UNIVERSITY OF TORONTO SCARBOROUGH
DEPARTMENT OF MANAGEMENT
MGEB02: Price Theory: A Mathematical Approach
Instructor: A. Mazaheri
Sample Final (Solutions)
Instructions:
This is a closed book test. You are allowed the use of a non-programmable
calculator
You have 150 minutes.
Good Luck!
Page 1 of 16
Answer all following 6 questions in the Exam Paper:
际 [35 Points] Answer the following short questions.
Final Question-1
optimal Consumption bundle
mhs
学
a) (8 Points) A consumer is considering choosing a calling plan for her cell phone. The
plan has a fixed monthly fee of $40, and it gives 200 free minutes per month and charges
$0.1 for each additional minute. The consumer has a monthly income of $100,2and she
spend it on cell phone and another composite good y, where py =$1. Her utility function
y
U ( x, y )
x
2 , where x is the minutes of cell phone she uses in a month.
is given by
Find her optimal consumption bundle. Graph your solution on a diagram – including the
budget line and a representative indifference curve.
队
xitiy
mas
Solution:
mas
器
0.5 x 0.5
MRS
0.5
! x 100
dx
̅
0.1
0.1
点 位
声
tin
irl_arners.tn
0.01
This is an interior solution but is not optimal since there is 200 free minutes.
ŸShould expect a corner solution, with x = 200 and y = 60.
ŸCheck MRS at this bundle (1/2000.5 < 0.1).
ŸShe tries to increase y and reduce x. But this is not feasible.
ŸThe in
corner solution is optimal.
we can
ci
ŸThe utility level at the corner solution is 2000.5 + 60/2 = 44.14 > 1000.5 + 60/2.
啖的
the graph
bundle
the
time
find
o
op
y
200
200
600
so3x
at9X
y 60
些
ucx y
Ucx y
200
号
44.14
60
200
600
Page 2 of 16
800
Answer all following 6 questions in the Exam Paper:
Question-1 [35 Points] Answer the following short questions.
a) (8 Points) A consumer is considering choosing a calling plan for her cell phone. The
plan has a fixed monthly fee of $40, and it gives 200 free minutes per month and charges
$0.1 for each additional minute. The consumer has a monthly income of $100, and she
spend it on cell phone and another composite good y, where py =$1. Her utility function
y
U ( x, y )
x
2 , where x is the minutes of cell phone she uses in a month.
is given by
Find her optimal consumption bundle. Graph your solution on a diagram – including the
budget line and a representative indifference curve.
xitiy
100 40
Δ
100 Solution:
60
L
60
L
0.5 x 0.5
MRS
10.5
! x 100
1
600
0.1
600
1
MRS
700
800
器
This is an interior solution but is not optimal since there is 200 free minutes.
ŸShould expect a corner solution, with x = 200 and y = 60.
ŸCheck MRS at600
this bundle
(1/2000.5 < 0.1).
800
2
ŸShe tries to increase y and reduce x. But this is not feasible.
ŸThe corner solution is optimal.
weŸThe
utility level at the corner solution is
2000.5 + 60/2 = 44.14 > 1000.5 + 60/2.
must
cost
I
mas
pay the fix
y
which is4 buck
per month
I when we Pay 40bucks
we can get 200free mints
i
0
0.1
tˊ__T
兴 主六
The optimelbundieat.si
x
100
X 100 however we cannot
make that in the question because
60
we will
get 20s free wins afterDay hefix lose
200
s100
irene is a
200
600
corner
solution
800
Page 2 of 16
cty 器
Answer all following 6 questions in the Exam Paper:
Question-1 [35 Points] Answer the following short questions.
a) (8 Points) A consumer is considering choosing a calling plan for her cell phone. The
plan has a fixed monthly fee of $40, and it gives 200 free minutes per month and charges
$0.1 for each additional minute. The consumer has a monthly income of $100, and she
spend it on cell phone and another composite good y, where py =$1. Her utility function
y
U ( x, y )
x
2 , where x is the minutes of cell phone she uses in a month.
is given by
Find her optimal consumption bundle. Graph your solution on a diagram – including the
budget line and a representative indifference curve.
Solution:
0.5 x 0.5
MRS
0.5
! x 100
0.1
This is an interior solution but is not optimal since there is 200 free minutes.
ŸShould expect a corner solution, with x = 200 and y = 60.
ŸCheck MRS at this bundle (1/2000.5 < 0.1).
ŸShe tries to increase y and reduce x. But this is not feasible.
ŸThe corner solution is optimal.
ŸThe utility level at the corner solution is 2000.5 + 60/2 = 44.14 > 1000.5 + 60/2.
y
60
200
600
Page 2 of 16
800
b) (8 Points) In a perfectly competitive market, there are 100 firm split equally between
the following short run cost functions:
C1 (q 1 ) 2q 10q 1 200
品 C (q ) 0.5q 10q 50
2
2
2
2
AT
ne
2
1
2
P
Find the total short run market supply curve and graph it.
ma Arce
Solution:
M4
10
49
10
29
AVC
shut down points
Avc
MC
SRMC 4q 10
, SRAVC 2 q 10
C1 (q 1 ) 2q 12 10q 1 200
1
P
1
10
49
4q 10(if P t 10) 22
SRMC
SRAVC ( q1
0, p
1
10 shoutdown
10 point)
SRMC
q 2 10, SRAVC
9
0
SRMC
SRAVC ( q 2
0, p
C 2 (q 2 ) 0.5q 22 10q 2 50
P
P
mC
q 2 10(if P t 10)
Market supply:
q1
q2
Q
Q
P 10
7 P 42 10
491 P 10
0.5q 2 10
10 shoutdown point)
本P
Marketsuppb.fi
P
2.5(ifP t 10)
4
P 10(ifP (t 10)
MCz
0
mLz
50(
A C
92
0
MC
p
lzt 10
10
P
A(Pv 10L) 0.592
q2 10
P
2.5) 50( P 10
v ) 62.5P
z - 625
4
P
10
( P t 10)
10
0.512 10
f2
10
shut douupoinec
market suppy
7921
S
10
Ī
7
10
p
10
Q
10
Q
fp
5
0
P 10Htip.I v5
Page 3 of 16
Final Part2 P10 14
7min
b) (8 Points) In a perfectly competitive market, there are 100 firm split equally between
5不
the following short run cost functions:
C1 (q 1 ) 2q 12 10q 1 200
吕 C (q ) 0.5q 10q 50
2
2
mCEA_vc.intdownpoint
p
Find the total short run market supply curve and graph it.
2
2
MC
Solution:
mLi
2
4q t10
Avc
AVC
29 10
shutdownpoints
C1 (q 1 ) 2q 12 10q 1 200
4q 10 2q.tl0
q 0 P MC 点
p 49 t10 4q p 10
2.54210
Marketsup以41
Market supply:
MCL q2 10 AVC2 0592 10
SRMC
4q 1 10, SRAVC
2 q1 10
SRMC
SRAVC ( q1
10 shoutdown point)
0, p
4q 1 10(if P t 10)
P
C 2 (q 2 ) 0.5q 22 10q 2 50
SRMC
q 2 10, SRAVC
0.5q 2 10
SRMC
SRAVC ( q 2
10 shoutdown point)
0, p
q 2 10(if P t 10)
P
q1
P
2.5(ifP t 10)
4
z
P 10(ifP (t 10)
2
0
10
92
P
Q
Q
50(
Shutdown points
Avec
Mq C
4
q
0.92
1s
( P 10)
2.5) 50( P 10) 62.5P - 625
p q2
10
YPC10
P210
10
( P t 10)
R tsnP.lt
92
Q
Q
17
s
s
印 12.5
10
Page 3 of 16
b) (8 Points) In a perfectly competitive market, there are 100 firm split equally between
the following short run cost functions:
C1 (q 1 ) 2q 12 10q 1 200
1 ㄟ ni
C 2 (q 2 ) 0.5q 22 10q 2 50
C
shutdownpoint P AUC
Find the total short run market supply curve and graph it.
Solution:
C1 (q 1 ) 2q 12 10q 1 200
SRMC
4q 1 10, SRAVC
2 q1 10
SRMC
SRAVC ( q1
10 shoutdown point)
P
4q 1 10(if P t 10)
0, p
1
Ef_Tp.mi. u
C 2 (q 2 ) 0.5q 10q 2 50
2
2
SRMC
q 2 10, SRAVC
0.5q 2 10
SRMC
SRAVC ( q 2
10 shoutdown point)
P
q 2 10(if P t 10)
Market supply:
q2
P
2.5(ifP t 10)
4
P 10(ifP (t 10)
Q
0
Q
50(
q1
0, p
( P 10)
P
2.5) 50( P 10) 62.5P - 625
4
( P t 10)
10
Page 3 of 16
c) (6 Points) Explain why a monopolist will never produce in the inelastic portion of its demand
curve, (say the demand of; P = 10 – 2Q.) Draw a properly-labeled graph to support your
argument.
Solution:
Pneyi.simiuniieii tic
d) A monopolist will never set a price at which the (linear) demand curve is inelastic because it is not the
profit-maximizing price. Profit is maximized at the output level where MR = MC. Since MC is positive
个 be associated with the elastic portion of the demand curve.
(MC>0), the profit-maximizing output will
niii.ci
Another way to conclude this is to first realize the relationship between TR and P. If P increases, TR
increases WHEN demand is inelastic. If P decreases, TR increases, WHEN demand is elastic. TR is
maximized when demand is unit-elastic. The figure below shows how this is illustrated using TR and TC
curves. As indicated, TR reaches a peak when demand is unit-elastic (Ed = -1), and falls when elasticity is
> -1 (less than 1 in absolute value). TC certainly decreases as output falls. Thus, starting in the inelastic
portion of the demand curve, profit will definitely increase (because TR rises and TC falls) as price is
increased & quantity is decreased. Maximum profit is where the vertical distance between TR and TC is the
greatest, which occurs on the upward sloping part of the TR curve (since that is where MR > 0 in order to =
MC > 0).
1
领
1 MR
iip
$/unit
Elastic portion K< -1
K= -1
imsnoplycluugsproduceprodnetul.eu
P
MC
Inelastic portion K> -1
M不
me
i As
$ [not $/unit
we
produce
in
pyuiil
Q
$
because this is a
uwn it isTC elastic
Total graph]
products
MR
can see
on
D
TR
S max
Q
Page 4 of 16
s
c) (6 Points) Explain why a monopolist will never produce in the inelastic portion of its demand
curve, (say the demand of; P = 10 – 2Q.) Draw a properly-labeled graph to support your
argument.
Solution:
简hiiast.ie
go n
1iniinCticb
d) A monopolist will never set a price at which the (linear) demand curve is inelastic because it is not the
profit-maximizing price. Profit is maximized at the output level where MR = MC. Since MC is positive
(MC>0), the profit-maximizing output will be associated with the elastic portion of the demand curve.
Another way to conclude this is to first realize the relationship between TR and P. If P increases, TR
increases WHEN demand is inelastic. If P decreases, TR increases, WHEN demand is elastic. TR is
maximized when demand is unit-elastic. The figure below shows how this is illustrated using TR and TC
curves. As indicated, TR reaches a peak when demand is unit-elastic (Ed = -1), and falls when elasticity is
> -1 (less than 1 in absolute value). TC certainly decreases as output falls. Thus, starting in the inelastic
portion of the demand curve, profit will definitely increase (because TR rises and TC falls) as price is
increased & quantity is decreased. Maximum profit is where the vertical distance between TR and TC is the
greatest, which occurs on the upward sloping part of the TR curve (since that is where MR > 0 in order to =
ifü
MC > 0).
$/unit
m不
Elastic portion K< -1
K= -1
P
MC
me Ac
Inelastic portionwhen
K> -1
i monopolist will produce product
c
i Point A inte graphCmc A
D
in tufuaph
A$/unitis elastic
[not
$
point$because
this is a
MR
的 we can see
Q
TC
Total graph]
TR
S max
Q
Page 4 of 16
照
me
c) (6 Points) Explain why a monopolist will never produce in the inelastic portion of its demand
curve, (say the demand of; P = 10 – 2Q.) Draw a properly-labeled graph to support your
argument.
Solution:
d) A monopolist will never set a price at which the (linear) demand curve is inelastic because it is not the
profit-maximizing price. Profit is maximized at the output level where MR = MC. Since MC is positive
(MC>0), the profit-maximizing output will be associated with the elastic portion of the demand curve.
Another way to conclude this is to first realize the relationship between TR and P. If P increases, TR
increases WHEN demand is inelastic. If P decreases, TR increases, WHEN demand is elastic. TR is
maximized when demand is unit-elastic. The figure below shows how this is illustrated using TR and TC
curves. As indicated, TR reaches a peak when demand is unit-elastic (Ed = -1), and falls when elasticity is
> -1 (less than 1 in absolute value). TC certainly decreases as output falls. Thus, starting in the inelastic
portion of the demand curve, profit will definitely increase (because TR rises and TC falls) as price is
increased & quantity is decreased. Maximum profit is where the vertical distance between TR and TC is the
greatest, which occurs on the upward sloping part of the TR curve (since that is where MR > 0 in order to =
MC > 0).
$/unit
是新和
mC
Elastic portion K< -1
nhvezptjg
P
K= -1
i130 02013
MC
Inelastic portion K> -1
P 130 Δ2C13
MR
$ [not $/unit
because this is a
Total graph]
DCID
Q
$
不 Iii
品
TC
úttǜm
TR
Rco
S max
i不
P.Q
M不交点
Q
i不 P.Q
Page 4 of 16
d) (7 Points – a bit challanging) A consumer spends all of his income on x and y. Last
year, he consumed 20 units of x at a price of $50 per unit, and 50 units of y at price $40
per unit. This year, he got some good news and some bad news. The bad news was that
the price of y had risen to $50 per unit. The good news was that the price of x has
declined to $25 per unit. Using budget constraints and indifference curves, show that the
consumer's utility can be increased following these changes.
Px
p1x
Py
50
Solution:
x 20
40
I 50, P
40, x 20, y 50
p xxtpyy
1
y
1
1
I 50 u 20 40 u 50 $3000
I 25, p50420
50
p x2
50
y
2
y
50X40
3000
50 x 40 y 25 x 50 y
X2 (intersection )
x 20, y 5025
P
Pyz 50
3000 25X
4254504
t50y
3000
50
50七 4oy 3000
1
yn
betteroff
20
4 50
20
g
__p
x
L
x
2
Page 5 of 16
d) (7 Points – a bit challanging) A consumer spends all of his income on x and y. Last
year, he consumed 20 units of x at a price of $50 per unit, and 50 units of y at price $40
per unit. This year, he got some good news and some bad news. The bad news was that
the price of y had risen to $50 per unit. The good news was that the price of x has
declined to $25 per unit. Using budget constraints and indifference curves, show that the
consumer's utility can be increased following these changes.
50 Py 40 X
Px
Solution:
I
p1x
50
x 20, y 50x2
P Xt40,Pyy
50, p1y
1
1
20
4s x50
y
50
3000
I 50 u 20 40 u 50 $3000
Pxz
p 25, p2550 Py2 50
2
x
50 x 40 y
2
y
25 x 50 y
3000
50y
on )
50(intersecti
x 20, y 25X
X 20 y f
Jsx 4sy
c intersection
better off
50
i
x
20
20
Page 5 of 16
d) (7 Points – a bit challanging) A consumer spends all of his income on x and y. Last
year, he consumed 20 units of x at a price of $50对
per unit, and 50 units of y at price $40
per unit. This year, he got some good news and some bad news. The bad news was that
the price of y had risen to $50 per unit. The good news was that the price of x has
declined to $25 per unit. Using budget constraints and indifference curves, show that the
consumer's utility can be increased following these changes.
Py
px
Solution:
p1x
50, p1y
40, x1
20, y1
50
I 50 u 20 40 u 50 $3000
p x2
25, p 2y
I
50
Poxtpyy
50 x 40 y 25 x 50 y
x 20, y 50(intersection )
off
50
rsbtter
old
Men
x
20
Page 5 of 16
e) (6 Points – More changing) A food coupon program requires families to pay a certain
amount for food coupons. Suppose all families can receive $150 in food coupons for a
payment of $50 (call this policy A). Also, assume all households have $250 of income
and the price of food is $1 per unit. With the composite consumption good (CCG) on the
y-axis and food on the x-axis, draw the original budget line and the budget line under this
policy.
Compared with an allocated grant of $100 in food coupons (call this policy B), would
policy A lead to more, less, or the same food consumption? Why? Assume well-behaved
indifference curves.
250
50
20s
350
100 250
i
Plan A: The new budget line is RR'Z' - subtract $50 from the purchase of other
Answer:
250
350
200 goods (point R) and moving horizontally ($150/Pfood150t200
) units from R to R'. The price
1it
of food has not changed, so from point R' on, the budget line falls at slope -Pfood = $1
ben
1
until Z' is reached.
i both Polin
le
1
4Lu i
Plan B: The $100 gift of food stamps generates budget line AA'Z'.
nor
Conclusion:
A will generate
350 more food consumption if the optimal
n 150 Policy
250
consumption point under Policy B is tangent to AA'Z' to the left of R'. For example,
U2 is the highest utility for budget line AA'Z', but U1 is the highest utility for budget
line RR'Z', and it has more food consumption. Otherwise, food consumption is the
same under both policies since the budget lines are the same from R’ to Z’. For
example, U3 is the highest utility for both the AA'Z' and RR'Z' budget lines.
U2
U1
Other
Goods
$250
$200
A
A'
R
R'
$100
0
U3
50
100
150
Z
Z'
Food (units) if Pfood = $1
Page 6 of 16
e) (6 Points – More changing) A food coupon program requires families to pay a certain
amount for food coupons. Suppose all families can receive $150 in food coupons for a
payment of $50 (call this policy A). Also, assume all households have $250 of income
and the price of food is $1 per unit. With the composite consumption good (CCG) on the
y-axis and food on the x-axis, draw the original budget line and the budget line under this
policy.
i
Compared with an allocated grant of $100 in food coupons (call this policy B), would
policy A lead to more, less, or the same food consumption? Why? Assume well-behaved
indifference curves.
II
台
150 50 100
250 poji1 150
每
25
Answer:
PlanzA: The new budget line is RR'Z' - subtract $50 from the purchase of other
100
goods (point R) and moving horizontally ($150/Pfood) units from
R to R'.250
The price
of food has not changed,
1 so1from point R' on, the budget line falls at slope -Pfood = $1
until Z' is reached.
Police all betelr
e
35
hr
i both
Plan B: The $100 gift of food stamps generates budget line AA'Z'.
than non pslicn
i
353
Conclusion: Policy A will generate
主 more food consumption if the optimal
Police
consumption point under Policy B is tangent to AA'Z' to the left of R'. For example,
U2 is the highest utility for budget line AA'Z', but U1 is the highest
utility forman
budgetUn
theSane
line RR'Z', and it has more food consumption. Otherwise, food consumption is the
which
same under both policies since the budget lines are the same from R’ to Z’. For
example, U3 is the highest utility for both the AA'Z' and RR'Z' budget lines.
both
hen
of food
is 350
U2
U1
Other
Goods
$250
$200
A
A'
R
R'
$100
0
U3
50
100
150
Z
Z'
Food (units) if Pfood = $1
Page 6 of 16
4s
e) (6 Points – More changing) A food coupon program requires families to pay a certain
amount for food coupons. Suppose all families can receive $150 in food coupons for a
payment of $50 (call this policy A). Also, assume all households have $250 of income
and the price of food is $1 per unit. With the composite consumption good (CCG) on the
y-axis and food on the x-axis, draw the original budget line and the budget line under this
policy.
A
B
Compared with an allocated grant of $100 in food coupons (call this policy B), would
policy A lead to more, less, or the same food consumption? Why? Assume well-behaved
indifference curves.
Answer:
Plan A: The new budget line is RR'Z' - subtract $50 from the purchase of other
goods (point R) and moving horizontally ($150/Pfood) units from R to R'. The price
of food has not changed, so from point R' on, the budget line falls at slope -Pfood = $1
until Z' is reached.
Plan B: The $100 gift of food stamps generates budget line AA'Z'.
Conclusion: Policy A will generate more food consumption if the optimal
consumption point under Policy B is tangent to AA'Z' to the left of R'. For example,
U2 is the highest utility for budget line AA'Z', but U1 is the highest utility for budget
line RR'Z', and it has more food consumption. Otherwise, food consumption is the
same under both policies since the budget lines are the same from R’ to Z’. For
example, U3 is the highest utility for both the AA'Z' and RR'Z' budget lines.
Bothpolicies are better
U2
than none
U1
Both policies can hae
250i 1 250units
Other
Goods
$250
$200
A
A'
R
DR'
foodwhich is 350
1 200units
zoo
$100
0
the same max units of
U3
50
100
150
Z
Z'
nj_i.in
Food (units) if Pfood = $1
Page 6 of 16
Question-2 [15 Points]: A consumer who derives his utility from two goods is
characterized by the following utility function
ㄥ
U(x,y)= min{2x, 5y}
Suppose that the consumer is endowed with an income of 100 (I = 100), and that px = 10
and py = 5.
a) (6 Points) Analytically derive this consumer optimal consumption point. Depict the
budget line and optimal consumption point in a well-labeled diagram.
Now suppose the price of declines to 5 (px = 5).
b) (4 Points) Derive analytically and show on the same diagram this consumer new
optimal consumption point.
c) (5 Points) Decompose the change in x consumption into a substitution and an income
effect.
5y
1)2 x 5 y
I
Pxxtpyy
2)10 x 5 y 100
u
aSolution:
a)
Blt00
20
(1), ( 2) ! x 8.33, y
10x
2ㄨ
5y
3.33
5y
20
TE = 5.96
9年
333
SE = 0
3.33
b
424
57
100
y
b)
14.286
4x
54 57
y 5.714
8.33
14.29
10
1)2 x 5 y
2)5 x 5 y 100
(1), ( 2) ! x 14.29, y 5.71
iii
No substitution effect.
8 335ii
TE= 14.29-8.33
SE=0, IE=14.29-8.33
No s E
LDL x
亨兰
淡
儷 尛
Page 7 of 16
20
Question-2 [15 Points]: A consumer who derives his utility from two goods is
characterized by the following utility function
LL
U(x,y)= min{2x, 5y}
BL
Suppose that the consumer is endowed with an income of 100 (I = 100), and that px = 10
and py = 5.
a) (6 Points) Analytically derive this consumer optimal consumption point. Depict the
budget line and optimal consumption point in a well-labeled diagram.
BL
Now suppose the price of declines to 5 (px = 5).
b) (4 Points) Derive analytically and show on the same diagram this consumer new
optimal consumption point.
c) (5 Points) Decompose the change in x consumption into a substitution and an income
effect.
4
57
1)2 x 5 y
4 los
l o x t5y
2)10 x 5 y 100
U
2x
Solution:
y 3.332
a)
b
(1), ( 2) ! x 8.33, y
2x
57
8.331为
X
20
1429
3.33
9
94
5x
y 5.71
100
t5y
TE = 5.96
iii
i
的
3.33
C
0
瓷 㸚
TE
b)
XB
SE = 0
8.33
XA
14.4 8.33 5.96
14.29
10
1)2 x 5 y
2)5 x 5 y 100
(1), ( 2) ! x 14.29, y 5.71
No SE
No substitution effect.
TE= 14.29-8.33
SE=0, IE=14.29-8.33
Page 7 of 16
20
P15
Final
Question-2 [15 Points]: A consumer who derives his utility from two goods is
characterized by the following utility function
U(x,y)= min{2x, 5y}
U
2x
5y
Suppose that the consumer is endowed with an income of 100 (I = 100), and that px = 10
and py = 5.
a) (6 Points) Analytically derive this consumer optimal consumption point. Depict the
budget line and optimal consumption point in a well-labeled diagram.
Now suppose the price of declines to 5 (px = 5).
ㄗ
b) (4 Points) Derive analytically and show on the same diagram this consumer new
optimal consumption point.
c) (5 Points) Decompose the change in x consumption into a substitution and an income
effect.
Solution:
a)
1)2 x 5 y
2)10 x 5 y 100
(1), ( 2) ! x 8.33, y
20
BL
TE = 5.96
3.33
SE = 0
3.33
yn
8.33
b)
14.29
10
1)2 x 5 y
2)5 x 5 y 100
(1), ( 2) ! x 14.29, y 5.71
No substitution effect.
TE= 14.29-8.33
SE=0, IE=14.29-8.33
IE XD XA
IE XB XD
X
20
感
ii
I
xniiiib
Page 7 of 16
207
20
Question-3 [10 Points]: Adam & Eve own separate farms. Depending on the market
conditions Adam has $10,000 profits and Eve only $900 or Eve has $10,000 profits and
Adam only $900. The probability of each state is 50%. Their satisfaction is characterized
by:
U
I
Adam & Eve think they should work together and merge their farms. If they were to
merge their farms, both would get half of the combined profits.
a) (6 Points) Do you think it is a good idea to merge? Use a graph to illustrate your
answer.
b) (4 Points) It turns out that because of all the lawyers involved, a merger is quite costly.
What is the maximum sum either one is willing to pay in lawyer fees?
Solution:
10000x503 900X5
Note with the merger the total income will be 10900 in each state and each will get
450
5000
5450, which is equal to the expected income. We also know
that:
ECI
a
IIiEEiiii
5450
fjjuIEcin.TT
0.5JT
73.82
E ( I ) 0.5 u10,000 0.5 u 900 5450
E (U ) 0.5 u 10,000 0.5 u 900
5450
U ( E ( I ))
U ( E ) ! EU
73.82 !
65
Ecu
1
100
65
UTE I I
Expected utility
73.82
0.5T
50415
usono.si
Therefore, they should merge.
9
nonrist
risk
Eu
iUtility
of Expected
need
to wage rick
65
b
30
900
5450
10,000
Page 8 of 16
averse
Insurance
Question-3 [10 Points]: Adam & Eve own separate farms. Depending on the market
conditions Adam has $10,000 profits and Eve only $900 or Eve has $10,000 profits and
Adam only $900. The probability of each state is 50%. Their satisfaction is characterized
by:
I
U
Adam & Eve think they should work together and merge their farms. If they were to
merge their farms, both would get half of the combined profits.
a) (6 Points) Do you think it is a good idea to merge? Use a graph to illustrate your
answer.
b) (4 Points) It turns out that because of all the lawyers involved, a merger is quite costly.
What is the maximum sum either one is willing to pay in lawyer fees?
Solution:
ELI
a
9
1000x503_
Note with the merger the total income will be 10900 in each state and each will get
5450, which is equal Joo0
to the expected income. We also know that:
450
5450
E ( I ) 0.5 u10,000 0.5 u 900 5450
T
E (U ) 0.5 u 10,000 0.5 u 900 65
KIEl III
5450
U ( E ( I ))
U ( E ) ! EU
El u
T
73.82 !
73.8L
0.5d
Therefore, they should merge.
65
吤定
30
ˇ__
周
900
Risk
n IE 211
Eul
Expected utility
73.82
65
05T
5
50
100
P.is le
No n
i need to
Utility of Expected
wef si.in
5450
10,000
Page 8 of 16
b
E
RP
I
10000
EI
CE
65
JE
CEE 652
RP 5450
900 x i
5450
4225
4225
三井
12
Question-3 [10 Points]: Adam & Eve own separate farms. Depending on the market
conditions Adam has $10,000 profits and Eve only $900 or Eve has $10,000 profits and
Adam only $900. The probability of each state is 50%. Their satisfaction is characterized
by:
I
U
Adam & Eve think they should work together and merge their farms. If they were to
merge their farms, both would get half of the combined profits.
a) (6 Points) Do you think it is a good idea to merge? Use a graph to illustrate your
answer.
b) (4 Points) It turns out that because of all the lawyers involved, a merger is quite costly.
What is the maximum sum either one is willing to pay in lawyer fees?
Solution:
Note with the merger the total income will be 10900 in each state and each will get
5450, which is equal to the expected income. We also know that:
E ( I ) 0.5 u10,000 0.5 u 900 5450
E (U ) 0.5 u 10,000 0.5 u 900
5450
U ( E ( I ))
U ( E ) ! EU
65
73.82 !
Therefore, they should merge.
100
Expected utility
73.82
Utility of Expected
Ecu
65
I
30
坏1
CE
900
5450
10,000
Page 8 of 16
b)
U ( I Cost ) EU
Cost 1225
5450 Cost
65
Which is equal to the risk premium because:
E (U ) 0.5 u 10,000 0.5 u 900
65
CE 65
U (CE )
CE 4225
RP 5450 4225 1225
RP
E I
CE
Page 9 of 16
Final2 P13
Question-4 [15 Points]: The market demand and supply functions for cotton are:
4P 10 2d
p Ī 本Qd
Qd = 10 - 4P
Qs = 6P + 5
a) (5 Points) Calculate the consumer surplus and producer surplus and show it on a graph
b) (5 Points) To assist cotton farmers, the government initiates a subsidy of $0.10 per
unit. Calculate the new level of consumer surplus and producer surplus.
c) (5 Points) Show that the combined increase in consumer and producer surplussuppg
is less
than the increased government spending necessary to finance the subsidy. Demonstrate
your answer in your graph for part (a).
a
Qd Qs
10 4P 617 5
品
管品
2.5
Q
0.5
4
8
8
ps
品
8xi
2.5 0.5
5
8
i2
5 8
0.5
13 4
3.25
Qd = 10 – 4P = Qs = 6P + 5.
=> P = 0.5. Q = 8
bCS = 1/2Ps(2.5 – 0.5)i 8Q=s8. ō
Pd Í_ iad
Ps Pd 0.1
Q = 10 - 4P
Ē Étia 0.1
6Q
PS = 0.5(5) + 1/2 (8 – 5) 1/2 = 3.25.
1
s
iii
88.2472
i 8.24 É 0.54
g Pd Ē 本 8.24 0 x4
Ps
b)
d
6.5
i 汉
iii 588
6 0.5 5
CS
1 3IL
s
0 5
18 8 24
d
卡
2
号
0.1
三
C
8.24 0.1 Page 10 of 16
824
xd
o 5 0.44
8
Question-4 [15 Points]: The market demand and supply functions for cotton are:
Qd = 10 - 4P
Qs = 6P + 5
a) (5 Points) Calculate the consumer surplus and producer surplus and show it on a graph
b) (5 Points) To assist cotton farmers, the government initiates a subsidy of $0.10 per
unit. Calculate the new level of consumer surplus and producer surplus.
c) (5 Points) Show that the combined increase in consumer and producer surplus is less
than the increased government spending necessary to finance the subsidy. Demonstrate
your answer in your graph for part (a).
2.5
ps
0.5
Pp
5
8
8.24
Qd = 10 – 4P = Qs = 6P + 5.
=> P = 0.5. Q = 8
CS = 1/2 (2.5 – 0.5) 8 = 8.
PS = 0.5(5) + 1/2 (8 – 5) 1/2 = 3.25.
b)
Qd = 10 - 4Pd
Page 10 of 16
Qs = 6Ps + 5
Pd +0.1 = Ps
Qd = 10 – 4Pd = Qs = 6(Pd +0.1)+ 5.
Ö Pd = 0.44.
Ö Ps = 0.54
Ö Q = 8.24
CS/ = 1/2 (2.5 – 0.44) 8.24 = 8.487
PS/ = 0.54(5) + 1/2 (8.24 – 5) 0.54= 3.58
c) Government spending is 0.1*8.24 = $0.824. The increase in consumer surplus is
$0.487. The increase in the producer surplus is 0.32 => Total change in the consumer
surplus and producer surplus is: 0.812. The increase in consumer and producer surplus is
less than government spending.
Page 11 of 16
Question-5 [15 Points]: The market demand is given as:
Qd = 1000 – 40P
75
>
@
2
The producers are characterized by q L0.5 K 0.5 . The cost of labor is $4 per hour and
the cost of capital is $1 per unit. In the short run the capital is fixed at 16 units.
r
n_p.me
Ave
a) (5 Points) Find the short run market supply curve. (Make sure to identify the short run
r
shut-down price).
b) (5 Points) Find the long-run supply curve? Identify the equilibrium quantity and price?
How many firms will be operating in the long run? Use a graph to demonstrate your
solution.
c) (5 Points) Suppose in addition to the firms identified above (parts a & b) there is a
single firm with a production function characterized by: q 8 L0.5 K 0.5 . Find the long-run
supply curve for this firm. Explain briefly how the industry adjusts to the emergence of
this firm in the long run?
不
Solution:
器
>
@
2 2
q ( L1 / 2 K 1 / ㄥ
) 05
!L
q
0.5
q
! SRTC
4
2
5
16 522
SRMC
> E @ 422
2
4 q 0.5 4 16
qi
i
L
>
qi
b
MRTS
卡
4
@4
SRAVC
qi_472
L
SPAvc
c
器
4
k
RTS
8cqit4 ciq i
4q
ª
4 º
P
! SRMC 4 «1 5
0.5 »
¬ q ¼
2
4
! SRAVC
5 q q 0.5 4
SRAVC SRMC ! P 0(Shutdown Price)
要 2 16
三毕
i
qit4
出 ii ˇ
ǎ19 _4
shutdomp.in
SRmC
05
5
号
0.5
7
0.5
5
P
16LEK.TT Eǎoǎ
4
Page 12 of 16
4
5
c
L
1164
5
4L
2
5
2
5
5L 514
25L
1
i
k
号
不 TC
w l
t r k
L
4
9
Ělt
告9
L不 MC
is
LRMC
LRAC
supply curves
shut down
学
Q
PA
54
1000
40 54
1000
1
i
968
me
a
32
5
C
q 8L
MR7s
器 5号
0.5
MRTS
4L
4
卡
告
k
k
0.5
5
c
k
0.5
L
1
t
k
Question-5 [15 Points]: The market demand is given as:
Qd = 1000 – 40P
>
@
2
The producers are characterized by q L0.5 K 0.5 . The cost of labor is $4 per hour and
the cost of capital is $1 per unit. In the short run the capital is fixed at 16 units.
a) (5 Points) Find the short run market supply curve. (Make sure to identify the short run
shut-down price).
vc shutdown
me
p
PA
b) (5 Points) Find the long-run supply curve? Identify the equilibrium quantity and price?
How many firms will be operating in the long run? Use a graph to demonstrate your
solution.
TMR is pk
c) (5 Points) Suppose in addition to the firms identified above (parts a & b) there is a
single firm with a production function characterized by: q 8 L0.5 K 0.5 . Find the long-run
supply curve for this firm. Explain briefly how the industry adjusts to the emergence of
this firm in the long run?
Solution:
q ( L1 / 2 K 1 / 2 ) 2
>q 4@
! SRTC 4>q 4@ 16
!L
0.5
2
0.5
2
ª
4 º
4 «1 0.5 » P
¬ q ¼
2
4 0.5
q 4
! SRAVC
q
SRAVC SRMC ! P 0(Shutdown Price)
! SRMC
>
@
4
ˋˋ
Page 12 of 16
b)
MRTS
L0.5
K 0.5
q [ L1 / 2 4 L1 / 2 ]2
4
K 0.5
! 0.5
1
L
25 L ! L
4 ! K 16 L
q
,K
25
16 q
25
expantion path
q 16 q 4
q
25 25 5
4
! LRMC LRAC
P(long run supply curve)
5
4
Q 1000 - 40 ( ) 968
5
! LRTC
4
The number of firms operating in the market is indeterminate.
LRMC=LRAC
4/5
r
968
Page 13 of 16
c)
MRTS
4
!K
1
K
L
4L
q
,K
16
16 L ! L
q 8 L0.5 2 L0.5
q 4q
16 16
! LRTC
4
! LRMC
LRAC
4q
16
8
q
16
1
2
P(long run supply curve)
This firm is endowed with a technology that allows it to produce at a lower long run
marginal cost that other firms. Other firms need to adopt to the new technology or exit
the market.
Question-6 [10 Points]: A monopolist has two factories for which costs are given by:
The firm faces the following demand curve:
MR
P = 700 - 5Q
700 102
where Q is total output, i.e. Q = Q1 + Q2.
a) [6 Points] Calculate the values of Q1, Q2, Q, and P that maximize profit.
b) [4 Points] On a diagram, draw the marginal cost curves for the two factories, the
average and marginal revenue curves, and the total marginal cost curve (i.e., the marginal
cost of producing Q = Q1 + Q2). Indicate the profit-maximizing output for each factory,
total output, and price.
a
MC
20Q
MCz
402
Solution:
QF 或
MC
mcz MG
am Q
22
MC'l
Q2 万MC2 Qm
Me
Tt TomL
The average revenue curve is the demand curve,
P = 700 - 5Q.
Qm
了
9Mci
MR 2D 700 10a
器 器 10Q
10 312
Q
700
Tmci
Page 14 of 16
30
P 700
MCi
3
5 30
550
30
400
9Q
22
b
mnmuMC Pon
Mci
tl. i
1
1
I
怀
Q2Q i Q m
D
器
古 4
1
10
c)
MRTS
4
!K
1
K
L
4L
16 L ! L
q 8 L0.5 2 L0.5
q 4q
16 16
! LRTC
4
! LRMC
LRAC
q
,K
16
4q
16
8
q
16
1
2
P(long run supply curve)
This firm is endowed with a technology that allows it to produce at a lower long run
marginal cost that other firms. Other firms need to adopt to the new technology or exit
the market.
Final2 P18
Question-6 [10 Points]: A monopolist has two factories for which costs are given by:
The firm faces the following demand curve:
P = 700 - 5Q
where Q is total output, i.e. Q = Q1 + Q2.
a) [6 Points] Calculate the values of Q1, Q2, Q, and P that maximize profit.
b) [4 Points] On a diagram, draw the marginal cost curves for the two factories, the
average and marginal revenue curves, and the total marginal cost curve (i.e., the marginal
cost of producing Q = Q1 + Q2). Indicate the profit-maximizing output for each factory,
total output, and price.
Solution:
The average revenue curve is the demand curve,
P = 700 - 5Q.
Page 14 of 16
For a linear demand curve, the marginal revenue curve has the same intercept as the
demand curve and a slope that is twice as steep:
MR = 700 - 10Q.
Next, determine the marginal cost of producing Q. To find the marginal cost of
production in factory 1, take the first derivative of the cost function with respect to Q:
Similarly, the marginal cost in factory 2 is
Rearranging the marginal cost equations in inverse form and horizontally summing them,
we obtain total marginal cost, MCT:
or
Profit maximization occurs where MCT = MR.
Calculate the total output that maximizes profit, i.e., Q such that MCT = MR:
, or Q = 30.
Next, observe the relationship between MC and MR for multiplant monopolies:
MR = MCT = MC1 = MC2.
We know that at Q = 30, MR = 700 - (10)(30) = 400.
Therefore,
MC1 = 400 = 20Q1, or Q1 = 20 and
Page 15 of 16
MC2 = 400 = 40Q2, or Q2 = 10.
To find the monopoly price, PM, substitute for Q in the demand equation:
PM = 700 - (5)(30), or
PM = 550.
b) See the following Figure for the profit-maximizing output for each factory, total
output, and price.
Page 16 of 16