Question

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【 4 】 Consider an individual who lives for two periods. The individual has no initial wealth and earns (exogenous) labor incomes of amounts Y1 and Y2 in the two periods. The individual can borrow and lend at a fixed interest rate r. The individual’s lifetime utility function is given by U = ln C1 +   1   ln C2, where ρ is the rate of time preference.

Also consider the government that “lives” for two periods.   The government finances     its spending tt1 either by imposing lump-sum tax T1 or by issuing bond B1 in the first period. In the second period, the government needs to redeem the debt (and cannot issue bond B2 = 0). The first and second period government budget constraints are therefore tt1 = T1 + B1 and tt2 + (1 + r)B1 = T2 respectively.

What is the individual’s lifetime budget constraint?

What is the government’s “lifetime” budget constraint?

Find the first-order condition (Euler equation) for the consumer?

Assume ρ = 0.1, r = 0.1, Y1 = 120 and Y2 = 330. Find the optimal consumption path (C1∗, C2∗) when tt1  = tt2  = T1  = T2  = 0.

Now suppose that the government spending in the first period becomes tt1 = 21, which is all financed by tax in the same period. Find the optimal consumption path in this case.

Suppose instead the government spending in the first period tt1 = 21 is all financed    by issuing bond. Find the optimal consumption path in this case.

Now consider a situation in which the individual faces a liquidity constraint. That is, he cannot borrow at all (but can lend). Repeat (4)-(6).

Answer 1

Please correct the utility function to include p for further parts to be solved.