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a) Given utility function U (C; R) = min (C; 2R); daily endowment of time 24h, price and wage pc= w = 1; nd optimal choice of C; relaxation time R and labor supply L. (three numbers, use secrets of happiness for perfect complements). b) Find optimal choice given quasilinear preferences U (x1; x2 ) = x1+50 ln x2; prices p1= $8; p2= $1 and income m = $40. Is your solution corner or interior? c) Assume Interest rate r = 10%. Choose one of the two: a consol (a type of British government bond) that pays annually $500 starting next year or cash $4; 000 now (compare PV). d) Your annual income when working (age 21-60) is $70; 000 and then you are are going to live for the next 35 years. Write down equation that determines constant (maximal) level of consumption during your lifetime. Assume annual interest rate r = 10%: Bonus question (Just for fun) a) Derive magic formulas for perfect complements U (x1; x2 ) = min (ax1 ; bx2 ) that give optimal choices x1; x2 as a function of a; b; p1 ; p2 ; m: b) Provide economic intuition for the magic formula for perfect complements (economic interpretation for the numerator and denominator)

Lila spends her income on two goods: food, x1; and clothing, x2. a) The price food is p1= 4 and one piece of clothing costs p2= 8 . Show geometrically Lilas budget set if her income is m = 120. Find the relative price food in terms of clothing (one number)? Where can the relative price be seen in the graph of a budget set? (one sentence) b) Lilas preferences are represented by utility function U(x1 ; x2 ) = (x1) 1 (x2) 2 : (3) We know that her preferences can be alternatively represented by function V (x1 ; x2 ) = ln x1+2 ln x2: (4) Explain the idea behind monotonic transformation (one sentence) and derive function V from U. c) Assuming utility function V (x1 ; x2 ) = ln x1+2 ln x2 -Find marginal rate of substitution (MRS) for all bundles (derive formula). For bundle (2; 2) nd the value of MRS (one number). Give economic interpretation of MRS (one sentence). Which of the goods is more valuable given consumption (2; 2)? -Write down two secrets of happiness that determine optimal choice given parameters p1; p2 and m. Explain economic intuition behind the two conditions (two sentences for each). -Using secrets of happiness derive optimal consumption x1,x2; given values of p1= 4; p2= 8 and m = 120: Is the solution corner or interior (chose one) d) (Harder) Using magic formulas for Cobb-Douglass preferences argue that the two commodities are 1) ordinary; 2) normal; and 3) neither gross complements nor gross substitutes (one sentence for each property). Problem 2 (15p) (Perfect substitutes) You are planing a budget for the state of Wisconsin. The two major budget positions include education, x1 and health care, x2: Your preferences over the two are represented by function U(x1 ; x2 ) = 3x1+3x2: a) Find marginal rate of substitution (give number). b) Find optimal consumption of x1 and x2, given prices p1= 2 and p2= 1 and available funds m = 100 (two numbers). c) Is solution interior? (yes-no answer). Is marginal utility of a dollar equalized? (give two numbers and yes-no answer) Problem 3 (20p) (Intertemporal choice) Zoe is a professional Olympic skier. Her income when youngis high (m1= 80) but her future (period two) is not so bright (m2= 40) a) Depict Zoes budget set, assuming that she can borrow and save at the interest rate r = 100%. Partition her budget set into three regions: the area that involves saving, borrowing and none of the two. 7 b) Find PV and FV of Zoes lifetime income (two numbers) and show the two values in the graph. Interpret economically PV. c) Zoes utility function is U(C1 ; C2 ) = ln C1+ 1 1+ ln C2 where discount rate is = 4. Using magic formulas, nd optimal consumption plan (C1 ; C2 ) (two numbers) and the corresponding level of savings/borrowing, S. d) Is Zoe tilting her consumption over time?

a) Given utility function U (C; R) = min (C; 2R); daily endowment of time 24h, price and wage pc= w = 2; nd optimal choice of C; relaxation time R and labor supply L. (three numbers, use secrets of happiness for perfect complements). b) Find optimal choice given quasilinear preferences U (x1; x2 ) = x1+50 ln x2; prices p1= $8; p2= $1 and income m = $60. Is your solution corner or interior? c) Assume Interest rate r = 10%. Choose one of the two: Choose one of the two: a consol (a type of British government bond) that pays annually $100; starting next year or $1; 200 now (compare PV). d) Your annual income when working (age 21-60) is $70; 000 and then you are are going to live for the next 35 years. Write down equation that determines constant (maximal) level of consumption during your lifetime. Assume annual interest rate r = 10%: Bonus question (Just for fun) a) Derive magic formulas for perfect complements U (x1; x2 ) = min (ax1 ; bx2 ) that give optimal choices x1; x2 as a function of a; b; p1 ; p2 ; m: b) Provide economic intuition for the magic formula for perfect complements (economic interpretation for the numerator and denominator).

1. (Chap 12, problem 1) Use the open economy IS/LM model to predict what would happen to aggregate income (Y), the exchange rate (e), and the trade balance (NX) under both floating and fixed exchange rates in response to each of the following shocks: a. A fall in consumer confidence about the future induces consumers to spend less and save more. (The MPC falls) b. The introduction of a stylish line of Toyotas makes some consumers prefer foreign cars over domestic cars (NX demand falls) c. The introduction of ATMs reduces the demand for money (L falls) 2. (Chap 12, problem 3) The open economy IS/LM model takes the world interest rate (r) as given.Consider what happens when this world rate changes. a. What might cause world interest rates to rise? b. In the open economy IS/LM model with a floating exchange rate, what happens to aggregate income (Y), the exchange rate (e), and the trade balance (NX) when the world interest rate rises? c. Answer part b again, except now with a fixed exchange rate. 3. (Chap 12, problem 4) Business executives and policymakers are often concerned about the "competitiveness" of American industry (the ability of U.S. industries to sell their goods profitably in world markets). a. How would a change in the exchange rate affect competitiveness? b. Suppose you wanted to make domestic industry more competitive but did not want to alter aggregate income (Y).According to the open economy IS/LM model, what combination of monetary and fiscal policies should you pursue? 4. (Based on Chap 11, problem 3 - with my own extensions)The economy is a described by a closed economy IS/LM model. a. Consumption is C = 200 + 0.75(Y - T) and investment is I = 200 - 25r.G = T = 0. Solve for the IS curve (i.e. an equation for Y in terms of r). b. Money demand is given by L = Y - 100r, the money supply is 1000 and the price level is P.Solve for the LM curve (i.e. an equation for Y in terms of r and P). c. Find the equilibrium interest rate of r and the equilibrium level of income (i.e. solve for where the IS and LM curves cross)

d. The LRAS curve is Y* = 975.What is the price level at which output is exactly equal to this? e. Now the money supply increases to 1200.What is the new equilibrium level of income (i.e. solve the IS/LM problem again with M = 1200). f. If prices remain at exactly the level you found in d), what is the new level of income?Does the increase in money supply cause an expansion or contraction in income? g. What would prices have to be so that income is exactly equal to Y* = 975 again?

Tony likes nuts x1 and berries x2, and his preferences are described by the following utility function U (x1; x2) = x a 1x b 2 : Find the following variables: 1) the optimal fraction (percentage) of income spent on berries; 2) the optimal amount of total cash (dollars) spent on berries; 3) the optimal quantity of nuts consumed; 4) the slope of the indierence curve at the optimal bundle for the following values of parameters: a) a = 4, b = 8, p1 = 5; p2 = 10, m = 60 b) a = 1 3 , b = 1 3 , p1 = 4; p2 = 1, m = 12 c) a = 1 2 , b = 3 2 , p1 = 5; p2 = 1, m = 20 Hint: Instead of calculating optimal choices using "two secrets of happiness," take advantage of the demand formulas for Cobb Douglas utility that we derived in the class. Problem 2 Benjamin spends his time either watching movies (x1) (he uses "on demand" option, cable TV) or listening to songs - MP3 downloaded from the Internet (x2) . His preferences are described by U (x1; x2) = ln x1+ ln x2 Answer the following questions: a) Derive Benjamins demand for movies and MP3 les as a function of prices p1, p2 and his income m: (do not use Cobb Douglas formula but rather derive demand using "two secrets of happiness"). b) Fix the price of MP3 at p2 = 1 and income on m = 10: Find the price oer curve (give an exact formula x2 = f (x1)) and plot it in the commodity space. Find the demand curve x1 = f (p1) and plot it in the graph (with p1 on vertical axis and x1 on horizontal axis). c) Is x1 an ordinary good or a Gien good? Explain. d) Now x p1 = 1 and p2 = 1: In the commodity space, plot the income oer curve. In addition, in two separate graphs, plot Engel curves for both movies and MP3 les. Argue that the two commodities are normal (not inferior). e) For the demand functions from point a), determine whether the two goods are gross complements, substitutes or neither. Problem 3 (Perfect Complements) Consider Trevor from our previous problem set who begins his day with strawberry milkshake. To prepare it, he mixes milk, x1; strawberries x2 and does so always in his favorite: proportion 1 glass of milk to 2 strawberries. What is his utility function? Answer all the questions from Problem 2, from a) to e) using these preferenes. Problem 4 (Perfect Substitutes) Kates preferences are dened over consumption of two types of apples: Red Delicious (x1) and Jonagold (x2) U (x1; x2) =2x1+x2: Answer all the questions from Problem 2, starting from a) to e) using these preferenes. Problem 5 (Quasilinear preferences) George is a stamp (x1) collector, but he also likes fancy clothes (x2). His utility function is given by U (x1; x2) = x1 + 10x2 1 2 x 2 2 : Each stamp costs p1 = 1 and a piece of his favorite clothing costs p2 = 2

1. Consider a homeowner with Von-Neumann and Morgenstern utility function u, where u (x) = 1 ex for wealth level x, measured in million US dollars. His entire wealth is his house. The value of a house is 1 (million US dollars), but the house can be destroyed by a flood, reducing its value to 0, with probability (0, 1). (a) What is the largest premium P is the homeowner is willing to pay for a full insurance? (He pays the premium P and gets back 1 in case of a flood, making his wealth 1 P regardless of the flood.) The homeowner's utility for getting 1 P always is (1P) u(1 P) = 1 e his utility in the outside option is 1 u(0) + (1 )u(1) = (1 )(1 e ) The largest premium P he is willing to pay is the P that makes him indifferent between buying and not buying insurance. 1 1 e ) (1P) = (1 )(1 e 1 P = ln(1 (1 )(1 e1 )) P = 1 + ln(1 (1 )(1 e1 )) (b) Suppose there is a local insurance company who has insured n houses, all in his neighborhood, for premium P. Suppose also that with probability there can be flood in the neighborhood destroying all houses (i.e., either all houses are destroyed or none of them is destroyed). Suppose finally that P is small enough that the homeowner has insured is house. Having insured his house, what is the largest Q that he is willing to pay to get the 1/n share of the company? (The value of the company is the total premium it collects minus the payments to the insured homeowners in case of a flood.) The company's value is nP with probability (1) and nP n with probability . His utility from buying insurance and not buying stock is (1P) u(1 P) = 1 e 1 And his utility from buying stock and insurance is (1 )u(1 P Q + P) + u(1 P Q + P 1) = (1 )(1 exp(1 + Q)) + (1 exp(Q)) We find the Q that makes him indifferent between buying and not buying: 1 e(1P) = (1 )(1 exp(1 + Q)) + (1 exp(Q)) exp(Q) exp(P) = (1 ) exp(0) + exp(1) Q = P ln(1 + e) We saw before that it must be true that exp(P) exp(1) + (1 ) exp(0) (this is a rearrangement of the indifference condition of part (a)), so we need exp(Q) 1, so Q 0. Thus, the homeowner is never willing to buy the stock. (c) Answer part (b) assuming now that the insurance company is global. It insured n houses in different parts of the world (all outside of his neighborhood), so that the destruction of houses by flood are all independent (i.e., the probability of flood in one house is independent of how many other houses has been flooded). The chance that i houses flood, out of n is i (1 ) ni Cn,i and the company's wealth is nP i. Thus, the buyer's expected utility from buying insurance and stock is i 1 i (1 ) ni Cn,i exp((1 P Q + P i/n)) Setting this equal to his outside option i 1 e(1P) = 1 i (1 ) ni Cn,i exp((1 P Q + P i/n)) i eQ = i (1 ) ni Cn,i exp((P i/n)) i Q = P ln( i (1 ) ni Cn,i exp(i/n)) n k/nk (1 ) nk e+(1)/(2n) (d) Assume that n is large enough so that k=0 Cn,ke = , discuss your answers to above questions (briefly). [Here, Cn,k denotes the number of k combinations out of n, and the sum is one minus the expected payoff from the loss due to the payments to the flooded houses.] Substituting in to our last result, we get Q = P (1 )/(2n) The expected value of a share of the company is P . You can see that the player's willingness to pay for a share of the company is the expected value minus (1)/(2n). 2 3 1 For this utility function and a lottery that is normally distributed (which is this case for large n), the agent's willingness to pay is always 2/2 In part b, we found that the agent would never buy into the company that only insures people like him. This is the equivalent of paying for a company to sell himself insurance. However, when the company insures many different people, the company's splits the risk of all of the people. Thus, a 1/N share of the company holds less risk as N increases. The agent that buys a share holds a part of everyone's risk, which is preferrable to holding only his own risk.