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?
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.
To determine for which values of α and β the production function exhibits decreasing marginal product of input 1 and input 2, we need to compute the partial derivatives of the production function with respect to each input.
1. Marginal product of input 1 (MP₁):
The marginal product of input 1 measures the additional output produced when input 1 is increased by a small incremental amount while keeping input 2 constant. To find MP₁, we need to compute the partial derivative of the production function with respect to 𝑥₁ while holding 𝑥₂ constant.
MP₁ = ∂𝑓/∂𝑥₁ = α(𝑥₁^(α-1))(𝑥₂^β)
For the marginal product of input 1 to be decreasing, the derivative MP₁ with respect to 𝑥₁ must be negative. Therefore, we need to determine the conditions under which:
α(𝑥₁^(α-1))(𝑥₂^β) < 0
This equation suggests two possible cases:
Case 1: α < 0
If α is negative, then (𝑥₁^(α-1)) will be negative, regardless of 𝑥₁'s value. Hence, there will always be decreasing marginal product of input 1.
Case 2: (𝑥₁^(α-1))(𝑥₂^β) < 0
If (𝑥₁^(α-1))(𝑥₂^β) is negative, then α must be such that 𝑥₁ > 0 and β should be such that 𝑥₂ > 0. In this case, the marginal product of input 1 will be decreasing for the given values of α and β.
2. Marginal product of input 2 (MP₂):
Similarly, to find the marginal product of input 2, we need to compute the partial derivative of the production function with respect to 𝑥₂ while keeping 𝑥₁ constant.
MP₂ = ∂𝑓/∂𝑥₂ = β(𝑥₁^α)(𝑥₂^(β-1))
For the marginal product of input 2 to be decreasing, the derivative MP₂ with respect to 𝑥₂ must be negative. Hence, we need to determine the conditions under which:
β(𝑥₁^α)(𝑥₂^(β-1)) < 0
Similar to the previous case, we have two possibilities:
Case 1: β < 0
If β is negative, then (𝑥₂^(β-1)) will be negative, regardless of 𝑥₂'s value. Thus, there will always be decreasing marginal product of input 2.
Case 2: (𝑥₁^α)(𝑥₂^(β-1)) < 0
If (𝑥₁^α)(𝑥₂^(β-1)) is negative, then β must be such that 𝑥₂ > 0 and α should be such that 𝑥₁ > 0. In this case, the marginal product of input 2 will be decreasing for the given values of α and β.
In summary, the production function will exhibit a decreasing marginal product of input 1 if α < 0 or for (𝑥₁^(α-1))(𝑥₂^β) < 0, and it will exhibit a decreasing marginal product of input 2 if β < 0 or for (𝑥₁^α)(𝑥₂^(β-1)) < 0.
To determine the values of α and β for which the production function exhibits decreasing marginal product of input 1 and input 2, we need to calculate the partial derivatives of the production function with respect to each input.
The partial derivative of the production function with respect to input 1 (𝑥1) can be found using the power rule, which states that the derivative of 𝑥^n with respect to 𝑥 is 𝑛𝑥^(𝑛-1):
∂f/∂𝑥1 = α(𝑥1^(α-1))(𝑥2^β)
Similarly, the partial derivative of the production function with respect to input 2 (𝑥2) is:
∂f/∂𝑥2 = β(𝑥1^α)(𝑥2^(β-1))
To determine if the marginal product of input 1 is decreasing, we need to evaluate the second derivative of the production function with respect to input 1:
∂^2f/∂𝑥1^2 = α(α-1)(𝑥1^(α-2))(𝑥2^β)
For decreasing marginal product of input 1, we need the second derivative to be negative, which means that α(α-1)(𝑥1^(α-2))(𝑥2^β) < 0.
Similarly, to determine if the marginal product of input 2 is decreasing, we evaluate the second derivative with respect to input 2:
∂^2f/∂𝑥2^2 = β(β-1)(𝑥1^α)(𝑥2^(β-2))
For decreasing marginal product of input 2, we need the second derivative to be negative, which means that β(β-1)(𝑥1^α)(𝑥2^(β-2)) < 0.
Since α and β are positive, we can simplify the conditions for decreasing marginal product of input 1 and input 2:
1. Input 1: The condition for decreasing marginal product of input 1 is α < 1. If α is less than 1, then the second derivative ∂^2f/∂𝑥1^2 will always be negative.
2. Input 2: The condition for decreasing marginal product of input 2 is β < 1. If β is less than 1, then the second derivative ∂^2f/∂𝑥2^2 will always be negative.
In summary, the production function exhibits decreasing marginal product of input 1 when α < 1, and decreasing marginal product of input 2 when β < 1.