Question 4 Solving Consumer's Choice Problem under CD Utility, Numerical The consumer with income Y has a preference represented by CD utility function 0.4 0.6 U(91,92) = 914926 Given prices for the two goods, denoted by P₁, P2, solve the consumer's optimal choice problem following the steps below. (a) Write down the consumer's maximization problem, i.e be clear about (1) the choice variables, (2) the objective function and (3) the constraint. (b) Write down the Lagrangian for this optimization problem. (c) Derive the three first-order conditions.
Question 4 Solving Consumer's Choice Problem under CD Utility, Numerical The consumer with income Y has a preference represented by CD utility function 0.4 0.6 U(91,92) = 914926 Given prices for the two goods, denoted by P₁, P2, solve the consumer's optimal choice problem following the steps below. (a) Write down the consumer's maximization problem, i.e be clear about (1) the choice variables, (2) the objective function and (3) the constraint. (b) Write down the Lagrangian for this optimization problem. (c) Derive the three first-order conditions.
Chapter1: Making Economics Decisions
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![### Solving Consumer’s Choice Problem under CD Utility, Numerical
**Introduction:**
The consumer, with an income \( Y \), has a preference represented by a Cobb-Douglas (CD) utility function:
\[ U(q_1, q_2) = q_1^{a_1}q_2^{a_2}. \]
**Problem Outline:**
Given prices for the two goods, denoted \( (p_1, p_2) \), tackle the problem by following the outlined steps:
1. **Choice Variables and Objective Function:**
- **(a)** Write down the consumer’s maximization problem:
- Define (1) the choice variables.
- Define (2) the objective function.
- Define (3) the constraint.
2. **Optimization Problem:**
- **(b)** Write down the Lagrangian for this optimization problem.
- **(c)** Derive the three first-order conditions.
- **(d)** Solve for the optimal consumption bundle.
3. **Indirect Utility Function:**
- **(e)** Plug the optimal consumption bundles \( q_1^*, q_2^* \) into \( U(q_1, q_2) \) to derive the expression for the indirect utility function, e.g., \( U(q_1^*, q_2^*) \) as a function of \( Y, p_1, p_2 \).
4. **Analysis of Responses:**
- **(f)** Now, analyze the properties of demand. Specifically, examine how \( q_1 \) responds to changes in:
1. \( p_1 \) increases?
2. \( p_2 \) increases?
3. \( Y \) increases?
- Investigate whether \( q_1 \) responds by increasing or decreasing under these scenarios.
**Note:**
For simplicity, this analysis only covers \( q_1^* \). Results for \( q_2 \) are similar. (Hint: Use comparative statics, calculate \(\frac{\partial q_1^*}{\partial p_1}\), etc.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2e6f77a4-2bfc-4d58-b6c2-577005b77990%2F10347404-937b-4e09-8b47-c87662f3996d%2F9quhxmj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Consumer’s Choice Problem under CD Utility, Numerical
**Introduction:**
The consumer, with an income \( Y \), has a preference represented by a Cobb-Douglas (CD) utility function:
\[ U(q_1, q_2) = q_1^{a_1}q_2^{a_2}. \]
**Problem Outline:**
Given prices for the two goods, denoted \( (p_1, p_2) \), tackle the problem by following the outlined steps:
1. **Choice Variables and Objective Function:**
- **(a)** Write down the consumer’s maximization problem:
- Define (1) the choice variables.
- Define (2) the objective function.
- Define (3) the constraint.
2. **Optimization Problem:**
- **(b)** Write down the Lagrangian for this optimization problem.
- **(c)** Derive the three first-order conditions.
- **(d)** Solve for the optimal consumption bundle.
3. **Indirect Utility Function:**
- **(e)** Plug the optimal consumption bundles \( q_1^*, q_2^* \) into \( U(q_1, q_2) \) to derive the expression for the indirect utility function, e.g., \( U(q_1^*, q_2^*) \) as a function of \( Y, p_1, p_2 \).
4. **Analysis of Responses:**
- **(f)** Now, analyze the properties of demand. Specifically, examine how \( q_1 \) responds to changes in:
1. \( p_1 \) increases?
2. \( p_2 \) increases?
3. \( Y \) increases?
- Investigate whether \( q_1 \) responds by increasing or decreasing under these scenarios.
**Note:**
For simplicity, this analysis only covers \( q_1^* \). Results for \( q_2 \) are similar. (Hint: Use comparative statics, calculate \(\frac{\partial q_1^*}{\partial p_1}\), etc.)
Expert Solution
Step 1
Disclaimer :- Since you asked for multipart question we are solving the first 3 subparts as per guidelines. If you want any specific question to be solved the please repost the question and mention them.
Optimal utilisation of resources by the consumer means attaining the situation where consumer is getting maximum utility with given level of income .
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