How does impact the household consumption savings decision

Question

Assignment:

Section I: Fiscal Theory with the Ricardian Equivalence.

Suppose that we have a two period endowment economy model where the household knows with certainty they will receive Y_t in period t and Y_t+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(C_t) + βU(C_t+1) where 0 < β < 1

subject to:

C_t + C_t+1 /1 + r_t = Y_t - T_t + Y_t+1/1 + r_t - T_t+1/1 + r_t

T_t are lump sum taxes issued by the government, and r_t 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 affect 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 affect the consumption function?

(c.) How does a large decit 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, G_t and G_t+1, exogenously, and must raise taxes, T_t and T_t+1, to satisfy:

G_t + G_t+1/(1 + r_t) = T_t + T_t+1/(1 + r_t)

(e.) Write down the denition of a competitive equilibrium for this economy.

(f.) Assuming that the household knows (and anticipates) that the government 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, D_t, from previous generations no longer present in this model. How does the present debt effect the present optimization problem? Does the dynamic consumption savings decision change? Does the level of consumption change?

Section II: Fiscal Theory without the Ricardian Equivalence.

Suppose that we have a two period endowment economy model where the household knows with certainty they will receive Y_t in period t and Y_t+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(C_t) + βU(C_t+1) where 0 < β < 1

subject to: C_t(1 + τ^c_t ) + B_t = Y_t

C_t+1(1 + τ^c_t+1) = Y_t+1 + B_t(1 + r_t)

Note that here, revenue for the government is collected through a distortionary 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, intertemporal budget constraint.

(b.) Find the first 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 optimization problem by solving the intertemporal budget constraint for C_t+1 and substituting this denition into the objective function.

(c.) Re-write the FOC's in modied 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 indierence curve with a single tangency point.

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, D_t, from previous generations no longer present in this model which the government decides they will pay off in period t + 1. How does the debt effect the present optimization problem? Does the dynamic consumption savings decision change? Does the level of consumption change?