Consider an investor who solves

$ma{x}_{c,}E{\displaystyle \underset{t=0}{\overset{T}{}}}{}^{t}log{c}_t$

subject to

$W_{t+1}=(1+r_{f,t+1})(W_t-c_t)+{}_{t+1}^{TT}(r_{t+1}-r_{f,t+1})$

$($a$)$ Rewrite this problem in the recursive form. Conjecture that $V(W_t,Y_t,t)=g(t)log{W}_t+k(Y_t,t).$

Verify this conjecture by induction. In particular, show that $g(t)={\displaystyle \underset{s=0}{\overset{T-t}{}}}{}^s.$

$($b$)$ Show that the consumption$-$wealth ratio is $\frac{c_t}{W_t}=\frac{1}{1++{}^2+\text{c}\text{d}\text{o}\text{t}\text{s}+{}^{T-t}}.$