Primal Problem: Suppose the firm wants to produce Q1 units of output. What combo of L and K will it choose to minimize cost? Min C =rK + wL subject to F(K,L) = Q1 Form the Lagrangian for the Problem: L = rK + wL+ 2 [Q1 - F(K,L)] To solve for the optimum, differentiate the Lagrangian wrt the variables K, L, and 2: FOCs (First Order Conditions): i) LK = 1- 2 = 0 so 2 = r/MPK ii) LL = W - 2 2 =0 so 2 = w/MPL iii) Lx = [Q1 - F(K,L)] = 0 i) and ii) > 1) MPk/r = MPL/w iii) gives us 2) F(K,L) = Q1 2 tells us how much one more unit costs (MC)Dual Problem: Suppose firm wants to spend Ci to hire inputs. What combination of L and K will it choose to maximize output? Max F(K,L) subject to rK + wL = C1 Form the Lagrangian for the Problem: [ = F(K,L) + u [Ci - rK - wL] To solve for the optimum, differentiate the Lagrangian wrt the variables K, L, and u: FOCs (First Order Conditions): i ) LK OF ak - ur = 0 so u = MPk/r ii) LL = 27 - HW = 0 so u = MPL/w iii) Lu = C1 - rK + wL =0 i) and ii) > 1) MPk/r = MPL/W iii) gives us 2) rK + wL = C1