economics assignment
Problem Set 4
Professor Jonathan Wol� Fall 2019
Section I: Fiscal Theory with the Ricardian Equivalence.
Suppose that we have a two period endowment economy model where the house- hold knows with certainty they will receive Yt in period t and Yt+1 in period t + 1. Assume utility is log. There is a representative household who seeks to maximize utility given by:
max Ct,Ct+1
U(Ct) + βU(Ct+1) where 0 < β < 1
subject to:
Ct + Ct+1 1 + rt
= Yt − Tt + Yt+1 1 + rt
− Tt+1 1 + rt
Tt are lump sum taxes issued by the government, and rt is the real interest rate. The household takes each as given.
(a.) Derive the Euler equation characterizing an optimal consumption plan. How does the presence of the government a�ect the Euler equation?
(b.) Use the Euler equation in consumption with the intertemporal budget constraint to derive the consumption function. How does the presence of a government a�ect the consumption function?
(c.) How does a large de�cit in period t impact the household's consumption savings decision? You may answer this graphically or with intuition.
(d.) What is the marginal propensity to consume out of period t+ 1 income (the change in consumption for a change in income)?
The government chooses its spending, Gt and Gt+1, exogenously, and must raise taxes, Tt and Tt+1, to satisfy:
Gt + Gt+1
(1 + rt) = Tt +
Tt+1 (1 + rt)
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(e.) Write down the de�nition of a competitive equilibrium for this economy.
(f.) Assuming that the household knows (and anticipates) that the govern- ment budget constraint will bind, you can combine that with the consumption function. Do this, then explain what happens to taxes in the consumption function. What is the intuition for this?
(g.) Finally, assume that the government carries a large debt, Dt, from previous generations no longer present in this model. How does the present debt e�ect the present optimization problem? Does the dynamic consumption savings decision change? Does the level of consumption change?
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Section II: Fiscal Theory without the Ricardian Equivalence.
Suppose that we have a two period endowment economy model where the house- hold knows with certainty they will receive Yt in period t and Yt+1 in period t + 1. Assume utility is log. There is a representative household who seeks to maximize utility given by:
max Ct,Ct+1,Bt
U(Ct) + βU(Ct+1) where 0 < β < 1
subject to:
Ct(1 + τ c t ) + Bt = Yt
Ct+1(1 + τ c t+1) = Yt+1 + Bt(1 + rt)
Note that here, revenue for the government is collected through a distor- tionary tax on consumption rather than through a lump sum tax. It is assumed that the tax rate can change across time as denoted by the time subscript. For the time being, assume that the household has perfect information concerning the future tax rate and future endowment.
(a.) Combine the two, intratemporal budget constraints into a single, intertem- poral budget constraint.
(b.) Find the �rst order optimization conditions for the household's problem using either 1.) a constrained optimization problem in the Lagrangian form (with the intertemporal budget constraint as, or 2.) as an unconstrained opti- mization problem by solving the intertemporal budget constraint for Ct+1 and substituting this de�nition into the objective function.
(c.) Re-write the FOC's in modi�ed Euler Equation notation. What impact would a correctly anticipated increase in the future value of the distortionary tax rate play in the household's dynamic consumption decision?
(d.) Create a two dimensional graph plotting the budget constraint as well as an indi�erence curve with a single tangency point.
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i.) What determines the slope of the budget constraint?
ii.) What determines the level of the budget constraint?
iii.) How would an increase (or a decrease) in one of the tax rates impact the household's optimal consumption bundle?
(e.) Does a change in the timing of taxes impact household consumption and therefore welfare? If so, what assumption is it that results in this violation in the Ricardian Equivalence?
(f.) Finally, assume that the government carries a large debt, Dt, from previous generations no longer present in this model which the government decides they will pay o� in period t + 1. How does the debt e�ect the present optimization problem? Does the dynamic consumption savings decision change? Does the level of consumption change?
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