Econ 510a (second half)Yale UniversityFall 2004Prof. Tony SmithHOMEWORK #6You do not need to turn in answers to this homework but you are responsible for the material it covers.1. Introduce a “pay-as-you-go” social security system into the Diamond overlapping gen-erations model that we developed in lecture on Wednesday, December 1. In particular,suppose that in every period the government taxes the labor income of the young atrate τ and gives the proceeds to the old.(a) Assume time-separable utility with logarithmic felicity function and discount fac-tor β, Cobb-Douglas production with capital’s share α, and full depreciation ofthe capital stock in one period. Solve explicitly for the equilibrium law of motionof the aggregate capital stock.(b) Show that the introduction of social security leads to a downward shift in the equi-librium law of motion: for any (positive) level of capital today, capital tomorrowis lower in a world with social security than in a world without.(c) Suppose τ = 0, β = 3/4, and α = 1/4. Is the steady state dynamically efficient?(d) For what value of τ is the steady-state capital stock equal to its “golden-rule”value?2. Introduce a government that borrows and lends into the neoclassical growth mode l (setthe growth rate equal to zero). Specifically, the government seeks to finance a fixed(deterministic) stream of expenditures {gt}∞t=0; these expenditures are not valued byconsumers. The government issues a sequence of one-period debt {bt}∞t=0, with b0= 0.This debt is a promise by the government to pay one unit of the consumption good inthe next period; let the price of such a promise in period t be qt. The government alsoengages in lump-sum taxation of consumers; let τtbe the lump-sum tax in period t.In every period, the government satisfies the following budget constraint:τt+ qtbt+1= bt+ gt.The left-hand side of this constraint is the government’s inflows in period t (measuredin terms of period-t consumption goods), while the right-hand side is the government’soutflows in period t (again, measured in terms of period-t consumption goods).The representative consumer takes prices as given and seeks to maximize the lifetimeutility of consumption subject to a lifetime budget constraint given by:∞Xt=0pt(ct+ kt+1+ qtat+1) =∞Xt=0pt((rt+ 1 − δ)kt+ wt+ at− τt),where ptis the price of period-t consumption goods in terms of period-0 consumptiongoods (whose price p0is normalized to 1). In this budget constraint, at+1is the amountof government debt (i.e., promises by the government to deliver consumption goodsin period t + 1) that the consumer purchases in period t (assume that a0= 0). Inequilibrium, the demand for government debt is equal to the supply of governmentdebt in every period: at= btfor all t.(a) Use the no-arbitrage condition ptqt= pt+1to derive the government’s consolidatedbudget constraint:∞Xt=0ptgt=∞Xt=0ptτt.(b) Use the result from part (a) to show that the consumer’s lifetime budget constraintcan be written as follows:∞Xt=0pt(ct+ kt+1+ gt) =∞Xt=0pt((rt+ 1 − δ)kt+ wt).Because the sequences {τt}∞t=0and {bt}∞t=0do not appear in this budget constraint,it is evident that the way in which the government finances its expenditure stream{gt}∞t=0is irrelevant to the consumer’s optimization problem. Instead, all thatmatters to the consumer is the net present value of government expenditures, i.e.,P∞t=0ptgt. This implies in turn that the government’s financing decisions do notaffect the determination of equilibrium prices and quantities. This result is knownas the Ricardian equivalence
View Full Document
Unlocking...