Introduction to Modern Economic Growth to express, and since we will make no use of the constrained maximization problems in this book, we will not state these theorems The vector-valued theorems are direct generalizations of the ones presented above and are useful in growth models with multiple capital goods In particular, let Z t1 f (t, x (t) , y (t)) dt (7.21) max W (x (t) , y (t)) ≡ x(t),y(t) subject to (7.22) x˙ (t) = g (t, x (t) , y (t)) , and (7.23) y (t) ∈ Y (t) for all t, x (0) = x0 and x (t1 ) = x1 Here x (t) ∈ RK for some K ≥ is the state variable and again y (t) ∈ Y (t) ⊂ RN for some N ≥ is the control variable In addition, we again assume that f and g are continuously differentiable functions We then have: Theorem 7.6 (Maximum Principle for Multivariate Problems) Consider the problem of maximizing (7.21) subject to (7.22) and (7.23), with f and g continuously differentiable, has an interior continuous solution y ˆ (t) ∈IntY (t) with corresponding path of state variable x ˆ (t) Let H (t, x, y, λ) be given by (7.24) H (t, x, y, λ) ≡ f (t, x (t) , y (t)) + λ (t) g (t, x (t) , y (t)) , ˆ (t) and the corresponding path of the where λ (t) ∈ RK Then the optimal control y state variable x (t) satisfy the following necessary conditions: (7.25) ∇y H (t, x ˆ (t) , y ˆ (t) , λ (t)) = for all t ∈ [0, t1 ] (7.26) λ˙ (t) = −∇x H (t, x ˆ (t) , y ˆ (t) , λ (t)) for all t ∈ [0, t1 ] (7.27) ˆ (t) , y ˆ (t) , λ (t)) for all t ∈ [0, t1 ] , x (0) = x0 and x (1) = x1 x˙ (t) = H (t, x Ô Proof See Exercise 7.10 Moreover, we have straightforward generalizations of the sufficiency conditions The proofs of these theorems are very similar to those of Theorems 7.4 and 7.5 and are thus omitted 329