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

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