Modeling Conjunctive Water Use as a Reciprocal Externality
R. Garth Taylor, Robert D. Schmidt, Leroy Stodick, and Bryce A. Contor
Corresponding author: Taylor, Department of Agricultural Economics, University of Idaho. Moscow ID
83843 [email protected] 208-885-7533. Stodick, Contor, and Schmidt; Idaho Water Resources
Research Institute, University of Idaho.
Technical and financial support provided by the US Bureau of Reclamation Pacific Northwest Regional
Office, Snake River Area Office, and Science and Technology Program. Funding for Taylor was
provided, in part, by the Idaho Agricultural Experiment Station and the USDA-NIFA.
Modeling Conjunctive Water Use as a Reciprocal Externality
Conjunctive hydrology gives rise to reciprocal externalities between hydraulically connected
ground and surface water. A partial equilibrium model was formulated for an n-node
groundwater surface water conjunctive use in which economic equilibrium was defined by a
system of complementary slackness equations. Using Mixed Complementary Programming, a
three node example was parameterized using site specific data and functional forms applied to
the general model. This example was solved for equilibrium water prices and quantities in the
respective surface and ground water markets. The model provided a functional framework for
cost/benefit analysis of three policy scenarios in the presence of reciprocal conjunctive use
externalities: (1) Pigouvian tax/subsidy, (2) elimination of the externality through conservation
infrastructure, and (3) aquifer recharge. The Pigouvian policy yielded the highest social welfare
followed by the payment for aquifer recharge. Conservation that eliminated the externality
decreased social welfare.
Key words: conjunctive use, reciprocal externalities, complementary slackness conditions, partial
equilibrium, cost benefit analysis, water conservation, Pigouvian tax, Pigouvian subsidy, water
policy, aquifer recharge.
JEL codes: C61, Q25, Q28, H23
2
Introduction
Exploding municipal, energy, agricultural and environmental water demands are
colliding with limited or diminishing supplies. Policies and projects that would increase water
supplies through cost prohibitive or environmentally unacceptable new infrastructure are being
replaced by those that would reduce demand and/or redistribute supplies. Agriculture, being the
least valued and largest water user (over 80% of water diversions in the West, Hutson et. al
2004) is the principal target of the imperative to allocate and use water more efficiently.
The conventional water planning approaches of supply or demand management fail to
adequately evaluate efficient water project or allocation. Supply management first forecasts a
perfectly inelastic water requirement then calculates the prices to recover costs (Howitt and
Lund, 1999). Demand management ignores water supply elasticity and focuses on reducing
water demand through regulation, conservation, or infrastructure. In rejecting the elasticity of the
opposing supply or demand blade of Marshall’s scissors, both approaches disregard the value of
water. Furthermore, conventional analyses are too often conducted piece-meal, for a specific
water project, irrigation district or municipality, detached from the basin-wide hydrology and
regional economic structure.
From a hydrologic perspective, groundwater and surface water form an integrated
hydrologic system. To manage surface and subsurface water resources efficiently requires
conjunctive water accounting and modeling tools capable of simulating the interactions between
surface water storage or conveyances (rivers, canals, and drains) and groundwater recharge from
those sources. Aquifers created or sustained by irrigation activities extend over vast areas in
every western state. In the aquifers of California’s Central Valley, virtually all recharge is from
“infiltration of irrigation water” (Alley et al. 2002). Aquifers in Idaho’s Eastern Snake River
Plain (Cosgrove, Contor, and Johnson, 2006), and Lower Boise Valley, Nebraska’s North Platte,
Colorado’s South Platte (Howe, 2002) and Washington’s Columbia River Basin, are also
sustained largely by seepage from canals, off-stream reservoirs, or on-farm infiltration. Shakir et
al. (2011) raised alarm concerning the impact of canal water shortages on the conjunctive
groundwater in the Punjab. Schmidt et al. (2013) estimated that about 45% of Boise Project
diversions in the Lower Boise River Basin are consumptively used by canal irrigated crops, with
the remainder either recharging the shallow aquifer or discharging to drains.
3
Externalities occur when the economic activities of one entity affect those of another and
pecuniary remuneration is wanting (Mishan 1971; Baumol and Oates, 1988). Meade (1952)
illustrated a positive reciprocal production externality using the classic example of bees
providing nectar for honey and pollination for apples. In contrast to Meade’s positive/positive
reciprocal externality, conjunctive (joined or connected) surface water and groundwater use is
capable of producing a positive/negative reciprocal externality. Groundwater pumpers near a
canal enjoy a positive externality of canal seepage while inflicting a negative externality of
pumping-induced seepage upon the canal user. The wells which extract tepid water from these
hydrologically connected canals have been aptly labeled “warm water wells” (Strauch 2009).
Water management decisions are often made without regard to exiting surface/subsurface
hydrologic connections or the benefits/costs to other water users in the basin (Booker et al.,
2012; Evans, 2010). In many Western states, groundwater and surface water conjunctive
property rights are poorly defined or non-existent and lack compensation mechanisms to sustain
or curtail conjunctive use, which promulgates externalities (e.g. Strawn, 2004; Evans 2010;
Blomquist et. al. 2001). The resulting conjunctive use externalities cause a divergence between
private and social benefit/cost, with price/market institutions failing to sustain desirable activities
or curtail undesirable activities (Bator, 1958). Omission of conjunctive use externalities also
compromises basin-wide cost/benefit analysis (CBA) as required for Federal water project
planning (U.S. Water Resources Council 1983; Council on Environmental Quality 2009).
The overarching objective of our study is to demonstrate a generalized coupled
hydrologic and economic modeling methodology for conducting basin-wide CBA that
recognizes conjunctive use externalities. The tasks to achieve that goal are threefold: (1)
demonstrate a functional model of conjunctive surface and ground water hydrologic interactions
as conjunctive use externalities, (2) formulate a spatial partial equilibrium model in which
economic equilibrium is defined by a system of complementary slackness equations, and (3)
perform cost/benefit analysis (CBA) of alternative water policies aimed at addressing the market
failures and inefficient water use resulting from conjunctive use externalities.
Methods
4
Within the taxonomy of hydro-economic models, conjunctive use of surface water and
groundwater is a distinct category of research (Harou et al. 2009). Burt’s (1964) foundational
research treated conjunctive groundwater and surface water use as substitutes in an inventory
problem with uncertain supply and controlled demand. The linear programming methods of
optimal recharge and conjunctive groundwater and surface water use (Milligan and Clyde 1970;
O’Mara and Duloy 1984) have been eclipsed by optimal control methods. Noel et al. (1980) used
an optimal control model to determine tax on pumping for the socially optimal spatial and
temporal allocation of groundwater and surface water. Tsur (1990, 1991), and Tsur and GrahamTomasi (1991) extended Burt’s optimal control framework with uncertain surface water supplies.
Azaiez (2002) developed a multi-staged decision model to optimize conjunctive use of
groundwater and surface water with artificial recharge. Burness and Martin (1988) determined an
efficient pumping rate to recharge a tributary aquifer to a steady-state aquifer condition from a
perennially losing and hydraulically connected stream. Pongkijvorasin and Roumasset (2007)
expanded upon the static framework of Chakravorty and Umetsu (2005) with an optimal control
model of canal water use and groundwater extraction. Groundwater users were recipients of two
positive externalities, canal conveyance loss and on-farm return flows to the aquifer. In
summary, conjunctive use research has largely focused on the optimal groundwater and surface
water use and optimal aquifer recharge. Despite the explicit label of conjunctive use as an
externality, the hydrologic connection has not been treated as an externality created by
interrelated supply functions.
Two approaches have been taken to the integration of hydrologic with economic models;
network simulation and optimization (Harou et al. 2009). Beginning with Maass et al. (1962),
hydrologic systems have been modeled as networks of water storage, with demand and supply
nodes linked by the conveyance structures of rivers, canals, and pipelines. The network
simulation approach conjoins a hydrologic network model e.g. a river-reservoir prior
appropriations water allocation model (MODSIM, Labadie et. al 1994) with a system-wide
agricultural optimization model (Hamilton et al., 1999 and Houck et al. 2008). While network
simulation models have the necessary basin spatial dimension, these models are limited to
addressing “what if” questions and cannot maximize social welfare, necessary to address
market failures of conjunctive use externalities and conduct CBA of water projects and policies.
5
The partial equilibrium optimization model was introduced in the water literature by
Flinn and Guise (1970), who adopted the concept of interregional trade modeling as developed
by Takayama and Judge (1964). Howe and Easter (1971) used the trading model concept to
evaluate large-scale inter-basin water transfers. Cummings (1974) introduced water benefit and
supply cost functions within a linear programming context representing nonlinear benefit
measures by piecewise linearization. Vaux and Howitt (1984) formulated the first regional
water-trading model linking water demand and supply sectors to demonstrate the costeffectiveness of reallocation of water from existing uses as opposed to development of new
supplies. Booker and Young (1994) elaborated upon the water-trading framework, to contrast
benefits of intrastate versus interstate water transfers with salinity and hydroelectric demand also
determining the benefits of water transfers. Further extensions used sequential and dynamic
aspects to analyze and climate or drought on interregional water allocation (Booker, 1995;
Booker and Ward, 1999).
Partial Equilibrium Model
Following Takayama and Judge (1964), partial equilibrium models have been cast as an
optimization problem, where a quasi-welfare or net social function is maximized subject to
constraints, and equilibrium is assumed to be the optimal point that maximizes this objective
function. When Takayama and Judge published their book in the early 1970’s (Takayama and
Judge, 1971) numerical optimization techniques were well understood but mixed complementary
programming (MCP) was in its infancy. With the advent of GAMS (Brooke et al., 1988) and
accompanying solvers, the equilibrium equations can be formulated as complementary slackness
equations in a mixed complementary problem and solved directly. The alternative is to formulate
an objective function optimization problem and assume that the Karush-Kuhn-Tucker (KKT)
conditions coincide with the equilibrium conditions (see Kjeldsen 2000).
The relationship between non-linear optimization used by Takayama and Judge and the
MCP approach is demonstrated by the KKT theorem. Given a function, F : R n  R n , the
nonlinear complementary problem (NCP) finds z  R n such that z  0 , F ( z )  0 , and
zi Fi  0i where zi is termed complementary to Fi . These conditions are written as,
0  z  F ( z )  0 and equations of this form, where the  operand denotes complementary
6
slackness, are termed complementary slackness equations (Ferris and Munson, 1999). The KKT
theorem demonstrates that a constrained optimization problem, in certain circumstances, can be
transformed into a nonlinear complementary problem. The KKT conditions (a set of
complementary slackness equations) are the first order NCP conditions for the constrained
optimization problem which can be solved using MCP. The Takayama and Judge restriction of
equilibrium using linear supply and demand functions can be generalized to convex function.
A partial equilibrium model is defined for a single commodity (or set of substitutable
commodities) exchanged between nodes where I = set of supply nodes, J = set of demand nodes
and IJ  I  J = set of allowable arcs between nodes. A supplier produces a commodity at node
i and ships a quantity X i , j to node j along arc i, j . A supplier at node i can ship to the same
node i, thus I  J is not necessarily empty. The exogenous variables are the demand functions
i
Pj ( q )  j  J , supply or marginal cost functions P (q) i  I and transportation cost function
i
ti , j (i, j )  IJ . The endogenous variables are: q j = quantities demanded  j  J , q = quantities
supplied i  I , X i , j = quantities shipped from node i to node j (i , j )  IJ ,  j = market
i
demand price  j  J and  = market supply price i  I . Per the notation of Takayama and
Judge, a variable superscript is a node index, not an exponent.
Absent non-convexities and assuming the equations have a unique solution; five sets of
equations define economic equilibrium (Takayama and Judge, definition 2.4.9).
(1a) qi  i  Pi (qi )i  J
i
i
i
i
(2a) q  P (q )   i  I
i
(3a) xi , j  ti , j   j   i, j  IJ
(4a)  i   x j , i  q i  i  J
i
i
i
(5a)   q  xi , j i  I
j
where the overscore operand indicates the equilibrium value and the  operand denotes
complementary slackness. Thus, x  f ( y ) means x  0 , f ( y )  0 and x f ( y)  0 . Equations 1a
state that, at equilibrium, if the quantity demanded is greater than zero, demand price must equal
marginal benefit. Equations 2a state that, at equilibrium, if the quantity supplied is greater than
7
zero, supply price must equal marginal cost. Together, equations 1a and 2a ensure the existence
of a Pareto optimum at equilibrium. If the quantity traded is greater than zero, Equation 3a
ensures locational price equilibrium, by equating the sum of supply price and transportation cost
with demand price. Equations 4a state that, at equilibrium, if the demand price is greater than
zero, quantity demanded must equal the sum of all shipments to that node from all supply nodes.
Equations 5a state that, at equilibrium, if the supply price is greater than zero, all shipments from
a supply node must equal the total quantity produced at that node. Equations 4a and 5a ensure
market quantity equilibrium.
Partial Equilibrium with Conjunctive Use Externalities
The conjunctive hydrologic connection creates a Meade production externality where
supply functions of groundwater and canal water users become interrelated. The conjunctive use
externality is one-way or reciprocal depending on the surface water and groundwater connection,
and positive (e.g. aquifer recharge) or negative (e.g. reduced canal flows) depending on the
recipient’s production function. When a leaky canal is not hydraulically connected to the aquifer,
the canal water user gifts the groundwater pumper with a one-way positive externality of passive
seepage (i.e. independent of pumping). The canal user’s water demand is thus an argument in the
supply function of the groundwater pumper but the pumper’s demand is not an argument in the
canal supply function. When a leaky canal is hydraulically connected to the aquifer and a
hydraulic gradient exists as a result of pumping, the groundwater pumper reciprocates by
inflicting a negative externality of pumping-induced seepage on the canal user. With a reciprocal
externality, passive and pumping-induced canal seepage enter into the supply function of the
groundwater pumper as a positive externality, and the pumper inflicts a negative externality of
induced seepage upon the canal water user as an additional transportation cost. Production
externalities necessitate the specification of interrelated supply equations of the equilibrium
conditions 2a, 3a, and 4a of the basic model. Equations 1a and 5a, which do not contain supply
expressions, are thus unchanged from the basic model. In contrast to 2a, where price equals only
pecuniary marginal costs ( Pi (qi )) , at equilibrium price equals pecuniary and externality
marginal cost ( P k (q k ,{ X i , j }) :
8
(2b) q k  P k (q k ,{ X i , j })   k k  I
With the reciprocal externality, the locational price equilibrium equations (3a) must account for
the marginal cost of seepage in equating supply price at node i with the demand price at node j
(i.e. the marginal cost of seepage at the head of the canal with demand price at the end of the
canal  i X i , j  ti , j X i , j   j ( X i , j   S{k },i , j ) . The transportation cost is the seepage along the arc of a
{k }
leaky canal: S{k }, i , j ( X i , j ,{q k }) where X i , j is the diversion at the head of the canal and qk {k}  I
is the total pumping rate of pumpers that induce canal seepage. Taking the derivative of both
sides of the equation with respect to X i , j results in the price linkage equation;
ti , j   j   i   j 
{k }
S{k},i, j
X i, j
 0 , so that 3a becomes:
i
(3b) X i , j  ti , j   j     j 
S{ k }, i , j ( X i , j ,{ q k })
 X i, j
{k }
i, j  IJ
Equation 3b thus states, that at equilibrium, if the quantity traded is greater than zero, the sum of
supply price and transportation cost (i.e. seepage cost) must equal demand price. If the demand
price at node j is greater than 0, then at equilibrium, the quantity demanded must equal total
quantity transported less seepage:
X
j
j ,i
 S{k }, j , i ( X j , i ,{q k })  qi i  J . With the reciprocal
j {k }
externality, the excess demand for canal water is eliminated by:
(4b)  i   X j , i  S{k }, j , i ( X j , i ,{q k })  qi i  J
i
j {k }
A reciprocal conjunctive use externality is illustrated with an example of a leaky canal
that recharges an aquifer used by a groundwater pumper (Figure 1). Three nodes define three
respective markets. Node 1 is a canal company at the head of the canal with an exogenous
marginal cost; P1 (q1 ) . Node 1 supplies water through a leaky canal to Node 2 (a farm at the
canal terminus) that has an exogenous demand; P2 (q2 ) . Node 2 lacks an exogenous supply
function but faces an effective supply of canal water from Node 1 minus the canal seepage. The
conveyance arc between Node 1 and 2 is the leaky canal. The exogenous seepage function S{3},1,2 ,
analogous to transportation shrinkage of canal water conveyed from the canal head (Node 1) to
9
canal end (Node 2), determines recharge of the aquifer used by Node 3. The farm at Node 3 is
supplied by an aquifer that is in part recharged by canal seepage and has a demand for pumped
water; P3 (q3 ) . Node2 sole water source is the canal. Node 3 water source is seepage from the
canal (both passive and induced) and aquifer recharge from sources other than the canal. The
marginal cost function for groundwater at Node 3 equals the pecuniary marginal costs ( P3 (q3 ))
and the externality of seepage gifted by Node 2, S{3},1,2 . The indices for the example are thus;
i  {1,3}, j  {2,3} and ij  {(1, 2), (3,3)} .
Economic equilibrium is defined by the system of complementary slackness equations
(2b, 3b and 4b). The endogenous equilibrium prices 1 , 2 , and 3 and quantities q1 , q2 , and q3
, can be determined by solving the complementary slackness equations using MCP or by
maximizing a social welfare objective function (e.g. Booker and Young 1994; Booker and Ward,
1999). In our example, complementary slackness conditions were programmed in GAMS and
solved using the PATH solver (Ferris and Munson 1999). Equilibrium prices and quantities are
then limits on the respective consumer and producer surplus integrals.
Exogenous Functions
The two exogenous demand functions (Node 2 demand at the end of the canal and Node
3 demand at the well head); two exogenous marginal cost functions (Node 1 supply at the head
of the canal and Node 3 supply of groundwater at the well head), and an exogenous
transportation cost of canal seepage are outlined below and detailed with data in the Appendix.
Typically, hydro-economic models represent hydrology through constraints (Booker et
al., 2012). In contrast, the conjunctive hydrologic connection of the seepage function is an
argument in both the groundwater and canal water supply functions and thus the seepage
function defines the reciprocal externality. The aquifer, from which Node 3 pumps, is recharged
by two sources, canal seepage and so-called far-field sources at the boundaries of the hydrologic
model. Seepage is a convex function of the amount of water conveyed in the canal from Node 1
to Node 2 (X(12)) and the amount of pumping by Node 3 (q3 ):

S{3},1,2  a0 1  e
 a 1 X 1,2
  a X 1  e  .
2
1,2
 a3q3
10
Total seepage ( S{3},1,2 ) is the sum of: (1) passive seepage (the first-term), a function only of
canal diversion (X(1,2)), (a0 is the maximum canal seepage possible), and (2) induced seepage (the
second term), a function of canal diversion (X(12)) and pumping (q3). Marginal seepage with
respect to canal diversion is:
MS X (1,2 ) 
S{3},1, 2
X 1,2
 0 and
 2 S{3},1,2
 2 X 1,2
 0.
MS X (1,2) is thus downward sloping. As more water is diverted into the canal seepage increases
and the water table beneath the canal rises. As the water table connects with the canal over an
increasing length of canal, the head gradient between the canal and aquifer is reduced, so that as
X 1,2 increases MS X (1,2) decreases.
Marginal seepage with respect to pumping is:
MSq3 
S{3},1,2
q3
 0 and
 2 S{3},1, 2
 2 q3
0.
MSq3 is thus downward sloping -- with increased pumping (ceteris paribus the canal head
condition) the water table declines, the aquifer becomes disconnected from the canal (over a
greater length of the canal), and the pumper is less able to induce seepage i.e. as MSq3  0
seepage is increasingly passive.
The locational price equilibrium (Eqn. 3a) is determined by equating the sum of supply
price and transportation cost (effective supply) with demand price. Effective cost of supply at
Node 2 ( P2 ) is the cost of water supplied at the head of the canal plus the marginal cost of
2
1
seepage; P  P  2 MS X1,2 where the exogenous marginal cost for Node 1 at the head of the
1
canal (in our example, P  k1 ) and 2 is the price at Node 2. In our example, the total charge at
1
the head of the canal equals k1X1 where k1 is a fixed O&M charge, thus P  k1 . P2 is thus the
sum of exogenous levy at canal head plus the externality of MS X1,2 which is determined in part
by the amount of pumping (q3) of Node 3. The quantity of water delivered to Node 2 at the end
of the canal is the quantity at the head of the canal net of canal seepage. Downward sloping
11
seepage costs are added to a fixed levy resulting in an effective supply cost at Node 2 which
declines as diversion increases.
The marginal cost of groundwater to Node 3 at the well head is: P3  k2  k3 lift ; where k2
is a fixed pumping cost, k3 is the power cost per foot of pumping lift, and pumping lift is;
lift  b0  e
( b1q3 b2 X1,2 )
. P3 is thus a function of the exogenous pumping cost parameters plus the
canal seepage externality, as determined by the actions of Node 2. As pumping increases, the
water table drops and become disconnected from the canal, thus MSq3  0 and seepage becomes
increasingly passive. Simultaneously, the drop in water table causes the marginal cost of
pumping to escalate.
The two exogenous irrigation demand functions correspond spatially to the respective
supply functions, the well head and head gate at the end of the canal for Nodes 3 and 2,
respectively. Demand is short-run, exhibiting decreasing marginal returns to the variable input of
water and with crop mix, acreage, and irrigation technology fixed. The general functional of the
demand is; Pi   0 (1  1qi ) , where β0, β1, and β2 and is parameterized using case specific
3
agronomic data on crop yield, evapotranspiration, irrigation application, and irrigation efficiency,
and commodity prices (see appendix, Contor et al., 2008).
ResultsandAnalysis
Externalities result in undesirable market behavior when market institutions fail to sustain
desirable activities or reduce undesirable activities (Bator, 1958). A Pigouvian wedge thus arises
between private and social cost/benefit, with positive externalities under produced and negative
externalities over produced. The reciprocal conjunctive use externalities result in failures in the
surface water and groundwater markets; the positive externality of passive seepage is under
produced and the negative externality of induced seepage is over produced.
Using the above three node example, three scenarios were developed to illustrate the
application of partial equilibrium modeling with conjunctive use externalities. The first scenario
is a Pigouvian tax/subsidy internalization of the reciprocal externalities. The two subsequent
scenarios are common management alternatives intended to mitigate conjunctive use
12
externalities; (1) eliminating the externality with water conservation infrastructure and (2)
augmenting the externality with managed aquifer recharge. Equilibrium prices, quantities and
surpluses for each scenario are reported in Table 1. The Pigouvian tax/subsidy and aquifer
recharge scenario results are displayed in Figure 2a and 2b and Figures 3a and 3b, respectively.
The figures illustrated three points: (1) the production externality of conjunctive use shifts only
supply the functions, (2) the CBA that is performed numerically and can be illustrated
graphically (Griffin 1998) and (3) the tax/subsidy that eliminates the wedge between the private
and social causes a feedback in the reciprocal groundwater or canal water market.
Baseline
The baseline is the “without” scenario in the with-versus-without evaluation required by
CBA. In the baseline, the leaky canal is connected to the aquifer. The groundwater pumper
receives a positive externality of reduced pumping lift from canal seepage and pumping inflicts a
negative externality upon canal user by inducing additional seepage. Evident in the equilibrium
prices and quantities is the rudimentary water budget for the baseline. In the canal water market,
Node 1 supplies 3,097AF priced at $15/AF of which 2,130AF reaches Node 2 who is willing-topay $21.65/AF for the water delivered at the canal end. Node 2 makes a payment to Node 1 in
the amount of $46,406  $15  3097  which includes $14,505  $15   3097  2130   for water
not received, i.e. canal seepage. In the groundwater market, Node 3 pumps 1,283AF (286AF of
induce seepage plus 681AF of passive seepage plus 316AF from sources other than seepage) (see
endnote 1) for which he pays $30.65/AF in pumping costs. The Node 3 pumper thus makes a
payment to the Node 3 supplier (i.e. the power company) in the amount of $39,325
 $30.65 1, 283AF  . Baseline surplus totals to $164,087; $90,900 Node 2 consumer surplus,
$64,242 Node 3 consumer surplus and $8,946 Node 3 producer surplus. The horizontal supply
function of Node 1 yields no producer surplus.
Pigouvian Tax/Subsidy
In a competitive equilibrium, the welfare of two agents depends only on consequences of
their own choices. An externality creates an asymmetry between social and private prices.
13
Internalization forces both agents to account for the consequences of their actions on the other’s
welfare by aligning prices. Absent internalization (i.e. the base case), the canal water user
responds only to the pecuniary supply cost of the canal company and the shrinkage cost of
seepage. Ignored is the reduction in costs that the pumper receives from canal seepage. Similarly,
the pumper is signaled only by the pecuniary cost of pumping, ignoring the cost to the canal
diverter of pumping-induced seepage. A Pigouvian tax/subsidy internalizes the externality, by
aligning marginal supply prices of canal diverters and groundwater pumpers. The price
alignment signals to decrease production of the negative externality (pumping induced seepage)
and increase production of the positive externality (canal diversions that create seepage).
To internalize the reciprocal conjunctive use externality, the canal water supply function
is redefined as the pecuniary marginal cost at Node 1 plus the negative externality of seepage in
the transportation of water from Node 1 to Node 2 plus a Pigouvian subsidy. The subsidy equals
the reduction in marginal cost that canal seepage provides the Node 3 groundwater pumper. To
determine the subsidy, the price linkage equation (3b) which equates canal water demand prices
with the sum of supply prices (including seepage and transportation costs) is replaced with
equation 3c:
(3c)
xi , j  ti , j   j   i   j
where,   MC
3
X1,2


X1,2
S{ k }, i , j ( xi , j ,{q k })
xi , j
  i, j  IJ
,
q3
 P (q,{X
3
1,2
})dq is the Pigouvian subsidy to the canal diverter, defined
0
as the Node 3 marginal cost of pumping with respect to canal diversion. Equation 3c states, that
at equilibrium, if the quantity of water diverted in the canal is greater than zero, the sum of
supply price, and seepage cost plus the Pigouvian subsidy equals the demand price.
Similarly, the groundwater supply function is redefined as the pecuniary cost of pumping
at Node 3 plus the marginal cost of pumping induced canal seepage to the canal diverter at Node
2. To determine the tax, the price linkage equation (2b) which equates groundwater demand price
with supply price is replaced with equation 2c:
(2c)
q k  P k ( q k ,{ X i , j })   k  
k  I
14
where   2 MSq3 is the Pigouvian tax imposed on the groundwater pumper, defined as the
marginal cost to Node 2 of pumping induced seepage. Equation 2c states that at equilibrium, if
the quantity of groundwater pumped is greater than zero, the pumping cost plus the Pigouvian
tax equals the demand price. Together, α and  monetize or price canal seepage in both the
surface water market and the groundwater market, and equations 2c and 3c ensure the existence
of a social optimum at equilibrium.
The Pigouvian tax/subsidy aligns prices to erase the wedge between social and private
costs in surface water and groundwater markets that arise from the canal seepage externalities.
However the reciprocal nature of the externalities means that the tax/subsidy accounts for only a
portion of the total wedge between social and private supply cost. A portion of the wedge in each
market is erased through feedback from the other market. The imposition of an optimal tax on
Node 3 pumping and an optimal subsidy for Node 2 canal diversion nevertheless insures that the
marginal benefit of canal seepage to groundwater pumpers matches the marginal cost of seepage
to the canal diverter.
In the surface water market, the Pigouvian subsidy shifts the supply function downward
from the baseline (Figure 2a). At equilibrium, Node 1 supplies 3,692AF to Node 2 (an increase
of 595AF over the base case) for which Node 2 is willing-to-pay $8.31/AF. The Pigouvian
subsidy (β) for canal diversion is $9.23/AF. Node 2 pays Node 1 $55,383 ($15*3,692) which
includes seepage costs of $17,250 (1,150*$15). Assuming the base case canal diverter supply
function without the subsidy, but the scenario 2 rate of diversion and pumping, the equilibrium
price at Node 2 is $21.61/AF (Figure 2a) (see endnote 2). The total wedge between social and
private cost in the surface water market is therefore $13.30 ($21.61 - $8.31). Of this, $9.23 is
erased by the canal diversion subsidy (A to B in Figure 1a) and $4.07 is erased by feedback from
the groundwater market (B to C in Figure 2a). The feedback is a consequence of the tax on
groundwater pumping which reduces pumping and thereby the induced component of canal
seepage.
In the groundwater market, the Pigouvian tax shifts the supply function upward from the
baseline (Figure 2b). At equilibrium, the Pigouvian tax (α) on pumping is $0.45/AF. However, as
a consequence of the combined tax and subsidy, pumping increases by 23AF (1,306 – 1,283),
and the price of groundwater declines from $30.65/AF to $25.72/AF. Again, assuming the base
15
case groundwater pumper supply function without the tax, but the rate of diversion and pumping,
the wedge between social and private cost in the groundwater market is $5.01 ($30.73-$25.72)
(figure 1b). The $0.45/AF tax, on pumping has the intended effect of reducing induced canal
seepage, although it actually increases the wedge between private and social groundwater supply
(A to B in Figure 2b). The wedge in supply is only erased by the $5.46 feedback from the surface
water market (B to C in Figure 2b) (see endnote 3). The reciprocal externality produces feedback
between the surface and groundwater markets that reinforces or cancels the tax/subsidy effect.
In the groundwater market, the subsidy increases canal diversion which then increases seepage
available for pumping which more than offsets the tax induced reduction in pumping. Despite
taxing the negative externality of pumping, the marginal cost of pumping declines (internalized
supply, Figure 2b) and pumping increases from the private equilibrium (1283) to the social
equilibrium (1306).
Internalization with a tax/subsidy increases the welfare of both groundwater pumper and
canal diverter. In the with-versus-without CBA comparison, total irrigator surplus increased by
21% over baseline. In the groundwater market, consumer surplus increased by 10% ($64,242 to
$70,625), and in the surface water market consumer surplus increases by 34% ($90,900 to
$121,986). The benefit of seepage received by Node 3 outweighs the tax levy and the subsidy
received by Node 2 outweighs the seepage loss. As tax and subsidy, and  generate an optimal
social equilibrium, neither can be increased or decreased without being offset by greater social
cost or reduced social benefit. A complete market now exists where trade is unimpeded, thus
ensuring by definition, the gains in efficiency from trade (Randall, 1983). The Pigouvian
incentive requires exogenous cash flows i.e. collected tax revenues are received and subsidy is
funded by an external agent. A tax paid to the injured party (canal user) or a subsidy paid by the
beneficiary (pumper) distorts the tax/subsidy incentive (Baumol and Oates, 1988).
Correcting the market failure by omniscient regulation of the reciprocal externality
requires pumping restrictions and canal diversion increases. Regulation of induced seepage
damage alone (e.g. a binding constrain on pumping) will simply reproduce the equilibrium
quantities and welfare of the baseline scenario. Prior appropriations regulations to protect senior
canal or river rights from the damage of induced seepage are usually enforced as well setbacks or
outright bans. For example, Oregon (Oregon Water Resources Department 2010) mandates a
16
quarter mile set back from the river for wells in a riparian aquifer. Absent damage to the senior
water users, prior appropriations regulation cannot compel canal users to “waste water” for
aquifer recharge (Fereday et. al 2006). Regulation is thus limited to pumping restrictions reduces
the benefit derived from passive seepage intended to prevent the damage of pumping-induced
seepage.
Eliminating the Externality
The externality can be eliminated by lining the leaky canal. Canal lining is promoted as
conservation panacea that maintains agricultural production, while liberating water for
environmental or municipal use (Rodgers 2008; Hanak et al. 2010). Millions have been
expended to subsidize agricultural water conservation by either reducing on-farm infiltration or
improving conveyance efficiency. CBA for water conservation infrastructure are required to
account for the costs and benefits, including those of the current recipients of the seepage
externality.
When the canal is lined the cost of seepage is no longer imposed upon the canal diverter.
The quantity supplied by Node 1 (2,327AF) equals the quantity demanded at Node 2, making
Node 1 supply price ($15) equal to Node 2 demand price. Canal diverter demand increases
196AF (2,327 - 2,130). Absent canal seepage, aquifer levels decline, pumping cost increases
from $30.65 to $95.24/AF and pumping declines by 875AF (1,283-408). Surface water market
consumer surplus increased by 16% ($90,900 to $105,709), offset by a decrease in consumer and
producer surplus in the groundwater market by 64%. Total surplus, the CBA of with-versuswithout comparison, declined 67% ($110,239 to $164,087). In our example, construction costs
are ignored and the sole beneficiary of canal lining is the short run increase in irrigation intensity
of the existing crop mix and acreage of the canal water user.
Aquifer Recharge
As illustrated in previous research, aquifer recharge is an accepted policy in conjunctive
use management (Blomquist et al. 2001). In contrast to the external Pigouvian tax/subsidy flows,
aquifer recharge payments are internal -- the junior appropriator pay to receive the benefit or
17
compensate for damages to the senior. As the case with most conjunctive hydrology situations,
the pumper has an adjudicated right that is junior to the canal user whose seepage created the
aquifer. Thus, in our example, the junior pumpers compensate the senior canal users for
increased diversions to recharge the aquifer. Diversions for recharge are assumed to occur during
the irrigation season, rival with canal demand.
Canal diversion is priced via a payment from Node 3 to Node 2 that matches the decrease
in total pumping cost attributable to the seepage resulting from canal diversion. The payment is
calculated by integrating the groundwater pumpers’ marginal cost function with respect to
pumping yields total pumping cost, and differentiating with respect to canal diversion to yield
the marginal cost of pumping with respect to diversion: MC
3
X1,2


X 1,2
q
3
 P (q,{ X
3
1,2
})dq . The
0
total value of canal seepage to groundwater pumpers is X 1,2  MC X3 1,2 , and the payment rate for
canal diversion per acre-foot of groundwater pumping is
X 1,2
q
3
 MC X3 1,2 . Groundwater pumpers
maximize their returns by paying the canal diverter this amount for each acre-foot of water
diverted down the canal. In turn, this payment is inserted into the canal water and groundwater
price linkage equations, (3b) and (2b). Equation 3d states, that at equilibrium, if canal diversion
is greater than zero, the sum of supply price, transportation and seepage costs, minus the
payment from groundwater pumpers equals the canal diverter’s marginal demand price.
(3d)
S{ k }, i , j ( xi , j ,{q k })
xi , j  ti , j   j     j
i
xi , j


k xi , j
qk
P
k
(q,{xi , j })dqi, j  IJ
0
Equations 2d states, that at equilibrium, if the quantity of groundwater pumping is greater than
zero, the marginal cost of pumping plus the seepage payment to canal diverters equals the
groundwater pumper’s marginal demand price.
(2d)
q  P ( q ,{ X i , j })    
k
k
k
k
i, j
X i, j
qk

X i , j
qk
P
k
( q,{ X i , j }) dqk  I
0
In the groundwater market, the transfer payment raises the marginal cost of pumping and
thus shifts the supply function upward from the baseline (Figure 3b). The transfer payment from
Node 3 to Node 2 increases Node 3 marginal cost by $21.04/AF ($51.68 - $30.65) relative to the
18
base case scenario. As a consequence, groundwater pumping decreases by 159AF (1,283 - 1124)
relative to the base case.
In the surface water market, the transfer payment lowers the marginal cost of deliveries to
the end of canal and thus shifts the supply function downward from the baseline (Figure 3a). In
the surface water market, the transfer payment reduces Node 2 marginal cost by $11.86 ($21.65$9.79) from the base case. As a result, the quantity of water shipped from Node 1 to Node 2
increases by 506AF (3604 - 3097) and Node 2 increases demand by 362AF (2492 - 2130). The
increase in diversion increases seepage by 145AF (1112 - 967). The Node 3 groundwater pumper
pays the Node 2 canal diverter $8.17/AF for canal water diverted at Node 1 which translates into
a payment of $26.21/AF of groundwater pumped. The total payment by Node 3 is $29,447
($26.21 × 1,112) which matches the total receipts of Node 2, $29,447 ($8.17 × 3,604) i.e. the
decrease in total pumping cost attributable to the seepage resulting from canal diversion.
In the surface water market, Node 2 consumer surplus increased by 30% ($90,900 to
$118,253); (A+B+C) – A in figure 3a). In the groundwater market, consumer surplus for Node 3
decreased by 40% ($64,242 to $38,737); (A+B+C+D) – A in figure 3b). Node 3 producer surplus
increases 282%, from $8,946 (E+F+G in figure 3b) to $34,214 (B+E in figure 3b). The big
increase in Node 3 producer surplus is attributable to the payment made to Node 2. The
groundwater pumper is better off paying for relatively cheap electricity than paying the canal
diverter for relatively costly seepage water. The managed recharge alternative lifts net social
welfare above the baseline by 17% ($164,087 to $191,204). Even after making payments for
canal seepage, the groundwater pumpers’ surplus is not exhausted. The surplus of Node 2 canal
diverters damaged by the negative externality of pumping induced seepage increases with receipt
of payment from groundwater pumpers. Although the managed recharge scenario does not
penalize pumping induced seepage, the CBA social benefit from this scenario exceeds that of the
baseline status quo.
In contrast to the Pigouvian tax and subsidy that corrects both sides of the reciprocal
externality, only the positive externality of seepage is addressed by the recharge payment. The
negative externality of induced seepage, addressed by a payment from the canal to the pumper to
curtail pumping, is left uncorrected. The recharge payment addresses the market failure in that
the senior appropriator canal user cannot be compel to “waste water” to sustain the positive
19
externality of seepage to the aquifer (Fereday et al. 2006). The property right establishes that the
pumper needs to pay for passive seepage that he previously received free. While payment is a
potential remedy for the market failure the necessity for property rights contravenes a long
standing principle of prior appropriation; the loss of physical control of water, whether through
surface returns or percolation to aquifers, returns the water to public ownership.
Conclusions
Aquifers created or sustained by surface water are a global phenomenon. Previous
research on conjunctive use failed to treat conjunctive ground and surface water use as an
externality of un-priced and interrelated supply functions. Reciprocal conjunctive use
externalities thus result in failures in both surface and groundwater markets.
Per our goal of functionality, an n-dimension generalized partial equilibrium model of
conjunctive groundwater and surface water use was formulated. Using MCP, a three node
example was parameterized with site-specific data and was solved for equilibrium water prices
and quantities in the respective surface water and groundwater markets. The three exogenous
variables are: (1) demand, for various crops, irrigation technologies and scalable to any acreage,
(2) supply, from pumping costs and canal company levies and (3) the transportation arc, a
seepage function estimated from a groundwater model. Functionality was further demonstrated
in the CBA performed on three conjunctive management policies; tax/subsidy, conservation, and
aquifer recharge. A basin water budget, which accompanies groundwater models, can provide an
external check on baseline scenario results. The partial equilibrium framework and data can be
replicated; calculation of the demands and marginal costs are well documented but seepage
functions deserve further research.
The Pigouvian tax/subsidy aligns private and social price to erase the wedge between
private and social cost and when imposed on a one way externality results in an unambiguous
decrease or increase of that externality. In contrast, the tax/subsidy on a reciprocal externality
produces feedback between the markets that reinforces or cancels the tax/subsidy effect. In our
example, the subsidy increased canal diversion which then increased seepage available for
pumping which more than offsets the tax induced reduction in pumping. Pumping quantity at the
social equilibrium (1306AF) increased over the private equilibrium (1283AF), despite taxing the
20
negative externality of pumping. The feedback effect improves the welfare of the recipient – e.g.
the tax discourages the negative externality of pumping to the benefit of canal and the subsidy
encourages canal diversions to the benefit of the pumper. The magnitude of the feedback is
determined by the relative magnitude and elasticity of exogenous supply, demand and
transportation variables.
Of the three policy scenarios, internalizing the externalities via a tax/subsidy generated
the greatest social benefit. Conservation failed the CBA -- the foregone benefit of canal seepage
to the pumpers by eliminating the externality outweighs the increase in canal benefit. In the
recharge scenario, canal seepage is priced via a payment from pumper to canal water user that
matches the decrease in total pumping cost attributable to the seepage resulting from canal
diversion. In our example, total surplus of the recharge policy was 97% of the Pigouvian
scenario. With the recharge payment, the equilibrium quantities, in all three markets, was only
slightly less than the Pigouvian scenario. However, the price of water at node 3, driven by the
payment, increased from $25.72 to $51.68.
Market failures arise from poorly assigned and/or enforced property rights. Governance
often fails to legally acknowledge conjunctive hydrologic connections, let alone recognized
conjunctive use as a reciprocal externality. The Prior Appropriation Doctrine defines and
protects property rights not correct the externality by maximizing social welfare as does partial
equilibrium framework. The lack of property right extends to both sides of the reciprocal
externality: junior groundwater users are not forced to pay for the seepage that they now freely
enjoy nor are senior canal users compensated for the damage of pumping induced seepage. Prior
appropriation removes the incentive of seepage payments because once the seepage leaves the
canal diverters control, the water reverts to the public domain to be appropriated for beneficial
use. On other side of the reciprocal, the beneficiary of seepage has no mechanism to invite
additional seepage. Absent damage to the senior water users, prior appropriations regulation
cannot compel canal users to “waste water” for aquifer recharge (Fereday et. al 2006).
Cheung’s (1973) rejoinder to Meade’s bee example, pointed out that the market failure of
a reciprocal externality can be corrected, without government intervention, by payments from
apple to bee farmers. Reciprocal conjunctive use externalities have not been similarly resolved
and as water supplies shrink and demands grow, the lawsuit and water calls etc. are symptoms of
21
the proliferation of market failure conflicts. Continued reliance upon regulation of conjunctive
externalities will increase inefficient water use that a water-scare society ill afford. A functional
model of reciprocal conjunctive use externalities is a first step in the investigation of conjunctive
water management policy alternatives.
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25
Endnotes
1. Induced seepage is 286 AF  0.1 3097(1  e( 0.0021283) ) and passive seepage is:
681AF  15, 000  (1  e( 0.0000153097) ) .
2. The wedge between private and social supply is β plus the difference between the
marginal cost of canal water to Node2 at the equilibrium price with internalization ($9.35)
and the base marginal cost of canal water evaluated at the equilibrium quantity of pumping
q3 and diversion X 1,2 .


$21.61 / AF  15.00  21.65 15, 000 1.5 E  05  e 1.5 E 053692   1.5 E  05  1  e 0.0021283

3. The wedge between private and social supply is α plus the difference between the
marginal cost of pumping at the equilibrium price with internalization ($25.72) and the base
marginal cost of pumping evaluated at the equilibrium quantity of pumping q3 and diversion

X 1,2 is: $30.73 / AF  9.50  .08 1000  e
26
 1.7 E  041283 5.0 E  43097 

Table 1. Equilibrium prices ($/AF) and quantities (AF) for model scenarios
symbol
Baseline
Pigouvian tax/subsidy
Elimination
Aquifer Recharge
Consumer Surplus node 2
$90,900
$121,986
$105,709
$118,253
Consumer Surplus node 3
$64,242
$70,625
$3,346
$38,737
Producer Surplus node 1
$0
$0
$0
$0
Producer Surplus node 3
$8,946
$5,555
$1,184
$34,214
$164,087
$198,166
$110,239
$191,204
$21.65
$8.31
$15.00
$9.79
$30.65
$25.72
$95.24
$51.68
Total Surplus
Demand Price node 2
Demand/Supply Price node 3
1
3  
3
Supply Price node 1
1
$15.00
$15.00
$15.00
$15.00
Demand Quantity node 2
q2
2,130
2,542
2,327
2,492
q3  q 3
1,283
1,306
408
1,124
Supply Quantity node 1
q1
3,097
3,692
2,327
3,604
Quantity Shipped node 1 to node 2
X12
3,097
3,692
2,327
3,604
Quantity Shipped node3 to node 3
X33
1,283
1,306
408
1,124
Seepage in Canal from node 1 to node 2
967
1,150
0
1,112
Transfer Payment per Quantity Pumped
n/a
n/a
n/a
$8.17
Transfer Payment per Canal Quantity
n/a
n/a
n/a
Total Transfer Payment
n/a
n/a
n/a
Demand/ Supply Quantity node 3
Pigiouvian Tax
Pigouvian Subsidy
α
β
$29,447
$0.45
$9.23
Absent constraints the demand/ supply quantity node 3 will equal quantity shipped node3 to node 3 and the supply
quantity node 1 will equal quantity shipped node3 to node 3. A constraints will also cause the Node 3 demand and
supply price to be unequal.
27
Figure 1. Schematic of 3 node example
28
ρ2
$21.65
Privateequilibrium
A
$21.61
Baselinesupply
$9.23subsidyeffect
B
$12.38
C
$8.31
$4.07feedbackeffect
Socialequilibrium
Internalizedsupply
2130
2542
q2
Figure 2a. Pigouvian tax/subsidy scenario in the surface water market.
29
ρ3
$35.40
$0.45tax
effect
A
$34.95
$30.65
Baselinesupply
B
$5.46feedbackeffect
Private
equilibrium
Internalizedsupply
C
$25.72
Socialequilibrium
1283
1306
Figure 2b. Pigouvian tax/subsidy scenario in the groundwater market.
30
q3
ρ2
P2
A
$21.65
Baselinesupply
B
C
$9.79
Rechargesupply
2492
2130
Figure 3a. Aquifer recharge scenario in the canal water market.
31
q2
ρ3
RechargeSupply
P2
A
$51.68
B
$30.65
E
C
F
BaselineSupply
D
G
1124
1,283
Figure 3b. Aquifer recharge scenario in the groundwater market.
32
q3
Appendix
Hydrologic response data was generated using analytic element methods (Strack, 1989).
Analytic element models represent each hydrologic feature by one or more analytic elements and
a head or flow specified boundary condition is assigned to each element. The model then
calculates the hydrologic response e.g. for head-specified canals, the response is the canal
seepage rate. For flow-specified wells, the response is the aquifer head condition at the well
radius, from which pumping lift can be derived. The hydrology of both one-way and reciprocal
externalities, canal diversion is a function of canal capacity and canal water elevation relative to
an aquifer datum. Canal seepage and pumping lift response data are generated for a range of
canal diversion and groundwater pumping rates. The data are fitted to response functions which
approximate the change in canal seepage rate and pumping lift as the canal transitions between
direct water table contact and a perched condition. Non-linear least squares was used to estimate
the site-specific parameter values for the Boise Basin sub-area pumping lift and canal seepage

response functions: S{3},1,2  15000 1  e
1.5 E  5 X 1,2
  0.1X 1 e
1,2
0.002 q3
 The seepage function is
limited to hydrologic settings where canal diversion and groundwater pumping are the two main
influences on canal seepage rate. All other hydrologic influences are represented by a far-field
hydrologic boundary condition. Applicability is further limited to the range of values for well
pumping and canal diversion that are input to the hydrologic model. Mention a steady state condition
as per the comment for R#1 and provide a citation for our Boise Valley hydro paper
Node 1 marginal cost of canal water is: P1 = $15. Node 3 marginal cost of groundwater
at the well collar is: P 3  9.50  0.08  lift ; where lift  1000  e (0.00017 q
3
 0.0005 X 1,2 )
.
The marginal benefit functions for Node 2 and Node 3 are: P2  2888(1  0.009 q20.8182 ) and
P3  1350(1  0.009 q32.3333 ) . The marginal benefit functions were calibrated for 825 acres of
alfalfa at Node 2 with 45 % efficiency and an application rate of 3.5 AF/acre, and for 600 acres
of alfalfa at Node 3 with 70 % irrigation efficiency and an application rate of 2.25AF/acre. The
differences in efficiency reflect surface versus sprinkler application.
33