Proposition 1.11.(The Le Chatelier Principle) ay(p, w)> ay(,u,22 p 之=z(p, ai(p, w)axi (p, w, 2) du 1.4. Production Plans with Multiple Outputs Lety≡(m,,…,ym) be a net output vector, YArn be a convex set,G:Y→R be twice differentiable Production possibility set:{y∈Y|G(y)≤0} Assumption 1.1. Gy (y)>0, Vi,yEY. Proposition 1. 12. Production frontier yEY G(y)=0 contains technologically efficient production plans Definition 1.1. Marginal rate of transformation MRT Assumption 1. 2. G is quasi-convex Proposition 1.13. Under Assumption 1. 1, G is quasi-convex iff <0. <0. ynyny1 yn yn Example 1.6. Let y= f(ar) be a production function, f: R4-R+. The production possibility set P={(-x,y)∈R+1|f(x)≥y} Let f(y1). TH P={(,y)∈R+|G(v,m)≤0
Proposition 1.11. (The LeChatelier Principle). ∂y(p, w) ∂p ≥ ∂y(p, w, z) ∂p z=z(p,w) , ∂xi(p, w) ∂wi ≥ ∂xi(p, w, z) ∂wi z=z(p,w) . 1.4. Production Plans with Multiple Outputs Let y ≡ (y1, y2,...,ym) be a net output vector, Y ⊂ Rm be a convex set, G : Y → R be twice differentiable. Production possibility set: {y ∈ Y | G(y) ≤ 0}. Assumption 1.1. Gyi (y) > 0, ∀ i, y ∈ Y. Proposition 1.12. Production frontier {y ∈ Y | G(y)=0} contains technologically efficient production plans. Definition 1.1. Marginal rate of transformation: MRTij ≡ − ∂yi ∂yj G=0 = Gyj Gyi . Assumption 1.2. G is quasi-convex. Proposition 1.13. Under Assumption 1.1, G is quasi-convex iff 0 Gy1 Gy2 Gy1 Gy1y1 Gy1y2 Gy2 Gy2y1 Gy2y2 ≤ 0, ..., 0 Gy1 ··· Gyn Gy1 Gy1y1 ··· Gy1yn . . . . . . . . . Gyn Gyny1 ··· Gynyn ≤ 0. Example 1.6. Let y = f(x) be a production function, f : Rn + → R+. The production possibility set: P = (−x, y) ∈ Rn+1 | f(x) ≥ y . Let y1 ≡ −x, y2 ≡ y, G(y1, y2) ≡ y2 − f(−y1). Then, P = (y1, y2) ∈ Rn+1 | G(y1, y2) ≤ 0, y1 ≤ 0, y2 ≥ 0 1—8
Gm(,2)=f(-)=f(x), nd G. GG 921291:9292 Thus, if f'>0 and f"<0, Assumptions 1.1 and 1.2 are satisfied Consider (p)≡maxp·y 2.9) (y G(y)≤0 (P)≡maxp·y LetC(A,y)≡p·y+G(y) FOC:p+入DG(y)=0 By the quasi-convexity of G, the Foc guarantees optimality
and Gy1 (y1, y2) = f0 (−y1) = f0 (x), Gy2 (y1, y2)=1, and 0 Gy1 Gy2 Gy1 Gy1y1 Gy1y2 Gy2 Gy2y1 Gy2y2 = f00(x). Thus, if f0 > 0 and f00 < 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 CR, the consumer has preferences over X Reflexivity.x≥x. Transitivity.x≥ y and y≥x→x≥z Completeness.Vx,y∈X, either a≥yorx<y Continuity.o∈X,{x∈X|x≥r}and{x∈X|x≤ao} are closed sets in X A function u: X-R represents the preferences if 1.u(x)≥u(y)ix≥y; 2. u()=u(y) iff x N y u is called a utility function on(X, 2) Proposition 1. 14.(Existence of Utility Function). If is reflexive, transitive complete, and continuous, then there exists a continuous utility function u on(, X) Proposition 1.15. (1) If u is a utility function, for any strictly increasing R++R, U=pou is also a utility function for the same preference relation (2)The utility function is unique up to a strictly increasing transformation
2. Consumer Theory 2.1. Existence of Utility Function Given consumption set X ⊂ Rk, the consumer has preferences over X. Reflexivity. x x. Transitivity. x y and y z ⇒ x z. Completeness. ∀ x, y ∈ X, either x y or x ≺ y. Continuity. ∀ x0 ∈ X, {x ∈ X | x x0} and {x ∈ X | x x0} are closed sets in X. A function u : X → R represents the preferences if 1. u(x) ≥ u(y) iff x y; 2. u(x) = u(y) iff x ∼ y. u is called a utility function on (X, ). Proposition 1.14. (Existence of Utility Function). If is reflexive, transitive, complete, and continuous, then there exists a continuous utility function u on (, X). Proposition 1.15. (1) If u is a utility function, for any strictly increasing ϕ : R → R, v ≡ ϕ ◦ 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 The consumers problem u(p, I) Rn t.p·x≤I * (p, I) is the ordinary demand function or Marshallian demand function v(p, I) is the indirect utility function. FOC x, ( r*) pi SO h·D2u(x")·h≤0forp·h=0. Remark 1.1. if u is continuous. a* exists Remark 1.2. Strict quasi-concavity of u unique Remark 1.3. T' is independent of utility representation The dual problem of utility maximization is expenditure minimization e(P,a)≡minp·x (x)≥t The solution I=i(p, u) is the compensated demand function or Hicksian demand function. e(p, u) is the expenditure function normal good a/ =0 inferior good dxs 0: luxury good r t al necessary go 01. inelasti ap
2.2. Consumer’s Problem The consumer’s problem: ⎧ ⎪⎪⎨ ⎪⎪⎩ v(p, I) ≡ max x∈Rn + 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 : uxi (x∗) uxj (x∗) = pi pj . SOC : h0 · D2 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 x¯ = ¯x(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 1, necessary good: 0 ≤ I x∗ i ∂x∗ i ∂I 0, gross complements: ∂x∗ i ∂pj 0, net complements: ∂x¯i ∂pj 0, usual good: ∂x∗ i ∂pi ≤ 0; elastic: − pi x∗ i ∂x∗ i ∂pi > 1, inelastic: 0 ≤ − pi x∗ i ∂x∗ i ∂pi < 1. 1 — 11
2.3. Properties Proposition 1.16.(Indirect Utility Function). u(p, I)is (1)decreasing in p, increasing in I; zero home ogeneous in (p, I) (3quasi-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)cD,v(P,D)=I; (2)tp,c(p,)]= (3)x(P,D)=,(P,D (4)x(P,)=x{p,c(P,u).■ Proposition 1.19(Shephard's Lemma) de(p, u) Proposition 1.20.(Roy s Identity). For p>>0,I>0, i(p, I) v(p, I) Example 1.7. For u(a1, 52)=(ai+a2)1/p, verify Roy's identit Proposition 1.21.(Slutsky Equation) ar (p, I) a,p, v(p, I)I ax(p, I)
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)=¯xi[p, v(p, I)]; (4) x¯i(p, u) = x∗ i [p, e(p, u)]. Proposition 1.19. (Shephard’s Lemma). x¯i(p, u) = ∂e(p, u) ∂pi , ∀ i. Proposition 1.20. (Roy’s Identity). For p >> 0, I> 0, x∗ i(p, I) = −vpi (p, I) vI (p, I) , ∀ i. Example 1.7. For u(x1, x2)=(xρ 1 + xρ 2)1/ρ, verify Roy’s identity. Proposition 1.21. (Slutsky Equation). ∂x∗ j (p, I) ∂pi = ∂x¯j [p, v(p, I)] ∂pi − ∂x∗ j (p, I) ∂I · x∗ i(p, I). 1 — 12