Final answer:
To find the long-run capital-to-labor ratio (K/L) when the cost of capital is three times the cost of labor in a Cobb-Douglas production function, one must set the marginal product per dollar spent on labor equal to the marginal product per dollar spent on capital, solve the equation, and find the optimal K/L ratio.
Step-by-step explanation:
To determine the long-run capital-to-labor ratio (K/L) given that the cost of a unit of capital (r) is three times the cost of a unit of labor (w), you can use the following approach based on the Cobb-Douglas production function provided, q=24 L0.65 K0.70.
Firstly, we acknowledge that in the long run, all factors are variable. Since r = 3w, firms will minimize cost by choosing K and L such that the marginal product per dollar spent on each input is equal, which leads to the equation:
MPL / w = MPK / r
Using the given production function, we can derive the marginal products of labor (MPL) and capital (MPK):
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- MPL = 0.65 * 24 L-0.35 K0.70
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- MPK = 0.70 * 24 L0.65 K-0.30
Setting MPL / w equal to MPK / r, we get:
(0.65 * 24 L-0.35 K0.70) / w = (0.70 * 24 L0.65 K-0.30) / (3w)
Simplifying, we solve for K/L:
K/L = (0.65 / (0.70 * 3))1/0.05
This gives us the optimal long-run capital-to-labor ratio a firm should use.