Theorem Consequences. The Main Result and its Consequences

2.3. THE MAIN RESULT AND ITS CONSEQUENCES 11

2.3 The Main Result and its Consequences

2.3.1 Theorem

. Let Ω be an open subset of R n . Let f : R n → R be a differentiable function. Let x ∗ be an optimal solution of the following decision-making problem: Maximize subject to f x x ∈ Ω. 2.10 Then ∂f ∂x x ∗ = 0. 2.11 Proof: Since x ∗ ∈ Ω and Ω is open, there exists ε 0 such that x ∈ Ω whenever |x − x ∗ | ε. 2.12 In turn, 2.12 implies that for every vector h ∈ R n there exits η 0 η depending on h such that x ∗ + δh ∈ Ω whenever 0 ≤ δ ≤ η. 2.13 Since x ∗ is optimal, we must then have f x ∗ ≥ f x ∗ + δh whenever 0 ≤ δ ≤ η. 2.14 Since f is differentiable, by Taylor’s theorem we have f x ∗ + δh = f x ∗ + ∂f ∂x x ∗ δh + oδ, 2.15 where oδ δ → 0 as δ → 0 2.16 Substitution of 2.15 into 2.14 yields 0 ≥ δ ∂f ∂x x ∗ h + oδ and dividing by δ 0 gives 0 ≥ ∂f ∂x x ∗ h + oδ δ 2.17 Letting δ approach zero in 2.17 and taking 2.16 into account, we see that 0 ≥ ∂f ∂x x ∗ h, 2.18 Since the inequality 2.18 must hold for every h ∈ R n , we must have 0 = ∂f ∂x x ∗ , and the theorem is proved. ♦ 12 CHAPTER 2. OPTIMIZATION OVER AN OPEN SET Table 2.1 Does there exist At how many points an optimal deci- in Ω is 2.2.2 Further Case sion for 2.2.1? satisfied? Consequences 1 Yes Exactly one point, x ∗ is the say x ∗ unique optimal 2 Yes More than one point 3 No None 4 No Exactly one point 5 No More than one point

2.3.2 Consequences.

Let us evaluate the usefulness of 2.11 and its special case 2.18. Equation 2.11 gives us n equations which must be satisfied at any optimal decision x ∗ = x ∗ 1 , . . . , x ∗ n ′ . These are ∂f ∂x 1 x ∗ = 0, ∂f ∂x 2 x ∗ = 0, . . . , ∂f ∂x n x ∗ = 0 2.19 Thus, every optimal decision must be a solution of these n simultaneous equations of n variables, so that the search for an optimal decision from Ω is reduced to searching among the solutions of 2.19. In practice this may be a very difficult problem since these may be nonlinear equations and it may be necessary to use a digital computer. However, in these Notes we shall not be overly concerned with numerical solution techniques but see 2.4.6 below. The theorem may also have conceptual significance. We return to the example and recall the N = R − C. Suppose that R and C are differentiable, in which case 2.18 implies that at every optimal decision α ∗ , p ∗ ∂R ∂α α ∗ , p ∗ = ∂C ∂α α ∗ , p ∗ , ∂R ∂p α ∗ , p ∗ = ∂C ∂p α ∗ , p ∗ , or, in the language of economic analysis, marginal revenue = marginal cost. We have obtained an important economic insight.

2.4 Remarks and Extensions