2. Let U=U(x,y) be the utility function of the agent. x and y_represent the goods. Assume positive marginal utilities. Let Px, Py be the prices and I the income. a. State the agent's utility maximization problem. b. Present the Lagrangian function. c. Derive and interpret the first order condition. In your analysis, you must include the Lagrange multiplier. d. Present a graphical interpretation of the optimality condition (include indifferent curves and budget sets). 3 دیا e. Present the second-order condition and describe the conditions under which one could secure a maximum. f. Solve the previous questions but assuming: (i) U(x,y)=xy where a+b<1, (ii) U(x,y)=ax+by, (iii) U(x,y)=Min{x,y}. ax. apx g. Now, using the general formulation U=U(x,y), express the comparative-static derivative as the sum of income and substitution effects.
2. Let U=U(x,y) be the utility function of the agent. x and y_represent the goods. Assume positive marginal utilities. Let Px, Py be the prices and I the income. a. State the agent's utility maximization problem. b. Present the Lagrangian function. c. Derive and interpret the first order condition. In your analysis, you must include the Lagrange multiplier. d. Present a graphical interpretation of the optimality condition (include indifferent curves and budget sets). 3 دیا e. Present the second-order condition and describe the conditions under which one could secure a maximum. f. Solve the previous questions but assuming: (i) U(x,y)=xy where a+b<1, (ii) U(x,y)=ax+by, (iii) U(x,y)=Min{x,y}. ax. apx g. Now, using the general formulation U=U(x,y), express the comparative-static derivative as the sum of income and substitution effects.
Chapter1: Making Economics Decisions
Section: Chapter Questions
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![### Advanced Microeconomics - Utility Maximization Problems
**Question 2: Utility Maximization**
Given a utility function \( U=U(x,y) \), where \( x \) and \( y \) represent goods:
- Assume positive marginal utilities.
- Let \( P_x \) and \( P_y \) be the prices, and \( I \) the income.
**Parts of the Question:**
a. **State the agent’s utility maximization problem.**
- This involves setting up the problem where the agent allocates income \( I \) across goods \( x \) and \( y \) to maximize their utility:
\[
\max U(x,y) \quad \text{subject to} \quad P_x x + P_y y \leq I
\]
b. **Present the Lagrangian function.**
- The Lagrangian function for the utility maximization problem is formulated as:
\[
\mathcal{L} = U(x,y) + \lambda (I - P_x x - P_y y)
\]
where \( \lambda \) is the Lagrange multiplier.
c. **Derive and interpret the first-order condition. Include the Lagrange multiplier in your analysis.**
- The first-order conditions for a maximum are obtained by taking the partial derivatives of the Lagrange function with respect to \( x \), \( y \), and \( \lambda \), and setting them to zero:
\[
\frac{\partial \mathcal{L}}{\partial x} = \frac{\partial U}{\partial x} - \lambda P_x = 0
\]
\[
\frac{\partial \mathcal{L}}{\partial y} = \frac{\partial U}{\partial y} - \lambda P_y = 0
\]
\[
\frac{\partial \mathcal{L}}{\partial \lambda} = I - P_x x - P_y y = 0
\]
Interpretation involves:
\[
\frac{\partial U/\partial x}{\partial U/\partial y} = \frac{P_x}{P_y}
\]
showing that the marginal rate of substitution (MRS) equals the price ratio.
d. **Present a graphical interpretation of the optimality condition (include indifference curves and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F362a762f-914f-42d1-b0fa-542c2b1764ea%2F8af33371-a618-449d-9ae8-4c5ad4309a30%2Fk5f2l1_processed.png&w=3840&q=75)
Transcribed Image Text:### Advanced Microeconomics - Utility Maximization Problems
**Question 2: Utility Maximization**
Given a utility function \( U=U(x,y) \), where \( x \) and \( y \) represent goods:
- Assume positive marginal utilities.
- Let \( P_x \) and \( P_y \) be the prices, and \( I \) the income.
**Parts of the Question:**
a. **State the agent’s utility maximization problem.**
- This involves setting up the problem where the agent allocates income \( I \) across goods \( x \) and \( y \) to maximize their utility:
\[
\max U(x,y) \quad \text{subject to} \quad P_x x + P_y y \leq I
\]
b. **Present the Lagrangian function.**
- The Lagrangian function for the utility maximization problem is formulated as:
\[
\mathcal{L} = U(x,y) + \lambda (I - P_x x - P_y y)
\]
where \( \lambda \) is the Lagrange multiplier.
c. **Derive and interpret the first-order condition. Include the Lagrange multiplier in your analysis.**
- The first-order conditions for a maximum are obtained by taking the partial derivatives of the Lagrange function with respect to \( x \), \( y \), and \( \lambda \), and setting them to zero:
\[
\frac{\partial \mathcal{L}}{\partial x} = \frac{\partial U}{\partial x} - \lambda P_x = 0
\]
\[
\frac{\partial \mathcal{L}}{\partial y} = \frac{\partial U}{\partial y} - \lambda P_y = 0
\]
\[
\frac{\partial \mathcal{L}}{\partial \lambda} = I - P_x x - P_y y = 0
\]
Interpretation involves:
\[
\frac{\partial U/\partial x}{\partial U/\partial y} = \frac{P_x}{P_y}
\]
showing that the marginal rate of substitution (MRS) equals the price ratio.
d. **Present a graphical interpretation of the optimality condition (include indifference curves and
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Continue from part d - part g. Thank you!
Transcribed Image Text:2. Let U=U(x,y) be the utility function of the agent. x and y_represent the goods. Assume positive
marginal utilities. Let Px, Py be the prices and I the income.
a. State the agent's utility maximization problem.
b. Present the Lagrangian function.
c. Derive and interpret the first order condition. In your analysis, you must include the Lagrange
multiplier.
d. Present a graphical interpretation of the optimality condition (include indifferent curves and
budget sets).
3
e. Present the second-order condition and describe the conditions under which one could secure
a maximum.
f. Solve the previous questions but assuming: (1) U(x,y)=xy where a+b<1, (ii) U(x,y)=ax+by, (iii)
U(x,y)=Min{x,y}.
əx+
apx
g. Now, using the general formulation U=U(x,y), express the comparative-static derivative as
the sum of income and substitution effects.
Solution
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