A consumer’s preferences are Cobb-Douglass. In the initial optimal bundle the consumer buys 4 and 9 units of x1 and x2 respectively. If income doubles from its initial level of $100 to $200, in the consumer's new optimal bundle they consumer units of good 1 and and of good 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?

A consumer with a limited income will maximize utility when each good is purchased in amounts such that theMultiple Choicetotal utility is the same for each good in a bundle.marginal utility of each good in a bundle is maximized.marginal utility per dollar spent on each of the final choices in a bundle is equal.marginal utility per dollar spent on each of the final choices in a bundle is maximized for each good.

Figure 22-14Refer to Figure 22-14. Suppose the price of good X is $8, the price of good Y is $10, and the consumer's income is $360. Then the consumer's optimal choice is to buyGroup of answer choices40 units of good X and 4 units of good Y.30 units of good X and 12 units of good Y.15 units of good X and 24 units of good Y.20 units of good X and 20 units of good Y.

Exercise 1. Consider an economy with two consumers (named Con-sumer 1 and Consumer 2) and two goods (called good 1 and good 2).Suppose that Consumer 1’s preferences are given by the Cobb-Douglasutility functionu1(x11, x21) = x11x21where x11 is the amount of good 1 that Consumer 1 consumes andx21 the amount of good 2 that Consumer 1 consumes. Suppose thatConsumer 1 is endowed with 9 units of good 1 and 2 units of good 2.Suppose that Consumer 2’s preferences are given by the Leontiefutility functionu2(x12, x22) = min{x12, x22}where x12 is the amount of good 1 that Consumer 2 consumes andx22 the amount of good 2 that Consumer 2 consumes. Suppose thatConsumer 2 is endowed with 1 units of good 1 and 8 units of good 2.Normalise the price vector so that p2 = 1.(1) The contract curve is defined as the set of Pareto optimal alloca-tions that make each consumer at least as well off as they are atthe initial allocation. Find the contract curve. [Hint: Think beforeyou start calculating! It may help to start with a draft answer topart (3).](2) Find the equilibrium price and the equilibrium allocation.(3) Represent this economy in an Edgeworth Box diagram. Clearlymark the initial endowment, the equilibrium allocation, and theequilibrium price vector. Draw the indifference curves through theinitial endowment for the two consumers, labelling clearly whichis which. Also draw the contract curve and clearly label it. Drawyour Edgeworth Box as neatly and accurately as you can.Date: First Semester, 2024.1

Below are the utility preferences for an individual: Apples Marginal Utility of Apples Marginal Utility of Apples per Dollar (PA=$1)PA=$1Oranges Marginal Utility of Oranges Marginal Utility of Oranges per Dollar (PO=$2)PO=$21 9 9 1 14 72 8 8 2 12 63 5 5 3 8 44 1 1 4 2 1 What is the utility-maximizing consumption bundle if the individual has $4 to spend?Multiple choice question.3 apples and 1 orange4 apples2 apples and 1 orange2 oranges