The purpose of this paper is to provide a unified treatment to find sufficient conditions for the existence of a subgradient of the value function associated with a convex optimization problem. We recall basic results in convex programming with linear constraints. In particular, the subdifferential of the value function is the opposite of the set of multipliers associated with a solution. We state two results on the non-emptiness of the subdifferential of the value function. The first one is known and the second one is original since we do not assume any continuity condition on the objective function. We apply these results to different cases arising in mathematical economics. The last part is devoted to the case with equality and inequality constraints. We provide a necessary and sufficient condition for the non-emptiness of the subdifferential of the value function which works even if the interior of the positive cone is empty.