Theorem A-3(Kuhn-Tucker). For differentiable f: R-R and G: R-R,let LAx=f()+A G(x) If x' is a solt Appendix: Math Preparation max f(r) G(x)≥0, then there exists AER such that(Kuhn-Tucker condition)A G(r)=0 and 1. Lagrange method for Constrained Optimization FOC:D,L(Ax)=0.■ The following classical theorem is from Takayama(1993, p. 114) Theorem A-d (Sufficiency). Let f and g, i=L., m, be quasi-concave, where Theorem A-1.(Lagrange). For f: R+R and G: R-R, consider the following G=(8-g). Let r'satisfy the Kuhn-Tucker condition and the FOC for(A 2). Then, x' (1)Df(x)=0, and f is locally twice continuously differentiable, or Let L(Ar=f(x)+A G(r) Lagrange function Kuhn-Tucker Theorem is not useful. We usually use Lagrange Theorem only J solves(4. I)and jf DG(r)has full rank, then there exists AER agrange e A mapping H: R+R is linear if it can be written as H()=Ax+B, where A is FOC: D L(A,r)=0. Theorem A-5. Let f:R”→ R and g:R"→R" be concave,andH:R"→ r be linear Also, Ex, s.t. G(x,)>0. Then, ris a solution of SONC: hDf(r)hs0, for h satisfying DG(rh=0 max f(r) s.G(x)≥0, If the FoC is satisfied, G(r)=0, and SC: hD f(r)h<0, for h=0 satisfying R then x is a unique local maximum. AG(r)=0 and xis a solution of What is the intuition for FOc and Soc? maxL(A,,x)≡f(x)+入G(x)+p·H(x).■ How to verify the SOC? Theorem a2LetA∈R" be symmetric,C∈R" has full rank, m<n,ad…,bbe xample A-2. Given utility function M(x, x,)=xx, consider B el. NP2m三四减x马) x'Ar<0forx≠0 satisfying Cx=0分(-1)b>0,k≥2m+1 Example A-1. For a>0 and b>0, consider-1- Appendix: Math Preparation 1. Lagrange Method for Constrained Optimization The following classical theorem is from Takayama (1993, p.114). Theorem A-1. (Lagrange). For n f : → and n m G : → , consider the following problem max ( ) s t ( ) 0. x f x .. = G x (Α.1) Let L x ( ) () () λ λ , ≡ f x Gx + ⋅ (Lagrange function). • If x∗ solves (A.1) and if DG x( )∗ has full rank, then there exists k λ ∈ (Lagrange multiplier) such that FOC ( ) 0 DL x x λ ∗ : , = , and 2 SONC ( ) 0 for satisf h D f x h h DG x h ying () 0 ′∗ ∗ : ≤ , = . • If the FOC is satisfied, G x() 0 ∗ = , and 2 SOSC ( ) 0 for 0 satisfying ( ) 0 h D f x h h DG x h ′∗ ∗ : <, ≠ = , then * x is a unique local maximum. What is the intuition for FOC and SOC? How to verify the SOC? Theorem A-2. Let n n A × ∈ be symmetric, m n C × ∈ has full rank, m n < , and 1, , m n b b … + be the principal minors of 0 T C B C A ⎛ ⎞ ⎜ ⎟ ≡⎜ ⎟⎟ ⎜ ⎟ ⎝ ⎠. Then, ' 0 for 0 satisfying 0 ( 1) 0, 2 1 k m k x Ax x Cx b k m − < ≠ = ⇔ − > ∀ ≥ + . Example A-1. For 0 a > and b > , 0 consider 1 2 2 2 1 2 1 2 ( ) max s t 1. x x F a b ax bx x x , , ≡ −− .. + = Is the solution indeed optimal? -2- Theorem A-3 (Kuhn-Tucker). For differentiable n f : → and n m G : → , let L x ( ) () () λ λ , ≡ f x Gx + ⋅ . If x∗ is a solution of max ( ) st ( ) 0 x f x . . G x ≥ , (Α.2) then there exists m λ ∈ + such that (Kuhn-Tucker condition) λ G x() 0 ∗ ⋅ = and FOC ( ) 0 DL x x λ ∗ : , =. Theorem A-4 (Sufficiency). Let f and i g , i m =, , , 1 … be quasi-concave, where 1 (… )T Gg g = ,, . m Let x∗ satisfy the Kuhn-Tucker condition and the FOC for (A.2). Then, x∗ is a global maximum point if (1) Df x() 0 ∗ ≠ , and f is locally twice continuously differentiable, or (2) f is concave. Kuhn-Tucker Theorem is not useful. We usually use Lagrange Theorem only. A mapping n k H : → is linear if it can be written as ( ) H x Ax B = +, where A is matrix and B is a vector. Theorem A-5. Let n f : → and n m G : → be concave, and n k H : → be linear. Also, 0 ∃ x s.t. 0 G x() 0 > . Then, x∗ is a solution of max ( ) . . ( ) 0, ( ) 0. x f x st G x H x ≥ = if and only if ( ) 0 ( ) 0 Gx hx ∗ ∗ ≥ , =, and there exist m λ ∈ + and k µ ∈ such that λ G x() 0 ∗ ⋅ = and x∗ is a solution of max ( ) ( ) ( ) ( ) x L x λ, , µ ≡ f x Gx Hx +λ⋅ +µ⋅ . Example A-2. Given utility function 1 1 2 12 ( ) a a ux x xx − ,= , consider 1 2 1 2 12 0 11 2 2 ( ) max ( ) s t x x vp p m ux x p x p x m , ≥ , , ≡ , .. + ≤