3. [30+10] A consumer who consumes two goods and has the following utility function: U(x₁, x₂) = log(x₁ + 1) + log(x2 + 1), where ₁ and ₂ denote the monthly consumption of two goods. The consumer maximizes her utility subject to the following constraints: P₁x1 + P₂x2 ≤m, x₁ ≥ 0, X₂ ≥ 0, where p₁ > 0, p2 > 0 and m > 0. [Hint: Note that log f(x) = f(df (2).] 1_df f(x) dx (a) Write down the Kuhn-Tucker conditions for a point (x₁, x2) to solve the problem, and find all points that satisfy the conditions. Do any of those points solve the problem? (b) Show the above results by drawing indifference curves and budget set on the x₁x2 plane.
3. [30+10] A consumer who consumes two goods and has the following utility function: U(x₁, x₂) = log(x₁ + 1) + log(x2 + 1), where ₁ and ₂ denote the monthly consumption of two goods. The consumer maximizes her utility subject to the following constraints: P₁x1 + P₂x2 ≤m, x₁ ≥ 0, X₂ ≥ 0, where p₁ > 0, p2 > 0 and m > 0. [Hint: Note that log f(x) = f(df (2).] 1_df f(x) dx (a) Write down the Kuhn-Tucker conditions for a point (x₁, x2) to solve the problem, and find all points that satisfy the conditions. Do any of those points solve the problem? (b) Show the above results by drawing indifference curves and budget set on the x₁x2 plane.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
![3. [30+10] A consumer who consumes two goods and has the following utility function:
U(x₁, x₂) = log(x₁ + 1) + log(x2 + 1),
where ₁ and ₂ denote the monthly consumption of two goods. The consumer maximizes
her utility subject to the following constraints:
P₁x1 + P₂x2 ≤m,
x₁ ≥ 0, X₂ ≥ 0,
where p₁ > 0, p2 > 0 and m > 0. [Hint: Note that log f(x) = f(df (2).]
1_df
f(x) dx
(a) Write down the Kuhn-Tucker conditions for a point (x₁, x2) to solve the problem, and
find all points that satisfy the conditions. Do any of those points solve the problem?
(b) Show the above results by drawing indifference curves and budget set on the x₁x2 plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd60261f3-72eb-4b33-bd74-516ec2297ca3%2Fd841539d-3b4b-4996-986f-059fd2fbe5b2%2Fobz3jqg_processed.png&w=3840&q=75)
Transcribed Image Text:3. [30+10] A consumer who consumes two goods and has the following utility function:
U(x₁, x₂) = log(x₁ + 1) + log(x2 + 1),
where ₁ and ₂ denote the monthly consumption of two goods. The consumer maximizes
her utility subject to the following constraints:
P₁x1 + P₂x2 ≤m,
x₁ ≥ 0, X₂ ≥ 0,
where p₁ > 0, p2 > 0 and m > 0. [Hint: Note that log f(x) = f(df (2).]
1_df
f(x) dx
(a) Write down the Kuhn-Tucker conditions for a point (x₁, x2) to solve the problem, and
find all points that satisfy the conditions. Do any of those points solve the problem?
(b) Show the above results by drawing indifference curves and budget set on the x₁x2 plane.
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