Exercise $1.[48$ Points.$]$ Consider a small open economy with $2$ periods and a single non$-$

storable good. Preferences $of$ the representative household are described $by$ the following

utility function:

$U(C_1,C_2)=ln{C}_1+C_2$

$In$ periods $1$ and $2$ households receive a constant endowment $of{Q}_1=Q_2=Q.$ Households

have access $to$ financial markets, where they can lend $or$ borrow $at$ the international interest

rate $r^{*}.$ The initial net foreign asset position $of$ the representative household $is{B}_0^{*}=0.$

Suppose that $Q>\frac{1}{2+r^{*}}.$

$(a)[10$ Points.$]$ Set $up$ the budget constraint $of$ households for periods $1$ and $2.$ Derive

the inter$-$temporal budget constraint.

$(b)[10$ Points.$]$ Find the equilibrium levels $of$ consumption and the current account for

periods $1$ and $2.$ Your answer should depend $on$ the parameters $r^{*}$ and $Q.$

$(c)[8$ Points.$]$ What happens $to$ consumption and the current account $in$ period $1$ when

the interest rate $r^{*}$ increases? Provide intuition for your results.

Now suppose households have uncertainty regarding their future endowment. $In$ par$-$

ticular, the endowment can $be{Q}_2=Q-with$ probability $\frac{2}{3}or{Q}_2=Q+2with$

probability $\frac{1}{3}.$ Agents preferences are now given $by$

$U(C_1,C_2)=ln{C}_1+E[C_2]$

where $E[*]$ indicates expected value.

$(d)[2$ Points.$]$ Compute the expected value $of$ the endowment $in$ period $2.$

$(e)[10$ Points.$]$ Set $up$ the maximization problem for the households and find equilibrium

levels $of$ consumption $in$ periods $1$ and $2(there$ should $be$ more than one consumption

$in$ period $2,$ depending $on$ the realization $of$ the endowment$).$

$(f)[8$ Points.$]$ Are there precautionary savings $in$ this economy? Why $or$ why not?

Provide intuition.