How does a change in d affect g2 think of the period-1

Question

Problem

The Person-Swenson model. (Person and Swenson, 1989.) Suppose there are two periods. Government policy will be controlled by different policymakers in the two periods. The objective function of the period-t policymaker is U + α_t[V(G₁) + V(G₂) 0020 where U is citizens' utility from their private consumption; αt is the weight that the period-t policymaker puts on public consumption; Gt is public consumption in period t; and V(•) satisfies V'(•) > 0, V'' (•) < 0. Private utility, U, is given by U = W -C(T₁) - C(T₂), where W is the endowment; Tt is taxes in period t; and C (•), the cost of raising revenue, satisfies C (•) ≥ 1, C (•) > 0. All government debt must be paid off at the end of period 2. This implies T₂ = G₂+D, where D = G₁- T₁ is the amount of government debt issued in period 1 and where the interest rate is assumed to equal zero.

(a) Find the first-order condition for the period^-2 policymaker's choice of G₂ given D. (Note: Throughout, assume that the solutions to the policymakers' maximization problems are interior.)

(b) How does a change in D affect G₂?

(c) Think of the period-1 policymaker as choosing G₁ and D. Find the first order condition for his or her choice of D.

(d) Show that if α1 is less than α₂, the equilibrium involves inefficiently low taxation in period 1 relative to tax-smoothing (that is, that it has T₁ <>₂). Explain intuitively why this occurs.

(e) Does the result in part (d) imply that if α₁ is less than α₂, the period-1 policymaker necessarily runs a deficit? Explain.