Question $1$

Continue with the setting $of$ Homework $3.$ But now add the assumption that the aggregate endowment $of$ the three consumption goods $is$ some vector $(y_1,y_2,y_3)in{R}_{++}^3$ that

satisfies the restriction There $is$ a mutual fund that owns these endowments,

and all consumers are shareholders, with everyone holding $an$ equal share.

$In$ this question, the goal $isto$ construct a competitive equilibrium under the assumption that the condition given $in$ part $aon$ Homework $3$ holds.

$a.$ Give the market clearing conditions for goods $1$ and $2,$ expressed $in$ terms $of{y}_1,y_2,T_1$and $T_2,$ where $T_1$ and $T_2$ are defined $in$ terms $of$ the relative prices $\frac{p_1}{p_3}$ and $\frac{p_2}{p_3}by(1).$

Explain why you will not need the market clearing condition for good $3to$ find potential equilibrium prices.

$b.$ Solve the market clearing conditions for goods $1$ and $2$ for $T_1$ and $T_{2}to$ show that

$T_n=\frac{y_n}{y_1+y_2}ln(\frac{1}{1-(y_1+y_2)}),$nin$\{1,2\}$

Hint: Start $by$ constructing expressions for $\frac{y_1}{y_2}$ and $y_1+y_2.$

$c.If$ there are $no$ further restrictions $on{y}_1$ and $y_2,$ what will you have $to$ assume about

$\frac{{}_1}{}$ and $\frac{{}_2}{}?$

$d.$ Given the solution $in$ part $b$ and the parameter restrictions $in$ part $c,$ you have candidate equilibrium prices $\frac{p_1}{p_3}$ and $\frac{p_2}{p_3}.$ Explain why, $in$ order $tobe$ sure that these are Question $1$

Continue with the setting $of$ Homework $3.$ But now add the assumption that the $ag-$

gregate endowment $of$ the three consumption goods $is$ some vector $(y_1,y_2,y_3)in{R}_{++}^3$ that

satisfies the restriction There $is$ a mutual fund that owns these endowments,

and all consumers are shareholders, with everyone holding $an$ equal share.

$In$ this question, the goal $isto$ construct a competitive equilibrium under the assump$-$

tion that the condition given $in$ part $aon$ Homework $3$ holds.

$a.$ Give the market clearing conditions for goods $1$ and $2,$ expressed $in$ terms $of{y}_1,y_2,T_1$

and $T_2,$ where $T_1$ and $T_2$ are defined $in$ terms $of$ the relative prices $\frac{p_1}{p_3}$ and $\frac{p_2}{p_3}by(1).$

Explain why you will not need the market clearing condition for good $3to$ find potential

equilibrium prices.

$b.$ Solve the market clearing conditions for goods $1$ and $2$ for $T_1$ and $T_{2}to$ show that

$T_n=\frac{y_n}{y_1+y_2}ln(\frac{1}{1-(y_1+y_2)}),$nin$\{1,2\}.$

Hint: Start $by$ constructing expressions for $\frac{y_1}{y_2}$ and $y_1+y_2.$

$c.If$ there are $no$ further restrictions $on{y}_1$ and $y_2,$ what will you have $to$ assume about

$\frac{{}_1}{}$ and $\frac{{}_2}{}?$

$d.$ Given the solution $in$ part $b$ and the parameter restrictions $in$ part $c,$ you have candi$-$

date equilibrium prices $\frac{p_1}{p_3}$ and $\frac{p_2}{p_3}.$ Explain why, $in$ order $tobe$ sure that these are

actual equilibrium prices, you will have $to$ assume that $y_3$ satisfies

maxactual equilibrium prices, you will have $to$ assume that $y_3$ satisfies

max