Download Problem Set 2 (Ch 4,5,6) 1. (4.12 in book). Let the supply curve be P

Problem Set 2 (Ch 4,5,6)
1. (4.12 in book). Let the supply curve be P = Q and the demand curve
be P = 24 − 2Q.
(a) If a tax of 3 per unit is imposed on buyers, what is the change in
the equilibrium price and quantity, producer and consumer surplus and
revenue?
First, figure out what happens if there is no tax. Equating supply and
demand, Ps = Qs so
Qs = 24 − 2Qd ,
so Q=8. In that case, using either demand or supply, P=8 too.
The producer surplus is .5(8-0)(8)=32 and the consumer surplus is .5
(24-8)(8) = 64. Total surplus is 96.
Next, figure out what happens if a tax of 3 is imposed on buyers. The
demand curve will shift in, towards zero– with the tax, they’re less eager
to buy. A good way to think about it is to ask at what price they’ll demand
exactly Q=0. Before, that was at P=24. Now, they have to pay the tax of 3,
so even if the price is 3 lower, at P=21, they’ll demand zero. Thus, it must
be that the new demand equation is Pd + 3 = 24 − 2Qd , so Pd = 21 − 2Qd ,
so if Q=0, P=21.
Equate supply and demand using the new demand curve, so
Qs = 21 − 2Qd
and Q=7. That means P=7 too, since Ps = Qs . As a result, PS = .5(7)(7)
= 24.5, and CS = .5(21-7)7 = 49. Tax revenue is the tax of 3 per unit
times the output of 7, so it equals 21. Adding those three things up yields
94.5 in total surplus. That’s down by 1.5 from the pre-tax surplus, so the
deadweight loss is 1.5.
1
(b) If a tax of 3 per unit is imposed on sellers instead, what is the result?
Now let’s put the tax on the seller. The supply curve will shift, because
sellers are less eager to sell. Before, if the price rose above 0, there’d
be positive supply. Now, since they have to pay the tax of 3, quantity
supplied would be zero at that price. The price has to rise to 3 before
quantity supplied becomes positive. So now, Ps − 3 = Qs , so Ps = Qs + 3
and if P=3, Q=0.
Equate supply and demand using the new demand curve so
Qs + 3 = 24 − 2Qd
and Q=7. Using the supply curve, P = 7 + 3 = 10.
Producer surplus is .5(10-3)(7) = 24.5. Consumer surplus is .5 (2410)(7) = 49. Government revenue is (3)(7)=21. So everything is the same
as with the tax being on buyers, including total surplus of 94.5 and deadweight loss of 1.5.
2. (5.12 in the book). Under what circumstances would a company prefer
immediate cash equal to 30% of its building costs instead of a tax break
equal to 5% of its cost each year for 20 years? (you do not need to solve out
for the values of the variables in the equation that is the answer)
.05C
Let C be the building cost. The cash is better if .3C > Σ20
t=1 (1+r )t where r
is the discount rate, or, slightly simplified:
This can also be expressed as .3C > (.05C/r )(1 − 1/(1 + r )20 ).
3. (not in book). There are 5000 fishermen working in various jobs on a
fishing boat that vary in their dangerousness. Regression analysis shows
2
that their wages can be predicted by the following equation:
wage = 24,000 + 1000*Experience - 800*past injury + 500*deathrate.
Experience is measured in years working in fishing. Past injury equals
1 if the worker was already injured in the past. Death rate is in deaths
per 10,000 workers per year. The government is thinking of investing $4.3
million per year in weather prediction technology that would reduces deaths
by 1 per year. Should it do so?
If reducing the rate of death by 1 in 10,000 per year is worth $500,
then the value of a statistical life is $500*10,000 = $5 million. That is
greater than $4.3 million, so the technology should be bought. Note that
the fact that there are 5,000 fishermen is not needed for the answer.
4. (6.2 in the book). Suppose the supply curve is Ps = 2Qs , the demand
curve is Pd = 48 − 3Qd , and each transaction creates an externality with
marginal cost X = Q. What is the optimal pollution tax?
What we need to do here is figure out the social marginal benefit and
marginal cost curves and find at what quantity they intersect. Then we
need to set the pollution tax equal to the marginal cost of the externality
at that quantity.
The social marginal benefit curve is the same as the demand curve
here. The social marginal cost curve adds up the private marginal cost—
the height of the supply curve— and the externality. Thus, we add 2Q
and Q and get MCsocial = 3Q. Equating that to social marginal benefit
yields
3Q = 48 − 3Q,
3
so Q=8. At Q=8, the externality marginal cost is also 8, so that should be
the amount of the tax. The equilibrium price will be 16, from the demand
curve, so buyers will buy 8 units.
5. (not in book) Companies A, B, and C emit amounts 10, 20, and 30 of air
pollution (amounts Ea , Eb , Ec ). The marginal benefit of pollution is Xa =
20 − 2Ea for company A, Xb = 30 − Eb for company B, and Xc = 30 − Ec
for company C. Each company is assigned the right to pollute 10 units. If
these rights can be bought and sold, how much will each company end up
polluting?
The Coase Theorem says they will trade emissions to get to the surplusmaximizing allocation. We need the marginal benefits to equal each
other and the total emissions to equal 30. Thus, we have 20 − 2Ea =
20 − Eb = 30 − Ec and Ea + Eb + Ec = 30. We can solve for Ea this way:
Ea = Eb /2 and Ec = Eb + 10 so Eb /2 + Eb + Eb + 10 = 30. Then 2.5Eb = 20,
so Eb = 8. Then we can solve for Ea = 4 and Ec = 18.
4