Ramsey Pricing

for i = 1, 2, ..., n. The total revenue is

The cost of producing these goods is C(q₁, ..., q_n). The regulator wants to choose outputs so as to maximize consumer surplus,

qi 0

pi(t)dt - C(q1, ..., qn),

subject to the constraint that revenue exactly equals cost (or that profit is a given constant). The first-order conditions are

p_i - C_i = (R_i - C_i)

for i = 1,... ,n, where C_i and R_i are the partial derivatives of C and R with respect to q_i and is the Lagrangian multiplier on the constraint. This condition may be rewritten as

pi - Ci ---------- pi

= -

k --- i

,

where k = /(1 + ) and _i is the elasticity of demand for q_i. That is, the price markup over marginal cost, (p_i - C_i)/p_i, is inversely proportional to the price elasticity of demand for that good. If k = 1, this condition is the standard monopoly price-discrimination condition. If k = 0, this condition is the same as in competition.