CHAPTER • The Cost of Production 277 (A7.21) is the expansion path Now use Equation (A7.21) to substitute for L in equation (A7.18): AKa a b br Kb - q0 = aw (A7.22) We can rewrite the new equation as: Ka + b = a aw b q0 b br A (A7.23) or b aw a + b q0 a + b K = a b a b br A (A7.24) (A7.24) is the factor demand for capital We have now determined the cost-minimizing quantity of capital: Thus, if we wish to produce q0 units of output at least cost, (A7.24) tells us how much capital we should employ as part of our production plan To determine the cost-minimizing quantity of labor, we simply substitute equation (A7.24) into equation (A7.21): b br br aw a + b q0 a + b L = K = £a b a b § aw aw br A a L = a (A7.25) br a + b q0 a + b b a b aw A (A7.25) is the constrained factor demand for labor Note that if the wage rate w rises relative to the price of capital r, the firm will use more capital and less labor Suppose that, because of technological change, A increases (so the firm can produce more output with the same inputs); in that case, both K and L will fall We have shown how cost-minimization subject to an output constraint can be used to determine the firm’s optimal mix of capital and labor Now we will determine the firm’s cost function The total cost of producing any output q can be obtained by substituting equations (A7.24) for K and (A7.25) for L into the equation C ϭ wL ϩ rK After some algebraic manipulation we find that a b/(a + b) a -a/(a + b) q 1/(a + b) §a b C = w b/(a + b)r a/(a + b) £ a b + a b (A7.26) b b A This cost function tells us (1) how the total cost of production increases as the level of output q increases, and (2) how cost changes as input prices change When a + b equals 1, equation (A7.26) simplifies to C = w br a[(a/b)b + (a/b)-a](1/A)q (A7.27)