Final answer:
The production function q=K¹/³ L²/³ represents a Cobb-Douglas production function. The marginal product of labor (MPL) is 2/3 * (K/L)¹/³ * L^-1/3. The marginal product of capital (MPK) is 1/3 * (K/L)^-2/³ * K^-2/3.
Step-by-step explanation:
The production function q=K¹/³ L²/³ represents a Cobb-Douglas production function, which is a widely used production function in economics. It is named after economists Charles Cobb and Paul Douglas.
The marginal product of labor (MPL) can be calculated by taking the partial derivative of the production function with respect to labor (L). In this case, MPL = 2/3 * (K/L)¹/³ * L^-1/3.
The marginal product of capital (MPK) can be calculated by taking the partial derivative of the production function with respect to capital (K). In this case, MPK = 1/3 * (K/L)^-2/³ * K^-2/3.
To determine the cost-minimizing bundles of labor and capital, we need to equate the marginal productivities of labor and capital to their respective factor prices. Given a wage rate of $20 and an interest rate of 4%, we set MPL/20 = MPK/(0.04 * K).
Simplifying the equation, we get 2/3 * (K/L)¹/³ * L^-1/3 / 20 = 1/3 * (K/L)^-2/³ * K^-2/3 / (0.04 * K).
Further simplifying, we find that (K/L)^(3/4) = 15.
Therefore, the cost-minimizing bundle of labor and capital ratio is K/L = 15^(4/3) = 54.25.