supply, the smaller the markup. Frank P. Ramsey discovered this principle in 1927 in the context of optimal taxation: the more elastic the demand or supply, the smaller the optimal tax. The rule was later applied by Marcel Boiteux (1956) . to natural monopolies (industries with decreasing average cost). A natural monopoly earns negative profits if it sets prices equal to marginal cost, so it must set prices for some or all of the products it sells above marginal cost if it is to remain viable without government subsidies. Ramsey pricing indicates that goods with the least elastic (that is, least price-sensitive) demand or supply should receive the highest markup.

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{q}), where q_{i} is the output of good i and p_{i} is the price. Suppose that the products are sold in separate markets so demands are independent, and demand for good i is q_{i}\left( p_{i}\right) , with inverse demand function p_i(q). Total revenue is R\left( \mathbf{p,q}\right) =\sum_i p_i q_i (p_i). Total welfare is given by 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 W\left( \mathbf{p,q}\right) by choice of the subject to the requirement that profit \Pi = R-C equal some fixed value \Pi_0 . 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{p,q}) -C( \mathbf{q}) \geq \Pi_0 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{q} are \begin{align} 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}{\mathrm{\epsilon}_i}\right) - C_i \left(\mathbf{q}\right) \right) \end{align} where C i (q) is the partial derivative of C(q) with respect to q i , evaluated at q, \mathrm{\epsilon}_i= \frac{\partial q_i}{\partial p_i}\frac{p_i}{q_i} is the elasticity of demand for good i. and \lambda is the Lagrange multiplier for the optimization problem. Note that, by the envelope theorem: \lambda=\frac{\partial W^*}{\partial \Pi_0} so, around the optimum, \lambda can be interpreted as the amount of welfare that, on the margin, the public monopolist is willing to trade off for a unit of profit, which coincides with the marginal rate of substitution between welfare and revenue for the monopolist. Dividing by p_i and rearranging yields \frac{p_i - C_i\left( \mathbf{q}\right) }{p_i}=\frac{k}{\mathrm{\epsilon}_i} where k=-\frac{\lambda }{1+\lambda}, \quad |k|\leq 1 . That is, the price margin compared to marginal cost for good i is again inversely proportional to the elasticity of demand. Thus the Ramsey mark-up is smaller or equal to the ordinary monopoly markup from the Lerner Rule (which has k=1 ). If \lambda \rightarrow \infty (i.e. the monopolist is willing to trade off an infinite amount of welfare for the last unit of profit, or, in other words, it is indifferent to the welfare implications of its pricing) will k\rightarrow 1 , resulting in the monopolist price. Alternatively, if the monopolist is not willing to sacrifice any welfare to obtain revenue, and thus sets \Pi_0 to a level for which the revenue restriction is not binding (which makes \lambda=0 ) then k\rightarrow 0 and, as in perfect competition: \frac{p_i-C_i(\textbf{q})}{p_i}\rightarrow 0, \quad p_i\rightarrow C_i(\textbf{q}) As a result, 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, in order to minimize deadweight losses, one must increase prices to rigid and elastic demands/supplies 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

References

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