Introduction to Modern Economic Growth Theorem 7.1 (Necessary Conditions) Consider the problem of maximizing (7.1) subject to (7.2) and (7.3), with f and g continuously differentiable Suppose that this problem has an interior continuous solution yˆ (t) ∈IntY (t) with corre- sponding path of state variable xˆ (t) Then there exists a continuously differentiable costate function λ (·) defined over t ∈ [0, t1 ] such that (7.2), (7.9) and (7.10) hold, and moreover λ (t1 ) = As noted above, (7.9) looks similar to the first-order conditions of the constrained maximization problem, with λ (t) playing the role of the Lagrange multiplier We will return to this interpretation of the costate variable λ (t) below Let us next consider a slightly different version of Theorem 7.1, where the terminal value of the state variable, x1 , is fixed, so that the maximization problem is (7.11) max W (x (t) , y (t)) ≡ x(t),y(t) Z t1 f (t, x (t) , y (t)) dt, subject to (7.2) and (7.3) The only difference is that there is no longer a choice over the terminal value of the state variable, x1 In this case, we have: Theorem 7.2 (Necessary Conditions II) Consider the problem of maximizing (7.11) subject to (7.2) and (7.3), with f and g continuously differentiable Suppose that this problem has an interior continuous solution yˆ (t) ∈IntY (t) with corresponding path of state variable xˆ (t) Then there exists a continuously differentiable costate function λ (·) defined over t ∈ [0, t1 ] such that (7.2), (7.9) and (7.10) hold Proof The proof is similar to the arguments leading to Theorem 7.1, with the main change that now x (t1 , ε) must equal x1 for feasibility, so xε (t1 , 0) = and λ (t1 ) is unrestricted Exercise 7.5 asks you to complete the details Ô The new feature in this theorem is that the transversality condition λ (t1 ) = is no longer present, but we need to know what the terminal value of the state variable x should be.2 We first start with an application of the necessary conditions 2It is also worth noting that the hypothesis that there exists an interior solution is more restrictive in this case than in Theorem 7.1 This is because the set of controls n o F = [y (t)]tt=0 : x˙ (t) = g (t, x (t) , y (t)) with x (0) = x0 satisfies x (t1 ) = x1 320