Jerry has a total of I dollars per year to spend on apples (x) and Polo-shirts (y). His preferences are represented by the utility function U(x,y) = x¹y. a. Write down Jerry's marginal utility of apples and Polo-shirts. b. What are Jerry's demand functions x*(Px, Py, I) and y*(Px, Py, I)? c. What is Jerry's optimal yearly consumption of apples and Polo-shirts if I = $1200, Px = $4, and Py = $24? Sketch an indifference curve-budget line diagram and show the optimal bundle graphically.
Jerry has a total of I dollars per year to spend on apples (x) and Polo-shirts (y). His preferences are represented by the utility function U(x,y) = x¹y. a. Write down Jerry's marginal utility of apples and Polo-shirts. b. What are Jerry's demand functions x*(Px, Py, I) and y*(Px, Py, I)? c. What is Jerry's optimal yearly consumption of apples and Polo-shirts if I = $1200, Px = $4, and Py = $24? Sketch an indifference curve-budget line diagram and show the optimal bundle graphically.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Utility Maximization Problem Set
#### Problem Statement
Jerry has a total of \( I \) dollars per year to spend on apples \((x)\) and Polo-shirts \((y)\). His preferences are represented by the utility function \( U(x,y) = x*y \).
**Questions:**
a. **Marginal Utility Calculation:**
Write down Jerry’s marginal utility of apples and Polo-shirts.
\[MU_x = \frac{\partial U}{\partial x}, \quad MU_y = \frac{\partial U}{\partial y}\]
b. **Demand Functions:**
What are Jerry’s demand functions \( x^*(P_x, P_y, I) \) and \( y^*(P_x, P_y, I) \)?
c. **Optimal Consumption:**
What is Jerry’s optimal yearly consumption of apples and Polo-shirts if \( I = \$1200 \), \( P_x = \$4 \), and \( P_y = \$24 \)? Sketch an indifference curve-budget line diagram and show the optimal bundle graphically.
#### Solution Outline
1. **Marginal Utilities:**
Given the utility function \( U(x,y) = x*y \):
\[ MU_x = \frac{\partial U}{\partial x} = y \]
\[ MU_y = \frac{\partial U}{\partial y} = x \]
2. **Demand Functions:**
Using the budget constraint \( I = P_x x + P_y y \) and the condition for utility maximization where the marginal rate of substitution (MRS) equals the price ratio:
\[ \frac{MU_x}{MU_y} = \frac{P_x}{P_y} \Rightarrow \frac{y}{x} = \frac{P_x}{P_y}\]
Solving for \(x\) and \(y\):
\[ y = x \frac{P_x}{P_y} \]
Substituting this into the budget constraint:
\[ I = P_x x + P_y \left(x \frac{P_x}{P_y}\right) \]
Simplifies to:
\[ I = P_x x + P_x x \]
\[ I = 2 P_x x \]
\[ x^* = \frac{I}{2 P_x} \]
Subsequently, for \( y \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1cae3b69-8ddd-4940-ab36-18050edbfcd8%2F154689e5-2db7-4310-9782-88cdf2d8d05f%2Fo7zj5wd_processed.png&w=3840&q=75)
Transcribed Image Text:### Utility Maximization Problem Set
#### Problem Statement
Jerry has a total of \( I \) dollars per year to spend on apples \((x)\) and Polo-shirts \((y)\). His preferences are represented by the utility function \( U(x,y) = x*y \).
**Questions:**
a. **Marginal Utility Calculation:**
Write down Jerry’s marginal utility of apples and Polo-shirts.
\[MU_x = \frac{\partial U}{\partial x}, \quad MU_y = \frac{\partial U}{\partial y}\]
b. **Demand Functions:**
What are Jerry’s demand functions \( x^*(P_x, P_y, I) \) and \( y^*(P_x, P_y, I) \)?
c. **Optimal Consumption:**
What is Jerry’s optimal yearly consumption of apples and Polo-shirts if \( I = \$1200 \), \( P_x = \$4 \), and \( P_y = \$24 \)? Sketch an indifference curve-budget line diagram and show the optimal bundle graphically.
#### Solution Outline
1. **Marginal Utilities:**
Given the utility function \( U(x,y) = x*y \):
\[ MU_x = \frac{\partial U}{\partial x} = y \]
\[ MU_y = \frac{\partial U}{\partial y} = x \]
2. **Demand Functions:**
Using the budget constraint \( I = P_x x + P_y y \) and the condition for utility maximization where the marginal rate of substitution (MRS) equals the price ratio:
\[ \frac{MU_x}{MU_y} = \frac{P_x}{P_y} \Rightarrow \frac{y}{x} = \frac{P_x}{P_y}\]
Solving for \(x\) and \(y\):
\[ y = x \frac{P_x}{P_y} \]
Substituting this into the budget constraint:
\[ I = P_x x + P_y \left(x \frac{P_x}{P_y}\right) \]
Simplifies to:
\[ I = P_x x + P_x x \]
\[ I = 2 P_x x \]
\[ x^* = \frac{I}{2 P_x} \]
Subsequently, for \( y \
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