.Answer all questions.
Consider a consumer who allocates her income m to the consumption of goods 1 and 2. Denote by p; the price of good i = 1, 2. The consumer's preferences are such that there exists a bundle r = (11, 12) such that for any bundle y = (y1, y2), y # r, we have x > y. (i) Does the consumer has monotone preferences over bundles of goods 1 and 2? (ii) Suppose r = (5, 10), p1 = $1, p2 = $4, and m = 60. Can the consumer afford his favorite bundle, r? In a graph, draw the budget line and locate the bundle r. (iii) Suppose p, and p2 are as in (ii ), but m = $100. Is the consumer better off after the increase in income?Each individual consumer takes the prices as given and chooses her consumption bundle, (21, 12) E R2, by maximizing the utility function U(r1, 12) = In(r,x2). subject to the budget constraint p1 . r, + p2 . 12 = 900. (a) (3 points) Write out the Lagrangian function for the consumer's problem. (b) (6 points) Write out the system of first-order conditions for the consumer's problem. (e) (6 points) Solve the system of first-order conditions to find the optimal values of c, and 12. Your answer might depend on p, and pa. (d) (Voluntary, 0 points) Check that the critical point satisfies the second-order condition.Question 1 (25 Marks) Consider the simple (two-variable) linear regression model for which the population regression equation can be written in conventional notation as: Y1 = Bo + BiXi + ui where Y, and X, are observable variables, Bo and B, are unknown (constant) regression coefficients, and u; is an unobservable random error. The Ordinary Least Squares (OLS) sample regression equation corresponding to regression equation above is P = Bot Bix,+ 0 where Bo is OLS estimator of the intercept coefficient Bo , , is OLS estimator of the slope coefficient B1, 0, is the OLS residual for the i-th sample observation and N is the number of observations in the sample. (a) Show that the OLS slope coefficient estimator , is a linear function of Y, sample values. Stating explicitly all required assumptions, prove that the OLS slope coefficient estimator , is an unbiased estimator of the slope coefficient , . (10) (b) Derive an expression for the variance of , . (10) (c) Now consider the following quadratic form Y1 = Bo + B , X, + B2 X, + 1 (i) What problems might be created if the equation, Y = Bo + B,X, + up , in the first model is estimated rather than the second equation, Y, = P, + B, X, + B,X2, + u,? (5) (ii) Derive an expression which includes S, and B2 from the equation, Y, = P. + B,X, + B2X", + up that captures that impact of x on y. (5)c. Graph an indifference curve that shows the bun- e. Connect the dots to create Josd's budget con- dles of X and Y for which U = 6 and U = 8. Is straint. What is the slope of the budget constraint? the "more is better" assumption satisfied for X f. Divide the price of fireworks by the price of and Y? music. Have you seen this number before while 7. Kelly's utility function is given by U= 3X + 21, working on this problem, and if so, where? where MU, = 3 and MU, = 2. R. Suppose that a holiday bonus raises Jose's in- a What is MRS ,? come temporarily to $360. Draw Jose's new b. What is MRS,, when X = 1 and Y - 57 When budget constraint. X = 2 and Y = 2.57 h. Indicate the new bundles of music and fireworks c. Draw a sample indifference curve. that are feasible, given Jose's new income. 8. Andrea loves to eat burritos with hot sauce, In fact, 1 1. Suppose that only one person in the world sells ice she cannot enjoy a bumto (8) unless it has three cream. She employs a strange pricing policy: You can servings of hot sauce (H ). She gets no additional en- buy I ice cream cone for $1, but if you buy 2 cones, joyment from more than three servings per burrito. you have to pay $2 each. If you buy 3. you have to Thus, her utility function is (= min [ B.- H ). Graph pay $3 each, cic., so that if you buy 10, you have Andrea's indifference curves for U= 1 and U= 2. to pay $10 each. You have $100 dollars to spend on ice 9. Suppose John's utility function is 4XY, where X is cream comes and chocolate milk, and chocolate milk consumption of beer and I is consumption of pizza. costs $1 per unit Draw your budget constraint. This For this utility function, the marginal utility of X is strange ice cream pricing, where buying more costs given by MU, = 41, the marginal utility of F is given you more, is called a quantity surcharge. by MU,= 4X. 12. John enjoys ordering out for pizza and renting mov- I Suppose Y= 3. Calculate John's utility for X - 2, ies online. He makes $30 each week at a part-time 3, 10, and I1. For a given level of Y, does good X job. If movies cost $2 per rental and pizza costs display diminishing marginal utility? $7.50 per slice. graph John's budget constraint. Then illustrate the effects of each of the following b. Suppose X - 3. Calculate John's utility for Y = 2. events; 3, 10, and 11. For s given level of X, does good F display diminishing marginal utility? a. John's mother finds a coupon good for one free pizza and gives it to John. C. Find three different bundles containing X and Y' that give John 48 wils of satisfaction. Plot the The company John rents movies from sponsors theme bundles and connect them with an indiffer. holiday weck promotion: Rent the first five move coe curve. What happens to the marginal rate ies at their regular price, and all movies after the of sbalitution between X and Y' as consumption of fifth are half off. X' increases? C. John's favorite plaza place increases the price of 4. Does the principle of diminishing MAS depend pizza from $7 50 to $10. on the diminishing marginal utility of X and 17 13. Good X sells for $4, and good F sells for $2. At your 10. Jose gets satisfaction from both music and fireworks. current level of consumption, the marginal rate of Jon's income is $340 per week. Music costs $12 per substitution between X and Fisd. CD, and fireworks cost $8 per bag. I Draw an indifference curve and budget constraint It Graph the budget constraint Jose faces, with that reflects the facts given above. (Mar: You music on the vertical axis and fireworks on the will have to choose an linkial income level and horizontal suit an initial bundle of X and F.) I Jost spends all his income on music, how b. Are you maximizing your utility? much music can be affordt Plod a point that c. If you are not maximizing your utility, are you illustrates this scenario, buying low much X or too much IT Explain. c.IT hard spends all his incommon fireworks, him 14. Andre gets utility from playing laver tag and read- many bags of fireworks can be afford? Plot a ing books. Each week. Andre spends his entire $100 point that Illustrates this scenario paycheck in both pools, One hour of laver rag cous If Joad spends half his income on fireworks and $30: s bick costs $10. half his income un music, how much of each a. Graph Andre's budget constraint. Put books on can be allland? Hot s point then illustrates this the horizontal axis and hour of laser tag on the vertical unit
