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Suppose a monopolist faces two independent markets. In the first market, demand is , and in the second market, demand is . The monopolist solves the typical problem of setting prices (in conjunction with volume of production) so as to maximize the total profit collected from both markets. We consider the discriminating monopolist, who sets the prices and , respectively, for the first and second markets.

We want to know the total costs , where is the total quantity of production. The marginal costs function is the derivative of the total costs function. This means that knowing only the marginal costs is not enough, as we cannot reconstruct the fixed part of the total costs (we have set the fixed costs equal to 2). We find the required optimal price levels from the condition , where is in fact a result of horizontal summation of individual marginal revenue functions [1, pp. 76–77]. Then we can find the optimal values of and separately for both markets. Inverse individual demand functions give us the sought-for optimal prices and followed by the profit calculation: . We model different functions ("linear", "quadratic", and "cubic"), including trivial cases that lead to .

Note that "slope" and "intercept" sliders control the curve reflected on the right plot. Use the tooltips on the plots to identify the respective curves. Pay attention to the fact that horizontal summation of different marginal revenues curves is not equivalent to the total marginal revenue . The monopolist uses this horizontal summation to derive the condition for optimization and then effectively deals with two independent markets with separate marginal revenues. (The case of a nondiscriminating monopolist dealing with total is considered in another Demonstration—see the Related Link.)

Reference

[1] O. Shy, Industrial Organization: Theory and Applications. Cambrige, MA: MIT Press. 1995.