Dirk BergemannDepartment of EconomicsYale UniversityEconomics 121b: Intermediate MicroeconomicsProblem Set 5: Expenditure Minimization, Slutsky,2/10/10Please put your name, section time and TA on your problem set.This problem set is due on Wednesday, 2/17/10.1. First, consider a consumer who maximizes the utility function U (x1; x2)subject to a b udge t constraint p1x1+ p2x2= I. This gives rise to uncom-pensated demand functions xi(p1; p2; I), i = 1; 2. Second, consider thesame consumer who minimizes the expenditures p1x1+ p2x2subject to autility constraint U (x1; x2) = U. This gives rise to compensated demandfunctions xci(p1; p2; U), i = 1; 2. (Note: Use the Lagrangean method (withequality, and thus disregarding the nonnegativity constraints on xifor thesubsequent problems in (1a) and (1b).)(a) Now let the utility function be U(x1; x2) =px1+px2. Formulatethe utility maximization problem, …nd the …rst-order conditions forutility maximization, and …nd the demand function for good 1.(b) Now formulate the corresponding expenditure-minimization problemand …nd the compensated demand function xc1(p1; p2; U).2. The expenditure function Ep; Uis the value of the objective at thesolution x to the expenditure minimization problem:Ep; U= minx(IXi=1pixi)subject toU (x)  Uwhere U is the utility target to be achieved, and x = (x1; :::; xI) andp = (p1; :::; pI) are the vectors of consumption and price.(a) Write the above expenditure minimization program as a Lagrangeanproblem L (x; ) (with a single equality constraint, and disregardingthe nonnegativity constraints on xi.) Hint: What is minimized andwhat is maximized in this Lagrangean problem?(b) Derive the …rst order conditions for the expenditure minimizationprogram.(c) Argue that at the optimal solution (x ;  ) of the Lagrangean prob-lem we haveEp; U= L (x ;  ) . (1)1(d) Finally, the optimal solution (x ;  ) clearly depends on the valuesof the exogenous parametersp; U. We may therefore write moreexplicitly x = x p; Uand  =  p; U. Now we want to knowhow the expenditure function Ep; Uchanges as we change someprice pj, in other words@Ep; U@pj.We can do this by using the relationship (1) and di¤erentiating theLagrangean at the optimal solutionx p; U;  p; U:L (x ;  ) = Lx p; U;  p; Uwith respect to pj. Using the …rst order conditions from (2b) showthat@Lx p; U;  p; U@pj= xjand thus conclude that@Ep; U@pj= xj.3. In class we made the observation th at the uncompensated demand xi(p; M) andthe compensated demand functions xcip; Uare related to each other byxi(p; M) = xcip; U(2)if the indirectly utility func tion at (p; M ) equals U :V (p; M ) = U (x (p; M)) = Uor conversely if the expenditure function atp; Uequals M :Ep; U= M. (3)The vector p = (p1; :::; pI) is the price vector for all the goods: i = 1; :::; I.Now using (3) to replace M we can write (2):xip; Ep; U= xcip; U. (4)(a) Now consider the price of good pjand derive the Slutsky equationby di¤erentiating with respect to pj:@xi(p; M)@pj=@xcip; U@pj@E@pj@xi(p; M)@M: (5)Then use your insights from (2d) to conclude that (5) can be rewrittenas@xi(p; M)@pj=@xcip; U@pj xj@xi(p; M)@M: (6)2(b) Interpret this equation (relating these demand functions to one an-other) in terms of the income and substitution e¤ect.(c) Using your solutions to parts (1a) and (1b) to verify that the Slutskyequation (6) holds in this case.4. Gi¤en goods are an interesting theoretical possibility, but are thought tobe rare. (Potatoes during the midst of the Irish potato famine are themost oft-cited example, but even this is debated.) Consider now a personwho consumes leisure (h =hours) and a consumption good (c), maximizingU(c; h) subject to the budget constraintpc = w(T  h) + M;or equivalentlypc + wh = wT + M:Here T is total number of hours, and h is number of hours spent on leisurerather than work. The utility function is increasing and concave in c andh. Write the Slutsky equation (which we derived in class) for this person.Under what conditions will leisure be a Gi¤en good, i.e., will an increasein the price w of leisure - not that w is the wage paid per hour on workingtime (T  h) - lead to an increase in the consumption of leisure (h)? Whyis this case di¤erent from that of parts (1)? (Look at how the prices pand w enter the budget constraints. We can without loss of generalitynormalize p = 1) In light of these results, explain why leisure is a likelycandidate to be a Gi¤en good.Reading Assignment:1. NS Chapter 2,5,6.2. Jensen (2007): The Digital