Dr. Donna Feir
Economics 313
Problem Set 3: Solutions
General Equilibrium: Productive efficiency
1. A firm operates two plants – plant A and plant B – in which it can produce two different goods
– good x and good y. Each plant can be operated for a maximum of ten hours per day. For every
hour that plant A operates, the firm can either produce 10 units of good x or it can produce 5
units of good y. For every hour that plant B operates, the firm can either produce 3 units of good
x or it can produce 6 units of good y. Assume throughout this question that hours can be divided
into fractions (for example, it is possible to spend 3.4 hours producing good x and 6.6 hour
producing good y, etc.).
a. Draw a diagram showing the PPF for plant A. Put units of good x on the horizontal axis
and units of good y on the vertical axis.
y
If all resources are devoted to producing y,
the firm can produce 50 units. Similarly, if
all resources are devoted to producing x, the
firm can produce 100 units. Marginal costs
are constant, so the PPF is linear, as drawn.
50
100
x
b. Draw a diagram showing the PPF for plant B. Put units of good x on the horizontal axis
and units of good y on the vertical axis.
If all resources are devoted to producing y,
the firm can produce 60 units. Similarly, if
all resources are devoted to producing x, the
firm can produce 30 units. Marginal costs
are constant, so the PPF is linear, as drawn.
60
30
x
c. Which firm has the comparative advantage in producing good x? Which firm has the
comparative advantage in producing good y?
Firm A’s MC of producing x is ½ y, while Firm B’s is 2 y. Hence Firm A has the comparative
advantage in producing x. Firm A’s MC of producing y is 2 x, while Firm B’s is ½ y. Hence Firm B
has the comparative advantage in producing y.
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Dr. Donna Feir
Economics 313
d. Now derive the aggregate PPF for this firm. Once again, put units of good x on the
horizontal axis and units of good y on the vertical axis.
y
See lecture material for the method for
deriving the aggregate PPF.
110
60
100
130
x
e. If both goods sell for $10/unit, how much of goods x and y will be produced in plant A?
What about in plant B?
MRTA = ½. That is, the MC of producing one x is ½y. If the firm produces and sells an extra unit
of x it will gain $10 in revenue. To do so, it must (given its PPF) produce and sell ½ fewer units
of y, and so would lose $5 in revenue from sales of good y. So the net gain in revenue from
producing one more x (and ½ less y) is $5. This tell us that it is profit maximizing to produce as
much x as possible (100 units) in plant A. That is, MRT is always < px/py which tells the firm to
all x and no y.
MRTB = 2. That is, the MC of producing one x is 2y, which tells us that the MC of producing one
y is ½x. If the firm produces and sells an extra unit of y it will gain $10 in revenue. To do so, it
must (given its PPF) produce and sell ½ fewer units of x, and so would lose $5 in revenue from
sales of good x. So the net gain in revenue from producing one more y (and ½ less x) is $5. This
tell us that it should produce as much y as possible (60 units) in plant B. That is, MRT > px/py
tell it to always produce less x and more y.
f.
Now suppose that good x sells for $10/unit and good y sells for $4/unit. How much of
goods x and y should be produced in plant A? What about in plant B?
Now we have MRT < px/py for both plants, and to maximize profits both plants should produce
as much x as possible.
g. Now suppose that good x sells for $4/unit and good y sells for $10/unit. How much of
goods x and y should be produced in plant A? What about in plant B?
Now we have MRT > px/py for both plants, and to maximize profits, both plants should
produce as much y as possible.
h. Your answers to the above questions show that firms will (usually) make different
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Dr. Donna Feir
Economics 313
production decisions when prices are different. In each of the cases considered here did
the production choices for each firm always ensure that the two firms were operating
on the aggregate PPF?
Yes, you can see that given the prices in part d), the production choices of the firm resulted in
aggregate output that is just on the kink of the aggregate PPF. Each plant is completely
specializing; plant A in the production of x and plant B in the production of y. Similarly, in part
e) (part f)) both plants are completely specializing in the production of x (y) so are operating at
the horizontal (vertical) intercept of the aggregate PPF. In all three cases, productive efficiency
was achieved.
2. Firm A and B’s PPFs for two goods (x and y) are given below:
PPFA: yA = 300 – 10xA
PPFB: yB = 400 – ¼ xB2
Suppose that we want to efficiently produce 30 units of good x.
a. How many units of good x should firm B produce? Given this, how many units of good x
should firm A produce?
Let’s start by equating the marginal costs of production across the two firms, and see if that
takes us where we need to go.
MRTA = 10; MRTB = ½ xB. Setting the MRTs equal to one another yields xB = 20. If xB = 20, xA =
10 (so that the total x produced is 30 units).
b. How many units of good y will firm B produce? How many units of good y will firm A
produce?
If xB = 20  yB = 400 – (¼)202 = 300.
If xA = 10  yB = 300 – (10)10 = 200.
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Dr. Donna Feir
Economics 313
c. Draw a diagram of each firm’s individual PPF illustrating your answers to parts a) and b).
yB
yA
400
A
B
MRT = MRT = 10
300
300
200
xA
30
10
20
40
xB
Now suppose that we want to efficiently produce 60 units of good x.
d. How many units of good x should firm B produce? Given this, how many units of good x
should firm A produce?
Again, let’s begin by equating the MRTs. MRTA = 10; MRTB = ½ xB. Setting the MRTs equal to
one another again yields xB = 20. If xB = 20, and we want total x of 60, the we would want xA =
40. But, wait: Firm A cannot produce more than 30 units of good x, so this won’t work. This
tells us that we need firm A to produce to capacity (xA = 30) and have firm B produce the rest
(xB = 30). At this point we have MRTA = 10 < MRTB = 15. This tells us that A’s MC of producing x
is less than B’s, so we would like A to produce more x, but A is already producing at capacity.
e. How many units of good y will firm B produce? How many units of good y will firm A
produce?
If xB = 30  yB = 400 – (¼)302 = 175.
If xA = 30  yB = 0.
f.
Draw a diagram of each firm’s individual PPF illustrating your answers to parts d) and e).
yB
yA
400
A
300
B
MRT = 10 < MRT = 15
175
30
xA
30
40
xB
Finally, suppose that we want to efficiently produce 10 units of good x.
g. How many units of good x should firm B produce? Given this, how many units of good x
should firm A produce?
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Dr. Donna Feir
Economics 313
MRTA = 10; MRTB = ½ xB. Setting the MRTs equal to one another again yields xB = 20. If xB = 20,
xA = -10. But this makes no sense, as we can’t have negative production! This tells us that we
want firm B to produce all the desired x (xB = 10) and have firm A produce no x. At this point
we have MRTA = 10 > MRTB = 5. This tells us that A’s MC of producing x is greater than B’s, so
we would like A to produce less x, but A isn’t producing any x. Another (equivalent) to think
about this is that MRTA > MRTB means that A’s MC of producing y is less than B’s, so we would
like A to produce more y. But if xA = 0, A is already producing as much y as possible.
h. How many units of good y will firm B produce? How many units of good y will firm A
produce?
If xB = 10  yB = 400 – (¼)102 = 375.
If xA = 0  yB = 300.
i.
Draw a diagram of each firm’s individual PPF illustrating your answers to parts g) and h).
MRTA = 10 > MRTB = 5
yA
400
375
300
30
xA
10
40
xB
5
Dr. Donna Feir
Economics 313
3. For each of the individual firm PPFs below, derive an expression for the aggregate PPF. Draw a
diagram illustrating your answer for each.
a. PPFA: yA = 250 – 5xA; PPFB: yB = 400 – 2xB.
y
650
250
MRTB = 2
PPF: y = 650 – 2x for x < 200
y = 1250 – 5x for x > 200
MRTA = 5
200 250
x
b. PPFA: yA = 200 – xA; PPFB: yB = 400 – xB.
y
600
MRTB = MRTA = 1
PPF: y = 600 – x
600
x
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Dr. Donna Feir
Economics 313
c. PPFA: yA = 100 – 0.04 xA2; PPFB: yB = 400 – 0.01 xB2.
y
500
250
x
MRTA = MRTB  0.08xA = 0.02xB.
 xB = 4xA.
x = xT = xA + xB  xA = xT – xB
 xB = 4(xT – xB)
 xB = (4/5)xT & xA = (1/5)xT
y = yT = yA + yB = (100 – 0.04 xA2) + (400 – 0.01 xB2)
 y = 500 – 0.04((1/5)xT) 2 – 0.01((1/5)xT) 2
 y = 500 – (4/2500)xT 2 – (16/2500)xT 2
 y = 500 – 0.008xT 2
4. Are the following statements true or false? Explain your reasoning carefully and illustrate your
answer with a diagram or numerical example where useful.
a. If MRTA > MRTB then we want firm A to produce more x and less y (and firm B to
produce less x and more y), as firm A has a lower marginal cost of producing x (where
this marginal cost is measured in units of y foregone).
False. If MRTA > MRTB then the MC of producing x is greater for firm A than for firm B. Thus we
want A to produce less x (and more y) and B to produce more x (and less y) if this is indeed
possible. (If this were an exam question, you should try to provide an example to show that
you really understand this: For instance you could assume that MRTA = 6 and MRTB = 2, and
show that aggregate output could increase by getting A to produce less x and B to produce
more x).
b. As long as each firm is operating on its production possibilities frontier then the
economy is achieving productive efficiency.
False: See lecture material.
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Dr. Donna Feir
Economics 313
5. Suppose that there are two firms, one operating in Canada, and one operating in
Hong Kong. Each firm can produce two goods, x and y. All output is sold in the Canadian
market. Each of the firm’s PPFs is given below:
Canadian firm’s PPF = PPFC: yC = 100 – 0.04 xC2
Hong Kong firm’s PPF = PPFHK: yHK = 400 – 0.01 xHK2.
Assume that good x sells for $2 per unit and good y sells for $1 per unit.
a) If each firm is maximizing its profits, how much of each good will the Canadian
firm produce? How much of each good will the Hong Kong firm produce?
The Canadian firm will produce:
𝑴𝑹𝑻𝑪 = 𝟐 → 𝟎. 𝟎𝟖𝒙𝒄 = 𝟐 → 𝒙𝒄 = 𝟐𝟓
𝒚𝒄 = 𝟏𝟎𝟎 − 𝟎. 𝟎𝟒(𝟐𝟓)𝟐 = 𝟏𝟎𝟎 − 𝟐𝟓 = 𝟕𝟓
The Hong Kong firm will produce:
𝑴𝑹𝑻𝑯𝑲 = 𝟐 → 𝟎. 𝟎𝟐𝒙𝑯𝑲 = 𝟐 → 𝒙𝑯𝑲 = 𝟏𝟎𝟎
𝒚𝑯𝑲 = 𝟒𝟎𝟎 − 𝟎. 𝟎𝟏(𝟏𝟎𝟎)𝟐 = 𝟒𝟎𝟎 − 𝟏𝟎𝟎 = 𝟑𝟎𝟎
b) Draw a diagram of each firm’s PPF illustrating its profit maximizing output
choice.
Slope = -2
𝑦𝐻𝐾
𝑦𝐶
400
300
Slope = -2
100
75
25
50
𝑥𝐶
100
200
𝑥𝐻𝐾
c) Does profit-maximizing behavior on the part of the firms result in productive
efficiency? Explain.
Yes it does because profit maximizing behavior equalizes the marginal
productive efficiency between the firms (𝑴𝑹𝑻𝑪 = 𝑴𝑹𝑻𝑯𝑲 = 𝟐).
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Dr. Donna Feir
Economics 313
d) Derive a mathematical expression for the aggregate PPF given these two
individual PPFs. Draw a carefully labeled diagram illustrating the aggregate PPF.
Use the fact 𝑴𝑹𝑻𝑪 = 𝑴𝑹𝑻𝑯𝑲 to derive the aggregate PPF in order to get an
expression for 𝒙𝑪 and 𝒙𝑯𝑲 in terms of total 𝒙𝑻 .
𝟏
𝒙
𝟒 𝑯𝑲
𝟏
→ 𝒙𝑻 = 𝒙𝑪 + 𝟒𝒙𝒄 = 𝟓𝒙𝑪 → 𝒙𝑪 = 𝒙𝑻
𝟓
𝑴𝑹𝑻𝒄 = 𝑴𝑹𝑻𝑯𝑲 → 𝟎. 𝟎𝟖𝒙𝑪 = 𝟎. 𝟎𝟐𝒙𝑯𝑲 → 𝒙𝑪 =
𝒙𝑻 = 𝒙𝒄 + 𝒙𝑯𝑲
𝟒
This implies 𝒙𝑯𝑲 = 𝟓 𝒙𝑻 .
𝟒
𝟏
Now use the fact 𝒚𝑻 = 𝒚𝒄 + 𝒚𝑯𝑲 → 𝒚𝑻 = 𝟏𝟎𝟎 − 𝟏𝟎𝟎 𝒙𝟐𝒄 + 𝟒𝟎𝟎 − 𝟏𝟎𝟎 𝒙𝟐𝑯𝑲 .
𝟐
𝟐
𝟒 𝟏
𝟏 𝟒
( 𝒙𝑻 ) −
( 𝒙𝑻 )
𝟏𝟎𝟎 𝟓
𝟏𝟎𝟎 𝟓
𝟒
𝟏𝟔
𝟐𝟎 𝟐
𝒚𝑻 = 𝟓𝟎𝟎 −
𝒙𝟐𝑻 −
𝒙𝟐𝑻 = 𝟓𝟎𝟎 −
𝒙
𝟐𝟓𝟎𝟎
𝟐𝟓𝟎𝟎
𝟐𝟓𝟎𝟎 𝑻
𝟏 𝟐
𝒚𝑻 = 𝟓𝟎𝟎 −
𝒙
𝟏𝟐𝟓 𝑻
𝒚𝑻 = 𝟓𝟎𝟎 −
𝑦𝑇
500
250
𝑥𝑇
Now suppose that the Canadian government has decided to impose a $0.20 per unit
tariff1 on good x, meaning that each foreign producer of good x must give $0.20 in tax
revenue to the Canadian government for each unit of good x sold.
e) After the imposition of the tariff, what will be each company’s profit-maximizing
output choices? (assume that the prices at which each firm can sell its output do
not change after the imposition of the tax).
The price that the Hong Kong firm receives for each unit of good x sold is now $0.20 lower:
1
A tariff is a tax on imported goods.
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Dr. Donna Feir
Economics 313
that is, the effective price of x for this firm is now $1.80. The effective price ratio is therefore
1.8. MRTHK = px/py  0.02xHK = 1.8  xHK =90. When xHK = 90, yHK = 319. Because the price
ratio is unchanged for the Canadian firm, this firm does not change its output.
f) How much tax revenue will the Canadian government raise from this tariff? By
how much will the Hong Kong firm’s profits decrease?
The government raises $0.20(90) = $18. The firm’s profits decrease from 500 to 481 so the firm
loses $19. (Note since costs are the same over the PPF we are counting revenue as profit).
g) After the tariff has been imposed, does profit-maximizing behavior on the part of
the firms result in productive efficiency? Explain.
No. Now MRTHK = 1.8 < 2 = MRTC, so we do not have productive efficiency. You can also see
that aggregate output given the tariff lies on the interior of the production possibilities set.
Note that the government has made the firm worse off by $19 in order to raise $18 in
revenue. An alternative policy (if the government needed the $18 in revenue) would be to tax
the Hong Kong firm a lump-sum amount of $18. A lump-sum tax is one that is not conditional
on the behavior of the agent in question, so in this case, does not depend on output, and thus
will not alter the price that the firm receives for it’s output. The firm’s output decision will
therefore not be distorted. This policy (the lump sum tax) makes the firm better off than the
per unit tax and raises the same revenue. Thus it is a Pareto improvement over the tariff. Note
that we would also end up with productive efficiency, precisely because prices are not altered.
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