Exercise $1.$ Consider a small open economy with $2$ periods and one non$-$storable good. Households receive endowments Y$1,$ Y$2$ in periods $1$ and $2,$ respectively, and are taxed $($lumpsum$)$ by the government. Let T$1,$ T$2$ be the lump sum taxes paid by households in periods $1$ and $2,$ respectively. Households cannot borrow or save. Governments can access credit markets and can decide in t $=2$ to repay or default $($entirely$)$ on its debt. Let D$1$ be the total face value of the bonds issued by the government in t $=1,$ to be paid in t $=2,$ and q$1$ be the price of a bond of one dollar of face value. International credit markets are populated by risk$-$neutral foreign investors that can access borrowing$/$lending at the international interest rate r$.$ The government is benevolent and chooses debt repayment to maximize consumption in t $=2$ for the households. If the government defaults, households receive a fixed endowment of Y$2=$ Y d $2=2.$ If the government repays households receive an endowment Y$2=$ Y r $2.$ From the perspective of period $1,$ Y r $2$ is a random variable Y$2$ uniformly distributed over $[3,5].$ A random variable Y is uniformly distributed over $[$a$,$ b$]$ if and only if Pr$($Y $$ y$)=$ y $$ a b $$ a for y in $[$a$,$ b$].($a$)$ Write down the budget constraints for the households and the government in t $=1.($b$)$ Write down the budget constraints for the households and the government in t $=2$ if the government decides to repay. Write down the budget constraints for the households and the government in t $=2$ if the government decides to default. $($c$)$ Which is the maximum value of D$1$ for which debt is risk$-$less $($i$.$e$.,$ the probability of repayment is one$)?($d$)$ Which is the minimum value of D$1$ for which debt has a price of q$1=0?$ Suppose the equilibrium level of debt in t $=1$ is given by D$1=1.($e$)$ Compute the probability of default and the price of debt in t $=1,$ q$1$ and the value of debt in t $=1,$ q$1$D$1.$