Benefit and Cost

For an industry producing a good for consumption, the optimum is to choose the quantity to maximize consumer benefit minus cost. The primal/dual pair below shows that this optimum is attained by a price that achieves market equilibrium. The primal chooses the quantity to maximize benefit minus cost, and the dual chooses the price to minimize the sum of consumers’ surplus plus profit. For consumption quantity x, let g denote the dollar benefit to consumers, and let f denote the dollar cost function. Choose x to maximize benefit minus cost: $$\sup _{x}\left [ g\left ( x\right ) -f\left ( x\right ) \right ] \!.$$ Consumer demand x maximizes consumers’ surplus, benefit minus expenditure: $$\sup _{x}\left [ g\left ( x\right ) -x^{\ast }x\right ] =-g_{\ast }\left ( x^{\ast }\right ) \!.$$ Inverting the first-order condition obtains $$\left \{x\right \}=\partial g_{\ast }\left ( x^{\ast }\right ) ,$$ demand as a function of price. Because benefit is concave and increasing, demand falls as the price rises. The conjugate of cost is the profit function, $$f^{\ast }\left ( x^{\ast }\right ) =\sup _{x}\left [ x^{\ast }x-f\left ( x\right ) \right ] \!,$$ the maximum value of revenue minus cost. Inverting the first-order condition obtains $$\left \{x\right \}=\partial f^{\ast }\left ( x^{\ast }\right ) ,$$ the profit-maximizing quantity as a function of price. A higher price raises the supply. The dual chooses the price to minimize the profit plus the consumers’ surplus, $$\inf _{y^{\ast }}\left [ f^{\ast }\left ( y^{\ast }\right ) -g_{\ast }\left ( y^{\ast }\right ) \right ] \!.$$ A solution $$\left ( x,y^{\ast }\right )$$ to the primal and the dual is on the demand curve and the supply curve, so market equilibrium maximizes benefit minus cost.