Equivalencies: Consider a representative agent whose preferences are $U(\{c_t{\}}_t=0^{})=$

${\displaystyle \underset{t=0}{\overset{}{}}}{}^{t}u(c_t).$ Assume that $u$:${}_{+}$ satisfies all the assumptions discussed in class.

The representative agent is the owner of technology $($firm$),$ the stock of physical capital,

and ineslastically supplies labor to the firm. The firm maximizes profits subject to the

technology $y_t=f(k_t).$ Assume that all the assumptions discussed in class are satisfied.

$($a$)$ Define the Sequential Equilibrium and provide the conditions characterizing

the equilibrium.

$($b$)$ Define the Planner's Problem and find the conditions that characterize the

solution.

$($c$)$ Compare the conditions obtained in $($a$)$ and $($b$).$ Are the welfare theorems satisfied

in this economy?