# Dynamic programming problem: Optimal growth with linear utility

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.

• Why don't you write question 5.3 so that we could know what it states and help you with your problem? – Joel Reyes Noche Apr 28 '17 at 7:47
• Hey. Definitely. The book is available online so I thought linking might help. But let me just put the question down. – user441293 Apr 28 '17 at 7:54