Let the production function be 𝑓(𝑥1, 𝑥2) = 𝑥1(𝑥1 + (𝑥2)^1/2). Find the technical rate of substitution of input 1 (TRS1) and provide an interpretation when 𝑥1=1 and when 𝑥2=4
To find the technical rate of substitution of input 1 (TRS1), we need to take the partial derivative of the production function with respect to input 1 (𝑥1) and divide it by the partial derivative of the production function with respect to input 2 (𝑥2).
Partial derivative of the production function with respect to 𝑥1:
∂𝑓/∂𝑥1 = 1 + 2𝑥2^(1/2)
Partial derivative of the production function with respect to 𝑥2:
∂𝑓/∂𝑥2 = 𝑥1 / (2𝑥2^(1/2))
Now we can calculate the TRS1:
TRS1 = (∂𝑓/∂𝑥1) / (∂𝑓/∂𝑥2)
TRS1 = (1 + 2𝑥2^(1/2)) / (𝑥1 / (2𝑥2^(1/2)))
Simplifying further:
TRS1 = (1 + 2𝑥2^(1/2)) * (2𝑥2^(1/2)) / 𝑥1
TRS1 = 2𝑥2 + 2𝑥2 / 𝑥1
Now let's evaluate the TRS1 when 𝑥1 = 1 and 𝑥2 = 4:
TRS1 = 2(4) + 2(4) / 1
TRS1 = 8 + 8
TRS1 = 16
Interpretation when 𝑥1 = 1:
When 𝑥1 = 1, the technical rate of substitution of input 1 (TRS1) is 16. This means that for every one unit increase in input 2 (𝑥2), input 1 (𝑥1) must increase by 16 units to maintain the same level of output.
Interpretation when 𝑥2 = 4:
When 𝑥2 = 4, the technical rate of substitution of input 1 (TRS1) is still 16. This suggests that regardless of the level of input 2 (𝑥2), input 1 (𝑥1) must increase by 16 units for every one unit increase in input 2 (𝑥2) to keep the output constant.
To find the technical rate of substitution of input 1 (TRS1), we need to calculate the derivative of the production function with respect to input 1, 𝑥1, holding 𝑥2 constant.
Let's start by finding the first-order partial derivative of the production function with respect to 𝑥1:
∂𝑓/∂𝑥1 = (𝑥1 + (𝑥2)^1/2) + 𝑥1(1/2)(𝑥2)^{-1/2}
Simplifying further:
∂𝑓/∂𝑥1 = 2𝑥1 + (𝑥2)^{1/2}
This represents the marginal product of input 1.
To find the technical rate of substitution of input 1 (TRS1), we divide the partial derivative of 𝑓 with respect to 𝑥1 by the partial derivative of 𝑓 with respect to 𝑥2:
TRS1 = (∂𝑓/∂𝑥1) / (∂𝑓/∂𝑥2)
Substituting the partial derivatives we calculated previously:
TRS1 = (2𝑥1 + (𝑥2)^{1/2}) / 𝑥1(1/2)(𝑥2)^{-1/2}
Simplifying further:
TRS1 = 2 + 2(𝑥2)^{-1/2} / 𝑥1
Now, let's interpret the TRS1 when 𝑥1 = 1 and when 𝑥2 = 4:
1. When 𝑥1 = 1:
TRS1 = 2 + 2(4)^{-1/2} / 1
= 2 + 2(2) / 1
= 2 + 4
= 6
When 𝑥1 = 1, the technical rate of substitution of input 1 (TRS1) is 6, which means that for every additional unit of input 2 (𝑥2), we need 6 units of input 1 (𝑥1) to maintain the same level of output. In other words, input 1 is relatively more substitutable for input 2, indicating that input 1 can be more easily replaced by input 2.
2. When 𝑥2 = 4:
TRS1 = 2 + 2(4)^{-1/2} / 𝑥1
= 2 + 2(2) / 𝑥1
= 2 + 4 / 𝑥1
When 𝑥2 = 4, the technical rate of substitution of input 1 (TRS1) is given by 2 + 4 / 𝑥1. The interpretation of this TRS1 value would depend on the specific value of 𝑥1.
To find the technical rate of substitution of input 1 (TRS1) in the given production function, we need to calculate the partial derivative of the production function with respect to input 1, holding input 2 constant.
First, let's find the partial derivative of the production function with respect to 𝑥1:
∂𝑓/∂𝑥1 = ∂/∂𝑥1 (𝑥1(𝑥1 + (𝑥2)^1/2))
To calculate this derivative, we use the product rule in calculus. Applying the product rule, we have:
∂𝑓/∂𝑥1 = (𝑥1) ∂/∂𝑥1 (𝑥1 + (𝑥2)^1/2) + (𝑥1 + (𝑥2)^1/2) ∂/∂𝑥1 (𝑥1)
Now, let's differentiate each term separately:
∂/∂𝑥1 (𝑥1 + (𝑥2)^1/2) = 1 + 0 = 1
∂/∂𝑥1 (𝑥1) = 1
Substituting these derivatives back into the expression, we have:
∂𝑓/∂𝑥1 = (𝑥1) (1) + (𝑥1 + (𝑥2)^1/2) (1)
Simplifying this expression, we get:
∂𝑓/∂𝑥1 = 2𝑥1 + (𝑥2)^1/2
This is the technical rate of substitution of input 1 (TRS1). It measures the rate at which input 2 can be substituted for input 1 in the production process while keeping output constant.
Now, let's interpret the TRS1 value when 𝑥1=1 and 𝑥2=4:
When 𝑥1=1 and 𝑥2=4, we substitute these values into the TRS1 expression:
TRS1 = 2(1) + (4)^1/2
= 2 + 2
= 4
The TRS1 value is 4 when 𝑥1=1 and 𝑥2=4. This means that, at this particular level of inputs, for each additional unit of input 2 used in the production process, we would need to reduce input 1 by 4 units to maintain the same level of output.
In other words, input 2 is substitutable for input 1 in a 4:1 ratio while maintaining the same level of output.