DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES
CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CALIFORNIA 91125
Design of Tradable Permit Programs under
Imprecise
Measurement
John O. Ledyard
California Institute of Technology
SOCIAL SCIENCE
WORKING PAPER
1438
March 2018
Design of Tradable Permit Programs under
Imprecise Measurement
John O. Ledyard
∗†
Abstract
If the measurement of production in a commons is accurate and
precise, it is possible to design a tradable permit program such that,
under a fairly general set of conditions, the market equilibrium is ef-
ficient for the given aggregate permit level and everyone is better off
after the permit program than before. Often, however, implementa-
tion of a tradable permit system is postponed or never undertaken be-
cause an inexpensive technology able to provide accurate and precise
measurements does not exist. However, there often is an inexpensive
technology which accurate but not precise. I study the possibilities
for the design of a tradable permit system when the measurement
technology involves an imprecise, indirect measure of production that
contains statistical uncertainty. To the best of my knowledge, this has
not been studied before.
As one might expect, imprecise measurement can lead to ineffi-
ciency and prevent voluntary participation. But there are positive
results. If measurement errors are proportional to use, it is possible
to design so that aggregate output is efficiently allocated. Also, it is
possible to calculate a set of individual firm lump-sum subsidies to
attain voluntary participation.
Keywords: quota, permits, rights, tradable, imprecise measurement, volun-
tary participation, imperfect enforcement
∗
I thank the Max Factor Family Foundation in partnership with the Jewish Community
Foundation of Los Angeles for its financial support of this project.
†
Address: Division of the Humanities and Social Sciences, California Institute of Tech-
nology, 1200 E. California Blvd., Pasadena, CA 91125. Email:jledyard@caltech.edu
1
1 Introduction
The formal approach to mechanism design began with Hurwicz’s 1960 paper.
He recognized that the information about the economic environment, such as
technological possibilities, preferences, and endowments, is dispersed among
economic agents and that the “informational tasks entailed by the mecha-
nism imply costs in real resources used to operate the mechanism” (Hurwicz
and Reiter 2006, p.1). He then focused on those informational tasks and
searched for mechanisms that produced an efficient resource allocation and
were informationally efficient in the sense that the messages were smaller
that those of other possible mechanisms that produced an efficient resource
allocation. In his 1972 paper, Hurwicz took mechanism design to the next
level by introducing the concept of incentive compatibility. Soon after, in
1973, Gibbard introduced the Revelation Principle which led theorists to fo-
cus on direct mechanisms, mechanisms whose messages are everything an
agent knows about the environment, and to ignore the “informational tasks”
entailed by such a mechanism. I believe more attention needs to be put on
the costs of acquiring the information necessary to attain a measure of re-
source efficiency and the trade off between the two types of efficiency. In this
paper, I take a very small step in that direction.
I look at one of the informational tasks involved in cap and trade programs.
Such programs possess a degree of informationally efficiency since they em-
ploy the price mechanism to communicate needed information about the
environment.
1
But there is another, usually ignored, part of the problem
−
identifying the size of the output decisions of producers in a commons. In
order to enforce the rules of a cap and trade program, it is necessary to com-
pare that output to the permits held by the agents. But it is often the case
in practice that it is difficult or impossible to achieve valid measurement of
that output. Instead imprecise measurement is possible and less expensive.
In this paper I pursue what is lost, if anything, by relying on that imprecise
measurement when using cap and trade to manage a commons.
The tragedy of the commons is well-known: an unmanaged, common resource
will be over used and the benefits from its use will be lower than would be
possible under a benevolent dictator. It is also well-known that, if use can
1
I will side-step adverse selection incentive problems by assuming competitive behavior
on the part of the economic agents.
2
be accurately and precisely measured, it is possible to design a tradable per-
mit program such that, under a fairly general set of conditions, the permit
market equilibrium allocation is efficient for the given aggregate permit level
and everyone is better off after the permit program than before. I will refer
to such a program as satisfactory.
In practice, however, the implementation of a tradable permit program is
often postponed or never undertaken because there does not exist an in-
expensive technology able to measure violations accurately and precisely.
However, sometimes there is an inexpensive technology available that can
measure violations imprecisely. That is, there is random measurement error.
Examples of imprecise measurement in commons problems are easy to find.
I look at only two: fisheries and ground water.
In the management of fisheries, it is the catch is that is often permitted. The
easiest and most direct way to measure the catch is at the landing where the
catch is unloaded and can easily be weighed. But that is actually only an
inaccurate and imprecise measure of what has really been caught. For exam-
ple, high grading, keeping the best and tossing the rest, would mean more
fish are caught than measured. The best place to measure the actual catch
is at the point of catch, but that is remote. Nevertheless, a variety of meth-
ods have been employed to try to get such measurements. Remote sensing
through satellites, onboard human observers and/or surveillance cameras are
some of the technologies used. While these seem to improve measurement,
there are still errors. The measurement is imprecise.
In the management of ground water, it is important to know the amount of
water pumped from an aquifer. The obvious way to measure it is to place a
meter at the pump. But in many situations, meters do not exist. Meters are
costly and the manpower to read them is expensive. There have been sev-
eral alternative measurement schemes proposed to be used in place of direct
metering. One is to measure the electricity used by the pumps. The Turlock
Irrigation District in California uses this method.
2
Another method is to use
remote sensing by satelite or unmanned drones, using evapotranspiration as
2
See also Zekri (2009) for a deeper discussion of this method.
3
an indirect measure of water use.
3
Of course, both these technologies have
errors in measurement. The measurement is imprecise.
To the best of my knowledge, the implications of imprecise measurement for
permit markets have not been studied before. In the literature, measurement
is always perfect - precise and accurate. Some studies have considered the
case when, although the measurement is perfect, it is only done with some
probability - thereby lowering the costs. This is often called random or im-
perfect monitoring.
4
The main conclusion of this research is that, with risk
neutral producers, if the penalty rate per unit violation is high enough then
the equilibrium allocations in a competitive permit market will be exactly
the same as under perfect measurement. As long as the expected cost of a
violation is high enough, producers in a commons will chose to hold permits
exactly equal to their planned output. Because of that, the combination of
perfect measurement and random monitoring can lead to a satisfactory pro-
gram.
In this paper, I study whether imprecise measurement could be the basis for
a satisfactory permit program. Unfortunately and perhaps not surprisingly,
the answer is no. It is useful to understand why. With imprecise measure-
ment, producers will buy more permits than their actual output in order to
insure against the potential that measurement will indicate higher produc-
tion than is actually the case. This leads to two types of resource inefficiency.
One occurs because aggregate output will be less than the aggregate amount
of permits. The second occurs if the errors in measurement are biased. If
errors are not proportional to production, then aggregate output will not
be efficiently allocated. Imprecise measurement can also make it difficult or
impossible to guarantee that all producers will be at least as well off as they
were before the program was put into place. This because the aggregate cost
of holding permits to insure against mis-measurement can be greater than
the increase in aggregate benefits from better management of the commons.
This can cause political difficulties.
3
See Water Education Foundation (2015) for a discussion of this method.
4
See Malik (1990), Malik (1992), Stanlund and Dhanda (1999), Stanlund (2007), Mur-
phy and Stanlund (2007), and Stanlund et. al. (2008).
4
In spite of these negative findings, there are some positive results. First, if
measurement errors are proportional to use, then it is possible to design a
tradable permit program with imprecise measurement such that the equi-
librium aggregate output is efficiently allocated. Second, there are easily
calculated subsidies that allow the design of a tradable permit program that
guarantees that producers will be better off than they were before the pro-
gram. Third, as the precision of the measurement approaches perfect accu-
racy, the equilibrium of the market with imprecise measurement approaches
what it would be with perfect measurement and the size of the subsidies
necessary to guarantee a Pareto-improvement decline to zero.
2 The Commons
A collection of producers, named
i
= 1
,...,N,
are involved in a commons.
Each chooses a level of production,
q
i
.
5
A producer’s economic profits are
b
i
(
q
i
,Q
) where
Q
=
∑
i
q
i
. The fact that aggregate output affects an indi-
vidual’s profits creates the externality that is at the heart of the commons
problem. There is a set of standard assumptions that guarantees this model
of the commons is well-behaved.
Assumption 1
(Regular Commons)
.
(i)
b
i
(
q
i
,Q
)
is strictly concave and increasing in
q
i
, decreasing in
Q
, and
continuously differentiable in
q
i
and
Q
.
(ii)
lim
x
→
0
b
iq
(
x,Q
) =
∞
.
6
(iii)
b
iqQ
≤
0
,
∀
i
.
Assumption 1(i) is standard. Assumption 1(ii) ensures that no producer will
ever want to drop out. This is not necessary for many of the results in this
paper but does make them a little cleaner. Assumption Assumption 1(iii) is
similar to a single crossing condition. It ensures that aggregate demand is
downward sloping.
5
The model would be essentially the same if the producers were choosing an input.
6
I use the following notation for derivatives of functions:
f
x
=
∂f
(
x,y,z
)
/∂x
and
f
xy
=
∂
2
f
(
x,y,z
)
/∂x∂y
. The index i is the name of a producer and is not a variable.
5
I assume throughout that producers behave competitively in the commons.
That is, each producer acts as if their individual choice of production level
will not affect aggregate production.
In the absence of collective management, autarky reigns.
Definition 1
(Autarkic Equilbrium)
.
An Autarkic Equilibrium is
(
q
a
,Q
a
)
where (i)
q
a
i
solves
max
q
b
i
(
q,Q
a
)
,
∀
i
and
(ii)
Q
a
=
∑
i
q
a
i
.
Economic efficiency is a standard benchmark and a desirable target for pub-
lic policy.
Definition 2
(Efficient Allocation)
.
An allocation
̃
q
is Efficient if and only if
̃
q
solves
max
q,Q
∑
i
b
i
(
q
i
,Q
)
subject
to
Q
=
∑
i
q
i
.
It is well known that Autarkic equilibria are generally not efficient. Thus,
there is the opportunity to design a policy that will guide the producers on
the commons to higher aggregate profits. A tradable permit program can be
one such policy.
In Section 3, I provide some basic results for a permit market with accurate
measurement. This material is reasonably well known and is provided to
introduce the reader to the notation, concepts and prior results that I will
later refer to. Section 4 contains the main results of this paper.
3 Tradable Permits with Valid Measurement
A tradable permit program specifies an aggregate level of permits, L, allocates
it to the producers, requires the producers to keep their production level to no
6
more than the permits than they hold, and allows trading of those permits.
The initial allocation of permits is
l
o
, where
∑
l
o
i
=
L
.
A valid measurement technology is both accurate and precise. With valid
measurement of
q,
a tradable permit program can be enforced by monitoring
and imposing a penalty for producing more than the permits held. I assume
that the penalty is a function only of the level of violation
v
i
=
q
i
−
l
i
. This is
standard in the literature.
7
The penalty to be paid by
i
is indicated by
P
(
v
i
).
One typical and simple form of the penalty function has
P
(
v
i
) = max
{
0
,av
i
}
where
a
is a positive constant. I allow a wider range of possibilities.
I assume producers behave competitively in the permit market.
8
Let
p
be
the market price of permits.
Definition 3
(Market Equilibrium under Valid Measurement)
.
A Permit Market Equilibrium under Valid Measurement is
(
q
∗
,v
∗
,p
∗
,Q
∗
)
such that,
(i) each producer
i
chooses
(
q
∗
i
,v
∗
i
)
to solve
max
(
q
i
,v
i
)
b
i
(
q
i
,Q
∗
)
−
p
∗
(
q
i
−
v
i
−
l
o
i
)
−
P
(
v
i
)
,
(ii)
Q
∗
=
∑
i
q
∗
i
,
and
(iii)
∑
i
q
∗
i
−
v
∗
i
=
L
.
In Definition 3, (
q,Q
) and
v
can be solved for independently since they are
separable. We use this fact to define a supply of output and a demand for
violations.
7
Of course,
P
could depend on more than just
v
i
. For example,
P
could depend on
the percentage violation so that the penalty is
P
(
q
i
/l
i
). But when
∂P/∂q
i
+
∂P/∂l
i
6
= 0,
permit market equilibria will generally not be efficient. Thus, the design choice is usually
P
(
v
i
).
8
This is rarely true in practice. Even if there is a large number of producers, most
extant permit markets are disorganized and thinly traded. They tend to violate the Law
of One Price and, therefore, traders’ behaviors are not really competitive. There are ways
to design a trading mechanism to avoid this, but that rarely happens.
7
Definition 4
(Supply and Demand under Valid Measurement)
.
(a) Supply under valid measurement is
[
q
V
(
p
)
,Q
V
(
p
)]
where
q
V
i
(
p
)
∈
arg max
q
i
b
i
(
q
i
,Q
V
(
p
))
−
pq
i
and
Q
V
(
p
) =
∑
N
i
=1
q
V
i
(
p
)
.
(b) Demand under valid measurement is
[
v
V
(
p
)
,V
V
(
p
)]
where
v
V
i
(
p
)
∈
arg max
v
pv
i
−
P
(
v
i
)
and
V
V
(
p
) =
∑
N
i
=1
v
V
i
(
p
)
.
The market equilibrium price satisfies
Q
V
(
p
∗
)
−
V
V
(
p
∗
) =
L
. To ensure the
market is well-behaved, I impose a regularity assumption.
Assumption 2
(Regular Accurate Measurement)
.
(i)
P
(
v
i
) = 0
on
v
i
≤
0
. P
(
v
i
)
is convex, increasing, and continuously differ-
entiable on
v
i
≥
0
.
(ii)
L > Q
V
(
P
v
(0
+
))
.
9
Assumption 2 (i) is a slight generalization of the linear penalty, max
{
0
,av
i
}
.
Assumption 2 (ii) is needed to ensure that the penalty is strong enough so
that producers do not want violations in equilibrium.
Result 1.
Under Assumptions 1 and 2,
Q
p
(
p
)
<
0
and
V
p
(
p
)
≥
0
.
10
The geometry of this market is displayed in Figure 1 in Section 7.1.
3.1 Efficiency and Political Viability
Permit markets mimic competitive markets which, under the appropriate
conditions, produce allocations that are efficient. But permit markets do not
necessarily produce efficient allocations unless the supply of permits,
L
, is
exactly right. Instead, permit markets produce allocations that are efficient
given the total amount of permits,
L.
9
P
v
(0
+
) =
lim
x
→
0
+
P
v
(
x
).
10
The proofs are omitted since they are standard and straight-forward. For the details
see, e.g., Ledyard (2018).
8
Definition 5
(Efficiency given X)
.
The allocation
̃
q
is Efficient given X if and only if
̃
q
solves
max
q
∑
i
b
i
(
q
i
,X
)
subject to
∑
i
q
i
=
X
.
If the total number of permits,
L
, is chosen appropriately, aggregate profits
after the program is implemented will be higher than before. But producers
won’t receive those increases unless the program is actually adopted and
implemented. A satisfactory program must be politically viable.
If the permit program can be designed in a way that all producers are better
off after the introduction of the permit program than they are in the au-
tarkic equilibrium, then the program will be more likely to be adopted. In
the language of mechanism design, such a property of the design is called
voluntary participation or individual rationality. The well known result is
that all producers will be better off at the permit market equilibrium than
in autarky as long as the distribution of the initial permits,
l
o
,
is such that
all producers are at least as well off at
l
o
i
as they are in autarky.
11
Result 2.
Let
(
q
∗
,v
∗
,p
∗
,Q
∗
)
be a permit market equilibrium. Under As-
sumptions 1 and 2:
A) Efficiency
(i)
q
∗
is efficient given
L
,
Q
∗
=
L
, and
v
∗
= 0
.
12
(ii) The permit market equilibrium is independent of the initial distri-
bution,
l
o
, and only depends on
L
.
B) Voluntary Participation
(i) For any initial allocation of permits,
l
o
, such that
b
i
(
l
o
i
,L
)
> b
i
(
q
V
,Q
V
)
,
∀
i,
it will be true that
b
i
(
q
∗
i
,Q
∗
)
−
p
∗
(
q
∗
i
−
v
∗
i
−
l
o
i
)
−
P
(
v
∗
i
)
> b
i
(
q
V
i
,Q
V
)
,
∀
i
.
(ii) Let
q
e
be efficient given
L
and
Q
e
=
∑
i
q
e
i
. There exists an
l
o
satis-
fying (i) if and only if
∑
i
b
i
(
q
e
i
,Q
e
)
>
∑
i
b
i
(
q
V
i
,Q
V
)
.
11
Since it is rare that such a distribution is unique, there may still be serious political
bargaining over the allocation of L.
12
If the penalty is weak so that
L < Q
V
(
P
v
(0
+
)), then
v
∗
i
>
0 and
Q
∗
> L
. In this case,
q
∗
is not efficient given
L
. But
q
∗
will be efficient given
Q
∗
. See, e.g., Malik (1990).
9
The proofs are omitted since they are standard and straight-forward.
13
Result (i) holds because the producer has been put into an initial position
at least as good as autarky and, by choosing
q
i
=
l
o
i
,
she can protect that
position. Anything she then decides to do will be at least as good. If the
commons effect of
Q
is large enough, as in fisheries, then letting
l
o
be based
on autarkic output, so that
l
o
i
=
q
a
i
Q
a
L,
will often be sufficient.
Result (ii) holds because of the quasi-linearity of profits. Locally, if
L < Q
V
,
there exists an
l
o
satisfying (i).
To summarize, with a valid measurement technology it is possible to design
a tradable permit program such that, under a fairly general set of conditions,
the market equilibrium is efficient for the given permit total and everyone is
better off than with autarky.
4 Tradable Permits with Imprecise Measure-
ment
As shown in the previous section, under ideal conditions, the benefits of using
a tradable permit system to manage an over-used commons are increased
aggregate profits and political viability. But, often implementation of such
a system is postponed or never undertaken because the conditions are not
ideal. One reason for this can be the absence of an inexpensive technology
able to provide valid measurements of violations. However, there often is
an inexpensive technology which is approximately valid. In this section, I
study the possibilities for the design of a tradable permit system when the
measurement technology involves an indirect measure of
q
i
that contains
statistical uncertainty. Such a measurement is imprecise.
I model imprecise measurement technologies as follows. The indirect measure
of producer i’s output is
w
i
=
q
i
+
where
is the measurement error. I
assume
is a random variable with density
f
(
,q
i
)
d.
Further, I assume that
E
(
|
q
i
) =
∫
f
(
,q
i
)
d
= 0; that is, the measurement technology is accurate.
This is pretty much without loss of generality since, if
E
(
|
q
i
) is not accurate
13
For details, see, e.g., Ledyard (2017).
10
but is a known function of
q
i
,
then one can adjust the penalty function to
account for the inaccuracy.
14
Finally, I assume that the measurement error is
the same for every producer. This is not entirely without loss of generality.
15
The measured violation is
w
i
−
l
i
=
+
q
i
=
v
i
+
.
The penalty to be
paid by the producer, based on the measured violation, is
ρ
(
w
i
−
l
i
)
.
When
the producer chooses her output, she faces an
expected penalty payment
of
P
(
v
i
,q
i
) =
∫
ρ
(
v
i
+
)
f
(
,q
i
)
d
.
With imprecise measurement, the definition of market equilibrium in Section
3 needs to be slightly altered to allow for the fact that the expected penalty
now depends on
q
i
.
Definition 6
(Market equilibrium under Imprecise Measurement)
.
A permit market equilibrium under imprecise measurement is
(
q
∗
,v
∗
,p
∗
,Q
∗
)
such that,
(i) each producer
i
, chooses
(
q
∗
i
,v
∗
i
)
to solve
max
(
q
i
,v
i
)
b
i
(
q
i
,Q
∗
)
−
p
∗
(
q
i
−
v
i
−
l
o
i
)
−P
(
v
i
,q
i
)
(ii)
Q
∗
=
∑
i
q
∗
i
,
and
(iii)
∑
i
q
∗
i
−
∑
i
v
∗
i
=
L
.
(
q,Q
) and
v
are no longer independent as they were under valid measure-
ment. But we can still define demand and supply functions.
Definition 7
(Supply and Demand under Imprecise Measurement)
.
Supply under imprecise measurement is
[
q
I
(
p
)
,Q
I
(
p
)]
and Demand under
imprecise measurement is
[
v
I
(
p
)
,V
I
(
p
)]
where
14
See Ledyard (2018).
15
Differences across producers might occur in water markets for different crops or dif-
ferent irrigation technologies, and in fishing markets for different species or different gear
types. These differences would affect the efficiency results below to some extent. I leave
it to the reader to work out those implications.
11
q
I
i
(
p
)
∈
arg max
q
i
b
i
(
q
i
,Q
I
(
p
))
−
pq
i
−P
(
v
I
i
(
p
)
,q
i
)
and
Q
I
(
p
) =
N
∑
i
=1
q
I
i
(
p
)
.
v
I
i
(
p
)
∈
arg max
v
i
pv
i
−P
(
v
i
,q
I
i
(
p
))
and
V
I
(
p
) =
N
∑
i
=1
v
I
i
(
p
)
.
The market equilibrium price satisfies
Q
I
(
p
∗
)
−
V
I
(
p
∗
) =
L
.
To ensure the market is well-behaved, I impose a regularity assumption.
Assumption 3
(Regular Imprecise Measurement)
.
A. Errors
(i) The indirect measure of output is
w
i
=
q
i
+
δh
(
q
i
)
,
∀
i,
where
δ
is a
random variable with density
g
(
δ
)
and
E
[
δ
] =
∫
δg
(
δ
)
dδ
= 0
.
(ii)There is a
δ
>
0
such that
δ
≥−
δ
and
q
h
(
q
)
≥
δ
for all
q
≥
0
.
16
(iii) The size of the errors,
h
(
q
)
, is positive, non-decreasing, continu-
ously differentiable, and convex in
q
. That is,
h
(
q
i
)
>
0
,h
q
(
q
i
)
≥
0
and
h
qq
(
q
i
)
≥
0
,
for all
q
i
>
0
.
Also
h
(0) = 0
.
B. Enforcement
(i)
ρ
(
m
i
) = max
{
0
,am
i
}
with
a >
0
,
where
m
i
=
w
i
−
l
i
is the measured
violation.
(ii)
L > Q
I
(
P
v
(0+)) =
Q
I
[
a
(1
−
G
(0))]
.
Assumption 3A provides a structure that is like a single crossing property.
It keeps the densities of the errors under control when
q
changes. Increasing
h
(
q
) is then like applying a mean-preserving spread to the error.
Assumption 3B is similar to Assumption 2 with a slight weakening. Assump-
tion 3B(ii) ensures that penalties are strong enough that producers will not
want violations in equilibrium.
16
This ensures that
w
i
≥
0. If errors are proportional to output with
h
(
q
) =
τq
for some
τ >
0, then this will be true if
τδ
≤
1
.
12
Under Assumption 3, the expected penalty function becomes
P
(
v
i
,q
i
) =
a
∫
−
v
i
h
(
q
i
)
[
v
i
+
δh
(
q
i
)]
g
(
δ
)
dδ.
(1)
The geometry of a permit market under imprecise measurement satisfying
Assumption 3 is illustrated in Figure 2 in Section 7.2. The crucial fact is
that
Q
I
(
p
) is downward sloping.
Result 3.
Under Assumptions 1 and 3,
Q
I
p
(
p
)
<
0
.
Proof.
Let
k
i
=
−
v
i
h
(
q
i
)
, and
R
(
k
i
) =
∫
k
i
δg
(
δ
)
dδ.
The first order conditions
for a solution to Definition 6(i) and (ii) for a given
p
are:
b
iq
(
q
i
,Q
)
−
p
−
h
q
(
q
i
)
aR
(
k
) = 0
(2)
p
−
a
(1
−
G
(
k
)) = 0
(3)
v
i
+
h
(
q
i
)
k
= 0
(4)
∑
i
q
i
−
Q
= 0
(5)
∑
i
q
i
−
∑
i
v
i
=
L
(6)
To solve for the partial equilibrium comparative statics, differentiate (2)-(5)
with respect to
p
to get:
b
iqq
q
I
ip
+
b
iqQ
Q
I
p
−
1
−
h
qq
aRq
I
ip
−
h
q
aR
k
k
I
p
= 0
(7)
1 +
agk
I
p
= 0
(8)
v
I
ip
+
h
q
kq
I
ip
+
hk
I
p
= 0
(9)
[
∑
i
q
I
ip
]
−
Q
I
p
= 0
.
(10)
From (8),
k
I
p
(
p
) =
−
1
ag
(
k
)
<
0. By definition
R
k
=
−
k
I
g.
Since
v
I
i
<
0,
k
I
>
0. Solving (7) for
q
I
ip
yields
q
I
ip
=
1 +
h
q
k
−
b
iqQ
Q
I
p
b
iqq
−
h
qq
aR
=
α
i
(
p
)
−
β
i
(
p
)
Q
I
p
(
p
)
.
13
From (10)
Q
I
p
=
∑
i
α
i
−
(
∑
i
β
i
)
Q
p
.
Solving gives
Q
I
p
=
∑
i
α
i
1 +
∑
i
β
i
.
Since
α
i
<
0 and
β
i
≥
0, the result follows.
4.1 Efficiency and Political Viability - Impossibility
Although the equilibrium equations for valid measurement and imprecise
measurement look very similar, the latter create serious problems for both
the efficiency and the political viability of permit programs.
4.1.1 Efficiency
Efficiency given
L
requires that, at a permit market equilibrium (
q
∗
,v
∗
,p
∗
,Q
∗
),
v
∗
i
= 0 for all producers. With imprecise measurement, under Assumptions
1 and 3, this will not be true. Instead,
v
∗
i
<
0 for all i and, therefore,
Q
∗
< L
in equilibrium.
Result 4.
Under Assumptions 1 and 3, at a permit market equilibrium
(
q
∗
,v
∗
,p
∗
,Q
∗
)
,
0
< p
∗
< a
[1
−
G
(0)]
,v
∗
i
<
0
,
∀
i,
and
Q
∗
< L
.
Proof.
(
p
∗
< a
[1
−
G
(0)]) Assume the contrary. Then
v
∗
i
>
0 and, so,
Q
∗
> L
.
But
Q
(
p
∗
)
≤
Q
(
a
[1
−
G
(0)])
< L
which is a contradiction.
(
v
∗
i
<
0) From profit maximization
v
∗
i
∈
arg max
v
i
p
∗
v
i
−
∫
−
v
i
h
(
q
i
)
a
(
v
i
+
δh
(
q
i
))
g
(
δ
)
dδ
.
The first order condition for this is
p
∗
=
a
∫
−
v
i
h
(
q
i
)
g
(
δ
)
dδ
=
a
[1
−
G
(
−
v
i
h
(
q
i
)
)].
Since
p
∗
< a
[1
−
G
(0)], it follows that
v
∗
i
<
0. This implies
Q
∗
< L.
With imprecise measurement, producers will pay a penalty even if
q
=
l
.
They can reduce that expected penalty cost by producing less than the per-
mits they hold. Because of this, if regulators issue a total of permits
L
equal
14
to the desired aggregate output target,
Q
, actual output will be less than
Q
and, therefore, not efficient given
Q
.
Knowing that equilibrium output is less than
L
, one might ask whether reg-
ulators could change
L
to some other
ˆ
L
so as to move
Q
I
(
p
∗
) to
L
. For the
regular case, the answer is yes. Consider Figure 2. The amount of permits
that works is
ˆ
L
=
Q
I
(
p
V
)
−
V
I
(
p
V
)
> L
. There must be enough permits
to allow the producer to use the extras to insure herself against a faulty
measurement. That is, an additional amount
ˆ
L
−
L
∗
=
−
V
I
(
p
V
) must be
added to
L
∗
.
Of course, to compute
ˆ
L
the regulator must know, prior to the
implementation of the program, the functions
Q
I
(
p
) and
V
I
(
p
) which they
do not.
4.1.2 Political Viability
With valid measurement, by Result 2B, there are distributions of the initial
permit,
l
o
, such that all producers will be at least as well off in the permit
market equilibrium as they would have been without the program. With
imprecise measurement, this may not be true because a producer’s expected
penalties are positive even if she chooses to buy licenses equal in number to
her production levels. That is,
P
(0
,q
i
) =
ah
(
q
i
)
∫
0
δg
(
δ
)
dδ >
0
.
To see the effect of imprecise measurement on political viability, consider the
following result which adapts Result 2B to imprecise measurement.
Result 5.
Under Assumptions 1 and 3,
(i) for any initial allocation of permits,
l
o
,
such that
b
i
(
l
o
i
,L
)
−P
(0
,l
o
i
)
>
b
i
(
q
a
i
,Q
a
)
, at the market equilibrium
(
q
∗
,v
∗
,p
∗
,Q
∗
)
b
i
(
q
∗
i
,Q
∗
)
−
p
∗
(
q
∗
i
−
v
∗
i
−
l
o
i
)
−P
(
v
∗
i
,q
∗
i
)
> b
i
(
q
a
i
,Q
a
)
,
∀
i,
and
15
(ii) there exists an
l
o
satisfying (i) if and only if there is a
ˆ
q
such that
∑
i
ˆ
q
=
L
and
∑
i
b
i
(ˆ
q,L
)
−
∑
i
b
i
(
q
a
,Q
a
)
>
∑
i
P
(0
,
ˆ
q
i
)]
.
Proof.
(i) Let (
q
∗
,v
∗
,p
∗
,Q
∗
) be the permit market equilibrium. Then
b
i
(
q
∗
i
,Q
∗
)
−
p
∗
(
q
∗
i
−
v
∗
i
−
l
o
i
)
−P
(
v
∗
i
,q
∗
i
)
≥
b
i
(
l
o
i
,Q
∗
)
−
p
∗
(
l
o
i
−
0
−
l
o
i
)
−P
(0
,l
o
i
)
≥
b
i
(
l
o
i
,L
)
−
P
(0
,l
o
i
)
> b
i
(
q
a
i
,Q
a
)
.
The first inequality follows from profit maximization.
The second follows because
L > Q
(
a
[1
−
G
(0)]) implies
Q
∗
≤
L
. The last
follows from the assumption on
l
o
.
Comparing this to Result 2 under valid measurement, it is easy to see that
imprecise measurement introduces a potential barrier to voluntary participa-
tion. If either the penalty rate or the errors are large, the expected penalty
with no violations,
ah
(
q
i
)
∫
0
δg
(
δ
)
dδ
, is large. Then, if the gains from improv-
ing the management of the commons are not very large, it may be difficult
or impossible to find an appropriate
l
o
i
.
4.2 Efficiency and Political Viability - Possibility
In spite of the difficulties described in the previous section, some positive
results can be found.
4.2.1 Efficiency
Although there is no general efficiency result when measurement is impre-
cise, if measurement errors are proportional to output, then even though
q
∗
is not efficient given
L
,
q
∗
is efficient given
Q
∗
. That is, production will be
organized efficiently given the aggregate output level.
Assumption 4.
(Errors are proportional to output)
h
qq
(
q
i
) = 0
,
∀
q,
∀
i.
This assumption along with Assumption 3 A(iii) imply that
h
(
q
) =
τq
for
some
τ >
0.
16
Result 6.
Under Assumptions 1, 3 and 4,
q
∗
is efficient given
Q
∗
.
Proof.
Under Assumptions 1 and 3, the FOC for a permit market equilib-
rium are found in (2)-(6). From (3) it follows that
k
∗
is the same for all i.
Therefore, from (2) and (4),
q
∗
is efficient given
Q
∗
if and only if the
h
q
(
q
∗
i
)
are equal for each i. This is true under Assumption 4 since
h
qq
(
q
i
) = 0.
Without Assumption 4, the equilibrium will not be efficient given
Q
∗
. It is
easy to see why. If
h
qq
>
0, then
h
q
(ˆ
q
)
> h
q
( ̃
q
) iff ˆ
q >
̃
q
. Let
̄
h
q
=
∑
h
q
(
q
∗
i
)
N
be
the average value of
h
q
in equilbrium. If
h
q
(
q
∗
i
)
>
̄
h
q
then in equilibrium
q
∗
i
is relatively smaller than desired for efficiency. If
h
q
(
q
∗
i
)
<
̄
h
q
then in equi-
librium
q
∗
i
is relatively larger than desired for efficiency. The fact that the
imprecise measurement errors are getting worse as q gets larger means that
those who produce a large amount will have more incentive to cut back on
their output to avoid the penalties from mis-measurement. Non-proportional
measurement errors interfere with efficiency given
Q
∗
.
4.2.2 Political Viability
There is a way to design around this problem by using the same insights em-
ployed for valid measurement. Put the producer in an initial position that is
at least as good as autarky and that she can protect. If the regulator gives
each producer an initial subsidy of
P
o
i
=
P
(0
,l
o
i
)
,
and if the producer then
chooses (
q
i
,v
i
) = (
l
o
i
,
0), she can guarantee that her expected penalty less
the subsidy will be zero. That plus the appropriate initial permit allocation
guarantees voluntary participation.
Result 7.
Under Assumptions 1 and 3, for any initial allocation of permit,
l
o
,
such that
b
i
(
l
o
i
,L
)
> b
i
(
q
a
i
,Q
a
)
for all i, there are lump-sum payments
P
o
i
=
P
(0
,l
o
i
)
such that the market equilibrium
(
q
∗
,v
∗
,p
∗
,Q
∗
)
satifies
b
i
(
q
∗
i
,Q
∗
)
−
p
∗
(
q
∗
i
−
v
∗
i
−
l
o
i
)
−P
(
v
∗
i
,q
∗
i
) +
P
o
i
> b
i
(
q
a
i
,Q
a
)
,
∀
i.
Proof.
Let (
q
∗
,v
∗
,p
∗
,Q
∗
) be the permit market equilibrium. Then
b
i
(
q
∗
i
,Q
∗
)
−
p
∗
(
q
∗
i
−
v
∗
i
−
l
o
i
)
−P
(
v
∗
i
,q
∗
i
) +
P
o
i
≥
b
i
(
l
o
i
,Q
∗
)
−
p
∗
(
l
o
i
−
0
−
l
o
i
)
−P
(0
,l
o
i
) +
P
o
i
≥
b
i
(
l
o
i
,L
)
> b
i
(
q
a
i
,Q
a
)
.
The first inequality follows from profit maximization.
17
The second follows because
L > Q
(
a
[1
−
G
(0)]) implies
Q
∗
≤
L
. The last
follows from the assumption on
l
o
and the fact that
P
(0
,l
o
i
) =
P
o
i
.
If the measurement technology is well understood, then
P
o
i
=
ah
(
l
o
i
)
∫
0
δg
(
δ
)
dδ
is easy to calculate.
Although the subsidies in Result 7 help solve the problem of generating vol-
untary participation, they create a new problem for the designers. If the
subsidies are deployed and equilibrium output is less than or equal to
L
,
then producers will be receiving a positive net aggregate subsidy. The per-
mit program will not be self financing in expected value.
Result 8.
Under Assumptions 1, 3, and Assumption 4 ,
∑
i
P
(0
,l
o
i
)
>
∑
i
P
(
v
∗
i
,q
∗
i
)
.
Proof.
Under Assumptions 3 and 4, in equilibrium
∑
i
[
P
(0
,l
o
i
)
−P
(
v
∗
i
,q
∗
i
)] =
aτl
o
i
∫
0
δg
(
δ
)
dδ
−
aτq
∗
i
∫
−
v
∗
h
(
q
∗
i
)
δg
(
δ
)
dδ.
By (3),
∫
−
v
∗
h
(
q
∗
i
)
δg
(
δ
)
dδ
=
p
∗
a
is the same for
every i. Therefore the aggregate net subsidy is
S
=
∑
i
[
P
(0
,l
o
i
)
−P
(
v
∗
i
,q
∗
i
)] =
aτ
[
∑
i
l
o
i
]
∫
0
δg
(
δ
)
dδ
−
aτ
[
∑
i
q
∗
i
]
∫
−
v
∗
h
(
q
∗
i
)
δg
(
δ
)
dδ
=
aτ
[
L
∫
0
δg
(
δ
)
dδ
−
Q
∗
∫
−
v
∗
h
(
q
∗
i
)
δg
(
δ
)
dδ
]
.
Under Assumption 3,
v
∗
i
<
0 which implies that
∫
0
δg
(
δ
)
dδ >
∫
−
v
∗
h
(
q
∗
i
)
δg
(
δ
)
dδ.
Also
L > Q
∗
. Therefore,
S >
0
.
The voluntary participation of the producers has been bought with funding
from outside of the market - presumably from taxpayers. This creates a
new friction against adoption. Nevertheless, such subsidies may be justified.
The rationalization of the management of the commons can lead to gains,
not only for producers, but also for the consumers of the products of the
commons. This is certainly true for fisheries, and probably true for many
other situations. Using some of these gains to ease the transition to permit
markets might be a very good bargain for all concerned.
4.3 Precision
In this section, I explore what happens to efficiency and political viability
as the measurements become more precise. The easiest way to do that is
18
to introduce a precision parameter,
η
and replace
h
(
q
i
) with
ηh
(
q
i
) in the
measurement model. Thus,
w
i
=
q
i
+
ηh
(
q
i
)
δ.
I will assume
η
≤
1
.
With this
small change, the measurement error is
w
i
−
q
i
=
zh
(
q
i
) where
z
=
δη
. The
expected value of z is 0 and the variance of z is
ηV ar
(
δ
). As
η
decreases,
precision increases.
The penalty that a producer now faces, given output
q
i
and actual violation
v
i
=
q
i
−
l
i
,
is
P
η
(
v
i
,q
i
) =
a
∫
−
v
i
ηh
(
q
i
)
[
v
i
+
δh
(
q
i
)
η
]
g
(
δ
)
dδ.
(11)
As one might expect, increased precision improves efficiency and eases polit-
ical viability. In the limit as
η
→
0, the equilibrium under imprecise mea-
surement approaches the equilibrium under valid measurement. The permit
market equilibrium becomes efficient given L and no subsidies are required
for voluntary participation.
Result 9.
Let
(
q
(
η
)
,v
(
η
)
,p
(
η
)
,Q
(
η
))
be the market equilibrium under im-
precise measurement with the expected penalty
P
η
(
v
i
,q
i
) =
a
∫
−
v
i
ηh
(
q
i
)
[
v
i
+
δh
(
q
i
)
η
]
g
(
δ
)
dδ
and let
(
q
∗
,v
∗
,p
∗
,Q
∗
)
be the market equilibrium under valid
measurement with the penalty
P
(
v
i
) = max
{
av,
0
}
.
Under Assumptions 1
and 3,
(i)
lim
η
→
0
(
q
(
η
)
,v
(
η
)
,p
(
η
)
,Q
(
η
)) = (
q
∗
,v
∗
,p
∗
,Q
∗
)
and
(ii)
lim
η
→
0
P
η
(0
,l
o
i
) = 0
.
Proof.
First note that, as precision increases, the penalty function under
imprecise measurement approaches the penalty function under valid mea-
surement. For any
q
i
such that 0
< q
i
<
∞
,
lim
η
→
0
P
η
(
v
i
,q
i
) =
av
if
v
≥
0
(12)
lim
η
→
0
P
η
(
v
i
,q
i
) = 0 if
v
≤
0
.
(13)
The first order conditions for a market equilibrium under imprecise measure-
19
ment are:
b
iq
(
q
i
,Q
)
−
p
−P
η
q
i
(
v
i
,q
i
) = 0
(14)
p
−P
η
v
i
(
v
i
,q
i
) = 0
(15)
∑
i
q
i
−
Q
= 0
(16)
∑
i
q
i
−
∑
i
v
i
=
L
(17)
It is easy to show that
P
η
q
i
(
v
i
,q
i
)
→
0 as
η
→
0. Also
P
η
v
i
(
v
i
,q
i
)
→
a
as
η
→
0
if
v >
0 and
P
η
v
i
(
v
i
,q
i
)
→
0 as
η
→
0 if
v
i
<
0. Result 9(i) follows from the
implicit function theorem.
P
η
(0
,l
o
i
) =
∫
0
δh
(
l
o
i
)
ηg
(
δ
)
dδ
=
ηh
(
l
o
i
)
∫
0
δg
(
δ
)
dδ
=
η
P
(0
,l
o
i
). Result 9(ii)
follows.
5 Final Comments
In this paper I have explored the geometry, efficiency and political viability of
permit market equilibria when enforcement can only use imprecise measure-
ment. I provide a set of conditions (Assumption 3) such that the geometry
corresponds to most economists’ intuitions. These conditions are similar in
spirit to those needed under accurate measurement (Assumption 2).
Unlike the standard results when there is accurate and precise measurement
of use, when measurement is imprecise permits will not be efficiently allo-
cated. There are two sources of the inefficiency: aggregate output is less
than the number of permits as producers overbuy to insure against mis-
measurement and there can be bias in the efficient allocation of output as
mis-measurement can have different marginal effects on producers depend-
ing on their size. Further, when measurement is imprecise, it may not be
possible to find an initial allocation of permits such that all firms are bet-
ter off than they were before implementation of the program. The reason is
that all firms will face positive expected penalty payments, even if they have
negative actual violations. This could seriously affect the political viability
of the program, even if measurement were free.
20
But there are also some positive results. First, if the errors are proportional
to output, then aggregate output will be efficiently allocated among firms.
The only inefficiency will be that aggregate output is less than the target,
the number of permits. This can be compensated for by issuing more per-
mits than the target output. Second, there are easily calculated individual
subsidies that will make it possible to find an initial allocation of permits
to guarantee voluntary participation. This will mean that taxpayers must
subsidize the implementation of the program. But if the benefits from the
program are large enough and some of these benefits accrue to consumers,
then the program may still be politically viable.
Finally, if the measurement errors are small, inefficiencies will be small and
the subsidies required for voluntary participation will be small. In this case,
it may well be reasonable to proceed with an inexpensive, imprecise mea-
surement system as if it were accurate.
21
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23