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October 7, 2014
Economics 203: Intermediate Microeconomics I
Lab Exercise #1
Applied Questions
Question 1) Suppose Sandy gets the same utility from each of the 6 combinations of goods Z
and W shown below. Good Z costs $10 per unit and good W costs $4 per unit.
Commodity Combination
1 2 3 4 5 6
Amount of good W
(units)
1
2
4
7
11
16
Amount of good Z
(units)
10
9
8
7
6
5
a) What is the minimum income she musts have to achieve the level of utility
associated with these commodity combinations?
b) If she has the income you found in part a, draw her budget line.
c) Draw her indifference curve.
Question 2: Suppose total income to spend on two products is $100. The price of the
first product is Px=$2 per unit and the price of the second product is Py=$4 per unit.
A) Determine the formula for the budget constraint.
B) What is the MRSy,x?
C) Illustrate the consumption decision with a diagram. (You will not be able to show
the exact consumption bundle.) Make sure you include the intercepts of the
budget constraint and an indifference curve. (2 Marks)
Solutions: Question 1) Suppose Sandy gets the same utility from each of the 6 combinations of goods Z
and W shown below. Good Z costs $10 per unit and good W costs $4 per unit.
Commodity Combination
1 2 3 4 5 6
Amount of good W
(units)
1
2
4
7
11
16
Amount of good Z
(units)
10
9
8
7
6
5
Total income 104 98 96 98 104 114
a) What is the minimum income she musts have to achieve the level of utility associated
with these commodity combinations?
Minimum income must be $96
b) If she has the income you found in part a, draw her budget line.
0 5 10 15 20 25
Good W (units)
c) Draw her indifference curve.
Z
10
5
0
Indifference Curve
Budget line
Question 2: Suppose total income to spend on two products is $100. The price of the
first product is Px=$2 per unit and the price of the second product is Py=$4 per unit. )
a) Determine the formula for the budget constraint.
100
100
4
2
4
25 05
P X P Y
Y X
Y X
x y
.
b) What is the MRSy,x?
MRSP
Py x
x
y
, . 2
405
c) Illustrate the consumption decision with a diagram. (You will not be able to show
the exact consumption bundle.) Make sure you include the intercepts of the
budget constraint and an indifference curve. (2 Marks)
The consumer will consume where MRSP
P
Y
Xy x
x
y
,
Quantity of Y
25
I
0 Quantity of X
50
MRSyx=
Y
X
October 14, 2014
Economics 203: Intermediate Microeconomics I
Lab Exercise #2
True or False?
1) A commodity’s income elasticity of demand may be positive or negative.
2) The income elasticity of demand for food is very high.
Section 2: Discussion
Suppose that each of the four corners of an intersection contains a gas station, and that
the gasoline is essentially the same. Do you think that the price elasticity of demand for
each station’s gasoline is above or below 1? Why?
Section 3: Applications:
1) The price elasticity of demand for a particular kind of screwdriver is 2, and the
marginal revenue is $2. What is the price of this screwdriver?
2) The Drink-It Distributors concludes that the demand function for its product
is:
Q=95-73P+82PR+ 0.25 M
where Q is the quantity demanded of its product, P is the price of its product, PR is
the price of its rival’s product, and M is per capita disposable income (in dollars).
Currently, P=$255, PR=$225 and M=$50,650.
a) What is the price elasticity of demand for the firm’s product? Use price elasticity
of demand formula for a point:
Q
P
P
Q
b) What is the income elasticity of demand for the firm’s product?
(Use income elasticity of demand formula for a point:
Q
M
M
Q
.)
c) What is the cross elasticity of demand between its product and its rival’s product.
(Use Cross elasticity of demand formula for a point:
Q
P
P
QR
R
.)
3) According to the U.S. Department of Agriculture, the income elasticity of demand for
coffee is about 0.23. If income rose by 1 percent, what effect would this have on the
quantity demanded of coffee?
4) Give the sign of the cross elasticity of demand for each of the following pairs of
commodities: a) tea and coffee
b) tennis rackets and tennis balls
c) whiskey and gin
Solutions:
Section 1
True or False?
1) A commodity’s income elasticity of demand may be positive or negative. TRUE
2) The income elasticity of demand for food is very high. FALSE
Section 2: discussion
Suppose that each of the four corners of an intersection contains a gas station, and that
the gasoline is essentially the same. Do you think that the price elasticity of demand for
each station’s gasoline is above or below 1? Why?
Elasticity is above 1 because the gasoline provided by the four gas stations are very close
substitutes. Very price sensitive.
Section 3: Applications
1) The price elasticity of demand for a particular kind of screwdriver is 2, and the
marginal revenue is $2. What is the price of this screwdriver?
Since MR= P(1-[1/η]), it follows that P=MR÷(1-[1/η]).
MR=2
η=2=price elasticity of demand
P=2÷(1-½)=4
2) The Drink-It Distributors concludes that the demand function for its product
is:
Q=95-73P+82PR+ 0.25 M
where Q is the quantity demanded of its product, P is the price of its product, PR is
the price of its rival’s product, and M is per capita disposable income (in dollars).
Currently, P=$255, PR=$225 and M=$50,650.
a) What is the price elasticity of demand for the firm’s product? Use price elasticity
of demand formula for a point:
Q
P
P
Q
Solving for Q=95-73(255)+82(225)+0.25(50,650)
Q=12,592.5
Q
P
P
Q
= -73(255/12592.5) = -1.4783
b) What is the income elasticity of demand for the firm’s product?
(Use income elasticity of demand formula for a point:
Q
M
M
Q
.)
0.25 (50,650/12592.5) =1.00556
c) What is the cross elasticity of demand between its product and its rival’s product.
(Use Cross elasticity of demand formula for a point:
Q
P
P
QR
R
.)
82 (225/12592.5) =1.4652
3) According to the U.S. Department of Agriculture, the income elasticity of demand
for coffee is about 0.23. If income rose by 1 percent, what effect would this have on the
quantity demanded of coffee
ηM=[%ΔQd / %ΔM]=0.23
if %ΔM=0.01
then, 0.01*0.23=0.0023
The effect is 0.23% increase in demand.
4) Give the sign of the cross elasticity of demand for each of the following pairs of
commodities:
a) tea and coffee -POSITIVE
b) tennis rackets and tennis balls --NEGATIVE
c) whiskey and gin--POSITIVE
October 21, 2014
Economics 203: Intermediate Microeconomics I
Lab Exercise #3
Section A: Test Your Understanding
True or False?
1) The long run refers to a period when all inputs are variable and none is fixed.
2) The average product of an input is the addition to the total output due to the
addition of the last unit of input used, the quantity of other inputs used being held
constant.
3) The law of diminishing marginal returns is not applicable to cases in which there
is a proportional increase in all inputs.
Section B: Questions
Question 1: Sketch the graph of a "standard " short-run production function, and identify
on it the points where the average product peaks(A), the marginal product peaks(B), the
marginal product reaches zero(C), and the average and marginal product intersect(D).
Question 2: Which of the following production functions exhibits increasing returns to
scale?
A) Q = K1/2
Ll/2
B) Q = Kl/2
L2/3
C) Q = Kl/4
Ll/3
D) Q = K/L
Question 3: A representative isoquant for perfect complements is
A) L-shaped.
B) a straight line.
C) a ray passing through the origin.
D) concave.
E) not necessarily any of the above.
Question 4: The production function for calculators is Q = 12L2
- L3, where Q is in
calculators /day and L is in labour-days /day. Show all your work for the following
question.
Give the equation for the marginal product of labour (MPL), and its value when L = 6
labour-days/day.
Question 5: The production function for gravel is Q = 10L
1/2K
1/2 (where Q is in units, L is in labour-
days, and K is in machine-days). With L on the horizontal axis and K on the vertical axis,
give the formula for the MRTS of the relevant isoquant for any given combination of
inputs (K, L), and give the value of the MRTS when L= 9 labour-days and K = 16
machine-days.
Solutions
Section A: Test Your Understanding
True or False?
1) The long run refers to a period when all inputs are variable and none is fixed.
True
2) The average product of an input is the addition to the total output due to the
addition of the last unit of input used, the quantity of other inputs used being held
constant.
False
3) The law of diminishing marginal returns is not applicable to cases in which there
is a proportional increase in all inputs.
True
Section A:
1. Sketch the graph of a "standard " short-run production function, and identify on it the
points where the average product peaks(A), the marginal product peaks(B), the marginal
product reaches zero(C), and the average and marginal product intersect(D).
Ans: The average product of labour peaks at the output level where the ray from the
origin is tangent to the total product curve, where it equals the marginal product of labour
(at A=D). The marginal product peaks at the output level corresponding to the inflection
point on the total product curve (at B), and the marginal product reaches zero when the
total product peaks (at C).
Question 2: Which of the following production functions exhibits increasing returns to
scale?
A) Q = K1/2
Ll/2
B) Q = Kl/2
L2/3
C) Q = Kl/4
Ll/3
D) Q = K/L
Ans: B
Question 3: A representative isoquant for perfect complements is
A) L-shaped.
B) a straight line.
C) a ray passing through the origin.
D) concave.
E) not necessarily any of the above.
Ans: A
Question 4: The production function for calculators is Q = 12L2
- L3, where Q is in
calculators /day and L is in labour-days /day. Show all your work for the following
questions.
Give the equation for the marginal product of labour (MPL), and its value when L = 6
labour-days/day.
Ans: MPL = 24L - 3L2
, which equals 36 calculators/labour-day when L = 6 labour-
days/day.
Question 5 : The production function for gravel is Q = 10L
1/2K
1/2 (where Q is in units, L is in labour-
days, and K is in machine-days).
With L on the horizontal axis and K on the vertical axis, give the formula for the MRTS
of the relevant isoquant for any given combination of inputs (K, L), and give the value of
the MRTS when L= 9 labour-days and K = 16 machine-days.
MRTSMP
MP
L K
L K
K
L
L
K
5
5
16
9
1 2 1 2
1 2 1 2
/ /
/ /
October 28, 2013
Economics 203: Intermediate Microeconomics I
Lab Exercise #4
Section 1: Discussion:
Explain the why the short-run minimum cost of producing a certain output may
differ from the long-run minimum cost. Illustrate your explanation with a diagram.
Section 2: Application:
For the Bridges-to-Shores Corporation, the relationship between output (Q) and
the number of hours of specialized manual labour (S) and machine-
operated labour (M) is:
Q=650S +145M - 0.45 S2 - 0.65M
2
The hourly wage of specialised manual labour is $42, and the hourly wage of
machine-operated labour is $25. The firm can hire as much labour as it wants at
these wage rates.
A) The vice president of manufacturing recommends that the firm hire 50 hours of
manual labour and 95 hours of machine-operated labour. Evaluate this
recommendation.
B) If the Bridges-to-Shores company decides to spend a total of $25,000 on inputs
(specialized manual and machine-operated labour), how many hours of each type of
labour should it hire?
Economics 203: Intermediate Microeconomics I
Lab Exercise # 4 Solutions
Section 1: Discussion:
Explain the why the short-run minimum cost of producing a certain output may
differ from the long-run minimum cost. Illustrate your explanation with a diagram.
In the short-run, at least one factor of production is assumed fixed. When this
happens and the firm must produce a specific quantity of output, the firm may not be
able to use the input combination that attains minimum cost relative to the long-run
situation. In the long run, all factors of production are variable. Hence, a firm will
produce a specific output where the isocost line is tangent to the isoquant (the specific
quantity that must be produced). In the short-run, this may not be possible for a
specific quantity.
Capital
K1
K*
L1 L2
If capital is fixed at K* units, and the firm must produce q1 units, L2 units of labour will
be used with this fixed amount of capital to produce q1 units of output. The total cost is
C2. If the firm were able to use any combination of K and L, it would use L1 and K1
units of labour and capital and lower its cost to C1. This would be the long-run solution.
Total cost is C1.
q1
C0 C1
C2
q0
Expansion Path
q2
Minimum cost input
combination to produce
q1 units.
Actual input combination to
produce q1 units when K is
fixed.
Section 2: Application:
For the Bridges-to-Shores Corporation, the relationship between output (Q) and
the number of hours of specialized manual labour (S) and machine-
operated labour (M) is:
Q=650S +145M - 0.45 S2 - 0.65M
2
The hourly wage of specialised manual labour is $42, and the hourly wage of
machine-operated labour is $25. The firm can hire as much labour as it wants at
these wage rates.
B) The vice president of manufacturing recommends that the firm hire 50 hours of
manual labour and 95 hours of machine-operated labour. Evaluate this
recommendation.
To find the optimal input combinations, choose where:
MP
P
MP
P
S
S
M
M
The marginal products are:
MPS= 650 - 0.9S
MPM =145 – 1.3M
MP
P
MP
P
S Mcross multiply
S M
S M
S M
S M equation
S
S
M
M
650 0 9
42
145 1 3
25
25 650 9 42 145 1 3
16250 22 5 6090 54 6
22 5 10160 54 6
4515556 2 426667
. .
( . ) ( . )
. .
. .
. .
Enter in S=50 and M=95 into either equation we see that this is not the optimal
input combination.
50 4515556 2 42667 95
50 682 08925
. . ( )
.
B) If the Bridges-to-Shores company decides to spend a total of $25,000 on inputs
(specialized manual and machine-operated labour), how many hours of each type of
labour should it hire?
TC=25,000
TC P S P M
25,000 S
25,000
25,000
6034.6648 = 126.92014M
M = 47.5469
S = 451.5556 + 2.42667M = 566.9363
s M
42 25
42 4515556 2 42667 25
189653352 10192014 25
M
M M
M M
( . . )
. .
Must use 567 hours of specialized labour and 47.5 hours of machine-operated labour.
Economics 203: Intermediate Microeconomics I
Lab Exercise #5
Section 1: Test Your Understanding
True or False?
1) No industry, now or in the past, has met all of the requirements of perfect competition.
2) Under perfect competition the product of any one seller must be the same as the product
of any other seller.
3) For a perfectly competitive firm, choose the output rate at which marginal cost is equal to
price.
4) If a firm’s marginal cost curve intersects its average variable cost curve at $4 per unit out
output, the firm will shut down in the short run if the price of its product falls below $4
per unit.
Section 2: Applications
1) Suppose that the total costs of a perfectly competitive firm are as follows:
Output Rate Total cost
0 40
1 60
2 90
3 130
4 180
5 240
a) If the price of the product is $50, what output rate should the firm choose?
b) Suppose the firm experienced an increase of $30 in its fixed costs. What is its new total
cost function?
c) What effect will this increase in its fixed costs have on the output it will choose?
d) After the increase in fixed costs, what does the firm’s marginal cost curve look like?
e) After the increase in fixed costs, what output rate would the firm choose if the price of
its product were $40?
2) Data are provided below concerned the Allied Peanut Company, a firm
producing peanut brittle.
A) Supposing that this firm is a member of a perfectly competitive industry, complete the table
below:
Output per
day
Price Total Cost Total
revenue
Profit Marginal
cost
0 200 100
1 200
2 310
3 500
4 700
5 1000
B) Assume that the output rate must equal an integer number per day. If price of peanut
brittle falls to $50, will Allied continue producing, or will it shut down?
C) What is the minimum price at which Allied will continue production?
D) If price is $200, what output rate will Allied choose?
Solutions:
Section 1
1) True
2) True
3) True
4) True
Section 2: Applications
Suppose that the total costs of a perfectly competitive firm are as follows:
Output Rate Total cost
0 40
1 60
2 90
3 130
4 180
5 240
a) If the price of the product is $50, what output rate should the firm choose?
Produce where P=MC
Output Rate Total cost Marginal cost
0 40
20
1 60
30
2 90
40
3 130
50
4 180
60
5 240
Should produce 3 or 4 units.
b) Suppose the firm experienced an increase of $30 in its fixed costs. What is its
new total cost function?
Output Rate Total cost +30 =New TC
0 40+30=70
1 60+30=90
2 90+30=120
3 130+30=160
4 180+30=210
5 240+30=270
c) What effect will this increase in its fixed costs have on the output it will choose?
None.
Output Rate Total cost +30 =New TC MC new
0 40+30=70
20
1 60+30=90
30
2 90+30=120
40
3 130+30=160
50
4 180+30=210
60
5 240+30=270
d) After the increase in fixed costs, what does the firm’s marginal cost curve look
like? As above. Same as before the increase in fixed cost.
e) After the increase in fixed costs, what output rate would the firm choose if the
price of its product were $40? 2 to 3 units.
Lab Exercise #7
Price Discrimination
Section 1: Discussion:
Some university bookstores give faculty a discount that students do not receive. Show
with a sketch graph why this practice is most likely a profit-maximizing strategy instead
of a college perk given to the faculty at college expense.
Section 2: Applications
1. The demand curve facing a monopolist is given by P = 350 – 7Q, and the short-run
total cost curve is given by TC = 500 + 70Q, where TC is in $s and Q is in tonnes.
What are the profit-maximizing price and quantity? Find the monopolist's economic
profit.
2. A monopolist faces two separate demand curves: P1 = 65 – 2 Q1 and P2 = 35 – 3 Q2.
The total cost curve is TC = 7 + 5Q. Find, P1, P2, Q1,and Q2.
3. Find the price elasticities of demand at equilibrium for Problem 2.
Solutions:
Section 1: Discussion:
Some university bookstores give faculty a discount that students do not receive. Show
with a sketch graph why this practice is most likely a profit-maximizing strategy instead
of a college perk given to the faculty at college expense.
Ans:
PricePrice Price
Quantity
Studentmarket
Ps
MR D
Faculty market
MR
D
PfHorizontal sum of
both markets
MC
GMR
The graph shows a situation where the student demand is more inelastic at any price level
than the faculty demand. (At any given price the location on the demand curve is
proportionately further down the demand curve for the students.) For this reason, they
can be charged a higher price. By observing where the MC is equal to the marginal
revenue received from both markets, the bookstore manager will discover that overall
profits are maximized when he charges Ps to the students and Pf to the faculty.
Section 2: Applications: 1. The demand curve facing a monopolist is given by P = 350 – 7Q, and the short-run
total cost curve is given by TC = 500 + 70Q, where TC is in $s and Q is in tonnes.
What are the profit-maximizing price and quantity? Find the monopolist's economic
profit.
Marginal cost is equal to the slope of the straight line total cost curve:
MC = 70.
Marginal revenue has a slope which is twice as large as the slope of the demand curve:
MR = 350 – 14Q.
Set MC = 70 = 350 – 14Q = MR:
Q = 280/14 = 20 tonnes.
P = 350 – 7(20) = 350 – 140 = $210/tonne.
Profit = PQ – TC = 210(20) – [500 + 70(20)] = 4200 – 1900 = $2300.
2. A monopolist faces two separate demand curves: P1 = 65 – 2 Q1 and P2 = 35 – 3 Q2.
The total cost curve is TC = 7 + 5Q. Find, P1, P2, Q1,and Q2.
MC = slope of TC = 5
MR1 = 65 – 4 Q1
MR2 = 35 – 6 Q2
Set MR1 = MR2 = MC.
First solve for Q1 and P1:
65 – 4 Q1 = 5
4 Q1 = 60
Q1 = 60/4 = 15 kg.
P1 = 65 – 2(15) = $35/kg.
Next solve for Q2 and P2:
35 – 6 Q2 = 5
6 Q2 = 30
Q2 =30/6 = 5 kg.
P2 = 35 – 3(5) = $20/kg.
3. Find the price elasticities of demand at equilibrium for Problem 2.
Elasticity l = (∆Q1/∆P1)(P1/Q1) = (–1/2)(35/15) = –7/6.
Elasticity 2 = (–1/3)(20/5) = –4/3.
Check:
MR= P(1 + 1/)
MR1 = 35(1 – 6/7) = 5
MR2 = 20(1 – 3/4) = 5
Economics 203: Intermediate Microeconomics I
Lab Exercise #8
Section 1: Test Your Understanding
The following payoff matrix represents the long-run payoffs for two duopolists faced
with the option of buying or leasing buildings to use for production. Determine whether
any dominant strategies exist and whether or not there is a Nash equilibrium.
Firm 1
Lease
Building
Buy
Building
Lease F1 = 500 F1 = 750
Firm 2 F2 = 500 F2 = 400
Buy F1 = 300 F1 = 600
F2 = 600 F2 = 200
Section 2: Discussion:
What are the fundamental differences among the Cournot, Bertrand, and Stackelberg
models of oligopoly?
Section 3: Application
The market demand curve for a pair of Cournot duopolists is given as P=36-3Q,
where Q=Q1+Q2. For each duopolists, the constant per unit marginal cost is $18/unit
and fixed costs are zero.
a) Find the Cournot equilibrium price, quantity and profits.
b) Solve the problem for Bertrand duopolists.
c) Find the equilibrium price, quantity and profit for each firm, assuming the
firms act as a Stackelberg leader and follower, with Firm 1 as the leader.
Solutions:
Section 1
The following payoff matrix represents the long-run payoffs for two duopolists faced
with the option of buying or leasing buildings to use for production. Determine whether
any dominant strategies exist and whether or not there is a Nash equilibrium.
Firm 1
Lease
Building
Buy
Building
Lease F1 = 500 F1 = 750
Firm
2
F2 = 500 F2 = 400
Buy F1 = 300 F1 = 600
F2 = 600 F2 = 200
The dominant strategy for firm 1 is to buy the building, since regardless of firm 2’s
strategy, firm 1 is better off buying the building. Firm 2, on the other hand, is better off
buying the building only if firm 1 leases; since firm 1 will not lease, firm 2 should lease.
Nash equilibrium occurs when firm 1 buys and firm 2 leases, since this is the best
strategy for each player given the strategy chosen by the other player.
Section 2: Discussion: What are the fundamental differences among the Cournot,
Bertrand, and Stackelberg models of oligopoly?
The models differ in their assumptions about firm behaviour.
The Cournot model assumes that firms choose quantities simultaneously (taking as
given the other firm’s quantity).
The Bertrand model assumes that firms choose prices simultaneously (taking as given the
other firm’s price.)
The Stackelberg model assumes that one firm (the leader) chooses its quantity before the
other firm (the follower) chooses its quantity, and that the leader takes into account how
the follower will respond to the leader’s quantity decision.
Section 3: Applications
a) P1 = 36 – 3Q = 36 – 3 (Q1 + Q2 ) = (36 – 3 Q2 ) – 3 Q1.
MR1 = (36 – 3Q2) – 6Q1 = MC = 18.
1’s reaction function: Q1 = 3 – (1/2) Q2.
Similarly for 2: Q2 = 3 – (1/2) Q1.
This solves for Q1 = Q2 = 2 units.
P = 36 – 3Q = 36 – 3(4) = $24/unit.
1 = TR – TC = 2(24) – 2(18) = $12 = 2.
= 1 + 2 = $24.
b) P = MC = $18/unit.
Since P = 36 – 3Q, we have Q = 6.
Thus, Q1 = Q2 = Q/2 = 3 units.
TR1 = TR2 = 3(18) = $54.
TC1 = TC2 = 3(18) = $54.
So 1 = 2 = = $0.
c) 2’s reaction function is the same as in the Cournot case: Q2 = 3 – (1/2)Q1.
1’s demand is
P = 36 – 3(Q2) – 3Q1 = 36 – 3(3– (1/2)Q1) – 3Q1 = 27 – (3/2) Q1.
MR1 = 27 – 3Q1 = MC = $18/unit.
This solves for Q1 = 3 units and Q2 = 1.5 units.
Therefore Q = Q1 + Q2 = 4.50 units and P = 36 – 3(4.5) = $4.50/unit.
1 = 3 (45/2 – 18) = $13.50.
2 = (3/2) (45/2 – 18) = $6.75.
: