Question $1.$ Consider the Lucas tree model. There $is$ a tree that yields

dividends $d_t$ every period. Dividends follow a Markov process and there are two

different households. The problem $of$ the household $iis$:

$max{E}_t{\displaystyle \underset{t=0}{\overset{}{}}}{}^{t}u(c_{it})s.t.$

$c_{it}+p_t{}_{it}=y_{it}+(d_t+p_t){}_{it-1}$

where $y_{it}isan$ income shock. $We$ have that $y_{1t}+y_{2t}=1.$ Households trade $in$

shares $of$ the tree ${}_{it}$ and $we$ have that ${}_{1t}+{}_{2t}=1.$

$a.$ What are the state variables that each household has $to$ take into account?

$b.$ Write the problem $as$ a dynamic programming problem and define $a$

recursive competitive equilibrium $(RCE)$ for this economy.

$c.$ Define a sequential markets equilibrium $(SME)$ for this economy and

verify that the first order conditions are the same $asin$ the RCE.

$d.$ Compare the optimality conditions from the SME and the date $0$ trade

equilibrium with a complete set $of$ contingent claims. Question $1.$ Consider the Lucas tree model. There $is$ a tree that yields

dividends $d_t$ every period. Dividends follow a Markov process and there are two

different households. The problem $of$ the household $iis$:

$max{E}_t{\displaystyle \underset{t=0}{\overset{}{}}}{}^{t}u(c_{it})s.t.$

$c_{it}+p_t{}_{it}=y_{it}+(d_t+p_t){}_{it-1}$

where $y_{it}isan$ income shock. $We$ have that $y_{1t}+y_{2t}=1.$ Households trade $in$

shares $of$ the tree ${}_{it}$ and $we$ have that ${}_{1t}+{}_{2t}=1.$

$a.$ What are the state variables that each household has $to$ take into account?

$b.$ Write the problem $as$ a dynamic programming problem and define $a$

recursive competitive Question $1.$ Consider the Lucas tree model. There $is$ a tree that yields

dividends $d_t$ every period. Dividends follow a Markov process and there are two

different households. The problem $of$ the household $iis$:

$max{E}_t{\displaystyle \underset{t=0}{\overset{}{}}}{}^{t}u(c_{it})s.t.$

$c_{it}+p_t{}_{it}=y_{it}+(d_t+p_t){}_{it-1}$

where $y_{it}isan$ income shock. $We$ have that $y_{1t}+y_{2t}=1.$ Households trade $in$

shares $of$ the tree ${}_{it}$ and $we$ have that ${}_{1t}+{}_{2t}=1.$

$a.$ What are the state variables that each household has $to$ take into account?

$b.$ Write the problem $as$ a dynamic programming problem and define $a$

recursive competitive equilibrium $(RCE)$ for this economy.

$c.$ Define a sequential markets equilibrium $(SME)$ for this economy and

verify that the first order conditions are the same $asin$ the RCE.

$d.$ Compare the optimality conditions from the SME and the date $0$ trade

equilibrium with a complete set $of$ contingent claims.equilibrium $(RCE)$ for this economy.

$c.$ Define a sequential markets equilibrium $(SME)$ for this economy and

verify that the first order conditions are the same $asin$ the RCE.

$d.$ Compare the optimality conditions from the SME and the date $0$ trade

equilibrium with a complete set $of$ contingent claims.