1. Consider the followingdeterministicmodel. There are two periods,t= 0,1. Households wish to maximize utility

1

^tu(c_t)

t=0

withustrictly increasing and concave, and(0,1). At periodt, the household receives endowment incomey_t, and consumesc_t. In the first period, the household can also purchase financial assetsa, which earn interestrfromt= 0tot= 1. In this question, it will be useful to work with the gross interest rateR1 +r, rather thanrdirectly. Unlike what we've seen before, assumeR(and thereforer) depends on the household's choice of assetsa. In particular, assume thatR=(a), with(a)>0and(a)0, so that the more assets you have, the lower is the interest rate on them (also, the moredebtyou're in, thehigherthe interest rate you pay).

1. Write down the budget constraints fort= 0andt= 1(substituting in(a)to replaceR= 1 +rwhere appropriate). Show all work in details.
2. Assuming it has a solution, set up the HH's problem and obtain the FOC that characterizes the HH's optimal choice. Explain the economic intuition for this condition. Show all work in details.
3. Assume the optimal choice featuresa >0. Recall that the elasticity of a functionf(x)is given byf(x)x/f(x), and can be interpreted as the percentage change inf(x)arising from a 1% increase inx. LetE(a)(a)a/(a)denote the elasticity of (a). In order for it to be possible for the optimality condition you found in part (b) to hold, do we needE(a)to be greater than or less than1? Explain the intuition in economic terms for why this is the case. Show all work in details.