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.
1. (Worth 15 points.) State the householdís Lagrangian.
2. (Worth 35 points.) State the Örst order conditions for the householdís (that is, the
economyís) utility maximization problem using s = t.
3. (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
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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.)
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