Question

Consider a production function given by 𝑓(𝑥1 , 𝑥2 ) = (𝑥1^α)(𝑥2^β), where α, β>0.

For which values of the parameters α and β does this production function exhibit decreasing marginal product of input 1? Input 2?

Answer 1

To determine when the production function exhibits decreasing marginal product of input 1, we need to calculate the marginal product of input 1, denoted by MP1:

MP1 = ∂f/∂x1 = α(x1^(α-1))(x2^β)

To determine when the marginal product of input 1 is decreasing, we need to find the conditions for which the derivative of the marginal product of input 1 with respect to x1 is negative:

d(MP1)/dx1 = α(α-1)(x1^(α-2))(x2^β)

Since α > 0, the factor α(α-1) is always positive. Therefore, for the production function to exhibit decreasing marginal product of input 1, we need the term (x1^(α-2))(x2^β) to be negative or zero. This means that either x1^(α-2) = 0 or x2^β = 0.

Since x1^(α-2) cannot be equal to zero for any positive value of x1, we can conclude that for the production function to exhibit decreasing marginal product of input 1, β must be either zero or negative.

Similarly, to determine when the production function exhibits decreasing marginal product of input 2, we need to calculate the marginal product of input 2, denoted by MP2:

MP2 = ∂f/∂x2 = β(x1^α)(x2^(β-1))

To determine when the marginal product of input 2 is decreasing, we need to find the conditions for which the derivative of the marginal product of input 2 with respect to x2 is negative:

d(MP2)/dx2 = β(β-1)(x1^α)(x2^(β-2))

Again, since β > 0, the factor β(β-1) is always positive. Therefore, for the production function to exhibit decreasing marginal product of input 2, we need the term (x1^α)(x2^(β-2)) to be negative or zero. This means that either x1^α = 0 or x2^(β-2) = 0.

Since x2^(β-2) cannot be equal to zero for any positive value of x2, we can conclude that for the production function to exhibit decreasing marginal product of input 2, α must be either zero or negative.

In summary:
- The production function exhibits decreasing marginal product of input 1 when β is zero or negative.
- The production function exhibits decreasing marginal product of input 2 when α is zero or negative.