Econ 510a (second half)Prof: Tony SmithTA: Theodore PapageorgiouFall 2004Yale UniversityDept. of EconomicsSolutions for Homework #6Question 1a) The problem of the new generation when they’re born at time t can be written as:cit,c2t1,stmax uc1t uc2t1s.t.wt1 −  c1t stc2t1 Rt1st wt1We substitute in for c1tand c2t1and take the f.o.c. w.r.t. st:− u′wt1 − − st u′Rt1st wt1Rt1 0Assuming that prices are competitively determined, that we have log utility, that st kt1and that fk.l kan1−a, we have (after normalizing n  1):−11 − 1 − akta− kt1akt1a−1akt1a−1kt1 1 − akt1a 0 akt1a−11 − 1 − akta− akt1a−1kt1 a  1 − akt1aakt1a−11 − 1 − akta− akt1a a  1 − akt1aakt1a−11 − 1 − akta a  a  1 − akt1aa  a  1 − akt1 a1 − 1 − aktakt1a1 − 1 − aa  a  1 − aktaand thus the steady state level of capital is:ka1 − 1 − aa  a  1 − a11−ab) If there was no social security (  0), then the law of motion would be given by:kt11 − a1  ktaand the steady state is equal to :k 1 − a1  11−aWe need to show that:a1 − 1 − aa  a  1 − a1 − a1  a1 − a  a  1 − a11  a1 − 1   a  a  1 − aa  a − a − a  a  a   − a − a  which is true since 0  ,a,  1.c) In a world with full depreciation the golden rule level of capital (in the steady state) isgiven by:kmax f k − k  f′k  1A steady state is therefore dynamically efficient if:f′k  1 aka−1 1Setting   0 and substituting in for the steady state level of capital we found in part a), wehave:f′k  a1  1 − 147/43/4  3/479 1and therefore for these parameter values the steady state is not dynamically efficient.d) We need:a  a  1 − aa1 − 1 − a 1 143/41/41/4 3/43/41/41 − 3/4 1 3/4 1  33/41 − 3/4 4 34 1  3 941 − 74 3 94−94 214 12 221Thus by setting  221, the government can push the economy to the golden rule level ofcapital, which is dynamically efficient.Question 2a) If we multiply both sides of the government budget constraint by pt,weget:ptt ptqtbt1 ptbt ptgtwhich holds for every t. Therefore if we sum up over all t’s we have:∑t0ptt∑t0ptqtbt1∑t0ptbt∑t0ptgtUsing the fact that ptqt pt1(no-arbitrage condition), the above equation becomes:∑t0ptt∑t0pt1bt1∑t0ptbt∑t0ptgtFinally, since b0 0, it will be the case that∑t0pt1bt1∑t0ptbt(notice that the onlydifference between the two sums is that the first sum starts from p1b1, whereas the second alsoincludes p0b0.But since b0 0, the sums are equal). Therefore we get:∑t0ptt∑t0ptgtb) Substituting in the that in equilibrium there is no excess demand for government debtat btfor all t , we can write consumer’s lifetime budget constraint as:∑t0ptct∑t0ptkt1∑t0ptqtbt1∑t0ptrt 1 − kt∑t0ptwt∑t0ptbt∑t0pttAgain using the no-arbitrage condition ptqt pt1we can write:∑t0ptct∑t0ptkt1∑t0pt1bt1∑t0ptrt 1 − kt∑t0ptwt∑t0ptbt∑t0pttAs we saw however in part a), it is the case that∑t0pt1bt1∑t0ptbt, so the twocancel out:∑t0ptct∑t0ptkt1∑t0ptrt 1 − kt∑t0ptwt∑t0pttFinally substituting in for∑t0pttfrom the government consolidated budget constraintwhich we derived in part a), we have:∑t0ptct kt1 gt ∑t0ptrt 1 − kt wtThus the way in which the government finances its expenditure stream is irrelevant to theconsumer’s optimization problem. This is an important result and it implies that for example atax cut is unable to boost consumer demand.Note: The arbitrage condition condition ptqt pt1simply says that the price of the bondof bond today qt is equal to what it pays next period relative to today pt1pt 1. If the pricewas anything different and the inequality didn’t hold, then there would be room fo arbitrage,since the either the bond would be too cheap compared to what it is really worth, or