I have a doubt regarding the next optimization problem with infinite horizon. My objetive function $U(x_t)\in C^{\prime}$ is a function $U:R^+ \rightarrow R^+ $ such that $U^{\prime}(x_t)>0$, $ U^{\prime\prime}(x_t)<0$, $ \lim_{_{x}\to\infty} U^{\prime}(x_t)=0$ and $ \lim_{x_{t}\to 0} U^{\prime}(x_t)=\infty$.
Furthermore, the constraints are: (i) $f(y_{t})\geq x_{t}+y_{t+1}$ where the function $f$ has the same properties as $U(x)$, (ii) $y_o > 0$ and (iii) $y_{t+1},x_t\geq 0$. Finally I have a parameter $\beta\in (0,1)$. So the maximization problem is,
$max \sum_{t=0}^{\infty}\beta^{t}U(x_t)$
$s.t \quad f(y_{t})\geq x_{t}+y_{t+1}$
$y_{t+1},x_t\geq 0$
$y_0>0$
So, with all this on hand it is clear that we have a concave programming problem (The Lagrangian is concave) and from the F.O.C we get the optimum.
My question is related to the objetive function. This function $U(X)$ must be bounded too or it is enough with the assumptions that I have posted?
