$(30$ points$)($Sovereign Default Model$)$ Le $i_t^{**}$ be a risk free rate on the T$-$Bills. Let $D_t$ be the amount of debt. When the government issues the amount of debt $D_t$ ahd defaults on it$,$ the household consumption with any given $in(0,1)$ in the next period is given by

$C_{t+1}=(1-)Y_{t+1},($ Default ${)}_{?}$

When the government does not default, the household consumption in the next period is given by

$C_{t+1}=Y_{t+1}-D_t,(No-$Default $)$

Finally, the income shock in the next period $Y_{t+1}$ is uniformly distributed from $0\mathrm{.}5$ to $1\mathrm{.}5.$ That is$,$ the probability density function for $Y_{t+1}$ is given by

$\frac{f(y_{t+1})}{f(y_{t+1})}=1=0,$ otherwise

a$)$ An investor bought the bond issued by the government at the price of $P.$ What is the return on this bond when the government defaults $($i$.$e$.,$ compute $i^d)?$

b$)$ Let $i_i^{**}=0$ and $=0\mathrm{.}4,$ and the investors are risk$-$neutral. $($i$)$ Find a range of $D_t$ such that the bond price for this debt is the same as that for the T$-$Bills. $($ii$)$ Find a range of $D_t$ such that the bond price is zero. $($Note that $D_{t}0).($iii$)$ Find a range of $D_t$ such that no investors would buy this bond in the government bond auction market.

c$)$ Let $i_i^{**}=0$ and $=0\mathrm{.}4,D_t=0\mathrm{.}4,$ and the investors are risk$-$neutral. What is the probability of default on this debt, $p_d?$ What is the sovereign spread on this debt when the government issues $D_t=0\mathrm{.}4$ amount of debt in the government bond auction market?