ECON 00 EXERCISES 4 EXCHNGE ECONOMY 4 Equilibrium in an ecange economy Tere are two consumers and wit te same utility function U ( ) ln H {, } Te aggregate endowment is tat prices sum to Tat is ( p, p) (, ) (,) Consider price vectors normalized so (a) Sow tat if eac consumer purcases bot commodities, market demand for commodity ( ) is ( p) (b) Hence solve for te equilibrium price vector (c) Wat is eac consumer s demand for commodity? (d) Use your answer to sow tat (b) is not te equilibrium price vector if (e) re tere any oter restrictions on tat must be satisfied? E Let ( p, ) be s demand Let p ( ) be te equilibrium value of p wen consumer s endowment is E s equilibrium consumption is terefore ( p ( ), ) Note tat as s endowment approaces zero, er demand must approac zero Tat is, if tere is a equilibrium, E lim ( p ( ), ) 0 0 In equilibrium markets must clear Terefore E lim ( p ( ), ) (,) 0 (f) Use tis observation to sow tat, in te limit, te equilibrium price vector p( ) approaces (, ) Tat is E lim p ( ) (, ) 0 E (g) ONUS: Is it also true tat lim p ( ) (, )?
4 Pareto Efficiency and Walrasian Equilibrium Consumer as utility function U ( ) ( 4)( ) Consumer as utility function U ( ) Te aggregate endowment is (6,38) Consider a PE allocation {, } were 0, H {, } Sow tat te set of suc allocation is a line troug one of te corners of te Edgewort o Caracterize te set of endowments for wic te WE is on tis line Is te WE price ratio te same for all tese endowments? Wat are te oter PE allocations? Eplain and provide a formal derivation by appealing to te K-T conditions Wat are te possible Walrasian Equilibrium price ratios in tis economy? 43 Pareto Efficiency and Walrasian Equilibrium wen consumers ave identical CES preferences Consumer as utility function U ( ) ( ), {, } H and (, ) 0 Te aggregate endowment is (00,00) (a) Solve for te PE allocations {, } were 0, H {, } (b) re tere also PE allocations were consumption by one consumer is not strictly positive? (c) s te endowment of consumer canges, ow does te WE price ratio cange? (d) How do te answers cange if te set of consumers is H {,, C}? 44 Pareto Efficiency and Walrasian Equilibrium Consumer as utility function U ( ) ( )( ) Were (, ) 0 Consumer as utility function U ( ) Te aggregate endowment is
(a) Consider a PE allocation {, } were 0, H {, } Sow tat te set of suc allocation is a line must go troug at least one of te corners of te Edgewort o Under wat condition does it go troug bot corners? Hencefort assume tat (4,) and tat te aggregate endowment is (6,38) (b) Caracterize te set of endowments for wic te WE allocation is in te interior of te Edgewort o Is te WE price ratio te same for all tese endowments? (c) Wat are te oter PE allocations? Eplain and provide a formal derivation by appealing to te K-T conditions (d) Wat are te possible Walrasian Equilibrium price ratios in tis economy? 45 Pareto Efficiency and Walrasian Equilibrium Consumer as utility function U ( ) ( )( ) H {, } were aggregate endowment is 0 Te (a) Define ˆ Ten preferences can be written as U ( ˆ ) ˆ ˆ for aggregate feasibility ˆ ˆ ˆ Confirm tat (b) Consider a PE allocation {, } were, H {, } Sow tat te set of suc allocation is a line (c) For eac suc PE allocation, solve for te supporting price ratio 46 Pareto Efficiency and Walrasian Equilibrium in a linear economy Consumer as utility function U ( ) Consumer as utility function U ( ) Te aggregate endowment is (00,00) (a) Using an Edgewort o diagram eplain wy no PE allocation can lie in te interior of te bo (b) Caracterize te PE allocations and sow tat for all PE allocations sufficiently favorable to consumer, te supporting price ratio p/ p is one of te following (i) (ii) 5 (iii) (c) Wat is te supporting price ratio for PE allocations tat are sufficiently favorable to consumer? (d) For wat endowments is te WE price ratio equal to (i) First Welfare Teorem a WE is Pareto Efficient (ii) Hint: y te
(e) For wat endowments is te WE price ratio neiter wat are te WE allocations? nor In suc cases 47 Pareto Efficient llocations Consumer, H {,, H} as utility function U ( ) ( ) ( ), 0, 0 and (, ) H Te aggregate endowment is For tis question assume tat H, and 0 n allocation is Pareto Efficient if tere is no oter feasible allocation in wic no consumer is worse off and at least one is strictly better off Tus te above two allocations are PE allocations (a) Prove tat te allocation { } H tat maimizes te utility of consumer (ev) given tat consumer (le) must ave a utility of at least is { } H {(0,),(8,9)} Solve also for te allocation { } utility of at least H tat maimizes te utility of le given tat consumer ev must ave a (e precise about te wy te Constraint Qualifications old and wy te necessary conditions are also sufficient) Te first figure below depicts te allocations to le and is indifference curves troug te two allocations
ev s allocation is sown below Tus we can depict ev s allocations and indifference curves as Superimposing te two diagrams we ave te Edgewort o diagram (see EM Capter 3) (b) Sow tat any conve combination of { } H and { } H is also a PE allocation Grapically, te conve combinations are te points on te line joining E and E
(c) re tere any oter PE allocations Eplain using te Edgewort o diagram (d) Consider te PE allocation { ˆ } H for any, Eplain wy { ˆ } H uniquely solves Ma U U U ˆ { ( ) { ( ) ( ), } { } H H (e) Let j be te sadow price associated wit te constraint H j Sow tat j 0 (f) Suppose tat te initial endowments { } H are Pareto Efficient Write down te necessary and sufficient conditions for utility maimization if te price vector is p (g) Compare tese wit te necessary and sufficient conditions for a PE allocation Hence sow tat no consumer will wis to trade at tese prices Since no consumer wises to trade, p is an equilibrium (market clearing) price vector 48 Pareto Efficient llocations* Consider te same economy as te previous question for any H,, (a) Consider te PE allocation { ˆ } H Eplain wy { ˆ } H uniquely solves Ma U U U ˆ { ( ) { ( ) ( ), } { } H H (b) Let j 0 be te sadow price associated wit te constraint H j Sow tat j (c) Suppose tat te initial endowments { } H are Pareto Efficient Write down te necessary and sufficient conditions for utility maimization if te price vector is p (d) Compare tese wit te necessary and sufficient conditions for a PE allocation Hence sow tat no consumer will wis to trade at tese prices Tat is, U( ) U( ˆ ) ˆ (e) Suppose tat te utility function is U ( ) u( ) u( ) were u( ), 0 /
(Tis is te same function if ) Sow tat all te arguments above continue to old 49 Walrasian Equilibrium and Pareto efficiency wen all consumers ave a strictly positive consumption (a) le as utility function U Solve for is demand for commodity if is endowment is (b) ev as utility function U Solve for er demand for commodity if er endowment is (c) Solve for te WE price ratio and sow tat it depends only on te ratio of aggregate endowments (d) n allocation is Pareto efficient if it is not possible to increase te utility of one consumer witout lowering te utility of anoter In tis person economy eplain wy te PE allocation { ˆ, ˆ } must ave te following property ˆ arg Ma{ U ( ) U ( ) U ( ˆ ) } (e) Sow tat te PE allocations satisfy HINT: Wat does efficiency imply about marginal rates of substitution? (f) Is te WE allocation a PE allocation? 40 Walrasian Equilibrium and Pareto efficiency U ( ) and U (7 ) Te endowments are (8,8) and (,) (a) Consider te price vector p (5,) Solve for te demands for te two commodities at tese prices and ence determine weter or not markets clear HINT: e careful wen solving for ev s demand
(b) Wat must be true if tere is a PE allocation ˆ 0? { ˆ, ˆ } in tis economy in wic ˆ 0 and (c) Use your answer to sow tat tere can be no suc PE allocation (d) Try to draw te Edgewort o diagram for tis economy sowing all te PE allocations 4 Edgewort o wit linear utility le as utility function U ( ) a and ev as utility function U ( ) b were Suppose tat a a b b (a) Write down te optimization problem tat must be solved to obtain a PE allocation Hence sow tat for an allocation to be PE it cannot be te case tat 0 and 0 (b) Use te Lagrange metod or a grapical argument to caracterize te set of PE allocations (c) Depict tese in an Edgewort o diagram (d) If le as a large endowment relative to ev and te endowments are not PE, do bot gain from trade in a WE? If so eplain If not, wy not (e) For every PE allocation wat is te no-trade price ratio (f) Hence or oterwise draw a conclusion as to te possible WE price ratios in tis economy as te endowments vary (g) For wat endowments if any do bot consumers consume just one commodity in a WE? 4 Equilibrium and efficiency wit L-saped preferences le and ev ave te following utility functions U ( ) Min{, ), U ( ) Min{, ) Te aggregate endowment is (6,5) Trougout normalize so tat te sum of te prices is (a) Draw te Edgewort o sowing wit dotted lines te kinks in te indifference curves Suppose tat le as an endowment of (,) so tat ev s endowment is (4,4) Eplain wy tis is a PE allocation Wat price ratios are supporting? Tat is, wat are te WE price vectors given tis endowment?
(b) For wat endowments is tis allocation a WE allocation? HINT: Tere are two areas of te Edgewort o (c) Wat are all te PE allocations in te Edgewort o? (d) Wat prices are supporting prices for all te PE allocations oter tan te one in (a)? Etra: (Do not and in) Wat are te possible equilibrium prices if (5,6)? 43 Pareto Efficient llocations U ln ln, U ln ln Suppose / Te aggregate endowment is (, ) (a) Eplain wy te PE allocations in te interior of te Edgewort o must lie below te diagonal (b) Wat condition must old for an allocation to be a PE allocation in te interior of te Edgewort o? (c) ppealing to te ratio rule sow tat at an interior PE allocation ( ) (d) Hence sow tat tat / rises along te map of te PE allocations as increases (e) Let ( p, p ) be a supporting price vector of a PE allocation Normalize so tat te prices sum to ( matematician would say tat te price vector lies on te unit simple) Sow tat p rises along te map of PE allocations as increases (f) Wat is te range of possible PE prices of commodity? (g) Wat is te range of possible WE prices of commodity (for all possible initial endowments)? Eplain () s le s endowment approaces zero (olding constant) wat will be te limiting WE price of commodity? Wat if le s endowment approaces?
44 WE wit different preferences Consumer (le) as utility function U ( ) ln ( )ln and endowment ( aa, ) Consumer (ev) as utility function U ( ) ln ( )ln and endowment ( bb, ) were 0 Trougout normalize by considering price vectors on te unit simple so tat p p (a) Wat is te WE price vector if (i) 0 b a (ii) 0 a b (b) Eplain wy te answer to (i) would be te same if tere were a group of consumers wit te same utility function as le and a total endowment of (, ) Hencefort suppose tat a and b are bot strictly positive (c) Solve for le and ev s demand for commodity (d) Hence solve for te WE price of commodity Hint: Wat is te definition of a WE allocation? (e) Wat is te WE price of commodity? (f) Sow tat if a rises te WE price of commodity rises and te WE price of commodity falls (g) Provide te intuition () Draw an Edgewort bo diagram and carefully depict te indifference curves troug te endowment ( aa, ) (i) Does it follow tat any PE allocation in te interior of te Edgewort bo lies below te diagonal? (j) Does it follow tat any WE allocation wit a 0 and b 0 in te interior of te Edgewort bo lies below te diagonal? aa
5 WLRSIN EQUILIRIUM 5 No trade economy (a) Suppose at te price vector p consumer does not wis to trade Write down te FOC and ence sow tat if utility is differentiable and strictly increasing, ten for some U ( ) p (b) Suppose tat in an endowment economy no individual wises to trade at price vector p Ten for eac demand ( p) and so aggregate demand is equal to total supply Te price vector is terefore market clearing If U ( ) ( ) ( ), 0 and (, ), H sow tat tere is a no trade equilibrium Wat is te market clearing price? (c) Suppose instead tat (, ), and solve for te equilibrium price H Sow tat tere is again a no-trade equilibrium 5 Local Non-satiation utility function satisfies te local non satiation property over te consumer s consumption set X n if for any ˆ X and any 0 tere eists ˆ N( ˆ, ) suc tat U( ˆ) U( ˆ) n Define e (,,) Te set H { ˆ e ˆ e} is a ypercube wit center ˆ (a) H N( ˆ, ) is a square Depict te square and te neigborood N ( ˆ, ) Eplain wy H (b) For all n sow tat n N( ˆ, ) H Suppose tat pˆ I Prove tat for all sufficiently small 0, N( ˆ, ) p I (c) Use (b) to sow tat if utility satisfies te LNS property, ˆ arg Ma{ U( ) p I} 0 53 Price taking consumers Consumer H {,, H} as continuous utility function U ( ) defined on endowment Te LNS property olds on n n and Te price vector is p 0 Suppose tat
arg Ma{ U ( ) p p } (a) Use your answer to eercise 3 to eplain wy p p (b) Eplain wy U ( ) U ( ) p p (c) Prove tat U ( ) U ( ) p p (d) Let { } H be an allocation tat is Pareto Preferred to { } and at least one consumer strictly better off) Use your answers to sow tat H (No consumer worse off H p H p (e) ppealing to (a) and (d) H p ( ) p ( ( ) 0 H Eplain wy it follows tat te Pareto preferred allocation { } feasibility constraint H does not satisfy te H H 54 Second Welfare Teorem in an ecange economy Consumer, as a strictly increasing quasi-concave utility function U ( ) and an n endowment (a) If te price vector is Confirm tat te Kun-Tucker conditions for preferred coice can be written as follows to be a most
U ( U ) 0 and ( ( ) ) 0 (b) Te Kun-Tucker conditions are sufficient for a maimum if certain conditions are satisfied Confirm tat tese conditions are satisfied (c) Sow tat te following conditions are necessary conditions for efficient { ˆ } to be Pareto U ( ˆ ) 0 and ˆ U ( ( ˆ ) ) 0, were (d) ppeal to (a) and (b) to sow tat te allocation { } { ˆ } { ˆ } is a WE if te endowment is (e) To acieve tis final allocation given some oter endowments { } ta/subsidy is needed?, wat lump sum 55 Equilibrium and efficiency Individual,,, H as a utility function U( ) jv ( ) (Note tat te utility functions may be different) Te aggregate endowment of commodity and of commodity is k (a) Sow tat for Pareto efficiency (b) Solve for te WE price ratio on tis economy and sow tat it is independent of individual endowments 3 3 (c) If (, ) (, ) confirm tat te price vector p (, ) is a WE price vector 4 4 3 (d) In a person economy, if (5,), (5,) and p (, ) to be subsidized for te final allocation to be equal? j 4 4 4 4, ow muc would le need 56 Walrasian equilibrium
consumer as a consumption set X {(, ) (, ) (,)} and utility function U( ) ln( ) ln( ) His endowment is (a) If te price vector is condition p ( ) p( ) p sow tat is consumption bundle must satisfy te following ppealing to te ratio rule it follows tat p ( ) p ( ) p p p (b) Hence sow tat demand is p ( p) (( ) ( )) p (c) Suppose tat tere are two consumers le as utility function U( ) 7ln( ) ln( ) and endowment (9, ) were [ 9,] ev as utility function U( ) ln( ) ln( ) and endowment (,9 ) 7 3 p Sow tat te aggregate demand function is ( p) 0 ( ) 4 p Hence sow tat p (,) is a WE price vector for all (d) Depict te WE in an Edgewort o diagram (e) Note tat te aggregate demand for commodity is a decreasing function of and an increasing function if 0 Wat can you say about WE if 0? p if 0 57 Walrasian Tatonnement consumer as a strictly increasing and strictly concave utility function U and consumption set X ( a) { a} ( p, I, a) arg Ma{ U( a) p I} X ( a)
Note tat tat tere is a solution if and only I pa 0 Note also tat if a = 0, ( p, I,0) arg Ma{ U( ) p I} (a) Sow tat if I pa 0 ten ( p, I, a) a ( p, I p a,0) HINT: Define te new vectors X a and a and rewrite te maimization problem in terms of X and (b) Hence sow tat if U( ) ln ln and I pa 0 ten I p a I p a ( p, I, a) a ( ) and ( p, I, a) a ( ) p p le as a utility function U ( ) ln( 0) ln( 0) and endowment 5 ( ) (30,0 ) Te aggregate endowment is (40,40) ev as a utility U ( ) 5ln( 0) ln( 0) (c) Solve for teir demands for commodity if 0 (ppeal to (b)) Define z( p, ) to be ecess market demand wit parameter 0 p (d) Hence sow tat z( p,0) ( ) 6 p (e) Discuss ow te tatonnement (trial and error) process migt work (f) If 7 sow tat for te intersection of eac consumer s budget set and consumption set p to be nonempty, te price ratio lies in some interval P { p } p (g) Solve for z( p, ) if 7 and p P () Depict te market ecess demand function z( p,7) for all price ratios suc tat p P You sould put te price ratio p/ p on te vertical ais and z on te orizontal ais (i) Comment on te tatonnement process in tis case EXTR CREDIT:
(j) Finally solve for z( p, ) Remark: Te result is striking 5 WLRSIN EQUILIRIUM 46 Two person economy wit production if 5and caracterize any equilibria in tis case Commodity is leisure/labor and commodity is food Eac consumer as an endowment of 6 units of leisure Tere is a single price taking firm wit production set Y z y y y z Y {(, ) 0, 6 0} {(, ) 0, 6 }, equivalently y y y y y le and ev eac ave te same Cobb-Douglas utility function U ( ), Eac owns as a 50% sare of te firm (a) Sow tat preferences are omotetic and ten analyze tis as a representative agent model Sow tat eac as a WE allocation (3,8) Note tat We net modify te production set as follows Y {( z, y ) y 6 z, 0 z, y ( z ), z }, equivalently p is te wage Y {( y, y ) y 6y 0, y 0, y ( y ), y } (b) Initially assume tat Depict te firm s production set (c) Wat is te firm s supply correspondence? (d) Continue to use te representative agent model to solve for te WE allocation HINT: Normalize by setting te wage p Since food is more scarce wit te new production set, te WE price of food must rise If p te firm will use units of input to produce units of output Te aggregate consumption vector is terefore (0,) Depict tis in a neat figure Eplain wy te WE price of food must be p 5 / 6 (c) Wat is te WE profit? (d) Suppose tat le as a 00% ownersip of te firm t te WE prices from (b) sow tat le s demand for leisure is not feasible
(Te representative consumer approac ignores te upper bound on leisure so works only if te solution to te relaed problem satisfies tis constraint ) (e) Wat ten would be te new WE price of food and te equilibrium allocations? HINT: If le is not working and ev as no dividend, wat is te aggregate supply of labor z( p, p ) ONUS (a) How does te answer to (e) cange if 5 TIME 5 Coice over time 0? Commodity t is period t consumption Tere are two periods le and ev ave linear preferences ot discount te future but ev discounts te future more U, U Te aggregate endowment is (0,30) 3 (a) Depict te PE allocations in an Edgewort o diagram (b) Wat is te unique supporting price ratio p/ p for a PE allocations tat are close to (i) O? (ii) O? (c) Consumers can lend and borrow at te interest rate of r Eplain wy te life-time budget constraint of consumer can be written as follows r r (d) Wat is te WE price ratio and ence interest rate if te endowment is close to (i) O? O? (ii) Wat is te WE price ratio and ence interest rate if te endowments are (,3) and (8, 7)? 5 Economy wit production
Commodity and are apple and coconut consumption in period wile commodity 3 and 4 are apple and coconut consumption in period Te period utility is utility is u(, ) 3 4 Te aggregate endowment is Te discount factor is so lifetime utility is U( ) u(, ) u(, ) 3 4 Suppose tat all consumers ave te same omotetic utility function U( ) ln 4ln (ln 4ln ) (,80,30,90), 3 4 u(, ) 3/ 4 and te period Coconuts can be stored but not planted pples can be planted Eac apple planted in period bears apples in period (a) Solve for te WE of tis economy if te period price of apples is Hencefort suppose tat 40 0 and You sould normalize so tat (b) Solve for te efficient number of apples planted for all (c) Solve for te WE of tis economy if 8 You sould normalize so tat te period price of apples is (d) Continuing wit te data of part (c), suppose tat tere are no futures markets Consumers can, owever borrow or lend at an interest rate Solve for te WE spot prices of apples and coconuts and te period WE spot prices of tese commodities r = 0 ( ) r = () i ii 53 Time and a production function s in te previous question, we define 3 to be te period consumption of commodity and 4 to be te period consumption of commodity Tere is no endowment of commodity 4 It is produced using commodity and commodity as inputs Let z ( z, z) be te period input vector Output is produced according to te constant returns to scale production function 8z z / / 4 ll consumers ave te same omotetic utility function u ( ) 4ln ln (4ln ln ) 3 4 3 4
Te aggregate endowment is (9,44,4,0) (a) Sow tat it is efficient to produce 48 units of commodity 4 (Note tat (b) Solve for a WE price vector z z ( zz) ) / / / Hencefort suppose tat z z were 4 0 (c) Discuss briefly (i) ow to solve for te efficient production plan and (ii) weter tere is a WE price vector for all 0
6 UNCERTINTY 6 Uncertainty on a Sout Pacific Island Witout a volcanic eruption le would ave a plantation wit 500 coconut palm trees and ev 00 However every year a volcano erupts and one or te oter plantations suffers damage State is te event tat le s plantation is damaged and state is te event tat ev s plantation is damaged Tus le as a state contingent endowment of (00,500) wile ev as a state contingent endowment of (00,700) Te probability of state is ¼ and te probability of state is ¾ le and ev ave VNM utility functions (a) Wat are te PE allocation in tis economy? (ppeal to your answer to question ) (b) If le and ev can trade in state claims markets (ie insurance markets) wat is te ratio of te WE state claims prices? (gain appeal to your answer to question ) (c) Depict te WE and te PE allocations in an Edgewort-o diagram (d) Wat would be te total value of eac plantation if te sum of te WE prices is? (e) Suppose tat instead of trading in state claims, le sells two tirds of is plantation to le Wat fraction of ev s plantation could e purcase? (Use te prices computed in (d)) (f) Given te new ownersip sares wat would be le s total dividend in eac state? 6 More uncertainty on a Sout Pacific Island
Witout a volcanic eruption le would ave a plantation wit 5 coconut palm trees and ev However every year a volcano erupts and one or te oter plantations suffers damage State is te event tat le s plantation is damaged and state is te event tat ev s plantation is damaged Tus le as a state contingent endowment of (,5) wile ev as a state contingent endowment of (,7) Te probability of state is ¼ and te probability of state is ¾ le and ev ave VNM utility functions (a) Wat are te PE allocations in tis economy? (ppeal to your answer to question ) (b) If le and ev can trade in state claims markets (ie insurance markets) sow tat p (/ 4,3/ 4) is a WE state claims price vector (gain appeal to your answer to question ) (c) Depict te WE and te PE allocations in an Edgewort-o diagram (d) Depict le s budget set in a neat figure and mark in is maimum feasible consumption of state claims and state claims (d) Sow tat ev s plantation is twice as valuable as le s at te WE prices (e) Suppose tat instead of trading in state claims le sells part of is plantation and purcases part of ev s plantation Let te asset prices be P (,) Let (, ) be le s asset oldings Since tere is one unit of eac plantation le as an endowment (, ) (,0) His portfolio constraint is terefore P P Terefore ( ) (f) Let D be te matri of dividends, tat is D 5 7 Ten wit portfolio olding (, ) is final consumption is D 5 7 5 7 ( ) Write out epression for and as functions of Eliminate and so obtain te feasible set of final consumption bundles wit asset trading Compare tis wit your answer to (d)
Remark: Te point of tis eample is tat if tere are many as linearly independent dividend vectors as states, ten trade in asset markets can perfectly substitute for trade in state claims markets 63 cceptable gambles Tere are two states Carles is a risk-averse consumer wit wealt w If e cooses to gamble e gains s in state s (Of course tis may be negative or positive) Te probability of state s is gamble (, ) is just acceptable to Carles if s v ( w ) v ( w ) v ( w) C C C (a) Suppose tat v( ) e Sow tat te set of acceptable gambles is independent of te consumer s wealt Define v v ( ) C w and v v ( ) C w and v v v vc ( w) (b) Suppose tat Denis as a concave utility function v ( w ) f ( v ( w )) were f is strictly increasing and strictly concave Eplain carefully wy D C v ( w ) v ( w ) f ( v ) f ( v ) f ( v) v ( w) D D D Tus a gamble just acceptable to Carles is not acceptable to Denis For every we can map out v ( w ) and v ( w ) as depicted below C D
Note tat in te figure te implied mapping f: from concave v C to v D is strictly increasing and strictly v ( w ) f ( v ( w )) D C (c) Differentiate wit respect to, ten take te logaritm of bot sides and differentiate again and ence sow tat v v f (*) D C v f D v C Te ratio of ( w ) v( w ) / v( w ) is called te degree of absolute risk aversion So we ave sown tat f D( ) C( ) f Ten if Denis as an everywere iger degree of absolute risk aversion, te implied mapping f is strictly concave and so is set of acceptable gambles is smaller Suppose tat te only difference between Denis and Carles is tat Denis as a wealt wd w Ten v ( w ) v ( w ) D D C Suppose tat Carles is indifferent between ( wwand, ) te gamble ( w, w ) Note tat v D ( wd ) v C ( w ) D ( wd ) C ( w ) v ( w ) v ( w ) D D C Tus if absolute aversion to risk is independent of wealt, it follows from (*) tat Denis will also be indifferent Typically owever, more wealty individuals are more willing to accept a gamble of fied size Tus economists usually assume tat absolute aversion to risk is lower for individuals wit a iger wealt Ten Denis will strictly prefer te gamble ( w, w ) Tus te set of acceptable gambles rises wit wealt 64 Purcasing insurance
In state a consumer wit wealt a as a loss L Te probability of te loss state is He can purcase claims in te loss state for claims in te no loss state at te ecange rate p/ p so tat is budget constraint is p p p ( a L) p a (a) If insurance is fair (zero epected profit) eplain wy ( a L) a (b) If tis is te case sow tat te consumer will purcase fill coverage, tat is ( a L) a (b) Suppose tat it costs more to purcase insurance so tat te optimal coice * * as depicted below Sow tat at te optimum, * satisfies v( ) * * * * * v( ) p M ( ) MRS(, ) p
second individual as wealt a b and faces te same loss If e purcases te same amount of coverage is final consumption is M ( b ) v( b ) ( ) * * * v b b * and so is MRS is Note tat if M b M te coice * * ( ) ( ) * b is optimal (c) Define m b M b Use te rules of logaritms to get a simple epression for * * ( ) ln ( ) m and ten differentiate by b and sow tat m b b b * ( ) ( ) ( ) Under te assumption of decreasing absolute risk aversion tis is negative for all Terefore * * * * * * m( b ) m( ) and so M( b ) M ( ) b 0 since (d) Wat does tis imply about te difference between te slope of te indifference curve at ( b, b ) and te slope of te budget line * * (e) Does it follow tat te more wealty individual will purcase more or less insurance? 65 Insurance market equilibrium le is risk averse wit VNM utility function v( ) ln ev is risk neutral Tere are equally likely states Te aggregate endowment is (00,50) (a) Sow tat if le as an endowment (60,0) te WE state claims price ratio will be and tat le as a WE allocation (40,40) Tus te risk neutral ev bears all te risk (b) Wat is te WE price ratio and allocation if le as an endowment (0,0) Does ev bear all te risk? (c) Depict bot equilibria in an Edgewort-o diagram More generally, suppose le and ev are bot risk averse wit individual endowments, and te probability vector is (, ) Te aggregate endowment is (, ) were
(d) Suppose tat claims p p Eplain carefully wy bot consumers will demand more state (e) Does it follow tat for a WE p p? Eplain