Question 1Joe’s preferences are described by the following utility functionU (x, y) = xαyβwith α > 0 and β > 0.(a) Let I denote Joe’s income, and px and py denote the prices of good x and y, respectively.Find Joe’s optimal consumption bundle.(b) Now, suppose α = 6, β = 2, px = 2, py = 3 and I = 24. Evaluate Joe’s optimal choice.(c) Suppose px increases by 50%. What is Joe’s new optimal consumption bundle? Calculateboth the Income Effect and the Substitution Effect.Question 2Collin likes milkshakes (m) and sushi (s). His preferenes over these two goods are representedby the following utility functionU (m, s) = 2√m + s.Collin’s income is $100 and the price of sushi is $10.(a) Suppose the price of milkshakes is initially $2. Find Collin’s optimal consumption bundle.(b) Draw a graph (with m on the horizontal axis and s on the vertical axis) to show Collin’sbudget line, his indifference curve, and his optimal bundle.(c) Suppose the price of milkshakes increases to $5. How many units of milkshake and sushiare in Collin’s new optimal consumption bundle?(d) Draw a new graph for the new optimal bundle (or add it to the graph you have drawnabove in part (b)).(e) What are the substitution and income effects that result from the increase in the price ofmilkshake? Calculate these effects algebraically and illustrate them on a new graph.(f) What is the amount of additional income needed for Collin to achieve the initial level ofutility? What is the amount of additional income needed for Collin to purchase the initialbundle? What is the ideal cost of living index?
Question
Question 1Joe’s preferences are described by the following utility functionU (x, y) = xαyβwith α > 0 and β > 0.(a) Let I denote Joe’s income, and px and py denote the prices of good x and y, respectively.Find Joe’s optimal consumption bundle.(b) Now, suppose α = 6, β = 2, px = 2, py = 3 and I = 24. Evaluate Joe’s optimal choice.(c) Suppose px increases by 50%. What is Joe’s new optimal consumption bundle? Calculateboth the Income Effect and the Substitution Effect.Question 2Collin likes milkshakes (m) and sushi (s). His preferenes over these two goods are representedby the following utility functionU (m, s) = 2√m + s.Collin’s income is 10.(a) Suppose the price of milkshakes is initially 5. How many units of milkshake and sushiare in Collin’s new optimal consumption bundle?(d) Draw a new graph for the new optimal bundle (or add it to the graph you have drawnabove in part (b)).(e) What are the substitution and income effects that result from the increase in the price ofmilkshake? Calculate these effects algebraically and illustrate them on a new graph.(f) What is the amount of additional income needed for Collin to achieve the initial level ofutility? What is the amount of additional income needed for Collin to purchase the initialbundle? What is the ideal cost of living index?
Solution
This is a complex set of questions that involve understanding of utility functions, budget constraints, and the effects of price changes on consumption. Here's how you can approach each question:
Question 1: (a) Joe's optimal consumption bundle is determined by maximizing his utility function subject to his budget constraint. This involves setting up the Lagrangian function and taking the first order conditions. The optimal bundle (x*, y*) is given by the solution to the following system of equations:
αx^(α-1)y^β = λpx (1) βx^αy^(β-1) = λpy (2) pxx + pyy = I (3)
where λ is the Lagrange multiplier.
(b) Substituting the given values into the above equations, we can solve for x* and y*.
(c) If px increases by 50%, we need to solve the system of equations again with the new px. The income effect is the change in x* and y* due to the change in real income (holding relative prices constant), and the substitution effect is the change in x* and y* due to the change in relative prices (holding real income constant).
Question 2: (a) Collin's optimal consumption bundle is determined by maximizing his utility function subject to his budget constraint. This involves setting up the Lagrangian function and taking the first order conditions. The optimal bundle (m*, s*) is given by the solution to the following system of equations:
1/√m = λpm (1) 1 = λps (2) pmm + pss = I (3)
(b) To draw the graph, you need to plot the budget line using the equation pmm + pss = I, and the indifference curve using the equation U(m, s) = constant. The optimal bundle is the point where the budget line is tangent to the indifference curve.
(c) If the price of milkshakes increases to $5, we need to solve the system of equations again with the new pm.
(d) The new optimal bundle can be added to the graph by plotting the new budget line and finding the new tangency point with the indifference curve.
(e) The substitution and income effects can be calculated by comparing the old and new optimal bundles.
(f) The amount of additional income needed for Collin to achieve the initial level of utility or to purchase the initial bundle can be found by setting the utility function equal to its initial level and solving for I, or by setting the budget constraint equal to the cost of the initial bundle and solving for I. The ideal cost of living index is the ratio of the new income to the old income.
Similar Questions
(c) Suppose px increases by 50%. What is Joe’s new optimal consumption bundle? Calculateboth the Income Effect and the Substitution Effect.
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