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

Answer 1

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.