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 π(π₯1, π₯2) = π₯1(π₯1 + sortπ₯2).
To find the MRTS1, we need to take the partial derivative of the production function with respect to π₯1 and divide it by the partial derivative of the production function with respect to π₯2.
βπ/βπ₯1 = 2π₯1 + π πππ‘π₯2
βπ/βπ₯2 = π₯1
MRTS1 = (βπ/βπ₯1) / (βπ/βπ₯2)
= (2π₯1 + π πππ‘π₯2) / π₯1
Now, let's evaluate the TRS1 when π₯1 = 1 and π₯2 = 4:
TRS1 = (2(1) + π πππ‘(4)) / (1)
= (2 + β4) / 1
= (2 + 2) / 1
= 4
Interpretation:
When π₯1 = 1 and π₯2 = 4, the technical rate of substitution of input 1 (TRS1) is equal to 4. This means that for every additional unit of input 1 (π₯1), the firm can reduce input 2 (π₯2) 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.
Let the production function be π(π₯1, π₯2) = π₯1(π₯1 + sortπ₯2). Find the technical rate of substitution of input 1 (TRS1) and provide an interpretation when π₯1=1 and when π₯2=4
3 answers
Let the production function be π(π₯1, π₯2) = π₯1(π₯1 + sqrtπ₯2). Find the technical rate of substitution of input 1 (TRS1) and provide an interpretation when π₯1=1 and when π₯2=4
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 π(π₯1, π₯2) = π₯1(π₯1 + βπ₯2).
To find the MRTS1, we need to take the partial derivative of the production function with respect to π₯1 and divide it by the partial derivative of the production function with respect to π₯2.
βπ/βπ₯1 = 2π₯1 + (1/2)βπ₯2
βπ/βπ₯2 = (1/2)π₯1 / βπ₯2
MRTS1 = (βπ/βπ₯1) / (βπ/βπ₯2)
= (2π₯1 + (1/2)βπ₯2) / [(1/2)π₯1 / βπ₯2]
= 4(π₯1 + (1/4)βπ₯2) / π₯1
Now, let's evaluate the TRS1 when π₯1 = 1 and π₯2 = 4:
TRS1 = 4(1 + (1/4)β4) / 1
= 4(1 + 1/2) / 1
= 4(3/2) / 1
= 6
Interpretation:
When π₯1 = 1 and π₯2 = 4, the technical rate of substitution of input 1 (TRS1) is equal to 6. This means that for every additional unit of input 1 (π₯1), the firm can reduce input 2 (π₯2) 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.
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 π(π₯1, π₯2) = π₯1(π₯1 + βπ₯2).
To find the MRTS1, we need to take the partial derivative of the production function with respect to π₯1 and divide it by the partial derivative of the production function with respect to π₯2.
βπ/βπ₯1 = 2π₯1 + (1/2)βπ₯2
βπ/βπ₯2 = (1/2)π₯1 / βπ₯2
MRTS1 = (βπ/βπ₯1) / (βπ/βπ₯2)
= (2π₯1 + (1/2)βπ₯2) / [(1/2)π₯1 / βπ₯2]
= 4(π₯1 + (1/4)βπ₯2) / π₯1
Now, let's evaluate the TRS1 when π₯1 = 1 and π₯2 = 4:
TRS1 = 4(1 + (1/4)β4) / 1
= 4(1 + 1/2) / 1
= 4(3/2) / 1
= 6
Interpretation:
When π₯1 = 1 and π₯2 = 4, the technical rate of substitution of input 1 (TRS1) is equal to 6. This means that for every additional unit of input 1 (π₯1), the firm can reduce input 2 (π₯2) 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.