Need help to Answer the following question (exercise 3.8)

Fido LTE 17:34 @12 31% columbia.edu Exercise 3.8 (Habit Formation and the Trade Balance) Consider a two-period, small open economy populated by a large number of identical households with preferences described by the utility function In C1 + In(C2 - aC1), where Ci and C2 denote consumption in periods 1 and 2, respectively, and a E (0, 1) is a parameter measuring the degree of habit formation. This preference specification nests the standard case of no habits when o is zero. The reason why these preferences capture consumption habits is that current consumption influences the marginal utility of future consumption. Specific cally, the marginal utility of period-2 consumption is given by 1/(C2 -aCi), which is higher for a > 0 than for a = 0. Intuitively, the more the household 112 Schmitt-Grohe, Uribe, Woodford eats in period 1, the hungrier it will wake up in period 2. Households are endowed with Q > 0 units of consumption goods each period and can borrow or lend at the world interest rate, r*, which, for simplicity, we assume is equal to zero. Households start period 1 with no assets or debts from the past (B; = 0). 1. Derive the household's intertemporal budget constraint. 2. Calculate the equilibrium levels of consumption and the trade balance in period 1 as functions of the structural parameters of the model, or and Q. Compare the answer to the one that would have obtained in the absence of habits and provide intuition. Exercise 3.9 (External Habit Formation) The specification of habits considered in exercise 3.8 is known as internal habit formation, because