Editing Ramsey problem (section)

==Formal presentation and solution==

Consider the problem of a regulator seeking to set prices <math>\left(p_1,\ldots,p_N\right) </math> for a multiproduct monopolist with costs <math>C(q_1,q_2,\ldots,q_N) =C( \mathbf{q}), </math> where <math>q_{i}</math> is the output of good ''i'' and <math>p_{i}</math> is the price.<ref>{{Cite journal|last=Ramsey|first=Frank P.|date=1927|title=A Contribution to the Theory of Taxation|journal=The Economic Journal|volume=37|issue=145|pages=47–61|doi=10.2307/2222721|jstor=2222721}}</ref> Suppose that the products are sold in separate markets so demands are independent, and demand for good ''i'' is <math>q_{i}\left( p_{i}\right) , </math> with inverse demand function <math>p_i(q).</math> Total revenue is <math>R\left( \mathbf{p,q}\right) =\sum_i p_i q_i (p_i).</math>

Total welfare is given by

:<math>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). </math>

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

:<math>R( \mathbf{p,q}) -C( \mathbf{q}) \geq \Pi^*</math>

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 <math>\mathbf{q} </math> are

:<math>\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{Elasticity}_i}\right) - C_i \left(\mathbf{q}\right) \right)
\end{align}</math>

where <math>\lambda </math> is a Lagrange multiplier, ''C''<sub>''i''</sub>('''q''') is the partial derivative of ''C''('''q''') with respect to ''q''<sub>''i''</sub>, evaluated at '''q''',  and <math>\mathrm{Elasticity}_i= -\frac{\partial q_i}{\partial p_i}\frac{p_i}{q_i} </math> is the elasticity of demand for good <math>i. </math>

Dividing by <math>p_i </math> and rearranging yields

:<math>\frac{p_i - C_i\left( \mathbf{q}\right) }{p_i}=\frac{k}{\mathrm{Elasticity}_i}</math>

where <math>k=\frac{\lambda }{1+\lambda}< 1. </math>. That is, the price margin compared to marginal cost for good <math>i</math> 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 index|Lerner Rule]] which has <math>k=1 </math>, since <math>\lambda=1 </math> (the fixed-profit requirement, <math>\Pi^* = R-C </math> is non-binding). The Ramsey-price setting monopoly is in a second-best equilibrium, between ordinary monopoly and perfect competition.