EEP 100: PROBLEM SET 3(1) (T,F,U) When looking at an indifference map, one can tell that the preferences arehomothetic if the marginal rates of substitution are constant along rays from theorigin.(2) You are the benevolent ruler of a very small kingdom consisting of of two subjects(indexed by i = 1, 2). The subjects in your kingdom consume two different goods,x1 and x2, which can be purchased at prices of p1 and p2, respectively (you yourselfconsume nothing).a) Suppose the two subjects have identical utility functionsUi(x1, x2) = α log(x1) + (1 − α) log(x2),with α > 0 for i = 1, 2; also subject 1 has income y1, while subject 2 has incomey2. Derive Marshallian demand functions for both goods. If the total income(GDP) of your kingdom (y1 + y2) is equal to $1500, what will total demand foreach good be?b) Say prices doubled last season, but the total income (GDP) of your kingdom(y1 + y2) was equal to $3000 for that season as well. If the utility functions werethe same as in part (a), what do you expect total demand was in that season?What property or pattern does the Marshallian demand function exhibit here?c) Now suppose that subject 1 has income y1 = $500 and subject two has incomey2 = $1000. Derive an expression which describes the level of utility each subjectwill have as a function of her income (i.e., what are their indirect utility functionsVi(y, p1, p2)?).d) Suppose that you’ve been intrigued by a mention of Bentham’s work in EEP100 and have taken the initiative to read his Introduction to the Principles ofMorals and Legislation (available on-line at http://www.econlib.org/library/Bentham/bnthPML1.html). This leads you to wonder whether you can improvematters in your kingdom through a system of redistributive taxation. Your readingof Bentham suggests that an optimal distribution of income would maximizethe sum of your subjects’ indirect utility functions, V1(y1 + τ, p1, p2) + V2(y2 −τ, p1, p2), where τ is a transfer from subject 2 to subject 1 (which could be negative).Given this criterion, what is the optimal value of τ?e) You hold a town-hall meeting to discuss the proposed new redistribution. At themeeting, subject two claims that your beliefs regarding her utility function areincorrect, and that in factU2(x1, x2) = 2[α log(x1) + (1 − α) log(x2)].Derive the Marshallian demand functions and indirect utility for subject 2 withthis correction. How does the correction affect your earlier calculation of individualdemands? Of indirect utility?Date: Due September 29, 2015.1f) With the new indirect utility function for subject 2 (subject one’s is unchanged)recalculate the optimal scheme of redistributive taxation. Comment. How doyour results depend on assuming that utility is cardinal rather than ordinal?(3) Consider the plight of two equally skilled economists, Ann and Bill. Ann lives inOhio, where she pays a monthly rent of $1000 for a 1000 square foot apartment. Billlives in Oakland in an 800 square foot apartment that is otherwise identical to Ann’sbut which costs him $2000 per month. Assume that the only other good that Annand Bill consume is food, which costs $10 per unit in Ohio and only $5 per unit inOakland.a) Suppose that Ann and Bill both have ordinal utility functionsU(h, z) = α log(h) + (1 − α) log(z),with 0 ≤ α ≤ 1, where h is housing services (measured in square feet occupied permonth) and z is food. Derive Ann and Bill’s Marshallian Demands for housingand food. Give an interpretation of the parameter α; how does it affect choicesbetween housing services and food?b) Suppose preferences are as in (a) but with α = 0.25. What are the locallyprevailing monthly salaries for economists in Ohio and Oakland?c) Now suppose that we don’t know what Ann and Bill’s incomes are equal to(though they each know, and also know each others’), and that further we don’tknow what the value of α is, and Ann’s α may be different from Bill’s α. EitherAnn or Bill could choose to move to the other’s location and get a job as aneconomist there at the prevailing local salary, yet they choose not to. What canyou say about the value of Ann’s α relative to Bill’s α? (Hint: Ann’s indirectutility in Ohio must be greater than her indirect utility would be in Oakland;similarly, Bill must be happier in Oakland than in Ohio.)d) A pollster surveys a random sample of Ohio residents and Oakland residents, andasks each respondent, “Would you say that you’re (1) Very happy; (2) Prettyhappy; or (3) Not so happy?” After tabulating the results, the pollster findsthat on average people in Ohio are just as happy as people in California. Notingthat the cost of living in Ohio is lower than in California, she argues that manyCalifornians should move to Ohio, where they could be just as happy but spendless money. Comment on the quality of this argument. Are you tempted to move?(4) Extra Credit: If you know how to estimate simple regressions, this one’s for you! Onbcoures you’ll find two files, happydata.txt and codebook.txt. The former containsvariables called HAPPY, REALINC, and REGION. Download these.a) Estimate the mean values of HAPPY and REALINC by region. Discuss. (Note:you will want to ignore any values for these variables which indicate differentmissing codes; see the codebook).b) Consider a simple regression of the formVHAPPYi = α + β log(REALINC) + i,where the subscripts i indicate the value of the variable for the ith observation,and VHAPPYi takes the value 1 if the respondent is “Very happy”, and zerootherwise. Estimate the regression. How would you interpret your estimates ofthe coefficients?2c) Try estimating the previous regression again, but replace the constant term witha set of dummy variables for the different regions. Interpret your results.