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(x)=={(y,−x):y≤x}whichiscertainlynotconvex,buttheinputre-√quirementsetisV(y)={x:x≥y}’²=aand²=(t)=f(tx).Thenn∑dy∂f(x)=x,idt∂xii=1sothatn∑1dy1∂f(x)=(x)∂xii==1,2togeti11ρρρρρρf(tx,tx)=[(tx)+(tx)]=t[x+x]=tf(x,x).12121212ThisimpliesthattheCESfunctionexhibitsconstantreturnstoscaleandhencehasanelasticityofscaleof1.′:ifg(x)>0,thenthefunctionmustbestrictlyincreasing,,forexample,thefunction3′g(x)=,butg(0)=0.′(x)=g(h(x))andsupposethatg(h(x))=g(h(x)).Sincegis′monotonic,itfollowsthath(x)=h(x).Nowg(h(tx))=g(th(x))and′′g(h(tx))=g(th(x))(h(x))whereh(x)
2ANSWERSform′∂h∂hg(h(x))∂x∂x11=.∂h∂h′g(h(x))∂x∂x22Thatis,[]1ρ1aa1ρ2ρρ(a+a)x++aa+a12121ρNowsimplydefineb=a/(a+a)andA=(a+a).11212′,wemustshowthatforallyandyinYand′0≤t≤1,wemusthavety+(1−t)′tyand(1−t)yareinY,,wemustshowthatifyisinY,ands>0,>0,letnbeanonnegativeintegersuchthatn≥s≥n−,nyisinY;sinces/n≤1,divisibilityimplies(s/n)ny=>0(ifweallowinputstobenegative).,(x,x),itissufficient(butnotnecessary),formamatrixusingthesecondderivativesoftheproductionfunction,,−1311∂f(x)1∂f(x)1−−2222=−xx=xx121224∂x∂x4∂x12121−3∂f(x)122=−xx1224∂x2
[]−3/21/2−1/2−1/211−xxxx121244Hessian=−1/2−1/21/2−3/211xx−xx1212441−3/21/2D=−xx<0112411−1−1−1−1D=xx−xx=,monotonic,,butthereisnowaytoproduceanyy>,writedowntheproduction√functionf(x)=ax−xx+bxandcompute1122∂f(x)1−1/21/2=a−∂x21√1x2Thisispositiveonlyifa>,,itsdeterminantiszero,:fisconvex;therefore,allsetsoftheform√{x,x:ax−xx+bx≤y}(theinequalitysigngoesinthewrongdirection).Ascomplementsofconvexsets(suchthattheborderlineisnotastraightline),soithasallofthepropertiespossessedbythesetwotypesoffunctions,includingbeingregular,monotonic,
,theKuhn-Tuckertheoremrequiresthefollow-ingthreeinequalitiestohold()∗∂f(x)∗p−wx=0,jj∂xj∗∂f(x)p−w≤0,j∂xj∗x≥∗∗Notethatifx>0,thenwemusthavew/p=∂f(x)/∂′-maximizingbundlewithpositiveprofits′pi(x)>′′f(tx)>tf(x),fort>1,wehave′′′′′′′pi(tx)=pf(tx)−twx>t(pf(x)−wx)>tpi(x)>pi(x).′Therefore,,theprofitfunctionisgivenby()a()1a−1a−1wwpi(p,w)=p−,notethat()a()1a−1a−1wwpi(tp,tw)=tp−tw=tpi(p,w),apapwhichimpliesthatpi(p,w),factortheprofitfunctioninthefollowingway:()1aa11a1−aa−11−a1−a1−aa−1pi(p,w)=pwa−a=pwφ(a),whereφ(a)isstrictlypositivefor0<a<()22∂pi(p,w)∂pi(p,w)22∂p∂p∂wDpi(p,ω)=22∂pi(p,w)∂pi(p,w)2∂w∂p∂w2a−1aa1aa1−aa−11−aa−1pw−pw22(1−a)(1−a)=φ(a).a112−aaa1−aa−11−aa−1−pwpw22(1−a)(1−a)
−1aa1−aa−1pwφ(a)>02(1−a),theHessianisapositivesemidefinitematrix,whichimpliesthatpi(p,w)isconvexin(p,w).,wehave∂fw∂x11|TRS|==.∂fw2∂x2Now,notethatln(wx/wx)=−(ln(w/w)+ln(x/x)).22111212Therefore,dln(wx/wx)dln(w/w)dln|TRS|221112=−1=−1=1/σ−(x/x)dln(x/x)dln(x/x),weknowthatln(wx/wx)=ln(w/w)+ln(x/x),22112121Differentiating,wegetdln(wx/wx)dln(x/x)221121=1−=1−σ.dln(w/w)dln|TRS|⊃Y⊃,themaximumofpyoverYOmustbelargerthanthemaximumoverY,−x−-orderconditionis20−2x−w=,thederivativeofprofitwithrespecttoxmustbenonpositiveatx=0:20−2x−w<0whenx=0,orw≥==10−w/2,or,tobemoreprecise,x=max{10−w/2,0}.
−x−wx=[20−w−x]=10−w/2tofind[]2wpi(w)=10−.−(10−w/2),whichis,ofcourse,′′′factorprices,weknowthatφ(w)≤0andφ(w)≥(x,x)=g(x)+g(x).-orderconditionsarep/x=w,whichgivesusthedemandfunctionx=p/wandthesupplyfunctiony=ln(p/w).Theprofitsfromoperatingatthispointarepln(p/w)−-wayschoosex=0andmakezeroprofits,theprofitfunctionbecomespi(p,w)=max{pln(p/w)−p,0}.-orderconditionsarepa−w=011x1pa−w=0,-orderconditionsarea−1a12paxx−w=01112a−1a21paxx−w=0,
,thetechnologymustexhibitdecreasingreturnstoscale,soa+a<,thefirmwillneverusemoreoffactorithanitineedsto,whichimpliesx=−wx−-orderconditionisa−1pax−(w+w)=(x)=x,butthefactorpriceisw+,a<∗-maximizinginputvectorforprices(p,w).This∗∗∗meansthatxmustsatisfypf(x)−wx≥pf(x)−wxforallpermissible∗∗(x);.,there∗∗∗∗∗∗∗∗existsavectorxsuchthatf(x)≥f(x)andw(x−x)<∗∗thentheprofitsachievedwithxmustbegreaterthanthoseachieved∗withx:∗∗∗∗∗∗∗pf(x)−wx≥pf(x)−wx∗∗>pf(x)−wx,∗()∗∂f(x)∗t−wx=0,jj∂xj∗∂f(x)t−w≤0,j∂xj∗x≥0,j∗(y−f(x))t=0,∗y−f(x)≤0,t≥0.∗∗If,forinstance,wehavex>0andx=0,theaboveconditionsimplyij∗∂f(x)w∂xii≥.∗∂f(x)wj∂xj
8ANSWERSThismeansthatitwoulddecreasecosttosubstitutexforx,butsinceijthereisnoxused,,,weequatemarginalcoststofindy=+y=y,sowecancombinethesetwoequationstoget12y=y−(y)=1/2+y−1=y−1/,onreflectionthiscan’tberight:itisobviouslybettertoproduceeverythinginplant1ify<,wehaveignoredtheimplicit1constraintthaty≥{2y/2ify<1c(y)=y−1/2ify>,wecanwritethecostfunctionforthefirstplantasc(y)=Ayandforthesecondplantasc(y)=By,whereAandB12dependona,b,w,(y)=min{A,B}+aw,andthecostofusingactivity1122bisbw+,so1122c(w,w,y)=ymin{aw+aw,bw+bw}.1211221122Thedemandfunctionforfactor1,forexample,isgivenbyayifaw+aw<bw+bw111221122byifaw+aw>bw+bw111221122x=1anyamountbetween+aw=bw+,√√c(y)=min{4y+2y:y+y≥y}.1212ItistemptingtosetMC(y)=MC(y)tofindthaty=y/5and11221y=4y/,ifyouthinkaboutitaminuteyouwillseethatthis2
’tmakesense—youareproducingmoreoutputintheplantwiththehighercosts!,ratherthanconvex,,youwillproducealloutput√atthecheaperplantsoc(y)=,,+x12xx=+y/-orderconditionis21−y/x,1√√whichimpliesx=,x=√√2y=4,soy=2,fromwhichitfollowsthaty=:22c(y)=miny+y12y,y12suchthaty+y==y=y/(y)=(y/2)+(y/2)=y/-orderconditionsare6y=2y,ory=+y==y/4121andy=3y/[]2[]22y3y3yc(y)=3+=.444
,thenweneedtohave2x+x=,thefirmwilltypicallyspecializeand12setx=yorx=y/{w/2,w}.Similarly,thecostfunction12fortheothertechniqueisymin{w,w/2}.Sincebothtechniquesmustbe34usedtoproduceyunitsofoutput,c(w,w,y)=y[min{w/2,w}+min{w,w/2}].+x=yandthenthelinex+2x=“cross.”Theslopeis−2totheleftofthediagonaland−1/
=2,=1,thefirst-orderconditionsuggestsy=1/2,=(p)=2max{p/4−1,0}.
,∂l/∂p<,∂l/∂p=−∂pi/∂p∂w=2−∂pi/∂w∂p=−∂y/∂∂y/∂w>:n∑∂c∂cdc=dw+∂w∂yii=1Itfollowsthat∑n∂cdc−dwii=1∂c∂wi=.∂ydyNowsubstitutethefirstdifferencesforthedy,dc,dwtermsandyou’,weknowwewilluseeitherx,,weknowthatifweusexandxtoproducey,,ifthecostofusing23oneunitofxislessthanthecostofusingoneunitofbothxandx,123thenwewilluseonlyx,:{3yifw<w+w123x=10ifw>w+w123{0ifw<w+w123x=23yifw>w+w123{0ifw<w+w123x=33yifw>w+w123ifw=w+w,thenanybundle(x,x,x)withx=xandx+x=3y1231232312(orx+x=3y)(w,y)=3ymin(w,w+w).:1/23/4c(tw,y)=y(twtw)123/21/23/4=t(y(ww))123/2=tc(w,y)No.
:∂c3∂c3−1/43/43/4−1/41/21/2=yww>0=yww>∂w4∂w412Concave:[]−5/43/4−1/4−1/431/291/2−ywwyww12121616Hessian=−1/4−1/43/4−5/491/231/2yww−yww12121616|H|<01981−1/2−1/2−1/2−1/2|H|=yww−yww2121225625672y=−√<0No256ww12Continuous::√c(tw,y)=y(tw+twtw+tw)1122√=ty(w+ww+w)1122=tc(y,w)YesMonotone:(√)(√)∂c1w∂c1w21=y1+>0=y1+>0Yes∂w2w∂w2w1122Concave:[]1/2−3/2−1/2−1/211−ywwyww212144H=−1/2−1/2−3/21/211yww−yww212144|H|<0111−1−1−1−1|H|=yww−yww=0Yes221211616Continuous:Yes
14ANSWERSProductionFunction:(√)1w2x(w,y)=y1+(1)12w1(√)1w1x(w,y)=y1+(2)22w2Rearrangingtheseequations:√yw2′x−y=(1)12w1√yw1′x−y=(2)22w22y′′Multiply(1)and(2):(x−y)(x−y)=.Thisisaquadraticequation124√2222whichgivesy=(x+x)±x+x+2−:−tw1c(tw,y)=y(twe+tw)12−tw1=ty(we+w)126=tc(w,y)NoMonotone:∂c−w−w−w111=y(−we+e)=ye(1−w)11∂w1Thisispositiveonlyifw<∂c=y>0No∂w2Concave:[]−w1y(w−2)e01H=0−w1|H|=y(w−2)e11Thisislessthanzeroonlyifw<|H|=0No2Continuous:Yes
:√c(tw,y)=y(tw−twtw+tw1122√=ty(w−ww+w)1122=tc(w,y)YesMonotone:√∂c1w2=y(1−)∂w2w11√1w2Thisisgreaterthan0onlyif1>2w1√∂c1w1=y(1−)∂w2w22√w2Thisisgreaterthan0onlyif2>w11√√w>w(bysymmetry)2w>w21124or1w<4ww>w121241Monotoneonlyifw<w<:[]−3/21/2−1/2−1/211yww−yww121244H=−1/2−1/21/2−1/211−ywwyww1212441−3/21/2|H|=yww>01124|H|=0No(itisconvex)2Continuous::√1c(tw,y)=(y+twtw)12y=tc(y,w)Yes
16ANSWERSMonotoneinw:√√∂c11w∂c11w21=(y+)>0=(y+)>0Yes∂w2yw∂w22w1122Concave:[]−3/21/2−1/2−1/21111−(y+ww(y+)ww12124y4yH=Butnotiny!−1/2−1/21/2−3/21111(y,)ww−(y+)ww12124y4y|H|<01Yes|H|=02Continuous:Notfory=0.√=ax+,exceptthat2thelinearcombinationofxandxwillproducey,,weknowthatifxisrelativelycheaper,wewilluseallxandnox,(w,y)=ymin(,).(x,x)=x+yifw<wijx=0>wjianyamountbetween0andyifw=(w,w,y)=x(w,w,y)=(x,x)=min{x,x}.,,,thecostfunctionmustbehomogeneousofdegree1,soa=1−b.
,exceptat(0,0).m/pifp<p112x=anyxandxsuchthatpx+px=mifp=p1121222120>p2Theindirectutilityfunctionisv(p,p,m)=max{m/p,m/p},andthe1212expenditurefunctionise(p,p,u)=umin{p,p}.(p,p,u)=umin{p,p}.Theutility1212functionisu(x,x)=x+x(oranymonotonictransformation),andthe1212demandfunctionism/pifp<p112x=anyxandxsuchthatpx+px=mifp=p1121122120>=m/(p+p),x=m/(p+p).(p,p,u)=(p+p+2)(x,x)=min{x,x}(1).(p,p,m)=max{u(1)−p+m,m}(p,u)=u/A(p)µ(p;q,m)=mA(q)/A(p),(p,m)≡h(p,v(p,m))≡∂e(p,v(p,m))/∂p.()jjj
18ANSWERS(Notethatthepartialderivativeistakenwithrespecttothefirstoccurrenceofp.)Differentiatingequation()withrespecttomgivesusj2∂x∂e(p,v(p,m))∂v(p,m)j=.∂m∂p∂u∂mjSincethemarginalutilityofincome,∂v/∂m,mustbepositive,-Douglasdemandsystemwithtwogoodshastheformam1x=1p1am2x=2p2wherea+a=()−2−2−1−12−amp−amp−aampp11211112.−1−1−2−222−aampp−amp−µ/dt=at+bµ+[]caccaaqb(q−p)µ(q,p,m)=em+++p−−−.22bbbbbbc+ap+=µat+bµ+c=µ−bµcate=,wehave−]]bµcatqqeee−=.bpapEvaluatingtheintegrals,wehavecbebµ(q;p,m)−bmapaqe=e−[e−e].a
(x,λ)=lnx+lnx−λ(3x+4x−100).12122(Besureyouunderstandwhywecantransformuthisway.)Now,equatingthederivativeswithrespecttox,x,andλtozero,wegetthreeequations12inthreeunknowns3=3λ,2x11=4λ,x23x+4x=,wegetx(3,4,100)=20,andx(3,4,100)=,,themarginalutilityofincomecanbemeasured∗∗inoriginal‘utils’orin‘lnutils’.Letu=lnuand,correspondingly,v=lnv;then∂v(p,m)∗∂v(p,m)µ∂mλ===,∂mv(p,m)v(p,m)whereµdenotestheLagrangemultiplierintheLagrangian32L(x,µ)=xx−µ(3x+4x−100).212132201Checkthatinthisproblemwe’dgetµ=,λ=,andv(3,4,100)=(x,λ)=xx−λ(px+px−m),112212takingderivatives,111−23xx=λp,1122121−23xx=λp,2123px+px=,weget3m2mx(p,m)=,x(p,m)=.125p5p12
20ANSWERSPluggingthesedemandsintotheutilityfunction,wegettheindirectutilityfunction()1()1()1()1()523233m2mm632v(p,m)=U(x(p,m))==.5p5p5pp1212Rewritetheaboveexpressionreplacingv(p,m)byuandmbye(p,u).Thensolveitfore(·)toget32()()p5p56125e(p,u)=,sinceh=∂e/∂p,theHicksiandemandsareii()2()2−p5p56125h(p,u)=u,132and()3()3−p5p56125h(p,u)=,let’ββ12L(x,µ)=px+px−µ((x−α)(x−α)−u);11221122thefirst-orderconditionsareβ−1β12p=µβ(x−α)(x−α),111122ββ−112p=µβ(x−α)(x−α),221122ββ12(x−α)(x−α)=βx−α1222=,pβx−α2111usingthelastequation1()−ββ12x−α=(x−α)u;2211substitutingandsolving,β()2β+β112pβ21β2h(p,u)=α+u,11pβ12
β()1β+β112pβ12β1h(p,u)=α+β21Verifythat()1()()β+βββ2112∂h(p,m)uββ∂h(p,m)1122==.∂pβ+βpp∂p212121Theexpenditurefunctionisββ()2()1β+ββ+β112112pβpβ2112ββ21e(p,u)=pα+u+pα+βpβ1221Solvingforu,wegettheindirectutilityfunctionv(p,m)=(())(())ββ12βm−αpβm−αp122211−α−α.12β+βpβ+βp121122ByRoy’slawwegettheMarshalliandemands()1m−αp11x(p,m)=βα+β,1122β+βp122and()1m−αp22x(p,m)=βα+β.2211β+β—amonotonictransformationofutilitydoesn’,theMarshalliandemandsx(p,m)maximizeφ(x)subjecttopx=ψ(φ(x)),therewouldexistsomeotherchoice′′′xsuchthatψ(φ(x))>ψ(φ(x(p,m)))andpx=−1thetransformationψ(·)tobothsidesoftheinequalitywillpreserveit,we′′wouldhaveφ(x)>φ(x(p,m))andpx=m,whichcontradictsourinitialassumptionthatx(p,m)maximizedφ(x)subjecttopx=∗x(p,m)=x(p,m).(Checkthatthereversepropositionalsoholds—.,∗thechoicethatmaximizesualsomaximizesuwhenthethesamebudgetconstrainthastobeverifiedinbothcases.)∗∗v(p,m)=ψ(φ(x(p,m)))=ψ(φ(x(p,m))=ψ(v(p,m)),
22ANSWERSthefirstandlastequalitiesholdbydefinitionandthemiddleonebyourpreviousresult;now∗∗∗e(p,u)=min{px:ψ(φ(x))=u}−1∗=min{px:φ(x)=ψ(u)}−1∗=e(p,ψ(u));again,we’reusingdefinitionsatbothendsandthepropertiesofψ(·)—namelythattheinverseiswelldefinedsinceψ(·)ismonotonic—togetthemiddleequality;finallyusingdefinitionsandsubstitutionsasoftenasneededweget∗∗∗∗∗∗∗h(p,u)=x(p,e(p,u))=x(p,e(p,u))−1∗−1∗=x(p,e(p,ψ(u)))=h(p,ψ(u)).(p,u)≡x(p,e(p,u))withrespecttopjjitoget∂h(p,u)∂x(p,m)∂x(p,e(p,u))∂e(p,u)jjj=+.∂p∂p∂m∂-tionprobleme(p,u)=min{p(x−x):u(x)=u}.Bytheenvelopetheorem,wehave∂e(p,u)=h(p,u)−x=x(p,e(p,u))−∂piTherefore,wehave∂h(p,u)∂x(p,m)∂x(p,e(p,u))jjj=+(x(p,m)−x),ii∂p∂p∂miiandreorganizingwegettheSlutskyequation∂x(p,m)∂h(p,u)∂x(p,e(p,u))jjj=+(x−x(p,m)).ii∂p∂p∂,—.,thesetsSB(x)={z:zÂx}andSW(x)={z:z≺x}.Whenpgoes1downthenewbudgetsetiscontainedinSW(x),whilewhenpgoesdown2there’saregionofthenewbudgetsetthatliesinSB(x).
—alsoknownas“ownrateofinterest”—ongoodxis(p/p)−,(2,4)(2,1),butrejectsit;therefore,(1,2)Â(2,1).Whenpricesare(6,3)(1,2)butrejectsit;therefore,(2,1)Â(1,2).,wehavee(p,u)=u/f(p).Substituting,wehaveµ(p;q,y)=v(q,y)/f(p)=f(q)y/f(p).+2x=20andx+2x=−p/,x=
24ANSWERS∗=,sincegood1isinferior,hewillchoosetoreduce1∗hisconsumptiontolessthanx=,,hewillacceptthe1grant,,hewouldspend(48+g)/≤(48+g)>(48+g)/4,
.(UseRoy’sidentitytocheckthatthisindirectutilityisindeedconsistentwithouroriginaldemandfunction.)∗Togetthedirectutilityfunctionwemustsolve2bqu(x,z)=min{v(q,m):qx+z=m}=min{m−aq−:qx+z=m};qq2usethebudgetconstrainttoeliminatemfromtheobjectivefunctionandgettheoptimalvalueofqx−a∗q=.bThus2(x−a)u(x,z)=z+.2bThisis,ofcourse,,solvemax{u(x,z):px+z=m}andx,’sthedemandforz?µ(p;q,m)=a+bp+cµ(p;q,m).-tiontothenonhomogeneousequationisgivenby(a+bp)c+bµ=−.2cThereforethegeneralsolutiontothedifferentialequationisgivenby(a+bp)c+bcpµ(p;q,m)=Ae−.2cSinceµ(q;q,m)=mweget()(a+bq)c+b(a+bp)c+bc(p−q)µ(p;q,m)=m+e−.22ccHence,theindirectutilityfunctionis()(a+bq)c+b−cqv(q,m)=m+∗Strictlyspeaking,weshouldbesaying“autilityfunctionconsistentwiththegivendemand,”butwe’lljustsay“theutilityfunction”withtheunderstandingthatanymonotonictransformationofitwouldalsogeneratethesamedemandfunction.
26ANSWERS(VerifythatusingRoy’sidentitywegettheoriginaldemandfunction.)Togetthedirectutilityfunction,wemustsolve()(a+bq)c+b−cqminm+e2qcsuchthatqx+z=−cz−a∗q=,b+cxwhichimpliesthat{}2b+cxac−cx+czu(x,z)=+cx(Again,substitutezbym−pxabove,equatethederivativeoftheresultingexpressionwithrespecttoxtozero,solveforxandrecovertheoriginaldemandfunction.)+px+px=∂x∂x12==⇒b=∂p∂p21Thenegativesemidefinitenessofthesubstitutionmatriximpliesb<0and12bb−b>0.(Provethatthesetwoconditionstogetherimplythatb<0122mustalsohold.)Wehavetosolvethefollowingsystemofpartialdifferentialequations∂µ(p;q,m)=a+bp+bp,1112∂p1and∂µ(p;q,m)=a+bp+∂p2Thefirstequationimpliesb12µ(p;q,m)=ap+p+bpp+C,µ(p;q,m)=ap+p+bpp+
,wemusthavebb1222µ(p;q,m)=ap+p+bpp+ap+p+C1112221222[][][]1abbp111=[p,p]+[p,p]+µ(q;q,m)=m,wehaveb122µ(p;q,m)=m+a(p−q)+(p−q)+b(pp−qq)1111212112b222+a(p−q)+(p−q).222222Theindirectutilityfunctionisgivenbybb1222v(q,m)=m−aq−q−bqq−aq−q1112221222[][][]1abbq111=m−[q,q]−[q,q].(x,z)=min{v(q,m):z+qx+qx=m}.1122qAfterafewminutesofalgebraicfun,wegetb(x−a)−b(x−a)21122∗q=,12bb−b12andb(x−a)−b(x−a)12211∗q=.22bb−b12Substitutingthesevaluesbackintov(·),wegetu(x,z)22b(x−a)+b(x−a)b(ax+ax−xx−aa)21112212211212=z++222(bb−b)bb−b1212[][]1b−bx−a211=z+[x−a,x−a].11222−bbx−a2(bb−b)(p)=v(q/m)anddifferentiatewithrespecttoqandm:i∂v∂v1=∂q∂pmiik∑∂v∂vqi=−2∂m∂pmii=1k∑∂v1=−∂pmii=1
28ANSWERS∂v∂vDividingbyyieldstheresult.∂q∂,-goodsareCobb-23Douglasdemands:bmzz=1b+cp2cmzz=.2b+cp3dµ=a−bp+cµµ(q,q,y)≡(x,y)=min{x,y},andU(V,z)=V+-goodisz=m/pifp<p+>p+p,thenthedemandforthex-goodandthey-goodisgivenbyzxyx=y=m/(p+p).Ifp=p+p,{}mmv(p,p,p,m)=max,.xyzp+−1/2solvefortheindirectutilityfunctiontogetv(p,p,m)=m(pp).1121112NowuseRoy’sidentitytocalculate:1m1x=12p11m1x=.-Douglasutility,wecanwritedownthedemandsimmediately:3m2x=13+ap1am2x=.23+=,23+awhichimpliesthata=3.
'(p)+(p,u)=u−v(p).Theexpenditurefunctionisnecessarilyaconcavefunctionofprices,whichimpliesthatv(p)’sdemandfunctionsforthex-goodandthey-goodtaketheform150x=y=p+,wefindthattheindirectutilityfunctiontakestheform150v(p,p,150)=.xyp+pxyHenceAisthesolutionto150−A150=1+11+2andBisthesolutionto150150+B=.1+11+2Solving,wehaveA=50andB=’stheoremestablishedthat122pi(t)≈r(w)σσtissimplythevarianceofthegamblet²˜.,wemustsolvethedifferentialequation′′′−rxu(x)/u(x)=−(x)=−e,,thedifferentialequationis′′′1−ru(x)x/u(x)=−(x)=x/(1−r)forr6=1andu(x)=lnxforr=,we’veseenthat
30ANSWERSconstantabsoluteriskaversionimpliesthattheutilityfunctiontakesthe−rwformu(w)=−e.′(w)=1−2bw;
(·)suchthatu(µ)=,wehave∫∞∂E[u(R)]1=u(s)(s−µ)f(s)ds>0,2∂µσ−∞sincetheterms[u(s)(s−µ)]andf(s)∫∞∂E[u(R)]122=u(s)((s−µ)−σ)f(s)ds23σ∂σ−∞∫∞1′22<u(µ)(s−µ)((s−µ)−σ)f(s)ds3σ−∞{∫∫}∞∞′u(µ)32=(s−µ)f(s)ds−σ(s−µ)f(s)ds3σ−∞−∞=(·)andthenormaliza-tionimposed;thelastequalityfollowsfromthefactthatRisnormallykdistributedand,hence,E[(R−E[R])]=α∈[0,1][u(αw(1+R)+(1−α)w(1+R))]0102∫∫=u(αw(1+r)+(1−α)w(1+r))f(r)f(r)drdr01021212∫∫>[αu(w(1+r))+(1−α)u(w(1+r))]f(r)f(r)drdr01021212∫∫=u(w(1+r))f(r)dr=u(w(1+r))f(r)dr=01110222=E[u(w(1+R))]=E[u(w(1+R))].0102Theinequalityfollowsfromtheconcavityofu(·).Forpartb,proceedasbeforereversingtheinequalitysincenowu(·)[u(w˜−pi)]=E[u(w˜+²˜)]:uE[u(w˜−pi)]=pu(w−pi)+(1−p)u(w−pi)u1u2u′′≈p(u(w)−u(w)pi)+(1−p)(u(w)−u(w)pi);11u22upE[u(w˜+²˜)]=(u(w−²)+u(w+²))+(1−p)u(w)1122()′′2u(w)²1≈pu(w)++(1−p)u(w).122
32ANSWERSCombining,weobtain1′′′′2−(pu(w)+(1−p)u(w))pi≈pu(w)²,12u12or1′′2−pu(w)²12pi≈.u′′pu(w)+(1−p)u(w)12′′′,theArrow-Prattmeasuresare−u/u=′′′a,and−v/v=>bandwewanttoshowthatavalueof(w−w)12largeenoughwilleventuallyimplypi>pi,thuswewanttogetvu−aw−bw11aebe<;−aw−aw−bw−bw1212pe+(1−p)epe+(1−p)ecross-multiplyingweget−w(a+b)−(aw+bw)−w(a+b)−(aw+bw)112112ape+a(1−p)e<bpe+b(1−p)e,whichimpliespa(w−w)b(w−w)1212(a−b)<be−−pThederivativeoftheRHSofthislastinequalitywithrespecttow−wis12()a(w−w)b(w−w)1212abe−e>0wheneverw>w;,this1212inequalitywilleventuallyholdfor(w−w)-Prattmeasure,’veshownthatvcouldimplyahigherriskpremiumthanutoavoidafairlotteryprovidedthere’sanadditionalrisk“big”,√√114+12+4+0=,√4+p=3.
,wehavep=(w+x)+(1−pi)ln(w−x).Thefirst-orderconditionispi1−pi=,w+xw−xwhichgivesusx=w(2pi−1).Ifpi=1/2,x=−p1+=.www12Aftersomemanipulationwehaveww12w=.pw+(1−p)α(p,α),′′(p)+pi(p)=’slawandHotelling’slemma,wehave−x(p)+y(p)=0,′′measureis−x(p)+y(p)whichisclearlypositive;hence,,thefirmwantstosupplyadifferentamountthantheconsumerwantstodemand;hence,the“welfare”’slawweknowthat∂pi(p,w)/∂p=y(p);therefore,∫p1y(p)dp=pi(p,w)−pi(p,w).10p0
(w,y)y+1y+2=w+√w/w+212y=.mw/w+112Thederivativeofywithrespecttow/wisnegative,sotheminimumm12oftheaveragecostshiftstotheleft(right)asw/wincreases(decreases).12√Infactitconvergesto1astheratioapproaches∞∂c(w,y)=2y(w+w),12∂ysoshort-runsupplyscheduleisgivenbypy(p)=.2(w+w)-runsupplycurveis{arbitrarilylargeamountifp>2y(w+w)m12Y(p)==y+1andx=y+,,12theinputrequirementsetforanindividualfirmis{{√√}}V(y)=(x,x)∈[1,∞)×[2,∞):y≤minx−1,x−(p)=p/(p)=50p∗∗∗(p)=Y(p),wegetp=2andy=1(Y=100).-fore,r=2−1=1.
,;thatis,chooset(s)=,theproducersreceivep−t(s).If∗∗yistoremainoptimal,wemusthavep−t(s)=∂c(w,r−s,y)/∂∗∗∂c(w,r−s,y)∂K(w,r−s,y)′t(s)==.∂y∂r∂y′(w,r−s,y)=K(w,r−s,1)y,theformulareducestot(s)=K(w,r−s,1).∂K/∂y<
,,however,themonopolisthad6unitstosell,,—thatis,aprofit-maximizinglevelofoutputdoesn’,wemusthave1=1,′′′2+yp(y)/p(y)or′′′yp(y)=−p(y).(y)=a−=a/b−p/b,’sprofitmaximizationproblemismaxp(y,t)y−-orderconditionforthisproblemis∂p(y,t)p(y,t)+y−c=0.∂yAccordingtothestandardcomparativestaticscalculations,thesignofdy/,2dy∂p∂psign=sign+∂t∂y∂tForthespecialcasep(y,t)=a(p)+b(t),thesecondtermontheright-handsideiszero.
-maximizinglevelofoutputsincetheelasticityofdemandisconstantat−,−1,=20,whichimpliesy=1/,thederivativeofconsumer’∂u/∂q−∂p/∂qx≡,thismeansthatwemusthave2∂u/∂q≡x∂u/∂x∂(x,q)=f(q)∫∫xx2∂u(z,q)∂p(x,q)dz<dz.∂z∂q∂q00Carryingouttheintegrationgives∂u(x,q)∂p(x,q)<x,∂q∂(x),thenthemostthatitcanchargeforentryistheconsumer’ssurplus,u(x)−p(x),thefirmmakesaprofitofp(x)−(x)−p(x)x+(p(x)−c(x))x=u(x)−c(x).xItfollowsthatthemonopolistwillchoosetheefficientlevelofoutputwhere′′u(x)=c(x).Theentryfeeissetequaltotheconsumer’=-demandconsumer’sinversedemandcurveisalwaysgreaterthantwicethelow-demandconsumer’sinversedemandcurve,thisconditioncannotbesatisfiedandthelow-demandconsumerwillbepushedtoazerolevelofconsumption.
=qandw=-to-onebasis;.,theconsumerwhovaluesqualitythemorehighlyendsupconsumingthesociallyoptimalamount,(p)−c(y(p)).Differentiating,pwehave′′′py(p)+y(p)−c(y)y(p)=′′p+y(p)/y(p)−c(y)=0,or′p[1+1/²]=c(y).()1(1−τ)P=1+²Undertheoutputtaxwehave()1P−t=1+.D²SolveeachequationforP,settheresultsequaltoeachother,andsolveDforttofindτkc1t=k=11−τ1+²’sprofitmaximizationproblemismaxp(y,t)y−-orderconditionforthisproblemis∂p(y,t)p(y,t)+y−c=0.∂yAccordingtothestandardcomparativestaticscalculations,thesignofdy/,2dy∂p∂psign=sign+∂t∂y∂t
(y,t)=a(y)+b(t),thesecondtermontheright-handsideiszero,sothat∂p/∂t=∂b/∂-orderconditionsintheusualwaygives∂x11=<0′′′∂tp−c111∂x12=<0.′′′′′∂t2p+px−=u(x)+u(x)−c(x)−112211c(x).Thetotaldifferentialis22′′′′dW=(u−c)dx+(u−c),weshouldtaxthecompetitiveindustryandsubsidizethemonopoly!Toseethis,combinetheanswerstothefirsttwoquestionstogetthechangeinwelfarefromataxpolicy(t,t).12dxdx12′′dW=(p−c)dt+(p−c)-dustryiszero,,priceexceedsmarginalcost,—.,+r12suchthatax−r≥0111ax−r≥0222ax−r≥ax−r111122ax−r≥ax−r222211x+x≤=randax−r=ax−r,111222211andx+x=+(2a−a)
,ourproblemistosolvemaxax+(2a−a)x22121subjecttotheconstraintthatx+x==10−xandsubstituteintotheobjectivefunctiontogettheproblem21max10a+2(a−a)>athecoefficientonthesecondtermisnegative,whichmeans21∗∗∗∗thatx=0and,therefore,x==10,wemusthaver=1222∗∗=0,wemusthaver=-maximizingchoicesofpandpare12p=a/2b111p=a/
,settheresultsequaltoeachother,andsolveDforttofindτkc1t=k=11−τ1+²=20andsolvingfordemand,wefindthatD(20)=,sop=(1)=<1,thenprofitsaremaximizedatp=3/2+c/+c/≥−x/2+x,thenshewillchoosetopay12122kif(a−p)/2>,shewillbuya−<(2−p)/2,22thendemandisN(4−p)+N(2−p).If(2−p)<k<(4−p)/2,then2demandisN(4−p).Ifk>(4−p)/2,=candk=(4−c)/(4−c)/,ifbothtypesofconsumersbuythegood,=(2−p)/((6−2p)(p−c)+(2−p)/2).Thisismaximizedwhenp=2(c+2)/
≥e,c≥g,b≥d,f≥≥e,b≥,(Swerve,Stay)and(Stay,Swerve).−.25=.9375..,+≤pi+.djrrRearranging,wefindpi−pijcr≤pi−,(Top,Left)and(Bottom,Right).(Top,Left)dominates(Bottom,Right).(Top,Left).,c,
∂F(p,u)/∂u=1−r/,,increasingtheratioofuninformedconsumersdecreasestheprobabilitythatlowpriceswillbecharged,δ=ββ−γ.Thenbydirectcalculation:a=(αβ−αγ)/δ,12iijjb=β/δ,andc=γ/δ.&22Vives(1984).Let∆=4ββ−γ,andD=4bb−−p=αγ/∆andq−q=ac/D,,−1meansthatwhenonefirmreducesitsoutputbyoneunit,itbelievesthattheotherfirmwillincreaseitsoutputbyoneunit,therebykeepingjointoutput—andthemarketprice—,suchanequalitycanonlybeestablishedwheny>
44ANSWERS′(Y)+P(Y)Y=nc+∑nt,andnotethatindustryoutputYcanonlydependonthesumofii=’tchange,∆ymustsatisfyi′P(Y)+P(Y)[y+∆y]=c+t+∆′Usingtheoriginalfirstordercondition,thisbecomesP(Y)∆y=∆t,orii′∆y=∆t/P(Y).==(p)=1000−===50×5==y+y=∗′∗′,weestablishedthatx∼,∗=ay/p=ap/p,x=xsofrombudgetconstraint,(p+1211ABB11p)x=p,sox=p/(p+p).Choosep=1annumeraireandsolve211121BBap+1/(1+p)==ay/p,x=by/py=y=p++x=
∂x∂hii=.∂p∂:
-profitconditionimpliesthatp2x−x==1/=1/-Douglas,=8,1gbbx=18,x=10,x=,,(1987),:(1+pi)pp+≥+rInequilibrium,thisconditionmustbesatisfiedasanequality,sothat1+rp=+r++rp=+r+.
-orderconditionsas′′˜˜˜Eu(C)R=Eu(C)Ra0′′˜˜˜Eu(C)R=Eu(C)˜˜R=V/p,wehaveaaa˜˜R=R−Rcov(F(C),R).,+1,(p),wehavedV(p)=−2z(p)Dz(p)p˙dt−1=−2z(p)Dz(p)Dz(p)z(p)=−2z(p)z(p)<∑∂hjθx=∂pij=1Multiplybothsidesofthisequationbytandsumtogetikk∑∑∑∂hjθR=θtx=∂piij=1i=1
48ANSWERSTheright-handsideofexpressionisnonpositive(andtypicallynegative)θ(p,m)k∑suchthat(p−c)x(p)==1Thisisalmostthesameastheoptimaltaxproblem,wherep−,+b=candb≤,thereare12iialsomanyinefficientequilibria,suchasb=b=,(w)=+=α’sreactionfunctionis11f(w+g)=max{α(w+g)−g,0}.Solvingf(w+αw)=0yields221211221w=(1−α)()wGG=α+.kkSolvingforG,wehaveG=αw/(k−α).Askincreases,,,theamountofthepublicgoodprovidedwillbetheParetoefficientamount:∑1unitifr>c,
(G+g)+w−gi−iiiigisuchthatg≥-orderconditionforaninteriorsolutionisai=1,GorG=,,aParetoefficientamountofthepublicgoodcanbefoundbymaximizingthesumoftheutilities:n∑alnG−G,ii=1∑n∗whichimpliesG==’sutilitymaximizationproblemismaxu(x)−p(x,x)c,11121x1whilethesocialproblemismaxu(x)+u(x)−p(x,x)[c+c].11221212x,x12Sinceagent1ignoresthecostheimposesonagent2,,agent1shouldbechargedafinet=,thenthetotalcostsbornbytheagentsinthecaseofanaccidentare2[c+c],’sobjectivefunctionis(1−p(x,x))u(x)−p(x,x)(x)−p(x,x)[u(x)+c].1112111Thisisjusttheformofthepreviousobjectivefunctionwithu(x)+=u(x)+
(u(s))≡′′′′showsthatf(u)u(s)=(s)>0,wemusthavef(u)>,wehave′′′′′′2f(u)u(s)+f(u)u(s)=0.′′′Usingthesignassumptionsonu(s),weseethatf(u)>,∂V/∂c=λ+µand∂V/∂c=µ.abThus,thesensitivityofthepaymentschemetothelikelihoodratio,µ,,theincentiveconstraintwillnotbebinding,whichimpliesµ=(x),theoriginalincentivescheme(s),∑max(x−s)piiiibi=1n∑suchthatspi−c≥uiibbi=1nn∑∑spi−c≥spi−=1i=1Assumingthattheparticipationconstraintisbinding,andignoringtheincentive-compatibilityconstraintforamoment,wecansubstituteintotheobjectivefunctiontowritem∑maxspi−c−=1Hence,theprincipalwillchoosetheactionthatmaximizesexpectedoutputminus(theagent’s)costs,-compatibilityconstraintbychoosings=x+F,andchooseiiFsothattheparticipationconstraintissatisfied.
−c(x)≥u,whichwecantttwriteass−(c(x)+u).Definec(x)=c(x)+u,,whichaddsanextracasetotheanalysis.′′(x)>c(x),wemusthave21∫∫xx22′′c(x)dx>c(x)=s−c(x)andu=s−c(x).1122Writetheseass=u+c(x)ands=u+c(x).Thedifferencebetween1122thesetwofunctionsisd(x)=u−u+c(x)−c(x),andthederivativeof2121′′′thisdifferenceisd(x)=c(x)−c(x)>,-costworkerstobeemployed,-2∗tracttoahigh-costworkermaximizesx−x,whichimpliesx=1/(1/2)=1/,s−1/4≥u,ors=u+1/,2222∗x≥
