Problem 1 (40p) (Well-behaved preferences) Lila spends her income on two goods: food, 21, and clothing. 12. a) The price food is p1= 4 and one piece of clothing costs p2= 8 . Show geometrically Lila's 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) bj Lila's preferences are represented by utility function (3) We know that her preferences can be alternatively represented by function (4) Explain the idea behind "monotonic transformation" (one sentence) and derive function | from U. ) Assuming utility function (,,,) = Inc, +2 Inc2 Find marginal rate of substitution (MRS) for all bundles (derive formula). For bundle (2, 2) find 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 pi, p, and m. Explain economic intuition behind the two conditions (two sentences for each). -Using secrets of happiness derive optimal consumption 21 ,12, given values of pi= 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. In and health care. Ty. Your preferences over the two are represented by function U(1 1: 1)) = 35 +32. a) Find marginal rate of substitution (give number). b) Find optimal consumption of I, and 22, given prices p1= 2 and py= 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 "young" is high (m1= 80) but her future (period two) is not so bright (my= 40) a) Depict Zoe's 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. b) Find PV and FV of Zoe's lifetime income (two numbers) and show the two values in the graph. Interpret economically PV . q Zoe's utility function is U (C,. C,) =InC+14 In C2 where discount rate is o = 4. Using magic formulas, find optimal consumption plan (C1, (2) (two numbers) and the corresponding level of say- ings/borrowing, S. d) Is Zoe tilting her consumption over time? (yes-no answer). Problem 4 (25p) (Short questions) a) Given utility function U (C, R) = min (C, 2R), daily endowment of time 24/, price and wage pe= w = 2. find 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 (21, 2,)= 21+50 In 12, prices p = $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 (21, 2, ) = min (ar, . br,) that give optimal choices Ti, Ty 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).Problem 1 (40p) (Well-behaved preferences) Lila spends her income on two goods: food, 21, and clothing. 12. a) The price food is p1= 4 and one piece of clothing costs p2= 8 . Show geometrically Lila's 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) bj Lila's preferences are represented by utility function (3) We know that her preferences can be alternatively represented by function (4) Explain the idea behind "monotonic transformation" (one sentence) and derive function | from U. ) Assuming utility function (,,,) = Inc, +2 Inc2 Find marginal rate of substitution (MRS) for all bundles (derive formula). For bundle (2, 2) find 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 pi, p, and m. Explain economic intuition behind the two conditions (two sentences for each). -Using secrets of happiness derive optimal consumption 21 ,12, given values of pi= 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. In and health care. Ty. Your preferences over the two are represented by function U(1 1: 1)) = 35 +32. a) Find marginal rate of substitution (give number). b) Find optimal consumption of I, and 22, given prices p1= 2 and py= 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 "young" is high (m1= 80) but her future (period two) is not so bright (my= 40) a) Depict Zoe's 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. b) Find PV and FV of Zoe's lifetime income (two numbers) and show the two values in the graph. Interpret economically PV . q Zoe's utility function is U (C,. C,) =InC+14 In C2 where discount rate is o = 4. Using magic formulas, find optimal consumption plan (C1, (2) (two numbers) and the corresponding level of say- ings/borrowing, S. d) Is Zoe tilting her consumption over time? (yes-no answer). Problem 4 (25p) (Short questions) a) Given utility function U (C, R) = min (C, 2R), daily endowment of time 24/, price and wage pe= w = 2. find 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 (21, 2,)= 21+50 In 12, prices p = $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 (21, 2, ) = min (ar, . br,) that give optimal choices Ti, Ty 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. LEE. economic data show that the recent increase in the Federal government decit has been accompanied by an increase in private sector saving. This is evidence in favor of Ricardian equivalence. 2. Firms operating constant returns to scale technologies cannot make positive prots. 3. Increasing the money supply is the most efcient 1.vay to pay for government expendi tures. 4-. Government policy should not be concerned with transitional pathsit is only tvhat happens in steadystate that matters. 5. Increasing the money supply causes ination. ti". Economies with high real interest rates are more likely to expand i". Diamond overlapping generations with specic functional fonns Time: discrete. innite horizon Demography: :1 mass 3;} E Id 1 + n}' of newborns enter in every period. Everyone lives for 2 periods except for the rst generation of old people. Preferences: For the generations born in and after period D: Uii;z.s+1i = liui + 3lizs+1l where cm is consumption in period t andst agei of life and In{ _]. is the natural logarithm function. For the initial old generation Lil-emu] = ln{(:3'n]l. Productive technology: The production function available to rms is F{K_ I] = ARI-\"If\" where K is the capital stock and I is the number of workers employed. It will be convenient to use the implied per worker production function. .rlkl: where k is the capital stock per worker. Capital is fully used up in production. Endowments: Everyone has one unit of labor services when young. {Old people cannot work.) Institutions: There are competitive markets every period for labor and capital. [You can think of a single collectively owned rm that takes wages and interest rates as given} {a} 1|I."|."rite out and solve the problems faced by generation t households for given prices if}: and R, {the wage and rental rate of capital}. {b} 1|I."|."rite out and solve the problem faced by the rm in period t. {c} 1|I."|."rite down the market clearing condition for capital. dene a competitive equi librium and solve for the implied law of motion for the peryoung person stock of capital. in. in the economy. {d} Obtain values for each steady-state capital stock: if. in terms of the model's parameters. For each determine its dynamic properties {i.e_ stability. oscillatory}. lie} Under what conditions 1' on parameters} could there be over accumulation of cap itll'ir Diamond-Mortensen-Pissarides with two types of worker Time: Discrete, infinite horizon Demography: A mass of 1 of workers with infinite lives. A proportion @ of the workers are high, h, productivity the rest are low, , productivity workers. A worker's type, i = h, l is observable by the firm when they meet and it never changes. There is a large mass of firms who create individual and identical vacancies. The number of vacancies is controlled by free-entry. Preferences: Workers and firms are risk neutral (i.e. u(#) =>). The common discount rate is r. To simplify the algebra the value of leisure for workers (usually referred to as b) is set to zero. The cost of holding a vacancy for firms is a utils per period. Productive Technology: A firm matched to a worker of productivity i produces pi > 0, i = h, I units of the consumption good per period ( PA > pr). With probability A each period, jobs (filled or vacant) experience a catastrophic productivity shock and the job is destroyed Matching Technology: With probability m() each period unemployed workers encounter vacancies. Here 8 = v/u, V is the mass of vacancies, u = un + up is the total mass of unemployed workers and wj is the mass of type i = h, i unemployed workers. The function m(.) is increasing concave and m() em'(8). The rate at which vacancies encounter unemployed workers is then m()/e which is decreasing in d. (Assume that job destruction and matching are mutually exclusive ao m(#) + >