Assuming that the production function is homogeneous of degree one and Euler's equation holds, if \( MP_L \) (marginal product of labour) is greater than \( AP_L \) (average product of labour), then:

Question

Match List-I with List-II \[\begin{array}{|c|c|} \hline \textbf{Types of Production Functions} & \textbf{Their Functional Forms} \\ \hline \text{(A) Translog Production Function} & \text{(I) \( q = \prod_{i=1}^{n} x_i^{a_i} \)} \\ \hline \text{(B) Generalised Leontief Production Function} & \text{(II) \( q = \sum_{i=1}^{n} a_i x_i^{\rho}, \ \rho \leq 1 \)} \\ \hline \text{(C) Cobb Douglas Production Function} & \text{(III) \( q = \sum_{i=1}^{n} a_{ij} x_i x_j \), where \( a_{ij} = a_{ji} \)} \\ \hline \text{(D) Constant Elasticity of Substitution Production Function} & \text{(IV) \( q = a_0 + \sum_{i=1}^{n} a_i \ln x_i + 0.5 \sum_{i=1}^n \ln x_i \)} \\ \hline \end{array}\] Choose the correct answer from the options given below:

Answer 1

Step 1: Production Function Properties.
A production function homogeneous of degree one follows Euler's equation: \[ F(K, L) = K \cdot \frac{\partial F}{\partial K} + L \cdot \frac{\partial F}{\partial L} \] Here, \( F(K, L) \) represents output, \( K \) is capital, and \( L \) is labor. The condition \( MP_L>AP_L \) indicates diminishing returns to labor.

Step 2: Evaluation of Options.
- (A) \( MP_L \) will be negative: Incorrect. The marginal product of labor is not negative if the production function yields positive output. - (B) \( MP_L \) will be zero: Incorrect. Since \( MP_L \) is greater than \( AP_L \), it cannot be zero. - (C) \( MP_K \) will be negative: Incorrect. Insufficient information is provided to conclude that \( MP_K \) is negative. - (D) \( MP_L \) and \( MP_K \) will both be negative: Correct. When marginal products are decreasing, both labor and capital can experience negative marginal returns at certain output levels.

Step 3: Conclusion.
The correct option is (D). When the marginal product of labor surpasses the average product, both \( MP_L \) and \( MP_K \) may be negative.

Assuming that the production function is homogeneous of degree one and Euler's equation holds, if \( MP_L \) (marginal product of labour) is greater than \( AP_L \) (average product of labour), then:

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The Correct Option is D

Solution and Explanation

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