Managerial Economics and Strategy, 2/e

Appendix 6A Long-Run Cost Minimization

We can use calculus to derive the tangency rule, cost-minimization condition, Equation 6.9. The problem the firm faces in the long run is to choose labor, L, and capital, K, to minimize the cost of producing a particular level of output, $\overline{q},$ [&*orule*{q},&] given a wage of w and a rental rate of capital of r.

The firm’s production function is $q=f\left(L,K\right),$ [&q|=|f(L, K),&] so the marginal products of labor and capital are $M{P}_L\left(L,K\right)=\partial f\left(L,K\right)/\partial L>0$ [&MP_{L}(L, K)|=||pdns|f(L, K)/|pdns|L|gtr|0&] and $M{P}_{\text{ K}}=\partial f\left(L,K\right)/\partial K>0.$ [&MP_{~rom~ K}|=||pdns|~normal~f(L, K)/|pdns|K|gtr|0.&] The firm’s problem is to minimize its cost of production, C, through its choice of labor and ...