statistics and probability 3

Problem 1 (40p) (Well-behaved preferences) Lila spends her income on two goods: food, x1; and clothing, x2. a) The price food is p1= 2 and one piece of clothing costs p2= 4 . Show geometrically Lilas budget set if her income is m = 30. 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) 2 (x2) 1 : We know that her preferences can be alternatively represented by function V (x1 ; x2 ) = 2 ln x1+ ln x2: Explain the idea behind monotonic transformation (one sentence) and derive function V from U. c) Assuming utility function V (x1 ; x2 ) = 2 ln x1+ ln x2 -Find marginal rate of substitution (MRS) for all bundles (derive formula). For bundle (8; 8) nd the value of MRS (one number). Give economic interpretation of MRS (one sentence). Which of the goods is more valuable given consumption (8; 8)? -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= 2; p2= 4 and m = 30: 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+x2: 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 = 50 (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= 30) but her future (period two) is not so bright (m2= 20) 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. 1 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 = 2. 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?

Problem 4 (25p) (Short questions) a) Given utility function U (C; R) = min (2C; R); 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+100 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: a consol (a type of British government bond) that pays annually $1000 forever, starting next year or cash $12; 000 now (compare PV). d) Your annual income when working (age 21-60) is $60; 000 and then you are are going to live for the next 30 years. Write down equation that determines constant (maximal) level of consumption during your lifetime. Assume annual interest rate r = 2%: 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).

Problem 1 (40p) (Well-behaved preferences) Lila spends her income on two goods: food, x1; and clothing, x2. a) The price food is p1= 2 and one piece of clothing costs p2= 4 . Show geometrically Lilas budget set if her income is m = 60. 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) 2 (x2) 1 : We know that her preferences can be alternatively represented by function V (x1 ; x2 ) = 2 ln x1+ ln x2: Explain the idea behind monotonic transformation (one sentence) and derive function V from U. c) Assuming utility function V (x1 ; x2 ) = 2 ln x1+ 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= 2; p2= 4 and m = 60: 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 ) = x1+x2: 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 = 50 (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= 40) but her future (period two) is not so bright (m2= 20) 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. 3 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 = 3. 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?

Problem 4 (25p) (Short questions) a) Given utility function U (C; R) = min (2C; R); 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+100 ln x2; prices p1= $8; p2= $1 and income m = $80. 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 $1000; starting next year or cash $8; 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= 2 and one piece of clothing costs p2= 4 . Show geometrically Lilas budget set if her income is m = 60. 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 : (1) We know that her preferences can be alternatively represented by function V (x1 ; x2 ) = ln x1+2 ln x2: (2) 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= 2; p2= 4 and m = 60: 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 ) = 2x1+2x2: 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. 5 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 = 3. 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?