Eco504, Part II Spring 2002 C. SimsOLG MODELS AND TAX BURDEN-SHIFTING1. RICARDIAN EQUIVALENCE: THE REPRESENTATIVE AGENTConsider a representative agent model in which the objective function is the expectationof some function U of current and future C and L and the constraint isf(Ct+τt+ Bt− Rt−1Bt−1,Kt,Kt−1,Lt,At) = 0. (1)Though this may look unfamiliar, it includes as a special case, for example,Ct+τt+ Kt−δKt−1+ Bt− Rt−1Bt−1= AtF(Kt−1,Lt) . (2)Hereτrepresents lump sum taxes, B government debt, and R the gross interest rate ongovernment debt. The agent chooses C, L, K, and B. The Euler equation FOC’s are∂C: EtDCtU =λtD1ft(3)∂L: EtDLtU =λtD4ft(4)∂K:λtD2ft= −βEt[λt+1D3ft+1] (5)∂B:λtD1ft=βRtEt[λt+1D1ft+1] . (6)2. RICARDIAN EQUIVALENCE: GOVERNMENTThe government budget constraint isBt− Rt−1Bt−1+τt= gt. (7)We will treat the stochastic process for g as exogenously fixed and consider the effects ofvaryingτand B. In order to avoid allowing B to explode upward, which will generally beruled out by the representative agent’s transversality condition,τwill have to be set so itreacts to B. For example, ifτt= −φ0+φ1Bt−1, withφ1chosen so that Rt−φ1< 1 for all t,then the government budget constraint (GBC) (7) becomes a stable difference equation in B.Substituting the GBC into the private constraint gives us the social resource constraint(SRC):f(Ct+ gt,Kt,Kt−1,Lt,At) = 0. (8)Note that the SRC, together with the three Euler equations (3-5), form a system of fourequations in the four unknowns C, L, K, andλ.τ, B and R do not appear in these equa-tions. In fact, these are the same four equations that would define the solution to a planner’sproblem in which the objective function was the same and the SRC itself was the constraint.Thus we can conclude that any solution to the planner’s problem is also an equilibrium forthe economy with traded government debt, and that the equilibrium stochastic process for C,K, and L is invariant to the policy that sets the time path forτand B, so long as the policykeeps B from exploding and is thus consistent with equilibrium.c°2002 by Christopher A. Sims. This document may be reproduced for educational and research purposes, solong as the copies contain this notice and are retained for personal use or distributed free.OLG MODELS AND TAX BURDEN-SHIFTING 23. WHAT THE RESULT DEPENDS ON, AND DOESN’T DEPEND ON• Doesn’t require complete markets, in this single-agent model. Asset markets arecompetitive in the model and government debt is freely traded, but a complete menuof assets is not present.• A single representative agent? If we had several agents with different tastes, but nouncertainty, the same result would hold. With uncertainty, the question is whetherprivate agents can issue debt with the same characteristics as government debt. Ifso, the result holds. If not, the choice of tax policy can create variation in the riskcharacteristics of government debt and thereby affect the ability of agents to traderisk. With complete asset markets, we would be back to Ricardian equivalence evenin this case.• Requires no new births. As we’ll see, new agents coming on the scene will undoRicardian equivalence.• Lump-sum taxes. We’ve assumed non-distorting taxes. If taxes are distorting, thentheir timing will matter.4. AN OLG MODELA representative agent from the generation born at time t lives two periods, consumingC1(t) at time t in the first period of life and C2(t + 1) at time t + 1 in the second period oflife. The constraints on the agent areC1(t) + S(t) ≤¯Y (9)C2(t + 1) ≤θS(t) (10)S(t) ≥ 0, (11)reflecting the fact that the agent is endowed with the single good in the first period of lifeand must save, earning real returnθon the saving, in order to consume in the second period.The agent’s problem ismaxC1(t),C2(t+1),S(t)EtU(C1(t),C2(t + 1)) . (12)The FOC’s, assuming that the S(t) ≥ 0 constraint is not binding, are∂C1: EtD1U(C1(t),C2(t + 1)) =λt(13)∂C2:D2U(C1(t),C2(t+1)) =µt+1(14)∂S:λt=θEtµt+1. (15)Eliminating the Lagrange multipliers, we arrive at the solved FOCEtD1Ut+1EtD2Ut+1=θ. (16)For the issues we will be studying with this model for now, uncertainty is not central, so wewill assume perfect foresight and drop the “Et”’s, which allows us to use a familiar diagramto characterize equilibrium (labeled for the caseθ< n = 1):OLG MODELS AND TAX BURDEN-SHIFTING 3C1C2private budget set, slope −θ Indifference curvesSRC, slope=−1 Y This equilibrium involves no trade. Every generation provides for its own “retirement” bysaving. This is called the autarchy solution. Under some conditions, a planner can do better.A planner would recognize that the social resource constraint at t isNtC1(t) + Nt−1C2(t) + NtSt≤ Nt¯Y +θNt−1St−1(17)St≥ 0. (18)For simplicity, assume Nt= N0nt. If the planner chooses to make C1(t) and C2(t) constantover time and sets S(t) ≡ 0, then the planner’s constraint becomesC1(t) + n−1C2(t) ≤¯Y . (19)This constraint and the implied optimum is also shown in the figure. Clearly if n >θ, theplanner can achieve an equilibrium that improves on that obtained by individual saving. Thissituation, in which the return on private saving is below the population growth rate, is calleddynamic inefficiency.5. GOVERNMENT DEBT AND TAXATIONWith government debt, the private constraints becomeC1(t) + St+BtPt+τt≤¯Y (20)C2(t + 1) ≤RtBtPt+1+θSt(21)Bt≥ 0, St≥ 0. (22)The government faces the budget constraintNtBtPt+ Ntτt= Nt−1Rt−1Bt−1Pt. (23)There is one new private FOC:∂B:λtPt= Etµt+1RtPt+1, (24)OLG MODELS AND TAX BURDEN-SHIFTING 4assuming the Bt≥ 0 constraint is non-binding. With no uncertainty, when both B and S arenon-zero, this FOC, together with the S FOC (15), impliesθ= RtPtPt+1, (25)i.e. equal real rates of return on private storage and government debt.6. EQUILIBRIUM WITH NO TAXESWith no taxes, the government simply rolls over the debt each period, settingBtNt= Bt−1Rt−1Nt−1. (26)We consider whether there can be an equilibrium in which C1(t),C2(t + 1) are constant overtime, no taxes are imposed, and there is no private storage. In such an equilibrium the amountof income saved in the form of bonds by the young would just match the amount of incomeconsumed by the old each period. This, together with the constancy of C1,C2, impliesNtBtPt= Nt−1Rt−1Bt−1Pt= Nt−1RtBtPt+1. (27)and thereforeRtPtPt+1=NtNt−1,