Microeconomic Theory -1- Uncertainty Choice under uncertainty A Introduction to choice under uncertainty B Risk aversion 11 C Favorable gambles 15 D Measures of risk aversion 0 E Insurance 6 F Small favorable gambles (017 skip) 8 G Portfolio choice 33 H Efficient risk sharing 43 55 slides
Microeconomic Theory -- Uncertainty A Introduction to choice under uncertainty (two states) Let X be a closed bounded set of possible outcomes ( states of the world ) An element of X might be a consumption vector, health status, inches of rainfall etc Initially, simply think of each element of X as a consumption bundle Let x element of X and let x be the least preferred element be the most preferred **
Microeconomic Theory -3- Uncertainty A Introduction to choice under uncertainty (two states) Let X be a closed bounded set of possible outcomes ( states of the world ) An element of X might be a consumption vector, health status, inches of rainfall etc Initially, simply think of each element of X as a consumption bundle Let x element of X and let x be the least preferred element be the most preferred Consumption prospects Suppose that there are only two states of the world X { x1, x} Let 1 be the probability that the state is 1 is the probability that the state is x 1 so that 1 x We write this consumption prospect as follows: * ( x; ) ( x, x ;, ) 1 1
Microeconomic Theory -4- Uncertainty A Introduction to choice under uncertainty (two states) Let X be a closed bounded set of possible outcomes ( states of the world ) An element of X might be a consumption vector, health status, inches of rainfall etc Initially, simply think of each element of X as a consumption bundle Let x element of X and let x be the least preferred element be the most preferred Consumption prospects Suppose that there are only two states of the world X { x1, x} Let 1 be the probability that the state is 1 is the probability that the state is x 1 so that 1 x We write this consumption prospect as follows: ( x; ) ( x, x ;, ) 1 1 If we make the usual assumptions about preferences, but now on prospects, it follows that preferences over prospects can be represented by a continuous utility function U( x1, x, 1, )
Microeconomic Theory -5- Uncertainty Prospect or Lottery L ( x, x,, x ;,, ) 1 S 1 (outcomes; probabilities) S Consider two prospects or lotteries, L A and L B L x x x L ( c, c,, c ;,, ) A A B B A ( 1,,, S; 1,, S ) B 1 S 1 S Independence Axiom (axiom of complex gambles) Suppose that a consumer is indifferent between these two prospects (we write L A L B ) Then for any probabilities 1 and ( L, L ;, ) ( L, L ;, ) A C 1 B C 1 Tree representation summing to 1 and any other lottery L C
Microeconomic Theory -6- Uncertainty This axiom can be used to prove the following Implication Suppose that a consumer is indifferent between the prospects L A and L B and is also indifferent between the two prospects ie L A L B and LC L D L C and L D, Then for any probabilities 1 and summing to 1, ( L, L ;, ) ( L, L ;, ) A C 1 B D 1 Tree representation
Microeconomic Theory -7- Uncertainty Expected utility Consider some great outcome x and bad outcome x and outcomes x 1 and x satisfying x x1 x and x x x Reference lottery R( v) ( w, w, v,1 v) so v R(0) x1 R (1) and R(0) x R (1) Then for some probabilities vx ( 1) and vx ( ) is the probability of the great outcome x R( v( x )) ( x, x; v( x ),1 v( x )) and x R( v( x)) ( x, x; v( x),1 v( x)) Then by the independence axiom ( x, x ;, ) ( R( v( x )), R( v( x ));, ) 1 1 1 1
Microeconomic Theory -8- Uncertainty We have just argued that by the independence axiom ( x, x ;, ) ( R( v( x )), R( v( x ));, ) 1 1 1 1 Define [ v] 1v( x1 ) v( x) Class exercise Show that ( R( v( x1 )), R( v( x)); 1, ) ( x, x; [ v],1 [ v]) Hence ( x1, x; 1, ) ( x, x; [ v],1 [ v]) Thus the expected win probability in the reference lottery is a representation of preferences over prospects
Microeconomic Theory -9- Uncertainty An example: A consumer with wealth ŵ is offered a fair gamble With probability and with probability 1 his wealth will be ŵ x that this is equivalent to a prospect with x 0 In prospect notation the two alternatives are ( w, w ;, ) ( wˆ, wˆ;, ) 1 1 1 his wealth will be ŵ x If he rejects the gamble his wealth remains ŵ Note and ( w, w ;, ) ( wˆ x, wˆ x;, ) 1 1 These are depicted in the figure assuming x 0 set of acceptable gambles Expected utility U( w, w,, ) [ v] v( w ) v( w ) 1 1 Class discussion MRS if vw ( ) is a concave function
Microeconomic Theory -10- Uncertainty Convex preferences The two prospects are depicted opposite The level set for U( w, w ;, ) through the riskless prospect N is depicted 1 set of acceptable gambles Note that the superlevel set U( w, w ;, ) U( wˆ, wˆ;, ) is a convex set 1 *
Microeconomic Theory -11- Uncertainty Convex preferences The two prospects are depicted opposite The level set for U( w, w ;, ) through the riskless prospect N is depicted 1 set of acceptable gambles Note that the superlevel set U( w, w ;, ) U( wˆ, wˆ;, ) is a convex set 1 This is the set of acceptable gambles for the consumer As depicted the consumer strictly prefers the riskless prospect N to the risky prospect R Most individuals, when offered such a gamble (say over $5) will not take this gamble
Microeconomic Theory -1- Uncertainty B Risk aversion Class Discussion: Which alternative would you choose? N : ( w1, w ˆ ˆ ; 1, ) ( w, w; 1, ) 50 ( w, w ;, ) ( wˆ x, wˆ x;, ) where 1 100 R : 1 What if the gamble were favorable rather than fair 55 60 75 ( w, w ;, ) ( wˆ x, wˆ x;, ) where (i) 1 (ii) 1 (iii) 1 100 100 100 R : 1 *
Microeconomic Theory -13- Uncertainty Class Discussion: Which alternative would you choose? N (, ;, ) ( ˆ, ˆ;, ) : w1 w 1 w w 1 50 ( w, w ;, ) ( wˆ x, wˆ x;, ) where 1 100 R : 1 What if the gamble were favorable rather than fair 55 60 75 R : ( w ˆ ˆ 1, w ; 1, ) ( w x, w x; 1, ) where (i) 1 (ii) 1 (iii) 1 100 100 100 What is the smallest integer n n such that you would gamble if 1? 100 Preference elicitation In an attempt to elicit your preferences write down your number n (and your first name) on a piece of paper The two participants with the lowest number n will be given the riskless opportunity n1 Let the three lowest integers be n1, n, n 3 The win probability will not be n3 the higher win probability 100 100 n or Both will get 100
Microeconomic Theory -14- Uncertainty B Risk preferences U( x, ) v( x ) v( x ) or U( x, ) [ v] Risk preferring consumer Consider the two wealth levels x x 1 and 1 x v( x x ) v( x ) v( x ) If vx ( ) is convex, then the slope of vx ( ) is strictly increasing as shown in the top figure Consumer prefers risk
Microeconomic Theory -15- Uncertainty U( x, ) v( x ) v( x ) Risk averse consumer v( x x ) v( x ) v( x ) In the lower figure ux () is strictly concave so that v( x x ) v( x ) v( x ) [ v] In practice consumers exhibit aversion to such a risk Thus we will (almost) always assume that the expected utility function vx ( ) is a strictly increasing strictly concave function Consumer prefers risk Class Discussion: If consumers are risk averse why do they go to Las Vegas? Risk averse consumer
Microeconomic Theory -16- Uncertainty C Favorable gambles: Improving the odds to make the gamble just acceptable New risky alternative: ( w, w ;, ) ( wˆ x, wˆ x;, ) 1 1 Choose so that the consumer is indifferent between gambling and not gambling ****
Microeconomic Theory -17- Uncertainty C Favorable gambles: Improving the odds to make the gamble just acceptable New risky alternative: ( w, w ;, ) ( wˆ x, wˆ x;, ) 1 1 Choose so that the consumer is indifferent between gambling and not gambling For small x we can use the quadratic approximation of the utility function v ( ˆ ) ( ˆ ) ( ˆ ) ( ˆ ) a w x v w v w x v w x 1 Class Exercise: Confirm that the value and the first two derivatives of equal at x 0 v( wˆ x) and v ( ˆ ) a w x are ***
Microeconomic Theory -18- Uncertainty C Favorable gamble: Improving the odds to make the gamble just acceptable New risky alternative: ( w, w ;, ) ( wˆ x, wˆ x;, ) 1 1 Choose so that the consumer is indifferent between gambling and not gambling For small x we can use the quadratic approximation of the utility function v ( ˆ ) ( ˆ ) ( ˆ ) ( ˆ ) a w x v w v w x v w x 1 Class Exercise: Confirm that the value and the first two derivatives of equal at x 0 v( wˆ x) and v ( ˆ ) a w x are The expected value utility of the risky alternative is ** ( ) v( wˆ x) ( ) v( wˆ x) ( ) v ( wˆ x) ( ) v ( wˆ x) a a
Microeconomic Theory -19- Uncertainty C Favorable gamble: Improving the odds to make the gamble just acceptable New risky alternative: ( w, w ;, ) ( wˆ x, wˆ x;, ) 1 1 Choose so that the consumer is indifferent between gambling and not gambling For small x we can use the quadratic approximation of the utility function v ( ˆ ) ( ˆ ) ( ˆ ) ( ˆ ) a w x v w v w x v w x 1 Class Exercise: Confirm that that the value and the first two derivatives of equal at x 0 u( wˆ x) and u ( ˆ ) a w x are The expected utility of the risky alternative is * ( ) v( wˆ x) ( ) v( wˆ x) ( ) v ( wˆ x) ( ) v ( wˆ x) a a ( )[ v( wˆ ) v( wˆ ) x v( wˆ ) x ] ( )[ v( wˆ ) v( wˆ )( x) v( wˆ )( x) ]
Microeconomic Theory -0- Uncertainty C Favorable gamble: Improving the odds to make the gamble just acceptable New risky alternative: ( w, w ;, ) ( wˆ x, wˆ x;, ) 1 1 Choose so that the consumer is indifferent between gambling and not gambling For small x we can use the quadratic approximation of the utility function v ( ˆ ) ( ˆ ) ( ˆ ) ( ˆ ) a w x v w v w x v w x 1 Note that the value and the first two derivatives of v( wˆ x) and v ( ˆ ) a w x are equal at x 0 The expected utility of the risky alternative is ( ) v( wˆ x) ( ) v( wˆ x) ( ) v ( wˆ x) ( ) v ( wˆ x) a a ( )[ v( wˆ ) v( wˆ ) x v( wˆ ) x ] ( )[ v( wˆ ) v( wˆ )( x) v( wˆ )( x) ] v( wˆ ) v( wˆ ) x v( wˆ ) x The utility of the riskless alternative is ˆ 1 1 v ( w) x v ( w) x 0 ˆ vw ( ˆ) Therefore the consumer is indifferent if ie v( wˆ ) x ( ) v ( wˆ ) 4
Microeconomic Theory -1- Uncertainty D Measure of risk aversion Absolute aversion to risk The bigger is v( w) ARA( w) v ( w) v( w) x x the bigger is ( ) ARA( w) v( w) 4 4 Thus an individual with a higher ARA( w ) requires the odds of a favorable outcome to be moved more Thus ARA( w ) is a measure of an individual s aversion to risk ARA( w) degree of absolute risk aversion
Microeconomic Theory -- Uncertainty Relative risk aversion Betting on a small percentage of wealth New risky alternative: ( w, w ;, ) ( wˆ(1 ), wˆ(1 );, ) 1 1 Choose so that the consumer is indifferent between gambling and not gambling Note that we can rewrite the risky alternative as follows: ** ( w, w ;, ) ( wˆ x, wˆ x;, ) where x wˆ 1 1
Microeconomic Theory -3- Uncertainty Relative risk aversion Betting on a small percentage of wealth New risky alternative: ( w, w ;, ) ( wˆ(1 ), wˆ(1 );, ) 1 1 Choose so that the consumer is indifferent between gambling and not gambling Note that we can rewrite the risky alternative as follows: * ( w, w ;, ) ( wˆ x, wˆ x;, ) where x wˆ 1 1
Microeconomic Theory -4- Uncertainty Relative risk aversion Betting on a small percentage of wealth New risky alternative: ( w, w ;, ) ( wˆ(1 ), wˆ(1 );, ) 1 1 Choose so that the consumer is indifferent between gambling and not gambling Note that we can rewrite the risky alternative as follows: ( w, w ;, ) ( wˆ x, wˆ x;, ) where x wˆ 1 1 From our earlier argument, v( wˆ ) x v( wˆ ) wˆ wv ˆ ( wˆ ) ( ) ( ) ( ) v( wˆ ) 4 v( wˆ ) 4 v( wˆ ) 4 Relative aversion to risk The bigger is wv( w) wv( w) RRA( w) the bigger is ( ) RRA( w) v ( w) v( w) 4 4 Thus an individual with a higher RRA( w ) requires the odds of a favorable outcome to be moved more Thus RRA( w ) is a measure of an individual s aversion to risk RRA( w) degree of relative risk aversion
Microeconomic Theory -5- Uncertainty Remark on estimates of relative risk aversion wv( w) RRA( w) Typical estimate between 1 and v ( w) Remark on estimates of absolute risk aversion v( w) 1 ARA( w) RRA( w) v( w) w Thus ARA is very small for anyone with significant life-time wealth
Microeconomic Theory -6- Uncertainty E Insurance A consumer with a wealth ŵ has a financial loss of L with probability 1 We shall call this outcome the loss state and label it state 1 With probability 1 1 the consumer is in the no loss state an label it state With no exchange the consumer s state contingent wealth is ( x ˆ ˆ 1, x) ( w L, w) This consumer wishes to exchange wealth in state for wealth in state 1 Suppose there is a market in which such an exchange can take place For each dollar of coverage in the loss state, the consumer must pay p p 1 dollars in the no loss state
Microeconomic Theory -7- Uncertainty Suppose that the consumer purchases q units x ˆ 1 w L q p x p ( wˆ L) p q p 1 ˆ p 1 x w q p p x p wˆ p q p 1 Adding these equations, * p x p x p ( wˆ L) p wˆ 1 Slope = p p 1
Microeconomic Theory -8- Uncertainty Suppose that the consumer purchases q units x ˆ 1 w L q p x p ( wˆ L) p q p 1 ˆ p 1 x w q p p x p wˆ p q p 1 Adding these equations, p x p x p ( wˆ L) p wˆ 1 Slope = p p 1 The consumer s expected utility is U v( x ) v( x ) We have argued that the consumer s choices are constrained to satisfy the following budget constraint p x p x p ( wˆ L) p wˆ 1 This is the line depicted in the figure Group Exercise: What must be the price ratio if the consumer purchase full coverage? (ie x1 x)
Microeconomic Theory -9- Uncertainty F Small favorable gambles (skip): New risky alternative: ( w, w ;, ) ( wˆ y, wˆ x;, ), where y(1 ) x for some 1 1 0 It is tempting to believe that a highly risk averse consumer would not accept such a favorable gamble unless is sufficiently large We show that this intuition is false
Microeconomic Theory -30- Uncertainty Small favorable gambles: Consider a small x and small y x where y(1 ) x Since x is small, we can then use the quadratic approximation of the utility function The expected value of the risky alternative is v( wˆ y) v( wˆ x) v ( wˆ y) v ( wˆ x) a a [ v( wˆ ) v( wˆ ) y v( wˆ ) y ] [ v( wˆ ) v( wˆ )( x) v( wˆ )( x) ] v wˆ v wˆ y x v wˆ y x ( ) ( )( ) 4 ( )( ) *
Microeconomic Theory -31- Uncertainty Small favorable gambles: Consider a small x and small y x where y(1 ) x Since x is small, we can then use the quadratic approximation of the utility function The expected value of the risky alternative is v( wˆ y) v( wˆ x) v ( wˆ y) v ( wˆ x) a a [ v( wˆ ) v( wˆ ) y v( wˆ ) y ] [ v( wˆ ) v( wˆ )( x) v( wˆ )( x) ] v wˆ v wˆ y x v wˆ y x ( ) ( )( ) 4 ( )( ) The value of the riskless alternative is v wˆ y x v wˆ y x ( )( ) 4 ( )( ) v wˆ x ARA wˆ x x ( )[ ( )((1 ) )] xv wˆ ˆ ( )[ xara( w)((1 ) )] 0 for any x that is sufficiently small vw ( ˆ) Therefore the consumer is better off taking the risk if
Microeconomic Theory -3- Uncertainty F Portfolio choice An investor with wealth ˆ W chooses how much to invest in a risky asset and how much in a riskless asset Let 1+ r 0 be the return on each dollar invested in the riskless asset and let 1 r be the return on the risky asset (a random variable) If the investor spends x riskless asset) her final wealth is ** W ( Wˆ x)(1 r ) x(1 r) 0 on the risky asset (and so ˆ W x on the
Microeconomic Theory -33- Uncertainty F Portfolio choice An investor with wealth ˆ W chooses how much to invest in a risky asset and how much in a riskless asset Let 1+ r 0 be the return on each dollar invested in the risky asset and let 1 r be the return on the risky asset (a random variable) If the investor spends q on the risky asset (and so ˆ W q on the riskless asset) her final wealth is * W ( Wˆ q)(1 r ) q(1 r) W ˆ (1 r ) q( r r ) 0 0 0
Microeconomic Theory -34- Uncertainty G Portfolio choice An investor with wealth ˆ W chooses how much to invest in a risky asset and how much in a riskless asset Let 1+ r 0 be the return on each dollar invested in the risky asset and let 1 r be the return on the risky asset (a random variable) If the investor spends q on the risky asset (and so ˆ W q on the riskless asset) her final wealth is W ( Wˆ q)(1 r ) q(1 r) W ˆ (1 r ) q( r r ) ˆ (1 ) 0 0 0 r r W r 0 q where 0 Class exercise: What is the simplest possible model that we can use to analyze the investor s decision?
Microeconomic Theory -35- Uncertainty Two state model Wealth in state s, s 1, ** W ( Wˆ q)(1 r ) q(1 r ) s Wˆ (1 r ) q ( r r ) 0 0 s 0 Wˆ (1 r ) q s where s rs r0 s
Microeconomic Theory -36- Uncertainty Two state model Wealth in state s, s 1, W ( Wˆ q)(1 r ) q(1 r ) s Wˆ (1 r ) q ( r r ) 0 0 s 0 ˆ (1 0 ) s W r q where s rs r0 q 0 W Wˆ (1 r ) s s * q Wˆ W Wˆ(1 r) W ˆ s s
Microeconomic Theory -37- Uncertainty Two state model Wealth in state s, s 1, W ( Wˆ q)(1 r ) q(1 r ) s Wˆ (1 r ) q ( r r ) 0 0 s 0 Wˆ (1 r ) q s r r 0 r r where 0 0 s q 0 W Wˆ (1 r ) s q Wˆ W Wˆ(1 r) W ˆ s Expected utility of the investor U( w, ) u( W ) u( W ) s When will the investor purchase some of the risky asset?
Microeconomic Theory -38- Uncertainty Two state model W ˆ s W (1 r ) q s where s rs r0 The steepness of the boundary of the set of feasible outcomes is ** 1
Microeconomic Theory -39- Uncertainty Two state model W ˆ s W (1 r ) q s where s rs r0 The steepness of the boundary of the set of feasible outcomes is 1 U( W, ) v( W ) v( W ) The steepness of the level set through the no risk portfolio is N MRS 1 *
Microeconomic Theory -40- Uncertainty Two state model W ˆ s W (1 r ) q s where s rs r0 The steepness of the boundary of the set of feasible outcomes is 1 U( W, ) v( W ) v( W ) The steepness of the level set through the no risk portfolio is N MRS 1 Purchase some of the risky asset as long as The risky asset has a higher expected return 1 1 ie 1 1 0
Microeconomic Theory -41- Uncertainty Calculus approach U( q) v( W ) v( W ) v( W(1 r ) q) v( W(1 r ) q) Where 0 1 0 1 0 r r and r r0 U( q) v( W(1 r ) q) v( W(1 r ) q) * 0 1 0
Microeconomic Theory -4- Uncertainty Calculus approach U( q) v( W ) v( W ) v( W(1 r ) q) v( W(1 r ) q) Where 0 1 0 1 0 r r and r r0 U( q) v( W(1 r ) q) v( W(1 r ) q) Therefore 0 1 0 U(0) v( W(1 r )) v( W(1 r ) ( ) v( W(1 r )) 0 0 0 Thus if q 0, then the marginal gain to investing in the risky asset is strictly positive if and only if, 0 ie ( r r ) ( r r ) r r r 0 0 0 0 ie the expected payoff is strictly greater for the risky asset Class Exercise: Is this still true with more than two states
Microeconomic Theory -43- Uncertainty H Sharing the risk on a South Pacific Island Alex lives on the west end of the island and has 600 coconut palm trees Bev lives on the East end and has 800 coconut palm trees If the hurricane approaching the island makes landfall on the west end it will wipe out 400 of Alex s palm trees If instead the hurricane makes landfall on the East end of the island it will wipe out 400 of Bev s coconut palms The probability of each event is 05 Let the West end landing be state 1 and let the East end landing be state Then the risk facing Alex (00,600;, ) while the risk facing Bev is (800,400;, ) is What should they do? What would be the WE outcome if they could trade state contingent commodities?
Microeconomic Theory -44- Uncertainty Let B v () be Bev s utility function so that her expected utility is line U ( x ) v ( x ) v ( x ) B B B B B B 800 where s is the probability of state s B In state 1 Bev s endowment is 1 800 B In state the endowment is 400 * 400
Microeconomic Theory -45- Uncertainty Let B v () be Bev s utility function so that her expected utility is line U ( x ) u ( x ) u ( x ) B B B B B B 800 where s is the probability of state s B In state 1 Bev s endowment is 1 800 slope = B In state the endowment is 400 B B The level set for U ( x ) through the 400 endowment point B is depicted At a point xˆ B in the level set the steepness of the level set is MRS B U MU x v ( xˆ ) B B B B 1 B 1 ( xˆ ) B MU B U ˆ v B x B x ( ) 1 Note that along the 45 line the MRS is the probability ratio (equal probabilities so ratio is 1)
Microeconomic Theory -46- Uncertainty The level set for Alex is also depicted At each 45 line the steepness of the line Respective sets are both 1 800 Therefore MRS ( ) 1 MRS ( ) B B A A Therefore there are gains to be made from trading state claims 400 The consumers will reject any proposed exchange that does not lie in their shaded superlevel sets line 00 600
Microeconomic Theory -47- Uncertainty Equivalently, Bev will reject any proposed exchange that is in the shaded sublevel set Since the total supply of coconut palms is 1000 in each state, the set of potentially 800 acceptable trades must be the unshaded region in the red Edgeworth Box * 400
Microeconomic Theory -48- Uncertainty Equivalently, Bev will reject any proposed exchange that is in the shaded sublevel set Since the total supply of coconut palms is 1000 in each state, the set of potentially 800 acceptable trades must be the unshaded region in the red Edgeworth Box Note also that 400 A B x x Thus if Bev s allocation is xˆ B then Alex has the allocation xˆ A B xˆ We can then rotate the box 180 to analyze the choices of Alex
Microeconomic Theory -49- Uncertainty The rotated Edgeworth Box A B A Note that and xˆ xˆ B 400 Also added to the figure is the green level set for Alex s utility function through A 800 ** 00 600
Microeconomic Theory -50- Uncertainty The rotated Edgeworth Box A B A Note that and xˆ xˆ B 400 Also added to the figure is the green level set for Alex s utility function through A 800 Any exchange must be preferred by both consumers over the no trade allocation (the endowments) Such an exchange must lie in the lens shaped region to the right of Alex s level set and to the left of Bev s level set * 00 600
Microeconomic Theory -51- Uncertainty The rotated Edgeworth Box A B A Note that and xˆ xˆ B 400 Also added to the figure is the green level set for Alex s utility function through A 800 Any exchange must be preferred by both consumers over the no trade allocation (the endowment) Such an exchange must lie in the lens shaped region to the right of Alex s level set and to the left of Bev s level set Pareto preferred allocations 00 600 If the proposed allocation is weakly preferred by both consumers and strictly preferred by at least one of the two consumers the new allocation is said to be Pareto preferred In the figure both x ˆ A and x ˆ A A (in the lens shaped region) are Pareto preferred to since Alex is strictly better off and Bev is no worse off
Microeconomic Theory -5- Uncertainty Consider any allocation such as xˆ A Where the marginal rates of substitution differ From the figure there are exchanges that 400 the two consumers can make and both have a higher utility 800 00 600
Microeconomic Theory -53- Uncertainty Consider any allocation such as xˆ A Where the marginal rates of substitution differ From the figure there are exchanges that 400 the two consumers can make and both have a higher utility 800 Pareto Efficient Allocations It follows that for an allocation 00 600 A x and x B x A to be Pareto efficient (ie no Pareto improving allocations) MRS A ( x A ) MRS B ( x B ) A A 1 B B Along the 45 line MRS ( x ) MRS ( x ) Thus the Pareto Efficient allocations are all the allocations along 45 degree line Pareto Efficient exchange eliminates all individual risk
Microeconomic Theory -54- Uncertainty Walrasian Equilibrium? Suppose that there are markets for state claims Let p s be the price that a consumer must pay for delivery of a unit in state s, ie the price of claim in state s A consumer s endowment ( 1, ), thus has a market value of p p1 1 p The consumer can then choose any outcome ( x1, x ) satisfying p x p Given a utility function u ( x ), the consumer chooses h h Max{ U ( x, ) p x p } x h h s h x to solve ie Max{ v ( x ) v ( x ) p x p } h x h h h h 1 h 1 h FOC: MRS h h MU v ( x ) ( x ) MU v ( x ) p p h h h h
Microeconomic Theory -55- Uncertainty We have seen that MRS h h MU v ( x ) ( x ) MU v ( x ) p p h h h h Thus in the WE for Alex and Bev MRS A A v ( x ) p ( x ) v ( x ) p A 1 A A A v ( x ) p B B 1 B and MRSB ( x ) B v B ( x ) p Class Question: What does the First Welfare Theorem tell us about the WE allocation? Given this, what must be the WE price ratio
Microeconomic Theory -56- Uncertainty Exercises (for the TA session) 1 Consumer choice (a) If u( x ) ln x what is the consumer s degree of relative risk aversion? s s (b) If there are two states, the consumer s endowment is and the state claims price vector is solve for the expected utility maximizing consumption p, p1 1 (c) Confirm that if then the consumer will purchase more state claims than state 1 claims p Consumer choice (a), (b), (c) as in Exercise 1 except that u( x ) s x 1/ s (d) Try to compare the state claims consumption ratio in Exercise 1 with that in Exercise (e) Provide the intuition for your conclusion
Microeconomic Theory -57- Uncertainty 3 Equilibrium with social risk Suppose that both consumers have the same expected utility function U ( x, ) ln x ln x h h h The aggregate endowment is ( 1, ) where 1 (a) Solve for the WE price ratio p1 1 (b) Explain why p 4 Equilibrium with social risk p p 1 Suppose that both consumers have the same expected utility function U ( x, ) ( x ) ( x ) h h 1/ h 1/ The aggregate endowment is ( 1, ) where 1 p1 (a) Solve for the WE price ratio p (b) Compare the equilibrium price ratio and allocations in this and the previous exercise and provide some intuition