Economic growth and economic development 295

Introduction to Modern Economic Growth
6.5.1. Basic Equations. Consider the functional equation corresponding to
Problem A2:
(6.18)

V (x) = max [U (x, y) + βV (y)] , for all x ∈ X.
y∈G(x)

Let us assume throughout that Assumptions 6.1-6.5 hold. Then from Theorem
6.4, the maximization problem in (6.18) is strictly concave, and from Theorem 6.6,
the maximand is also differentiable. Therefore for any interior solution y ∈IntG (x),
the first-order conditions are necessary and sufficient for an optimum. In particular,

optimal solutions can be characterized by the following convenient Euler equations,
where we use ∗’s to denote optimal values and ∇ for gradients (recall that x is a
vector not a scalar, thus ∇x U is a vector of partial derivatives):
(6.19)

∇y U (x, y ∗ ) + β∇y V (y ∗ ) = 0.

The set of first-order conditions in equation (6.19) would be sufficient to solve for
the optimal policy, y ∗ , if we knew the form of the V (·) function. Since this function
is determined recursively as part of the optimization problem, there is a little more
work to do before we obtain the set of equations that can be solved for the optimal
policy.
Fortunately, we can use the equivalent of the Envelope Theorem for dynamic
programming and differentiate (6.18) with respect to the state vector, x, to obtain:
(6.20)

∇x V (x) = ∇x U(x, y ∗ ).

The reason why this is the equivalent of the Envelope Theorem is that the term
∇y U (x, y ∗ )+β∇y V (y ∗ ) times the induced change in y in response to the change in x
is absent from the expression. This is because the term ∇y U (x, y ∗ )+β∇y V (y ∗ ) = 0

from (6.19).

Now using the notation y ∗ = π (x) to denote the optimal policy function (which
is single-valued in view of Assumption 6.3) and the fact that ∇x V (y) = ∇x V (π (x)),

we can combine these two equations to write
(6.21)

∇y U(x, π (x)) + β∇x U (π (x) , π (π (x))) = 0,
281