Consider the social planning problem of choosing sequences $c_t$, $x_t$, $l_t$, $k_t$ to solve
$\max \sum_{t=0}^{\infty} \beta^t (\log c_t + \gamma \log x_t)$
s.t.
$c_t + k_{t+1} \le \theta k_t^{\alpha} l_t^{1-\alpha}$
$x_t + l_t \le 1$
$c_t, x_t, l_t, k_t \ge 0$
$k_0 \le \bar{k}_0$
a) Write down the Bellman equation for this problem.
b) Guessing that the value function $V(k)$ has the form $a_0 + a_1 \log k$ and that the policy function
for labor $l(k)$ is constant, find analytic solutions for the policy function $V(k)$ and the policy
functions $c(k)$, $x(k)$, $l(k)$, $k'(k)$.
c) Describe a sequential markets market structure for this world. Define a sequential markets
equilibrium. Use the answer to part (b) to calculate the unique sequential markets equilib-
rium.
d) Describe an Arrow-Debreu market structure for this world. Define an Arrow-Debreu equilib-
rium. Use the answer to part (b) to calculate the Arrow-Debreu equilibrium.
