The Calculus of Social Gains
A firm's profit is given by the function , where is the market price, is output, and is its private cost of production. The firm chooses its output level, , to maximize this profit. The firm's production also imposes a damage cost on a third party, described by the function .
- At the firm's profit-maximizing output level, , what is the value of the derivative of the firm's profit with respect to output, ?
- What is the expression for the derivative of the total social surplus (defined as the firm's profit minus the external damage cost) with respect to output, evaluated at ?
- Using your answers from parts 1 and 2, explain why a small reduction in output from must increase the total social surplus, assuming the marginal external damage, , is positive.
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