High School

We appreciate your visit to II Consider two friends Alfred A and Bart B with identical income IA IB 100 they both like only two goods ₁ and x₂ That. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

II. Consider two friends Alfred (A) and Bart (B) with identical income IA = IB = 100, they both like only two goods (₁ and x₂). That are currently sold at prices p₁ = 1 and P2 = 4. The only difference between them are preferences, in particular, Alfred preferences are represented by the utility function: UA (x1, x₂) = x.5x2.5 while Bart's preferences are represented by: UB (x₁, x₂) = min{x₁,4x2} 2. Assume that a new technology is discovered that makes the production of good 2 cheaper, and thus prices are now p2 = 2. Do the following: a) What quantities are going to be consumed in this new scenario. b) Determine the Hicksian demand curve (as a function of utility and prices) for each good for Alfred and Bart. c) How much of an increase in income (keeping prices as ((p₁ = 1, P2 = 4)) is equivalent to the drop in prices? (estimate for each consumer separately)³. d) Can you tell who benefited more from the price drop?

Answer :

a) After the discovery of the new technology, good 2 becomes cheaper, and the price is reduced to p2 = 2. In this new scenario, the quantities consumed by Alfred and Bart are obtained as follows; Alfred: His utility function is UA (x1, x2) = x1⁰.⁵x2⁰.⁵

Given the new prices, Alfred's budget line is given as 1x1 + 2x2 = 100. At optimal consumption, the marginal rate of substitution (MRS) must equal the price ratio (Px₁/Px₂). Thus, MRS = MUx₁/MUx₂ = (px₁/px₂) = 0.25. Solving for the optimal quantities, we get x₁ = 80 and x₂ = 10. Bart: His utility function is UB(x₁, x₂) = min{x₁, 4x₂}. Given the new prices, Bart's budget line is given as 1x1 + 2x2 = 100. At optimal consumption, the MRS must equal the price ratio. Thus, MRS = MUx₁/MUx₂ = (px₁/px₂) = 0.5. Solving for the optimal quantities, we get x₁ = 40 and x₂ = 20.

b) Hicksian demand curves are obtained by solving the utility maximization problem using prices as constraints. The first-order conditions of utility maximization for Alfred and Bart are given as follows; Alfred: px₁ = λx₁⁻⁰.⁵x₂⁰.⁵ and px₂ = λx₂⁻⁰.⁵x₁⁰.⁵, where λ is the Lagrange multiplier. Solving this system of equations yields the Hicksian demand for Alfred as x₁ = 2√(2)p₁⁻¹p₂⁰.⁵I⁰.⁵ and x₂ = 5√(2)p₁⁰.⁵p₂⁻¹I⁰.⁵. Bart: At optimal consumption, the marginal utility of good 1 must equal the marginal utility of good 2. The demand for good 1 is given by x₁ = λ, and the demand for good 2 is given by x₂ = λ/4. Solving for λ gives the Hicksian demand for Bart as x₁ = min{I/2, 2I/p₁} and x₂ = min{I/8, (p₂I)/4}.

c) To calculate how much of an increase in income is equivalent to the price drop, we will equate the consumer surplus before and after the price drop. Using the initial prices, Alfred's consumer surplus is given as; CS1 = 0.5(80)⁰.⁵(10)⁰.⁵ + 0.5(10)⁰.⁵(80)⁰.⁵ = 56.6. After the price drop, the new optimal consumption is (80, 20). Thus, the new consumer surplus is given as; CS2 = 0.5(80)⁰.⁵(20)⁰.⁵ + 0.5(20)⁰.⁵(80)⁰.⁵ = 70.7. The increase in consumer surplus is, therefore, 14.1. Equating this to the percentage increase in income, we get; (14.1/56.6) × 100% = 25%. Thus, Alfred needs an increase in income of 25% to maintain his initial level of utility. Bart's initial consumer surplus is given as; CS1 = 0.5(40)⁰.⁵(20)⁰.⁵ + 0.5(20)⁰.⁵(40)⁰.⁵ = 35.4. After the price drop, the new optimal consumption is (40, 20). Thus, the new consumer surplus is given as; CS2 = 0.5(40)⁰.⁵(20)⁰.⁵ + 0.5(20)⁰.⁵(40)⁰.⁵ = 44.7. The increase in consumer surplus is, therefore, 9.3. Equating this to the percentage increase in income, we get; (9.3/35.4) × 100% = 26%. Thus, Bart needs an increase in income of 26% to maintain his initial level of utility.

d) To determine who benefited more from the price drop, we compare the percentage increase in consumer surplus for Alfred and Bart. Alfred's percentage increase in consumer surplus is (14.1/56.6) × 100% = 25%. Bart's percentage increase in consumer surplus is (9.3/35.4) × 100% = 26%. Therefore, Bart benefited more from the price drop.

Know more about Alfred's budget line here:

https://brainly.com/question/29764556

#SPJ11

Thanks for taking the time to read II Consider two friends Alfred A and Bart B with identical income IA IB 100 they both like only two goods ₁ and x₂ That. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada