CHAPTER 7: UNCERTAINTY

Transcription

1 Eenial Microeconomic - ecion 7-3, 7-4 CHPTER 7: UNCERTINTY Fir and econd order ochaic dominance 2 Mean preerving pread 8 Condiional ochaic Dominance 0 Monoone Likelihood Raio Propery 2 Coninuou diribuion 5 Principal-gen problem 2

2 Eenial Microeconomic -2 ecion 7-3, 7-4 ochaic dominance Three ae cae wih x x x. Preference over propec π = ( π, π, π ) are illuraed below Conider wo propec π and ˆπ. The difference in expeced uiliy i Δ U = π u π u = Δπ u ˆ = = = lo 3 3 ˆ π = π = and o Δ π + ( Δ π +Δ π ) = = = Then Δ U =Δπ u ( Δ π +Δ π ) u +Δ π u = Δπ u u + Δπ u u ( ) ( ) Thu any change in probabiliy ha hif weigh from he wor Fir order ochaic dominance o he be oucome raie expeced uiliy. lo noe ha ince Δ π = ( Δ π +Δ π ) 3 2 Expeced uiliy rie if boh Δ π < 0 and Δ π +Δ π < 0. 2 Tha i, he probabiliy ma in each lef ail decline.

3 Eenial Microeconomic -3 ecion 7-3, 7-4 Fir order ochaic dominance Definiion: Fir Order ochaic Dominance Le x and x be wo random variable wih realizaion in Pr{ x x} Pr{ x x}, hen n X. If for each x x exhibi (fir-order) ochaic dominance over X, x. Propoiion 7.3.: If u() i increaing, x <... < x and he propec ( x; π ) exhibi fir-order ochaic dominance over he propec ( x; π ), hen propec i preferred o propec. Exercie: Prove hi reul for he 4 ae cae. Hence kech an argumen for he ae cae.

4 Eenial Microeconomic -4 ecion 7-3, 7-4 econd Order ochaic Dominance Definiion: econd-order ochaic Dominance Le π = ( π,..., π ) and π = ( π,..., π ) be wo probabiliy diribuion. Le and be he aociaed cumulaive probabiliie or lef ail probabiliie. Diribuion exhibi econd order ochaic dominance over if he um of lef ail probabiliie are lower for han for., =,...,. τ τ= τ= τ We now aume ha x x = + δ Three ae example. ( x, x, x ) = ( w, w+ δ, w+ 2 δ ). 2 3

5 Eenial Microeconomic -5 ecion 7-3, 7-4 Rik-neural individual Three ae example. ( x, x, x ) = ( w, w+ δ, w+ 2 δ ). 2 3 lope x x = x x herefore he lope of an indifference curve m x x = =. x x ccepable gamble for a rik-neural individual n equal increae in he probabiliy of ae and 3 doe no change he expeced payoff.

6 Eenial Microeconomic -6 ecion 7-3, 7-4 Rik avere individual ppealing o concaviy ux ( ) ux ( ) > ux ( ) ux ( ) Then he lope of an indifference curve indifference line for a rik-avere individual u u u m u u 2 = > 3 2 lope = Then any propec in he haded region i preferred over π, regardle of he degree of rik averion. Figure 7.3-: Preferred gamble for a rik-avere individual Noe ha in he haded region Δπ 0 and Δπ Δ π. 3 Equivalenly, Δ π < 0 and Δ ( π +Δπ ) Δ π. 2 Tha i, ( Δπ ) 0 and ( Δ π ) + ( Δ π +Δπ ) 0. 2 Each erm in parenhee i a change in a lef ail probabiliy. Thu he new propec i preferred if he um of all he lef ail probabiliie are all negaive. Define = π, = π + π. Then he ufficien condiion i 2 2 ˆ and ˆ +ˆ + 2 2

7 Eenial Microeconomic -7 ecion 7-3, 7-4 Propoiion 7.3.2: uppoe ha uc () i increaing and concave and ha c c = δ. If + ochaic dominance over π hen he propec ( c, π ) i preferred over ( c, π ). π exhibi econd-order We will prove hi in he coninuou cae. ee EM for he dicree cae.

8 Eenial Microeconomic -8 ecion 7-3, 7-4 Mean Preerving pread limiing pecial cae of OD i when he wo Diribuion have he ame mean a depiced oppoie. Noe ha Δ π < 0 and Δ π =Δ π. (*) 3 Thu, aring wih diribuion, indifference line for a rik-avere individual lope = an equal probabiliy ma i aken from Figure 7.3-: Preferred gamble for a rik-avere individual each ail. To creae diribuion. Hence i a mean preerving pread of. We can rewrie (*) a follow. Δ π < 0 and Δ ( π +Δ π ) =Δ π. 2 ha i, ( Δ π ) < 0 and ( Δ π ) + ( Δ π +Δ π ) = 0. 2 ha i, ˆ and ˆ + ˆ = + 2 2

9 Eenial Microeconomic -9 ecion 7-3, 7-4 Oher Dominance Relaion For many applicaion, he aumpion of fir order ochaic dominance need o be renghened o achieve unambiguou analyical reul. We now conider wo ronger aumpion.

10 Eenial Microeconomic -0 ecion 7-3, 7-4 Condiional ochaic Dominance For any pair of probabiliy diribuion π = ( π,..., π ) and π = ( π,..., π ), uppoe we runcae he diribuion a and conider he lef ail diribuion. The c.d.f. of hee lef ail diribuion are and. The probabiliy diribuion π exhibi condiional ochaic dominance over π if he fir- order ochaic dominance propery hold for every lef ail diribuion. Definiion: Condiional ochaic Dominance The probabiliy diribuion π exhibi condiional ochaic dominance (CD) over π if, for every lef ail diribuion he fir-order ochaic dominance relaion hold. Tha i,, and <.

11 Eenial Microeconomic - ecion 7-3, 7-4 Condiional ochaic Dominance, and < Noe ha an implicaion of CD i ha π π = =. π π Repeaing, = = Thu he CD relaion implie ha each righ runcaed diribuion ha more weigh in he exreme righ ail. We now how ha he convere i alo rue. Propoiion 7.3-3: Condiional ochaic Dominance The probabiliy diribuion π exhibi CD over π if and only if π π,. The direc proof for he coninuou cae i omewha differen. ee Exercie

12 Eenial Microeconomic -2 ecion 7-3, 7-4 Proof: We begin by rewriing he condiion π π π a π. Define k =. Then = k and π kπ and o π k( π ). Therefore π = k = π. ecaue hi argumen hold for all i follow ha and <. Rearranging hi inequaliy, and <. Q.E.D.

13 Eenial Microeconomic -3 ecion 7-3, 7-4 Monoone Likelihood Raio Propery Two probabiliy diribuion are aid o aify he monoone likelihood propery if for ome ordering of ae π π π, <. π Inuiively if he raio of he probabiliie (he likelihood raio ) i decreaing acro ae, hen here mu be more weigh in he lef ail of he diribuion of π. In fac he decreaing likelihood propery i a ufficien condiion for condiional ochaic dominance.

14 Eenial Microeconomic -4 ecion 7-3, 7-4 Le and be he cumulaive probabiliie, ha i = π and = = π. = Propoiion 7.3-4: The monoone likelihood raio propery implie condiional ochaic dominance. Proof: ppealing o he monoone likelihood raio propery, π π ( ) π, and o ha π π π = ( ) = ( ),. π π = π = π Rearranging hi inequaliy, π π,. ppealing o Propoiion The condiional ochaic dominance relaion hold. Q.E.D.

15 Eenial Microeconomic -5 ecion 7-3, 7-4 Coninuou Diribuion c and c have uppor (poibly differen) wihin he inerval [, β ], p.d.f. f (), c j {, }. j c.d.f. F() c and inegral of he c.d.f. T () c = F () θ dθ j j. j (Thee inegral are he coninuou equivalen of he um of lef ail.) c β E{( v c )} E{( v c )} = v() c f () c dc v() c f () c dc. β Inegraing by par and noing ha F ( ) = F ( ) = 0 and F ( β ) = F ( β ) =, β E{( vc )} Evc {( )} = v cf() cdc v cf() cdc. Hence, β β E{( vc )} Evc {( )} = v ()[ c F() c F()] c dc. (7.3-) Thu, if v i increaing and Evc {( )} Evc {( )} 0. Thi i Propoiion c exhibi fir-order ochaic dominance over c, hen

16 Eenial Microeconomic -6 ecion 7-3, 7-4 econd order ochaic dominance β E{( vc )} Evc {( )} = v ()[ c F() c F()] c dc. (7.3-2) Inegraing again, E{ v( c )} E{ v( c )} = v ( β)[ T ( β) T ( β)] v ( c)[ T ( c) T ( c)] dc. β Thu, if v() i increaing and concave and he lef ail um are maller for diribuion han, hen Evc {( )} Evc {( )}. Thi i Propoiion

17 Eenial Microeconomic -7 ecion 7-3, 7-4 We nex eek condiion under which wo diribuion have he ame mean. eing vc () β equaion E{( vc )} Evc {( )} = v ()[ c F() c F()] c dc (7.3-3) β = [ F ( c) F ( c)] dc= T ( β ) T ( β ). Hence, we have he following propoiion. = cin Propoiion The diribuion c and c wih uppor in [, β ] have he ame mean if F () c dc = F () c dc β β, ha i, he area under each c.d.f. i he ame.

18 Eenial Microeconomic -8 ecion 7-3, 7-4 Conider he fir of he wo diagram. If he wo haded area are equal, hen he area under each c.d.f. are equal. Noe ha he lope of F () c i decreaing o he lef of γ while he lope of F () c i increaing. Tha i, he deniy of c i decreaing and he deniy of c i increaing. The oppoie i rue o he righ of γ. The probabiliy deniy funcion mu herefore be a depiced. Noe alo ha for any c < β, c T () c = F () x dx> F () x dx= T () c c. ecaue he wo diribuion have he ame mean, Figure 7.3-2: Mean preerving pread i i naural o decribe c, wih i greaer weigh in he ail, a a mean-preerving pread of c.

19 Eenial Microeconomic -9 ecion 7-3, 7-4 Definiion: Mean-preerving pread uppoe ha c and c are random variable wih uppor in [, β ]. The random variable c i a mean-preerving pread of c if for all c [, β ] c T () c = F () x dx F () x dx= T () c c and T ( β ) T ( β ) =. From (7.3-) β E{( vc )} Evc {( )} = v ()[ c F() c F()] c dc. Inegraing again by par, β E{ vc ( )} Evc { ( )} = v ( c)[ T ( c) T ( c)] v ( c)[ T ( c) T ( c)] dc. β ppealing o he definiion of a mean preerving pread, we have he following propoiion.

20 Eenial Microeconomic -20 ecion 7-3, 7-4 Propoiion 7.3-6: If c i a mean preerving pread of c, and v() i concave, hen Evc { ( )} Evc { ( )}. If v() i convex hen Evc { ( )} Evc { ( )}.

21 Eenial Microeconomic -2 ecion 7-3, 7-4 PRINCIPL GENT PROLEM Key idea: conracing wih hidden acion, incenive conrain, inurance wih moral hazard Conider he following 2 peron economy. lex chooe an acion x X = { x,.. x n }, where x <... < xn. Penny own he ingle firm. The e of poible oupu i Y = { y,..., y } where y <... < y. For any acion x he firm oupu i a propec ( y, π ( x)). Higher acion hif probabiliy ma o higher oupu. If x > x hen π ( x ) exhibi FOD over π ( x). Laer we will need o aume ha he monoone likelihood raio propery hold. Tha i, for any x > x, π ( x ) π ( x ) <, < π ( x) π ( x) Le w= ( w,..., w ) be he ae coningen allocaion o lex and le r = ( r,..., r ) be he ae coningen allocaion o Penny. Then w + r y. Finally le Cx ( ) be he uiliy co o lex of aking acion x.

22 Eenial Microeconomic -22 ecion 7-3, 7-4 Then expeced uiliie are U ( x, w) = π ( x) v( w) C( x) and = U P = π ( x) v( r ) where = r + w = y ubiuing for r, U ( w, x) = π ( x) v( y w ) P = Efficien allocaion For any fixed acion x we can in principle olve for The efficien allocaion by olving one of he following Pareo efficien oucome wih 3 acion problem. (i) Maximize Penny expeced uiliy given ha lex mu have an expeced uiliy of a lea (ii) Maximize lex expeced uiliy given ha Penny mu have an expeced uiliy of a lea U U P

23 Eenial Microeconomic -23 ecion 7-3, 7-4 Efficien llocaion Conider problem (i). Fix U and x and olve for he efficien ae coningen paymen wx ( ) = MaxU { ( wx, ) U ( wx, ) U}. w P Then Penny expeced uiliy i Pareo efficien oucome wih 3 acion U ( w( x), x) = π ( x) u( y w ( x)). P = Repea for each x and hu olve for * x = Max U P w x x x arg { ( ( ), )}

24 Eenial Microeconomic -24 ecion 7-3, 7-4 ymmeric informaion (hidden acion) The Principal gen problem Thu far we have aumed ha all apec of he conrac are colely verifiable and hence enforceable. u wha if i prohibiively coly o verify he acion choen by lex (he agen)? We aume ha oupu i colely verifiable by a hird pary. ince oupu i higher in higher numbered ae, i follow ha he ae i verifiable. uppoe hen ha he efficien conrac i igned. lex i paid uiliy if he ake acion x i w x in ae and o hi expeced * ( ) * * ( ( ), ) = π ( ) ( ( )) ( ) = U w x x x v w x C x. we hall ee, excep in pecial cae no feaible. x x arg Max{ U( w( x ), x)}. Thu he efficien conrac i * * x Then Penny (he principal) mu deign a conrac in uch a way ha he agen he hire will chooe he acion in he conrac even wihou verificaion. We conider wo pecial cae

25 Eenial Microeconomic -25 ecion 7-3, 7-4 Rik-Neural gen uppoe ha he principal i rik avere and he agen i rik neural. n efficien conrac olve Max{ U ( w, x) U ( r, x) U }. (*) xrw,, P P For any acion x, efficiency require ha all he rik be borne by he agen. Tha i, he principal receive a fixed ren r and he agen receive he reidual w = y r. The conrain i hen U (, r x) = π ()() x u r = u() r U. P P = Then he efficien conrac i a fixed ren conrac wih ren To aify he conrain in (*), hi ren mu be choen o ha * r aifying ur * ( ) = UP π ( x) u( y w) = u( r) = UP. = We can rewrie (*) a follow:

26 Eenial Microeconomic -26 ecion 7-3, 7-4 * * = π = Max{ U ( y r, x) ( x) y C( x) r. x Hence * * = arg { π ( ) ( )} x = x Max x y C x r ymmeric Informaion uppoe ha he agen acion i no verifiable. The principal fixed ren conrac i verifiable ince i i independen of he oucome. The rik neural agen chooe * * = arg { (, ( )) = arg { π ( ) ( )} x x = x Max U x w x Max x y C x r Then x = x *

27 Eenial Microeconomic -27 ecion 7-3, 7-4 Rik-Neural Principal uppoe ha he principal i rik neural and he agen i rik avere. n efficien conrac olve Max{ U ( w, x) r + w = y, U ( r, x) U }. (*) xrw,, P 0 For any acion x, i i efficien for he principal o bear all he rik. Thereforew ( x) = w, =,...,. he opimum he conrain mu be binding herefore 0 w aifie U w x = v w C x = U. 0 0 (, ) ( ) ( ) Le v () be he invere funcion, ha i 0 w mu aify problem of he principal can be rewrien a follow: 0 w = v ( U + C( x)). Then he maximizaion 0 xw, 0 { π( ) = π( ) ( + ( )) = = Max x y w x y v U C x Then * π P P x = x = Max{ ( x) y C ( x)}, where C ( x) v ( U + C( x)). C ( ) P x i he virual co funcion for he principal.

28 Eenial Microeconomic -28 ecion 7-3, 7-4 The efficien conrac hu maximize expeced revenue le he virual co. Conider wo acion x and x > x. Define ΔC C ( x ) C ( x) P P P Thi i he marginal co of paying he agen o ake he more coly acion x. I i he increae in he wage needed o compenae he agen o ha he payoff remain a U. Pareo efficien oucome wih 3 acion Nex uppoe ha he agen payoff i increaed o U + Δ U. ecaue v i concave, he marginal uiliy of he agen i lower a he higher uiliy level. Thu he dollar compenaion needed o pay for he more coly acion i higher. Tha i, he marginal co of providing a big enough incenive for he agen o ake a more coly acion rie wih he uiliy level of he agen. Therefore, he higher he expeced uiliy of he agen, he lower will be he efficien acion. Thi i he cae depiced above. he expeced uiliy of he agen rie, he efficien acion change fir from he highe co acion x 3 o x 2 and hen from x 2 o he lowe co acion x.

29 Eenial Microeconomic -29 ecion 7-3, 7-4 Wih a fixed wage he agen opimal choice i he low co acion. Therefore unle * x = x he efficien conrac i no enforceable.

30 Eenial Microeconomic -30 ecion 7-3, 7-4 olving he Principal-gen problem Coninue wih he pecial cae in which he principal i rik neural and he agen i rik avere. Three ep approach ep : For any acion x, and expeced uiliy for he agen U characerize he ae coningen paymen which induce he agen o ake acion x raher han any oher x X ep 2: Of all hee paymen cheme chooe he one ha i be for he principal. ince he i rik neural hi i he cheme wih he lowe expeced co. Wrie her expeced payoff a U ** ( x ) P ep 3: For each x X olve for U ** ( x ) and finally chooe p x = arg Max{ U ( x)}. ** ** P cually we will no ake he hird ep bu learn wha we can by examining he fir wo ep.

31 Eenial Microeconomic -3 ecion 7-3, 7-4 ep : Incenive conrain For he acion x, he ae coningen paymen vecor w i incenive compaible if he agen ha an incenive o ake he acion x raher han any oher acion, ha i, Le U ( x, w) U ( x, w), x X. Incenive Conrain U be he conraced expeced uiliy of he agen. Then ( xw, ) mu alo aify he following conrain U ( x, w) U Paricipaion conrain The be conrac for he principal, given he elecion of acion x, hu olve he following opimizaion problem: ** w MaxUP U xw U U xw U xw x X w = arg { (, ), (, ) (, ), }. (**)

32 Eenial Microeconomic -32 ecion 7-3, 7-4 ep 2: Inuiively, he problem i one of deigning a conrac o deer he agen from aking ale coly acion. We will aume hi o ha (**) can be rewrien a follow. ** w MaxUP U xw U U xw U xw x x w = arg { (, ), (, ) (, ), < }. uppoe ha in fac here i a ingle binding conrain. 2 The opimizaion problem can hen be rewrien more imply a follow. Max U U x w U U x w U x w x < x. w * * * { P (, ), (, ) (, ), forome } The Lagrangian of hi opimizaion problem i L = U + U x w U + U x w U x w * * P λ( (, ) ) μ( (, ) (, )) = U + + U x w U x w + a conan * P ( λ μ)( (, ) μ (, )) * * π( x )( y w) ( λ μ) π( x ) v( w) μ π( x) v( w) a conan = = =. = I i no imporan ha here be one binding conrain, only ha he binding conrain are all aociaed wih lower co acion. General ufficien condiion o enure hi are quie ringen. However for numerical example i i ypically he downward conrain ha are binding.

33 Eenial Microeconomic -33 ecion 7-3, 7-4 * * π( x )( y w) ( λ μ) π( x ) v( w) μ π( x) v( w) a conan = = = L = The fir-order condiion are herefore L w = π + λ + μ π μπ = * * * * ( x ) ( ) ( x ) v( w) ( x) v( w) 0, =,...,. Hence v w π ( x) = λ + μ μ π * * ( ) ( x ) where x * < x. y hypohei, he likelihood raio π π ( x) * ( x ) of hi expreion increae wih. Therefore, becaue v() i concave, efficiency, he higher he oupu, he higher i he paymen o he agen. i increaing in he oupu ae. Thu he righ hand ide w i increaing in. Thu for

34 Eenial Microeconomic -34 ecion 7-3, 7-4 pplicaion: Inurance wih Moral Hazard n individual houe burn down wih probabiliy π ( x), where x i he effor he owner make o avoid uch a caarophe. The houe can be rebuil a a co of L. If he inurance company doe no oberve he homeowner acion bu he home owner i perfecly moral, he ell he ruh and he efficien conrac i unaffeced. However if a homeowner i no o ruworhy, he inurance company face a moral hazard problem. Wih full coverage he incenive o ake care i eliminaed o he homeowner be choice i he cheape acion x. To give he homeowner he incenive o ake appropriae care he homeowner mu be offered a conrac in which he i ufficienly penalized in he lo ae.