The Slutsky equation implies Substitution effect: aii (p, u Income effect a*(p, I) Proposition 1. 22(Compensated Demand). (1)i(p, u) is zero homogeneous in p (2) substitution matrix: Dpi(p, u)<0 (3)symmetric cross-price effects: 23i(p 2=2z(p, u) (4) decreasing:=n≤0.■ Proposition 1.23.(Ordinary Demand) (1)Homogeneity: (P, I) is zero homogeneous in(P, I) (2)Compensated Symmetry: x trial a (3) Adding-up:∑px(P,D)=I.■ 3. Uncertainty Theory 3.1. Introduction Example 1.8. Flip a coin once if tail, get SO: if head, get $100 i. e, lottery I=(0, 3; 100, 3). The expected income is E(game)=50. Let u(a)=v be the personal value of a. The expected utility is Eu(a=5. Let u(e)=Eu(a). Then, e=25. e is the certainty equivalent
The Slutsky equation implies Substitution effect: ∂x¯i(p, u) ∂pi ; Income effect: − ∂x∗ i(p, I) ∂I · x∗ i . Proposition 1.22. (Compensated Demand). (1) x¯(p, u) is zero homogeneous in p. (2) substitution matrix: Dpx¯(p, u) ≤ 0. (3) symmetric cross-price effects: ∂x¯j (p,u) ∂pi = ∂x¯i(p,u) ∂pj . (4) decreasing x¯ : ∂x¯i(p,u) ∂pi ≤ 0. Proposition 1.23. (Ordinary Demand). (1) Homogeneity: x∗(p, I) is zero homogeneous in (p, I). (2) Compensated Symmetry: ∂x∗ j ∂pi + x∗ i ∂x∗ j ∂I = ∂x∗ i ∂pj + x∗ j ∂x∗ i ∂I . (3) Adding-up: Spix∗ 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 xh ≡ (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(xh)=5. Let u(e) = Eu(xh). Then, e = 25. e is the certainty equivalent. 1 — 13
3.2. Expected Utility Lottery a=(x1,1;x2,p2;……;xn,pn):r; with probability p;and∑p=1. Axiom 1.(, 1;y, 0)=2 Axiom 2.(, p; 9, 1-p)=(y, 1-p; P Axiom 3(RCLA).z,p; g, 1-p), y, 1-g=(a, pq; y, I-pq Using Axioms 1-3, the lottery space L is well defined. Suppose the consumer has preference relation on L Axiom 4. is complete Axiom 5(Continuity). PE0, 1(,p; y, 1-p)2a) and p E 0, 1(, p; y,I- P)< 2) are closed sets in 0, 1 Axiom 6(Independence). n y (x,p;z,1-p)~(y,p;z,1-p) The expected utility property u(x,D;y,1-p)=p(x)+(1-p)u(y) Theorem 1.1.(Expected Utility Representation). If(, L)satisfies Axioms 1-6, there exists a utility representation u: L-R that has the expected utility property. L Example 1.9. For x =(a1, Pi;...; n, Pn), the expected utility is ()=∑l(x With a perfect capital market, the firm cares only about the expected money value ()=∑ Any monotonic linear transform v: L-R preserves the expected utility property v(r)=au(ar)+b, 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 iff there exist a>0 and bER such that v(=au()+b. 14
3.2. Expected Utility Lottery x = (x1, p1; x2, p2; ... ; xn, pn) : xi with probability pi and Spi = 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 on L. Axiom 4. is complete. Axiom 5 (Continuity). {p ∈ [0, 1] | (x, p; y, 1−p) z} and {p ∈ [0, 1] | (x, p; y, 1− p) 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 (,L) satisfies Axioms 1—6, there exists a utility representation u : L → R that has the expected utility property. Example 1.9. For x ≡ (x1, p1; ... ; xn, pn), the expected utility is u(x) ≡ [piu(xi). With a perfect capital market, the firm cares only about the expected money value: u(x) ≡ [pixi. 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 iff there exist a > 0 and b ∈ R such that v(·) = au(·) + b. 1 — 14
Example 1.10.(Allais Paradox). Consider the following four lotteries 0.10.010.89 lm 1r 5m0 In 1m1m1 Example 111.(Common Ratio Effect). Consider the following four lotteries x=(30001), y=(400,0.8:0,0.2), z=(3000.25:;0,0.75) U=(4000,0.2;0,0.8) 3.3. Risk Aversion ue()> eu(s) if risk averse ue())= Eu(i) if risk neutral ue()< eu(a) if risk loving For i =(a, p; y,I-P), y>a, he can buy an insurance z: he pays 2 in any vent, and is paid y-a when the bad event happens. The maximum insurance P that he is willing to buy is the insurance premium, satisfyin E[u()]=(y-P) The E()>y-P if risk averse E()=y-P if risk neutral e()<y-P if risk loving
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 Effect). 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 verse u is linear if risk neutral u is convex if risk loving The more concave u is, the higher P is, or the more risk averse the consumer is. Thus, we propose two measures of risk aversion absolute risk aversion Ra(c)=_u(a) u(a relative risk aversion: R( a'() They are for two types of fluctuations:z士εor(1±e) Let I=E(. Define the risk premium Ta ux-丌 )=E( (3 Letx≡+E.Then a≈52Ra() Similarly, define the relative risk premium Tr (1-x)=E[u(2) Let x=i(1+e). Then OER(T) Example 1. 12. 14. The Demand for Insurance. For(w-L, p; w, 1-P), the consumer can buy insurance q at price T Prob 1-P Before After-I+q-Tq The consumer's problem is max pu(w-L+q-T9)+(1-pu(w-q)
Therefore, u is concave if risk averse u is linear if risk neutral u is convex if risk loving The more concave u is, the higher P is, or the more risk averse the consumer is. Thus, we propose two measures of risk aversion: absolute risk aversion: Ra(x) ≡ − u00(x) u0 (x) , relative risk aversion: Rr(x) ≡ − xu00(x) u0 (x) . They are for two types of fluctuations: x¯ ± ε or x¯(1 ± ε). Let x¯ ≡ E(˜x). Define the risk premium πa : u(¯x − πa) = E[u(˜x)]. (3.1) Let xh ≡ x¯ + ε. Then, πa ≈ 1 2 σ2 εRa(¯x). Similarly, define the relative risk premium πr : u[¯x(1 − πr)] = E[u(˜x)]. Let xh = ¯x(1 + ε). Then, πr ≈ 1 2 σ2 εRr(¯x). 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 − l w After w − l + q − πq w − πq The consumer’s problem is max q pu(w − l + q − πq) + (1 − p)u(w − πq). 1 — 16
Assume a competitive market: T=p. With u"<O, we have g*=l Alternatively, let I1=W-q and I2=w-I+(1-q. Then, the problem becomes max(1-p)u(Ii)+pu(I2) st.(1-丌)l1+丌l2=(1-丌)+x(- In a competitive insurance market, T=p, implying I= 12, i.e., full insurance. L 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: economic agents take prices as given 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 (2) n consumers 2=1,2,…,n,with(ua,t2), where a2:R+→ IR and wi∈R Any z=(1, T2, .. n),T; E R+, is an allocation. An allocation z is feasible if The Edgeworth box contains all the allocations satisfying 4+ r24+ It has two key features Each point represents a feasible allocation Both persons share the same budget line
Assume a competitive market: π = p. With u00 < 0, we have q∗ = l. Alternatively, let I1 ≡ w −πq and I2 ≡ w −l + (1−π)q. Then, the problem becomes ⎧ ⎪⎨ ⎪⎩ max (1 − p)u(I1) + pu(I2) s.t. (1 − π)I1 + πI2 = (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: economic agents take prices as given. 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 (ui, wi), where ui : Rk + → R and wi ∈ Rk +. Any x = (x1, x2,...,xn), xi ∈ Rk +, is an allocation. An allocation x is feasible if Sn i=1 xi ≤ Sn i=1 wi. The Edgeworth box contains all the allocations satisfying x1 A + x1 B = w1 A + w1 B, x2 A + x2 B = w2 A + w2 B, xj i ≥ 0. It has two key features: • Each point represents a feasible allocation. • Both persons share the same budget line. 1 — 17