Consider a two-period, small open economy populated by a large number of identical households with preferences described by the utility function ln(�!) + ln(�" − ��!). Where �! and �" denote consumption in periods 1 and 2, respectively, and � in (0,1) is a parameter measuring the degree of habit formation. This preference specification nests the standard no habits when � = 0. The reason why these preferences capture consumption habits is that current consumption influences the marginal utility of future consumption. Specifically, the marginal utility of period 2 consumption is given by 1/(�" − ��!), which is higher for � > 0 than for � = 0. Intuitively, the more the household eats in period 1, the hungrier it will wake up in period 2.
Households are endowed with � > 0 units of consumption goods for each period and can borrow or lend at the world interest rate, �∗, which, for simplicity, we assume is equal to zero. Households start period 1 with no assets or debts from the past (�$ = 0).
1. Derive the household’s intertemporal budget constraint.
2. Calculate the equilibrium levels of consumption and the trade balance in period 1 as a function of the structural parameters of the model, � and �. Compare the answer to the one that would have been obtained in the absence of habits and provide intuition.