Post on 03-Jun-2018
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Economics01AProblemSetNumber8DUE: Tuesday ovember 9(afterThanksgiving)
L A competitiveirm hasa cost unctionC(V): ll2. Supposehat heownerof the irm is riskneutral nd s given he ollowingoptions:Option 1: (nopricestabilization)he sellingpriceof output s a random ariablehatequals 1withprobability5and$2 withprobability.5.Option 2: (pricestabilization)he sellingpriceof output s $1.50 or sure.(a)Supposehat he irm canmake ts output hoiceafter hepriceof output s known. Showthat heownerpreferso haveanunstabilizedrice, Explainwhy,
hint: Show hat he irm'sprofit unctionn(p): max{ p'y * C(V) is convexExplainwhy thismeanshat he irm likesa variable rice,
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(c) Suppose ow that heowner s risk averse,with a vonNeumannMorgenstern tility function
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ProblemSet#8Continued(b) Supposehat hefirm hiresan economisto setup their compensationackage.Theeconomist bserveshat obsareavailable lsewheren the economyhatpay$y*,andhavenorisk of layoff. Theeconomist roposeso choose and such hat hepackagey,z)minimizestheexpected ostof compensation,ubject o the constrainthat workershave he sameexpectedutility as heycanachieve n the alternativeriskless)ob. Solve he constrainedptimizationproblem. What s true abouty andz7 How do ttreyvary with theprobabilityof layofl?
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qx Wv.= (-r)rf---p. Assumeheconsumer'son-NeumannMorgensterntility function s:
u(z) : - exp - Rz)whereR>0.a) For an arbitraryvon-NeumannMorgenstern tility functionv, the "coefficientof riskaversion" t ncome is defined s r(z)=-u'rrrlv'(z). Show hat or u(z) defined bove'r(z):R.
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b) find theoptimalamount f coverageor theconsumer ndshow hat heuninsuredoss.L- C---
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4. An assistantrofessor tBerkeleys trying o decide ow hard o work. If shegets enure thercurrentjob, ersalarywill ey,. Ifshe sdeniedenure, he angetajob atSecond hanceStateCollege or a lowersalary r. Theprobability f tenures p(a),wherea s thenumber farticles ublished.Herexpectedtility dependsn ncome andeffortspent n writing: u(y) -fia, whereu(y) sconcave nds > 0. Note hatarticles ave o writtenbeforehe enuredecision.a) Develop nexpressionor here*p"cted ttttrr^ *.ri";;
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--4b) Assume '(a)>0andp"(a)< 0. Findanexpressionor a*, neopiinarcnorce raShow hata rise n thesalary , leadso an ncreasen thenumber f articles.HINT: lookat the .o.c. or a*,anddrawa diagam llustratingheoptimum.e u (o.)
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liec) Supposeome eople aveahigh valueof s (s:4"), andothers avea low valueof a (a:ar).Berkeleywants o keep he "low oo'professors,ecauset thinks hat heywill work harder venafter heyget enure Show hatBerkeley ando his by requiring minimumnumber f articlesfor tenure,
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5. Hereare2 "shoitanswer uestions"nauctions f the ype hatare ikely to appear n thefinal exam, Foreach uestion,ndicatef the statements true, alseor uncertain, ndexplainwhy,usingdiagrams,quations,tc 'a. n aprivate aluesEnglish ascendingid) auction, ach iddershould eepbiddingup to hisor herownvaluation.
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