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.

c) Suppose the price of milkshakes increases to $5. How many units of milkshake and sushiare in Collin’s new optimal consumption bundle?

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) = (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)

There are two goods in the economy, wine and cheese. The price of wine is p1 dollars per litre of wine. The price of cheese is p2 per kilogram of cheese. The consumer has wealth M dollars to spend on wine and cheese. Let x1 denote the quantity of wine in litres she chooses to buy and let x2 denote the quantity of cheese in kilograms she chooses to buy. Suppose throughout that she can only purchase non-negative quantities of either good. Solve the following utility maximization problems when the preferences of a consumer are characterized by the following utility functions. For each part clearly specify what is the quantity of each good the consumer wishes to purchase given her budget constraint. (a) (10 points) U (x1; x2) = x1 x 4 2 [Hint: MU1 (x1; x2) = x 4 2 and MU2 (x1; x2) = 4x1x 3 2 .] (b) (15 points) U (x1; x2) =  x1 + 2x2 ￾ x 2 2 if x2  1 x1 + 1 if x2 > 1 [Hint: MU1 (x1; x2) = 1 and MU2 (x1; x2) = 2 (1 ￾ x2). First work out the solution for the case in which x 1 > 0 and x 2 > 0. Then show if p2 > 2p1 it is optimal for our consumer to spend her entire wealth on wine.] (c) (5 points) U (x1; x2) = maxfx1; x2g That is, U (x1; x2) =  x1 if x1  x2 x2 if x1 < x2 . Warning: In this case MU1 (x1; x2) and MU2 (x1; x2) do not exist! [Hint: Reason from Örst principles how the individual should allocate her wealth between the two goods if her aim is to maximize a utility function of this form. To do this, I recommend drawing a diagram to graph the budget set for di§erent price ratios. Can you Ögure out which 2 bundle or bundles maximise her utility for each particular budget set? To complete your answer you should now be able to work out for general prices and income levels which bundle or bundles maximize her utility.]

Mary (consumer 1) and Lucy (consumer 2) are the only two consumers in the economy.Each of them consumes only two goods, fish (good x) and chips (good y), which they also own.Consumer 1’s utility function is given byU 1(x, y) = x2y3She has 25 units of x and 5 units of y. Consumer 2’s utility function is given byU 2(x, y) = ln x + ln y.She has 6 units of x and 20 units of y. Let p be the price of x and normalise the price of y to 1.(a) Draw the Edgeworth Box of this economy, marking clearly the endowment point. For eachconsumer, sketch the indifference curve passing through the endowment point.(b) Calculate the marginal rate of substitution for each consumer at the endowment point.(c) Who is going to sell x and buy y? Why?(d) Find the demand of consumer 1 for x—denote it as x1—and the demand of consumer 1for y—denote it as y1.(e) Find the demand of consumer 2 for x—denote it as x2—and the demand of consumer 2for y—denote it as y2.(f) Find the equilibrium price of good x.(g) Find the equilibrium consumption bundle of each consumer.(h) Mark your answer to part (g) in the Edgeworth Box you have drawn for part (a). Drawthe budget line under the equilibrium price. For each consumer, draw the indifferencecurve passing through the equilibrium consumption bundle.1