Question Analysis

The question asks about the implication of the marginal product of labour (MPL) being greater than the average product of labour (APL), given a production function that is homogeneous of degree one and satisfies Euler's equation. We need to determine the state of the marginal product of capital (MPK) under these conditions.

Key Economic Concepts

• Homogeneous Production Function of Degree One: A production function $f(K, L)$ exhibits constant returns to scale if $f(\lambda K, \lambda L) = \lambda f(K, L)$ for any positive scaling factor $\lambda$. This means doubling inputs doubles output.
• Euler's Equation: For a production function that is homogeneous of degree one, Euler's theorem states that the total product equals the sum of the inputs multiplied by their respective marginal products. Mathematically: $Q = K \cdot \frac{\partial Q}{\partial K} + L \cdot \frac{\partial Q}{\partial L}$ Using economic notation, this is: $Q = K \cdot MPK + L \cdot MPL$
• Relationship between MPL and APL:
  • When MPL > APL, the average product of labour (APL) is increasing.
  • When MPL < APL, APL is decreasing.
  • When MPL = APL, APL is at its maximum.

Applying the Condition

We are given the condition that MPL > APL. This tells us that the APL curve is rising.

Let's start with Euler's equation:

$Q = K \cdot MPK + L \cdot MPL$

To relate this to APL, we divide the entire equation by the quantity of labour ($L$):

$\frac{Q}{L} = \frac{K}{L} \cdot MPK + \frac{L}{L} \cdot MPL$

Recognizing that $\frac{Q}{L}$ is the Average Product of Labour (APL) and $\frac{K}{L}$ is the capital-labour ratio (let's denote it by $k$), the equation becomes:

$APL = k \cdot MPK + MPL$

Deriving the Implication for MPK

Now, we use the given condition: MPL > APL.

Substitute the expression for APL from the modified Euler's equation into the condition:

$MPL > (k \cdot MPK + MPL)$

Subtract MPL from both sides of the inequality:

$MPL - MPL > k \cdot MPK$ $0 > k \cdot MPK$

We know that the capital-labour ratio, $k = \frac{K}{L}$, must be positive, as we assume positive amounts of capital ($K$) and labour ($L$).

For the inequality $0 > k \cdot MPK$ to hold true, given that $k > 0$, the Marginal Product of Capital ($MPK$) must be negative.

$MPK < 0$

Conclusion

Therefore, if the production function is homogeneous of degree one, Euler's equation holds, and MPL is greater than APL, then the Marginal Product of Capital (MPK) must be negative.