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8/12/2019 Taller Carrin Pinzn

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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,

14t/

f t il"l" lp-'t f * f"*t*th '^^

trl

{t^}'"ttr"l

tt^"^"*^fI

&h4,t"{-th

t

tl-

b,,ft{Jt"" I

C.Ota&

+I'tt--n(r)

(ri-)"i-({.s)

I

I n ( t ) "f;(a),,fi'u-_

{+r

f1

h.*t S paslr'(r-;^ Ir l

9' t / l \e =Qn (p) =. fl '1"

' tn.*.cff.^'cluu =7

(tt^. ul.clrd fh't ('4r, uf" w^*1({r) (r' 'u1"'}'d-'6 op{'o^f

rr^ p4a c,ph'o . *L--1"S*p4^CP -*,


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(c) Suppose ow that heowner s risk averse,with a vonNeumannMorgenstern tility function

-T1,.-u(n):nY,where0(l^( n .c- *'T h @xcr.'*Ly-z tlr

*),r3

tt*)


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= dft{^^l u'

Ly-{'c

(h.^". ,"=, h L*r qT-'b

+ J$ o^T ^ + ' lL or^^{.. rt^ {{*^ stL, blr


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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?

F fea + lt-i lT,t.t. fl wt') + (\-r)f 0,.0)-d a rQ):) 4( lr"rJ*) E- f r=F(- e)tr- ).f n -ta=)(t-iltnra)--fOCs l J'? rs\") : o

,I (*-r)u (l) - o- f tl.\t) -t] {- U(1 ):o

Ktr tlu1*

(*-r)| n,1)

fnl'hJ'f {, CObCzll&

b? Mfy i^t'o tt-- to-h^:*J" Uke b&lL^ rnl^-fto-) I(r-Du = tr(,6, + u(r) - utl) . (-/>

W,,* l't^^*,- ) \L \E) : t (? )t^^r." ).ot unr-o\ *

:/l *

d t') --u' (?)

t tt)


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31 ddetL^^-h"h7 +-p.t l* I bolt ndr. :-t

Lr- r

&a 6>o U0")^hdhtn\ .)z o (a- "fi{^hna"YQu7 *; ^o"""?B

Cnf.ho^:l lt- frl&tlth^Jx- o-c-.{b o- lovtq

i"+'^,cs t* t[d^^t W

u.' l) ( a.' U' ^A b a.t( q/"('co"^l*-'b) 'hrllt

'Ls0y

:) Y---" c fi- wV\ - "tt ivc


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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.

r,.' -) =.r'1 \r) =

T[:s ta 6.+L W-cot^o-t)

co$ue^r { abs"L^}-

* u\[qrri[ arMio, i

* ';rtt

,^\t =


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b) find theoptimalamount f coverageor theconsumer ndshow hat heuninsuredoss.L- C---

is smaller,hebigger s R,and s larger,he argers n/phint: youshould eable o show hat L - C : (l/R) log( n(l -p) / p(l-n) )u(c) = r \L(.I L -...c.c)+(-r)ul?

e' r' i \ '[t-nr0

d(.A-t+ (^*))u't? rf

u\*) - -% (- D'

" - . )

Trq"T -or r(*)i

4-lc(^ ' f

- i1--r-1rI{-0

L,t-?f FTrv?Yn >(

>\-* ) i ' oYL' ,

f $- ' tbWr*rt(r- ol^'-t; trL ";"'*

9 b^,wWco 'L* '

1I-i-1rp^- f

rf

ry (-rW-NO{ f-rt.-D =-

{*( [. t '-q) L

'1i-ffi.e4-1

L(r--.)= tnvffi)zL


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L -c tn* n(ffi--i *aT*)="*nf{H *, H^^ o^rotL;rL o'^f4L W f' L | {L an""' *- [

hl.i'*aaal lPYt *

a,a l*1 , L-a l,>l

V2 >o

aQ ')n S\t) t+Lnlr c- Dbs('-n)o(nDt=6 a((t

["^ 't b6L &(* u(,rI L-#

= { 0- 0 bfl ,'()o nlr),t 4

? (."

t -y= - -t -y

oFtilpu)r - l- - rrf

-L( -t LR.1r\-c--,

>Q

r E-{{ ' jr",*F--*> 2o

"* I rl^,- ryt leI pn'ut*.


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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";;

ll^.o.G.

--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.)

g'

e

( ')' u(?,)u(1")

r(?Dl-- N* \'o t*ghan

drh-;i- b2 Ioay,^I;ot^

IlaIT(',(t') ,^' t)ir'.[?"I

p'(r) = wtl,) U .1)

tJ'\Yr)2


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? rdg,*--?"*dru'(r)p"(^)u-*J

1c(n ui >o

fo t o.,brS a/rl )

\/ ' u$l t r l r \ ' lL r ' o f t ' ( " )

3[") u*-tAn(r)-ult)

+ z-o, t

^n flr t o.(

r\ll, & 't*r "

b- *1,.*-l* t"(r o u, '1,=

I ntr,) \^t i{

?Y\d*\' ruf ,lt(

G*z '&te [r ' l a w(1,) :)

4' u(l) - \I(bt v 'rr oft ("-) r t{* ra'"-f (") ft*"ra- d+c,t'ts ' ASf (^) h corco''xr-VV a'ot,.*1: ) A-* t > d.

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,

3LA^tuA rd.a o*x)e s.b. Pc*)= r t so-> o'-a- < &tx

o- *fa..+,hi1 qnf:{ih^'^"L n *7"- A,E, Low a- dU-


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o+cL. &^cL o\-

b+ - /- dtt.' (tpr^- l.of ^r.+,h,^n q1^r^bn^ta,r-b. wer.L$ v,oaL "ll bV

ScJc .hrc ,t-d-t"utqt'e-,


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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.

Tl-rte , A^ W't^ (;tJ'"*UYd-ea.J( ,

a^.-c-ln'oz- Iwt, O1-z- u^


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