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Solved: Cobb-Douglas production function The output Q of
Chapter 12, Problem 58E(choose chapter or problem)
Cobb-Douglas production function The output Q of an economic system subject to two inputs, such as labor L and capital K, is often modeled by the Cobb-Douglas production function \(Q(L, K)=c L^{a} K^{b}\). Suppose \(a=\frac{1}{3}\), \(b=\frac{2}{3}\), and c=1.
(a) Evaluate the partial derivatives \(Q_{L}\) and \(Q_{K}\).
(b) If L=10 is fixed and K increases from K=20 to K=20.5, use linear approximation to estimate the change in Q.
(c) If K=20 is fixed and L decreases from L=10 to L=9.5, use linear approximation to estimate the change in Q.
(d) Graph the level curves of the production function in the first quadrant of the LK-plane for Q=1, 2, 3.
(e) If you move along the vertical line L=2 in the positive K-direction, how does Q change? Is this consistent with \(Q_{K}\) computed in part (a)?
(f) If you move along the horizontal line K=2 in the positive L-direction, how does Q change? Is this consistent with \(Q_{L}\) computed in part (a)?
Questions & Answers
QUESTION:
Cobb-Douglas production function The output Q of an economic system subject to two inputs, such as labor L and capital K, is often modeled by the Cobb-Douglas production function \(Q(L, K)=c L^{a} K^{b}\). Suppose \(a=\frac{1}{3}\), \(b=\frac{2}{3}\), and c=1.
(a) Evaluate the partial derivatives \(Q_{L}\) and \(Q_{K}\).
(b) If L=10 is fixed and K increases from K=20 to K=20.5, use linear approximation to estimate the change in Q.
(c) If K=20 is fixed and L decreases from L=10 to L=9.5, use linear approximation to estimate the change in Q.
(d) Graph the level curves of the production function in the first quadrant of the LK-plane for Q=1, 2, 3.
(e) If you move along the vertical line L=2 in the positive K-direction, how does Q change? Is this consistent with \(Q_{K}\) computed in part (a)?
(f) If you move along the horizontal line K=2 in the positive L-direction, how does Q change? Is this consistent with \(Q_{L}\) computed in part (a)?
ANSWER:Solution 58EStep 1 of 6The output Q of an economic system subject to 2 inputs, such as labor L and capital K, for .(a)Treating K as a constant and differentiating with respect to L, Holding L fixed and differentiating with respect to K,