This is a question from Stokey-Lucas-Prescott "Recursive Methods in Economic Dynamics", question 5.3: Optimal Growth with Linear Utility. Q. Consider the following model-
(1) $\underset{x_{t+1}}{max} \sum_{t=0}^{\infty} \beta^t U[f(k_t) - k_{t+1}]$
st $0 \leq k_{t+1} \leq f(k^t), t=0,1,2 \dots$
given $k_0 \geq 0$
(2) $v(x) = \underset{0 \leq y \leq f(x)}{max} U[f(x) - y] + \beta v(y)$
We are given U(c) = c, so $U[f(k_t) - k_{t+1}] = f(k_t) - k_{t+1}$.
We are also given that:
(T1) f is continuous
(T2)f(0) = 0 and for some $\bar{x} >0: x \leq f(x) \leq \bar{x}$, all $0 \leq x \leq \bar{x}$ and $f(x) < x$, all $x > \bar{x}$
(T3) f is strictly increasing
(T4) f is weakly concave
(T5) f is continuously differentiable
b. Define $k* = \underset{k \geq 0}{max} [\beta f(k) - k]$. Show that for some $\epsilon > 0$, $|k - k*| < \epsilon$ implies $v(k) = f(k) - k* + \beta\frac{[f(k*) - k*]}{1-\beta}$
I'm stuck at part b. I've worked out that if I set up my $\hat{v}= \frac{f(k) - k}{1-\beta} $, then I can get $v* = \lim_{n \rightarrow \infty} T^n\hat{v}$. But I am having trouble theoretically explaining why $\hat{v}$ should be of this form for $|k - k*| < \epsilon$.
Any help would be appreciated.