1、Allocative Eciency andthe Valuation of StateContingent SecuritiesRef: H; !2) Individuals Endowment fei;o;ei;!g3) Aggregation Constraints fCo;C!gIndividuals Objective:maxci;o;fci;!gX!2i;!ui;!(ci;o;ci;!)s.t.oci;o+X!2!ci;!= oei;o+X!2!ei;!Note: State-Dependent Utility: Compare Withui;o(ci;o)+X!2i;!ui(ci
2、;!)Lagrangian:L =X!2i;!ui;!(ci;o;ci;!)+i24oei;o;ci;o+X!2!ei;!; ci;!352First-Order Conditions: 8i 2 (1;I)Lci;o= 0 =X!2i;!ui;!(ci;o;ci;!)ci;o;ioLci;!= 0 = i;!ui;!(ci;o;ci;!)ci;!;i!8!Li= 0 = oei;o;ci;o+X!2!ei;!; ci;!Aggregation Constraints:Xici;0= C0;Xici;!= C!8!How Many Equations/Constraints?I( +2)+(
3、+1)How Many Unknowns (Individuals, Prices)?I(ci;o;ci;!;i)+( +1) = I( +2)+( +1)3However, The Economy Constraints + (Individual Constraints) Linearly Dependent) Only Relative PricingTypically Choose o= 1 (Numeraire)vs. Partial EquilibriumFor Individual Problem (Prices f!g Given):( +2) Equations( +2) U
4、nknownsci;o; fci;!g; i4FOC:i;!ui;!(ci;o;ci;!)ci;!= i!8!X!2i;!ui;!(ci;o;ci;!)ci;o= io()i;!ui;!(ci;o;ci;!)ci;!P!2i;!ui;!(ci;o;ci;!)ci;o=!o8! 2 (1; )RHS Independent of i ) LHS Independent of i) LHS ratio same for all agents.5For Time-Additive, State-Independent Utility:U = ui;o(ci;o)+X!2i;!ui(ci;!)Simp
5、li es to (MRS):i;!ui(ci;!)ci;!ui;o(ci;o)ci;o=!o8! 2 (1; )If Assume Homogeneous Beliefs i;!= !8i; 8! : ui;!(ci;!)ci;!ui;o(ci;o)ci;o=1o!8! 2 (1; )RHS Independent of i ) LHS Independent of iRHS = State Price Per Unit of Probability6Recall 2ndWelfare Thm: For each Pareto-optimalallocation, 9 (positive)
6、fg s.t. same allocation canbe achieved by social planner maximizing linearcombination of individuals utility functions.U =IXi=1i24X!=1i;!ui;!35I(1+ ) Choice Parameters (I fci;o;ci;!g)( +1) Market Clearing ConstraintsL =IXi=1i24X!=1i;!ui;!35+024C0;IXi=1ci;o35+X!=1!24C!;IXi=1ci;!35Here, fo; !g are Lag
7、range Multipliers.Solve, Set i=1i) get Eq.().7Implications()i;!ui;!(ci;o;ci;!)ci;!P!2i;!ui;!(ci;o;ci;!)ci;o=!o8! 2 (1; )1) An allocation is ecient, or Pareto Optimal (PO),i for all states ! the MRS between present andfuture consumption is equal across individuals.2) Competitive equilibrium is alloca
8、tionally ecientwhen markets are complete.) PO allocation obtains in a competitiveeconomy if markets are complete.8In Practice, Dont Have A/D SecuritiesHave Complex SecuritiesQ? Have Pareto-Ecient Allocation?A! Typically, Only if Market Complete# Independent Securities = # States of NatureIntuition:
9、With Complete Markets:Max EU s.t. Wealth ConstraintWith Incomplete Markets:Max EU s.t. Wealth, Market Constraints9Securities Markets Economy1 Unit Each of N Securities j = f1;2;NgSecurity j Pays xj;!in State !) xj= RV, xj!= Number ! 2 (1; )Endowment of Individual i = fei;o, i;jgEconomy has 1) Spot C
10、ommodities Market2) Securities MarketSj Time-0 Price of Security j10Individuals Objective:maxci;o;f i;jgX!=1i;!ui;!(ci;o;ci;!)s.t.ci;o+NXj=1i;jSj= ei;o+NXj=1 i;jSjWhere:ci;!= Wi;!=NXj=1i;jxj;!Note:ui;!(ci;o;ci;!) i;j=ui;!(ci;o;ci;!)ci;!ci;! i;j=ui;!(ci;o;ci;!)ci;!xj;!11Solve FOCX!=1i;!ui;!ci;!xj;!=
11、SjX!=1i;!ui;!ci;0= Hence, in contrast to (*):X!=1i;!ui;!ci;!P!=1i;!ui;!ci;0xj;!= SjDoes Not Indicate that MRS Between Present andFuture Consumption Are Identical Across Individuals) No Pareto Optimality) Due toP!=112De nemi! i;!ui;!ci;!P!=1i;!ui;!ci;0ThenX!=1xj;!mi!= SjOr, in Matrix NotationXmi= SIf
12、 Markets Complete ) X Invertible:mi= X;1SRHS Indep. of i ) LHS Indep. of iHave Pareto Eciency) Wealth Only ConstraintHave Solved for Ratio of Marginal Utilities) mi!= m!= !13ImplicationsA securities Market is a complete market i # linearly independent securities = # of states) Pareto optimal allocat
13、ion achieved in competitiveequilibrium with individuals trading only incomplex securities.Why? Because A/D securities can be replicated ifsecurities market is completeExample: Replicate, and Price A/D 1;0;0Choose f g s.t.Xjjxj;!= 1(!=1)14Matrix Notation0X = 1;0;0Since Inverse 90= 1;0;0X;1To Price:V
14、=XjjSj 0S= 1;0;0X;1S 1;0;0Again we see that !is price per unit consumptionin state !15In an incomplete market and arbitrary preferencesPareto Eciency not obtained in general.How can markets be made complete?1) Assume can create a State Index Portfolio,where payo s x are s.t. x!6= x!08 !; !02 2) Crea
15、te calls with strikes x! = 1; 2; :; ;1(Place in descending order) Payo matrix looks like (Prove X Invertible)0BBBBBBBx!1x!2x!3: x!0 x!2;x!1x!3;x!1: x!;x!10 0 x!3;x!2: x!;x!2.0 0 0 : x!;x!;11CCCCCCCAClearly, such a market does not exist in practice.16Restrict Economy To:1) Homogeneous Beliefs: i;!= !
16、8i; !2) State-Independent, Time Additive UtilityV = ui;o(ci;o)+X!=1!ui(ci;!)Then: (Set o= 1 as Numeraire)L = ui;o(ci;o)+X!=1!ui(ci;!)+i0Wi;0;ci;0;X!=1!ci;!1AIf markets complete, then FOCs give:u0i;o= i; !u0i;!= i!8 i17Hence, for any agents i; k:u0i;oi= 1 =u0k;oku0i;!i=!=u0k;!kConsider states !; !0s.
17、t. C! C!0Since u00ci;!0 8 iHence, given assumptions above, 9 1-1relation between aggregate consumption andagents optimal allocation for each state !:ci= fi(C) f0i() 0Can show analogously that:ci0= fi0(C0) f0i0() 0ffi; fi0g Pareto Optimal Sharing Rules18Implication: If 9 two states (!; !0) s.t. C!= C
18、!0,then dont need A/D securities for each for PO, sinceeveryone will want same payo in those two states.) Options on market portfolio, rather thanon every stock, sucient for PO.Note: ffi(C)g typically non-linear functions.However, if assume the formci;! fi(C!) = ai+biC!8 iThen only need market portf
19、olio and risk-freesecurity for PO:Purchase bishares of MPPurchase aishares of risk-free bondNote, if risk-free security in zero net-supply:IXi=1ci;!= ai+biC!)IXi=1ai= 0;IXi=1bi= 119Can show that necessary and sucient conditionson utility functions for ffg to be linear for allinitial wealth distribut
20、ions is:Ti(c) ;1RA(c)= ;u0iu00i= Ai+BcCall: Ti(c) Risk Tolerance at cT0i(c) Cautiousness at cNote: Bi= B 8 iSolution:B 6= 0 : u0i(c) = i;Ai+Bc;1BB = 0 : u0i(c) = ie;cAiFrom previous slide, we see that 2-Fundseparation obtains.20Recall:!= !u0i;!u0i;oFor Any Security-j:Sj=X!=1xj;!=X!=1xj;!u0i;!u0i;o=
21、E024u0i;!u0i;oxj35If 9 !;!0s.t. C!= C!0, can write in another form:Note: does not imply x!= x!0 for all assets.Def: k subset of s.t. C!= k if ! 2 k(k) price of claim that pays 1 unitof consumption i ! 2 k21(k) =X!2k!=X!2k!u0i;!u0i;o=X!2k!u0i;ku0i;o=u0i;ku0i;o(k)Hence, for Any Security j:Sj=X!=1xj;!=XkX!2kxj;!=Xku0i;ku0i;oX!2kxj;!=Xk(k)u0i;ku0i;oX!2kxj;!(k)=Xk(k) E0hxjjC = kiWhere Introduced Conditional Expectation E0 j (Important for Filtration (Multi-Period)22
