Equilibrium of a Discriminating Monopolist:

Let us consider a monopolist who sells her output at two different markets.

Let X₁ be the amount of output sold in the 1^st market and X₂ is the amount of output sold in the 2^nd market. Therefore, the total amount sold by the monopolist is (X₁ + X₂)

Let TR₁ and TR₂ are the total revenues obtained from the two markets.   

Hence, the condition of profit maximisation for a discriminating monopolist is given by marginal revenue in 1st market = marginal revenue in 2nd market= marginal cost  If MR₁ > MR₂ , it is profitable for the monopolist to sell more in 1^st market and this continues till MR₁ = MR₂.  Therefore, at equilibrium MR₁ = MR₂ = MC  The equality of marginal revenues does not mean the equality of prices in the two markets as long as the price elasticity of demand is different in the both
markets.

The second order condition requires that the principal minors of the Hessian determinant,

alternate in sign beginning with the negative sign. This requires

The equilibrium of a discriminating monopolist can be explained with the help of the following diagram.

In the first two panels of the diagram, we have drawn the MR curves of two markets. In the third panel, we obtained the combined MR curves by horizontally summing over MR1 and MR₂. The combined MR and MC curves meet at the point E. We have drawn a horizontal line passing through E which cuts MR₁ and MR₂ at the point E1 and E2. At E₁, MR₁ = MC and at E₂, MR₂ = MC. Therefore, the conditions of profit maximisation for a discriminating monopolist is satisfied at E₁and E₂, Accordingly, the monopolist will sell X₁* amount of output in market 1 at price P₁* and X₂* amount of output in market 2 at price P₂*