If we want to maximize $$\sum _{t=0}^\infty f(x_t,u_t), \text { s.t. } x_{t+1}=g(x_t,u_t)$$
Then the value function $$V(x_t)=\max_{u_t,u_{t+1},...}\sum _{t=0}^\infty f(x_t,u_t)$$
Often has approximately the same functional form as $f$.
For example, in a problem I was working on, $f$ had the form $\ln u_t$, and $V(x_t)$ had the form $A+B\ln(x_t)$
What is the reason that the value function often has the same functional form as the instantaneous target function $f$?