Econ 510a (second half)Yale UniversityFall 2007Prof. Tony SmithHOMEWORK #4This homework assignment is due at 4PM on Friday, November 30. Please put your assign-ment in Evrim Aydin’s mailbox.1. In lecture on Wednesday, November 14, we studied an ove rlapping generations modelin which consumers live for two periods and are endowed with one unit of the con-sumption goo d when young and zero units when old. A consumer born in period tvalues consumption goods according to: c1t+ c2,t+1, where c1tis his consumption whenyoung and c2,t+1is his consumption when old.Suppose that “fiat” money is introduced into this economy. In particular, the initialold in period 0 are endowed with M > 0 units of money. In period t, consumerscan exchange qtunits of the date-t consumption good for one unit of money (or,equivalently, 1/qtunits of money for one unit of the date-t consumption good); in otherwords, qtis the price in period t of a unit of money in terms of date-t consumptiongoods. A consumer born in period t solves the following problem:maxc1t,c2,t+1,mt+1c1t+ c2,t+1subject to:c1t+ qtmt+1= 1c2,t+1= qt+1mt+1mt+1≥ 0,where mtis the amount of money that a consumer owns at the beginning of period t.Because the initial old in period 0 are endowed with M units of money, the consumptionof the initial old is given by c20= q0M.A competitive equilibrium in this economy is a set of allocations c20and {(c1t, c2,t+1)}∞t=0and a sequence of prices {qt}∞t=0such that: one, consumers behave optimally takingprices as given; two, the goods market clears in every period: c1t+ c2t= 1 for t ≥ 0;and, three, the money market clears in every period: mt= M for t ≥ 0.1(a) Show that this economy has a multiplicity of competitive equilibria. Specifically,show that, for any q ∈ [0, M−1], qt= q for all t ≥ 0 is a competitive equilibriumsequence of prices.(b) For which (if any) of the equilibria in part (a) is the competitive equilibriumallocation of consumption Pareto optimal?2. Consider a Diamond overlapping-generations economy in which consumers supply (in-elastically) one unit of labor when young and λ ∈ [0, 1] units of labor when old. Thepreferences of a consumer born in period t are given by:log(c1t) + β log(c2,t+1),where c1tis consumption when young, c2,t+1is consumption when old, and β ∈ (0, 1).Assume that the aggregate production function is Cobb-Douglas with exponent oncapital equal to α ∈ (0, 1).(a) Find the equilibrium law of motion of the aggregate capital stock.(b) Show that the steady-state of this economy is dynamically efficient provided thatλ is greater than some “cutoff” value λ∗∈ (0, 1).3. Consider a Diamond overlapping-generations economy like the one described in thesecond problem, but set λ = 0. Introduce a “pay-as-you-go” social security system intothis economy: in every period, the government collects d units of the consumption goodfrom the young and gives the proceeds of this lump-sum tax to the old. Suppose that dis zero initially and that the government increases d by a small amount. Show that thisincrease reduces the steady-state aggregate capital stock in competitive equilibrium.Under what c onditions can the introduction of a pay-as-you-go social s ecurity systembe Pareto-improving (i.e., under what conditions can it improve the welfare of allgenerations, including the initial
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