Short Answer

The Calculus of Social Gains

A firm's profit is given by the function π(Q)=PQCp(Q)\pi(Q) = P \cdot Q - C_p(Q), where PP is the market price, QQ is output, and Cp(Q)C_p(Q) is its private cost of production. The firm chooses its output level, QQ^*, to maximize this profit. The firm's production also imposes a damage cost on a third party, described by the function Ce(Q)C_e(Q).

  1. At the firm's profit-maximizing output level, QQ^*, what is the value of the derivative of the firm's profit with respect to output, dπ/dQd\pi/dQ?
  2. 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 QQ^*?
  3. Using your answers from parts 1 and 2, explain why a small reduction in output from QQ^* must increase the total social surplus, assuming the marginal external damage, Ce(Q)C'_e(Q), is positive.

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Updated 2025-08-09

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