Exercise $4.[42$ points$]$ Consider a small open economy with $2$ periods. Households receive

$an$ endowment $Y_{2}in$ period $2$ and are taxed $T_2(lump-$sum$)by$ the government. Households

don't have any savings $or$ borrowing.

The government has $an$ outstanding stock $of$ government debt $B$ issued $in$ period $1,$ which

implies repaying $B(1+r)in$ period $2(if$ the government decides $to$ repay$),$ where $ris$ the

interest rate $on$ government debt. Suppose this debt $is$ held $by$ risk$-$neutral foreign investors

that can access borrowin$\frac{g}{l}$ending $at$ the risk$-$free international interest rate $r^{*}=0\%.$

The government can $beof$ two types: with commitment $or$ strategic. From the perspective

$of$ period $1,$ the probability $of$ each type $isp=\frac{1}{2}.$ The government with commitment

repays its debt always. The strategic government chooses $to$ repay $or$ default $on$ its debt

$to$ maximize consumption $in$ period $2$ for the households. $If$ the government repays, its

available endowment $is{Y}_2^r=2.If$ the government defaults, the available endowment for

consumption $is$ given $by{Y}_2^d=1.$

Suppose first that the government can choose $to$ default fully $on$ its debt, $i.e.$ not repay

anything $to$ investors.

$(a)[10$ points$]$ Write down the budget constraints for the households and the government

$in$ period $2if$ the government decides $to$ repay. Write down the budget constraints for

the households and the government $in$ period $2if$ the government decides $to$ default.

$(b)[10$ points$]$ For what levels $of$ debt $B$ does a risk$-$less debt equilibrium exist? The

answer should provide a range $ofB.$

$(c)[10$ points$]$ For what levels $of$ debt $B$ does a risky debt equilibrium $(i.e.,an$ equilibrium

$in$ which the strategic government defaults$)$ exist? The answer should provide a range

$ofB.$

$(d)[2$ points$]Is$ there a region $of$ multiple equilibria? $Ifso,$ for what values $ofB?$

Suppose now that the strategic government can only choose $to$ fully repay $or$ partially

default $on$ its debt. $In$ particular, assume that $if$ the government chooses $to$ partially

default $it$ only repays $0\mathrm{.}5$ dollars for every $1$ dollar that $it$ owes, $i.e.,it$ repays half $of$ the

full amount $of$ repayment.

$(e)[10$ points$]$ What $is$ the interest rate $r$ lenders will ask $if$ they expect the strategic

government $to$ partially default?