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An
Economic
Model
of
Soil
Conservation
Kenneth E. McConnell
By introducingsoil depthandsoil loss into a simplemodelof agriculturalproduction,this
paperseeks to determinewhenthe privatepathof erosiondiffersfromthe socially
optimalpath. Consideringthe depletionof soil only andabstractingfromthe
environmentaldisruptioncausedby erosion,the paperarguesthatundermost
institutionalarrangementsthe social andprivateratesof erosionarethe same. The paper
concludesthatpublicpolicy shouldbe directedtowardreducingerosiononly whenit
leads to significantpollutionexternalities.
Key words: asset markets,erosion,optimalcontrol,publicpolicy, soil depth.
During the 1970s the agriculturaloutput expansion caused a substantialincrease in soil
erosion. The resultingconcern stimulatedtwo
kinds of studies, one showing the economic
impact of reduced erosion (Wade and Heady
1978) and less technical papers sounding an
alarm about how current erosion reduces future agriculturalproductivity (National Association of Conservation Districts). Both
imply the need for collective action to curtail
soil loss.
The Soil ConservationService has provided
technical and financial assistance to curtail
erosion for almostfifty years. Despite substantial expenditures($311millionin 1981),thereis
little evidence that soil erosion has been curtailed. About 33% of U.S. cropland suffers
annualerosion rates in excess of five tons per
acre (table 3, USDA). There are several explanations of farmers' tolerance of erosion.
For example, "Farmers are simply too busy
with the many other problems" (Renard,
Heineman,Williams,p. 1278). Anotherexplanationis that farmersdo not observe erosion's
effects on land productivity. Burt suggests
that farmers are unconcerned about soil loss
because they can substitute other inputs for
soil depth. A consensus is that conservation
practices are not adopted because "even
though the practices may provide higher long
KennethE. McConnellis an associate professorof agricultural
and resourceeconomics, Universityof Maryland.
This is ScientificArticleNo. A-3278,ContributionNo. 6350of
the MarylandAgriculturalExperimentStation.
Withoutimplicatingthemconcerningthe featuresof this paper,
the authorthanksOscarBurtand BruceGardnerfor helpfulcomments on a previousdraft.
run profits . . . the savings are simply not
worth the change . . ." (Wade and Heady
1979, p. 1281).
The difference between conservationists'
goals and farmers' behavior suggests market
failure. One type of failure clearly exists-water pollution caused by soil erosion. However, the soil conservationliteraturesuggests
another failure concerning intertemporalsoil
use. For example, Wade and Heady state that
there are two externalitiesfrom soil erosion,
"potentially reduced agriculturalproduction
capacity and the pollutionof the eroded soil"
(Wade and Heady 1979, p. 1281). Similarly,
Lee expresses concern that increasing soil
losses are "intensifyingair and waterpollution
problemsand reducingthe productivitypotential of a cropland" (p. 1070).
However, there is little formal analysis of
social intervention to curtail soil loss. Burt
presents a formal intertemporalmodel of soil
use for farms in the Palouse area. CiriacyWantrupanalyzes soil as a renewableresource
with a threshold level below which resource
use becomes irreversible. Bunce attacks the
problem systematically but only in a static
framework; Swanson and coworkers have
studied soil erosion but without an intertemporal link throughthe asset market.
This paperdevelops an economic model for
the optimal private and social utilization of
soil. The focus is on the intertemporalpath of
soil use includingthe conditions under which
privateand social optimadiverge. It also gives
insight about effective instrumentsof erosion
control.
Copyright 1983 American Agricultural Economics Association
84
February 1983
Amer. J. Agr. Econ.
A Model of Private Decisions
trientsand moistureto absorb. We expect that
0. Eventually, additionalsoil depth adds
f,
This section presents a simple model of the nothing to current productivity, so that for
privaterate of soil loss over time. For simplic- some values of input use, soil loss, and soil
ity, suppose that the entrepreneurproduces depth,fxx 5 0 andf, = 0.
This model ignores two important items,
only one crop. Hence, decisions about crop
sequencing and variety are suppressed. The pollution from runoff and soil quality. The
contributionof erosion to water pollution is
single crop productionfunction is
well documented(Swanson, Wadeand Heady
(1)
q = g(t)f(s, x, z),
1978, and USDA). Swanson and his coworkwhere q(t) is output; s(t), soil loss; x(t), soil ers have shown that the social costs of water
depth; z(t), an index of variable inputs, and pollution from soil erosion are substantialin
g(t) is neutraltechnicalchange.1This model is some areas. However, pollution is ignored
a substantial, but plausible, simplificationof here to sharpen the focus on intertemporal
the complex process determiningsoil quality depletion.
The quality of the soil is assumed constant.
and depth.
Other specifications also capture the es- Cultivationnot only results in soil loss, it resence of the dynamic problem of agricultural duces the remainingsoil's fertilitybecause orproduction and soil depletion. The approach ganic materialand smallerparticles which fatakenby Burt, when the control variableis the cilitate nutrientexchange are lost. This probproportionof land in wheat, is more suitable lem is not treated in this paper because ferfor numerical analysis because it introduces tilizer can partiallysubstitutefor reduced soil
the more realistic problem of the choice of quality.
Left alone, soil regeneratesslowly. Natural
crops. Anotheralternativewould be to specify
the vector of inputs as of two types: z, is rebuildingcontributestwo to five tons per acre
productive inputs, and z2 is inputs used to per year dependingon soil type, weather, and
prevent soil depletion. Then one could other variables. Let k be this exogenous addimaximize the present discounted value of tion to the soil base. Assuming k to be conpgf(x, zJ) - cIz1 - c2z2, where c, is the cost of stant is an approximationbecause productivthe ith input subject to the transitionequation ity is more vulnerable to erosion as the soil
x = h(z1, z2), where h is the change in soil depth is reduced. The relationshipbetween k
depth, productive inputs increase soil loss and s determines how x changes:
(ah/lazl < 0), and ameliorativeinputs reduce (2)
x(t) = k - s(t).
soil loss (ah/az2 > 0). The results of this paper
are not affected by such a change in the spec- If soil losses occur at the accretionrate, s = k
andx = 0. Soil loss at the rate of k sustainsthe
ification of the model.
soil depth. Thus, k is the toleratedsoil loss. It
Soil
is
establishment
loss,
q.
Output per
s(t), influences output because, other things is generallyin the neighborhoodof two to five
equal, output expansion per farm in a given tons per acre, though it can be more or less
time period requiresmore soil loss. For exam- dependingon the climate and parentmaterial.
ple, output can be increased by cultivating The behavior of the entrepreneurtowards
land with greater slope, increasing soil loss. soil is determined by the soil's impact on
We expect thatf, - 0 andfs, 5 0. Outputcan profits.Assume that the farmerworks his land
increase with increasingsoil loss; but, eventu- to maximize the present value of the profits
ally, additionalsoil loss will not increase out- streamplus the value of the farmat the end of
put. For example, when all grasswaysand ter- the planning period. This is equivalent to
races have been plowed and the steepest maximizingthe present value of the consumpslopes cultivated, additionalproduction can- tion stream, if the farmer has access to
smoothly working capital markets. The presnot be gained by losing more soil.
The depth of soil, x, has a beneficial effect ent value of the stream of profits for T years
on crop production. More soil depth gives the is2
plant roots more room to grow and more nu-
I The assumption of neutral technical change makes matters
seem worse than they really are because, in fact, actual technical
change seems to be generally soil saving.
2
Optimal depletion plans under various assumptions about the
marginal value of the resource are derived in Dasgupta and Heal,
chapter 10.
McConnell
A Soil Conservation Model
85
unit value of soil lost equals the foregone
profits from having the soil on the farm. The
where r is the farmer'sdiscount rate, p is the implicit cost of soil loss, X, must grow at the
rate of discount less the soil's contributionto
per unit output price and c is an input price currentreturns. The term
pgf, is soil depth's
index. The time index forp, z, s, and c will be
increment
to
current
If additionalsoil
profits.
suppressed.To maximizethe present value of
has
no
on
current
production
impact
depth
his consumption stream, the farmer will
maximizeJ plus the present value of the farm (fx = 0), the value of the soil would grow at
in year T. Let the terminal value be R(x), the rate of discount reflecting only capital
where R stands for the resale value of the gains. Equation (11) makes it uneconomical
farm. MakingR a functionofx impliesthat the for the farmerto deplete the farm's value near
value of farm real estate depends, in part, on the end of his career. These conditions would
hold if farmerswere fully aware of soil's consoil fertility. The farmerwill maximize
tribution to both current production and the
(3) the net value of the farm
farm's resale value.
= J + R[x(T)]e-T, Under what conditions does the rate of soil
loss increase or decrease over time? Condisubject to i = k - s and the initial depth of tions
(6)-(11) hold for 0 ? t T, so that
soil:
_
+ gp)f, +
- X
+
(Pg
(4)
x(O) = x0.
f.zi)+
pg(fsS
(k - s)pgf,,= 0
The problem of maximizing (3) subject to
(3ig + gp)fz + Pg(fszi + fzzi) - c
(2) and (4) is an optimal control problem
+ (k - s)pgf,, = 0
which results in common sense rules for resource allocation. The input z will be used Solving for S and i gives
until the value of its marginalproduct (pgf,)
A12
equals its costs. Soil loss will be incurreduntil (12)
the value of returns obtained from additional
i ] [A11 A12 A22
soil loss (pgf,) equals the implicitcost of using
the soil. The cost of soil loss in foregonefuture
(k- s)
fr -f
profits is the change in the productivity and
P
g
f•fs
f
s
sale value of the farm caused by having less
soil. Soil is an asset. In equilibrium,this asset
fz - (k - s)fz
S
must earn a rate of returnequal to returnson
other assets. The return for holding soil is where
made up of capital gains and contributionsto
currentprofits.
A=[;S fs y
The undiscounted Hamiltonian associated
with (1), (3), and (4) is
Supposethatf is concave in s andz for fixedx.
(5) H = [pgf(s, x, z) - cz] + A(k- s).
Then Aj 0 for all i, j. Solving for S gives
to
the
maximum
According
principle, the op- (13) s = <_
All f,[r - pl/p - g/g - fx/fs
timal paths of z, s, x, and X satisfy:
- (k - s)fxs/fs] +
A12fx[C/C - pIP - g/g
(6) 8H(z, s, x, X)/8s = pgf,(s, x, z) - X = 0,
- (k - s)fx/lfz].
(7) 3H(z,s, x, X)/az = pgf (s, x, z) - c = 0,
It is possible that S 0, as we have observed
(8) X = rX = aH/lax = rX - pgfr(s, x, z),
_ suggests that there are
recently. Equation(13)
circumstances
underwhich ramany
plausible
x = k- s,
(9)
tional farmers,who consider the productivity
of their soil, would increasetheir soil loss over
(10)
x(0) = xo, and
time.
(11)
X(T) = aR[x(T)]/ax(T).
Equation (13) contradicts the notion that
Condition (7), for variable inputs, requires increased soil loss implies that farmers are
that the value of the marginalproductequal its ignoring the future. Increased soil loss can
cost. Soil loss is tolerated until the marginal occur throughrationalfarm managementwith
J=
e-rt[pg(t)f(s, x, z) - cz]dt,
s
86
Amer. J. Agr. Econ.
February1983
finite resources and complete knowledge of
soil productivity.Farmersdo not have perfect
foresight about markets nor complete knowledge of all technologicalrelationshipsfor their
crops and soil. But even if they did, they
would not necessarily maintainthe same soil
depth over time.
A strongbequest motive or a smoothlyfunctioning capital market induces the farmer to
value R(x) as part of his income stream. The
impact of soil depth on the resale value requires that the implicit cost of soil loss, h,
evolve over time until in the last period this
implicit cost just equals the return in farm
value from greater soil depth. Under these
conditions,the rationalfarmerwill exhaust his
soil only if it compensates him for the loss
which occurs because of the decline in the
farm's resale value. Anything which reduces
the impact of soil depth on resale value,
(&R/ax), will lower Xat each t. This decreases
the implicitsoil cost, inducingmore loss each
period. Anythingwhich increases the present
value of foregone profits because of less soil
will reduce loss. Thus, high future prices,
lower discount rates, movementup the marginal product schedule for soil depth will discourage excessive current soil loss. In this
model, only the movement up the marginal
product schedule is endogenous. In an aggregate model, output price also would rise in
response to declining soil productivity and
lower output.
A crucial element in the soil conservation
debate concerns the farmer's lack of knowledge about various technical and economic
relationships. By making assumptions about
farmer'sknowledgeof technicalrelationships,
we can assess impliedpaths of soil depletion.
First, suppose that soil depth does not
influence crop production (f, = 0). For
simplicity also assume that prices, costs, and
(T)= R/ax(T)
X(t)=X(O)"
?
x(O) WrT
Figure 1. User cost whenf, = 0
does not discount futurebenefits and believes
that soil is essential but changes in soil availability have no impact on the farm's resale
value (8R/8x
0) or current productivity
will
be
Profits
(f - 0).
T[pgf(s, x, z) - cz] - y[sT-
xo-
kT],
where y is the multiplieron the constraintthat
more soil cannot be used than exists or is
created (sT - xo - kT < 0). Conditions for
profit maximizationinclude
s[pgf -y] = 0; s, y - pgf
y[sT-
0,
x - Tk] = 0; y, kT + xo - sT > 0.
The same amount of soil will be lost each
period. Wheny = 0, the farmerwill maximize
output with respect to soil loss (f = 0). Thus,
assuming that the discount rate is zero; that
prices, costs, and technical change are constant; and that soil depth does not contribute
to plant growth ensures that soil loss is constant over time. These conditions will lead to
total soil depletion because depth has no impact on currentprofitsor the marketvalue of
the farm.
A more likely set of conditions is that soil
depth influencescurrentand futureproductivity but not the sale price of the farm. Thus,
= g = 0). fx > 0, but &R/&x= 0. These assumptions
j =
technology are constant (p
Equations (8) and (11) imply
requirethat X(T)= 0. Soil loss will rise to its
maximumwhen user costs are zero in the last
(14)
X(t) = e-r(Tr-tR[x(T)]/ x(T).
period. The exact path will depend on price
The implicitcost path of soil loss whenf, = 0 and cost paths and the technical relationships
is pictured in figure 1.
among variable inputs, soil loss, and soil
The farmer's discount rate determines the depth.
rising marginal cost of soil loss over time. The
increasing marginal cost of soil loss leads to its
reduction over time. This problem solved with
a higher interest rate will yield a lower current
value of X, but it will rise more rapidly. Consequently, higher interest rates imply more
rapidly declining soil losses.
Consider the simplest case, when the farmer
Impact of Land Tenure Arrangements
People believe that farm structure effects ero-
sion. For example, Lee states, "the changing
structureof agriculturehas led to the hypothesis that a larger, more corporate agricultural
McConnell
A Soil Conservation Model
structurewill have unfavorableconsequence
for soil conservation" (p. 1070).
Three tenure arrangementsprevail: owned
family farms, rented family farms, and corporate farms. One distinction among these tenure arrangementsis the planning horizon. A
corporationmay plan indefinitelyinto the future. Family farm's planning horizon is
adopted by the head of the household. The
distinctionbetween owningand rentingaffects
the farm's resale value. The family's horizon
is assumed to be known and constant. The
horizon's length will influence user cost and
the erosion rate. The family farmer, with a
planninghorizon of T years, maximizes
(15)
o
V=
.o
faces the same discount rate as private individuals, the currentperiod user cost is
(0) =
(19)
subject to (2) and (4). For owners 0 = 1, and
0 = 0 for renters. Taxes are ignored in this
model. The first-orderconditionsfor maximizing (15) are the same as for equations(6)-(11)
except that
(16)
X(TO)= 0aR[x(To)]/ax(To).
t=o p(t)g(t)f,(t)
(1 + r)t
This equation is not directly comparablewith
equation (17) for the family farmerbecause of
the differencein planningperiods and the terminal value function. However, when asset
marketswork smoothly, the value of the farm
at T with soil depth of x(T) = x, over a planning period of T* years is
R(xT) = Max
T+T*
Z
[pgf(s, x, z) - cz)(1 + r)r-t
t=T
[pgf(s, x, z) - cz]e-rtdt
+ e-rT0 R[x(To)],
87
+ R(xT.) (1 + r)-T*
subject to x(t + 1) - x(t) = k - s
x(T)= XT.
Reducingx, by one unit meansthatf(s, x, z) is
reduced by the marginalproductof x in each
period. Thus
T+T*
p(t)g(t)fx(t)(1 + r)T-t
For the renter, soil loss occurs so that user
t=T+ 1
cost is zero in the last period. This resultsfrom
+ (1 +
usingup soil in earlierperiods. Initialuser cost
r)-rT*R(xr,)/xT*,.
can be calculated approximatelyby restating This reasoningfor several generationsof farm
expression (8) in finite time and iterating:
owners implies that the currentuser cost for
family-ownedand operated farms is
aR(xT)/aXT
(17)
0
(0) =
(1 + r)ro
R[(x)To]
8x(To)
TO
t=
p(t)g(tO)f(t)
((1 + r)t
when f (t) means f,(s, x, z) evaluated at the
values of s, x, and z in period t. Expression
(17) shows how rental (versus ownership)
influences current soil use decisions. Institutional arrangementsinfluence soil loss and
depth by influencinguser costs. The current
user cost, X(0) when T, = To, is lower for
renters because farm resale value is unimportant. The only reason for renters to conserve
soil is for its productivecapacity. If soil depth
=
T
X(0) =
p(t)g(t)fx(t)(1 + r)-t
t=o
+
>p(t)g(t)fx(t)(1
+ r)-t,
t=T+l
which is equal to that of a corporatefarm.
The two ownershiparrangements,using the
same technology and inputs and facing the
same input prices and capital markets, will
have the same user cost. Hence, the corporate
farm will use up soil at the same rate as the
family farm. Hence, more specific behavioral
assumptionsare needed to show that increasing
corporateownershipof farmswill increase
does not affect production, the renter will igerosion.
nore the soil loss equation.
A corporation's status as a legal entity suggests a very long planning horizon. Consequently, the corporation will maximize
(18)
V* = I e-rt[pgf(s, x, z) - cz]dt,
subjected to x = k - s. If the corporation
Socially Optimal Soil Use
Optimal intertemporal soil use is that which
maximizes the chosen social welfare function.
For the single farm, the return to society is the
88
February 1983
Amer. J. Agr. Econ.
rent accruingto the farm. This ignores pollution externalities from loading streams with
waste material. However, the consequences
of ignoringthese externalitiesare well known
and necessary corrections can be made, at
least in principle.
The value of a single farm to society is
V** =
{
e-'[pgf(s,
x, z) - cz]dt.
This objective function differs from that of
corporate and family farms if the discount
rates differ. For the private sector, the discount rate reflects the capital market. For society, 8 is a measure of how we value the
welfare of distant generations. The private
farmplan for soil loss and the socially efficient
plan will be the same if the private discount
rate, r, equals the social discount rate, 8. If
capital markets work efficiently and 8 = r,
then both the corporate and the family farm
will mimic the intertemporalpath of soil use
chosen by a wise social planner.
The calculation of social benefits from a
farm implicitlyassumes that it is ethically acceptable to weigh future generations'welfare
lower the further in the future they extend.
There are many arguments against such a
myopic view (Dasgupta and Heal). The consequence of treating all generations equally
depends upon the importanceof the resource.
A steady-stateresult emerges when 8 = 0 and
the marginal value of the resource goes to
infinityas the resource goes to zero.2 One of
two conditions will cause the soil's marginal
value to go to infinity:(a) the marginalproduct
goes to infinityas the soil goes to zero, or (b)
the marginalvalue of the good (its price) goes
to infinityas the quantitygoes to zero. These
conditions are both met in agricultureat the
aggregatelevel. However, in a farmmodel the
marginalvalue of output measuredby price is
constant. Hence, the optimal policy even
when the social discount rate is zero may
include some soil exhaustion.
The use of tolerance values (i.e., fixed values of soil loss) for social policy requiresthat
the soil loss per year not exceed expected
annual replenishment. The socially optimal
steady state emerges only when the marginal
value of soil goes to infinity as depth goes to
zero and the social discount rate is zero. When
the rate is positive, socially determined tolerances for soil loss might exceed annual soil
replenishment.
Several conclusions follow. First, if finan-
cial and real-asset markets work efficiently,
private farmers may follow the socially
efficient path. Second, it may be optimal to
exhaust the soil on some farms, even when all
generations are treated the same. Third, national policy that requires annual soil loss to
be less than the tolerance values can be justified if all generations are treated identically
and if we cannot rely on technical change or
capitalaccumulationto substitutefor lost soil.
This reasoningextends logically to groundwater stocks, pesticide resistance, and other agricultural resources. The conservative and
conserving ideals which imply a constant soil
depth have radical consequences in that
achieving a constant soil depth would require
radical changes in cultivation practices and
policy instruments.
Conclusion
This paperexamines the economics of private
and optimalsoil use. One resultis that increasing soil loss does not imply that farmersignore
physical productionrelations. A second implication is that if farmersknow that the soil base
affects farm resale values, they will conserve
it. This result suggests that informationabout
soil depth and its economic value be disseminated. It also suggests that the impact of soil
depth on the value of farms be investigated.
A thirdconclusion concerns soil use policy.
Requiringthat soil removed each year be no
more than natural replenishmenthas radical
consequences for policy and for cultivation
practices. This conclusion is derived from
maximizingthe minimumwelfare of all generations. This strategymay be ethically defensible for croplands,but it is also ethicallydefensible elsewhere, and we do not pursue it with
vigor elsewhere.
The conclusions of this paperare consistent
with Swanson's-the majorimpactof soil erosion is water pollution. His research has
shown, at least for parts of Illinois, that the
impact of soil erosion on agriculturalproductive capacity is small. This papersuggests that
the asset market will account for this change
whether it is large or small. Either way, the
problem of water pollution is paramount, not
agriculture's future productive capacity.
[Received June 1981; revision accepted
September 1982.]
McConnell
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