Compute the equilibrium tax rate and the equilibrium levels

Question

Exercise - Distortionary Taxation

Consider a two-period, small, open, endowment economy. Assume that households' preferences are described by the following utility function

ln(C₁) + ln(C₂),

-where C₁ and C₂ denote consumption in periods 1 and 2, respectively. House-holds are endowed with 10 units of goods in each period and pay proportional taxes on consumption. Let r₁ and r₂ denote the consumption tax rates in periods 1 and 2, respectively. Finally, households are born with no financial asset (B₀^p = 0) and can borrow or lend in the international financial market at the world interest rate r^* = 0.1.

The government starts period 1 with no outstanding assets or liabilities (B₀^g = 0). It taxes consumption at the same rate in both periods (r₁ = r₂) and consumes 1 unit of goods in each period. That is, G₁ = G₂ = 1, where G₁ and G₂ denote government consumption in periods 1 and 2, respectively. Like the household, the government has access to the world financial market. In answering the following, questions, show your work.

1. Compute the equilibrium tax rate and the equilibrium levels of consumption, the trade balance, private savings, the primary and secondary is deficits, and the current account in periods 1 and 2.

2. Suppose now that the government implements a stimulus package consisting in reducing the tax rate by half in period 1, with government consumption unchanged in both periods. How does this expansionary fiscal policy affect private consumption, the trade balance, the current account, and the primary and secondary fiscal deficits in period 1 and the tax rate in period 2? Briefly explain your result.