Econ 510a (second half)Yale UniversityFall 2007Prof. Tony SmithHOMEWORK #3This homework assignment is due at 4PM on Friday, November 16. Please put your assign-ment in Evrim Aydin’s mailbox.1. In lecture on Monday, November 5, we studied a two-period exchange economy withtwo types of consumers, A and B, who differ in their endowment streams. This prob-lem considers several variations on this economy and asks you to ascertain in eachcase whether the first welfare theorem holds (i.e., whether the competitive equilibriumallocation is Pareto optimal).(a) Suppose that, in each period, a government taxes the consumption of a typicaltype-i consumer at rate τi. The government uses its tax proceeds in each periodto give each consumer an equal lump-sum transfer of the consumption good. Atypical consumer’s lifetime budget constraint, therefore, reads:p0(1 + τi)ci0+ p1(1 + τi)ci1= p0ωi0+ p1ωi1+ p0¯S0+ p1¯S1,where¯Stis the lump-sum transfer in period t. The left-hand side of the budgetconstraint is the consumer’s total expenditures (including taxes that he pays,which in turn depend on how much he consumes); the right-hand side is theconsumer’s total resources (including lump-sum transfers from the government).The government’s budget constraint in period t is:¯St= θ τA¯cAt+ (1 − θ) τB¯cBt,where ¯citis the equilibrium consumption of a typical type-i consumer in period t.Although, in equilibrium, all consumers of a given type will choose the same levelof consumption in period t, each consumer is free to choose any consumption levelwhen solving his individual maximization problem, taking as given both pricesand the behavior of other consumers in the economy (i.e., ¯cAtand ¯cBt).For this economy, you have two tasks. First, carefully define a competitive equi-librium and display a set of conditions that determine the competitive equilibriumallocations and prices. Second, determine whether the competitive equilibriumallocation is Pareto optimal and justify your answer appropriately.1(b) Suppose instead that the government imposes both date- and type-specific taxeson consumption: the tax rate on the consumption of a type-i consumer in periodt is τit. As in part (a), the government’s tax proceeds are used to fund equallump-sum transfers to consumers. For this economy, complete the same two tasksas in part (a).(c) Suppose now that the government eliminates consumption taxes but instead im-poses both date- and type-specific taxes on consumers’ endowments: in period t,the government confiscates fraction τitof the endowment of a typical type-i con-sumer. As in parts (a) and (b), the government’s tax proceeds are used to fundequal lump-sum transfers to consumers. For this economy, complete the same twotasks as in part (a).(d) Eliminate government altogether and instead assume that consumers’ preferencesexhibit consumption externalities: each consumer’s utility in period t is increasingin his own consumption, cit, but decreasing in average (or per capita) consumption,θ¯cAt+(1− θ)¯cBt. Recall that each consumer takes ¯cAtand ¯cBtas given when choosinghis own consumption cit. For such an economy, complete the same two tasks as inpart (a).2. For a neoclassical growth model in which consumers value leisure, carefully define:(a) a competitive equilibrium with date-0 trading;(b) a competitive equilibrium with sequential trading; and(c) a recursive competitive equilibrium (hint: you need two functions to describe thebehavior of the aggregate economy).In addition, use the the recursive competitive equilibrium formulation to show that thecompetitive equilibrium allocation is Pareto optimal.3. Consider a neoclassical growth model in which consumers live for only two time periods,0 and 1. There are two types of consumers: fraction θ of consumers have initial capitalholdings equal to kL0> 0 and fraction 1 − θ of consumers have initial capital holdingsequal to kH0> kL0(where ‘L’ and ‘H’ denote, ‘low’ and ‘high’, respectively).(a) Carefully define a competitive equilibrium for this economy (assume that con-sumers have time-separable preferences that depend on consumption but not onleisure).(b) Show that if the felicity function u exhibits constant relative risk aversion, thenredistributions of the initial endowments of capital (i.e., changes in kL0and kH0thatleave the total amount of capital in p eriod 0 unchanged) have no effect either onequilibrium aggregate savings in period 0 or on equilibrium prices in either period.2This is a version of an aggregation theorem for this economy: holding the totalamount of capital in period 0 constant, the behavior of the aggregates in thiseconomy does not depend on the distribution of capital in period 0.(c) Suppose now that consumers do value leisure. Can you find a felicity function(that now depends on both consumption and leisure) for which the aggregationtheorem in part (b) holds? How large is the class of such felicity