The Slutsky equation implies Substitution e?ect: ?ˉx i (p,u) ?p i ; Income e?ect: ? ?x ? i (p,I) ?I · x ? i . Proposition 1.22. (Compensated Demand). (1) ˉx(p, u) is zero homogeneous in p. (2) substitution matrix: D p ˉx(p,u) ≤ 0. (3) symmetric cross-price e?ects: ?ˉx j (p,u) ?p i = ?ˉx i (p,u) ?p j . (4) decreasing ˉx : ?ˉx i (p,u) ?p i ≤ 0.  Proposition 1.23. (Ordinary Demand). (1) Homogeneity: x ? (p,I) is zero homogeneous in (p,I). (2) Compensated Symmetry: ?x ? j ?p i + x ? i ?x ? j ?I = ?x ? i ?p j + x ? j ?x ? i ?I . (3) Adding-up: null p i x ? i (p,I)=I.  3. Uncertainty Theory 3.1. Introduction Example 1.8. Flip a coin once: if tail, get $0; if head, get $100. i.e., lottery nullx ≡ (0, 1 2 ;100, 1 2 ). The expected income is E(game)=50. Let u(x)= √ x be the personal value of x. The expected utility is Eu(nullx)=5. Let u(e)=Eu(nullx). Then, e =25.eis the certainty equivalent.  1—13 3.2. Expected Utility Lottery x =(x 1 ,p 1 ; x 2 ,p 2 ; ...; x n ,p n ): x i with probability p i and null p i =1. Axiom 1. (x,1; y,0) = x. Axiom 2. (x,p; y,1?p)=(y,1?p; x,p). Axiom 3 (RCLA). [(x,p; y,1?p),q; y,1?q]=(x,pq; y,1?pq). Using Axioms 1-3, the lottery space L is well defined. Suppose the consumer has a preference relation null on L. Axiom 4. null is complete. Axiom 5 (Continuity). {p ∈ [0, 1] | (x,p; y,1?p) null z} and {p ∈ [0, 1] | (x,p; y,1? p) null z} are closed sets in [0, 1]. Axiom 6 (Independence). x ～ y =? (x,p; z,1?p) ～ (y,p; z,1?p). The expected utility property: u[(x,p; y,1?p)] = pu(x)+(1?p)u(y). Theorem 1.1. (Expected Utility Representation). If (null,L) satisfies Axioms 1—6, there exists a utility representation u : L → R that has the expected utility property.  Example 1.9. For x ≡ (x 1 ,p 1 ;...;x n ,p n ), the expected utility is u(x) ≡ null p i u(x i ). With a perfect capital market, the firm cares only about the expected money value: u(x) ≡ null p i x i .  Any monotonic linear transform v : L → R preserves the expected utility property: v(x)=au(x)+b, a > 0. Proposition 1.24. (Uniqueness). Expected utility is unique up to a monotonic linear transformation: u(·) and v(·) are expected utility representations of the same preference relation i? there exist a>0 and b ∈R such that v(·)=au(·)+b.  1—14 Example 1.10. (Allais Paradox). Consider the following four lotteries: 0.1 0.01 0.89 x 5m 0 0 y 1m 1m 0 z 5m 0 1m w 1m 1m 1m Example 1.11. (Common Ratio E?ect). Consider the following four lotteries: x = (3000,1), y = (4000,0.8; 0,0.2), z = (3000,0.25; 0,0.75), w = (4000,0.2; 0,0.8). 3.3. Risk Aversion Definition: u[E(?x)] >Eu(?x) if risk averse u[E(?x)] = Eu(?x) if risk neutral u[E(?x)] <Eu(?x) if risk loving For ?x ≡ (x, p; y, 1?p),y>x,he can buy an insurance z : he pays z in any event, and is paid y?x when the bad event happens. The maximum insurance P that he is willing to buy is the insurance premium, satisfying E[u(?x)] = u(y?P). Then, E(?x) >y?P if risk averse E(?x)=y?P if risk neutral E(?x) <y?P if risk loving 1—15 Therefore, u is concave if risk averse u is linear if risk neutral u is convex if risk loving Themoreconcaveu is, the higher P is, or the more risk averse the consumer is. Thus, we propose two measures of risk aversion: absolute risk aversion: R a (x) ≡? u 00 (x) u 0 (x) , relative risk aversion: R r (x) ≡? xu 00 (x) u 0 (x) . They are for two types of fluctuations: ˉx ± ε or ˉx(1 ± ε). Let ˉx ≡ E(?x). Define the risk premium π a : u(ˉx?π a )=E[u(?x)]. (3.1) Let nullx ≡ ˉx + ε. Then, π a ≈ 1 2 σ 2 ε R a (ˉx). Similarly, define the relative risk premium π r : u[ˉx(1?π r )] = E[u(?x)]. Let nullx =ˉx(1 + ε). Then, π r ≈ 1 2 σ 2 ε R r (ˉx). Example 1.12. 14. The Demand for Insurance. For (w?l, p; w, 1?p), the consumer can buy insurance q at price π : Prob. p 1?p Before w?lw After w?l + q?πqw?πq The consumer’s problem is max q pu(w?l + q?πq)+(1?p)u(w?πq). 1—16 Assume a competitive market: π = p. With u 00 < 0, we have q ? = l. Alternatively, let I 1 ≡ w?πq and I 2 ≡ w?l+(1?π)q. Then, the problem becomes ? ? ? ? ? max (1?p)u(I 1 )+pu(I 2 ) s.t. (1?π)I 1 + πI 2 =(1?π)w + π(w?l). In a competitive insurance market, π = p, implying I ? 1 = I ? 2 , i.e., full insurance.  4. Equilibrium Theory See MWG (1995), Chapters 15—17 and 19. Arrow-Debreu world: 1. Complete market: any consumption bundle is obtainable. 2. Perfect market: no frictions such as transaction costs, taxes, etc. 3. Perfect competition: economicagentstakepricesasgiven. 4. Symmetric Information: same information for all. 5. Private consumption. 4.1. Equilibrium in a Pure Exchange Economy A pure exchange economy: (1) k commodities j =1, 2,...,k; (2) n consumers i =1, 2,...,n, with (u i ,w i ), where u i : R k + → R and w i ∈R k + . Any x =(x 1 ,x 2 ,...,x n ),x i ∈R k + , is an allocation. An allocation x is feasible if n null i=1 x i ≤ n null i=1 w i . The Edgeworth box contains all the allocations satisfying x 1 A + x 1 B = w 1 A + w 1 B ,x 2 A + x 2 B = w 2 A + w 2 B ,x j i ≥ 0. It has two key features: ? Each point represents a feasible allocation. ? Both persons share the same budget line. 1—17