Help with least cost production decision for firm using calculus solution

Question

Suppose a firm’s production function is given by F(K, L) = K^3/5 * L^2/5. The rental rate of

capital (r) is $8 per unit and the wage rate (w) is $20 per unit. What is the firm’s optimal capital-labor ratio?

Answer 1

TheThe marginal rate of technical substitution is the ratio of the marginal product of labor to the marginal product of capital when you take the derivatives of the two would take the quotient you wind up with the mrts equal to 3/2 x k / l. Technical efficiency requires the mrts to equal the ratio with the wage to the rental cost of capital. So, 3 / 2 x k / UL must be equal 20 / 8. If we divide both sides by 3/2 we get k / l equal to 40 / 24 which equals 10 / 6 or 5 / 3 depending upon what country you're from