Show the derivation process that leads to the final solution for all parts of the question.

Assume that there are 1,000 identical consumers and the equilibrium real interest rate is 20%. Each consumer receives a current income of 100 units and a future income of 120 units and consumes 80 units and 92 units in the current and future periods, respectively. Each consumer pays a current tax of 10 units and a future tax of 40 units. The government purchases amount to 20,000 units in the current period and 28,000 units in the future period.

(a) Show that the consumption bundle satisfies the consumer's lifetime budget constraint. (2 points)

(b) How much does the government need to borrow from the public in the current period? Show that the aggregate private saving equals government borrowing (minus government saving). (2 points)

(c) What is the government's present-value budget constraint? Does it hold in this setting? (2 points)

(d) Show that the aggregate income equals the aggregate consumption and government purchases in the current period. Why does this mean that the credit market has cleared? (2 points)

(e) Assume that the government announces an exemption of tax in the current period while keeping government expenditure plan unchanged. By how much does the government need to increase the future period tax to satisfy its intertemporal (present value) budget constraint? If Ricardian Equivalence holds, please draw a diagram in the in c-c' space for a representative consumer, identify his endowment points and optimal consumption bundle before and after the tax reform. Explain the intuition on the consumer's optimal response to the current tax exemption. (2 points)