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Question 9 1pts1 \mathrm{pts} Consider a firm with a production function given by...
May 14, 2024
Solution by Steps
step 1
The production function is given by F(L,K)=L+K+ln⁡(LK)F(L, K) = L + K + \ln(LK)
step 2
To find the marginal rate of technical substitution (MRTS), we need to compute the partial derivatives of FF with respect to LL and KK
step 3
The partial derivative of FF with respect to LL is ∂F∂L=1+1L\frac{\partial F}{\partial L} = 1 + \frac{1}{L}
step 4
The partial derivative of FF with respect to KK is ∂F∂K=1+1K\frac{\partial F}{\partial K} = 1 + \frac{1}{K}
step 5
The MRTS is given by the negative ratio of these partial derivatives: MRTS=−∂F∂L∂F∂K\text{MRTS} = -\frac{\frac{\partial F}{\partial L}}{\frac{\partial F}{\partial K}}
step 6
At the point (4,1)(4,1), we substitute L=4L = 4 and K=1K = 1 into the partial derivatives: ∂F∂L=1+14=1.25\frac{\partial F}{\partial L} = 1 + \frac{1}{4} = 1.25 and ∂F∂K=1+11=2\frac{\partial F}{\partial K} = 1 + \frac{1}{1} = 2
step 7
Therefore, the MRTS at the point (4,1)(4,1) is MRTS=−1.252=−0.625\text{MRTS} = -\frac{1.25}{2} = -0.625
Answer
The marginal rate of technical substitution at the point (4,1)(4,1) is −0.625-0.625.
Key Concept
Marginal Rate of Technical Substitution (MRTS)
Explanation
The MRTS is the rate at which one input can be substituted for another input while keeping the output level constant. It is calculated as the negative ratio of the marginal products of the inputs.
Solution by Steps
step 1
The utility function is given by U(x1,x2)=x12x22+x1+x2U(x_1, x_2) = x_1^2 x_2^2 + x_1 + x_2
step 2
To find the marginal utility with respect to x1x_1, we compute the partial derivative of UU with respect to x1x_1: ∂U∂x1=2x1x22+1 \frac{\partial U}{\partial x_1} = 2x_1 x_2^2 + 1
step 3
We are given that the marginal utility with respect to x1x_1 at the point (x1,3)(x_1, 3) is 10. Therefore, we substitute x2=3x_2 = 3 and set the partial derivative equal to 10: 2x1(3)2+1=10 2x_1 (3)^2 + 1 = 10
step 4
Simplifying the equation: 18x1+1=10 18x_1 + 1 = 10 18x1=9 18x_1 = 9 x1=918=12 x_1 = \frac{9}{18} = \frac{1}{2}
Answer
x1=12x_1 = \frac{1}{2}
Key Concept
Marginal Utility Calculation
Explanation
The marginal utility with respect to a variable is found by taking the partial derivative of the utility function with respect to that variable and then solving for the given conditions.
Question 11
step 1
The utility function is given by U(x1,x2)=x12x22+x1+x2U(x_1, x_2) = x_1^2 x_2^2 + x_1 + x_2
step 2
To find the marginal utility with respect to x2x_2, we compute the partial derivative of UU with respect to x2x_2: ∂U∂x2=2x12x2+1 \frac{\partial U}{\partial x_2} = 2x_1^2 x_2 + 1
step 3
We are given that the marginal utility with respect to x2x_2 at the point (5,x2)(5, x_2) is 5. Therefore, we substitute x1=5x_1 = 5 and set the partial derivative equal to 5: 2(5)2x2+1=5 2(5)^2 x_2 + 1 = 5
step 4
Simplifying the equation: 50x2+1=5 50x_2 + 1 = 5 50x2=4 50x_2 = 4 x2=450=225 x_2 = \frac{4}{50} = \frac{2}{25}
Answer
x2=225x_2 = \frac{2}{25}
Key Concept
Marginal Utility Calculation
Explanation
The marginal utility with respect to a variable is found by taking the partial derivative of the utility function with respect to that variable and then solving for the given conditions.
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