1. Consider the following economy X with an infinitely-lived agent where the representative consumer has preferences giv

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1 Consider The Following Economy X With An Infinitely Lived Agent Where The Representative Consumer Has Preferences Giv 1 (90.34 KiB) Viewed 15 times

1. Consider the following economy X with an infinitely-lived agent where the representative consumer has preferences given by at {log (c) + Blog h₁}, t=0 where c is consumption, h is leisure, and B> 0 and 0 < 3 < 1. The consumer has an endowment of one unit of time. The consumer has an access to a financial market where he can save or borrow at real interest rate rt. Thus, the consumer's budget constraint at time t is Ct +St+1=w₁(1ht) + (1 + rt)st - Tt where st is the asset holdings (i.e, savings) by the consumer at time t consumption. Assume that so = 0. The representative firm has a technology for producing consumption goods, given by Yt = zelt where y, is output, z is productivity, is the labor input. The government purchases 9 units of consumption goods each period, and finances these purchases through lump-sum taxes T, and by issuing bonds bt. The government's budget constraint is 9t + (1+rt)bt = It + bt+1 and bg = 0. (a) Write down the problem of the consumer and explain the optimality condition(s) (b) From the firms' optimality condition, 2+ = wt, and the total output y = c + gt in each period. Solve for the equilibrium labor supply and consumption. (c) Suppose the economy is at the steady state: z = z and gt = g for all t. What is the equilibrium interest rate? Explain what determines the steady state (long-run) interest rate.