ECO 37 Economic of Uncertainty Fall Term 2009 Week 8 Precept Novemer 8 Financial Market - Quetion Tere are S tate of te world laeled =, 2,... S, and H trader laeled =, 2,... H. Eac trader i a price-taker. Te proaility of tate occuring i π. Tere i one pyical good, and te endowment of trader in tate i denoted y C 0,. Tere are complete market for Arrow-Dereu ecuritie (contingent claim); a ecurity for tate i a promie to deliver a unit of te pyical good if tate occur, and noting in any oter tate. Eac trader i a price-taker in te market for contingent claim. Denote te price of a tate- Arrow-Dereu ecurity y P. Te price in te ex ante market for a promie to pay one unit of te pyical good regardle of wic tate occur i, o S = P =. (a) Write down te udget contraint for trader. () Eac trader a a utility-of-conequence function V (C) = C /2. (i) Write down te expreion for trader expected utility. (ii) Find expreion for trader optimal coice C of te contingent claim. (iii) Write down te equilirium condition for contingent claim market. (iv) Hence ow tat in equilirium, P = π ( Ĉ ) /2 t π t (Ĉt) /2, were te index of ummation t i ued to denote tate in te denominator, o a to ditingui it from te particular tate, and Ĉ = H H = C 0,. i te average endowment of te good in tate. Otain an economic explanation for ti reult. (c) Now uppoe tat eac trader a a linear-quadratic utility-of-conequence function V (C) = a C 2 C2, were a and are poitive contant. Aume tat te value of a and are uc tat every trader V function i increaing for all C in te range tat i relevant in ti analyi. Following te ame tep a in part () it can e own (ti will e upplied in te olution to e poted) tat in equilirium, P = π ( a Ĉ ) t π t [ a Ĉt ], were t and Ĉ ave te ame interpretation a in part (). Do tee pricing equation te one you derived in () and te one tated in (c) ugget any general teme aout efficient rik-earing?
ECO 37 Economic of Uncertainty Fall Term 2009 Week 8 Precept Novemer 8 Financial Market - Solution (a) Denoting trader conumption in tate y C, te udget contraint i P C P C 0, W 0, were I ave introduced te areviation W 0, for te rigt and ide of te contraint. () (i) Hi expected utility i π (C ) /2 (ii) Terefore te firt-order condition for te optimal coice are, for all tate : 2 π (C ) /2 = λ P or Ten Adding over tate, Eliminating te λ term, ( ) C π 2 = = (π 2 λ P 4 λ 2 ) 2 (P ) 2 P C = 4 λ 2 W 0, = 4 λ 2 C = (π ) 2 (P ) (π ) 2 (P ) (π ) 2 (P ) 2 t (π t ) 2 (P t ) W 0, () were I ave ued te index of ummation t for tate in te denominator to ditingui it from te particular tate. (iii) Adding over trader, te equilirium condition i C = C 0, C 0, uing te areviation defined y te equivalence ign. Terefore C 0 = (π ) 2 (P ) 2 t (π t ) 2 (P t ) W 0, (iv) Write te complicated denominator a D, and W 0 = W 0,, for revity. Ten (P ) 2 = (π ) 2 (C 0 ) W 0 /D
or Summing over tate, Terefore Finally, uing te definition P = π (C 0 ) /2 (W 0 /D) /2 = π (C 0 ) /2 (W 0 /D) /2 P = π (C 0 ) /2 t π t (Ct 0 ) /2 ti eaily convert to Ĉ = H H = C 0, = H C0, P = π (Ĉ) /2 t π t (Ĉt) /2 Oerve carefully te equence of logical tep in te argument. Tey follow te tandard logic of microeconomic: [] coice, [2] equilirium. [] Firt we mut find eac individual demand function quantity demanded on te left, and price and endowment on te rigt. In deriving ti, we ave to olve te firt-order condition we get from te Lagrangian, and te udget contraint. If you failed to ue te udget contraint, you mut ave mied ometing. [2] Having found te individual demand function, we can um tem to get te aggregate demand. If you did not ave ti tep, again you mut ave mied ometing. [3] Ten we mut ave te equilirium condition: et te aggregate quantitie demanded equal to te aggregate quantitie upplied for eac good. [4] Finally, olve te equilirium condition for te price. Tere i anoter way to tink aout ti. Eac individual demand function for tatey-tate conumption () are proportional to tat individual aggregate value of tate-ytate endowment W 0,, and te contant of proportionality i te ame for all individual. Terefore we can regard te wole group jut like one individual wo a te um total of everyone endowment, or an average or repreentative individual wo a te average endowment. Ten te equilirium price mut e uc tat te average individual i appy to go on olding i average endowment, and not wi to engage in any trade. Ti i exactly wat Roert Luca did in i famou paper on aet price (Econometrica Novemer 978), wic wa part of te foundation of te modern approac to rational expectation in macroeconomic. (c) (i) Invetor expected utility i π [ a C 2 (C ) 2 ] (ii) Terefore te firt-order condition for te optimal coice are π [ a C ] = λ P 2
or Ten Adding over tate, C = a λ P C = a P λ W 0, = a P λ P π (P ) 2 π (P ) 2 Note tat P =, and areviate (P ) 2 /π = Z. Ten and ten λ = ( a W 0, )/Z. C = a a W 0, Z (iii) Adding over invetor and uing te equilirium condition P π π C 0 = a H a H W 0 Z P π (iv) Alo note tat Ĉ = C 0 /H. Ten Add over tate again: Hence a W 0 /H Z a W 0 /H Z P = π [ a Ĉ ] = π [ a Ĉ ] P = π [ a Ĉ ] t π t [ a Ĉt ] To derive a general principle, oerve tat in (), (Ĉ) /2 i te marginal utility of te average conumption (equal average endowment) in tate. In (c), a Ĉ a te ame interpretation. Hence te general idea i tat price of Arrow-Dereu ecuritie will e proportional to te product of te proailitie of te repective tate and te marginal utilitie of conumption in tem, i.e. te expected marginal valuation of wealt in toe tate. Here i ome furter elaoration. Wit a general utility-of-conequence function V, we ave firt-order condition π V (C ) = λ P for invetor optimal coice of final conumption of contingent claim in te different tate. So te price will e proportional to a uitale average of tee marginal utilitie, 3
te average eing taken acro invetor. But te average of te marginal utilitie doe not in general ave to equal te marginal utility of average conumption. Tat require ome pecial aggregation propertie tat appen to old in tee example. If marginal utility i a linear function of conumption a in part (c), te evidently te average of marginal utilitie i te marginal utility of te average. Te cae in () i omewat more complicated, involving aggregation wen preference are identical and omotetic. 4