Advanced Microeconomics(ECH 6) Homeork --- Soutions Expected Utiity Teory On p Jee and Reny say tat AXIOM G4 (Monotonicity) impies a an Prove tis We prove tis by contradiction Suppose a an, ten a a n and a n a Furtermore, suppose e take α > β Ten by MON ( α a,( α) a n ) ( β a,( β) a n ) but aso ( α an,( α) a) ( β an,( β) a ) Te atter can be rearranged (( α) a, α a n ) (( β) a, β a n ) ic by MON impies α β ic impies β α ic contradicts te assumption Wic of te fooing utiity functions ave te DARA property, tat is, te Arro-Pratt measure of absoute risk aversion R is decreasing in eat (i) u( ) = n (ii) u( ) = a( b) ; a, b > (iii) u( ) = e (iv) u( ) = a α ; a >, α > (v) Discuss y or y not DARA is a convincing property a u( ) u u R = u / u DARA u( ) = n( ) ; > ( ) ( ) u = a b ; a > a( b) a u( ) = e u( ) = a α ; a >, α > e α a α b e a Yes No for b No ( α ) α a α ( α ) Yes for α <, ese no (v) A decision maker for om DARA is vioated oud be iing to pay iger risk premiums te ricer e gets Tat is not very convincing
A sports fan as an expected utiity function u( ) = n Se as subjective probabiity p tat er favourite cub Ajax i in te next matc and probabiity p tat tey i not in Se cooses to bet x on Ajax suc tat if Ajax ins, se ins x, but if Ajax does not in se oses x Te fan s initia eat is (a) Determine te fan s degree of absoute and reative risk aversion (b) Ho can you determine te fan s subjective odds p / ( p) by observing te size of is bet? (c) Ho muc oud se bet if p = 5 and = 9? No suppose for te next game te fan as received a betting ticket from a friend as gift If Ajax ins, se receives x, but if Ajax does not in, er oss is aready paid for Tis time p = / (d) For o muc oud te fan se er betting ticket depending on x and? (e) Cacuate te subjective money vaue of te gift en = 9 and x = a) Te agent is risk averse R = ; R = a r p b) A risk neutra payer oud bid any amount if > A risk averse agent baances trust and risk aversion Hence te agent maximizes p + x V = pu( + x) + ( p) u( x) From te first order conditions e obtain = p p x 9 c) Using te atter formua, e can use te numbers to obtain x = 5 d) No e need to cacuate te certainty equivaent vaue c of te vaue of te ottery ticket t en eventua osses are paid for c = n( + t) = p n( + x) + ( p)n( ) Using p = / e get n( + t) = (n( + x) + n( )) n( + t) = n( + x) ( + t) = + x Soving for t gives t = + + x e) Using te numbers e can cacuate t = 9 + 8+ 8 95 Vaue of Information 4 Consider a cemica Its use gives a benefit B( x) = x β, ere x is te eve of exposure, β ( < β < ) is a substance specific parameter In addition te use of te substance gives a damage D( x, τ ) = xτ tat depends on exposure and te substance s toxic potentia τ Suppose, troug adequate measures te reguator can contro exposure
(a) Ho soud te reguator set exposure of a substance it knon toxic potentia? (b) No suppose te toxic potentia is unknon but knon to be eiter o τ or ig τ it probabiities p and p, respectivey Ho soud exposure be reguated? (c) Consider case (b) and assume τ =, τ =, β = 5, and p = / Cacuate optima exposure and te expected efare (benefits minus damage cost) (d) Using te specification in (c), at is te reguators iingness to pay for a test tat provides perfect information about te substance s toxic potentia? (a) Equate te margina benefit and te margina cost: x = τ ( β ) β (b) No e equate margina benefit it te expected margina damage β β x ( p ( ) ) p β = τ + τ and sove for x (c) Using te specification e obtain x = = Te associated expected 5 exp ( ) 6 efare is B( xexp ) D( xexp, τ ) = 8 (d) If te substance ere knon to be τ =, ten te optima emission eve is x = 5 and te associated efare is B( x ) D( x, τ ) = 5 If te substance ere knon to be τ =, ten te optima emission eve is x = 5 and te associated efare is B( x ) D( x, τ ) = 5 So, if e can act on perfect information e get eiter 5 or 5 Te former situation oud occur it p = / and te atter it p = / Te resuting expected vaue is 675 Te expected vaue of acting itout furter information, ie acting on prior beiefs as 8 as cacuated before Te vaue of information is tus VOI = 675 8 = 457 5 Let A = { a, a, a} ere a a a A gambe g offers a it certainty Prove tat if a ( α a,( α ) a ), ten α must be stricty beteen zero and one Proof: By Continuity e kno tat α [,] Suppose α = Ten a ( a, a) = a ic contradicts te assumption No suppose suppose α = Ten a ( a, a) = a ic aso contradicts te assumption Hence α (,) Games it incompete Information 6 Te utimatum game Consider te fooing to-payer game Payer receives x coins of and x is a stricty positive even number Payer offers none or a positive number of tese coins to payer If payer rejects te offer, a te money is itdran If payer accepts te offer, se receives at is offered to er and payer keeps te rest (a) Describe in a forma ay te strategy spaces of te payers and dra a game tree
(b) Wat type of game is tis and at is an appropriate soution concept? In experiments many peope reject stricty positive offers in te utimatum game if tese are considered unfair Suppose of te popuation as strong fairness preferences suc tat if tey ere in te roe of payer, tey oud reject any offer ic is ort ess tan x (c) Wat type of game is tis and at is an appropriate soution concept? Dra te game tree (Hint: Harsanyi introduced a metod to represent tis kind a game) (d) Derive te equiibrium (e) Determine te tresod fraction of fair payers in te popuation suc tat ony fair offers occur in equiibrium Te utimatum game is a famous game from te eary days of experimenta economics (W Güt, R Scmittberger, B Scarze (98) An Experimenta Anaysis of Utimatum Bargaining Journa of Economic Beavior and Organisation, 67-88) a) strategy space of payer is {,,, x } and for payer {accept, reject} b) Sequentia perfect information game, Soution concept: Subgame Perfect Nas Equiibrium c) Sequentia incompete information game Sequentia Equiibrium Harsanyi introduced te idea to represent te uncertainty about te type of te payers it a move by nature So it probabiity / payer is of type fair and it probabiity / te payer is of type rationa In te game tree Nature moves first coosing fair or rationa Tis eads to to nodes in te same information set of payer o determines te offer from {,,, x } Payer, fair or rationa and knoing er on type sees te offer and accepts or rejects d) first note tat never offers more tan x/ If te type of ere knon, oud offer x/ to a fair payer and to a rationa payer (o stricty prefers over noting and oud accept) Hence offers eiter or x/ Te former yieds + ( x ), te atter yieds x/ x regardess of type Hence, pay x/ if ( ) 4 x x Ese pay e) Generay if p is s beief tat payer is fair, ten offering yieds p + ( p) ( x ) and offering x/ yieds x Te condition for offering x/ is ( p) ( x ) p x x ( x ) 7 A principe-agent game: Tere are to agents: a risk neutra and oner and a risk averse farmer Harvest is subject to risk Te risk is impacted be te farmer s effort Te and oner typicay cannot observe te farmer s effort Ony arvest is observabe a) Argue y or y not te and oner soud pay a fixed age to te farmer Assume no tat effort e is eiter o e = or ig e = Wit o effort arvest is eiter o x = 5 it probabiity or ig x = 6 it probabiity Wit ig effort arvest is 4
o it probabiity or ig it probabiity Furtermore assume te farmer can earn an off-farm age = 5 it effort e = 5 Te andoner maximises profits (assume te price of te crop is ) Te farmer s utiity function is u(, e) = e b) Give a forma description of te and oner s maximisation probem Wat are te reevant constraints te and oner faces en offering a contract to te farmer? Describe te contract tat te and oner offers to te farmer c) Wi tis contract be accepted? Cacuate te and oner s profit if te contract is accepted a) Argue y or y not te and oner soud pay a fixed age to te farmer Wen te andoner pays a fixed age e bears a te risk ic is te efficient risk aocation Hoever, in tat case, tere is no incentive for te farmer to ork ard Effort i be o and te andoner receives oer returns from te and tan it an incentive contract Assume no tat effort e is eiter o 5 or ig 6 x = it probabiity e = or ig e = Wit o effort arvest is eiter o x = it probabiity Wit ig effort arvest is o it probabiity or ig it probabiity Furtermore assume te farmer can earn an off-farm age = 5 it effort e = 5 Te andoner maximises profits (assume te price of te crop is ) Te farmer s utiity function is u(, e) = e b) Give a forma description of te and oner s maximisation probem Wat are te reevant constraints te and oner faces en offering a contract to te farmer? Describe te contract tat te and oner offers to te farmer Note tat te reservation utiity is 5 Wit o effort of te agent te andoner i ave to pay at east = 5 Te returns are 5 + 6 =, so te and oner i make a profit of 5 en te agent s effort is o Next e examine te and oner s profits from ig effort He maximises return from ig effort minus age subject to a participation and an incentive constraint max 5 + 6, s t PC : + 5 IC : + + Reriting te constraints e obtain te fooing Lagrangian function L = 5 + 6 λ(8 ) µ ( + ) From te first order necessary conditions e obtain: L () = = λ µ 5
L () = = λ + µ Since λ, µ by definition, and e require λ(8 ) = µ ( + ) =, at east IC must be binding If e rue out negative ages, PC must aso be binding Wit binding constraints e cacuate = 6; = 49; and expected profits are 7 c) Wi tis contract be accepted? Cacuate te and oner s profit if te contract is accepted Te contract i be accepted because te IC ods It is better tan a fixed age contract for o effort ic yieds profit 5 + 6 = 5 = 5 8 Market for emons Consider te fooing market for used cars Tere are many seers of used cars Eac seer as exacty one used car to se and is caracterised by te quaity of te used car e ises to se Let θ [,] index te quaity of a used car and assume tat θ is uniformy distributed on [, ] If a seer of type θ ses is car (of quaity θ) for a price p, is utiity is u ( p, θ ) If e does not se is car, ten is utiity is Buyers of used cars receive utiity s θ p if tey buy a car of quaity θ at price p and receive utiity if tey do not purcase a car Tere is asymmetric information regarding te quaity of used cars Seers kno te quaity of te car tey are seing, but buyers do not kno its quaity Assume tat tere are not enoug cars to suppy a potentia k buyers a) Argue tat in a competitive equiibrium under asymmetric information, e must ave E(θ p) = p b) So tat if us ( p, ) p θ θ =, ten every p [, ] is an equiibrium price c) Find te equiibrium price en us ( p, θ ) = p θ Describe te equiibrium in ords In particuar, ic cars are traded in equiibrium? d) Find an equiibrium price en us ( p, θ ) = p θ Ho many equiibria are tere in tis case? (a) Suppose not Coud demand for used cars equa suppy? Tere are to cases to consider If E( θ p) < p, ten te conditiona average quaity of cars on te market is beo te price and buyers receive negative expected utiity So tere is no demand and te market cannot cear And if E( θ p) > p ten buyers receive positive expected utiity and a potentia buyers oud ike to buy but not a seers oud ike to se (tose it ig quaity car i not se as te price is ess tan average quaity) So tere is excess demand Te equiibrium outcome is driven by positive seection: a iger price increases te average quaity of te cars avaiabe on te market So tere are potentiay many equiibria soving te pricing condition E( θ p) = p Tere migt exist a ig price equiibrium ere seers put teir ig quaity cars on te market and buyers are iing to pay te ig price Tere migt aso exist a o price equiibrium ere te seers remove te best cars from te market and buyers 6
are ony iing to pay a o price (b) Fixing te price p, if us ( θ p) = p θ, a seers it quaity θ p i prefer seing teir car to not seing Tus te average quaity of cars conditiona on price p is given by E( θ p) = min(, ) = p p ic is aso te competitive equiibrium outcome Tus any price p [, ] is an equiibrium it ony cars of quaity θ p traded, so tat te average quaity is just equa to te price (c) Wen u ( θ p) = p θ, ony seers it quaity s quaity tresod θ p i prefer to se Tus te θ * = p ic yieds te conditiona average quaity of cars p E( θ p) = min(, ) = p and p = is te ony possibe soution (d) Simiar to cases (b) and (c) e ave / p E( θ p) = min(, ) = p Suc tat e obtain p = 8 Anoter equiibrium is p = Ceary tere is market faiure because ig quaity cars do not se Pareto improvements are not possibe uness e cange te game One possibiity is tat seers give guarantees 7