ECON 100 Tutorial: Week 6
www.lancaster.ac.uk/postgrad/alia10/
a.ali11@lancaster.ac.uk
office hours: 3:45PM to 4:45PM tuesday LUMS C85
Past exam questions
If an indifference curve is smooth
and convex to the origin, then:
a) The two goods are said to be convex
combinations of each other
b) There is a diminishing marginal rate of
substitution
c) The indifference curve is said to be normal
d) None of the above
Q5
From Tutorial 4 worksheet: Question 1
Assuming an indifference curve which is convex to the
origin, what can this tell us about a consumer’s marginal
rate of substitution between coffee and muffins?
A profit maximizing firm would like
to produce at least the number of
units which minimises short run:
a)
b)
c)
d)
Average total cost
Average fixed cost
Average variable cost
Marginal cost
Note: A profit-maximizing firm produces at the efficient
scale: the quantity of output that minimizes ATC. We
can find this quantity where MC = ATC.
Q18
Long Run Exit Condition
• In the long run, firms will continue if there is a
profit, so the exit condition is:
Profit < 0
TR – TC < 0
TR < TC
AR < ATC
P < ATC
Short Run Exit Condition
• In the short run, fixed costs are sunk costs and firms
will run if there is greater profit from continuing than
from exiting. The firm pays the fixed cost whether it
continues or exits the market, so the exit condition is:
TR – (VC+FC) < -FC
TR-VC-FC < -FC
TR – VC < 0
TR < VC
AR < AVC
P < AVC
• So a firm’s short run exit
condition is P < AVC
P, C
• Since a firms supply
curve is equal to its
Marginal Cost Curve,
and since MC = AVC at
the minimum of AVC, if
0
Q is less than the
quantity that minimizes
AVC, P will be less than
AVC for that Q.
MC=S
AVC
q
Suppose demand curve written
D=120-2P, and the supply curve is
S=20+2P. What is the equilibrium
price and quantity?
a)
b)
c)
d)
P*=70 and Q*=25
P*=25 and Q*=70
P*=50 and Q*=35
P*=35 and Q*=50
Note: Set the two equations equal to each other and
solve for P. Plug that value back in to either equation to
solve for Q.
Q22
Suppose a product has a demand curve written D = 120 – 2P, and
the supply curve is S = 20 + 2P. What is the equilibrium price and
quantity.
Equilibrium occurs where D = S, i.e.
120 – 2P = 20 + 2P
100 = 4P
P = 25
Then, substitute P = 25 into either the D or S equation:
D = 120 – 2 * 25 = 70
or
S = 20 + 2*25 = 70
a.
b.
c.
d.
P* = 70 and Q* = 25
P* = 25 and Q* = 70
P* = 50 and Q* = 35
Q* = 35 and P* = 50
Suppose demand is given by D=120-2P
and supply is originally S=20+2P but
the government imposes a tax of 10
on this good. What happens to the
equilibrium price?
a)
b)
c)
d)
Rises by 10
Rises by 8
Rises by 5
Rises, but it’s not possible to say by how much
Q23
We have D=120-2P and S=20+2P
Then a tax of 10 is imposed on this good.
What happens to the equilibrium price?
There are two ways we can solve this.
1. By assuming that the tax is placed on
consumers, thus affecting the Demand curve
(shifting it to the left)
2. By assuming that the tax is placed on
suppliers/sellers, thus affecting the Supply curve
(shifting it to the left).
I’ll work through both methods in the following
slides.
We have D=120-2P and S=20+2P
Then a tax of 10 is imposed on this good.
By assuming that the tax is placed on consumers, thus affecting the Demand
curve (shifting it to the left)
The new demand curve can be written as:
D = 120 – 2(P+T), where T = 10.
D = 120 – 2P -20
D = 100 – 2P
We then need to find where this new
demand curve crosses the supply curve.
D=S
100 – 2P = 20 + 2P
80 = 4P
P = 20
This gives us the new market equilibrium price. It is the price that the
consumers will give to the suppliers for each good purchased.
On top of this, the consumers must pay the tax of 10, so the total cost to the
consumers will be: P + T = 30.
So the actual price consumers pay will rise by $5 because of this tax.
We have D=120-2P and S=20+2P
Then a tax of 10 is imposed on this good.
By assuming that the tax is placed on suppliers, thus affecting the Supply curve
(shifting it to the left)
The new Supply curve can be written as:
S = 20 + 2(P-T), where T = 10.
S = 200 + 2P -20
S = 2P
We then need to find where this new supply
curve intersects with our original demand curve.
D=S
120 – 2P = 2P
120 = 4P
P = 30
This gives us the new market equilibrium price. It is the price that the
consumers will give to the suppliers for each good purchased. From this, the
sellers have to pay the government a tax of 10, so the total cost to the
consumers will be: P = 30 and the total amount that sellers receive will be 20.
So the actual price consumers pay will rise by $5 because of this tax.
Suppose D=10/P, work out the
price elasticity at P=10 and P = 20
and P=30.
a) Not possible to say without knowing what the
corresponding level of demand is.
b) -1, -2, -3
c) -3, -2, -1
d) -1, -1, -1
Q25
Suppose D=10/P, work out the price elasticity at P=10
and P = 20 and P=30.
Because we are asked to find the price elasticity at a specific
point, we will use the point elasticity method. The equation for
𝑑𝐷
𝑃
𝐷
point elasticity is: 𝜀 =
∗
Step 1: we can solve
𝑑𝑝
𝑄
𝑑𝐷
for :
𝑑𝑝
D = 10/P = 10𝑃−1
𝑑𝐷
= −1 10 𝑃 −1−1
𝑑𝑝
𝑑𝐷
= − 10 𝑃 −2
𝑑𝑝
𝑑𝐷
10
=− 2
𝑑𝑝
𝑃
Now that we have dD/dp, we can plug it into our point
elasticity equation for any value of P and the corresponding Q.
Q25
We have solved for
𝑑𝐷
𝑑𝑝
=
10
− 2.
𝑃
Step 2: we need to solve for Q when P = 10 by plugging into
the demand equation that was given.
D = 10/P
D = 10/10
(remember, in equilibrium D = S= Q, so when P = 10, Q = 1))
D=1
Step 3: we can plug all of these parts into the elasticity
𝑑𝐷
𝑃
𝐷
equation : 𝜀 =
∗
𝑑𝑝
𝑄
10 10
𝜀 𝐷 =− 2 ∗
𝑃
1
10
10
𝐷
𝜀 =−
∗
2
(10)
1
10
10
𝐷
𝜀 =−
∗ =
100
1
100
−
100
= -1
To find elasticity when P = 20 and P = 30, repeat steps 2&3.
Q25
Suppose supply is perfectly elastic at a
price of £10 and the government imposes
a tax of £2 on a good whose demand curve
is given by D=100-5P. Compute the
amount of tax revenue raised, the
deadweight loss of the tax, and the change
in consumer surplus.
a)
b)
c)
d)
10, 80, 90
80, 10, 90
10, 90, 100
10, 75, 85
Q26
Suppose supply is perfectly elastic at a price of £10 (i.e. the S
curve is horizontal) and the government imposes a tax of £2 (so
the S curve shifts upward by 5) on a good whose demand curve is
given by D = 100 – 5P. Compute the amount of tax revenue raised,
the deadweight loss of the tax, and the change in consumer
surplus.
P = 20 – 1/5 D
20
P
To find horizontal intercept:
0 = 20 – 1/5 D
1/5 D = 20
D = 100
D
12
10
S’
S
0
D
40 50
100
If P = 10,
10 = 20 – 1/5 D
D = 50
If P = 12,
12 = 60 – ½ D
D = 40
Continued:
Compute the amount of tax revenue raised, the deadweight loss
of the tax, and the change in consumer surplus.
Tax Revenue:
£2 * 40 = 80
20
DWL:
½ * 10 * 2 = 10
P
D
12
10
S’
S
Tax
0
40 50
D
100
CS = ½ * 50 * 10 = 250
CS’ = ½ * 40 * 8 = 160
CS – CS’ = 90
a)
b)
c)
d)
10, 80, 90
80, 10, 90
10, 90, 100
10, 75, 85
Suppose the TC curve for a firm where
TC=12+4Q+Q2 and MR=8. What level of
output will the firm produce in order to
maximise profit (ie where MC=MR)?
a)
b)
c)
d)
0
2
4
8
Q30
Suppose the TC curve for a firm where TC = 12 + 4Q + Q^2 and
MR = 8. What level of output will the firm produce in order to
maximise profit (i.e. where MC = MR)?
Remember the rule
slope of Y = b.Xc is c.b.Xc-1
a.
b.
c.
d.
0
2
4
8
TC  12  4Q  2Q 2
MC  4  2Q
MR  8
MC  MR :
2Q  4  8
2Q  4
Q2
Exam this Friday
• 50 minutes:
– 30 questions; 20 Caroline, 10 Ian
• Check your timetable for exam time and location.
• Don’t forget to bring the following items:
– Library Card Number
– Pencil and Eraser
– Basic calculator (no programmable calculators or cell
phones will be allowed.)
Good Luck!
(For next week, check Moodle for a worksheet)
This Thursday:
Martin Ravallion
Edmond D. Villani Chair in Economics, Georgetown University;
Research Associate NBER; Non-Resident Fellow CGD;
Formerly Director of the World Bank’s Research Department
will deliver the
Esmée Fairbairn Lecture
Entitled
The Idea of Anti-Poverty Policy
Lecture Theatre 1, Leadership Centre,
Management School,
6.00pm Thursday 14th November 2013
Question 1
If the industry under perfect competition faces a
downward-sloping demand curve, why does an
individual firm face a horizontal demand curve?
In a perfectly competitive market, each firm is
quite small and unable to affect price on it’s own.
We say that firms are price takers.
This means that in a perfectly competitive
market, P = MR = MC.
Question 2
If supernormal profits are competed away under
perfect competition, why will firms have an
incentive to become more efficient?
Improving efficiency can lower costs and lead to
positive short run profits, based on the time it takes
for competing firms to adopt the more efficient
methods.
In the long run, profits will go back to zero, however.
Question 3
Why is the marginal cost curve of a competitive firm
its supply curve?
Question 4(a)
The following table contains information about the revenues and costs for
Ernst’s Golf Ball Manufacturing. All data are per hour. Complete the first
group of columns which correspond to Ernst’s production if P = £3.
Quantity
0
1
2
3
4
5
TR
(P=£3)
TC
Profit
MR
(P=£3) (P=£3)
MC
1
2
4
7
11
16
Total Revenue = Price X Quantity;
Profit = Total Revenue – Total Cost;
Marginal Revenue = Change in Revenue/Change in Quantity;
Marginal Cost = Change in Cost/Change in Quantity;
TR = (P)(Q)
Profit = TR – TC
MR = (TR2-TR1)/(Q2-Q1)
MC = (TC2-TC1)/(Q2-Q1)
Question 4(a)
The following table contains information about the revenues and
costs for Ernst’s Golf Ball Manufacturing. All data are per hour.
Complete the first group of columns which correspond to Ernst’s
production if P = £3. (TR = total revenue, TC = total cost, MR =
marginal revenue, MC = marginal cost)
Quantity
TR
(P=£3)
TC
Profit
MR
(P=£3) (P=£3)
MC
0
0
1
-1
3
1
1
3
2
1
3
2
2
6
4
2
3
3
3
9
7
2
3
4
4
12
11
1
3
5
5
15
16
-1
Question 4(b)
If the price is £3 per golf ball, what is Ernst’s optimal level
of production? What criteria did you use to determine the
optimal level of production?
Quantity
TR
(P=£3)
TC
Profit
MR
(P=£3) (P=£3)
MC
0
0
1
-1
3
1
1
3
2
1
3
2
2
6
4
2
3
3
3
9
7
2
3
4
4
12
11
1
3
5
5
15
16
-1
To find the optimal level
of production, we find
where MR = MC.
Optimal production is
either two or three golf
balls per hour. This level
of production maximizes
profit (at £2) and it is the
level of output where MC
= MR (at £3).
Question 4(c)
Is £3 per golf ball a long-run equilibrium price in the market for golf
balls? Explain. What adjustment will take place in the market for golf
balls and what will happen to the price in the long run?
Answer: No, because Ernst is earning positive economic profits of £2.
These profits will attract new firms to enter the market for golf balls,
the market supply will increase, and the price will fall until economic
profits are zero.
Quantity
TR
(P=£3)
TC
Profit
MR
(P=£3) (P=£3)
MC
0
0
1
-1
3
1
1
3
2
1
3
2
2
6
4
2
3
3
3
9
7
2
3
4
4
12
11
1
3
5
5
15
16
-1
Question 4(d)
Suppose the price of golf balls falls to £2. Fill out
the remaining three columns of the table above.
Quantity
0
1
2
3
4
5
TR
(P=£3)
0
3
6
9
12
15
TC
1
2
4
7
11
16
Profit
MR
(P=£3) (P=£3)
-1
3
1
3
2
3
2
3
1
3
-1
MC
1
2
3
4
5
TR
Profit
MR
(P=£2) (P=£2) (P=£2)
0
-1
2
2
0
2
4
0
2
6
-1
2
8
-3
2
10
-6
Question 4(d)
What is the profit-maximizing level of output when the
price is £2 per golf ball? How much profit does Ernst’s Golf
Ball Manufacturing earn when the price of golf balls is £2?
Answer: Optimal production is either one or two golf balls
per hour. Zero economic profit is earned by Ernst.
Quantity
0
1
2
3
4
5
TR
(P=£3)
0
3
6
9
12
15
TC
1
2
4
7
11
16
Profit
MR
(P=£3) (P=£3)
-1
3
1
3
2
3
2
3
1
3
-1
MC
1
2
3
4
5
TR
Profit
MR
(P=£2) (P=£2) (P=£2)
0
-1
2
2
0
2
4
0
2
6
-1
2
8
-3
2
10
-6
Question 4(e)
Is £2 per golf ball a long-run equilibrium price in the
market for golf balls? Explain. Why would Ernst
continue to produce at this level of profit?
Answer: Yes. Economic profits are zero, therefore
firms will neither enter nor exit the industry.
Zero economic profits means that Ernst doesn’t
earn anything beyond his opportunity costs of
production but his revenues do cover the cost of his
inputs and the value of his time and money.
Question 4(f)
Describe the slope of the short-run supply curve for
the market for golf balls. Describe the slope of the
long-run supply curve in the market for golf balls.
The slope of the short-run supply curve is positive
because when P = £2, quantity supplied is one or two
units per firm and when P = £3, quantity supplied is two
or three units per firm.
In the long run, supply is horizontal (perfectly elastic) at
P = £2 because any price above £2 causes firms to enter
and drives the price back to £2.
Question 5(a)
Draw the isoquant corresponding to the following table, which shows
the alternative combinations of labour and capital required to
produce 100 units of output per day of good X.
Capital 16
Labour 200
20
160
26.67
120
40
80
60
53.33
80
40
100
32
120
100
80
60
40
20
0
0
50
100
150
200
250
Question 5(b)
Assuming that capital costs are £ 20 per day and the wage rate is £10 per day, what is the
least-cost methods of producing 100 units? What will the daily total cost be?
Given:
Solve for:
Capital
Labour
cost of capital
cost of labour
total cost
16
200
320
2000
2320
20
160
400
1600
2000
26.67
120
533.4
1200
1733.4
40
80
800
800
1600
60
53.33
1200
533.3
1733.3
80
40
1600
400
2000
100
32
2000
320
2320
Price of Capital: £20/day
Price of Labor: £10/day
The least-cost method of production uses 40 units
of Capital and 80 units of Labor. This method
costs £1600 per day.
Question 5(b)
Use Excel to graph, or graph by hand, the isoquant curve and the Isocost lines:
120
100
80
Isoquant
Series2
60
Series3
Series4
40
20
0
0
50
100
150
200
250
Question 5(c)
Now assume that the wage rate rises to £20 per day. Draw a new series of isocosts. What
will be the least-cost method of producing 100 units now? How much labour and capital
will be used?
Given:
Solve for:
cost of capital cost of labour total cost
Capital
Labour
16
200
320
4000
4320
20
160
400
3200
3600
26.67
120
533.4
2400
2933.4
40
80
800
1600
2400
60
53.33
1200
1066.6
2266.6
80
40
1600
800
2400
100
32
2000
640
2640
Price of Capital: £20/day
Price of Labor: £20/day
The least-cost method of production uses 60 units
of Capital and 53.33 units of Labor. This method
costs approximately £2,267 per day.
Question 6
In a downturn firms want to layoff some workers.
This has an effect on productivity (output per
employee). On the one hand, it frees up some
machinery that the remaining workers can use
more flexibly – you don’t have to hang around so
much waiting for a machine to become free. On the
other hand, workers have to do a wider ranges of
tasks because there are fewer workers – so the firm
loses some of the advantages of specialisation.
Suppose the output of the firm, Q, depends on the
number of workers, L, and the number of machines,
K, in such a way that Q=LaKb Suppose a=0.4 and
b=0.6.
Question 6(a)
Suppose the output of the firm, Q, depends on the number of
workers, L, and the number of machines, K, in such a way that Q=LaKb
Suppose a=0.4 and b=0.6.
Write down an expression for the average product of labour, APL.
HINT: APL=Q/L.
We’ll start with:
And plug in Q=LaKb for Q:
APL=Q/L
APL= LaKb /L
This can be simplified to: APL = La-1Kb
Now we can plug in a=0.4 and b=0.6:
APL = L-0.6K0.6
APL = (K/L)0.6
Question 6(b)
Now Suppose L=10 and K=10. What is the firm’s output? And its APL?
To find the firm’s output, we plug in L=10 and K=10 into our Output
function from part (a):
Q = LaKb
From part (a), we know that a=0.4 and b=0.6, so:
Q = L0.4K0.6
Plugging in L=10 and K=10:
Q = 100.4100.6
Q = 101
Q = 10
So the firm’s output is 10 units.
We solved for APL in part (a):
APL = (K/L)0.6
So, we can plug in L=10 and K=10
APL = (10/10)0.6
APL = 1.
So the firm’s average product of labor is 1.
Question 6(c)
If the number of workers is reduced by 1 (i.e 10%) what happens to output?
And the APL?
So, now L = 9 and K = 10. We will go through the same steps in part (b).
From part (a), we know that Q = LaKb and that a=0.4 and b=0.6, so:
Q = L0.4K0.6
Plugging in L=9 and K=10:
Q = 90.4100.6
Q = 9.6
So the firm’s output has fallen from 10 to 9.6, or it has fallen by 4%.
We solved for APL in part (a):
APL = (K/L)0.6
So, we can plug in L=9 and K=10
APL = (10/9)0.6
APL = 1.065
So the firm’s average product of labor has gone from 1 to 1.065, or it has
risen by 6.5%.
Question 7
Cost functions depend on the nature of the firm’s
technology (i.e. its production function) and input
prices.
Suppose beer is produced according to the
production function:
Q=1.5 L0.4 K0.6
Assume that K is fixed at 100 units in the short run.
The price of a unit of K is £8. So fixed cost is £800.
Question 7(a)
Suppose beer is produced according to the production function: Q=1.5 L0.4 K0.6
Assume that K is fixed at 100 units in the short run. The price of a unit of K is £8. So
fixed cost is £800.
If K is fixed then, in the short run, show how Q depends on L, and L depends on Q.
Q=1.5 L0.4 1000.6
Q = 1.5L0.4(15.8)
Q = 24L0.4
To show how L depends on Q, we can solve the above equation for L:
Q = 24L0.4
𝑄
24
= L0.4
𝑄 1/0.4
=L
24
𝑄 1/0.4
L=
24
≈ 0.00035Q2.5
Question 7(b)
Suppose beer is produced according to the
production function: Q=1.5 L0.4 K0.6
Assume that K is fixed at 100 units in the short run.
The price of a unit of K is £8. So fixed cost is £800.
The only variable factor is L in the short run.
Suppose the wage rate is £25 per unit of L. What is
the relationship between VC and output?
VC = wL = w L(Q)
From part (a) we solved for L and can plug that in:
VC = 25
𝑄 1/0.4
24
VC = 0.0089 Q2.5
Question 7(c)
Suppose beer is produced according to the production
function:
Q=1.5 L0.4 K0.6
Assume that K is fixed at 100 units in the short run. The
price of a unit of K is £8. So fixed cost is £800.
The only variable factor is L in the short run. Derive AVC
We know that:
AVC=VC/Q
We solved for VC in part (b) and can plug that in here:
AVC = 0.0089 Q2.5/Q
AVC = 0.0089 Q1.5
Question 7(d)
Suppose beer is produced according to the production function: Q=1.5 L0.4 K0.6
Assume that K is fixed at 100 units in the short run. The price of a unit of K is £8. So
fixed cost is £800.
Derive MC. HINT: You will need to find the slope of
the VC function.
Marginal cost is the slope of variable cost curve, or
the derivative of VC:
From (b) we know: VC = 0.0089 Q2.5
MC = slope of VC = dVC/dQ
dVC/dQ = 2.5*0.0089 Q(2.5-1)
=0.022 Q1.5
Question 7(e)
Suppose beer is produced according to the
production function:
Q=1.5 L0.4 K0.6
Assume that K is fixed at 100 units in the short run.
The price of a unit of K is £8. So fixed cost is £800.
Use Excel to graph MC, AFC, AVC and AC against Q
(from a range of Q from 0 to, say 300)