Economic growth and economic development 279

Introduction to Modern Economic Growth though the value function is unique) This may be the case when two alternative feasible sequences achieve the same maximal value As in static optimization problems, non-uniqueness of solutions is a consequence of lack of strict concavity of the objective function When the conditions are strengthened by including Assumption 6.3, uniqueness of the optimum will plan is guaranteed To obtain this result, we first prove: Theorem 6.4 (Concavity of the Value Function) Suppose that Assumptions 6.1, 6.2 and 6.3 hold Then the unique V : X → R that satisfies (6.1) is strictly concave Combining the previous two theorems we have: Corollary 6.1 Suppose that Assumptions 6.1, 6.2 and 6.3 hold Then there exists a unique optimal plan x∗ ∈ Φ (x (0)) for any x (0) ∈ X Moreover, the optimal plan can be expressed as x∗ (t + 1) = π (x∗ (t)), where π : X → X is a continuous policy function The important result in this corollary is that the “policy function” π is indeed a function, not a correspondence This is a consequence of the fact that x∗ is uniquely determined This result also implies that the policy mapping π is continuous in the state vector Moreover, if there exists a vector of parameters z continuously affecting either the constraint correspondence Φ or the instantaneous payoff function U, then the same argument establishes that π is also continuous in this vector of parameters This feature will enable qualitative analysis of dynamic macroeconomic models under a variety of circumstances Our next result shows that under Assumption 6.4, we can also establish that the value function V is strictly increasing Theorem 6.5 (Monotonicity of the Value Function) Suppose that Assumptions 6.1, 6.2 and 6.4 hold and let V : X → R be the unique solution to (6.1) Then V is strictly increasing in all of its arguments Finally, our purpose in developing the recursive formulation is to use it to characterize the solution to dynamic optimization problems As with static optimization 265