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33 = B t=o 1 If the economy is inifially on its balanced growth path and if households preferences are given by U, what are the effects of a temporary increase in government purchases on the paths of consumption, capital, and the interest rate? (z) Show that the saving rate on the balanced growth path, (y*  c*)/y*, is decreasing in . (h ) Do citizens in low , highk*, highsaving countries have any incentive to invest in lowsaving countries? Why or why not? (e) Does your answer to part (c) imply that a policy of subsidizing investment (that is, making < 0), and raising the revenue for this subsidy through lumpsum taxes, increases welfare? Why or why not? (f) How, if at all, do the answers to parts (a) and (b) change if the government does not rebate the revenue from the tax but instead uses it to make government purchases? 2.9. Using the phase diagram to analyze the impact of an anticipated change. Consider the policy described in Problem 2.8, but suppose that instead of announcing and implementing the tax at time 0, the government announces at time 0 that at some later time, time ti, investment income will begin to be taxed at rate . (a) Draw the phase diagram showing the dynamics of and after time ti. (b) Can change discontinuously at time ti? Why or why not? (c) Draw the phase diagram showing the dynamics of and before fi. (d) In light of your answers to parts (a), (b), and (c), what must do at time 0? (e) Summarize your results by sketching the paths of and as functions of time. 2.10. Using the phase diagram to analyze the impact of unanticipated and anticipated temporary changes. Analyze the following two variations on Problem 2.9: (a) At time 0, the government announces that it will tax investment income at rate t from time 0 until some later date thereafter investment income will again be untaxed. ib) At time 0, the government announces that from time fi to some later time t2, it will tax investment income at rate ; before ti and after 2, investment income will not be taxed. 2.11. The analysis of government policies in the RamseyCassKoopmans model in the text assumes that government purchases do not affect utility from private consumption. The opposite extreme is that government purchases and private consumption are perfect substitutes. Specifically, suppose that the utility function (2.14) is modified to be
2.12. Precautionary saving, nonlumpsum taxation, and Ricardian equivalence. (This follows Leland, 1968, and Barsky, Mankiw, and Zeldes, 1986.) Consider an individual who lives for two periods. The individual has no initial wealth and earns labor incomes of amounts Yi and in the two periods. Yi is known, but is random; assume for simplicity that £[ 2] = 1. The government taxes income at rate n in period 1 and in period 2. The individual can borrow and lend at a fixed interest rate, which for simpUcity is assumed to be zero. Thus secondperiod consumption is Cz = [(ln)yiCi] 1 (1 2) . individual chooses Ci to maximize expected lifetime utility, UiQ) + ElUiCz)]. (a) Find the firstorder condition for Ci. (b) Show that E[Cz] = Ci if is not random or if utility is quadratic. (Hint: ifutility is quadratic, UiCz) is a linear function of C2, so EWiCz)] = U{E[C2]). (c) Show that if [/"(•) > 0 and Yz is random, E[Cz] > Ci. (Such saving due to uncertainty is known as precautionary saving. See Section 7.6.) (d) Suppose that the government marginally lowers n and raises tz by the same amount, so that its expected total revenue, n Yi + 2£[ 2], is unchanged. Implicitly differentiate the firstorder condition in part (a) to find an expression for how Q responds to this change. (e) Show that Ci is unaffected by this change if is not random or if utility is quadratic. (f) Show that Cl increases in response to this change if [/"(•) > 0 and Yz is random. (g) If the utility function is constantrelativeriskaversion, what is the sign of [/"(•)? 2.13. Consider the Diamond model with logarithmic utility and CobbDouglas production. Describe how each of the following affects kt+i as a function of kt. (a) Arise in n. (b) A downward shift of the production function (that is, f(k) takes the form Bk", and falls). (c) A rise in a. 2.14. A discretetime version of the Solow model. Suppose Y, = E(K,,A,Lt), with f (•) having constant returns to scale and the intensive form of the production function satisfying the Inada conditions. Suppose also that At+i = (1 1 g)A,, Lf+i = (1 I n)Lt, and iff+i = Kt + sYt  SK,. (a) Find an expression for kt+i as a function of kt. (b) Sketch kf+i as a function of kt. Does the economy have a balanced growth path? If the initial level of differs from the value on the balanced growth path, does the economy converge to the balanced growth path? (c) Find an expression for consumption per unit of effective labor on the balanced growth path as a function of the balancedgrowthpath value of k. What is the marginal product of capital, f{k), when maximizes consumption per unit of effective labor on the balanced growth path?
Note that this is the same as the Diamond economy with 0 = 0, F(Kt,AL,) = AL, + xK,, (d) Assmne that the production function is CobbDouglas. (i) What is kt+i as a function of kt? (zi) What isk*, the value of on the balanced growth path? (zzz) Along the lines of equations (2.62)(2.64), in the text, linearize the expression in subpart (i) around k, = k*, and find the rate of convergence of to k*. 2.15. Depreciation in the Diamond model and microeconomic foundations for the Solow modeL Suppose that in the Diamond model capital depreciates at rate S, so that n = fik,) S. (a) How, if at all, does this change in the model affect equation (2.60) giving kf+i as a function of kt? ib) In the special case of logarithmic utility, CobbDouglas production, and S = 1, what is the equation for kt+i as a function of k,? Compare this with the analogous expression for the discretetime version of the Solow model with <5 = 1 from part (a) of Problem 2.14. 2.16. Social security in the Diamond model. Consider a Diamond economy where g is zero, production is CobbDouglas, and utility is logarithmic. (a) Payasyougo social security. Suppose the government taxes each young individual amount T and uses the proceeds to pay benefits to old individuals; thus each old person receives (1 i n)T. (z) How, if at all, does this change affect equation (2.61) giving kt+] as a function of kt? (ii) How, if at all, does this change affect the balancedgrowthpath value ofk? (zzz) If the economy is initially on a balanced growth path that is dynamically efficient, how does a marginal increase in T affect the welfare of current and future generations? What happens if the initial balanced growth path is dynamically inefficient? (b) Fully funded social security. Suppose the government taxes each young person amount T and uses the proceeds to purchase capital. Individuals born at f therefore receive (1 + rt+i)T when they are old. (z) How, if at all, does this change affect equation (2.61) giving kt+i as a function of kt? (ii) How, if at all, does this change affect the balancedgrowthpath value ofk? 2.17. The basic overlappinggenerations modeL (This follows Samuelson, 1958, and AUais, 1947.) Suppose, as in the Diamond model, that Nt 2periodlived individuals are born in period f and that Nt = (11 n)Nti. For simplicity, let utility be logarithmic with no discounting: Ut = In(Cit) i ln(C2t+i). The production side of the economy is simpler than in the Diamond model. Each individual born at time is endowed with A units of the economys single good. The good can either be consumed or stored. Each unit stored yields x > 0 units of the good in the following period.
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