e) Now, suppose that the price of butter changes, whereas the price of the sour cream and Matt's income remain the same. The new price of Butter is P ′ B = βPB. What is the new Matt's optimal bundle? Does it depend on the value of β?
Question
e) Now, suppose that the price of butter changes, whereas the price of the sour cream and Matt's income remain the same. The new price of Butter is P ′ B = βPB. What is the new Matt's optimal bundle? Does it depend on the value of β?
Solution
The new price of Butter is P'B = βPB = β*3 = 3β. The price of Sour Cream and Matt's income remain the same, so PS = 2 and M = 100.
The new budget constraint is 3βB + 2S = 100.
Matt's optimal bundle is still determined by the point where his budget constraint is tangent to the highest possible indifference curve. This occurs when the MRS equals the ratio of the prices of the two goods.
The MRS is still β, and the new ratio of the prices is P'B/PS = 3β/2.
(i) If β > 3β/2, then Matt's MRS is greater than the new price ratio. This means he values Butter more than its price suggests, so he will consume only Butter until his income is exhausted. His optimal bundle will be (B, S) = (100/(3β), 0).
(ii) If β < 3β/2, then Matt's MRS is less than the new price ratio. This means he values Sour Cream more than its price suggests, so he will consume only Sour Cream until his income is exhausted. His optimal bundle will be (B, S) = (0, 100/2).
(iii) If β = 3β/2, then Matt's MRS equals the new price ratio. This means he values Butter and Sour Cream equally in terms of their prices, so he will consume a mix of both goods. His optimal bundle will be any combination of B and S that satisfies the new budget constraint 3βB + 2S = 100.
So yes, the new optimal bundle does depend on the value of β.
Similar Questions
Matt considers Butter (B) and Sour Cream (S) as substitutes. His utility function has the form u(B, S) = βB + S, where β > 0. Suppose the price of Butter is PB = 3, the price of Sour Cream is PS = 2, and Matt has an income of M = 100 dollars to spend between Butter and Sour Cream. (a) In a well-labeled diagram, draw Matt's indifference map, putting Butter in the x-axis and Sour Cream in the y-axis. Make sure to indicate the direction in which the indifference curves increase.
(d) Obtain Matt's optimal bundle when: - (i)β > 3/2 (ii)β < 3/2 (iii)β = 3/2
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?
c) Suppose the price of milkshakes increases to $5. How many units of milkshake and sushiare in Collin’s new optimal consumption bundle?
(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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