Cobb-Douglas Preferences Specific Example: Consider the util-ity function U (x, y) = x 0.5 y 0.5 that is defined on the consumption setR 2+ .(a) Find the equation of the indi↵erence curve that corresponds toU = 40.(b) What is the slope of the indi↵erence curve for U = 40 for anygiven value of x?(c) What is the equation of an arbitrary indi↵erence curve for thisutility function?(d) What is the slope of an arbitrary indi↵erence curve for this util-ity function at any given value of x?

Obtain the equation of the indifference curve for the utility level U = 20 (set min the vertical axis). Is this indifference curve steeper than the indifference curveyou derived in part (a)? Explain.3

U(x, y) = (3x + 2y)2 .The price of x is px = $10 per unit and his income is $200.(a) Obtain the equation of Johnathan’s indifference curve for the utility level U = 100. Drawthis indifference curve. (2 marks)(b) The price of y is py = $8 per unit. Obtain the marginal rate of substitution (MRS) and theequation of the budget line. Using a graph, find Johnathan’s optimal consumption bundle.In this graph, show the budget line, the optimal bundle, and the corresponding indifferencecurve. Make sure to label carefully all the curves. (3 marks)(c) Suppose that the price of y drops to py = $6 per unit (the price of x remains the same,at px = $10 per unit, and the income remains the same). Obtain the equation of the newbudget line. Using a new graph, find Johnathan’s optimal bundle with this new price. Inthis graph, show Johnathan’s new budget line, new optimal bundle, and the correspondingindifference curve. Make sure to label carefully all the curves. (3 marks)(d) Now suppose that the price of y is py = $8 per unit if Johnathan buys less than 10 units ofthis product, and py = $6 per unit if he buys 10 units of y or more (as an example, 20 unitsof y would cost $120). Assume that the price of x remains the same, at px = $10 per unit.Derive the equation of the budget line and draw it in a separate graph. (3 marks)(e) Using a new graph, find the optimal bundle(s) for the problem in part (d). In this new graph,show the budget line, the optimal bundle and corresponding indifference curve. Make sureto label carefully all the curves. (3 marks)

Collin 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 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?

U(x, y) = 2xy.(a) Obtain the marginal rate of substitution. Obtain the equation of the indifference curves forU = 16 and U = 36 (solving for y as a function of x). Draw these indifference curves. Dothese indifference curves intersect the y-axis and/or the x-axis? Explain in your own wordswhy this is the case. (4 marks)(b) Are Julian’s preferences homothetic? Justify mathematically your answer. (2 marks)(c) At the current prices (which Julian knows, but you don’t), initially, Julian spends 30 dollarsin x and 20 dollars in y. Suppose that Julian’s income increases by 5 dollars (and pricesremain the same). Compute Julian’s expenditure in x and Julian’s expenditure in y afterthis increase in income. (2 marks)(d) There is a decrease in the price of x. Is the (total) price effect on Julian’s demand for xlarger, the same, or smaller than the substitution effect? Explain. (1 mark)