Consider the dynamic programming pmblem;

$ma{x}_{\text{i}\text{i}\text{n}\text{R}}[{\displaystyle \underset{l=0}{\overset{T-1}{}}}-e^{-T{M}_i}=d{e}^{-T}*],x_{1+1}=2x_i=y_t,t=0,1,$dots,$T-1$

where $a,x>0$ and $x_0$ is given exogennusly.

$($a$)$ Compute $V_1(x),V_{-1}(x)$ and $V_{T-2}(x)$ using the recursive formulation method.

$($b$)$ Solve the dyamic programming problem using the Euler equation method.