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Introduction to Modern Economic Growth This assumption imposes conditions similar to those used in many economic applications: the constraint set is assumed to be convex and the objective function is concave or strictly concave Our next assumption puts some more structure on the objective function, in particular it ensures that the objective function is increasing in the state variables (its first K arguments), and that greater levels of the state variables are also attractive from the viewpoint of relaxing the constraints; i.e., a greater x means more choice Assumption 6.4 For each y ∈ X, U(·, y) is strictly increasing in each of its first K arguments, and G is monotone in the sense that x ≤ x0 implies G(x) ⊂ G(x0 ) The final assumption we will impose is that of differentiability and is also common in most economic models This assumption will enable us to work with firstorder necessary conditions Assumption 6.5 G is continuously differentiable on the interior of its domain XG Given these assumptions, the following sequence of results can be established The proofs for these results are provided in Section 6.4 Theorem 6.1 (Equivalence of Values) Suppose Assumption 6.1 holds Then for any x ∈ X, Problem A2 has a unique value V (x), which is equal to V ∗ (x) defined in Problem A1 Therefore, both the sequence problem and the recursive formulation achieve the same value While important, this theorem is not of direct relevance in most economic applications, since we not care about the value but we care about the optimal plans (actions) This is dealt with in the next theorem Theorem 6.2 (Principle of Optimality) Suppose Assumption 6.1 holds Let x∗ ∈ Φ (x (0)) be a feasible plan that attains V ∗ (x (0)) in Problem A1 Then we have that (6.3) V ∗ (x∗ (t)) = U(x∗ (t) , x∗ (t + 1)) + βV ∗ (x∗ (t + 1)) for t = 0, 1, with x∗ (0) = x (0) 263