Water Resources Development, Vol. 19, No. 1, 37–53, 2003
A Theoretical Analysis of Economic Incentive Policies
Encouraging Agricultural Water Conservation
RAY HUFFAKER & NORMAN WHITTLESEY
Department of Agricultural Economics, Washington State University, Pullman,
WA 99164-6210, USA. Email: [email protected]
ABSTRACT A conceptual model of a representative irrigated farm is formulated to study
farm responses to two economic policies commonly suggested to encourage agricultural
water conservation, and to characterize the hydrological and economic circumstances in
which these responses provide the desired conservation. The economic policies studied are
to increase the irrigator’s cost of applied water and to subsidize the irrigator’s cost of
investing in improved on-farm irrigation efficiency. Comparative statics results demonstrate that increasing the cost of applied water may be a more effectual water conservation policy than subsidizing the cost of improved on-farm irrigation efficiency.
Introduction
An alarming portion of the world’s population lives in water-stressed river
basins (Johnson et al., 2001). Proposed conservation measures focus on water use
in irrigated agriculture for two major reasons. First, irrigated agriculture accounts for about 70% of total water withdrawals (Johnson et al., 2001), and
60–80% of total consumptive water use (Tiwari & Dinar, 2001), in producing
about 40% of the world’s food crops. Moreover, the demand for irrigation water
is expected to grow with the food needs of increasing populations. Secondly,
water use efficiency in irrigated agriculture tends to be low world-wide, with
national averages in the range of 25–50% (Tiwari & Dinar, 2001). One study
recently noted that “more than half the water entering irrigation distribution
systems never makes it to the crops because of leakage and evaporation”
(Johnson et al., 2001, p. 1071).
A common policy prescription is to encourage irrigators to conserve water
with economic incentives (Willey & Diamant, 1995; Johnson et al., 2001; Tiwari
& Dinar, 2001). A number of countries have instituted, or are considering
instituting, higher agricultural water prices to discourage wasteful water use
(Dinar & Subramanian, 1997). Scholars also have proposed irrigator subsidy
programmes to encourage improvements in on-farm irrigation efficiency (Tiwari
& Dinar, 2001).
The authors investigate the conceptual circumstances in which higher water
prices and subsidies for improved on-farm irrigation efficiency encourage an
irrigator to conserve agricultural water. A stylized model of a representative
irrigated farm is formulated that extends past conceptual work by incorporating
a wider range of possible irrigator responses to these conservation policies. This
0790-0627 Print/1360-0648 On-line/03/010037–17  2003 Taylor & Francis Ltd
DOI: 10.1080/0790062032000040764
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R. Huffaker & N. Whittlesey
permits a more complete and insightful economic analysis of the linkages among
agricultural water pricing, agricultural water use efficiency and agricultural
water conservation.
The authors focus their attention in this paper on short-term agricultural water
conservation policies (i.e. those limited to a single irrigation season). This is
consistent with recent agricultural water conservation policies formulated in the
Pacific north-west by the Bonneville Power Administration, the US Fish and
Wildlife Service and the US National Marine Fisheries Service to increase
in-stream flows for endangered aquatic species while protecting the long-term
health of the impacted region’s agricultural economy. The behavioural implication is that the representative producer responds to these policies by undertaking only short-term marginal adjustments to make an existing irrigation system
more efficient in meeting the crop’s water needs. The authors do not consider
the longer-term adjustment of switching to a more efficient irrigation technology.
Past Work
Past studies typically have focused on how agricultural water pricing schemes
can generate water conservation. The typical response of a representative
irrigator to a higher water price is limited to reducing the demand for delivered
water. One such analysis is exemplified in Figure 1, which formulates a marginal
cost-based tiered water-pricing scheme (Willey & Diamant, 1995). The first tier
assigns price P(1) ($/acre foot) (all prices in US dollars) to water deliveries up
to quantity T(1) (acre feet/acre). The first-tier price typically is set to ensure that
water delivery costs are recovered if farmers complete seasonal irrigation with
first-tier water only. The second-tier price is set somewhat higher at P(2) and
covers water deliveries between tiering levels T(1) and T(2). Finally, the thirdtier price P(3) covers water deliveries beyond T(2), and is designed to encourage
efficient water use by equating the marginal value product of irrigation water
with the marginal supply cost.
A traditional ‘flat rate’ scheme prices all water deliveries at the first-tier price
P(1). Each irrigator subject to the scheme demands a water delivery of Q(1). If
the marginal cost-based tiered system is invoked, each irrigator is charged the
tier three price P(3), and consequently cuts back demanded water deliveries to
the optimal level Q(3). The reduction in demanded applied water measured by
Q(1)–Q(3) is presumed to represent the amount of water conserved under the
marginal cost-based pricing system.
This common presumption is oversimplified and potentially inaccurate in
wide-ranging circumstances. The relationship between water pricing and water
conservation is complicated by the additional responses that the irrigator can
make to increased water costs beyond reducing the demand for applied water.
These responses might include investment in improved irrigation efficiency, and
adjustments in crop mix and acreage. The conceptual relationship is complicated
further by the hydrological circumstances governing the fate of irrigation water
that is unconsumed in crop production.
The authors formulate a conceptual model of a representative irrigated farm
that introduces some of the above complications. The authors apply it to
investigate the interplay of optimal farm-level responses to two economic
policies intended to produce agricultural water conservation (i.e. higher water
Policies Encouraging Agricultural Water Conservation
39
Figure 1. Marginal cost-based pricing. Source: Willey & Diamant (1995).
prices and irrigation efficiency subsidies), and to characterize the circumstances
for which these responses conserve water.
An Irrigated Farm Model
The authors examine first the technical issue of how irrigation water is converted
into crop yield, and assume briefly that the producer’s irrigation technology
is100% efficient. This means that every unit of applied water is utilized by the
crop to satisfy its demand for evapotranspiration (ET). (A crop’s ET demand
measures its capacity to transpire water, and consequently its consumptive use
of water.) The agronomic literature indicates that crop yield is linearly related to
ET, as depicted by the dashed curve Y ⫽ f(C) in Figure 2 (Doorenboos & Kassam,
1979; Bernardo & Whittlesey, 1989; Klocke et al., 1990). The initial yield, Y0, on
the curve is generated by ET from rain and stored soil moisture (‘field water
supply’). Subsequent yields increase linearly from Y0 to the ‘maximum attainable
yield’, Ym (determined by external factors including climate, prevalence of insect
pests and plant diseases and so on), in response to ET from irrigation. Beyond
Ym, the crop is incapable of further ET and the yield response is flat.
In practice, irrigation systems are less than 100% efficient in converting
irrigation water to ET and ultimately to yield (Doorenboos & Kassam, 1979;
Bernardo & Whittlesey, 1989; Klocke et al., 1990). The solid curvilinear function,
40
R. Huffaker & N. Whittlesey
Figure 2. Hypothetical crop yield response (Y) to applied and consumed water.
Source: Adapted from Bernardo & Whittlesey (1989, figure 1).
Y ⫽ f(A), in Figure 2 represents the yield response per acre to applied water
when the irrigation system is less than 100% efficient. The response exhibits
marginal productivities that diminish at a rate governed by the farm’s irrigation
efficiency. It approaches a maximum attainable yield, ym, that depends on both
climatic factors and the farm’s irrigation efficiency, and that rests below the
maximum attainable yield for a 100% efficient system, Ym.
The horizontal difference between the yield response to consumptive water
use, Y ⫽ f(C), and the yield response to applied water, Y ⫽ f(A), represents water
that is applied but not consumed by the crop (‘irrigation losses’). For example,
A1 is the water application required for the crop to achieve the crop’s maximum
ET demand under the given irrigation technology, ET1m, and irrigation losses are
measured by A1 ⫺ ET1m (Figure 2). Irrigation losses increase as yields approach
ym due to the diminishing marginal productivity of applied water.
An irrigator can reduce irrigation losses by increasing on-farm irrigation
efficiency. For example, irrigators may increase the on-farm efficiency of surface
irrigation by an estimated 20% by increasing the labour needed to monitor
run-off, reduce set time and cut back stream size (Bernardo & Whittlesey, 1989).
This increases the marginal productivities of applied water, and has the impact
in Figure 2 of increasing the slope of the yield response to applied water,
Y ⫽ f(A), for each level of applied water, A. It also may increase the maximum
attainable yield associated with the given irrigation technology, ym, but the
model assumes away this possibility to focus on other impacts. By improving
Policies Encouraging Agricultural Water Conservation
41
on-farm irrigation efficiency, the curvilinear applied water–yield curve, Y ⫽ f(A),
approaches the linear ET–yield curve, Y ⫽ f(C).
Model Specification
It is assumed that the producer’s objective is to maximize profits from selecting
optimal levels of water application and investment in on-farm irrigation
efficiency required to achieve the maximum attainable per acre yield, ym, of a
single crop farm-wide. Consequently, ym is a parameter in the specification, and
not a variable to be optimized. In this way, the authors’ conceptualization is
similar to a conventional micro-economic cost-minimization formulation (e.g. see
Varian, 1992). Yield ym provides a feasible target during normal years for farms
receiving water from Bureau of Reclamation irrigation projects (the largest
supplier of irrigation water in the western United States). Project capacity is
designed with prevailing irrigation technologies and the ET demands of area
crops in mind, and water is priced to ensure that the quantity demanded by
irrigators “is sufficient to meet most typical grower water needs” (Michelsen et
al., 1999, p. 232).
The authors abstract away from the possibility that the producer changes
crops in response to agricultural water conservation policies in order to focus on
other important responses in an analytically tractable model. The authors also do
not consider deficit irrigation strategies that a farm might employ in water-short
years in which less than the maximum attainable yield is produced farm-wide.
The mathematical formulation of the model is:
max (A, I, L, Z) ⫽ R(L, Z) ⫺ wA A ⫺ wl I ⫺ wL L ⫺ FC
A, I, L
(1)
subject to:
ymL ⫽ f[A, E(I), Z]
(2)
The factors that we assume can be adjusted within the course of an irrigation
season are applied water, A (acre feet), investment in improved on-farm irrigation efficiency, I ($) and farm acreage, L (acres). Policies employing increased
water costs to encourage conservation rely on applied water being an adjustable
factor within an irrigation season. On-farm irrigation efficiency can be adjusted
significantly within a single season by shifting the amount of labour dedicated
to supervising irrigation for a given technology (Bernardo & Whittlesey, 1989).
Consequently, farm labour enters the model implicitly as a means of adjusting
the farm’s investment in irrigation efficiency, I. Finally, farm acreage can be
adjusted within a single irrigation season, e.g. by seasonal leasing. Other factors
of production are assumed to be fixed within a single irrigation season (e.g. farm
equipment and irrigation conveyance facilities), and are represented by Z in
equations (1) and (2).
Model parameters in addition to ym are: wA (the unit cost of applied water,
$/acre foot); wI (the unit cost of investment in on-farm irrigation efficiency); wL
(the unit cost of land, $/acre); and FC (the total cost of the fixed factors, $). The
unit cost of investment is one (wI ⫽ 1) when each dollar of investment is paid
entirely by the farm, and some fraction of one (wI ⬍ 1) when the public subsidizes some portion of the cost.
The objective function (equation (1)) measures profits ($) as the difference
42
R. Huffaker & N. Whittlesey
between the revenue from crop production, R(L,Z), and the sum of the total
wages paid to variable inputs and the fixed costs. The crop revenue function is
specified as:
R(L, Z) ⫽ pym L ⫺ h(L, Z)
(3)
m
where p ($/ton) is the unit price received for the crop. The first term, py L,
measures the revenue that the farm receives from producing the maximum
attainable yield per acre, ym, over the entire farm acreage, L. The second term,
h(L,Z), nets out adjustment costs associated with increasing farm acreage while
keeping fixed factors Z constant. It is assumed to increase at an increasing rate
with additional acreage (i.e. hL ⬎ 0, hLL ⬎ 0, where subscripts denote derivatives
with respect to L). Thus, marginal revenue decreases:
RLL ⫽ ⫺ hLL ⬍ 0
(4)
The production constraint (equation (2)) requires the farm to select levels of
applied water (A) and on-farm irrigation efficiency E(I) that produce the maximum attainable yield per acre farm-wide, ymL. The farm-wide production
function, f[A,E(I),Z], is assumed to be restricted by positive but diminishing
marginal productivities in each input (consistent with the applied water–yield
response in Figure 2). The joint marginal product between the two inputs is
restricted to be positive (indicating that an incremental increase in irrigation
efficiency increases the marginal productivity of applied water, and vice versa):
fA ⬎ 0, fE ⬎ 0, fAA ⬍ 0, fEE ⬍ 0 and fAE ⫽ fEA ⬎ 0
(5)
Solution
The model specified in equations (1)–(5) poses a classical constrained programming problem whose solution is initiated with the Lagrangian function:
L(A,I,L,Z,) ⫽ R(L,Z) ⫺ wAA ⫺ wII ⫺ wLL ⫺ FC ⫹ {f [A,E(I),Z] ⫺ ymL}
(6)
Optimal choices of applied water (A*), investment in irrigation efficiency (I*) and
acreage (L*) must satisfy the following first-order necessary and sufficient
conditions:
⭸L
⫽ f [A, E(I), Z] ⫺ ym L ⫽ 0
⭸
(7)
⭸L
⫽ ⫺ wA ⫹ fA [A, E(I), Z] ⫽ 0
⭸A
(8)
⭸L
⫽ ⫺ wI ⫹ fE [A, E(I), Z] EI ⫽ 0
⭸I
(9)
⭸L
⫽ RL (L, Z) ⫺ wL ⫺ ym ⫽ 0
⭸L
(10)
Equation (7) requires the farm to select levels of applied water and on-farm
irrigation efficiency that produce the maximum attainable yield per acre farmwide for the existing irrigation technology. Manipulation of equations (8) and (9)
yields the familiar tangency condition wA/wI ⫽ fA/fI (where fI ⫽ fEEI), requiring
that the economic rate at which investment in improved irrigation efficiency can
Policies Encouraging Agricultural Water Conservation
43
be substituted for applied water in production while keeping costs constant
(wA/wI) be equated with the technical rate at which such substitution can occur
while keeping farm-wide yield constant at ymL (fA/fI). Equation (10) requires that
the optimal choice of acreage equates marginal revenue (RL) with marginal
factor cost (wL), and an additional term (ym) reflecting the marginal cost of
adjusting applied water and investment in irrigation efficiency to ensure that the
maximum attainable yield is produced on the marginal acre of land.
By the implicit function theorem, equations (7)–(10) can be solved for the
demand functions for applied water, A*(wA,wI,wL), investment in irrigation
efficiency, I*(wA,wI,wL), and acreage, L*(wA,wI,wL), if the determinant of the
bordered hessian matrix associated with L(A,I,L,Z,), H̄ , is non-singular when
evaluated at A*, I*, L* and *, i.e.:
 0
 fA
H̄ ⫽
 fI
 ⫺ ym
fA
*fAA
*fIA
0
fI
*fAI
*fII
0
⫺ ym
0
0
RLL

 ⫽0


(11)
and fI ⫽ fE EI ⬎ 0, fII ⫽ fEE E2I ⫹ fE EII ⬍ 0, fAI ⫽ fAE EI ⬎ 0 and fIA ⫽ fEA EI ⬎ 0.
The second order condition guaranteeing that A*, I*, L* and * constitute a
strict local maximum is that H̄ be negative definite when evaluated at (A*, I*, L*,
*). The conditions for negative definiteness are that the last two bordered
principal minors of 兩H̄兩 alternate in sign beginning with the positive:
冷
0
fA
fI
fA
*fAA
*fIA
fI
*fAI ;
*fII
|H̄| ⬍ 0
冷
(12)
(13)
Condition (12) can be shown to hold given the restrictions set out in equations
(4) and (5). This excludes the possibility that (A*, I*, L*, *) represents a
minimizing solution.
Comparative Statics
The authors determine the optimal responses of the representative farm to
policies intended to conserve agricultural water via comparative statics (e.g. see
Varian, 1992). To analyse farm response to a differential increase in the cost of
applied water (wA), the authors look at how the farm adjusts its demands for
applied water, investment in irrigation efficiency and acreage as reflected in the
partial differentials dA*/dwA, dI*/dwA and dL*/dwA. To analyse farm response
to subsidies for improved irrigation efficiency, the authors look at how the farm
adjusts factor demands to a differential decrease in its share of investment costs
(wI ⬍ 1) as reflected in the partial differentials dA*/dwI, dI*/dwI and dL*/dwI.
The authors investigate the conditions in which these responses produce the
intended water conservation in a later section.
Comparative statics commence by substituting the demand functions
A*(wA,wI,wL), I*(wA,wI,wL) and L*(wA,wI,wL) into the system of first-order condi-
44
R. Huffaker & N. Whittlesey
tions (equations (7)–(10), and taking the total differential of the resulting
identities:
 0
 fA
 fI
 ⫺ ym
fA
*fAA
*fIA
0
fI
*fAI
*fII
0
⫺ ym
0
0
RLL




d*   0 
dA*  dwA 
 dI*  ⫽  dwI 
dL*  dwL 
(14)
The authors rely on Cramer’s rule and the non-singularity of to solve for the
differentials in the factor demands as functions of differentials in the exogenous
factor costs:


dA*  
dI*  ⫽ 
dL*  



* (ym)2 |fII| ⫹ f 2I |RLL|
⬍0
|H̄|
* (ym)2 fAI ⫺ fA fI |RLL|
|H̄|
* ym (fA |fII| ⫹ fI fAI)
⬍0
|H̄|
* (ym)2 fIA ⫺ fA fI |RLL|
|H̄|
* (ym)2 |fAA| ⫹ f 2A |RLL|
⬍0
|H̄|
* ym (fA fIA ⫹ fI |fAA|)
⬍0
|H̄|
* ym (fA|fII| ⫹ fI fIA)
⬍0
|H̄|
* ym (fA fAI ⫹ fI |fAA|)
⬍0
|H̄|
* (2fA fI fIA ⫹ f 2A|fII| ⫹ f 2I |fAA|)
⬍0
|H̄|


 dwA 
 dwI 
 dwL 



(15)
where |fAA|, |fII| and |RLL| are absolute values.
The substitution matrix in equation (15) is symmetric with diagonal elements
representing the marginal responses in factor demands with respect to their own
unit costs, and off-diagonal elements representing the symmetric cross-cost
marginal responses. The signs of all responses except for one are determinate
due to the restrictions imposed on R(L,Z) in equation (4), f[A,E(I),Z] in equation
(5) and |H̄| in sufficient condition (13). The signs of the own-cost responses are
negative as expected, implying that the factor demands are downward sloping
in their own costs. The symmetric cross-cost responses between the demands for
land and applied water, dL*/dwA ⫽ dA*/dwL, and between the demands for
land and investment in irrigation efficiency, dL*/dwI ⫽ dI*/dwL, are also negative. This implies that land is a complementary factor in production with applied
water and investment in irrigation efficiency. The sign of the symmetric crosscost response between the demands for applied water and investment in
irrigation efficiency, dA*/dwI ⫽ dI*/dwA, is indeterminate. The authors discuss
the economic conditions governing the direction of this relationship below.
Farm Response to an Increase in the Cost of Applied Water
Farm response to an increase in the cost of applied water (dwA ⬎ 0) is governed
by the first column of partial differentials in the substitution matrix (equation
(15)). The farm decreases its demand for applied water (dA*/dwA ⬍ 0), and
consequently has less water than previously required to produce the maximum
attainable yield per acre over the acreage prevailing before the cost increase. To
satisfy the production constraint after the cost increase, the farm decreases its
demand for acreage (dL*/dwA ⬍ 0), and either increases or decreases the demand
for investment in irrigation efficiency depending upon the ambiguous sign of
dI*/dwA.
The economic conditions governing the direction of the adjustment in investment can be determined by taking the total differential of production constraint
(equation (2)) with respect to wA:
Policies Encouraging Agricultural Water Conservation
ym
45
dL*
dA*
dI*
dwA ⫽ fA
dwA ⫹ fI
dwA
dwA
dwA
dwA
(16)
Equation (16) requires that the differential shift in the production target due to
a decrease in acreage (left-hand side of the equation) equals the weighted sum
of the decrease in applied water and the ambiguous change in investment
(right-hand side). The weights, supplied by the marginal productivities of
applied water and investment, translate the marginal adjustments in applied
water and investment into the marginal adjustments in yield required to meet
the differential shift in the production target of obtaining maximum attainable
yields per acre farm-wide.
Solving equation (16) for the required adjustment in investment demand
yields:
(17)
冉 冏 冏 冏 冏冊 dw
dA*
dL*
where 冏
and 冏
冏
冏 are absolute values. After dividing through by dw , we
dw
dw
dI*
dA*
dL*
1
dwA ⫽ fA
⫺ ym
dwA
fI
dwA
dwA
A
A
A
A
can infer:
冏 冏
冏 冏
dI*
dA*
dL*
ⱖ ( ⱕ ) 0 as fA
ⱖ ( ⱕ ) ym
dwA
dwA
dwA
(18)
Equation (18) generates a number of implications regarding the direction of the
marginal adjustment in investment given a differential increase in the cost of
applied water. First, the farm does not adjust investment (dI*/dwA ⫽ 0) when the
weighted marginal decrease in applied water is equal in absolute value to the
weighted marginal decrease in acreage:
dA*
冉f 冏dw
冏 ⫽ y 冏dwdL* 冏冊
m
A
A
A
Acreage takes the full brunt of the profit-maximizing adjustments required to
reacquire the production target with marginally less applied water.
Secondly, the farm increases investment in irrigation efficiency (dI*/dwA ⬎ 0)
when the weighted marginal decrease in applied water is greater in absolute
value than the weighted marginal decrease in acreage:
dA*
冉f 冏dw
冏 ⬎ y 冏dwdL* 冏冊
m
A
A
A
Profit-maximization dictates that the burden of reacquiring the production target
with marginally less applied water be distributed between a marginal decrease
in acreage and a marginal increase in investment. Compared to the first case (i.e.
the farm does not adjust irrigation efficiency), it pays to reduce acreage relatively
less severely and allow increased efficiency to distribute the reduced applied
water supply over the increased acreage as needed to reacquire the production
target.
Finally, the farm reduces investment in irrigation efficiency (dI*/dwA ⬍ 0)
when the weighted marginal decrease in applied water is less in absolute value
than the weighted marginal decrease in acreage:
dA*
冉f 冏dw
冏 ⬍ y 冏dwdL* 冏冊
m
A
A
A
46
R. Huffaker & N. Whittlesey
Such a marginal reduction would not require necessarily that the farm revert to
less efficient technology (e.g. revert from sprinkler to flood irrigation), but might
be accomplished simply by less farm time spent supervising the irrigation
process (e.g. monitoring run-off, reducing set time and curtailing back stream
size (Bernardo & Whittlesey, 1989)). Compared to the first case, it pays to reduce
acreage relatively more severely so that the irrigation efficiency existing before
the increase in the cost of applied water is no longer required to satisfy the
production target.
Farm Response to Subsidies for Improved Irrigation Efficiency
The authors investigate farm response to efficiency improvement subsidies by
analysing the comparative statics of a differential decrease in the farm’s share of
investment costs (dwI ⬍ 0). The results are governed by the second column of
partial differentials in the substitution matrix (equation (15)). Given the inverse
relationships between the demand for investment and investment cost (dI*/
dwI ⬍ 0), and between the demand for acreage and investment cost (dL*/
dwI ⬍ 0), the farm increases its demands for investment and acreage in response
to a differential decrease in wI. Whether these adjustments require an accompanying increase or decrease in the demand for applied water to satisfy the
production target in constraint (2) after the cost decrease depends upon the
ambiguous sign of dA*/dwI.
The economic conditions governing the adjustment in the demand for applied
water given a differential decrease in the investment cost can be determined by
taking the total differential of production constraint (2) with respect to wI:
ym
dL*
dA*
dI*
dwI ⫽ fA
dwI ⫹ fI
dwI
dwI
dwI
dwI
(19)
Solving for the adjustment in applied water required to satisfy production
constraint (2) yields:
冉冏 冏
冏 冏冊 dw
dA*
1
dI*
dL*
dwI ⫽
fI
⫺ ym
dwI
fA
dwI
dwI
I
(20)
After dividing through by dwI, we can infer that:
冏 冏
冏 冏
dA*
dI*
dL*
ⱖ ( ⱕ ) 0 as fI
ⱖ ( ⱕ ) ym
dwI
dwI
dwI
(21)
Equation (21) generates a number of implications regarding the required marginal adjustment in applied water in response to a differential decrease in the
farm’s share of the cost of investment. First, the farm does not adjust the demand
for applied water (dA*/dwI ⫽ 0) when the weighted marginal increase in investment is equal to the marginal weighted increase in acreage:
dI*
dL*
fI
⫽ ym
dwI
dwI
冉冏 冏
冏 冏冊
The marginal increase in the demand for investment in irrigation efficiency
allows the farm to meet the differential shift in the production target due to an
increase in acreage (left-hand side of equation (19)) with the status quo level of
applied water.
Secondly, the farm increases the demand for applied water given a differential
Policies Encouraging Agricultural Water Conservation
47
decrease in the cost of investment (dA*/dwI ⬍ 0) when the weighted marginal
increase in investment is less than the weighted marginal increase in acreage:
冉f 冏dwdI* 冏 ⬍ y 冏dL*
冏冊
dw
m
I
I
I
Given the relatively lower increase in irrigation efficiency compared to the first
case, the farm applies more water less efficiently to reacquire the production
target.
Finally, the farm decreases the demand for applied water given a differential
decrease in the cost of investment (dA*/dwI ⬎ 0) when the weighted marginal
increase in investment is greater than the marginal increase in acreage:
冉f 冏dwdI* 冏 ⬎ y 冏dL*
冏冊
dw
m
I
I
I
Given the relatively larger increase in irrigation efficiency compared to the first
case, the farm is able to reacquire the production target by applying a smaller
quantity of water more efficiently.
Farm Response to an Increase in the Cost of Land
Farm response to an increase in the cost of land (dwL ⬎ 0) is governed by the
third column of partial differentials in the substitution matrix (equation (15)).
The farm decreases its demand for acreage (dL*/dwL ⬍ 0) as expected; and
consequently does not have to produce as much farm-wide to obtain the
maximum yield per acre. The farm responds by reducing the demand for
applied water (dA*/dwL ⬍ 0) and for investment in irrigation efficiency (dI*/
dwL ⬍ 0).
The farm responses discussed above are summarized in Table 1.
Agricultural Water Conservation
Under what circumstances will higher applied water costs and subsidies to
improve irrigation efficiency conserve agricultural water in the representative
irrigated farm model? Agriculture water conservation can be measured in two
ways depending upon the hydrology of the river basin providing the irrigation
water. One possible measure is the reduction in applied water, and another is
the reduction in consumptive water use. First, the authors consider the hydrological circumstances under which each measure is relevant. Subsequently, they
discuss how higher applied water costs and subsidies to improve irrigation
efficiency affect these measures.
Basin Hydrology
A reduction in applied water measures conserved water when the portion of
applied water unconsumed by crops (irrigation tail water) is irretrievably lost to
the river basin, e.g. by surface evaporation (Huffaker & Whittlesey, 1995).
Figures 3(a) and 3(b) provide a simple illustration. Figure 3(a) represents the
status quo, in which the flow of water into the basin at the beginning of a single
irrigation season is 10 units, and the single farm in the basin has legal rights to
divert 8 of those units into irrigation. The farm operates a flood irrigation system
48
R. Huffaker & N. Whittlesey
Table 1. Representative farm responses to economic water conservation policies
Factor
Marginal
response
Increase in the cost of applied water
Applied water (A)
Reduction
Condition
Impact on water supply
Response
unambiguous
Full reduction in A
conserved in absence of
return flow
Acreage (L)
Reduction
Response
unambiguous
Full reduction in
consumptive use
conserved in presence of
return flow
Investment in irrigation
efficiency (I)
Increase
No change
Reduction
WMI in A ⬎ WMI in L
WMI in A ⫽ WMI in L
WMI in A ⬍ WMI in L
I adjusts to provide for
reduced level of
consumptive use given
reduction in applied water
WMI in I ⬍ WMI in L
Water use expanded by
full increase in A in
absence of return flow
No impact
Full reduction in A
conserved in absence of
return flow
Efficiency improvement subsidy
Applied water (A)
Increase
No change
Reduction
WMI in I ⫽ WMI in L
WMI in I ⬎ WMI in L
Acreage (L)
Increase
Response
unambiguous
Water use expanded by
full increase in
consumptive water use in
the presence of return flow
Investment in irrigation
efficiency (I)
Increase
Response
unambiguous
I increases to provide for
increased level of
consumptive use given
adjustments in A
WMI, Weighted marginal increase.
with an on-farm irrigation efficiency (E) of 25%, so that the 8 units of applied
water (A) produce 2 units of consumptive use (C) by crops (i.e.
C ⫽ (E)(A) ⫽ 0.25 ⫻ 8 ⫽ 2 units). The difference between applied water and consumptive use (irrigation tail water) is 6 units (A–C ⫽ 8–2 ⫽ 6 units), which is
assumed to be irretrievably lost to the basin. The flow of water below the farm’s
point of diversion is 2 units—the difference between basin inflow (10 units) and
the farm’s diversion (8 units). Figure 3(b) demonstrates the impact of a reduction
in applied water from 8 to 4 units with irrigation efficiency held constant. This
increases the flow below the farm’s point of diversion from 2 to 6 units. In sum,
basin water supply is conserved in an amount equal to the 4 unit reduction in
applied water when all irrigation tail water is irretrievably lost to the river basin.
Alternatively, a reduction in applied water does not measure conserved water
when irrigation tail water returns to the river basin to supply a portion of the
water supply (Huffaker & Whittlesey, 1995). Irrigation return flows are an
essential component of water systems in many basins. For example, one study
estimated that 63% of the water diverted from the Snake River in the western
United States for irrigation eventually returns to the river (Hydrosphere
Policies Encouraging Agricultural Water Conservation
49
Figure 3. Escape flow hydrology.
Resource Consultants, 1991). Another study estimated that irrigation return
flows constitute 47% of all water diverted for irrigation in the western United
States (Pulver, 1988).
Figures 4(a) and 4(b) illustrate this point. Figure 4(a) depicts a status quo
identical to that in Figure 3(a), except that irrigation tail water is assumed to
return to river. The flow below the point of return flow (8 units) is equal to the
sum of the difference between basin inflow and the farm’s diversion (10–8 ⫽ 2
units) and the return flow (6 units). Figure 4(b) shows the impact of a reduction
in applied water from 8 to 4 units with irrigation efficiency held constant. This
increases flow below the farm’s point of diversion from 2 to 6 units. Of the 4
units of water diverted, crops consume 1 unit, which results in a return flow of
3 units. Consequently, the flow below the point of return flow increases from 8
to 9 units. The unit increase in basin flow is due to the unit reduction in
50
R. Huffaker & N. Whittlesey
Figure 4. Return flow hydrology.
consumptive use—not the 4 unit reduction in applied water. In sum, water is
conserved in an amount equal to the reduction in the farm’s consumptive use of
water when all irrigation tail water returns to augment basin flow. In intermediate cases in which some portion of irrigation tail water is irretrievably lost and
some portion returns to the river basin, water conservation requires reductions
in either applied water irretrievably lost or consumptive water use, or both.
The authors now consider the impacts that higher applied water costs and
irrigation technology subsidies have on applied water and consumptive use.
These impacts and their implication for water conservation are summarized in
Table 1.
The Impact of an Increase in the Cost of Applied Water
The representative farm reduces its demand for applied water in response to an
increase in the cost of applied water (Table 1). The full reduction measures
Policies Encouraging Agricultural Water Conservation
51
conserved water in the absence of irrigation return flows, but overestimates
conserved water when some portion of irrigation tail water returns to the river.
The farm also reduces its demand for acreage. This has the impact of reducing
consumptive water use. To show this, the authors refer back to the linear curve
measuring the consumptive water use required by the crop to produce output
Y, i.e. Y ⫽ f(C) (Figure 2). This curve can be represented algebraically as:
Y(C) ⫽ C ⫹ (ym ⫺ C)
(22)
where is an indicator function such that:
Y(C) ⫽
再Cy
m
as ⫽ 0
as ⫽ 1
冎
(23)
and is a parameter measuring the marginal response of yield to consumptive
water use per acre. Focusing on the portion of Y(C) for which yield responds to
adjustments in consumptive water use ( ⫽ 0), we can solve for the consumptive
use (cm) required to produce the maximum attainable yield per acre (ym):
1
cm ⫽ ym
(24)
and the total consumptive use (Cm) required to produce the farm-wide target of
the maximum attainable yield on each acre (ymL):
1
Cm ⫽ cm L ⫽ ym L
(25)
The total differential of equation (25) measures how consumptive use farm-wide
responds to differential changes in the factors of production:
1
dCm ⫽ cm dL ⫽ ym dL
(26)
We see from equation (26) that farm-wide consumptive use (dCm) adjusts
directly to a differential shift in acreage (dL). As a result, the reduction in
acreage that the farm makes in response to an increase in the price of applied
water reduces consumptive water use. The full reduction in consumptive water
use measures conserved water in the presence of return flow, but inaccurately
measures conservation when some portion of irrigation tail water is irretrievably
lost.
Finally, the farm adjusts irrigation efficiency in the direction needed to
reacquire the production target after the increase in the cost of applied water.
The Impact of Subsidies for Improved Irrigation Efficiency
The representative farm increases its demand for acreage in response to a
decrease in its share of the cost of investing in improved on-farm irrigation
efficiency (Table 1). This increases the farm’s demand for consumptive water use
by equation (26). In the presence of irrigation return flows, water use is
expanded by the full increase in consumptive use. The subsidy policy has the
exact opposite impact of the intended conservation. In the absence of return
flows, the increase in consumptive use has no impact on basin water supply.
The farm adjusts its demand for applied water in the direction dictated by the
relative marginal adjustments required in acreage and irrigation efficiency to
52
R. Huffaker & N. Whittlesey
satisfy production constraint (2). The demand for applied water increases when
the relative marginal adjustment in land exceeds that in irrigation efficiency. In
the absence of irrigation return flows, water use is expanded by the full increase
in applied water. Once again, the subsidy policy has the exact opposite impact
of the intended conservation.
The demand for applied water decreases when the relative marginal adjustment in irrigation efficiency exceeds that in acreage. In the absence of irrigation
return flows, the full reduction in applied water measures conserved water. In
the presence of return flows, the reduction in applied water overestimates
conserved water.
Finally, the demand for applied water remains unchanged when the relative
adjustments between acreage and irrigation efficiency are equal. The irrigation
technology subsidy policy has no impact on basin water supply.
Summary
The spectre of increasing world-wide water scarcity has generated interest in
economic policies encouraging the conservation of water in agriculture. These
policies include assessing irrigators’ higher applied water costs, and establishing
subsidy programmes to encourage irrigators to improve the efficiency of onfarm irrigation technologies. Past work has focused on the impact of higher
applied water costs on agricultural water conservation, but has not accounted
for a wide range of economic and hydrological factors that may have an impact
on conservation. The authors formulate a conceptual model of a representative
irrigated farm to account for a wider range of farm responses to both increased
applied water costs and efficiency improvement subsidies, and consequently to
provide for a more complete economic analysis of the circumstances in which
these economic policies produce the desired agricultural water conservation.
The authors’ representative irrigated farm selects levels of applied water,
investment in irrigation efficiency and farm acreage to maximize the profits from
producing a single crop over one irrigation season. Feasible factor levels must
generate the agronomic maximum attainable yield per acre farm-wide. The
method of comparative statics is applied to study farm responses to higher
applied water costs (by differentially increasing the cost of applied water in the
model), and to subsidies for improved irrigation on-farm efficiency (by differentially decreasing the farm’s share of the cost of investment in the model). The
authors then investigate the impact that these responses have on two measures
of water conservation that are relevant in different hydrological circumstances:
namely, applied water use (relevant in the absence of irrigation return flows);
and consumptive water use (relevant in the presence of irrigation return flows).
The representative farm responds to increased applied water costs by reducing both applied water use and farm acreage. The reduction in farm acreage
leads directly to a reduction in consumptive water use farm-wide. The reduction
in applied water use is an accurate measure of conserved water in the absence
of irrigation return flows, but overestimates conservation in the presence of
return flows. The reduction in consumptive water use accurately measures water
savings in the presence of return flows, but inaccurately measures conservation
in the absence of return flows. The farm adjusts its investment in irrigation
efficiency to produce the maximum attainable yield farm-wide with reduced
levels of applied water use and acreage.
Policies Encouraging Agricultural Water Conservation
53
The representative farm responds to a decrease in its share of the cost of
investing in irrigation efficiency by increasing irrigation efficiency and acreage.
The increase in farm acreage increases consumptive water use farm-wide. Water
use is expanded by the full increase in consumptive use in the presence of
irrigation return flows, and is unaffected by the increase in the absence of return
flows. The impact of the subsidy policy on applied water use is ambiguous and
depends on the relative marginal adjustments made to acreage and investment
in irrigation efficiency. When the marginal adjustment to acreage is large relative
to that in investment, the farm must increase applied water use to satisfy the
maximum-attainable-yield production constraint farm-wide. Water use is expanded by the full increase in applied water use in the absence of return flows.
Alternatively, when the marginal adjustment to investment is greater than that
to acreage, the farm must reduce applied water use to satisfy the production
constraint. The full reduction in applied water use measures conserved water in
the absence of return flows, but overestimates conservation in the presence of
return flows.
These results indicate that increasing an irrigator’s cost of applied water may
be a more efficacious water conservation policy than subsidizing the irrigator’s
cost of investing in improved irrigation efficiency. Increasing the cost of applied
water conserves irrigation water in both the presence and absence of irrigation
return flows. Alternatively, subsidizing investment costs always expands water
use in the presence of return flows, and may potentially expand water use in the
absence of return flows. In these cases, a subsidization policy backfires and
accomplishes the exact opposite of its conservation purpose.
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