Monopoly with Two Plants

This Demonstration shows how a monopolist chooses the price and the distribution of production quantity among several plants. For the sake of simplicity, only the case with two plants is considered. So holds, where stands for the production volume of the respective plant. Once understood, this task can be easily extended to an arbitrary number of plants with more complex marginal cost functions. This model can be used in conjunction with other monopolist's problems—that is, when she faces several markets.
  • Contributed by: Timur Gareev
  • (The Skolkovo Institute of Science and Technology)

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Consider a monopolist with two plants. Each plant is characterized by its marginal costs: lines and , respectively. The monopolist also deals with sloped linear demand with respective marginal revenue curve, . Her goal is to set the optimal output from the traditional condition . In order to get total marginal costs, sum up and horizontally (technically an inverse function is used for such a summation).
In the linear case, we generally get a piecewise function. The economic sense of such a summation is that for each level of marginal cost, a monopolist gets total production as a sum of production at both plants because each plant produces according to that cost level, . This last expression is used to find the division of total optimal output between both plants. For that we just put the value of marginal costs at the optimal point value in the inverted functions . These values are calculated automatically in the bottom of the panel. The optimal quantities lines show exactly those values: the horizontal line is and the vertical lines are and (the color corresponds to the color of the respective costs curve), and (the color is black). Point the mouse to the lines you want to study to see a tooltip, or just switch the checkboxes off and on to customize the view.
Initial settings are taken from the exercise entitled "Determining the Optimal Output, Price, and Division of Production for a Multiplant Monopolist" [1, p. 465].
We also encourage you to study the author's other Demonstration to compare the horizontal summation of marginal costs curves with the horizontal summation of demand curves, which happens when a monopolist faces different markets.
Reference
[1] D. A. Besanko and R. R. Braeutigam, Microeconomics, 4th ed., Hoboken: John Wiley, 2010.
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