Proposition 1.11. (The LeChatelier Principle). ?y(p,w) ?p ≥ ?y(p,w,z) ?p null null null null z=z(p,w) , null null null null ?x i (p,w) ?w i null null null null ≥ null null null null null ?x i (p,w,z) ?w i null null null null z=z(p,w) null null null null null .  1.4. Production Plans with Multiple Outputs Let y ≡ (y 1 ,y 2 ,...,y m ) be a net output vector, Y ?R m be a convex set, G : Y → R be twice di?erentiable. Production possibility set: {y ∈ Y | G(y) ≤ 0}. Assumption 1.1. G y i (y) > 0, ? i, y ∈ Y. Proposition 1.12. Production frontier {y ∈ Y | G(y)=0} contains technologically e?cient production plans.  Definition 1.1. Marginal rate of transformation: MRT ij ≡? ?y i ?y j null null null null G=0 = G y j G y i .  Assumption 1.2. G is quasi-convex. Proposition 1.13. Under Assumption 1.1, G is quasi-convex i? null null null null null null null null null null null 0 G y 1 G y 2 G y 1 G y 1 y 1 G y 1 y 2 G y 2 G y 2 y 1 G y 2 y 2 null null null null null null null null null null null ≤ 0, ..., null null null null null null null null null null null null null null null 0 G y 1 ··· G y n G y 1 G y 1 y 1 ··· G y 1 y n . . . . . . . . . G y n G y n y 1 ··· G y n y n null null null null null null null null null null null null null null null ≤ 0.  Example 1.6. Let y = f(x) be a production function, f : R n + → R + . The production possibility set: P = null (?x,y) ∈R n+1 | f(x) ≥ y null . Let y 1 ≡?x, y 2 ≡ y, G(y 1 ,y 2 ) ≡ y 2 ?f(?y 1 ). Then, P = null (y 1 ,y 2 ) ∈R n+1 | G(y 1 ,y 2 ) ≤ 0,y 1 ≤ 0,y 2 ≥ 0 null 1—8 and G y 1 (y 1 ,y 2 )=f 0 (?y 1 )=f 0 (x),G y 2 (y 1 ,y 2 )=1, and null null null null null null null null null null null 0 G y 1 G y 2 G y 1 G y 1 y 1 G y 1 y 2 G y 2 G y 2 y 1 G y 2 y 2 null null null null null null null null null null null = f 00 (x). Thus, if f 0 > 0 and f 00 < 0, Assumptions 1.1 and 1.2 are satisfied.  Consider π(p) ≡ max y p · y s.t. G(y) ≤ 0. (2.9) By Assumption 1.1, π(p) ≡ max y p · y s.t. G(y)=0. Let L(λ,y) ≡ p · y +λG(y). FOC: p +λDG(y ? )=0. By the quasi-convexity of G, the FOC guarantees optimality. 1—9 2. Consumer Theory 2.1. Existence of Utility Function Given consumption set X ?R k , the consumer has preferences null over X. Reflexivity. x null x. Transitivity. x null y and y null z ? x null z. Completeness. ? x, y ∈ X, either x null y or x ? y. Continuity. ? x 0 ∈ X, {x ∈ X | x null x 0 } and {x ∈ X | x null x 0 } are closed sets in X. Afunctionu : X → R represents the preferences if 1. u(x) ≥ u(y) i? x null y; 2. u(x)=u(y) i? x ～ y. u is called a utility function on (X,null). Proposition 1.14. (Existence of Utility Function). If null is reflexive, transitive, complete, and continuous, then there exists a continuous utility function u on (null,X).  Proposition 1.15. (1) If u is a utility function, for any strictly increasing ? : R → R,v≡ ??u is also a utility function for the same preference relation. (2) The utility function is unique up to a strictly increasing transformation.  1—10 2.2. Consumer’s Problem Theconsumer’sproblem: ? ? ? ? ? ? v(p,I) ≡ max x∈R n + u(x) s.t. p · x ≤ I. x ? = x ? (p,I) is the ordinary demand function or Marshallian demand function. v(p,I) is the indirect utility function. FOC : u x i (x ? ) u x j (x ? ) = p i p j . SOC : h 0 · D 2 u(x ? ) · h ≤ 0 for p · h =0. Remark 1.1. If u is continuous, x ? exists. Remark 1.2. Strict quasi-concavity of u ? unique x ? . Remark 1.3. x ? is independent of utility representation. The dual problem of utility maximization is expenditure minimization: ? ? ? ? ? e(p,u) ≡ min p · x s.t. u(x) ≥ u The solution ˉx =ˉx(p,u) is the compensated demand function or Hicksian demand function. e(p, u) is the expenditure function. normal good: ?x ? i ?I ≥ 0, inferior good: ?x ? i ?I < 0; luxury good: I x ? i ?x ? i ?I >1, necessary good: 0 ≤ I x ? i ?x ? i ?I < 1; gross substitutes: ?x ? i ?p j > 0, gross complements: ?x ? i ?p j < 0; net substitutes: ?ˉx i ?p j > 0, net complements: ?ˉx i ?p j < 0; Gi?en good: ?x ? i ?p i > 0, usual good: ?x ? i ?p i ≤ 0; elastic: ? p i x ? i ?x ? i ?p i > 1, inelastic: 0 ≤? p i x ? i ?x ? i ?p i < 1. 1—11 2.3. Properties Proposition 1.16. (Indirect Utility Function). v(p,I) is (1) decreasing in p, increasing in I; (2) zero homogeneous in (p,I); (3) quasi-convex in p.  Proposition 1.17. (Expenditure Function). e(p,u) is (1) increasing in p; (2) linearly homogeneous in p; (3) concave in p.  Proposition 1.18. (1) e[p, v(p,I)] = I; (2) v[p, e(p,u)] = u; (3) x ? i (p,I)=ˉx i [p, v(p,I)]; (4) ˉx i (p,u)=x ? i [p, e(p,u)].  Proposition 1.19. (Shephard’s Lemma). ˉx i (p, u)= ?e(p,u) ?p i , ? i.  Proposition 1.20. (Roy’s Identity). For p>>0,I>0, x ? i (p, I)=? v p i (p,I) v I (p,I) , ? i.  Example 1.7. For u(x 1 ,x 2 )=(x ρ 1 + x ρ 2 ) 1/ρ , verify Roy’s identity. Proposition 1.21. (Slutsky Equation). ?x ? j (p,I) ?p i = ?ˉx j [p, v(p,I)] ?p i ? ?x ? j (p,I) ?I · x ? i (p,I).  1—12