We consider an optimal control problem arising in the context of economic theory of growth, on the lines of the works by Skiba and Askenazy-Le Van. The framework of the model is intertemporal infinite horizon utility maximization. The dynamics involves a state variable representing total endowment of the social planner or average capital of the representative dynasty. From the mathematical viewpoint, the main features of the model are the following: (i) the dynamics is an increasing, unbounded and not globally concave function of the state; (ii) the state variable is subject to a static constraint; (iii) the admissible controls are merely locally integrable in the right half-line. Such assumptions seem to be weaker than those appearing in most of the existing literature. We give a direct proof of the existence of an optimal control for any initial capital k(0) >= 0 and we carry on a qualitative study of the value function; moreover, using dynamic programming methods, we show that the value function is a continuous viscosity solution of the associated Hamilton-Jacobi-Bellman equation.