Non-point-source pollution regulation as a
multi-task principal-agent problem
Robert G. Chambers and John Quiggin
This paper considers a multi-task, principal-agent problem where risk-averse
farmers possessing private information have two tasks, pollution control and corn
production, but only direct incentive for corn production. Using a highly
tractable reformulation of the standard uncertain production model, a general
method for solving the associated constrained Paretian problem is developed and
analyzed. The optimal solution is shown to obey a generalized inverse-elasticity
rule, and the optimal solution is characterized under a number of assumptions
about the underlying technology and the role that pollution emission plays in
reducing farmer risk.
Keywords : Multi-task principal-agent; Non-point-source solution; Mechanism
design
Non-point-source pollution regulation as a multi-task principal-agent problem 1
1
Introduction
The problem this paper tackles is illustrated by the following example: N
independent, risk-averse farmers use a stochastic technology to produce
two outputs — corn and chemical runoff. Because chemical runoff is a
byproduct of producing corn and farmers receive direct bene…ts from producing corn, chemical runoff has a positive shadow value to farmers, even
though they do not value it directly. Runoff is socially undesirable, however, because it pollutes water and other ecosystems. Therefore, although
the farmer has private incentives to emit runoff, the rest of society wants to
control runoff. The sticking point is that runoff is only detectable and measurable (if at all) after it has entered the ecosystem. By then, identifying
its original source (i.e. the emitting farmer) is impossible. Chemical runoff
is a non-point-source pollutant: its effects can be felt, but its source cannot
be identi…ed.1 Hence, runoff emission involves an externality between the
farmer and rest of society.
While pollution control is socially desirable, it is also desirable, if possible, to spread the production risk farmers face across the rest of society.
But policies to mitigate farmer risk can also affect pollution incentives.
For example, fertilizer is usually thought of as a risk-increasing input. If
true, insuring farmers, thus making them marginally less risk-averse, should
increase fertilizer use and, thereby, nitrogen runoff. The insurance objective and runoff control push in different direction. Other treatments of
non-point-source pollution (e.g. Segerson, 1988), which ignore production
uncertainty and hence the legitimate social role of insurance and the concomitant moral hazard and externality problems, thus do not consider an
important channel through which public policy can affect runoff emission.
In designing schemes to cope jointly with the social need for insurance
and the associated pollution externality, one should recognize that monitoring all the actions that farmers take in organizing production is not feasible.
In particular, ‘effort’ to promote corn growth and to control runoff cannot
be disentabled from one another and are unobservable. Because the farmer
is the residual claimant and …rst handler of the corn harvested, he or she
can always divert corn to own consumption. Hence, ex post output is
1
Precisely this type of pollution problem has been reported and analyzed in Brossier
et al. (1992). Brossier et al. (1992) considered how farming systems might be changed in
order to prevent the percolation of nitrogen runoff from farmers in France’s Vittel plateau
into the Vittel mineral water supply.
Non-point-source pollution regulation as a multi-task principal-agent problem 2
only observable with constant (and prohibitively costly) monitoring. As a
result, output-contingent insurance schemes would lead farmers to understate their output, either by consuming it directly or by misrepresenting
their actual output. Thus, any mechanism to induce runoff control cannot
depend upon either the level of effort committed or the level of output.
This differs from, for example, more traditional models of non-point-source
pollution (Segerson, 1988) and closely related models of sharecropping relations (Stiglitz, 1974; Bliss and Stern, 1982) which assume that output is
observable.2 The sharecropping models, in particular, consider a different
contracting environment where the principal (here the planner) is the residual claimant and ‘owner’ of the output. In an agricultural context, this may
be plausible for a developing country but it is implausible for a developed
country where farmers are typically independent entrepreneurs. If farmers
are independent entrepreneurs, it is much more likely that the exogenous
forces (weather, humidity, etc.) characterizing a state of nature will be
observable to the planner than output. The social problem, therefore, is
to design an insurance mechanism that provides appropriate incentives for
both corn culture and runoff control. Thus, the corn-pollution problem is
a ‘multi-task’ principal-agent problem in the sense de…ned by Holmstrom
and Milgrom (1991).
This paper develops a simple, but exact, method for solving this problem. Although the model is speci…ed in terms of corn production and pollution, the method applies to all problems of this class. Our primary technical
contribution is an exact solution of this class of hidden-action problems by
simple nonlinear programming methods that resembles other reductions of
principal-agent problems to simpler non-linear programs (Guesnerie and
Laffont, 1984; Weymark, 1986; Chambers, 1989).
In what follows, the model is …rst introduced. Our representation of production uncertainty is somewhat novel and is based upon a state-contingent
production model formalized in Chambers and Quiggin (1992). The key
difference between this representation of stochastic production and more
traditional representations of production uncertainty is that both an input
vector and a state-contingent output vector are chosen by farmers. After
the model is set up, a brief discussion of the …rst-best problem is then
2
In the Segerson (1988) model, the observability of output and the farmer’s indifference
to risk implies, by standard results in the incentives literature (Harris and Raviv, 1979),
that the …rst best is achievable.
Non-point-source pollution regulation as a multi-task principal-agent problem 3
presented. The corn-pollution, hidden-action problem is formulated mathematically and shown to be equivalent to the solution of an unconstrained
nonlinear programming problem. The remaining sections discuss the properties of the optimal incentive scheme. Among other results we establish
that: the optimal mechanism obeys a generalized version of the ‘inverse
elasticity’ rule, and pollution control generally requires leaving the farm
higher returns in those states where the ability to emit pollution lowers
the marginal cost of production and a lower return in those states where
pollution increases the marginal cost of production.
2
The model
There are three types of individuals: N corn farmers who produce both
runoff and corn under conditions of uncertainty; an aggregate individual
‘Society’; and a planner. Only farmers engage in productive activity. Uncertainty is modeled by ‘Nature’making a choice from among a …nite set of
alternatives. Each of these alternatives is called a ‘state’ and is indexed by
a …nite set of the form = {1, 2, 3, ..., S }. Once the index is given, all possible factors determining production and contracting conditions (weather,
etc.) are known.
2.1
Preferences and technology
Society is presumed to be risk-neutral over different wealth levels, while
each farmer’s preference structure does not depend upon pollution directly.
Rather it is de…ned over returns (y ) in corn units and the effort vector x.
The k th farmer’s preferences over returns and effort are given by
wk (y, x) = uk (y )
gk (x).
Here uk : <+ → <+ (k = 1, 2, ..., N ) is strictly increasing, strictly concave, and at least twice differentiable. gk : <n+ → <+ (k = 1, 2, ..., N ) is
non-decreasing, continuous, and convex. Both uk and gk satisfy the von
Neumann-Morgenstern postulates. All (the planner, the farmer and Society) believe that all states are equally probable.
Non-point-source pollution regulation as a multi-task principal-agent problem 4
Production relations are governed by the state-contingent technology
set T <n+ <S+ <S+ de…ned by
3
T = {(x, p, z ) : x can produce (z, p); x ∈ <n+ , p ∈ <S+ , z ∈ <S+ }.
Here x is an input vector committed prior to the realization of the state
of nature, p is a vector of state-contingent pollution levels, and z is a vector of state-contingent (ex post) outputs of corn. The most appropriate
interpretation of T is as an ex ante technology: (x, p, z ) ∈ T implies that,
if input x is committed and ‘Nature’ chooses the j th state of nature, then
the j th elements of p and z represent, respectively, the pollution and corn
output levels realized ex post. In line with most hidden-action models, all
inputs are chosen prior to the resolution of uncertainty.4 It is assumed that
x contains any inputs which are used solely for ‘abatement activities’. However, unlike more standard models of the relationship between production
and pollution which assume that abatement activities can be captured by
a single scalar variable, we recognize that abatement actually represents
a complex interaction between a number of inputs which cannot be separated from other production activities without trivializing the problem.
Consider, for example, the problem of nitrogen runoff. Nitrogen runoff depends not only on the farmer’s direct abatement activities, such as building
catchments to prevent runoff, but also upon the amount of applied nitrogen, the care with which it is applied, the application of irrigation, and a
host of other activities. Each of the latter have an abatement component,
which cannot be trivially separated from their other roles in the production
process.
Our general representation allows both pollution and corn output levels
to be subject to uncertainty. However, for notational simplicity we shall
always treat p ∈ <+ (one pollution level occurs in all states). Ex post,
pollution in the amount p imposes a burden upon society of m(p), where
m : <+ → <+ is strictly increasing, strictly convex, and differentiable.5
3
The properties and advantages of state-contingent technology sets over traditional
representations of uncertain production technologies is discussed at length in Chambers
and Quiggin (1992).
4
The technology can easily be generalized to encompass such phenomena as sequential
resolution of uncertainty (Chambers and Quiggin, 1992). Some might …nd it more intuitive
to think of pollution itself as a state-contingent input.
5
Results for nonscalar p of course are somewhat less clear cut, but the same basic
principles apply. The main difference that would emerge in our analysis would be that
Non-point-source pollution regulation as a multi-task principal-agent problem 5
2.2
Two cost functions
Two indirect representations of T are useful. The …rst is the effort-cost
function for farm k de…ned by
ck (p, z ) =min {gk (x) : (x, p, z ) ∈ T }.
x
If T is convex and exhibits free disposability of z , ck (p, z ) is non-decreasing
in z , and convex and continuous for both p and z restricted to the strictly
positive orthant (Chambers and Quiggin, 1992). We strengthen these technical requirements to permit the use of the calculus, by assuming that
ck (p, z ) is strictly convex in z and p, increasing in z , and twice differentiable.
De…ne the farmer’s private-cost function by
C k (z ) =min ck (p, z ).
p
Assume a minimum exists for all farmers for all relevant state-contingent
output vectors. The strict convexity of ck (p, z ) then guarantees that the
minimizer denoted
pk (z ) = arg min ck (p, z )
is unique and that C k (z ) is convex. C k (z ) is also assumed to be at least
twice-differentiable, and Cik (z ) denotes the partial derivative of C k (z ) with
respect to zi .
The …rst-order condition of the private-cost minimization problem requires the farmer’s private marginal bene…t from pollution to equal zero for
an interior solution:
ckp (pk (z ), z ) = 0.
Given the convexity of ck (p, z ) in p, how pk (z ) reacts to changes in z is
determined by the vector of second-partial derivatives with typical element
ckpi (pk , z k ) (i = 1, ..., S ). If a small increase in zi increases the farmer’s marginal bene…t from pollution, i.e. ckpi (pk , z k ) < 0, it will be associated with
an increase in pk (z ). Alternatively, by Young’s theorem pki (z ) ∂pk (z )/∂zi
is positive (negative) if and only if an increase in p decreases (increases)
the marginal cost of producing zi.
ˆ
pk (z ) and pk de…ned below would be vectors and the discussions of marginal-cost reducing
pollution and risk-increasing and risk-reducing pollution would need to be modi…ed to take
account of that fact.
Non-point-source pollution regulation as a multi-task principal-agent problem 6
2.3
The planner
The planner has sufficient legal authority to enforce contracts and can commit to a corn-insurance scheme. The planner’s task is to design a mechanism that appropriately shares the production risk farmers face with Society, while providing farmers with the appropriate incentive for pollution
control. In doing so, the planner is assumed to know T, uk (k = 1, 2, ..., N ),
gk (k = 1, 2, ..., N ), and m, and can observe ex post the state of nature. Only
the individual farmer can observe effort and pollution. Therefore, although
the planner, ex post, knows the technology and observes the physical state
of nature (amount of rainfall, temperature-degree days, etc.), the planner
does not observe the exact conditions under which production takes place.
In short, hidden action exists. We assume explicitly that total pollution resulting from all farmers’ activities is either unobservable or noncontractible.
This assumption removes the possibility of strategic interactions between
farmers. (Hence, with little loss of generality, one could always model the
present problem as involving only one farmer.) Under these assumptions
the only contractible variable is the state of nature. The planner thus speci…es a mechanism represented by the vector of state-contingent net premia
I k ∈ <S for each individual k = 1, ..., N . [Iik can be either positive or
negative (i = 1, ..., S ).]
3
The …rst best
For later comparison, a representation of the …rst best is convenient. Here
the term ‘…rst best’ is reserved for the situation where the planner can
observe output, effort, and runoff pollution. The …rst best solves:
max
(
X
p,z,I
S
1
X
!
[Iik
+
u(zik
Iik )]
i
k
k
k
k
c (p , z )
m
X
!)
k
p
,
k
where pk , z k , and I k ∈ <S are, respectively, the pollution level, statecontingent corn vector, and state-contingent, net-insurance-premium vector
for the k th farmer.
Optimizing with respect to the k th pollution level gives
m
X
k
!
k
p
+ ∂ck (pk , z k )/∂pk = 0
Non-point-source pollution regulation as a multi-task principal-agent problem 7
(k = 1, 2, ..., N ) for an interior solution. The marginal private bene…t to the
k th farmer of increasing pollution equals marginal societal damage which
in turn must be equal across farmers.
Optimizing with respect to Iik gives, for an internal solution,
1
uk (zik
Iik ) = 0
(1)
(i = 1, 2, ..., S ) and (k = 1, 2, ..., N ). Expression 1 is Borch’s rule for optimal
risk sharing between risk-averse and risk-neutral individuals: risk-neutral
Society absorbs all the risk and provides the risk-averse farmers with full
insurance. The strict concavity of uk implies that each farmer’s ex post
return must be the same in every state of nature. The following de…nition
is convenient:
De…nition 1 The …rst-best ex post return for farmer k is denoted sk and
is de…ned as the solution to the following implicit equation:
1
uk (sk ) = 0.
Substituting sk into the societal objective function and choosing the
state-contingent outputs optimally then gives
∂ck (pk , zk )/∂zik = S
1
(i = 1, 2, ..., S ) and (k = 1, 2, ..., N ). The marginal cost of each statecontingent output equals its expected bene…t.
To implement the …rst best, one can set a state-contingent price vector
consisting of an output price ri = 1 (i = 1, ..., S ), a pollution tax (given
the ability to observe pollution t = m(k pk ) (where pk is evaluated at the
optimal pollution level), and a state-contingent payment ai from Society
to the producer, such that xki ri + aki = zik + aki = sk (i = 1, ..., S ), (k =
1, ..., N ). Thus, for each producer k , the planner has 2S +1 objectives; the S
state-contingent output levels, the S -dimensional optimal sharing objective
and the pollution level. Given 2S + 1 instruments, all of these objectives
can be attained exactly. However, it is well known from the incentives
literature that, in the presence of complete information, achieving the …rst
best often only requires using a subset of that information. For example,
in the standard moral-hazard model, information on both output and the
ex post state of nature is sufficient to support achievement of the …rst best
Non-point-source pollution regulation as a multi-task principal-agent problem 8
even in the absence of information on effort (Harris and Raviv, 1979). A
similar result applies here as well: achieving the …rst best only requires
complete information on pollution and the ex post state of nature.
Result 1. If the planner observes pollution, the …rst best can be achieved
even in the absence of ex post observations on output. (All proofs are in
the Appendix.)
Because farmers are the residual claimants, departures from the …rst
best generally only arise when pollution is unobservable. Moreover, if pollution is unobservable, the …rst best is only attainable when private-cost
minimizing pollution is zero. (This latter statement is proven formally as
Result 2 below.) Given information on pollution, information on output is,
therefore, super‡uous.
4
An algorithm for the hidden-action, corn-pollution
problem
The planner seeks a mechanism that maximizes social welfare when only
the state of the world is ex post observable and contractible.
Any implementable ex post pollution-output allocation must be consistent with the self-interested maximizing behavior of the farmer. Given a
vector of state-contingent premia for farmer k , I k ∈ <S , the only statecontingent output vectors (and implicitly the pollution level) that are implementable satisfy
(
z ∈max S
1
k
X
z
)
u k (z i
Iik )
k
C (z ) .
i
Formally stated, the planner’s problem is,6 therefore,
Problem I. Choose I k ∈ <S , z k ∈ <S+ (k = 1, 2, ..., N ) to
max
z,I
6
X
k
S
1
X
i
!
[Iik + u(zik
Iik )]
C k (z k )
m
X
!
pk (z k )
k
If the state of nature were not observable by the planner, ex post, the planner’s
problem would be characterized by both hidden action and hidden information. Because
nothing is observable, nothing is contractible. Hence, only a …xed payment (i.e. the same
charge in each state of nature) would be feasible. If output were observable ex post but
not the state, the current model becomes a standard moral hazard problem. Quiggin and
Chambers (1992) show that that problem can be reduced to a hidden information problem.
Non-point-source pollution regulation as a multi-task principal-agent problem 9
subject to
(
z k ∈ arg max S
X
1
)
u k (z i
Iik )
C k (z ) .
i
Our next result establishes that in most cases the solution to Problem
k
ˆk ˆ
p
z
I will not correspond to the …rst-best solution, which we denote as ( , )
(k = 1, ..., N ). In what follows we always presume that the following Inada
conditions are in force:
lim Cik (z ) = 0
z→0
(i = 1, ..., S ), and
lim uk (y ) = ∞.
y→0
Result 2. The solution to Problem I yields the …rst-best outcome with
respect to farmer k if and only if
ˆk
pk (z ) = 0.
ˆk
So long as the private-cost minimizing pollution amount at z is positive,
the unobservability of farmer k ’s pollution is costly to Society.
From Result 2 it is also easy to conclude
ˆk
Corollary 2 At the solution to Problem I unless pk (z ) = 0 (k = 1, ..., N )
a social gain would exist if farmer k reduced pollution by a small amount
provided incentive effects are ignored.
To understand the corollary, notice hat if all farmers pollute, then the
social gain from them all reducing pollution by a small amount (ε > 0) is
(ignoring incentive effects)
Nm
X
!
k
p
k
+
X
!
ckp (pk , z k )
ε,
k
which, by the farmers private-cost minimization, reduces to
Nm
X
k
!
pk ε > 0.
Non-point-source pollution regulation as a multi-task principal-agent problem 10
Hence, at the solution to Problem I, the planner should always want to
reduce pollution further. However, the presence of the incentive constraints
makes this further reduction in pollution suboptimal.
We are now ready to consider Problem I directly. Given the strict concavity of uk and the convexity of C k , once I k ∈ <S is speci…ed a unique
solution to the farmer’s expected-utility maximization problem exists and
is completely characterized by the farmer’s …rst-order conditions. Our
next result, reminiscent of results in the literature on nonlinear pricing
under asymmetric information (Guesnerie and Laffont, 1984; Weymark,
1986; Chambers, 1989) and Grossman and Hart (1983), exploits this fact
to establish that solutions to Problem I can be obtained through solving
an alternative unconstrained optimization problem. However, unlike the
results of Guesnerie and Laffont (1984), Weymark (1986) and Chambers
(1989), our result is novel in that our model formulation allows us to use
a ‘…rst-order approach’ to the agent’s problem to concentrate the objective function. The approach is based on the recognition that the order of
optimization in Problem I is irrelevant. Therefore, it is always possible
…rst to …x z k (k = 1, 2, ..., N ) in Problem I and to choose the premia I k
(k = 1, 2, ..., N ) conditional on z k , which we denote as I k (z k ). Once the
conditional optimization problem is solved, the z k can be chosen optimally.
Thus,
Result 3. If z k (k = 1, 2, ..., N ) is a solution to Problem I, then z k (k =
1, 2, ..., N ) also maximizes the following concentrated objective function (a):
P
P
max Rk (z k ) C k (z k ) m( pk (z k )),
(a)
k
k
where´´
Rk (z k ) =max S
1
X
I
[Iik + uk (zik
Iik )]
i
subject to
(
z ∈ arg max S
k
X
)
Iik )
C (z ) .
[Iik (z k ) + uk (zik
Iik (z k ))],
1
u (z i
k
i
Moreover,
Rk (z k ) = S
1
X
i
Non-point-source pollution regulation as a multi-task principal-agent problem 11
with
Iik (z ) = zik
hk (Cik (z k )),
where
hk (Cik ) = S/uk (zik
Iik ).
Rk (z k ) is the maximum social return obtainable from the state-contingent
vector z k when the farmer k ’s state-contingent premia are chosen to rationalize z k being picked by an expected utility maximizing farmer. Put another way, it is the maximum social return attainable from z k with hidden
action.
From Result 3:
∂Iik (z k )
Ciik (z )/Cik (z )
1,
=
1
+
∂zik
<( z 1 I i )
∂Ijk (z k )
Cjik (z )/Cjk (z )
,
=
∂zik
<(z j I j )
where <k (zi Ii) is the Arrow-Pratt coefficient of absolute risk aversion for
the k -farmer.
One cost structure that yields especially sharp results on I k (z k ) occurs
when farmers lose nothing by preparing for distinct states of nature separately. We refer to this as no effort economies of scope. Effort economies
of scope are present at a if
C k (z1, 0, ..., 0) + C k (0, z2, 0, ..., 0) + C k (0, ..., 0, zS ) > C k (z ).
Effort economies of scope are absent when this inequality is replaced by an
equality, implying an additively separable cost structure
k
C (z ) =
S
X
xki (zi ),
i=1
where xki (zi ) (i = 1, ..., S ) is increasing, convex, and twice differentiable.
With this de…nition, the following corollary is immediate.
Non-point-source pollution regulation as a multi-task principal-agent problem 12
Corollary 3 If there are no effort economies of scope and
C k (z ) =
S
X
xki (zi ),
i=1
where xki (zi) = exp(hi zi ) for hi ∈ <++ (i = 1, ..., S ) and uk (z ) exhibits
constant absolute risk aversion with <k (zi Ii ) = <k , then
!
Iik (z k )
=
aki
<k + hi
+
zi .
<k
Several points should be noticed about Result 3 and its corollaries. First,
a simple closed-form solution to the farmer’s optimization problem always
exists. This follows from the Inada conditions, and contrasts strongly with
other principal-agent problems (Grossman and Hart, 1984; Holmstrom,
1979). However, if the Inada conditions are relaxed, a set of expressions
based on the Kuhn-Tucker conditions replaces those in Result 3 and Corollary 3.1. The decomposition of the problem still applies. Second, the
responsiveness of Iik (z k ) to changes in z k has two key determinants: the
farmer’s attitudes toward corn risk and the effort-cost interdependences
(as measured by Cijk ) between different states of nature. These effort-cost
interdependences measure the cost of self insuring, and hence the responsiveness of Iik (z k ) is determined by the farmer’s desire for insurance versus
the cost of self-insurance.
All else constant, the more risk-averse the farmer is, the less responsive
is Iik (z k ) to changes in the state-dependent corn vector. In particular, as
risk aversion gets very large ∂Iik (z k )/∂zik approaches one and ∂Iik (z k )/∂zjk
zero: changes in zik must be exactly balanced by premium changes keeping
the farmer’s ex post return constant across states of nature. When farmer
risk aversion is very important, the optimal incentive contract should approach full insurance even in the presence of runoff. The reason is simple:
hidden-action contracts must always balance gains in efficiency from manipulating the incentive scheme against losses in risk sharing. When risk is
of overwhelming importance, the losses in risk sharing always outweigh efficiency gains. However, absent in…nitely large farmer risk aversion, changes
in zik are always at least matched by changes in Iik , i.e. the optimal incentive
contract will require farmers to self-insure.
Non-point-source pollution regulation as a multi-task principal-agent problem 13
5
The optimal incentive scheme and public pricing
For an interior optimum, i.e. zik > 0 (i = 1, 2, ..., S ) the …rst-order necessary conditions for Problem I after using the farmer’s …rst-order conditions
require:
S
1
X
(1
u(zjk
j


X
Ijk ))∂Ijk /∂zik = m pj (z j ) pik (z )
(2)
j
i = (1, ..., S ) and (k = 1, 2, ..., N ).
The right-hand side of 2 is the marginal pollution damage (bene…t)
caused by altering state-i contingent production. Because the private marginal cost of zi can be increasing in pollution, pki (z ) can be negative. The
expression on the left-hand side is the inner product of the vector of differences between Society’s marginal utility of corn and the farmer k ’s ex post
marginal utility of corn and the vector {∂Ijk (z )/∂zik }.
Because social efficiency in the absence of hidden action requires all
farmers to be fully insured, i.e.
1 = u(zjk
Ijk (z ))
for every j and k , the expression (1 u(zjk Ijk ) equals (u(sk ) u(zjk Ijk ))
in 2 and measures the departure from the …rst best. Hence, expression 2
generalizes the well-known ‘inverse-elasticity’ rule familiar from the publicpricing literature by requiring that divergences from perfect risk sharing
should be inversely related to the ∂Ijk(z )/∂zi in a generalized vector sense.
Some interesting results follow immediately. If there is no pollution
externality, i.e. m0() = 0 for all p (thus violating our assumptions), the divergences from complete risk sharing must be orthogonal to {∂Ijk(z k )/∂zik }
for all i. Because the origin is orthogonal to all vectors, complete risk sharing is then consistent with the …rst-order conditions for optimality. This
is as it should be: Results 1 and 2 together demonstrate that absent the
pollution externality there is no hidden-action problem for Society to resolve. Because the farmers are the residual claimants, the only individuals
directly affected by the farmers’ actions when no pollution externality exists are the farmers themselves, and Society’s sole role is as an insurer.
The social optimum then requires farmer ex post income to be the same
in each state of nature, and that farmers choose p to minimize cost of the
associated ex post output vector.
Non-point-source pollution regulation as a multi-task principal-agent problem 14
If all state-contingent outputs are gross substitutes in the private-cost
minimization problem, i.e. Cijk 0 for all i 6= j , and at least one statecontingent output’s private marginal cost is increasing in pollution, at an
interior optimum there must exist a (at least one) state j for which sk > zjk
Ijk . If all state-contingent outputs are gross substitutes in the private-cost
minimization problem and at least one state-contingent output’s private
marginal cost is decreasing in pollution, at an interior optimum there must
exist a (at least one) state j for which sk > zjk Ijk .
If pollution increases some marginal costs but not others and the privatecost problem is characterized by gross substitutability among state-contingent
outputs, the farmer receives a smaller ex post return in some states (and
a higher in others) than in the …rst best. Typically, therefore, the optimal
response by the planner to hidden action is to expose farmers to some upside and downside risk. This relationship is particularly stark when there
are only two states of nature.
Result 5. Suppose S = 2, all state-contingent outputs are gross substitutes and that pollution decreases the marginal cost of zi but increases the
marginal cost of z2, at an interior optimum z1k I1k > sk > z2k I2k .
Result 5 shows that the farmer’s ex post return is highest for the state in
which pollution decreases marginal cost and lowest in the state in which pollution increases marginal cost. When pollution decreases a state-contingent
output’s marginal cost, providing marginal incentives for producing more of
that output also encourages pollution because it raises the farmer’s shadow
return from polluting. The incentive problems associated with fully insuring farmers arise because insurance may give inappropriate incentives for
pollution control. Thus, some of the bene…ts from optimal risk sharing
must be traded for marginal gains in pollution control. Because the farmer
is risk-averse and the residual claimant, this entails allowing the farmer a
larger ex post return in those states where output raises the shadow return from polluting thus blunting the marginal incentive to increase that
state-contingent output (and hence pollution). Conversely, providing the
farmer with a smaller ex post return for states in which pollution tends to
increase marginal cost provides a greater marginal incentive for increasing
that state-contingent output (thereby diminishing pollution).
The connection between the inverse-elasticity rule and the optimal mechanism is most apparent in the absence of effort economies of scope.
Non-point-source pollution regulation as a multi-task principal-agent problem 15
Result 6. If there are no effort economies of scope, an interior optimal
mechanism satis…es
∂Ijk (z k )/∂zjk (u(zjk
Ijk (z k )))
= m
X
!
i
p (z ) pkj (z )
i
(k = 1, 2, ..., N ) and (j = 1, 2, ..., S ).
A trade-off exists between risk sharing and pollution abatement that
any mechanism must accommodate. The absence of economies of scope
crystallizes this trade-off. By Result 6, the divergence from optimal risk
sharing and the marginal externality associated with zi have the same sign.
Moreover, the marginal pollution externality is always at least as large
as the divergence from optimal risk sharing (in absolute value). (These
relationships follow from Corollary 3.1 with Result 6.)
The more responsive Ijk (z ) is to changes in zj , the smaller is the departure from optimal risk sharing required to accommodate a given marginal
pollution externality. The intuition is apparent. Absent effort economies of
scope, ∂Ijk (z k )/∂zjk is the reciprocal of the change in zjk that a risk-averse
farmer makes to respond to small changes in Ijk . So, when ∂Ijk (z k )/∂zjk is
large, changes in the I k schedule are relatively ineffective in changing production (and hence pollution) behavior. No matter how I k is tilted in the
j state, zj remains relatively unaffected. Marginal departures from optimal risk sharing in those states bring little if any gain in pollution control.
Thus, only in states where tilting the incentive scheme affects production
behavior signi…cantly will one expect signi…cant departures from optimal
risk sharing.
Corollary 4 Under the conditions of Result 6, sk < zjk Ijk if pollution
decreases the private marginal cost of zj ; and sk > zjk Ijk if pollution
increases the private marginal cost of zj . If pollution is invariant to changes
in z k , then sk = zjk Ijk .
Corollary 6.1 extends Result 5 to cover the absence of effort economies
of scope. If increasing a state’s ex post output increases pollution, then absent effort economies of scope the best policy is to leave the farmer with a
higher return for that state than in the …rst best. Absent of effort economies
of scope, the farmer effectively prepares for each state of nature independently. So, tilting the incentive scheme in any state only affects that state’s
Non-point-source pollution regulation as a multi-task principal-agent problem 16
contingent output. Thus, if Society decides to charge a very high premium
in state j , the strictly risk-averse farmer responds by expanding zj alone.
If zj encourages pollution, the result is an added pollution incentive to the
farmer.
6
Risk-reducing pollution and the optimal return
The preceding section showed that how pollution interacts with the ex post
outputs in the private-cost minimization problem is a crucial determinant
of the optimal mechanism. This section takes up a related issue: how
the ability to pollute affects the riskiness of the ex post output trajectory.
Different production inputs affect the riskiness of output (to the farmer)
in different ways. For example, chemical pesticides are usually seen as
reducing the riskiness of output because they preserve output even in the
worst states (severe pest infestations). Chemical fertilizers, on the other
hand, are often seen as increasing the riskiness of output because they
can actually decrease production in the event of severe moisture shortfalls.
Because chemicals are the original source of much of the runoff pollution,
reason then suggests that the degree of pollution will be related to the risk
characteristics of the ex post output pro…le.
To formalize the discussion, some new notation is convenient. For any
z ∈ <S+ , let z = (z[1] , ..., z[S ]) denote its decreasing rearrangement, i.e.
z[1] z[2] ... z[S ].
The curve labelled z in Fig. 1 represents the decreasing rearrangement of
farmer k ’s optimal state-contingent corn vector given I k (for notational simplicity the farmer superscript is dropped in the remainder of this section).
In Fig. 1, z is riskier in he Rothschild-Stiglitz sense than the trajector
labelled z0. (Both have the same mean but z is more disperse than z0 .)
Therefore, if p(z 0 ) > p(z ) one might intuitively characterize pollution as
risk-reducing because a higher private-cost minimizing pollution level is associated with a less risky state-contingent output vector. Conversely, when
p(z 0 ) < p(z ), pollution might be thought of as risk-increasing because
a lower private-cost minimizing pollution level is associated with the less
risky ex post output vector.
Our formal de…nition of risk-increasing pollution and risk-reducing pollution relies on the intuition summarized by Fig. 1: pollution is riskP
P
reducing (risk-increasing) over Z = {z : i zi = i zi } if for all z 0 m z ,
Non-point-source pollution regulation as a multi-task principal-agent problem 17
z 0 ∈ Z , p(z 0 ) p(z )(p(z 0 ) p(z )). Here z denotes the optimal ex post
output trajectory, and the notation ‘k m q ’ is to be read ‘k majorizes q ’
(Dasgupta et al., 1979; Marshall and Olkin, 1979). Vector k majorizes vector q if they share common means but k is riskier in the Rothschild-Stiglitz
(1970) sense than q .7 Several things should be noted about these de…nitions. First, the notion of risk-increasing and risk-reducing introduced here
P
P
is local (only applying to the set Z = {z : i zi = i zi}). Second at z ,
pollution is risk-reducing if p(z ) behaves as though it were Schur-concave
(Dasgupta et al., 1979; Marshall and Olkin, 1979) at that point. Similarly, over Z , pollution is risk-reducing if p(z ) behaves as though it were
Schur-convex (Dasgupta et al., 1979; Marshall and Olkin, 1979) over Z . [A
function f : <S → < is Schur-convex if a m b implies f (b) f (a). A
function f : <S → < is Schur concave if — f is Schur convex.] By this
observation:
Lemma 5 If pollution is risk-reducing, p(z ) is symmetric over Z and
p[1] (z ) p[2] (z ) ... p[S ] (z ).
If pollution is risk-increasing, p(z ) is symmetric over Z and
p[1] (z ) p[2] (z ) ... p[S ] (z ).
The economic content of the lemma is best understood by returning to
Fig. 1. There pollution is risk-reducing if rotating the state-contingent
output vector from z to z 0 requires increasing pollution. Notice, in particular, that the rotation requires the largest positive change to be the
increase in z[S ] and the smallest change to be the decrease in z[1]
as the
distribution of outputs is smoothed. The …rst part of the lemma captures
this same phenomenon. If increasing pollution is to smooth the output
distribution, any increases in the highest state-contingent outputs must be
at least matched by increases in lower-contingent outcomes.
Whether pollution is risk-reducing or risk-increasing can have important implications for both the optimal incentive scheme and the pattern of
farmer returns as the following demonstrate.
7
The concept of majorization is de…ned in Marshall and Olkin (1979). Our notation
differs slightly from theirs.
Non-point-source pollution regulation as a multi-task principal-agent problem 18
Result 7. If pollution decreases the private marginal cost of z[1] , and pollution is risk-reducing, pollution must decrease the private marginal cost of
all other z[1] (i = 2, ..., S ), i.e.
0 p[1] (z ) p[2] (z ) p[S ] (z ).
Corollary 6 If there are no economies of scope, pollution decreases the
private marginal cost of z[1] and pollution is risk-reducing; the farmer’s
optimal return is always greater than the …rst-best return.
Result 7 and its corollary extend Results 3, 5, and 6 to cover riskreducing pollution. Again the intuition is similar to that associated with
Results 5 and 6. Because providing extra incentives to the production of
any ex post output encourages further pollution emission by the farmer, the
farmer’s production is curtailed compared with the second best. Obvious
extensions of Result 7 and its corollary to cover the cases of risk-increasing
pollution are left to the reader.
Our last result shows how the inverse-elasticity relationships respond to
the presence of risk-reducing pollution and risk-increasing pollution:
Result 8. If pollution is risk-reducing [i] > [j ] implies
X
(1
u(zk
Ik ))(∂Ik /∂z[i]
∂Ik /∂z[j ]) 0.
k
If pollution is risk-increasing [i] > [j ] implies
X
(1
u(zk
Ik ))(∂Ik /∂z[i]
∂Ik /∂z[j ]) 0.
k
7
Conclusion
This paper considers a multi-task, principal-agent problem where farmers
have two tasks, pollution control and corn production, but only direct incentives for corn production. A general method for solving this problem
is developed and applied to the problem. The method ultimately involves
only the solution of an unconstrained non-linear programming problem. It
is shown that the solution to the problem obeys a generalized version of
the ‘inverse-elasticity’ rule familiar from the theory of public pricing, and
the optimal solution is characterized under a number of assumptions about
Non-point-source pollution regulation as a multi-task principal-agent problem 19
the underlying technology and the role that pollution emission plays in
reducing farmer risk.
Acknowledgment
We wish to thank several anonymous reviewers and Erik Lichtenberg
for comments that bene…ted this paper.
Appendix: Proofs of results
Result 1. Let circum‡exes (ˆ) over variables denote their …rst-best values. In the …rst best for individual k
k
ˆ
Iˆik =z i
sk
and
X
m
ˆk
!
p
=
ˆ k ˆk
ckp (p , z ).
k
Consider the two-part, state-contingent payment depending only upon the
state of nature and observed pollution:
I˜ik
=
Iˆik
m
X
ˆk
!
ˆk
k
p
p ).
(p
k
Farmer k facing this individual speci…c two-part, state-contingent payment
chooses z k and pk to give
max S
1
X
u
ˆk
zi
zik
X
k
+s + m
i
ˆk
!
p
ˆk
k
!
ck (pk , z k ).
p )
(p
k
The …rst-order conditions for an interior solution are
S
1
u
zik
ˆk
zi
k
+s + m
X
ˆk
!
k
p
(p
X
ˆk
ˆk
!
cki (pk , z k ) = 0,
p )
k
S
1
X
i
k
ˆk
u
zik
ˆk
zi
k
+s + m
k
p
!
k
(p
ˆk
!
p ) m
X
ˆk
p
!
+ ckp (pk , z k ) = 0.
k
ˆ
z and p solve these equations. The convexity and concavity properties
of uk and z k ensure that this solution is unique. For this two-part scheme
Non-point-source pollution regulation as a multi-task principal-agent problem 20
Society’s welfare is given by k i I˜ik
S
1
ˆk
m( p ).
sk )
k i (z i
m(pk ) which at the …rst best equals
ˆk
Result 2. Only if: First-best optimality requires
ˆ k ˆk
m+ ckp (p , z ) 0
with equality for an interior optimum. By assumption, m > 0, so an interior
ˆk
optimum cannot arise with pk = p for any farmer solving the private-cost
ˆk
minimization problem. Sufficiency: Suppose for the …rst best that pk (z = 0.
ˆk
A state-contingent contract specifying Iik = z i
the farmer solving
max S
1
X
u(zik
Ikk )
sk (i = 1, ..., S ) would lead
C k (z )
i
ˆk
to choose z . The convexity and concavity properties of u and C k (z ) ensure
that this is the unique …rst-best optimum.
Result 3. That Problem I can be rewritten as
max
z
X
k
k
k
R (z )
k
C (z )
m
X
k
!
k
k
p (z ) ,
k
where
Rk (z k ) =max S
1
X
I
[Iik + uk (zik
Iik )]
i
subject to
(
z ∈ arg max S
k
1
X
)
u(zik
Iik)
k
k
C (z )
i
follows by the principle of conditional optimization. Under the Inada conditions on the utility and cost functions, the rest of the result then follows
from the …rst-order conditions for z k in the farmer’s optimization problem.
Non-point-source pollution regulation as a multi-task principal-agent problem 21
Result 4. By expression (2),
S
1
X
(1


X
Ijk ))∂Ijk /∂zik = m pj (z j ) pik (z ).
u(zjk
j
j
If all state-contingent outputs are gross substitutes for one another in the
private-cost minimization problem, Result 3 implies ∂Ijk /∂zik 0 for all i
and j . If pollution increases the marginal cost of state i’s contingent output,
the right-hand side is negative. Thus, at least one expression under the
summation on the left-hand side must also be negative, whence for at least
one j , 1 u(zjk Ijk ) < 0. The strict concavity of u then implies zjk Ijk < sk .
The second part of the result is proved similarly.
Result 5. When S = 2, expression (2) implies
(1/2)(1
u(z1k
I1k ))∂Iik /∂zik + (1/2)(1
u(z2k
ki (z ),
I2k ))∂I2k /∂zik = mp
i = 1, 2. Solving yields
u(z1k
u(z2k
1
1
I1k ) = D 1 m[p1k (z )∂I2k /∂z2k
I2k ) = D 1 m[p2k (z )∂I1k /∂z1k
p2k (z )∂I2k /∂z1k ],
p1k (z )∂I1k /∂z2k ],
where D > 0 by the strict concavity of uk and the convexity of C k . Applying
Result 3 and the de…nition of sk under the conditions stated yields the
result.
Result 6. If there exist no effort economies of scope, the private-cost function can be written (dropping superscripts):
C (z )
S
X
x i (z i ).
i=1
Hence, applying Result 3 here gives
∂Ij (z )/∂zi = 0,
i 6= j . Expression (4) for zj (j = 1, 2, ..., S ) reduces to
S
1
{[1
u(zj
i (z ).
Ij (z ))][∂Ij /∂zj ]} = mp
Non-point-source pollution regulation as a multi-task principal-agent problem 22
Corollary 6.1. Apply the de…nition of sk and the conditions of the corollary.
Lemma 1. Follows by applying Theorem 3.A.4 in Marshall and Olkin (1979)
to the de…nitions of risk-increasing pollution and risk-reducing pollution.
Result 7. Use the de…nition and apply Lemma 1.
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