Consider a small open economy that lives for two periods, only, (t and t + 1; for

all purposes to some extent if it makes it easier you can think of this as t =

0 and t = 1) and is inhabited by a continuum of identical individuals grouped into an

aggregate risk-sharing household. Each period aggregate output Yt

is produced via the

function Yt = ZtLt where Zt

is exogenous productivity and Lt

is labor. If output is not

consumed or loaned out in any one period, then it spoils and cannot be carried over to

the following period (this just means that the economy cannot save internally, itís only way

to save would be to make international loans). This is a small open economy, so it can

borrow and lend freely at the constant-across-periods international interest rate r. The

householdís instantaneous utility is given by Ct=Lt and the household discounts the future

at rate , where: 2 (0; 1) is the householdís (constant) subjective discount factor (i.e.,

0 < < 1, where 2 means ìbelongsî and is the Greek letter ìbetaî); C is consumption;

L is labor. Moreover, the household ìownsî the production function, so its thinking with

regards to its maximization problem is akin to that of a benevolent social planner. Now,

consider the following version of the householdís intertemporal utility maximization problem,

where A denotes the (endogenous) state variable ì(internationally traded) assets.î The

household chooses consumption, labor, and assets to maximize lifetime utility (hint: this

involves Ps=t+1

s=t

, for all purposes, as noted earlier, to some extent if it makes it

easier you can broadly think of this as t = 0 and t = 1; the explicit changes just

involve setting up certain things using s instead of our usual t, but at the end of

the day things should look entirely familiar: trust me!) such that

Cs + As+1 (1 + r) As + ZsLs

Let s denote the time-s Lagrange multiplier.

- (Worth 15 points.) State the householdís Lagrangian.
- (Worth 35 points.) State the Örst order conditions for the householdís (that is, the

economyís) utility maximization problem using s = t. - (Worth 50 points.) Is the following claim true, false, or uncertain? When the

economy is open, if =

1

1+r

, then the solution to the householdís period-t intertemporal

1

utility maximization problem implies that Ct+1 =

Zt+1

Zt

Ct

. (Note: again, if it is

easier to think about this, you can just imagine that t = 0, in which case

the question is asking whether it is true C1 =

Z1

Z0

C0:) Justify your answer with

thorough mathematical detail. (Huge hint: There are three ways through which Ct+1,

Ct, Zt+1, and Zt can meet in this problem. The Örst way is really straightforward and

involves combining two FOCs. The second will be revealed by going through our “usual

thing” in the small open economy case, which is equating the slope of an indi§erence

curve with the slope of the lieftime budget constraint. Iím not saying that equating the

slope is the solution, Iím saying that in going through these steps youíll see the second

instance in which all of these variables meet. The last is a potential combination

between the Örst and second ways. Thatís it, you do those three things and they are

exhaustive to be able to answer the claim.)

Sample Solution