Problem 1
a)
Consider a …rm that emits a pollutant to the environment. Explain
what is meant by the abatement cost function of the …rm, and discuss brie‡y its
properties.
The abatement cost function gives the costs of the …rm’s abatement (its costs
of reducing its own emissions) as a function of the …rm’s abatement level. It is
usually reasonable to assume that abatement costs are increasing in the abatement level: Reducing emissions may require costly inputs (cleaning equipment,
extra labor), may make the production process less e¢ cient, or may require reducing the …rm’s output. Moreover, to determine a "reduction" we must know
how large emissions "would have" been without the abatement.
Assume that the value of the …rm’s production y is a concave and increasing
function of its emission level m (of course there must also be other inputs (labor,
capital); for simplicity, we keep those …xed and disregard them):
y = f (m)
where f 0 0, f 00 0. Assume that there exists a m
^ > 0 such that for for every
m m;
^ f 0 = f 00 = 0. Assume further that for every (weakly positive) m < m;
^
f 0 > 0 and f 00 < 0. Assume that the …rm is maxmizing its pro…ts (disregard
the costs of other inputs, which we have assumed …xed), and that without
an environmental regulation, it is costless for the …rm to emit the pollutant.
Maximizing f (m) wrt m yields the …rst-order condition for an interior solution
f 0 = 0:Then an unregulated …rm will emit up to m = m
^ (after that, there is
nothing to be gained from emitting even more). We can thus consider m
^ as a
"base emission level" from which to compare abatement. (Note: with the above
speci…cation, the …rm is actually indi¤erent between emitting m
^ and m > m.
^
We could have made the solution less ambiguous by assuming that f 00 < 0
always, and that when m > m;
^ f 0 < 0. In the reasoning below, it is implicitly
assumed that the …rm does not emit more than m
^ since it cannot strictly earn
anything by doing so.)
Abatement can now be de…ned as the reduction in emission compared to the
"base case" level m:
^
a=m
^ m
The abatement cost function c(a) = c(m
^ m) gives the cost of abatement as
a function of a. If f gives the maximal production for any emission level, given
the amount of other inputs, f also implicitly gives the cost of reducing emissions.
Hence, the abatement cost function "mirrors" the production function (that is,
c0 (m
^ m) = f (m)), and with the properties of f given above, we will have that
c0 (0) = 0, c0 (a) > 0 for a > 0, and c00 > 0.
Figure 1
b)
Assume that there is only one …rm that emits the pollutant we are
considering. The regulator knows the damage function of emissions. The regulator knows also that the …rm’s marginal abatement cost function is linear, and
1
knows the slope of this function, but is uncertain about its level. The regulator considers two policy instruments: i) direct regulation of the …rm’s maximal
emission, ii) a tax on the …rm’s emissions. Analyse under what conditions direct regulation is to be preferred, and under what conditions an emission tax is
to be preferred.
Let the marginal abatement cost for any a > 0 be strictly increasing (linarly)
in a, and thus decreasing in m (as long as m > m).
^ Assume (for simplicity) that
marginal damages of emissions are linear and increasing in m.
Assume that the regulator wants to minimize the total costs associated with
emissions (c0 (m
^ m) + D(m)) with respect to m, where D(m) is the damage
function. Di¤erentiating the total costs wrt m, setting the result = 0, gives
c0 = D0
Hence, total costs are minimized when marginal abatement cost equals marginal damage, corresponding to the emission level M in the …gure below (note:
…gures below are from the Perman et al. book; capital M corresponds to small
m in the model above).
If the regulator knew the …rm’s marginal abatement cost function, it could
achieve the desired emission level simply by setting a maximial emission M <
m.
^ Alternatively, it could set a tax per unit emission corresponding to t in the
…gure: With an emission tax, the pro…t of the …rm would be given by f (m) tm,
and maximizing pro…ts wrt m would then yield the …rst order condition f 0 = t.
These two instruments would be perfectly equivalent in the case with perfect
information.
Figure 2.
However, the regulator does not know the level of the MC curve (that is,
marginal abatement costs as a function of emissions, not abatement) in the
above …gure. If it uses direct regulation, emissions are controlled directly, but
marginal abatement costs may be higher or lower than the regulator thought;
if it uses a tax, …rm pro…t max. will ensure c0 = t, but emissions may be higher
or lower than the regulator thought:
Figure 3.
(Note: Figure from the Perman book for the case of many …rms; the "marketable" can be disregarded here.)
The optimal choice of instrument depends on the potential consequences
of "wrong" marginal abatement costs compared to the potential consequences
of "wrong" emission levels. If the marginal damage function is very steep, it
increases marginal damages a lot if emissions are too high; hence this should be
avoided. On the other hand, if the marginal abatement cost function is steep,
marginal costs can become very high if emission levels are set too low; hence
this should ideally be avoided.
Thus: A tax is preferred when the marginal abatement cost curve (MC=B’)
is steeper (absolute slope is greater) than the marginal damage curve (MD=D’).
A quantity restriction is preferred when the marginal abatement cost curve
(MC=B’) is ‡atter (absolute slope is lower) than the marginal damage curve
2
(MD=D’). This can be shown formally; here I just show two graphical illustrations from the Perman et al. book:
Figure 4
Figure 5
c)
Assume that the regulator asks the …rm directly about its abatement
cost functions, without being able to verify the answer. Assume that the …rm can
be of either a high cost or a low cost type. Assume also that the …rm thinks the
regulator will believe the …rm’s answer. What would the …rm have an incentive
to answer if it were truly high cost, respectively low cost, in the following cases?
i)
The …rm knows that a direct regulation will be used.
ii)
The …rm knows that an emission tax will be used.
d)
Would your conclusions in c) be a¤ ected if there were many …rms,
and marginal abatement cost functions may di¤ er between …rms in a way not
known to the regulator? Discuss brie‡y.
Since we know the slope of the marginal abatement cost function, it is reasonable to assume that a "high" or "low" cost type refers to the level of the
abatment cost curve.
Consider …rst the case where the …rm knows that a direct regulation will be
used, and assume that the emission cap will always be binding (less than m).
^
The …rm will then always want the cap to be as high (as close to m)
^ as possible.
If the regulator thinks the …rm is a high cost type, MC=MD implies a higher
optimal emission level than if the regulator thinks it is a low-cost type (see, for
example, …gure 8.1. above, and consider the two MC curves as alternative
beliefs by the regulator). Thus, the …rm prefers that the regulator thinks it is a
high-cost …rm, and will report to be so, regardless of its true type. If the …rm is
truly a low cost …rm, emissions will be too high; if the …rm is truly a high cost
…rm, it will report honestly and the emission level will be socially optimal.
Consider then the case where the …rm knows that a tax will be used. Regardless of its true costs, the …rm will always want the tax to be low. If the
regulator believes that the …rm has high marginal abatement costs, MC=MD
implies a higher optimal tax than if the marginal abatement costs were low (see
the …gures above for an illustration). Thus, the …rm will report to have low
marginal abatement costs, regardless of its true costs.
3
Figure 1:
Production and abatement cost
f (mj)
c(aj)
m^
mj
a=0
(mj=m^ )
a=m^-mj
Figure 2:
Shaded area: Max net
benefits. Can be achieved
through tax t* or quantity
restriction M*.
Figure 8.2 Target setting under perfect information.
MD = D’
(total marg.
WTP to
avoid
emissions)
t*
MC = B’ (marg.cost
of abatement=
marg. benefits of
emissions)
M*
Emissions, M
Figure 3:
Taxes
Permits
t*
P*
MC
M*
MC
L*(=
M
M*)
M
PH
t*
P*
MCH
MCH
PL
MC
MC
MCL
MCL
ML
M*
MH
M
L*(= M*)
M
Figure 8.1 A comparison of emissions taxes and marketable emissions permits when abatement costs
are uncertain.
Figure 4:
Figure 8.3 Uncertainty about abatement costs – costs overestimated.
MD
Loss when
licenses used
tH
t*
MC
(assumed)
Loss when
taxes used
MC (true)
Mt
M*
LH
Emissions, M
Target mistake: Costs of setting the ”wrong” target is larger with tax than permits
Figure 5:
Figure 8.5 Uncertainty about abatement costs – costs overestimated.
MD
MD
tH
t*
MC
(assumed)
MC (true)
Mt
M*
LH
Emissions, M
Target mistake: Costs of setting the ”wrong” target is smaller with tax than permits