Final Answer:
When 12 units of labor and 14 units of capital are invested, the marginal productivity of labor is approximately 7.02 and the marginal productivity of capital is approximately 9.03.
Step-by-step explanation:
Cobb-Douglas Production Function: The given function defines the relationship between output (P), labor (L), and capital (K).
Marginal Productivity of Labor (MPL): This measures the additional output produced by employing one more unit of labor, holding capital constant. It is calculated as: MPL = ∂P/∂L = (0.4) * P(L, K) / L.
Marginal Productivity of Capital (MPK): This measures the additional output produced by employing one more unit of capital, holding labor constant. It is calculated as: MPK = ∂P/∂K = (0.6) * P(L, K) / K.
Calculation with specific values: Substituting L = 12, K = 14, and the given production function, we get:
MPL ≈ (0.4) * (16 * 12^0.4 * 14^0.6) / 12 ≈ 7.02
MPK ≈ (0.6) * (16 * 12^0.4 * 14^0.6) / 14 ≈ 9.03
Therefore, with the given investment, one additional unit of labor is expected to increase output by approximately 7.02 units, while one additional unit of capital is expected to increase output by approximately 9.03 units.