Dirk BergemannDepartment of EconomicsYale UniversityEconomics 121b: Intermediate MicroeconomicsProblem Set 3: Demand Functions1/27/10Please put your name, section time and TA name on your problem set.This problem set is due on Wednesday, 2/3/10.1. Choice and Demand. Consider the general budget linepxx + pyy = m:For the following utility functions …nd the demand for goods x and y asa function of px; pyand m: For the Cobb-Douglas utility function, solvethe problem using both, the method of sub stitution and the method ofLagrange. For the Leontie¤ utility fu nc tion, we saw in class that we canonly employ the method of substitution.(a) Cobb-Douglas utility:u(x; y) = xayb:where a; b > 0;(b) “Leontie¤ Utility”:u(x; y) = minfax; by);where a; b > 0;2. Budget and Demand. The federal government provides grants to stateand local governments to assist in funding all sorts of activities, like policeprotection and education. Two particular types of grants that the federalgovernment uses are: Non-Matching Grants: the federal government provides the state gov-ernment with a quantity of income, which the state can use for anypurpose it likes; Matching Grants: for every dollar the state spends on the provisionof some good or service the federal government provides the stategovernment with an additional dollar to be spent on that good.You are a high-ranking o¢ cial at the U.S. Department of Education, andhave been given the task of allocating grant money to the Connecticutstate government to improve its public school education. The governor ofConnecticut, whom we ass ume has preferences that re‡ect the preferencesof her constituents, receives utility from the allocation of Connecticutincome between two commodities:1 x = education y = composite bundle of all other private consumption,where “y” re‡ects income of Connecticut’s residents that the resi-dents spend themselves. The governor’s preferences over x and y aregiven by the following utility fu nction:u(x; y) = 2px + 2pyThe income of the state is m = $100 million. The price of a unitof composite consumption (py) is $1, while the price of a unit ofeducation (px) is $2.(a) Graphically describe:i. the governor’s preferences for education and private consumptionandii. the state’s budget constraint.(b) Find the optimal provision of education that the governor wouldchoose in the absence of any federal grant. You may assume aninterior solution and that preferences are convex (which they are).(c) What is the governor’s MRS at (x; y ) = (10; 80), where magnitudesare expressed in millions of dollars? Interpret. Why is this not anoptimal choice of education and private consumption.(d) Suppose that you, as a high-ranking federal government o¢ cial, o¤erConnecticut a non-matching grant of $50 million.i. Graphically describe the e¤ect of this grant on the state’s budgetconstraint.ii. Calculate Connecticut’s optimal level of education provision withthis grant. What level of utility does Connecticut’s governorreach with the grant?(e) Repeat part (2d) under the premise that you o¤er Connecticut adollar-for-dollar education matching grant.Reading Assignment: NS Chapter 2,4 and