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ECON701 2017 Applied Microeconomics

1. Consider the following exchange economy. There are two goods (1 and 2) and two consumers (A and B). Preferences and endowments are as follows: uAxA 1 ; xA 2= 2xA 1 + xA 2 xA =1 2 ; 1 2 uBxB 1 ; xB 2= xB 1 + 2xB 2 xB =1 2 ; 1 2 (a) Depict the economy in an Edgeworth Box diagram. Your diagram shouldincludetheautarkicallocationandsomeindi⁄erencecurves for each consumer. [3 marks] (b) Identify the set of Pareto e¢ cient allocations. [Hint: No calculus is required to answer this question! Just use your Edgeworth Box diagram and explain your reasoning.] [4 marks] (c) Identify the set of allocations in the core. [2 mark] (d) There is a market for each good and a Walrasian auctioneer who announces a price vector p = (p1;p2). i. Suppose the auctioneer announces prices p1 = 3 and p2 = 4. Find each consumers optimal consumption plan and verify that these are not equilibrium prices. [Hint: Use a picture rather than calculus to think about the consumersoptimal consumption plans.] [4 marks] ii. Suppose the auctioneer announces prices such that p1 p2 = 1.

1

Show that markets will clear at such prices. What is the WEA? [4 marks]

2. Consider the following variation on the Dinner Game studied in class. There are TWO diners and each diner chooses between Chicken and Fish. Each diner values the Chicken at $4 and the Fish at $5. Therefore, if a diner orders the Chicken, her consumer surplus is equal to $4 less the amount she contributes towards the bill. If she orders the Fish, her consumer surplus is $5 less the amount that she contributes towards the bill. The Chicken costs $1 and the Fish costs $2:50. In particular, if the diners each paid for their own meals, a diner who orders the Chicken would get a consumer surplus of $3 and a diner who orders the Fish would get a consumer surplus of $2:50, so all would order the Chicken. Each diner starts with $12. At the end of the evening, each diners payo⁄(nancial consequence) is equal to $12 plus her consumer surplus from the meal.

(a) Suppose the diners plan to split the bill equally (i.e., each diner pays half of the total). Explain why each should order the Fish. [3 marks] (b) Suppose that the diners plan to settle the bill using the roulette method. That is, each diner writes her name on a piece of paper and puts it in a jar. At the end of the meal, the waiter draws a name at random and that person pays the entire bill. Thus, each diner has probability 1 2 of paying the whole bill and probability 1 2 of paying nothing. For example, if both diners order the Chicken, then each diner faces a lottery in which the nancial consequence is $14 (= $12 + $(42)) with probability 1 2 and $16 (= $12 + $(40)) with probability 1 2. Suppose that each diner has vNM utility function v (x) = ln(x10), where lnis the natural log function.1 This function is concave, so each diner is risk averse.2 i. Suppose you are one of these diners and you believe that the otherdinerwill orderthe Chicken. Use asuitable HYdiagram 1Note that the function ln(x10) is only dened if x 10 > 0 (i.e., x > 10). In this game, the lowest nancial consequence a diner can experience is $12, since the lowest possible consumer surplus is 0 (i.e., if both order the Fish and this diner ends up paying the whole bill). 2If you know some calculus you can check this by calculating the second derivative of v: v0(x) = 1 x10 2

to depict the two lotteries corresponding to your two choices (Chicken or Fish). Your diagram should include the fair-odds lines through each of these two points. [3 marks] ii. By calculating the expected utility of each lottery in (i), show thatyouprefertheFish.3 Sketchtheindi⁄erencecurvethrough the Chickenoption in your HY diagram from (i). It need not be perfectly accurate, but it should be compatible with the fact that you prefer the Fish, and with the fact that you are risk averse. [2 marks] iii. Suppose, instead, that you believe the other diner will order the Fish. Draw a fresh HY diagram to depict the two lotteries correspondingtoyourtwochoices(ChickenorFish), including the fair-odds lines through each point. [3 marks] iv. Bycalculatingtheexpectedutilityofeachlotteryin(iii), show thatyounowprefertheChicken. Sketchtheindi⁄erencecurve through the Fishoption in your HY diagram from (iii). It need not be perfectly accurate, but it should be compatible with the fact that you prefer the Chicken, and with the fact that you are risk averse. [2 marks]

and

v00(x) = 1 (x10)2

.

Since v00 < 0 the function is concave. 3Its ne to use a calculator to answer this question, but its not necessary its possible to get the answer by handif you know some properties of the natural log function. If youd like to try, here are the properties you need: (i) the natural log function is strictly increasing; and (ii) for any numbers a > 0, b > 0 and c,

cln(a) = ln(ac)

and

ln(a) + ln(b) = ln(ab).