Theorem 2.13.(global maximum). Let f and gi, i=1,., m, be quasi-concave, where G=(g1, ...,9m). Let a' satisfy the Kuhn-Tucker condition and the FOC for (2. 1). Then, I'is a global maximum point if (1) Df(r")+0, and f is locally twice continuously differentiable, or (2)f is concave H: Rn-R is linear if H()=Az+B, where A is matrix and B is a vector Proposition24.Letf:Rn→ R and g:Rn→ rh be concave,,andH:R→Rm be linear. Also, 3 o s.t. G(o>>0. Then, I* is a solution of max f(a) st.G(x)≥0 (x)=0 iG(x)≥0,H(x*)=0, and there exist入∈ Rn and A∈ RR Such that入(x”)=0 and x* is a solution of maxL(x,),)≡f(x)+入.G(x)+p·H(x).■ Example 2.15. For a E(0, 1), consider U(p1,p2,m)≡max。Arin2-a t 6. Envelope Theorem Theorem 2. 14.(Envelope). For differentiable f: XXA+R, XCR, A CRK and x*(a) being an interior maximum point of F(a)≡maxf(x,a), we have dF(a af(a, aTheorem 2.13. (global maximum). Let f and gi, i = 1, . . . , m, be quasi-concave, where G = (g1,...,gm)T . Let x∗ satisfy the Kuhn-Tucker condition and the FOC for (2.1). Then, x∗ is a global maximum point if (1) Df(x∗) 9= 0, and f is locally twice continuously differentiable, or (2) f is concave.  H : Rn → Rk is linear if H(x) = Ax + B, where A is matrix and B is a vector. Proposition 2.4. Let f : Rn → R and G : Rn → Rk be concave, and H : Rn → Rm be linear. Also, ∃ x0 s.t. G(x0) >> 0. Then, x∗ is a solution of maxx f(x) s.t. G(x) ≥ 0 H(x)=0 iff G(x∗) ≥ 0, H(x∗)=0, and there exist λ ∈ Rm + and μ ∈ Rk such that λ ·G(x∗)=0 and x∗ is a solution of max x∈Rn L(x, λ, μ) ≡ f(x) + λ · G(x) + μ · H(x).  Example 2.15. For a ∈ (0, 1), consider v(p1, p2, m) ≡ max x1, x2≥0 Axa 1x1−a 2 s.t. p1x1 + p2x2 ≤ m.  6. Envelope Theorem Theorem 2.14. (Envelope). For differentiable f : X × A → R, X ⊂ Rn, A ⊂ Rk, and x∗(a) being an interior maximum point of F(a) ≡ max x∈X f(x, a), we have dF(a) da = ∂f(x, a) ∂a     x=x∗(a) .  2 — 10