Question 1 - Consider an individual who lives two periods and whose utility function is given by

U(c₁, c₂) = [lnc₁ + γg₁] + β[lnc₂ + γg₂], 0 < β < 1 and γ > 0.

where β is the discount factor, γ indicates the individual's preference for per capita government spending g_t, and c_t denotes consumption in period t. Government expenditures in both periods are financed with taxes proportional to consumption. More specifically, suppose that for each unit of c_t consumed, our individual must pay the amount τ_tc_t to the government, where τ_t denotes the consumption tax rate in period t.

(a) Write down the budget constraints the consumer faces in periods 1 and 2 and then use these constraints to derive the consumer's lifetime budget constraint (the PVBC) as we did in class.

(b) Derive the consumer's Euler equation.

(c) Interpret the Euler equation derived in (2).

(d) Is government consumption (g₁ and g₂) directly affecting the optimal bundle (c₁, c₂) chosen by the consumer? Explain why or why not. [HINT: You don't need to derive optimal c₁, c₂ to answer this question, just focus on the Euler equation.]

Question 2 - Consider the two period economy we studied in class. Suppose that households live for two periods and have a per-period utility as

u(c_t) = c^½_t for t = 1, 2.

(a) Derive the Euler equation and briefly interpret it.

(b) Derive the household's optimal consumption in both periods and saving.

(c) Solve for the equilibrium interest rate. [Hint: Recall that the household consumes everything that it saves.]

(d) What is the effect on the equilibrium interest rate from a permanent income shock, i.e., when both y₁ and y₂ change equally?