Lagrangian method to find long run cost minimization. Problem: min l, k wl + rk such that f (l, k) = q. = wl + rk + [ q f (l, k) ] / l = w - f (l, k) / l = 0. / k = r - f (l, k) / k = 0. / = q f (l, k) = 0. W / r = ( f (l, k) / l) / ( f (l, k) / k) W / r = mp l / mp k. Q = f (l, k) plus the second-order condition: mrts diminishing hidden condition, l 0, k 0. Example of cost minimization using lagrangian f (l, k) = 2l 1/3 k 2/5. Given w > 0, r > 0, calculate c lr (q) = wl + rk + [q 2l 1/3 k 2/5] / l = w - 2/3 l -2/3 k 2/5 = 0.