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 ∗ ) = 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 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 ∗ ) = 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