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Berkeley ECON 202A - Economics 202A, Problem Set 3

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Economics 202A, Problem Set 3Maurice Obstfeld1. OG Model for the Open Economy. Consider the overlapping genera-tions model with the following twists: population is constant, the pathof output fyyt; yotgis exogenous with yot 0, the country may borrowfrom foreigners or lend to them at the …xed real interest rate r; andUcyt; cot+1= log cyt+  log cot+1: (a) If taxes on the young (old) areyt(ot), calculate the consumption functions of the young and the old.(b) What is the intertemporal budget constraint of the government.(c) Assume that initially government debt and taxes are zero. Nowconsider the following …scal p olicy: on date 0, the government makesa gift of d=2 in government bonds to the date-0 young and the samegift to the date-0 old. These bonds begin to pay interest (at rate r)on date 1. Taxes on date 0 population do not change, but taxes oneveryone rise by rd=2 from t = 1 onward. Show that this policy isconsistent with the government’s intertemporal budget constraint. (d)Calculate the e¤ect of the …scal policy (as a function of d) on aggregateconsumption for every date t = 0; 1; 2; 3; etc. (e) What is the e¤ecton long-run aggregate consumption? This e¤ect cannot be due to thecrowding out of capital, because there is no capital. Explain intuitivelywhat has happened.2. Barro versus Feldstein. In a critique of Barro’s famous paper "AreGovernment Bonds Net Wealth?" Martin Feldstein (JP E, April 1976)argued that government debt may be net wealth in a growing economy.His case went as follows: Suppose the government gives an amountof debt D0to people and taxes them for all interest paid on this andother government debt issued in the future. Let the interest rate inthe economy be …xed at r, the growth rate of total (not per capita)output g, and suppose r > g (ruling out Ponzi games). Suppose thegovernment taxers people to cover (r  g)D(t) each period but simplyissues new debt in the amount gD(t) to cover the balance of the totalinterest bill rD(t). Then, Feldstein argued, the public debt-to-outputratio will remain constant, the government’s intertemporal budget con-straint will still hold, but the people who receive the gift D0from the1government will enjoy a net increase in wealth equal to (g=r)D0, whichis proportional to the portion of the debt rolled over each p eriod. [Usethe continuous-time government budget constraintD0=Z10ertT (t)dtfor this one, where T (t) denotes total taxes collected on date t.] (a)Do you agree with Feldstein’s analysis for an economy with a constantnumber of identical immortal agents? (b) How would this argument fareif the demographics were di¤erent? Speci…cally, assume that people areimmortal, but that new immortal people are born each period and aretaxed to pay for past government liabilities as well as those incurredduring their own lifetimes.3. Intertemporal Tax Smoothing. Here is a problem related to F. P. Ram-sey’s other great contribution to economics, his paper on optimal tax-ation. (The rami…cations of the Ramsey tax principle throughout eco-nomics are many.) The government seeks to maximize the utility of atypical consumer,U =Z10ertflog[c(t)]  `(t) + v [g(t)]g dt;where ` is the labor that the individual devotes to production and g isa public good that the government provides (perhaps national defense).The path of g is exogenous. (I am assuming here that the subjectivediscount rate equals the market real rate of interest r.) Output is givenby y = A`; however, the government uses a tax on labor to …nance itself,so that the worker’s perceived return to working is insteady = (A   ) `:This results in a divergence between the private and social returns toe¤ort, a tax distortion. Assume that initially the government has nodebt (this is not an essential assumption, of course). One can view thegovernment as solving the intertemporal problemmaxfc(t); (t)gU2subject to the worker’s …rst-order condition for labor supply [wheredoes it come from? –this is part (a) of this problem],A  (t)c(t)= ; (1)the output constraintc(t) + g(t) = y(t) = A`(t); (2)and the intertemporal government budget constraintZ10ert[g(t)  (t)`(t)] dt = 0:This last budget condition can, alternatively, be represented by the setof equations for government debt_d(t) = rd(t) + g(t)   (t)`(t); d(0) = 0, limt!1ertd(t) = 0: (3)(b) Explain how equation (1), written asc =A  ;allows the government to control consumption through tax policy. Whydoesn’t g enter this consumption expression, and do you think thissimpli…cation would hold in general? [Hint: See the answer to part (c)below.](c) Show why (2) implies that labor supply can be controlled by` =A  A+gA:(d) Now show that the government’s problem can be expressed withoutdirect reference to the labor-income tax rate  asmaxf (t)gZ10ertlog [c(t)]  c(t) + g(t)A+ v [g(t)]dtsubject to_d(t) = rd(t) + g(t)  [A  c(t)]c(t) + g(t)A:3plus initial and terminal conditions on government debt.(e) Apply the Maximum Principle to the preceding problem, in whichconsumption is the only state variable. Show that the costate variable is constant over time. The variable has the interpretation of theshadow price of resources to the government. With lumpsum taxation = 1: With distorting taxes  > 1. The constancy of this marginaldistortion (the marginal deadweight burden of taxation) across di¤erenttime periods is Ramsey’s principle.(f) Show that the …rst-order condition for the control c takes the form1c= a decreasing linear function of c.Graph the two sides of this equation in a diagram where c is on thehorizontal axis. What do you make of the possibility of two intersec-tions? (Hint: look up "La¤er curve.") Which intersection representsan optimal tax rate?(g) Let’s imagine that A = 1;  = 0:1; g = 0:5;  = 1:2: (We basicallycan validate any level of  by adjusting the value of d(0) – can yousee why?) What values of consumption satisfy the …rst-order conditionof part (f)? Use your analytical …rst-order condition for consumptionto calculate numerically the optimal response dc=dg at the "good" taxsolution. Then use the expression  = Ac to calculate the derivatived=dg. Do taxes rise by as much as the increase in g, or by


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