Answer the following questions correctly

1 Precautionary Savings in General Equilibrium Let utility be given by: EX Bu (c) where u (c) = - exp {-ye} . Assume the standard intertemporal budget constraint Atti = (1 +r) (A, + y: - c) . (note: we do not necessarily impose 8 (1 + r) = 1 here). Assume that y, is ii.d. across time and agents. Let y = y + : where & is i.i.d. and Et-16: = 0. We do not impose a borrowing constraint on this problem, A, can take any value, although a no-Ponzi condition should be thought as being implicitly imposed for the problem to be well defined (you will not have to think about this no-Ponzi condition explicitly for solving the problem though). (a) Show that the consumption function, = Itr Art yet - 1 - T ( r, 7 ) for some * (r, 7) implies, Aa = 173 - g] + rx (r, 7) [note that the functional form of the consumption function, as a function of A, and current and expected income, is like the CEQ-PIH except for the constant a (r, 7)] (b) Use the Euler equation and your results in (a) to show that the consumption function in (a) is optimal for some a (hint: use the Euler equation to guess and verify the optimality of the above consumption function) which depends on r and the distribution of z. (c) Show that # (r, 7) > 0 if r is such that S (1 + r) = 1. Compare this to the CEQ-PIH case. How does a depend on the uncertainty in y? (d) Assume there is a constant measure 1 of individuals in the population. Argue that for aggregate consumption and assets to be constant and finite in the long run (in the limit as t - co) we require that a (r, 7) = 0. What does this imply about average long-run asset holdings as a function of r and ydenote this by A (r, 7))? What is happening to the cross-section of consumption? Does this distribution converge? (e) Use your results in (d) to compute the equilibrium interest rate re as a function of y for 3 = .97, y = 1. Assume & distributed normal with mean zero and standard deviation equal to o (this distributional assumption allows you to find an explicit expression for Eexp (-=)). Compute re for two cases o = 0.2 and 0.4 , fixing y =1. Compare this to the interest rate that prevails without uncertainty. (f) Briefly discuss how you would think of calibrating y if you really believed that pref- erences are CRRA el-s/ (1 -o) for some known value of o, but you want to work with this model as an approximation (because of its analytical tractability). Speculate on whether this is likely to be a good approximation for learning about me. Can this be a good approximation for learning about the long-run (invariant) distribution of asset holdings?2 Income Fluctuation Problem Numerical Computa- tion This exercise asks you to compute numerically an income uctuations problem. The problem is '3\" 1s- c, MEDEEl _ E, (=1) subject to: Ig+1 =(1 + T} (I; C1] + yf+1 where It represents "cash in hand" and y, labor income. Censider the following two alter- native income processes: Process. 1 y; is assumed to be i.i.d. with only two possible realizations 3.1 = Li's: and y}. = .519 with U2 probability each; Process. 11 3;, follows a two-state Markov process with transition matrix 11 Fan Fret Nu: FLL Where again y; = 1171;? and m. = .540 and FI'HH = 19 and :Iru = .51 Setp=.97,-r=2%auuw= 1. 1i'ou are given the basic matlab codes that should allow you to understand how to compute the solution to the income fluctuation problem. In fact you are given two codes, one that uses the simple grid method (explained in SL ch.4] and the other which uses splines which sometimes is more eicient, but you are not required to look at it. You are supposed to modify the simple code in order to match the new parameters and the two different income processes. You are required to hand in your code. Perform all calculations below Eor both the alternative income processes and for or = .8 and 0 =1.8. We solve this problem by iterating on the Bellman equation \"'\"yl=mu{w ago 10" +EE[v(',y'l | HI} Notice that in the and. we could have simplied the problem by reducing the state at cash- in-hand only. but since we are working also with a permanent process for income1 we need a two dirnensional state vector anyway. (a) Solve the optimal consumption problem obtaining the consumption function c(r]. Plot the function for consumption, asset holdings o' (2} = z E (z) , and oash-in-liand for to- morrow (for both realizations of tomorrow's income shock}, that is. z' (n, y'] = (1 + r} n' (z}+ y' for both possible values of y'. How do your results dier between the two alternative in- come processes? (b) Use this policy function together with random shocks to simulate the evolution of cash-in-hand, income1 consumption and assets for 200 periods; Plot the simulated series for consumption, income and assets. Is consumption smoother than income? How high are asset holdings? For what fraction of periods is the agent liquidity constrained (is. n, q = 0}? How do your results depend on 0'? Compare the two alternative income pIOCi (c) Compute a longer simulation (around 11001 periods}, throw away the rst 1001 periods (that is, from t = [II to t = 1000 included] and calculate the average asset holdings with the remaining periods. That is. based on the simulated sequence for {ALIEN compute 1; :Eiii'hl o.' (a) where o' (z) is the policy function found in a. (ma) (Carroll. 1997) Modify the income process to have the following characteristic: there is a small probability rr of income being zero. If this event does not occur income is drawn from the same distribution as before. Argue that, with the preferences above, the borrowing constraint will never bind: we always have n; = r, q \"2 0 (you do not have to compute; If we allow for some borrowing. so that we replace the constraint a; 2 0 with as E z, q 2 h for some positive 5 3 0, argue that this condition will never bind and that in fact a, b (l. (e) Now you will compute the equilibrium for this model as in Aiyagari (1994}. Using the Cobb-Douglas technology F (K, I.) = KLl solve the following system for K (1") and w (r): Egan5 (1] w = FrU'll a II Here K (r) and w(r) represent the level of capital demanded by firms and the associated wage if the interest rate at a steady state were equal to r and labor supply and demand are equal. For any proposed value of r (equivalently one can propose a value for K and obtain the implied proposed r using (1)) we can use the implied value for the wage w (r) and solve the individual's income fluctuations problem given r and w (r). Then using the method in (c) we can calculate the average asset holdings, denote this by A (r). Think of A (r) as representing the steady state supply of capital by individuals when the interest rate equals to r and the wage is w (r). We are looking for a value for r such that K (r) = A (r), i.e. that the quantity demanded and supplied of capital at a steady state are equal. Using a = .33, 6 = 8% and the parameters given above plot A (r) against K (r) by computing A (r) for several values of r (pick values in the interval 0