Answers to Homework 4 Summer 2014

7/21/2019 Answers to Homework 4 Summer 2014

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Economics 101Summer 2014Answers to Homework #4Due Thursday, June 12, 2014

Directions: The homework will e collected in a o! before the lecture" lease $lace your name, TAname and section numer on to$ o% the homework &le'ily(" )ake sure you write your name as it a$$earson your *D so that you can recei+e the correct 'rade" ate homework will not e acce$ted so make $lansahead o% time" Please show your work.-ood luck.

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1" Su$$ose you are told that the $roduction %unction %or wid'ets is 'i+en y the %ollowin' e/uation where is the /uantity o% wid'ets, is the numer o% units o% ca$ital used in $roducin' wid'ets, and is thenumer o% units o% laor used in $roducin' wid'ets

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5ou are also told that total cost %or this %irm can e %ound y usin' the %ollowin' e/uation where T6 istotal cost, kis the $rice $er unit o% ca$ital, and l is the $rice $er unit o% laor

T6 3 k 7 l

a" 8sin' this in%ormation, %ill out the %ollowin' tale de$ictin' the $roduction %unction and the cost%unctions %or this %irm" )ake sure you $ro+ide the units o% measurement in your answers" *% necessary,round your answers to the nearest hundredth"

96 :6 T6 A96 A:6 AT6 )60 2;


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)l0 2; 2; ?0 =>0 =;0 =140 =1";0 $er unit o%out$ut

=0"CB $erunit o%out$ut

=2"BB $erunit o%out$ut

=2";0 $erunit o%out$ut

1? 2; C0 =1?0 =;0 =210 =2"00 $erunit o%out$ut

=0"?B $erunit o%out$ut

=2"?B $erunit o%out$ut

=B";0 $erunit o%out$ut

2; 2; 100 =2;0 =;0 =B00 =2";0 $er unit o%out$ut


=B"00 $erunit o%out$ut


" aor is the +ariale in$ut %rom the tale we can see that the amount o% laor +aries as the le+el o%out$ut chan'es while the le+el o% ca$ital is constant and does not chan'e as the le+el o% out$ut chan'es"

c" To %ill in the tale you must %i'ure out the $rice o% laor and the $rice o% ca$ital"

:rom the tale we know that the a+era'e +ariale cost is e/ual to =2"00 $er unit o% out$ut and that thisA96 occurs when the %irm hires 1? units o% laor and $roduces C0 units o% out$ut" ecall that A96 396 and in this case 96 3 l" 8sin' this in%ormation we ha+e =2"00 $er unit o% out$ut 3 l&1? unitso% laor(&C0 units o% out$ut( or l 3 &=2"00 $er unit o% out$ut(&C0 units o% out$ut(&1? units o% laor( 3 =10$er unit o% laor" e+iew this calculation care%ully and note in $articular how the units o% measurementwork out to 'i+e you = $er unit o% the +ariale in$ut"

:rom the tale we know that the a+era'e %i!ed cost is e/ual to =1"2; $er unit o% out$ut and that this A:6

occurs when the %irm hires 2; units o% ca$ital and $roduces 40 units o% out$ut" ecall that A:6 3 :6and in this case :6 3 k" 8sin' this in%ormation we ha+e =1"2; $er unit o% out$ut 3 k&2; units o%ca$ital(&40 units o% out$ut( or k 3 &=1"2; $er unit o% out$ut(&40 units o% out$ut(&2; units o% ca$ital( 3 =2$er unit o% ca$ital" e+iew this calculation care%ully and note in $articular how the units o% measurementwork out to 'i+e you = $er unit o% the %i!ed in$ut"

d" This %irm does e!$erience diminishin' mar'inal returns to laor since as the amount o% laor hiredincreases, out$ut increases ut at a diminishin' rate" Fe can see this y e!aminin' the tale elow that

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shows the mar'inal $roduct o% laor" Gotice that as we mo+e down the tale, the +alues o% the mar'inal$roduct o% laor decrease each additional unit o% laor we hire results in out$ut increasin', ut at adecreasin' rate"

)l0 2; 0 3 2"22 units o% out$ut $er unit o% laor

Gote that the +alues o% were actually calculated in &a("

2" 6onsider the $er%ectly com$etiti+e industry %or 'ad'ets where there are initially 10 identical %irms$roducin'" Each %irm has the same )6 and T6 cur+es and these cur+es are 'i+en y the %ollowin'e/uations where T6 is total cost, / is the amount o% 'ad'ets $roduced y a %irm, and )6 is the mar'inalcost

T6 %or each %irm T6 3 4 7 ;/ 7 /2

)6 %or each %irm )6 3 ; 7 2/5ou also know that the market demand cur+e is 'i+en y the %ollowin' e/uation where is the $rice $er'ad'et and is the total amount o% 'ad'ets demanded in the market

)arket Demand %or -ad'ets 3 1 I

a" -i+en this in%ormation, the %irst thin' you need to do is to determine the short


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Answera" Fe are 'i+en the %irms )6 cur+e )6 3 ; 7 2/" This e/uation tells us that the %irms )6 cur+e has ay


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Demand cur+e %or $roduct 3 11< &11000()6 o% $roduction %or %irm )6 3 2 7 &11000(Total 6ost o% $roduction %or %irm T6 3 2 7 &12000(27 1000

a" Determine the $ro%it ma!imiKin' /uantity and $rice i% this %irm acts as a sin'le $rice mono$olist" Showyour work"

" Determine the %irms total re+enue &T(, total cost &T6( and $ro%it i% it acts as a sin'le $ricemono$olist" Show your work"

c" Determine the +alue o% consumer sur$lus &6S(, $roducer sur$lus &S(, and total sur$lus &TS( i% this %irmacts as a sin'le $rice mono$olist" Show your work"

d" Determine the +alue o% deadwei'ht loss i% this %irm acts as a sin'le $rice mono$olist" Show your work"

Su$$ose that instead o% actin' as a sin'le $rice mono$olist, this %irm acted as i% it was o$eratin' in a$er%ectly com$etiti+e industry" Assume the %ollowin'

)arket demand cur+e %or $roduct 3 11< &11000(

)arket su$$ly cur+e %or $roduct 3 2 7 &1000(Total cost o% $roduction o% 'ood T6 3 2 7 &12000(27 1000

e" *% this %irm o$erated as i% it was a $er%ectly com$etiti+e market, what would e the market e/uilirium$rice and market e/uilirium /uantity@ Show your work"

%" *% this %irm o$erated as i% it was a $er%ectly com$etiti+e market, what would e the %irms total re+enue&T(, total cost &T6(, and $ro%it under these assum$tions@ Show your work"

'" *% this %irm o$erated as i% it was a $er%ectly com$etiti+e market, what would e the +alue o% consumersur$lus &6S(, $roducer sur$lus &S(, and total sur$lus &TS( under these assum$tions@ Show your work"

h" *% this %irm o$erated as i% it was a $er%ectly com$etiti+e market, what would e the +alue o% deadwei'htloss under these assum$tions@ E!$lain your answer"

Answer

a" To %ind the $ro%it ma!imiKin' out$ut and $rice %or the sin'le $rice mono$olist we need to set )6 3)" Fe know that )6 3 2 7 &11000( , ut we need to %ind )" 8se the demand cur+e to 'et )recall that ) %or the mono$olist will ha+e the same y

3 B000To %ind the e/uilirium $rice %or the sin'le $rice mono$olist, sustitute 3 B000 into the demande/uation 3 11 I &11000(&B0000( 3 =C $er unit o% out$ut"

" T 3 3 &=C $er unit o% out$ut(&B000 units o% out$ut( 3 =24,000T6 3 2 7 &12000(27 1000 3 2&B000( 7 &12000(&B000(&B000( 7 1000 3 =11,;00ro%it %or sin'le $rice mono$olist 3 T I T6 3 =24,000 < =11,;00 3 =12,;00

c" 6S 3 &12(&=11 $er unit o% out$ut < =C $er unit o% out$ut(& B000 units o% out$ut( 3 =4;00

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S 3 &12(&=; $er unit o% out$ut < =2 $er unit o% out$ut(&B000 units o% out$ut( 7 &=C $er unit o% out$ut < =;$er unit o% out$ut(&B000 units o% out$ut( 3 =4;00 7 =>000 3 =1B,;00TS 3 6S 7 S 3 =4;00 7 =1B,;00 3 =1C,000

d" DF 3 &12(&=C $er unit o% out$ut < =; $er unit o% out$ut(&4;00 units o% out$ut I B000 units o% out$ut(3 =22;0

e" To %ind the market e/uilirium set the market demand cur+e e/ual to the market su$$ly cur+e11 I &11000( 3 2 7 &11000(> 3 &1;00( 3 4;00 units o% out$ut8se either the market demand cur+e or the market su$$ly cur+e to %ind the market e/uilirium $rice i% the%irm acts as i% this is a $er%ectly com$etiti+e market 3 11 I &11000( 3 11 I &11000(&4;00 units o% out$ut( 3 =?";0 $er unit o% out$utLr, 3 2 7 &11000( 3 2 7 &11000(&4;00 units o% out$ut( 3 =?";0 $er unit o% out$ut

%" T 3 3 &=?";0 $er unit o% out$ut(&4;00 units o% out$ut( 3 =2>,2;0T6 3 2&4;00( 7 &12000(&4;00(&4;00( 7 1000 3 =20, 12;

ro%it i% the %irm acts like this is a $er%ectly com$etiti+e market 3 =>12;

'" There is no DF i% the %irm $roduces 4;00 units o% out$ut and sells them %or =?";0 $er unit o% out$utsince at this le+el o% $roduction the )6 o% $roduction is e/ual to the $rice that the 'ood sells %or" ecallthat when 3 )6 %or the last unit sold, this tells us that the socially o$timal amount o% the 'ood is ein'$roduced"

4" Su$$ose you are 'i+en the %ollowin' 'ra$h o% a mono$oly"

a" -i+en this 'ra$h, write an e/uation %or this mono$olists mar'inal re+enue cur+e &)("

" *% this mono$olist is a sin'le $rice mono$olist, what $rice and out$ut will this mono$olist $roduce@Fhat will e the mono$olists $ro%its at this $rice and out$ut comination@ Show how you %ound youranswers"

?


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c" *% this mono$olist is instead re'ulated to $roduce the socially o$timal amount o% the 'ood, what $riceand out$ut will the mono$olist $roduce@ Fhat will the re'ulatory authority need to do in order to 'et themono$olist to $roduce at this $rice and out$ut le+el@ E!$lain your answer %ully"

d" *% this mono$olist is instead re'ulated to $roduce at an out$ut le+el where the %irm reaks e+en, what$rice and out$ut will the mono$olist $roduce@ *s there a deadwei'ht loss associated with this le+el o%

$roduction@ E!$lain your answer +erally and then $ro+ide a mathematical calculation o% this area o%deadwei'ht loss"

Answer

a" :rom the 'ra$h it is relati+ely easy to write the mono$olists demand cur+e 3 100 I &11000(" Lncewe ha+e the demand cur+e, then we can write the mono$olists ) y recallin' that ) %or themono$olist will ha+e the same y


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;" Su$$ose a mono$olist can e descried y the %ollowin' e/uations)ono$olists demand cur+e 3 2;0 I ;)6 %or the mono$olist )6 3 B0Assume that the mono$olist has no %i!ed costs

a" Determine the $ro%it ma!imiKin' le+el o% out$ut %or this mono$olist i% it char'es a sin'le $rice"Determine the $ro%it ma!imiKin' /uantity and the le+el o% the mono$olists $ro%it" Show your work" Also,

enter your answers in the tale elow &you will e addin' to this tale throu'hout your work on this$rolem("

ricin' Scheme rice 6har'ed uantity roduced ro%it

a( Sin'le rice )ono$olist

( )ono$olist char'es two$rices

c( )ono$olist $ractices$er%ect $rice discrimination

" Su$$ose this mono$olist decides that it will char'e 2 $rices =140 %or the %irst 22 units and =100 %or thene!t C units" Determine the $ro%it the mono$olist will earn when it char'es two $rices and then enter your

answers into the ao+e tale" Show how you %ound your answers"

c" Su$$ose this mono$olist decides to e a $er%ect $rice discriminator" Fhat /uantity o% the 'ood will themono$olist choose to $roduce@ E!$lain your answer" Then, calculate the +alue o% the mono$olists $ro%iti% the mono$olist is a $er%ect $rice discriminator" Show your work and then enter your answers in theao+e tale"

d" :rom your analysis in &a( throu'h &c(, what conclusion do you draw aout $rice discrimination@E!$lain your answer %ully"

Answera" To determine the $ro%it ma!imiKin' /uantity and $rice i% this %irm char'es a sin'le $rice %or its $roduct

we need to e/uate ) to )6 and then use the determined /uantity in the mono$olists demand cur+e to%ind the $rice the mono$oly will char'e" Fe are 'i+en the )6 cur+e %or the %irmM we need to %ind the )cur+e %or the %irm" Fe can use the demand cur+e to %ind the ) cur+e %or the %irm 3 2;0 I ; andthus, ) 3 2;0 I 10 since the ) cur+e shares the y00, or =2>C0"

C


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c" *% the mono$olist is a $er%ect $rice discriminator then the mono$olist will $roduce that amount o% the'ood where 3 )6 %or the last unit $roduced" This will e the socially o$timal amount o% the 'ood" To%ind this /uantity use the demand cur+e and the )6 cur+e B0 3 2;0 I ; or 3 44 units" The %irm will$rice each o% these units at a di%%erent $rice asically the %irm will $rice alon' its demand cur+e with the%irst unit ein' sold at 3 2;0 I ;&1( 3 =24; $er unitM the second unit ein' sold at 3 2;0 I ;&2( 3 =240$er unitM and on and on" Gotice howe+er that this $ro+ides a ste$ y ste$ oriented to this T calculation

&re+iew this in your te!t i% you %ind this hard to understand( and we would $re%er to use a smoothorientation to this T calculation" So, lets ima'ine that we can $rice this 'ood at really small /s so thatwe asically take the T as the sum o% the area o% the trian'le ao+e the )6 cur+e and the rectan'leelow the )6 cur+e" Thus, T 3 &12(&=2;0 $er unit < =B0 $er unit(&44 units( 7 &=B0 $er unit(&44 units( 3=4C40 7 =1B20 3 =?1?0" T6 3 &=B0 $er unit(&44 units( 3 =1B20" &Gotice, we could ha+e sa+e some ste$sy reco'niKin' that the rectan'le &=B0 $er unit(&44 units( is $art o% T and also e/ui+alent to T6 this areathere%ore is Nnetted outO and does not need to e e+en calculated in order to %ind the %irms $ro%its"( ro%it%or the $er%ect $rice discriminatin' mono$olist is e/ual to =4C40"

d" *% we e!amine the tale we see the more $rice discrimination there is the 'reater the $ro%it %or the %irm"Gote, also that the more $rice discrimination there is the closer we 'et to $roducin' the socially o$timalamount o% the 'ood its Pust with $er%ect $rice discrimination the mono$olist is ale to Nca$tureO all o%

the consumer sur$lus and the consumer 'ets none o% this sur$lus"

Heres the com$leted tale

ricin' Scheme rice 6har'ed uantity roduced ro%it

a( Sin'le rice )ono$olist =140 $er unit 22 units =2420

( )ono$olist char'es two$rices

=140 $er unit %or the %irst 22 units and=100 $er unit %or the ne!t C units

B0 units =2>C0

c( )ono$olist $ractices$er%ect $rice discrimination

A di%%erent $rice %or e+ery unit$roduced

44 units =4C40

?" 6onsider a mono$olist who sells her $roduct to two di%%erent classes o% customers" Su$$ose this

mono$olist knows the %ollowin' in%ormationDemand %or $roduct %rom class one 3 10 I Demand %or $roduct %rom class two 3 ; I &12()6 3 =2 $er unitM and assume there are no %i!ed costs %or this mono$olist

a" :ind the market demand cur+e %or the mono$olists $roduct i% there are only these two classes o%uyers" )ake sure you identi%y any ran'es &or domains( where a$$ro$riate"

" *% the mono$olist char'es a sin'le $rice to all the uyers o% its $roduct, what $rice will she char'e andhow many units will the mono$olist $roduce@ Show how you %ound this $rice and /uantity" Then,calculate the amount o% $ro%its the mono$olist will earn with this comination o% $rice and /uantity"

c" Su$$ose this mono$olist decides to en'a'e in third de'ree $rice discrimination" Fhat $rice and/uantity will she $ro+ide to class one uyers@ Fhat $rice and /uantity will she $ro+ide to class twouyers@ Fhat will $ro%its %rom class one e %or this $rice discriminatin' mono$olist@ Fhat will $ro%its%rom class two e %or this $rice discriminatin' mono$olist@ Fhat will total $ro%its e %or this $ricediscriminatin' mono$olist@ *s it $ro%itale to en'a'e in third de'ree $rice discrimination@ E!$lain youranswers %ully"

Answer

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a" To %ind the market demand cur+e %or this mono$olist we will need to sum the indi+idual 'rou$ demandcur+es to'ether horiKontally" :or $rices that are 'reater than or e/ual to =; $er unit, class two does notha+e any demand %or the $roduct since the $roduct is too e!$ensi+e %or them" So, %or $rices 'reater thanor e/ual to =; $er unit, the market demand cur+e is Pust class ones demand cur+e or 3 10 I " :or$rices less than or e/ual to =; we need to do a it more work" * sketched the two 'rou$s demand cur+eson two se$arate 'ra$hs arran'ed horiKontally ne!t to each other and then %ound at =; $er unit, only ; units

o% the 'ood were demanded y class one and Kero units y class two so, one $oint on the market demandcur+e would there%ore e &, ( 3 &; units, =; $er unit(" Then * thou'ht aout the two !1B &youwill see in a minute why * am kee$in' this as a %raction(" T6 3 2&( 3 14" ro%it with this $rice and/uantity is e/ual to >1B < 42B 3 4>B 3 =1?"BB" This is a hi'her le+el o% $ro%its than the %irm 'ets when itsells only to the %irst se'ment o% the market demand cur+e"

So, the %irm char'es =4"BB $er unit and $roduces units"

c" *% this %irm $ractices third de'ree $rice discrimination we will need to %i'ure out what total /uantity o%the 'ood the %irm will $roduce and then how the %irm will allocate this $roduction etween the two

classes o% uyers" :rom &( we know that the lower se'ment o% the market demand cur+e intersected the)6 cur+e at units" ets call this the total out$ut that the %irm will $roduce" Then, lets consider how toallocate these units o% out$ut"

*n class one, the )6 o% $roducin' the 'ood is =2 $er unit" The ) cur+e %or class one uyers is ) 3 10I 2" Settin' ) 3 )6 %or this class o% uyer we 'et 10 I 2 3 2 or 3 4 units" 6lass one should 'et 4units o% the total out$ut o% units allocated to the uyers in this class that im$lies that class two will 'etthe remainin' three units &we will check that in Pust a it(" Fhat $rice will class one $ay@ Sustitute 3 4into the demand cur+e %or this class 3 10 I 3 10 I 4 3 =? $er unit" That im$lies that T %rom class

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one will e =24" T6 %or this le+el o% $roduction %or class one will e &=2 $er unit(&4 units( 3 =C" ro%it%rom class one will e =1?"*n class two, the )6 o% $roducin' the 'ood is =2 $er unit" The ) cur+e %or class two uyers is ) 3 ;

Answers to Homework 4 Summer 2014

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