Consider a monopolist with two plants. Each plant is characterized by its marginal costs: lines
, 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
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
(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.
 D. A. Besanko and R. R. Braeutigam, Microeconomics
, 4th ed., Hoboken: John Wiley, 2010.