Let the production function be 𝑓(𝑥1, 𝑥2) = 𝑥1(𝑥1 + sort𝑥2). Find the technical rate of substitution of input 1 (TRS1) and provide an interpretation when 𝑥1=1 and when 𝑥2=4

Question

Let the production function be 𝑓(𝑥1, 𝑥2) = 𝑥1(𝑥1 + sort𝑥2). 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 find the marginal rate of technical substitution (MRTS1).

The MRTS1 measures the rate at which one input can be substituted for another while keeping the level of output constant.

In this case, we have the production function 𝑓(𝑥1, 𝑥2) = 𝑥1(𝑥1 + sort𝑥2).

To find the MRTS1, we need to take the partial derivative of the production function with respect to 𝑥1 and divide it by the partial derivative of the production function with respect to 𝑥2.

∂𝑓/∂𝑥1 = 2𝑥1 + 𝑠𝑜𝑟𝑡𝑥2
∂𝑓/∂𝑥2 = 𝑥1

MRTS1 = (∂𝑓/∂𝑥1) / (∂𝑓/∂𝑥2)
= (2𝑥1 + 𝑠𝑜𝑟𝑡𝑥2) / 𝑥1

Now, let's evaluate the TRS1 when 𝑥1 = 1 and 𝑥2 = 4:

TRS1 = (2(1) + 𝑠𝑜𝑟𝑡(4)) / (1)
= (2 + √4) / 1
= (2 + 2) / 1
= 4

Interpretation:
When 𝑥1 = 1 and 𝑥2 = 4, the technical rate of substitution of input 1 (TRS1) is equal to 4. This means that for every additional unit of input 1 (𝑥1), the firm can reduce input 2 (𝑥2) by 4 units while keeping the level of output constant. In other words, input 1 is 4 times more productive than input 2 in this scenario.

Answer 2

Let the production function be 𝑓(𝑥1, 𝑥2) = 𝑥1(𝑥1 + sqrt𝑥2). Find the technical rate of substitution of input 1 (TRS1) and provide an interpretation when 𝑥1=1 and when 𝑥2=4

Answer 3

Apologies for the previous mistake in the interpretation. Let's correct the calculation and interpretation.

To find the technical rate of substitution of input 1 (TRS1), we need to find the marginal rate of technical substitution (MRTS1).

The MRTS1 measures the rate at which one input can be substituted for another while keeping the level of output constant.

In this case, we have the production function 𝑓(𝑥1, 𝑥2) = 𝑥1(𝑥1 + √𝑥2).

To find the MRTS1, we need to take the partial derivative of the production function with respect to 𝑥1 and divide it by the partial derivative of the production function with respect to 𝑥2.

∂𝑓/∂𝑥1 = 2𝑥1 + (1/2)√𝑥2
∂𝑓/∂𝑥2 = (1/2)𝑥1 / √𝑥2

MRTS1 = (∂𝑓/∂𝑥1) / (∂𝑓/∂𝑥2)
= (2𝑥1 + (1/2)√𝑥2) / [(1/2)𝑥1 / √𝑥2]
= 4(𝑥1 + (1/4)√𝑥2) / 𝑥1

Now, let's evaluate the TRS1 when 𝑥1 = 1 and 𝑥2 = 4:

TRS1 = 4(1 + (1/4)√4) / 1
= 4(1 + 1/2) / 1
= 4(3/2) / 1
= 6

Interpretation:
When 𝑥1 = 1 and 𝑥2 = 4, the technical rate of substitution of input 1 (TRS1) is equal to 6. This means that for every additional unit of input 1 (𝑥1), the firm can reduce input 2 (𝑥2) by 6 units while keeping the level of output constant. In other words, input 1 is 6 times more productive than input 2 in this scenario.

Answer 4

To find the technical rate of substitution of input 1 (TRS1), we need to find the derivative of the production function with respect to input 1 (𝑥1) holding the level of output constant.

Step 1: Calculate the derivative of the production function with respect to 𝑥1:

𝑓(𝑥1, 𝑥2) = 𝑥1(𝑥1 + sort𝑥2)

∂𝑓/∂𝑥1 = 𝑥1(1) + sort𝑥2

Step 2: Simplify the derivative:

∂𝑓/∂𝑥1 = 𝑥1 + sort𝑥2

Now we have the TRS1, which is ∂𝑓/∂𝑥1 = 𝑥1 + sort𝑥2.

To interpret the TRS1 when 𝑥1 = 1 and 𝑥2 = 4:

When 𝑥1 = 1 and 𝑥2 = 4:

TRS1 = 1 + sort4
= 1 + 2
= 3

The interpretation of TRS1 is that for every 1 unit increase in input 1 (𝑥1), the firm needs to increase input 2 (𝑥2) by 3 units to keep the level of output constant.

Answer 5

To find the technical rate of substitution of input 1 (TRS1), we need to calculate the partial derivative of the production function with respect to input 2 (𝑥2) divided by the partial derivative of the production function with respect to input 1 (𝑥1).

First, let's calculate the partial derivatives of the production function 𝑓(𝑥1, 𝑥2) = 𝑥1(𝑥1 + sort𝑥2) with respect to 𝑥1 and 𝑥2:

∂𝑓/∂𝑥1 = 𝑥1 + sort𝑥2

∂𝑓/∂𝑥2 = 𝑥1

Now, we can calculate the technical rate of substitution (TRS1) by taking the ratio of the partial derivative with respect to 𝑥2 to the partial derivative with respect to 𝑥1:

TRS1 = (∂𝑓/∂𝑥2) / (∂𝑓/∂𝑥1)

TRS1 = (𝑥1) / (𝑥1 + sort𝑥2)

To interpret the TRS1 when 𝑥1=1, we substitute 𝑥1=1 into the TRS1 formula:

TRS1 = (1) / (1 + sort𝑥2)

To interpret the TRS1 when 𝑥2=4, we substitute 𝑥2=4 into the TRS1 formula:

TRS1 = (𝑥1) / (𝑥1 + sort4)

Please note that the "sort" notation you provided in the production function is not a mathematical operation or notation that I recognize. To provide a more accurate interpretation and results, please provide the correct mathematical expression.