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:
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
= 0.1
0.5
=> x = 100
MRS =
This is an interior solution but is not optimal since there is 200 free minutes.
⇒ Should expect a corner solution, with x = 200 and y = 60.
⇒ Check MRS at this bundle (1/2000.5 < 0.1).
⇒ She tries to increase y and reduce x. But this is not feasible.
⇒ The corner solution is optimal.
⇒ The utility level at the corner solution is 2000.5 + 60/2 = 44.14 > 1000.5 + 60/2.
y
60
Page 2 of 16
200
600
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 12 + 10q 1 + 200
C 2 (q 2 ) = 0.5q 22 + 10q 2 + 50
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 = 0, p = 10 − shoutdown point)
P = 4q 1 + 10(if P ≥ 10)
C 2 (q 2 ) = 0.5q 22 + 10q 2 + 50
SRMC = q 2 + 10, SRAVC = 0.5q 2 + 10
SRMC = SRAVC ( q 2 = 0, p = 10 − shoutdown point)
P = q 2 + 10(if P ≥ 10)
Market supply:
P
− 2.5(ifP ≥ 10)
4
q2 = P − 10(ifP (≥ 10)
q1 =
Q=0
( P < 10)
P
Q = 50( − 2.5) + 50( P − 10) = 62.5P - 625
4
( P ≥ 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:
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
Elastic portion η< -1
η= -1
P
MC
Inelastic portion η> -1
D
MR
$ [not $/unit
because this is a
Total graph]
Q
$
TC
TR
π max
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.
Solution:
p1x = 50, p1y = 40, x1 = 20, y1 = 50
I = 50 × 20 + 40 × 50 = $3000
p x2 = 25, p 2y = 50
50 x + 40 y = 25 x + 50 y
x = 20, y = 50(intersection )
50
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.
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.
U2
U1
Other
Goods
$250
A
A'
R
R'
$200
$100
U3
0
50
100
150
Z
Z'
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
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.
Solution:
a)
1)2 x = 5 y
2)10 x + 5 y = 100
(1), ( 2) => x = 8.33, y = 3.33
20
TE = 5.96
SE = 0
3.33
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
Page 7 of 16
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:
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 ×10,000 + 0.5 × 900 = 5450
E (U ) = 0.5 × 10,000 + 0.5 × 900 = 65
U ( E ( I )) = 5450 = 73.82 >
U ( E ) > EU
Therefore, they should merge.
100
Expected utility
73.82
Utility of Expected
65
30
900
5450
10,000
Page 8 of 16
b)
U ( I − Cost ) = EU = 65 = 5450 − Cost
Cost = 1225
Which is equal to the risk premium because:
E (U ) = 0.5 × 10,000 + 0.5 × 900 = 65
U (CE ) = CE = 65
CE = 4225
RP = 5450 − 4225 = 1225
Page 9 of 16
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
0.5
5
8
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.
 Pd = 0.44.
 Ps = 0.54
 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
[
]
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).
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?
Solution:
q = ( L1 / 2 + K 1 / 2 ) 2
[
=> L = q 0.5 − 4
[
]
2
]
2
=> SRTC = 4 q 0.5 − 4 + 16

4 
=> SRMC = 4 1 − 0.5  = P
 q 
2
4
=> SRAVC = q 0.5 − 4
q
SRAVC = SRMC => P = 0(Shutdown Price)
[
]
Page 12 of 16
b)
L−0.5 4
K 0.5
=
=>
= 4 => K = 16 L
K −0.5 1
L0.5
16 q
q
q = [ L1 / 2 + 4 L1 / 2 ]2 = 25 L => L = , K =
25
25
q 16 q 4
= q
=> LRTC = 4 +
25 25 5
4
=> LRMC = LRAC = = P(long run supply curve)
5
4
Q = 1000 - 40 ( ) = 968
5
MRTS =
The number of firms operating in the market is indeterminate.
LRMC=LRAC
4/5
968
Page 13 of 16
c)
MRTS =
K 4
= => K = 4 L
L 1
q = 8 L0.5 2 L0.5 = 16 L => L =
4q
q
,K =
16
16
q 4q 8
+
= q
16 16 16
1
=> LRMC = LRAC = = P(long run supply curve)
2
=> LRTC = 4
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:
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