problem sessions before midterm exam

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PROBLEM SESSIONS BEFORE MIDTERM EXAM_30-31.10.2013 & 06-07.11.2013

1. Suppose that due to more stringent environmental regulation it becomes more expensive for steel production firms to operate. Also, recent technological advances in plastics have reduced thedemand for steel products. Use Supply and Demand analysis to predict how these shocks willaffect equilibrium price and quantity of steel. an we say with certainty that the market price for steel will fall! "hy!

Solution: #he increase in the cost of production of steel will shift the supply curve to the left. #his effectalone on the market will influence the market price to rise while the market quantity will fall. #hisis shown above by a movement from the original supply curve S$ to a new supply curve such asS1. #he decrease in demand will cause the demand curve to shift to the left. #his effect alone onthe market will influence the market price and quantity of steel to fall. %ote that the supply and

demand effects on price work in opposite directions. &f the supply effect dominates the demandeffect, the equilibrium prices will rise. #his is exhibited by the decrease in demand to D'(. )n thisdemand curve, the net effect is for prices to rise from * $ to *'(. )n the other hand if the demandeffect dominates, equilibrium prices will rise. #his is exhibited by the decrease in demand to D'((.)n this demand curve, the net effect is for prices to fall from * $ to *'((. As we don(t know giventhe current information which effect dominates, we can(t perfectly predict the change in price.#he change in quantity is unambiguously decreased.

2. +idcontinent *lastics makes $ fiberglass truck hoods per day for large truck manufacturers.-ach hood sells for /$$. +idcontinent sells all of its product to the large truck manufacturers. &f the own price elasticity of demand for hoods is 0$.1 and the price elasticity of supply is './.

a. ompute the supply and demand for truck hoods.

b. &f the local county government imposed a per unit tax of 2/.$$ per hood manufactured,what would be the new equilibrium price of hoods to the truck manufacturer!

c. "ould a per unit tax on hoods change the revenue received by +idcontinent!

Solution:

3iven4 *5 6 /$$ 75 6 $ hoods per day

-d 6 0$.1$ -s 6 './

a. Demand4 7d 6 a) 8 a'* Supply4 7s 6 b) 8 b'*

S.D._2013-2014_Fall



Use4

*

7

7

*-

∆

∆×=

to compute a' and b'.

'a

,$

/$$1.$ =−

' b

,$

/$$/.' =

a1 = -0.064 b1 = 0.24

Solve for a$ and b$

7d 6 a) 8 a'* 7s 6 b) 8 b'*

$ 6 a) 8 0$.$91:/$$; $ 6 b) 8 $.21:/$$;

a = 112 b = -40

!" = 112 - 0.064P !# = -40 $ 0.24P

b. #he tax represents a price increase to the purchaser regardless of the current price. #hus, thesupply curve will be ad<usted vertically upward by 2/.

7s 6 01$ 8 $.21* or

* 6 '99.9= 8 1.'= 7s, then

P% = P $ 2' = 166.67 $ 2' $ 4.17!#

P% = 1(1.67 $ 4.17!# )

!# = -4'.(6 $ 0.24P

#he new equilibrium price will be4

%ew Supply 6 Demand

!#= -4'.(6 $ 0.24P = 112 - 0.064P = !"

S*+, /,*"# P = '1(.60 ) %)c "

c. Since the new selling price in :c; is '1(.60 and the tax is 2' per hood, +idcontinentwould receive only 4(4.6 per hood. As quantity sold has fallen too, revenues wouldfall.

S.D._2013-2014_Fall



3. Suppose that the long0run world "5a" a" #*/ *a#%,c,%,# c)" ,* a) -0.(06 a"

0.'1', respectively. #he current long0run equilibrium price is 30 per barrel and the equilibriumquantity is 16. billion barrels per year. D),+ % *,a) *-) "5a" a" #*/

8a%,#. N9%: ## % *-) #*/ c)+ / "),+" ab+ c#,#%#

c5%,%,+ #*/ a" OPE; #*/. I % *-) c5%,%,+ #*/ 8a%, ,#<

=.= $.2> ,= +C S P

what must be )*-?s level of production in this long0run equilibrium!

Solution: &f the demand curve is linear, it is in the form of

.= + DQ a bP

Also, we know that

'9.$.>$9 $./'$.

@$

= ⇔ = = − = − ÷ ÷ ÷

P Q E b b E

Q P

earranging the linear expression for

demand allows us to solve for a as follows4

'9. $./'$:@$; @2.'$.= − ⇒ = + = D

a Q bP a

"e

may now write the linear expression for demand as

@2.' $./'$ .= − DQ P

&f the supply curve is

linear, it is in the form of

.

= +S

Q c dP

Also, we know that

'9.$./'/ $.2>$.

@$

= ⇔ = = = ÷ ÷ ÷

P Q E d d E

Q P

earranging the linear expression for demand

allows us to solve for c as follows4

'9. $.2>$:@$; .'.= − ⇒ = − =S c Q dP c

"e may now

write the linear expression for supply as

.' $.2>$ .= +S Q P

OPE;# #*/ ,# % ",)c

b%> % >)*" #*/ a" c5%,%,+ #*/ a% 30. ? > %a% >)*" #*/ a% 30

,# 16.. ;5%,%,+ #*/ a% 30 ,#

( )=.= $.2> @$ '9.1.+ =

T,# ,5*,# %a% OPE;##*/ ,# 0.4 b,**, ba))*# ) /a) a% 30 , %,# *-) 8,*,b),5.

4. Barding -nterprises has developed a new product called the 3illooly Shillelagh. #he marketdemand for this product is given as follows4

7 6 21$ 0 1*

a. At what price is the price elasticity of demand equal to Cero!b. At what price is demand infinitely elastic!c. At what price is the price elasticity of demand equal to one!". &f the Shillelagh is priced at 1$, what is the point price elasticity of demand!

Solution:

#he demand curve given in this problem is linear. #he intercepts of the inverse demand curve onthe price and quantity axes are 9$ and 21$ respectively. #he price elasticity of demand variesalong the length of this demand curve. D5a" ,# ,,,%*/ *a#%,c a% % ,%)c% % ),c

a9,#. D5a" ,# c5*%*/ ,*a#%,c a% % ,%)c% % 8a%,%/ a9,#. D5a" ,# ,%

*a#%,c a% % a*->a/ ,% b%> %# %> 9%)5#.

a. #hus, the price elasticity of demand equals Cero :is completely inelastic; at a price of Cero.

b. Demand is infinitely elastic at a price of 9$.c. #he price elasticity of demand equals one at a price of @$.

S.D._2013-2014_Fall



". #he price elasticity of demand equals

* 7

7 *

∆ ÷ ÷∆

. &f * equals 1$, 7 equals $.

7

*

∆ ÷∆

isconstant along a linear demand curve. &n this case it equals 01. #herefore, the price elasticityof demand equals :1$$;:01; 6 02.

'. Suppose that a small market +a<or Eeague Faseball team currently charges '2 for a ticket. Atthis price, they are able to sell '2,$$$ tickets to each game. &f they raise ticket prices to '/, theywould sell '',$/@ tickets to each game. "hat is the price elasticity of demand at '2! &f thedemand curve is linear, what is the algebraic expression for demand!

Solution: #he price elasticity of demand is

'2 >1=$.@'9.

'2, $$$ @

∆ − = = = − ÷ ÷ ÷ ÷∆

P Q E

Q P

&f the

demand curve is linear, it is in the form of

. DQ a bP = + Also, we know that

'2,$$$$.@'9 @'9.

'2

= ⇔ = = − = − ÷ ÷ ÷

P Q E b b E

Q P

earranging the linear expression for

demand allows us to solve for a as follows4

'2,$$$ @'9:'2; '/, =>2.= − ⇒ = + = Da Q bP a

"e

may now write the linear expression for demand as

'/,=>2 @'9 .= − D

Q P

6. #he U.S. Department of Agriculture is interested in analyCing the domestic market for corn. #heUSDA?s staff economists estimate the following equations for the demand and supply curves4

7d 6 ',9$$ 0 '2/*

7s 6 11$ 8 '9/*7uantities are measured in millions of bushelsG prices are measured in dollars per bushel.

a. alculate the equilibrium price and quantity that will prevail under a completely free market.b. alculate the price elasticities of supply and demand at the equilibrium values.c. #he government currently has a 1./$ bushel support price in place. "hat impact will this

support price have on the market! "ill the government be forced to purchase corn under a program that requires them to buy up any surpluses! &f so, how much!

Solution:

a. Set 7d 6 7s to determine price.

'9$$ 0 '2/* 6 11$ 8 '9/*''9$ 6 2>$*

P = 4

)btain 7 by substituting into either expression.

7d 6 '9$$ 0 '2/:1;

7d 6 '9$$ 0 /$$

7 6 ''$$

S.D._2013-2014_Fall



P@ = 4: !@ = 1100

b.

Hor the )wn *rice -lasticity of Demand

$.1/''$$

1'2/- −=×−=

:approximately;

Hor the )wn *rice -lasticity of Supply

$.9$''$$

1'9/- =×=

c. alculate 7d and 7s at the 4.'0 price

7d 6 '9$$ 0 '2/:4.';

!" = 1037.'

7s 6 11$ 8 '9/:4.';

!# = 112.'

#)*# = !# - !" = 112.' - 1037.' = 14'

T #)% ),c >*" c)a% a 9c## #*/ 14' 5,**, b#*# %a% %+)5% >*" b )c" % b/.

7. #he current price charged by a local movie theater is per ticket. #he concession stand at thetheater averages / in revenue for each ticket sold. At the current ticket price, the theater typicallysells @$$ tickets per showing. &f the theater raises ticket prices to >, the theater will sell 2=$tickets. "hat is the price elasticity of demand at ! "hat happens to ticket revenue if the theater increases ticket prices to > from ! "hat happens to concession revenue if the theater increasesticket prices! &f the theater wants to maximiCe the sum of ticket and concession revenue, should

they raise ticket prices to >!

Solution: #he price elasticity of demand at is

@$$..

@$$ '

∆ − = = = − ÷ ÷ ÷ ÷∆

P Q E

Q P

&nitially, ticketrevenue is *57 6 :@$$; 6 2,1$$. &f ticket prices are raised to >, ticket revenue becomes *576 >:2=$; 6 2,1@$. #hus, if ticket prices are raised to >, ticket revenue increases by @$. At ,the concession stand will average ',/$$ per movie showing. &f ticket prices are raised to >, theconcession stand will average ',@/$. #hus, concession stand revenues will fall on average by'/$. I % %a%) >a%# % 5a9,5, % #5 %,c% a" cc##, )+: %/ #*"

% )a,# %,c% ),c# % (.

. Donald derives utility from only two goods, carrots :7c; and donuts :7d;. Bis utility function is as follows4

U:7c,7d; 6 :7c;:7d;

#he marginal utility that Donald receives from carrots :+Uc; and donuts :+Ud; are given as follows4+Uc 6 7d +Ud 6 7c

Donald has an income :&; of '2$ and the price of carrots :*c; and donuts :*d; are both '.a. "hat is Donald?s budget line!

S.D._2013-2014_Fall



b. "hat is Donald?s income0consumption curve!c. "hat quantities of 7c and 7d will maximiCe Donald?s utility!

". Bolding Donald?s income and *d constant at '2$ and ' respectively, what is Donald?s demand curve for carrots!

. Suppose that a tax of ' per unit is levied on donuts. Bow will this alter Donald?s utility maximiCing market basket of goods!

. Suppose that, instead of the per unit tax in :e;, a lump sum tax of the same dollar amount is levied on Donald."hat is Donald?s utility maximiCing market basket!

. #he taxes in :e; and :f; both collect exactly the same amount of revenue for the government, which of the twotaxes would Donald prefer! Show your answer numerically and explain why Donald prefers the per unit taxover the lump sum tax, or vice versa, or why he is indifferent between the two taxes.

S.D._2013-2014_Fall



Solution:

a.

Fudget line4 '2$ 6 7c 8 7d

b.

#he income consumption curve must satisfy4+Ud+Uc 6 *d*c

Substituting for +Ud, +Uc, *d, and *c yields4

7c7d 6 ' or 7c 6 7d

c.

Substituting the information in :b; into the budget line4

'2$ 6 7c 8 7c 6 27c

7c 6 9$

7d 6 9$

d.

ewriting the budget line4

'2$ 6 *c7c 8 7d

Substituting the information in :b; into the budget line4

'2$ 6 *c7c 8 7c 6 7c:*c 8 ';

7c 6 '2$:*c 8 ';

e.

#he ' tax on donuts raises the after0tax price to 2. #he income0consumption curve becomes4

+Ud+Uc 6 *d*c

Substituting for +Ud, +Uc, *d and *c yields4

7c7d 6 2 or 7c 6 27d

#he budget line is4

'2$ 6 7c 8 27d

Substitute the income0consumption curve into the budget line to eliminate 7c4

'2$ 6 27d 8 27d 6 17d

7d 6 @$

7c 6 9$

S.D._2013-2014_Fall



f.

Donald buys @$ donuts, so he pays @$ in tax. &f Donald paid @$ in a lump sum tax, his incomewould be >$. esolve the utility maximiCation problem with & 6 >$, *c 6 *d 6 '.

#he utility maximiCing market basket is 7c 6 7d 6 1/. g.

Donald prefers the lump0sum tax to the excise tax. Use the utility function to show which market

basket is preferred.

U:7c, 7d; 6 7c7d

lump0sum tax U:1/, 1/; 6 1/ x 1/ 6 2,$2/

excise tax U:9$, @$; 6 9$ x @$ 6 ',$$

(. #, % 5%" La)a 5*%,*,)#

-ach day *aul, who is in third grade, eats lunch at school. Be only likes #winkies :#; and )range Slice :S;, andthese provide him a utility of

Utility=U (T , S )=√ TS

a. &f #winkies cost .'$ each and Slice cost .2/ per cup, how should *aul spend the ' his mother gives him inorder to maximiCe his utility! b. &f the school tries to discourage #winkie consumption by raising the price to .1$, by how much will *aul(smother have to increase his lunch allowance to provide him with the same level of utility he received in part :a;!Bow many #winkies and cups of Slice will he buy now :assuming that it is possible to purchase fractional amountsof both of these goods;!

S*%,<

a.

L=T 0.5

S0.5+ λ( I − PT T − PS S)

∂ L

∂T =0.5T −0.5 S0.5− λ PT =0

∂L

∂S=0.5 S

−0.5T

0.5− λ P

S=0

∂L

∂ λ= I − P

T T − P

SS=0

S

T =

PT

PS

=0,100,25

5S=2T

1=0.10T +0.25S

S.D._2013-2014_Fall



S=2 T =5

b.

TS=10

S

T =

PT

'

PS

=0,40

0,25

5S=8T

S' =4 T

' =5 /2

I ' =0.40

(

5

2

)+0.25.4=2

10. #he demand curves for steak, eggs, and hot dogs are given in the table below. #he current price of steak is /.#he price of eggs is 2./$, and the price of hot dogs is $.=/. Hill in the remaining columns of the table using thisinformation. &ndicate which goods are substitutes and which goods are complements.

C" D5a" E8a%,

S%a P),c

E*a#%,c,%/

D5a"

E P),c

E*a#%,c,%/

D5a"

%" P),c

E*a#%,c,%/

D5a"

S%a ' '

/$$ 2'$ '$$

= − − +S S E H

D P P P

E'

=/ @'$

= − − + E E S H

D P P P

%"' '

@$$2 '$

= − + + H H S E D P P P

Solution:

C" D5a" E8a%,

S%a P),c

E*a#%,c,%/

D5a"

E P),c

E*a#%,c,%/

D5a"

%" P),c

E*a#%,c,%/

D5a"

S.D._2013-2014_Fall



S%a ' '

/$$ 2'$ '$$

= − − +S S E H D P P P

0$.$2$ 0$.$$$/' './@

E'

=/ @'$

= − − + E E S H D P P P

0$.$=> 0$.21 '.2$-01

%"' '

@$$2 '$

= − + + H H S E D P P P $.$'9 $.$$$2 0$.$$'2

Steak and eggs are complements. Steak and hotdogs and eggs and hotdogs are substitutes.

11. Answer both parts of the following question.

a. #he San Hrancisco hronicle reported that the toll on the 3olden 3ate Fridge was raised from 2 to @.

Hollowing the toll increase, traffic fell by / percent. Fased on this information, calculate the point priceelasticity of demand. &s demand elastic or inelastic! -xplain.

b. Stephen Eeonoudakis, chairman of the bridge?s finance auditing committee, warned that the toll increase couldcause toll revenues to decrease by 2. million per year. &s this statement consistent with economic theory!-xplain.

Solution:

a.

&ncreasing the toll on the bridge form 2 to @ is a /$ percent increase. #raffic is expected todecrease by / percent as a result of the toll increase. #herefore, the point price elasticity of demandis 0//$ or 0$.'. Demand is inelastic.

b.

Stephen Eeonoudakis? statement is not consistent with economic theory. "hen demand is inelastic,

an increase in price will increase total expenditures on a good :the total expenditure on the good isthe total revenue of the firm;. Since demand is inelastic here, toll revenues will increase rather thandecrease.

12. #he wheat market is perfectly competitive, and the market supply and demand curves are given by the followingequations4

7D 6 2$,$$$,$$$ 0 1,$$$,$$$*

7S 6 =,$$$,$$$ 8 2,/$$,$$$*,

where 7D and 7S are quantity demanded and quantity supplied measured in bushels, and * 6 price per bushel.

a. Determine consumer surplus at the equilibrium price and quantity.b. Assume that the government has imposed a price floor at 2.2/ per bushel and agrees to buy any resulting excess

supply. Bow many bushels of wheat will the government be forced to buy! Determine consumer surplus with the price floor.

Solution:

a.

#he first step is to determine the equilibrium price

( )e P

and quantity

( )eQ

by equating 7D and7S.

S.D._2013-2014_Fall



2$,$$$,$$$ 1,$$$,$$$ =,$$$,$$$ 2,/$$,$$$− = +e e P P

'@,$$$, $$$ 9,/$$,$$$= e

P

2.$$.=e

P

Substitute into 7D or 7S( )2$,$$$,$$$ 1,$$$,$$$ 2= −e

Q

'2,$$$,$$$.=e

Q

#o find consumer surplus, we must also determine the choke price :

C P

;. #hat is, the price atwhich quantity demanded is Cero. solve for * in terms of 7D and 7S.

$ 2$,$$$,$$$ 1,$$$,$$$= − C P

1,$$$,$$$ 2$,$$$,$$$=C

P

/.=C P

( ) ( )'

$./ / 2 '2,$$$,$$$ ',$$$,$$$.2

= − = − =C e eCS P P Q

b.

At price of 2.2/7D 6 2$,$$$,$$$ 0 1,$$$,$$$:2.2/;7D 6 '',$$$,$$$7S 6 =,$$$,$$$ 8 2,/$$,$$$:2.2/;7S 6 '2,92/,$$$

-xcess supply is 7S 0 7D.'2,92/,$$$ 0 '',$$$,$$$ 6 ',92/,$$$

3overnment should expect to buy ',92/,$$$ bushels.

s 6 :$./;:*c02.2/;7D 6 :$./;:/02.2/;:'',$$$,$$$;6'/,'2/,$$$.

S.D._2013-2014_Fall



.S. fell from ',$$$,$$$ to '/,'2/,$$$

13. #he following table gives the current price, quantity, and price elasticities of the linear demand curves for

pencils, paper and scissors. #he columns

rc E

under the *rice -lasticities heading are calculated as

. ∆

= ÷∆ cr

rc

c r

P Q E

P Q

#he terms r and c refer to the row of the table and the column under the price elasticities heading,

respectively. Hor example, if r is one and c is two, the value

'2 E

is the responsiveness of pencil demand to changesin the paper price :i.e., a cross0price elasticity;. #he demand curves for each good are in the form

' 2 @.= + + +r r r r r

Q a b P c P d P

Using the information in the table, derive the demand curve for each good.

D5a" I%5 O> P),c !a%,%/

P),c E*a#%,c,%,#

'r E

2r E

@r E

Pc,*# $.@/ 2/,$$$ 0'.2 $.2/ $

Pa) 2.$$ >$,$$$ $.$' 0$./ $.1/

Sc,##)# @.'/ ',/$$ $ '.2$ 0'.=/

S.D._2013-2014_Fall



Solution: Using the information in the table allows us to solve for the coefficients on prices as follows4

'

'

r r r

Qb E

P

=

G

2

2

r r r

Qc E

P

=

G and

@

@

.r r r

Qd E

P

=

Substituting these coefficient values into the demand

equation allows us to solve for

.r a

#his is done by setting

( )' 2 @ .

r r r r r a Q b P c P d P = − + +

*erforming thesecalculations for the first row of the table gives us the demand for pencils

' ' 2 @1, =/$ /, ='1.@ @,'2/ $ .Q P P P = − + + #he demand for paper is 72 6 '2/,$$$ 8 2,/='.1@*' I @,2/$*2 8

'2,/=.'1*@. #he demand for scissors is

@ ' 2 @2, @2/ $ >$$ @@.@@ .Q P P P = + + −

---------------------------------------------------------------------------------------------------------------------

---------------------

14. #he demand for hamburgers is estimated from this theoretical model4

7 6 k*a

& b

Ac

e,

where 7 6 units per day, * 6 price per unit, A 6 advertising budget per month by sellers, & 6 percapita income of consumers, and e 6 a random error. &n a recent study, one researcher estimatedthe log0linear form of this equation with regression analysis as4

log 7 6 2./ 0 $.@@log * 8 $.'/log & 8 $.2log A.

-xplain what the coefficients of log *, log &, and log A reveal about this product.

Solution:

#he coefficients of the variables are the respective elasticities of demand. #he price elasticity is :0$.@@;, income elasticity is $.'/, and advertising elasticity is $.2. #hese coefficients indicate thatthe product is relatively price inelastic, is a normal good, and is responsive to advertising outlays by sellers.

1'. Adriana is in charge of setting the price on basketball tickets for the local team(s home games.Hrom previous experience, she has estimated demand to be

* 6 /$ 0 $.$$'997,

where * represents price in dollars per seat, and 7 represents seats that could be sold per game.

#he seating capacity is 2/,$$$ seats. Determine the number of tickets that would be sold

at a ticket price of '/ each. Also, determine the consumer surplus that could be absorbed

from these consumers if Adriana were able to set ticket prices so that each customer :who

values the ticket at least at '/; pays the entirety of his or her actual valuation of

the ticket.

Solution:

At * 6 '/, the quantity sold is

/$ '/

$ $$'99

−.

6 2',$1 tickets.

S.D._2013-2014_Fall



#he consumer surplus that could be absorbed is represented by the area under demand and abovethe price line at '/. Area 6 :'2;b5h.

b 6 2',$1 0 $ 6 2',$1

h 6 /$ 0 '/ 6 @/

onsumer surplus 6 :./;:2',$1;:@/; 6 @9,>=$

16. +ay en<oys spending her free time with her friends at the mall and solving problems from her microeconomics text. She has '9 hours per week of free time. Diagram +ay?s time constraint. &f

'

1@

1

= ÷

F

P MU

F

and

@

1'

1

= ÷

P

F MU

P

where F is her time spent with friends at the mall and P isher time spent working problems, how much time should +ay spend at each activity!

Solution:

#he time constraint is '9 6 H 8 *.

Since the price of each activity is equivalent, +ay?s optimal choice will be to set the marginal

utilities of each activity to be equal. Doing so will allow us to solve for time spent withfriends as a function of time spent working problems.

' @

1 1@ '@ .

1 1

= = = ⇒ = ÷ ÷

F P

P F MU MU F P

F P

Hrom +ay?s time constraint, we know that'9 .= + F P

Substituting the optimal choice of F as a function of P into the time constraint

gives us

1'9 1 .

'2

== ⇔

= P

P F

17.

a. )n a given evening J.*. -n<oys the consumption of cigars :; and brandy :F; according to the

function

U:, F;62$028'F0@F2

Bow many cigars and glasses of brandy does he consume during an evening! :ost is no ob<ect to

J.*.;

b. Eately, however, J.*. has been advised by this doctors that he should limit the sum of brandy

and cigars consumed to /. Bow many glasses of brandy and cigars will he consume under

these circumstances!

S*%,<

a.

MU C =∂U

∂C =20−2C =0

S.D._2013-2014_Fall



MU B=∂U

∂B=18−6 B=0

C =10

B=3

b.

B+C =5

S.D._2013-2014_Fall

problem sessions before midterm exam

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