Econ 210, Microeconomic Theory
HW 7
Taxes, Subsidies, Tariffs, Quotas
Professor Guse
November 12, 2015
1. Rent Control. Suppose there are two adjacent nearly identical communities – Soho
and Greenwich Village. These two towns share the same demand curve for rental
housing units. (In other words, all potential renters in both towns are perfectly
indifferent about whether to live in Soho or the Village).
Qd (p) = 1000 − p
All rental housing units in both towns are identical and both towns have the same
supply.
Qs,Soho (p) = Qs,GV (p) = max{0, −250 + p}
Read “Qs,i (p)” as the quantity supplied by firm i (or in this case, town i) as a function
of price.
(a) Find the market equilibrium for apartments. What is the price and quantity?
How much is being supplied by Soho landlords? How much by Village landlords?
(b) Soho Rent Control. Suppose that Soho decides to impose a ceiling on rent.
In particular the law states that landlords may not charge more than $375
per month and that landlords who continue to offer their units for rent must
maintain their property at the same quality. However they are free to choose
whether or not to offer their units for rent. Assume that the law is perfectly
enforced. Greenwich Village passes no such law and continues to let the market
determine the price.
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i. Graph the new market being sure to accurately depict what is happening
to aggregate supply around $375.
ii. What will be the excess demand for controlled rental housing in Soho?
(P = 375) = 125. Therefore
ANSWER Qd (P = 375) = 625 while QSoho
s
there are 500 more people who would like Rent Control Apartments (either
instead of a non-RC apartment or instead of no apartment) than there are
such apartments available.
iii. Assume that an allocation rule for the controlled units in Soho is implemented such that these units end up going to renters with the highest willingness to pay. (ironic given the probable intent of the law) ANSWER
See Figure ??.
iv. Discuss how your surplus (CS, PS, SS) figures would change under the
opposite allocation rule. (i.e. controlled rental units go to those with the
lowest willingness to pay (above $375).) ANSWER First consider what
would happen to price in the non-rent-controlled market. With low-WTP
folks getting the RC’d apts, there would be higher-WTP people bidding
up the price in the non-rent-controlled district. So while producer surplus
in the RC-district would be unchanged, producer surplus in the non-rentcontrolled district would be higher. For CS, there would be a variety of
effects. First the low WTP people would have otherwise had no apartment
are clearly better off. However, overall, scarce apartments would be less
efficiently allocted across consumers in this case puutting overall downward
pressure on CS. Finally DWL would be higher.
2
fig:rcHigh
This portion of the Demand
Curve represents the 125 tenants
who arer willing to pay \$875 or more
for an apartment.
\$ / Apt
Loss by tenants who
do not get RC apts.
Greenwich Village Supply
Soho Supply
(identical supply from each town)
Loss by Soho
Landlords
111111111
000000000
000000000
111111111
0000000
1111111
0000000
1111111
0000000
1111111
Gain to GV
Landlords
\$562
\$500
\$375
\$250
111
000
000
111
0000000
1111111
0
1
000
111
0000
1111
000
0000000
1111111
0
1
000
111
0000111
1111
0
1
000
111
0000
1111
0
000
111
0000 1
1111
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
125
250 312
Aggregate Supply: Q=2p−500
(pre Rent Control)
Gain to Tenants
who get
RC apts.
DWL =
1111
0000
00000
000011111
1111
00000
11111
0000
1111
00000 12
000011111
1111
+
Residual Demand. This is what’s
left of the demand curve after
removing the 125 HIGHEST willing
to pay tentant from the original
demand: Q = 875−p.
Demand: Q = 1000−p
500
625
Apartments
2. (Optional Warm-up for Next Problem) Suppose that demand and supply are described by the following marginal willingness to pay and marginal cost functions.
M W T P = a − bQ
M C = c + dQ
Assume that 0 < c < a and b > 0 and d > 0. For each question below, where it
says “describe the welfare effects”, a well-labeled graph is a good start. Don’t bother
to make exact calculations. Focus on qualitative changes and identify winners and
losers. If you making additional assumptions, be specific about what they are.
(a) Describe the welfare effects of a quota, Q̄ set somewhere below a−c
b+d Do you
have to make any assumptions on who is producing or consuming Q̄ in order to
accurately describe the welfare effects?
ANSWER If we assume that the only unit being produced after the quota
goes into effect are those represented on the Supply curve below M C(Q̄) then
the welfare effects are as depicted below.
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Quota of Q̄
Minimum Dead-Weight Loss
a
B
D
Supply
Consumer Welfare Loss
M W T P (Q̄)
A
B
C
D
P0
M C(Q̄)
A
B
Producer Welfare Gain?
A
c
Demand
Q̄
Minus
D
C
Q0
=
a−c
d+b
(b) How would things change if the government auctioned off the quota rights to the
highest bidder. (Imagine a stack of Q̄ certificates each of which entitle the bearer
to produce one unit of the good.) If the quota rights sold for M W T P (Q̄) −
M C(Q̄), then area A+C depicted above would be a gain in Government Revenue
instead and the change in Producer Surplus (w.r.t no quota) would simply be a
loss of area D.
(c) Describe the welfare effects of a unit tax τ < a − c.
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Tax of τ
Minimum Dead-Weight Loss
a
B
D
Supply
Consumer Welfare Loss
Pd
τ
A
B
C
D
P0
Ps
A
B
Producer Welfare Loss?
C
D
c
Demand
Qd (Pd )
=
Qs (Ps )
Q0
=
Government Revenue
A
a−c
d+b
C
(d) Describe the welfare effects of a unit subsidy
Subsidy of σ
Minimum Dead-Weight Loss
a
Supply
Consumer Welfare Gain
Ps
σ
P0
Producer Welfare Gain
Pd
c
Demand
Q0
=
a−c
d+b
Qd (Pd )
Government Expenditure
=
Qs (Ps )
Note that in contrast to a tax, the equilibrium price paid by consumers,
Pd is lower than the price received by suppliers Ps .
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(e) Describe the welfare effects of an un-supported price floor.
Price Floor of F loor > P0
Minimum Dead-Weight Loss
a
B
D
Supply
Consumer Welfare Loss
F loor
A
B
C
D
P0
M C(Qd (F loor))
A
B
Producer Welfare Gain?
A
c
Demand
Minus
D
C
Qd (F loor) Q0
=
a−c
d+b
Note that the welfare effects of an unsupported price floor can be very
similar to those of a quota. In fact in order for the welfare effects to be as
indicated in this picture we have to again make the assumption that only
the lowest cost units are produced (even though suppliers with higher
costs may have incentive to produce if they can get the floor price).
3. The Softwood Lumber Agreement (SLA) between the US and Canada imposes imposes certain measures (tariffs, quotas) to protect the US lumber industry from
cheaper imports from Canada. Suppose the supply and demand equations for the
softwood lumber market in the US and the Canada can be characterized as:
S
QU
d = 20 − 10p
S
QU
s = 5 + 5p
QCanada
= 10 − 5p
d
QCanada
= 15p
s
In all parts, drawing a graph might be helpful. Where surpluses are requested,
calculate them, if you like, or just label the appropriate areas in a graph.
(a) Find the equilibrium in the US and Canada under autarky (no trade). In which
country is the equilibrium price higher? Find US consumer and producer surplus
under autarky.
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Autarky
U.S.
Canada
QUS S
2.00
2.00
QCan
D
CS
QCan
S
1.00
PS
CS
QUDS
0.50
10
20
PS
7.5
10
When the two markets are isolated, the price in the U.S. is $1.00 while
the price in Canada is $0.50. Output and consumption in the U.S. is
higher at 10 bf, while in Canada it is 7.5 bf.
(b) Suppose free trade is allowed. What will be the market clearing price in the
US? How much of the US market is served by domestic producers? How much
is served by foreign producers? Find US consumer and producer surplus. What
is social surplus in the US?
ANSWER. Under FT, all markets must clear at the same price. So let P =
PU S = PCan
S
Can
US
Can
QU
S (P ) + QS (P ) = QD (P ) + QD (P )
5 + 5P + 15P
= 20 − 10P + 10 − 5P
35P = 25
Solving this equation for P we get a world price of
7
5
7
or about $0.71 / bf.
Free Trade
U.S.
Gain in U.S. CS
QUS S
2.00
Loss in U.S. PS
1.00
QUS S + QCan
S
.71
QUDS
8.5
13
QUDS + QCan
D
20
30
Imports
Notice that when the two markets are combined as one, the price settles between the old U.S. price and the old Canadian price at around
$0.71/bf. The nation which had the lower price before free trade,
Canada, becomes an exporter. The U.S. becomes an importer, accepting from Canada the difference between U.S. demand at the new price
(about 13) and the U.S. supply at the new price (about 8.5). In the U.S.
Consumer Surplus increases and producers’ surplus decreases as price
falls. Note that consumers’ gains more than offsets producers’ losses.
(c) Suppose the SLA imposes an import quota of 3 board-feet of lumber. Compare
this case with the free trade case in part B. What is the resulting price in the US
market? What are imports now? What are US conusmer and producer surplus
now? What is the deadweight loss in the US? What is social surplus in the US?
Is it more or less than in part B?
ANSWER. With the quota, the quantity supplied to the U.S. is the U.S.
domestic supply plus the 3 bf allowed in from Canada. We can find the new
price in the U.S. after the quota is imposed by setting U.S. demand equal to
this. 1
= QU.S.
+3
QU.S.
S
D
20 − p = 5 + 5p + 3
1
Note that we can be sure that the full limit of 3 bf would be imported since under FT more than 3 bf
was imported
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Solving this equation yield a U.S. price of $.80.
Quota of 3 bf
U.S.
QUS S
2.00
QUS S + 3
Loss in CS compared to FT
Gain in PS compared to FT
.80
.71
QUDS
9
12
20
With a quota on imports imposed, the price in the U.S. will be higher
compared to the Free Trade case, but still lower than autarky. Here
we see it settles at $0.80 / bf. The changes in CS and PS are shown
w.r.t Free Trade. Note that whenever you show effect of a policy,
it is always necessary to be clear about the counter-factual.
Here we are thinking of Free Trade as the counterfactual. If
instead we were comparing the quota to Autarky, the signs on
the changes in PS and CS would be flipped.
(d) Suppose the SLA imposes a specific tariff of $.05 per board-foot of lumber.
Compare this case with the free trade case in part B. What is the resulting
price in the US market? What are imports now? What are US consumer and
producer surplus now? What is the tariff revenue? What is the deadweight loss
in the US? What is social surplus in the US? Is it more or less than in part B?
Explain.
ANSWER. With a tariff, the new equilibrium price in the U.S. will have to be
higher than the price in Canada by exactly the tariff amount. 2 Hence
PU.S. = PCan + .05
Now consider the market clearing condition. It must still be the case that total
2
Note a tariff is just a tax on some suppliers, in the case Canadian suppliers, therefore it creates a
difference between the price those supplier get for their output and consumers (in the U.S.) pay.
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quantity demanded (across both countries) will equal total quantity supplied.
Therefore we have
S
Can
US
Can
QU
D (PU S ) + QD (PCan ) = QS (PU S ) + QS (PCan )
Substituting in PCan + .05 for PU S , we get the follow single equation with one
unknown - the price in Canada.
S
Can
US
Can
QU
D (PCan + .05) + QD (PCan ) = QS (PCan + .05) + QS (PCan )
⇒ 20 − 10(PCan + .05) + 10 − 5PCan = 5 + 5(PCan + .05) + 15PCan
1
= 35PCan
⇒ 24
4
Therefore the price in Canada after the tariff is imposed is roughly $0.693 / bf
while it will be about $0.743 in the U.S. Compared to the Quota, this will create
a similar welfare effects, except that the Government now earns tariff revenue
and Canadian suppliers lose surplus.
Tariff
U.S.
QUS S
2.00
Loss in CS compared to FT
QUS S +IMPORTS
Gain in PS compared to FT
.71
.74
.69
Gain in Government Revenue
QUDS
.55
8.7
12.6
20
To see graphically how the new Tariff Equilibrium looks from the US
side, we again need to construct a Supply to US curve composed of
domestic US supply and imports. In this case, imports are a function of
U.S. prices and are given by QCan
− QCan
= 20PCan − 10 or 20PU S − 11.
S
D
Note that if prices in the U.S drop below $.55 / bf,imports will go to
zero.
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(e) In the mid 1990s, international courts ruled (under significant pressure and
opposition from the Canadian and US governments, respectively) that “predrilled studs” were not covered by the Softwood Lumber Agreement. A“predrilled stud” is essentially a 2 x 4 with holes drilled in it - for use in homebuilding.
The pre-drilled holes are drilled to accommodate wiring, plumbing, etc. How do
you think this decision affected the softwood lumber market in the US and in
Canada - that is how would this decision change your answers to parts C. and
D.?
ANSWER. Essentially such a decision should have the effect of liberalizing
trade. Canadian suppliers will to at least some extent be able to circumvent
the quota (in part c) or the tariff (part d) by drilling holes in their lumber
and re-labeling it for its trip through customs. This should bring prices in the
two countries closer together with welfare effects somewhere between FT and
the restrictive regimes (c and d). In other words, consumers in the US and
producers in Canada should cheer the ruling, while consumers in Canada and
producers in the US should boo it.
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