The Gains from Agricultural Groundwater Trade and the

THE GAINS FROM AGRICULTURAL

GROUNDWATER TRADE AND THE POTENTIAL

FOR MARKET POWER: THEORY

AND APPLICATION

ELLEN M. BRUNO, AND RICHARD J. SEXTON

This article models and estimates the efficiency gains from using market-based instruments relative tocommand and control to manage groundwater. A theoretical model of an imperfectly competitivegroundwater market is developed to show how the magnitude and distribution of the gains from tradechange as market structure varies. Market structure is a key consideration because future groundwatermarkets will likely feature geographic limitations to trade, large agricultural players, and a legal environ-ment that is conducive to forming cartel-like coalitions. Application of the model to a groundwater-dependent agricultural region in southernCalifornia shows the existenceof large gains from trade, despitethe potential for market power, with benefits up to 36% greater than that under command and control.Distributional impacts, however, can be sizable even for small degrees of market power. Simulations thatvary market conditions show that results likely generalize to other groundwater basins.

Key words: Groundwater, irrigation, market power, market structure, water markets.

JEL codes: Q15, Q25.

Improved management of water resources isbecoming increasingly important in the face ofclimate change. Climate models predict highertemperatures and more variable precipitation,with droughts and other extreme climate

events occurring more frequently (Kunkelet al. 2013; Swain et al. 2018). Shortages of sur-face water during times of drought are oftenmet with increased groundwater pumping(Howitt et al. 2015). However, many ground-water basins worldwide have seen declines ingroundwater storage over time, as groundwa-ter is extracted at a rate faster than it can bereplenished (Rodell, Velicogna, and Fami-glietti 2009). Declining groundwater levelsincrease pumping costs and reduce availabilityof the underground reserve in times of drought.Economists have espoused the merits of

market-based instruments to manage the envi-ronment (e.g. Goulder and Parry 2008), andgrowing empirical work points to substantialcost savings from market-based instruments rel-ative to command-and-control policies for airpollution (Fowlie, Holland, and Mansur 2012;Schmalensee and Stavins 2017). Less is known,however, about the performance of economicinstruments for managing water. Previous

Ellen M. Bruno is an assistant cooperative extension specialist inthe Department of Agricultural and Resource Economics at Uni-versity of California, Berkeley (Email: [email protected]).Richard J. Sexton is a distinguished professor in the Departmentof Agricultural and Resource Economics at University of Califor-nia, Davis (Email: [email protected]). The authors thankGraham Fogg, Michael Hanemann, Richard Howitt, Katrina Jes-soe, Stephen Maples, Katherine Markovich, Pierre Mérel, DanielSumner, Jim Wilen, and Jeffrey Williams for helpful commentson this work. Valuable comments and suggestions also came fromseminar participants at the Agricultural and Applied EconomicsAssociation annual meeting, UCBerkeley, UCRiverside, Purdue,Resources for the Future, the EPA National Center for Environ-mental Economics, the University of Western Australia, and theUniversity of Adelaide. A special thanks goes to Ivory ReyburnandRobert Cheng at theCoachella ValleyWaterDistrict. Fundingfor this research was provided by The UC Giannini Foundation ofAgricultural Economics, UC Water Security and SustainabilityResearch Initiative, and the National Science Foundation ClimateChange, Water, and Society IGERT (DGE No. 1069333).*Correspondence may be sent to: [email protected]

Amer. J. Agr. Econ. 00(00): 1–27; doi:10.1111/ajae.12031© 2020 TheAuthors.American Journal of Agricultural Economics published byWiley Periodicals, Inc. on behalf ofAgri-cultural and Applied Economics AssociationThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribu-tion and reproduction in any medium, provided the original work is properly cited.

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literature has focused on the gains from surfacewater transfers, suggesting that there exist largebenefits from the reallocation ofwater from low-value to high-value users (Vaux and Howitt1984; Howitt 1994; Hearne and Easter 1997;Sunding et al. 2002; Jenkins et al. 2004; Petersonet al. 2004; Grafton and Horne 2014; Grafton,Horne, and Wheeler 2016). However, theseconclusions are often drawn from programmingmodels that rely upon strong assumptionsregarding competition, information, and theavailability of substitutes and have yet to beempirically corroborated.This paper studies groundwater trading in an

environment where pumping is restricted toachieve long-run sustainability of the groundwa-ter basin. Relatively few studies have discussedgroundwater trading, with notable exceptionsfocused on the handful of jurisdictions wheregroundwater is actively regulated. In the U.S.,key examples include Thompson et al. (2009),Kuwayama and Brozovi�c (2013), Brozovi�c andYoung (2014), and Palazzo and Brozovi�c(2014) for Nebraska, and Guilfoos et al. (2017)for Kansas. Elsewhere, key examples includeGao et al. (2013) for Australia and Raffensper-ger, Milke, and Read (2009) for New Zealand.Unlike previous work, our model allows eitherbuyers or sellers to exercise market power inthe groundwater market. We contribute to thiswater markets literature by quantifying the effi-ciency gains to groundwater trading relative toa command-and-control regime and showinghow the gains and the distribution of benefitsfrom trading are influenced by the presence ofmarket power.Weapply ourmodel to a ground-water basin in California that faces pumpingrestrictions in light of landmark legislationpassed in California in 2014 to achieve sustain-ability of groundwater resources.This analysis is important because, prior to the

widespread implementation of groundwater mar-kets, it will be necessary to understand themagni-tude of the potential gains from trade and howthese gains are influenced by market conditions,including market structure. The limited historyof surface water trading is unlikely to providemuch insight into how groundwater markets willevolve. Surface-water trading has, to date, beenlargely bilateral, involving exchanges amongwater-supply organizations operating on behalfof their membership, often characterized by hightransaction costs, and limited by physical convey-ance infrastructure (Brown 2006; Hagerty 2019).Groundwater markets, however, are likely to

be localized in nature and involve participants

pumping water from a shared groundwaterbasin. The rights to pump groundwater are notshared collectively and, instead, under mostlegal doctrines, accrue to landowners withinthe geographic bounds of the basin. These dis-tinguishing features of groundwater trade haveimportant implications for how well the marketwill function. Market structure in particular islikely to be a key consideration in these settingsbecause of the geographic barriers to trade,presence in many instances of large-scale agri-cultural operations, and a legal environmentthat is conducive to forming cartel-like coali-tions of sellers or buyers.The economic literature on groundwater

management has to date been dominatedlargely by studies that evaluate the differencein social welfare between open-access ground-water use and socially optimal groundwateruse. Early research showed that high discountrates diminish the importance of higher futurepumping costs caused by depletion of basinresources, and, when extraction is small relativeto the total storage of an aquifer, the gains frommanagementmay be negligible (Gisser and San-chez 1980; Brill and Burness 1994). More recentliterature has found larger gains from manage-ment by developing spatially explicit models tocapture aquifer heterogeneity and spatial pump-ing externalities between wells (Brovozi�c, Sund-ing, and Zilberman Brozovi�c, Sunding, andZilberman 2010; Pfeiffer and Lin 2012; Edwards2016; Merrill and Guilfoos 2017) and byaccounting for drought reserve value andavoided capital costs (MacEwan et al. 2017).Ourmodel abstracts from these dynamic and

spatial dimensions to answer a different ques-tion: given a fixed cap on pumping, what arethe price, quantity, andwelfare impacts of trad-ing conditional on any degree of buyer or sellermarket power? The model begins with uncon-strained groundwater demand functions forheterogeneous farmers. Then property rightsfor pumping are allocated such that the aggre-gate use is restricted relative to open access.We then derive excess pumping permitdemand and supply functions, which are usedto derive trading equilibria and quantify thegains from trade. Using a flexiblemodel frame-work that can reflect any degree of buyer orseller market power in the permit market forgroundwater, we identify the relationshipbetween market power and the efficiency anddistributional impacts of water trading.Results show that the efficiency impacts of

market power are relatively small even forsubstantial market power, but the

2 Month 2020 Amer. J. Agr. Econ.

distributional impacts are large even for mod-erate levels of market power; traders withmarket power (whether as buyers or sellers)may be able to capture large shares of thegains from trade. Such impacts are importantfrom a policy perspective because they mayinfluence the political feasibility of implement-ing groundwater markets.

The contribution of the theoretical model istwofold. First, it extends a branch of literaturethat evaluates the impacts of market power inpermit markets to include groundwater. Stem-ming from the seminal paper by Hahn (1984),this literature considers the initial distributionof property rights, strategic behavior of com-petitors, the role of storable permits, andimpacts in the final product market, with appli-cations to fisheries (Anderson 2008), pollution(e.g. Misiolek and Elder 1989; Montero 2009;Hintermann 2011, 2017; Liski and Montero2011), and surface water (Chakravorty et al.2009; Ansink and Houba 2012).

Second, this analysis relaxes assumptions ofprior work regardingmarket structure to morebroadly characterize the impacts of imperfectcompetition. Whereas previous literature hasmade rigid assumptions on market structure,such as Cournot competition or dominantfirms with a competitive fringe (e.g. Hahn1984; Westskog 1996; Montero 2009; Hinter-mann 2011, 2017; Ansink and Houba 2012),we use a flexible framework for imperfectcompetition in the permit market. This allowsthe model to depict the entire range of possi-ble market power settings for either buyer orseller power and enables us to see how thegains from trade and distribution of benefitsvary with market power.

The model is then applied to a groundwa-ter basin in southern California that under-lies the Coachella Valley, a majorproduction region for citrus, dates, grapes,and vegetable row crops. Water supply orga-nizations in California are charged to adoptregulations to correct groundwater overdraftand achieve sustainability of groundwaterresources under the Sustainable Groundwa-ter Management Act (SGMA) of 2014. Thegroundwater basin serving the CoachellaValley is among those designated as over-drafted and subject to SGMA. Moreover,the structure of agriculture in the CoachellaValley exhibits features that could inducemarket power in groundwater trading, mak-ing it an ideal setting for application of thegroundwater trading model.

We estimate the gains from groundwatertrade for the water district serving CoachellaValley. An essential feature of this applica-tion is that all model parameters are eitherconstructed or estimated econometricallyfrom observational data for this region.Results show that the economic surplus withcompetitive trade is almost 40% greater thanunder command and control, given a 20%reduction in basin-wide use that is neededfor groundwater sustainability under SGMA.Either seller or buyer market power is shownto have a relatively small impact on the over-all gains to trade, but even moderate marketpower can significantly skew the gains totrade in the direction of entities with marketpower, either as buyers or sellers. As the firstmodel of California groundwater trade, thiswork brings new evidence and a new per-spective on the cost effectiveness ofincentive-based instruments for watermanagement.

Modeling Framework

We develop a theoretical model for studyingagricultural groundwater use and trading, withthe goal of understanding the magnitude ofthe gains from trade, the distribution of benefitsamong traders, and how both are affected bymarket power. Our model has the advantagethat, when expressed in its linear form, theimpacts of groundwater trade can be revealedvia a few pure-number parameters, most ofwhich can be estimated with commonly avail-able data for any groundwater basin. Tosharpen focus on our objective of analyzingthe gains from trade in a flexible setting regard-ing buyer or seller market power while main-taining tractability, the model necessarilyabstracts from some of the dynamic and spatialcharacteristics of groundwater pumping.Herein the groundwater basin defines the

geographic scope of the market, and permittrading occurs among farmers. We begin withan unmanaged, open-access groundwater set-ting and then introduce tradable propertyrights for pumping. For simplicity, we assumethere are two types of farmers pulling from acommon aquifer, low (L) and high (H), whoare homogeneous within their type. Each pro-duces a single output. Farmers of type L growa low-value crop, such as rice or cotton, withproduction functions for an individual

Bruno and Sexton The Gains from Agricultural Groundwater Trade and the Potential for Market Power 3

j denoted qLj = fLj(xLj, yLj). Farmers of type Hgrow a high-value crop, for example a producecommodity or tree nut, with individual pro-duction functions denoted qHj = fHj(xHj,yHj).

1 The variable x represents appliedgroundwater, and y represents a compositeof other inputs to production, such as laboror fertilizer. Production functions are assumedto be differentiable and exhibit diminishingmarginal productivity to variable inputs.There are Ni identical farmers within each

type i∈H,L. H-type growers have a higherwill-ingness to pay at any given quantity of ground-water. To account for system irrigationefficiency, pumpedgroundwater is distinguishedfrom the amount of water applied to the crop.Aggregate water quantities are denoted byX and x, where uppercase is for pumpedgroundwater and lowercase is for appliedgroundwater. That is, X = XH + XL andx = xH + xL, where Xi =

PNij = 1Xj, and

xi =PNi

j = 1xj for i ∈ H, L. The relationshipbetween pumped and applied water is givenby the efficiency parameter δ, with 0 < δ < 1such that xi = δXi.The marginal pumping cost of water is

denoted by c(X) > 0. Marginal pumping costsare assumed to be increasing and differentia-ble, i.e. c

0(X) > 0; marginal pumping costs

increase due to reduction in the water tableas more water is pumped. We assume farmersface the same pumping costs and that individ-ual pumping is small relative to the basin total,so farmers take the marginal pumping cost asgiven, but collectively their decisions deter-mine basin-wide pumping costs.2

Open-Access Groundwater Use

Consider the profit-maximization problem forfarmers in the unmanaged, open-access case.In this setting, the price of groundwater inany period equals the marginal pumping cost,which individual users regard as constant andis denoted by c. Firms choose inputs (xi, yi)to maximize farm profits, where pi is the

output price for the crop produced by type i,i∈H,L, andwy is price of the composite input,y, both of which are taken as given. Farmers oftype i face the following profit-maximizationproblem:

ð1Þ maxxi ≥ 0,yi ≥ 0

πi = pif i xi,yið Þ−cxiδ−wyyi:

The first-order conditions are pi∂f i xi,yið Þ

∂xi= c

δ

and pi∂f i xi,yið Þ

∂yi=wy. In any period, optimizing

farmers equate the marginal value product ofan additional unit of groundwater to its price,which under open access is the marginalpumping cost adjusted by the efficiencyparameter. Solving for xi and yi yields theinput demand curves for each farmer as func-tions of crop output price, price for input y,and the marginal pumping cost of groundwa-ter: xH pH ,wy, cδ

� �, xL pL,wy, cδ

� �.

The assumption that output prices arefixed implies that individual farmers are per-fect competitors for the sale of their output.An extension of this work is to introducethe potential for pumpers to exercise marketpower in their output sales, in which casethe parametric prices, pL, pH, are replacedwith marginal revenue functions, MRij(qij),MRij

0 ≤ 0, i ∈ H, L that reflect a farmer’sperceptions of the impact of her sales onthe price received. Hintermann (2011,2016) shows that seller power in the outputmarket can impact a player’s optimal behav-ior in a permit market by creating an incen-tive for the firm to inflate the permit price ifthose higher costs can be passed forward toconsumers.3

Open-Access Equilibrium

The open-access equilibrium in a given timeperiod comes from equating the aggregatewater demand relationship, attained by sum-ming demands across both types, with theaggregate water supply relationship, which issimply the marginal pumping cost function, c

1 This formulation assumes farmers have already preselectedinto producing certain crops, for example based on heterogeneousability levels or land quality. Changes in cropping patterns or land-holdings that might occur over time due to alternative groundwa-ter management regimes are not considered in this model.

2 One extension of this work is to expand on this farm-levelgroundwater optimization problem by allowing individual firmsto account for their influence on their own pumping costs, forexample because they are big enough to impact the water tablewith their consumption, or they pump enough to create a cone ofdepression at the site of a well (Theis 1940).

3 Hintermann’s extensions of Hahn’s (1984) seminal results formarket power in permit trading to account for a trader’s sellerpower in its output market are important in the pollution permitcontext they each studied, wherein firms subject to cap-and-traderegulations are often large and powerful. For example, Hinter-mann applies his theory to European electric power generators.Conversely, groundwater trading will most often be amongfarmers who are unlikely to have market power as sellers of farmproducts.


(X), adjusted by irrigation efficiency. If wedefine equilibrium groundwater extraction,applied water quantities, and marginal pump-ing cost in this period by X*,x*H ,x

*L,c

*� �

, thenthe equilibrium conditions in any period are:

x*H = xH pH ,wy, c*

δ

� �, x*L = xL pL,wy, c

*

δ

� �, X* =

x*L + x*Hð Þδ , and c(X*) = c*.4

To obtain analytical solutions and enablequantification of gains from establishment ofgroundwater markets requires explicit func-tions, so we assume aggregate demands ofH and L types for applied water are linearand parallel such that the H type demandcurve is greater than that of the L type by aconstant amount for all quantities of appliedwater. Although this approach entails someloss of generality, it has the advantage thatwe can define differences in H andL demands in terms of a single parameter α,0 < α < 1, which measures the vertical differ-ence between H and L water demands at anyquantity and is useful for comparative staticspurposes. Given these assumptions, we canexpress aggregate demands as xH = γ− β

2c andxL = αγ− β

2c. Aggregate applied water demandis the sum of the total water demands fromeach type: x = xH + xL = (α + 1)γ − βc.

We also assume marginal pumping costs arelinear and increasing in X: c(X) = ω + μX, withω, μ > 0. The intersection of the aggregatedemand function with the supply relationshipfor applied water, c xð Þ=ω+ μ

δx, reveals thecompetitive, open-access equilibrium mar-ginal pumping cost (c*) and applied waterquantity (x*) in a given period for the linearmodel:5

ð2Þ x* =δγ α+ 1ð Þ−δβω

δ+ βμ,c* =ω+ μ

γ α+ 1ð Þ−βωδ+ βμ

:

In what follows, we invoke normalizationsfor pumping cost and water quantity such this

open-access equilibrium price (i.e. marginalpumping cost) and quantity are each equal toone: (c*, x*) = (1, 1). Evaluated at this per-fectly competitive equilibrium, the supplyelasticity is given by ε= ∂xS

∂ccxS =

δμ where

xS cð Þ= δμc−

δμω is the direct form of the mar-

ginal pumping cost function. Similarly, the abso-lute value of the elasticity of demand is given byη= j ∂xD∂c c

xD j = β evaluated at the equilibrium(1, 1), with xD(c) = γ(α + 1)− βc. These rela-tions imply the following substitutions, whichare used to rewrite the original slope and inter-cept parameters in terms of the pure-numberelasticities: β = η,γ = 1+ η

1 + α ,ω= 1− 1ε ,μ=

δε.

Given these substitutions, the demands forL and H types and the aggregate supply rela-tionship are expressed in terms of (a) the sup-ply and demand elasticities evaluated at theperfectly competitive, open-access equilib-rium, ε and η respectively, (b) the demand shiftparameter, α ∈ (0, 1), reflecting differences inwater demands between H and L types, and(c) the irrigation efficiency parameter, δ ∈(0, 1). Restating aggregate H and L demandsand inverse groundwater supply with respectto these pure-number parameters yields:

ð3Þ xH =1+ η

1+ α−η

2c, xL = α

1+ η

1 + α

� �−η

2c,

ð4Þ c xð Þ= 1−1ε

� �+1εx:

Establishing Property Rights for Groundwater

Now assume that a regulatory agency estab-lishes non-tradable property rights for pump-ing. To induce conservation, the regulatormust set an aggregate endowment that is lessthan the amounts being pumped under openaccess. Without the ability to trade, farmersof both types must limit pumping to no morethan their assigned allocations. Although intheory it is possible to arrive at the sociallyoptimal solution through a discriminatory setof water allocations where each type isassigned the amount it would pump in thesocially optimal setting, we assume the regula-tor lacks necessary information, political abil-ity, and/or legal right to implement such anallocation.To simplify the exposition and focus on the

long-run sustainability of an aquifer withinour static model framework, we assume that

4 In the absence of regulation, a basin that is subject to over-pumping would face higher pumping costs in subsequent periods,affecting pumping decisions. This dynamic path has been studiedby Gisser and Sanchez (1980), Brill and Burness (1994), andothers. Consistent with a static model, our strategy instead is toconsider the implementation of regulations designed to put thebasin on a sustainable path and compare results across thecommand-and-control and cap-and-trade regimes under differentcompetition scenarios.

5 One way to introduce dynamics into our formulation would beto make ω a function of net pumping (pumping less recharge) inprior periods. Thus, a basin subject to overdraft, defined by pump-ing exceeding recharge in any given period, would face higherpumping costs through time until a steady state was reached.


regulators set a constant allocation based on arule, such as on a pro-rata basis by land hold-ings, that is designed on average to balanceaquifer extractions and recharge over time.6

Allocations set in this type of regulatory envi-ronment are highly unlikely to equate mar-ginal value products across user types,opening the door to possible welfare improve-ments achieved through water markets.Suppose each farmer receives an initial

groundwater allocation, denoted A0i , that is

the same across homogeneous farmers withineach farmer type and constant across time. Inthe absence of markets, each farmer is con-strained to choose xi �ð Þ≤ δA0

i . An individualof type i faces the following constrained opti-mization problem, where λi is the Lagrangemultiplier associated with the constraint:

ð5Þ maxxi ≥ 0,yi ≥ 0,λi ≥ 0

πi = pif i xi,yið Þ−cxiδ

−wyyi−λi xi−δA0i

� �:

The Kuhn-Tucker conditions for theinequality-constrained problem are:

ð6Þ pi∂f i xi,yið Þ

∂xi−cδ−λi ≤ 0 and

xi pi∂f i xi,yið Þ

∂xi−cδ−λi

� �= 0

ð7Þ pi∂f i xi,yið Þ

∂yi−wy ≤ 0 and

yi pi∂f i xi,yið Þ

∂yi−wy

� �= 0

ð8Þ xi−δA0i ≤ 0and λi xi−δA0

i

� �= 0:

Given that the aggregate endowment is lessthan the open-access pumping volume, theallocation must bind on pumping for at leastone type, that is, x*i = δA0

i . Therefore, in equi-librium we must have a strictly positiveshadow price, λ*i > 0, for at least one type. Aconstrained farmer must reduce pumpingbelow the unconstrained optimum by, forexample, fallowing acreage or applying waterless intensively to crops.

Whether the constraint binds for either typedepends both on demand and on the initial allo-cation of permits. In what follows, we assumethe allocation is binding for both types and thatboth types apply some portion of their alloca-tion to their farming operation. This assump-tion avoids analytical complexities thatemerge if the allocation does not bind for onetype or if one type idles its full acreage andmar-kets its entire allocation.7 The subsequentapplication to Coachella Valley, CA, relaxesboth of these assumptions, and a completeexposition of the analyticalmodel that accountsfor nonbinding allocation constraints and themarketing of one type’s total allocation is con-tained in the online appendix File S1.When allocations bind for both types, we

get the following equilibrium expressions forthe shadow prices:

ð9Þ λ*i = pi∂f i δA

0i ,y

*i

� �∂xi

−c0

δ> 0 for i∈ H,Lð Þ

where c0 = c X0H +X0

L

� �and X0

i =NiA0i for i ∈

(H, L). A necessary condition for water mar-kets to emerge is that the shadow prices forthe H and L types differ at the constrainedequilibrium. Applying functional forms toequation (9), we characterize the necessarycondition in terms of aggregate demands foreach type. The equilibrium shadow prices are

λ*H = 2η

1 + η1 + α −δX

0H

� �− c0

δ and λ*L = 2η

α 1 + ηð Þ1 + α

�−δX0

LÞ− c0δ , where c0 = 1− 1

ε

� �+ δ

ε X0H +X0

L

� �is

the marginal pumping cost at the constrainedequilibrium.Equation (10) expresses the difference in

shadow values between types (at the con-strained optimum, x*i = δX0

i ), which yields thenecessary condition for trading to occur:

ð10Þ j λ*H −λ*L j

=2η

1−αð Þ 1+ ηð Þ1 + α

+ δX0L−δX

0H

� > 0:

We define Ω= 1−αð Þ 1 + ηð Þ1 + α + δX0

L−δX0H for

simplicity. When equation (10) holds, that iswhen |Ω| > 0, a set of positive permit pricesexists where trading will occur. In the

6 A binding cap that is sustainable and consistent over timeallows us to abstract away from aquifer dynamics due to year-to-year variations in economic and hydrologic conditions. Thisassumption is also consistent with the legislative intent of Califor-nia’s SGMA, which mandates implementation of policies toachieve long-run sustainability of groundwater resources.

7 In particular, the supply of permits follows the horizontal axisup to the point where the allocation starts to bind and thenbecomes vertical at the fixed allocation quantity, creating kinks(and nondifferentiability at the kink point[s]) in the excess supplyfunction of permits.


subsequent analysis we assume (10) is satisfiedand focus on the case were λ*H > λ*L, so that Htypes are net demanders, which requires

ð11Þ x*H c0� �

−x*L c0� �

=1−αð Þ 1+ ηð Þ

1 + α> δ X0

H −X0L

� �,

that is, the difference in equilibrium quan-tity demanded between the two types, givenmarginal pumping cost c0, exceeds the differ-ence in their initial endowments.

Tradable Property Rights

We now introduce trade by using the ground-water demand functions and the exogenousallocation of pumping rights to derive excessdemand and excess supply functions for pump-ing permits. Selling or supplying groundwaterin this context does not rely on access to phys-ical infrastructure to transmit water to buyersbut, rather, is simply agreeing not to pump upto one’s allocation of groundwater.

Let P denote the full groundwater price in atrading regime, which consists of permit price,ρ, plus marginal extraction costs, c0, that aredetermined by the aggregate constrainedpumping allocation. If the difference betweeninput demand for pumped water at priceP and individual water allowance, A0

i , exceedszero, then that farmer has excess demand atgroundwater price P = ρ + c0. Otherwise, thatfarmer has excess supply at price P.

Given that (11) holds, the H types willbe net demanders and the L types will be netsuppliers in the water market in this model.We obtain the following excess demand andexcess supply curves as functions of thedemand elasticity and other parameters fromthe profit-maximization problems:

Excess Demand :X ρð Þ= xH ρð Þδ

−X0H =

σH −X0H

� �−

η

2δρwhere σH =

1δ

1 + η

1 + α

� �−

η

2δc0,

ð12Þ

Excess Supply :X ρð Þ=X0L−

xL ρð Þδ

=

X0L−σL

� �+

η

2δρwhere σL =

1δ

α 1+ ηð Þ1 + α

−η

2δc0:

ð13Þ

Trading Market Equilibrium with PossibleBuyer or Seller Market Power

We focus onwithin-basinmarket power for twocases: (1) sellers exercise oligopoly power overcompetitive buyers, and (2) buyers exercise oli-gopsony power over competitive sellers. Eithercase encompasses perfect competition as a lim-iting case.8

We introduce buyer or seller market powerthrough market-power parameters—ξ to mea-sure seller power, and θ to measure buyerpower. Both ξ and θ lie on the unit intervaland are interpreted as indexes of market com-petitiveness. Several papers have used thisapproach to study market power for trade ofagricultural products (e.g. Suzuki et al. 1994;Alston, Sexton, and Zhang 1997; Zhang andSexton 2002; Çakir and Balagtas 2012). Itallows for the complete range of competitiveoutcomes among buyers and sellers to be repre-sented. For example, ξ = θ = 0 gives the per-fectly competitive solution, whereas ξ = 1,θ = 0 depicts seller monopoly, and θ = 1, ξ = 0depicts buyer monopsony. Various degrees ofoligopoly power can be described by 0 < ξ < 1,θ = 0 and various degrees of oligopsony powerby 0 < θ < 1, ξ = 0.These market-power parameters can be

related to conjectural variations models of oli-gopoly or oligopsony and are sometimes inter-preted as conjectural elasticities, whichcapture firms’ expectation about how rivalswill react to a change in the firm’s purchases(θ) or sales (ξ) (Kaiser and Suzuki 2006; Perl-off, Karp, and Golan 2007).9 In this article,the market power parameters are related toperceived marginal revenue (PMR) and per-ceived marginal factor cost (PMFC) curvesbecause these interpretations of θ and ξ areparticularly conducive to graphical represen-tations of the market equilibrium. PMR(X) isrelevant to seller power and is expressed as alinear combination of the monopoly marginalrevenue curve, MR(X), and the marketinverse excess demand curve ED−1(X) for H

8 We do not consider situations where both buyers and sellersmay exercise market power. These cases of bilateral oligopolypower are most often studied in a setting of multilateral bargain-ing, a problem that is fundamentally intractable without imposingstrong assumptions on the bargaining environment (e.g. Inderstand Wey 2003; Dobson and Waterson 2007).

9 The market power parameters ξ and θ do not need to be inter-preted within the conjectural variations framework. They can beinterpreted simply as summary measures of market competitive-ness, that is, as the realizations at any point in time of an unob-served dynamic game among players in the groundwater market.


types (i.e. the marginal revenue curve underperfect competition) with weights given by ξ:

ð14Þ PMR Xð Þ= ξMR Xð Þ+ 1−ξð ÞED−1 Xð Þ:

Similarly, PMFC(X) applies in settings ofbuyer power and is a linear combination of theperfect competitor’s marginal factor cost curve,that is, the inverse supply curve (ES−1(X)), andthe monopsonist’s marginal factor cost curveMFC(X), with weights given by θ:

ð15Þ PMFC Xð Þ= θMFC Xð Þ+ 1−θð ÞES−1 Xð Þ:

To provide a benchmark for comparison tomarket-power solutions, we solve first for theperfectly competitive trading equilibrium byequating L types’ aggregate excess supplyfunction with H types’ aggregate excessdemand function to yield XT ,ρT

� �=

Ω2δ ,

δη σH + σL− X0

H +X0L

� �� . The gains from

trade under perfect competition, GT, calcu-lated as the sum of consumer and producersurplus in the permit market, are GT = Ω2

2δη.

Seller Market Power

From the excess demand curves for permits inequation (12), we derive the PMR curve forthe linear version of the model:

ð16Þ PMR Xð Þ= 2δη

σH −X0H

� �− 1 + ξð Þ2δ

ηX:

Then equating PMR(X) with the sellers’excess inverse supply from (13), we derive theequilibrium volume of permits traded underseller power and express it as a function of theequilibrium quantity under perfect competi-

tion, XSP = 1δ

Ω2 + ξ =

22+ ξ

� �XT . Plugging that

result back into the excess demand curvereveals the equilibrium permit price, ρSP, whichcan be written as a function of the perfectly

competitive groundwater price, ρSP = 2δη

�σH −

X0H − ½σH −σL +X0

L−X0H

2 + ξ ��= ρT + 1− 2

2 + ξ

� �Ωη . These

results are completely summarized in termsof the demand elasticity, the demand shiftand irrigation efficiency parameters, the initialassignment of property rights, and the degreeof seller market power, ξ. If ξ = 0, the equilib-rium outcome reverts to the perfect competi-tion solution. For ξ > 0, the equilibrium

quantity traded is lower and equilibrium pricehigher than under perfect competition.Figure 1 illustrates the model for the case of

seller oligopoly (0 < ξ < 1, θ = 0). The inter-section of the PMR curve with sellers’ excesssupply curve determines the equilibrium per-mit market volume, XSP, which yields equilib-rium groundwater permit price, ρSP. Relativeto traded quantity and price (XT, ρT) at the per-fectly competitive equilibrium (i.e. ξ = θ = 0),seller market power reduces trading, increasesthe permit price, and causes a deadweight lossequal to the shaded area in figure 1.Differentiating XSP and ρSP with respect to

ξ reveals how seller market power affects mar-ket outcomes:

∂XSP

∂ξ= −

1δ

Ω2 + ξð Þ2 < 0,

∂ρSP

∂ξ=2η

Ω2 + ξð Þ2

!> 0:

ð17Þ

The greater the market power exercised bythe sellers, the fewer the permits that aretraded and the higher the groundwaterprice.10 This creates an inefficiency relativeto a competitive permit market, with the dead-weight loss (DWL) due to the exercise of mar-ket power expressed as:

DWL ξð Þ=ðXT

XSP ξð ÞED−1 τð Þ−ES−1 τð Þdτ

=Ω2

2δηξ

2 + ξ

� �2

=GT ξ

2+ ξ

� �2

> 0:ð18Þ

Equation (18) shows that DWL is strictlypositive for ξ > 0 and increases in ξ at anincreasing rate. The expression also showshow the gains from competitive trade, GT,are diluted by deadweight loss as ξ increases.We can use the expression forDWL to char-

acterize the gains from trading under sellermarket power, GSP ξð Þ=GT −DWL ξð Þ=1− ξ

2 + ξ

� �2� �Ω2

2δη = 1− ξ2 + ξ

� �2� �GT . Consis-

tent with the result that DWL is increasing inξ at an increasing rate, gains to trading are adecreasing function of seller market power,and they decrease at an increasing rate.We can also express the welfare loss due to

seller power in the permit market relative to

10 These inequalities always hold because Ω > 0 based onequation (10).


perfect competition in percentage form, % 4GSP, as follows:

ð19Þ %4GSP = −ξ

2+ ξ

� �2

*100,

that is, for the linear model the relative wel-

fare change is solely a function of ξ, makingthe result robust to assumptions on parametersα, δ, η, X0. Figure 2 shows how equation (19)varies over the range of possible market powervalues. As ξ converges to 1 (monopoly case),the surplus from trading is 11.1% smaller thanunder perfect competition.

To assess the distributional impacts of mar-ket power, we study how the gains fromgroundwater trade for buyers and sellers

change as a function of ξ. Equation (20)depicts the percent change in consumer(buyer) surplus,%4 CS, due to seller marketpower relative to that under perfect competi-tion:

The percentage increase in consumersurplus is decreasing in ξ because fewerpermits are traded and at a higherprice, meaning buyers are increasinglyworse off with increasing seller power.Sellers’ relative surplus gains fromtrade, % 4 PS, are, conversely, increas-ing in their market power as equa-tion (21) shows11:

Figure 1. Groundwater permit market with seller market power.

Notes:PMRξ(X) represents the perceivedmarginal revenue curve when sellers exercise market power, ξ. (XSP, ρSP) denote the equilibrium solutions under sellerpower. The figure depicts consumer surplus, producer surplus, and deadweight loss when suppliers exercise market power, ξ, relative to the perfectly competitivecase indicated by (XT, ρT).

ð20Þ %4CS=

ÐXSP ξð Þ0 ED−1 τð Þ−ρSP ξð Þ� �

dτ−ÐXT

0 ED−1 τð Þ−ρT� �dτÐXT

0 ED−1 τð Þ−ρT� �dτ

*100:

ð21Þ %4PS=

ÐXSP ξð Þ0 ρSP ξð Þ−ES−1 τð Þ� �

dτ−ÐXT

0 ρT −ES−1 τð Þ� �dτÐXT

0 ρT −ES−1 τð Þdτ� � *100:


Unlike the efficiency impacts of marketpower, the distributional impacts dependon all of the model parameters. To give asense of these impacts, we set these param-eter values at their levels for our Coachellaapplication, with one exception. As we dis-cuss in detail in the application, Coachellaappears to represent a setting wherein Lproducers would sell their entire alloca-tions in a water market primarily becausethe base value for α is only 0.26, thus con-forming to the special case we discuss indetail in the online appendix File S1.13Togenerate model solutions consistent withthe underlying assumption of thissection that trading equilibria occur in theupward-sloping portion of L types’ excesssupply function, meaning that they applysome of their allocation to own crop pro-duction in the trading equilibrium, weadjust α above its base value in the Coa-chella case to α = .6 for purposes of thissimulation.Figure 3 shows the percent change in buyer/

consumer and seller/producer surplus relativeto that under perfect competition as a functionof ξ, given the adjusted Coachella parametervalues. Consumer surplus declines more rap-idly than the overall gains to trade shown infigure 2. At ξ = 1, consumer surplus is over

55.5% smaller than it would be under perfectcompetition, with most of the loss to buyerscaptured by sellers, given the relatively smalldeadweight loss. At ξ = 1, seller surplus is33.3% greater than that under perfectcompetition.Even a small degree of market power gen-

erates large distributional differences in sur-plus relative to competitive levels. Figure 3shows that an oligopoly index of ξ = 0.2,which is equivalent to that produced in a fivefirm symmetric Cournot equilibrium, resultsin buyer surplus losses of 17.4% and sellersurplus gains of 15% relative to perfect com-petition. A two firm symmetric Cournot equi-librium, which translates to an oligopolyindex of ξ = 0.5, generates surplus changesof −36% and 28% for buyers and sellers,respectively.These distributional impacts are important

because relative winners and losers from atrading environment will help determine thepolitical feasibility of implementing ground-water markets. Although all are absolutelybetter off than under no trade, these distribu-tional impacts may be undesirable from anequity perspective because a large share ofthe gains to trade accrue to those with marketpower, either as buyers or sellers, which couldbe large agribusiness enterprises with signifi-cant landholdings.

Buyer Market Power

In the same way, we can alternatively intro-duce buyer market power (ξ = 0, θ > 0) intothe framework. The perceived marginal factorcost curve for the linear version of themodel is:

ð22Þ PMFC Xð Þ= −2δη

X0L−σL

� �+ 1+ θð Þ2δ

ηX :

Equilibrium traded quantity, denoted XBP,is determined by the intersection of PMFC(X) with buyers’ excess demand and can beexpressed as a function of the perfectly com-petitive outcome: XBP = 1

δΩ

2 + θ =2

2+ θ

� �XT . The

equilibrium groundwater permit price, ρBP,is determined where XBP intersectsthe excess supply curve: ρBP =2δη

σH −σL +X0L−X

0H

2 + θ −X0L + σL

� �= ρT + 2

2+ θ −1� �Ω

η .

These equilibrium outcomes, which are

Figure 2. The effect of seller market power(ξ > 0) on total gains from trade.

Notes: The figure shows how the gains from trade, expressed as a percentagechange from the benefits under perfect competition (PC), change as marketstructure varies.

11 Equations (20) and (21), expressed as functions of parame-ters from the linear model, can be found in Appendix B.


symmetric to the seller power scenario, showthat buyer power will depress trade in thewater market and reduce price relative to theperfectly competitive equilibrium, with botheffects increasing as a function of θ. In a similarfashion, we can derive the welfare loss due tobuyer power in the permit market relative toperfect competition in percentage form: %4GBP = − θ

2 + θ

� �2*100. Under either buyer orseller market power, a larger market powerparameter implies fewer permits traded, effi-ciency loss relative to the perfectly competi-tive equilibrium, and transfer of substantialportions of the gains to trade into the handsof the entities exercising market power.

Application to California Agriculture

Groundwater management is at the forefrontof water policy debates in California. Ground-water accounts for 40% of the agriculturalwater supply on average (DWR 2016), and

several areas throughout the state have seensignificant declines in groundwater storage(Faunt, Belitz, and Hanson 2009; Famiglietti2014). In an effort to maintain a reliablegroundwater supply, California’s SustainableGroundwater Management Act (SGMA) of2014 provides a statewide framework for localagencies to manage groundwater. SGMArequires overdrafted basins throughout Cali-fornia to reach and maintain long-term stablegroundwater levels. However, SGMA is silentas to how groundwater agencies shouldachieve sustainability, even though the costeffectiveness of different policy instrumentsmay vary substantially.In what follows, we apply the model to esti-

mate the gains from groundwater trade for theCoachella Valley in Riverside County, CA,under pumping restrictions likely to beimposed under SGMA. The Coachella Valleyis an ideal setting for application of our modelbecause it is both subject to mandates underSGMA and exhibits structural elements thatmay give rise to imperfectly competitivegroundwater trade. In particular, the Valley

Figure 3. Distributional effects of market power.

Notes: The top panel shows how buyers’ gains from trade, expressed as a percentage change from consumer surplus under perfect competition, change as marketstructure varies. Likewise, the bottom panel shows how sellers’ gains from trade change as market structure varies. Parameter values are from the Coachellaapplication except for α as explained in the main text. Appendix A contains the linear model expressions for (20) and (21).


is home to large grower-shippers and growers’organizations, suggesting the potential formarket power in an emergent permit market.The Coachella Valley receives only four

inches of rain a year on average, making itsagriculture dependent on groundwater andimported surface water for irrigation. In thelate 1940s, a concrete-lined aqueduct was con-structed to transport surface water over120 miles from the ColoradoRiver to the Coa-chella Valley.13 Enabled by irrigation, thisregion now exhibits roughly 65,000 acres incrop production with an annual value of overhalf a billion dollars. In addition to producing95% of the nation’s dates, the area also pro-duces table grapes, citrus fruits, bell peppers,and other vegetables (ACO 2016).14

This section first provides background onSGMA and the specific mechanisms by whichmarket power may arise under SGMA-inspired groundwater trading. We then pre-sent the model parameters for Coachella,which is followed by estimation of the gainsfrom groundwater trade and a sensitivity anal-ysis. Results show that the economic surpluswith competitive trade is almost 40% largerthan under the no-trade equilibrium. Further-more, the gains remain large in the presence ofmarket power and over a reasonable range ofother model parameter values, indicating thatresults are likely to generalize to other basinswhere trading might occur.

Background

Under SGMA, groundwater agencies govern-ing overdrafted basins must reduce basin-widepumping to achieve groundwater sustainabilitytargets. SGMA affects 127 of 515 basins in Cal-ifornia, which account for 96% of the

groundwater pumping in the state (DWR2019). A logical means to achieve this goal istometer pumping and establish individual prop-erty rights for groundwater that restrict aggre-gate pumping volumes below open-accessoutcomes.However, the assignment of propertyrights for groundwater use will cause efficiencylosses in the absence of water trading if the reg-ulating agency lacks perfect information and/orfaces political or legal restrictions to setting allo-cations. Thus, groundwater trading representsan avenue for reaching basin sustainability tar-gets while minimizing efficiency losses.Given that trading will likely be restricted to

the boundaries of a given hydrologic region(Green Nylen et al. 2017), the structure of Cali-fornia agricultural production and marketing,and the legal environment regarding coalitionformation may give rise to consolidation ofgroundwater rights when they become properlydefined under SGMA, making market poweran important consideration in the evaluationof groundwater markets. First, it seems likelythat these markets will evolve in settings wherebuyer or seller coalitions can emerge withoutlegal impediments, which may lead to groupsof players coordinating interests in cartel-likefashion. For example, in many groundwaterbasins in California, multiple groundwateragencies are emerging to jointly manage thegroundwater on a shared basin (Conrad et al.2018). These agencies may be able to operateas joint buyers or sellers on behalf of farmersin their jurisdictions (Rosen and Sexton 1993).Other coalitions could take the form of

growers’ associations, cooperatives, or largedownstream processors who purchase inputson behalf of their growers. Both horizontaland vertical coordination of farmers throughsuch coalitions is common in agriculture. Forexample, dairy cooperatives have been shownto exercise market power by coordinatinginterests of dairy producers (Çakir and Balag-tas 2012). Downstream processing or packingfirms also commonly provide inputs to farmerssupplying raw products to their operations. Ifsuch vertical coordination were extended tothe purchase or sale of groundwater, the rele-vant concentration of buyers and sellers for awater market would be the processing/pack-ing/shipping stage rather than the farm pro-duction stage of the market chain.Finally, in some instances grower-shippers

themselves may be large enough to exercisemarket power as either buyers or sellers. Recentresearch points to some evidence of insider trad-ing occurring among water market participants

13 Although between 1/3 and 1/2 of the Coachella Valley’s irri-gation needs are met with Colorado River water in a given year,the fact that the valley has no natural surface water flow meansgroundwater pumping is unlikely to have a meaningful effect onsurface water hydrology. A spatially explicit aquifer model couldaccount for this physical groundwater-surface water interactionin settings where it is relevant. See, for example, Kuwayama andBrozovi�c (2013) for an illustration.

14 Although the Valley produces most of the nation’s dates, theCoachella production is only about 0.3% of world productionbased on UN Food and Agriculture Organization (FAO) statistics(FAO 2019). The relevant geographic market for dates appears tobe worldwide; for example, the U.S. is both an importer andexporter of dates. Given its tiny share in the world market, Coa-chella producers as individual sellers or as a group have no influ-ence on world date prices, meaning impacts on the outputmarket of the type considered by Hintermann (2011, 2017) arenot an issue. Our analysis suggests that this conclusion also appliesto the other commodities produced in Coachella.


in Australia’s Murray-Darling Basin, suggestingthat individual players can have impacts onmar-ket outcomes (de Bonviller, Zuo, and Wheeler2019). In California, the agricultural sector hasseen significant structural change over the lastseveral decades, leading to fewer and larger ver-tically integrated farming-shipping operations(Rogers 2001).Most groundwater rights in Cali-fornia are overlying rights based on ownershipof the land above the aquifer,15 so permits forpumping based on land holdings may directlyconcentrate permits in the hands of a relativelyfew large landowners.

Model Parameters

Our model characterizes the gains from ground-water trade as a function of six market parame-ters: the heterogeneity of demand forgroundwater across users (α), the price elasticityof groundwater demand (η), the total allowableextraction (X0), the irrigation efficiency (δ), theprice elasticity of groundwater supply (ε), andthe degree of buyer (θ) or seller (ξ) marketpower. All except X0 are pure numbers, and X0

is converted to that formbyexpressing it as a per-centage reduction from the open-access solutionrequired to achieve sustainability of the aquifer.In addition, a rule for apportioning X0 amongusers is needed. An allocation of pumping

permits that is proportional to land holdings isthe most likely scenario and the one we assumefor purposes of this application. Table 1 outlinestheparameters, all ofwhichwere estimatedusingdata from the Coachella Valley, and includeseither thedata source for theparameter or abriefsummaryof how theparameterwas estimated.Adetailed discussion of the methods used to esti-mate each parameter is provided inAppendixB.We encounter two complications relative to

the framework presented in the conceptualmodel in applying the model to Coachella Val-ley based on the parameter estimates summa-rized in table 1. First is that the measures ofgains to trade derived in the conceptual modelreflect the case when permit allocations con-strain pumping for both types. However, forthe baseline parameter values, the allocationconstraint is not binding for the L types inCoachella for some positive quantity range.In this case, the inverse excess supply curve isflat (ES−1(X) = 0) up to the sales volume,�X =X0

L−σL, where the allocation binds forthe L types, so they are only willing to sellquantities in excess of this amount at a positiveprice. Second is that under the baseline param-eters, L types sell their entire allocations inequilibrium, which corresponds to the casewhere acreage is idled in favor of rightsholders becoming exclusively water mar-keters.16 ES−1(X) is vertical at the level ofrights, X0

L, allocated to L types. ES−1(X) is

Table 1. Coachella Valley Parameter Estimates

Parameter Symbol Estimate Source

Estimated by comparing interceptsDemand shift index 0 < α < 1 .26 of H and L type water demands, which

were proxied with estimates of waterdemands for Coachella’s top ten crops.

Demand elasticity η .17 Absolute value of estimate from Brunoand Jessoe (2018).

Total allowable Calculated by comparing averageextraction X0

L +X0H

.8 annual basin total pumping withCVWD’s annual overdraft estimates.

Irrigation efficiency δ .85 Rogers et al. (1997)Calculated with a point on the supply

Supply elasticity ε 1.03 curve, an engineering formula thatrelates costs to depth to the water table,and an estimate of aquifer storativity.

Portion of cap - .53 Assumed to be equivalent to theto H types proportion of acreage in H-type crops.

15 California’s correlative rights doctrine gives landowners theright to use the groundwater underneath their land, making itlikely that land ownership will determine how property rights areallocated under the SGMA. For cases that clarified the legal tiesof groundwater to the land, see City of Barstow v. Mojave WaterAgency (2000) and City of Pasadena v. City of Alhambra (1949).

16 This outcome would represent a groundwater case of theland-fallowing equilibrium for surface-water trades discussed, forexample, in Howitt and Sunding (2003).


thus a piecewise linear function for L types inCoachella, as shown in figure 4, which depictsthe competitive equilibrium when ED−1(X)intersects ES−1(X) in its vertical portion. Thegains from perfectly competitive trade underthese conditions are depicted by the green-shaded area. An interesting aspect of the casewhere the competitive equilibrium occurs onthe vertical portion of ES−1(X) is that a rangeof prices clears the market, making it impossi-ble to render a sharp prediction as to themarket-clearing price. This range is illustrated

in figure 4 by ρ∈ ρ,�ρh i

.

Results for Coachella Valley, California

Equation (23) expresses the percentagechange in economic surplus from allowingtrade assuming perfect competition in thegroundwater market. D−1

i Xð Þ denotes theaggregate inverse demand curves for pumpedwater for type i, and ED−1(X) and ES−1(X)denote inverse excess demand and supplyfunctions for the H and L types, respectively.The numerator of (23) expresses total surplus

gains as the difference between ED−1(X) andES−1(X) of the H and L types (i.e. the shadedarea in figure 4), whereas the denominatorrepresents economic surplus generated fromgroundwater pumping in the no-trade settingwhen the initial allocation is nonbinding forthe L-type producers, who would, in theabsence of trading, apply their unconstrainedoptimum, X*

L:17

ð23Þ %4

=ÐXT

0 ED−1 τð Þ−ES−1 τð Þ� �dτÐX0

H0 D−1

H τð Þ−c0� �dτ +

ÐX*L

0 D−1L τð Þ−c0� �

dτ*100:

Given the aforementioned model parame-ters, the percent change in surplus due togroundwater trading in Coachella Valley rela-tive to command and control is 36% or $30.42million annually. In equilibrium, 86,430 AF ofgroundwater would be traded at a pricebetween ρ=$227 and �ρ =$235 per acre-foot.Details of these calculations can be found in

Figure 4. Market-level gains from trade (constraint non-binding for L types).

Notes: The figure denotes excess supply and excess demand curves when α is small, allocations differ between types, c0 = 0, and the constraint is nonbinding for the

L types at the initial endowment, X0L . ES

−1 and ED−1 are derived according to Equations (12) and (13). �X denotes the trade volume when sellers’ allocation

constraint binds. The shaded area illustrates the gains from trade. Here, the L types sell their entire allocation of pumping rights, that is,X0L =XT , and themarket-

clearing price resides between ρ and �ρ.

17 X*L is computed to reflect reduced pumping costs caused by

H types’ pumping being constrained by their binding allocations.


Appendix A. The quantity traded is 47% ofthe total water available for trade (or 37.6%of open-access equilibrium quantity), and themarket-clearing price is more than twice aslarge as the average marginal extraction costsprior to the regulation.18

Dividing the estimated gains of $30.42 mil-lion by the quantity traded reveals an annualaverage value per acre-foot traded of $352.These average gains to trade are larger thanthe $0-148/AF gains from groundwater tradein the Republican Basin, Nebraska, estimatedby Thompson et al. (2009), with the disparitydue in large part to the heterogeneity of cropsgrown in Coachella and differences in theirvalue relative to the cropping pattern in theRepublican Basin, which consists primarily ofcorn and wheat. Our estimate falls near themidpoint of the range of values reported byHagerty (2019) of $88-697/AF for the state-wide wholesale surface water market inCalifornia.

The Effect of Market Power on the Gainsfrom Trade

The theoretical model lends insight regardingthe efficiency and distributional impacts ofmarket power in a water market for the casewhen equilibrium occurs along the upward-sloping portion of sellers’ excess supply func-tion. In this case, as illustrated in figure 2, thegains from trade in the linear model can bereduced by at most 11% under monopoly(ξ = 1, θ = 0) or monopsony (θ = 1, ξ = 0),regardless of the initial allocation of permitsor the specific estimates for the otherparameters.

This result is not directly relevant to theCoachella application because, given our baseparameters, the competitive equilibriumoccurs in the vertical portion of excess supply,with L types selling their entire allocations toH types. This means that there is a range ofseller market power, ξ > 0, where PMR(X|ξ)

intersects ES−1(X) along its vertical portion.For this range, there is no deadweight lossfrom the exercise of seller power because thetrade volume remains atX0

L:A similar conclu-sion holds for buyer power. Buyers’ PMFC(X|θ) function becomes vertical at X0

L, mean-ing that a range of values exist for θ > 0 wherePMFC intersects buyers ED−1 function in itsvertical portion. Magnitudes of buyer powerin this range do not cause a deadweight loss.Further, predictions regarding the impacts ofmarket power on the distribution of welfareare not sharp, given that the competitivemodel does not yield a clear prediction forprice, as discussed earlier and as illustrated infigure 4.19

We can solve for the range of values of ξ andθ for which the trading equilibrium remains onthe vertical portion of excess supply to deter-mine the range of market power realizationsfor which some positive DWLoccurs. For bothseller and buyer market power, values of θ andξ ∈ [0, 0.0223] result in a trading equilibriumthat remains on the vertical portion of excesssupply, given other parameter values esti-mated for Coachella. Market power realiza-tions of ξ or θ greater than 0.0223 will resultin reduced trade volumes and positive DWL.Thus, any significant exercise of market powerin Coachella (i.e. ξ, θ > 0.0223) will reducetrading below the competitive equilibriumand cause a DWL, with the result from thegeneral model of a maximum 11.1% surplusreduction from pure monopoly or monopsonyrepresenting a close approximation forCoachella.

Sensitivity Analysis

We perform a sensitivity analysis to gauge therobustness of results to plausible alternativevalues for the market parameters. This exer-cise also helps in understanding how the Coa-chella results might generalize to othergroundwater trading environments. Agricul-ture fed by different groundwater basins willfeature different crops and groundwater con-ditions than Coachella, meaning that theirgroundwater market environments will18 These estimates assume that trading is costless. In reality, a

trading platform would need to be established, which would likelyinvolve both fixed and variable (per trade) costs. Variable costs totrading would create a wedge between sellers’ price paid andbuyers’ price received and decrease trading volume and surplusto trading below the estimates reported here. Other costs thatmight be associated with water markets, such as means to monitorextraction and enforce compliance are already in place for Coa-chella and, in general, will need to be implemented under SGMAwhether or not an agency establishes a trading mechanism.

19 In online appendix File S1 where this case is studied in detail,we argue that the presence of seller power makes it likely thatequilibrium price is set where the trade volume, X0

L intersectsbuyers’ inverse excess demand. Conversely the presence of buyerpower makes it more likely that equilibrium price is set at the kinkpoint where ES−1 becomes vertical.


feature distinct values for most or all of themodel parameters.Figure 5 shows for the perfectly competitive

markets case how the gains from groundwatertrade change as other conditions of the marketvary, where the surplus with trade is expressedas a percentage change in surplus from the no-trade (command-and-control) scenario.Panels A - E respectively show how the gainschange as we vary the demand shift parame-ter, α; the demand elasticity η; the total allow-able extraction on the basin relative to openaccess, X0; the groundwater supply elasticity,ε; and the initial allocation between types.The base parameter values are shown with avertical line in each panel of figure 5.The first panel of figure 5 shows to no sur-

prise that the gains to groundwater decrease

as α increases, making water demands moresimilar between types. The gains from tradeconverge to zero as α ! 1. Coachella Valleygrows a large diversity of crops, with no morethan 15% of total acreage in any single crop,a level of cash crop diversification that is notuncommon in California but is rarer else-where. The estimate of α for Coachellaremains small (and the gains to trade substan-tial) when different bundles of crops are con-sidered for the H and L types. The αparameter ranges between .21 and .43 acrossfive different plausible classifications of thecrops into low and high crop bundles.The second panel of figure 5 shows the per-

centage increase in surplus from trade as afunction of the demand elasticity. More elasticdemands lead to a greater percentage increase

Figure 5. Gains from trade sensitivity analysis.

Notes: Panels A through E in this figure show, respectively, the sensitivity of the surplus gains result to changes in five parameters: the demand shift index, theprice elasticity of groundwater demand, the total allowable extraction, the allocation of permits between types, and the supply elasticity. Each panel shows thegains from trade, expressed as a percent increase from the benefits under a no-trade regime. The vertical black lines denote the Coachella Valley parameterestimates of table 1.


in the gains from trade. More trading occurswith more elastic demands. We see from panelB that the percent increase in benefits is largefor a wide range of elasticity values.

Notably, the demand elasticity plays an addi-tional role in determining economic surplus totrade in the presence of seller market powerbecause the distortion from a given exercise ofmarket power as measured by ξ depends onthe demand elasticity, with more inelasticdemands increasing the distortion for a given ξ.This effect reinforces the impact shown in panelB because more inelastic demands will exacer-bate the surplus lost for a given value of ξ.

Panel C depicts the gains with trade as afunction of the total endowment of propertyrights for pumping, that is, the total allowable

extraction on the basin as a percentage of theopen-access extraction. As expected, the per-cent gains with trade are decreasing as theendowment increases. However, even asthe total allowable extraction approaches theaggregate quantity pumped in open access(i.e. X0

L +X0H ! 1), there are gains from trade

because we retain an inefficient allocation ofpermits between types. Thus, even if the totalrestriction on pumping is small relative to theopen-access consumption, an endowment ofproperty rights that does not equate marginalvalue products across types will reduce eco-nomic surplus relative to a trading scenariothat enables an efficient allocation.Panel D of figure 5 shows the percentage

change in surplus from trade as a function

Table 2. Top Ten Crops Grown in the Coachella Valley, CA

Revenue Applied RevenueCrop Acreage Revenue per acre water (AF) per AF Type

Dates 7,964 $40,110,000 $5,036 8.0 $630 LowGrapes 7,379 $143,222,000 $19,409 3.0 $6,470 HighBell peppers 5,288 $77,700,000 $14,693 2.0 $7,347 HighLemons 5,200 $110,605,000 $21,270 2.9 $7,334 HighCarrots 4,777 $28,700,000 $6,007 2.5 $2,403 LowBroccoli 2,475 $14,561,000 $5,883 1.7 $3,461 LowSweet corn 1,883 $11,174,000 $5,934 5.0 $1,187 LowLettuce 1,600 $12,480,000 $7,800 1.2 $6,500 HighWatermelon 1,525 $14,860,000 $9,744 3.0 $3,248 LowMandarins 1,475 $19,721,000 $13,370 2.5 $5,348 High

Notes: Revenue and acreage data come from the Coachella Valley 2016Acreage andAgricultural CropReport. Applied water by crop in acre feet (AF) per yearcomes fromUCCECost andReturn Studies. Revenue per acre-foot of water is calculated by dividing per-acre revenues by the average acre feet of applied water.

Figure 6. Surplus under command and control.

Notes: The right figure depicts the aggregate surplus for farmers of type i under command and control with a binding constraint of δX0i . Surplus is shown by the

shaded area. The left figure depicts the market-level outcomes, whereD−1(x) represents aggregate inverse demand. Open-access equilibrium quantity and price(x*, c*) were normalized to (1, 1). Marginal extraction costs at the binding constraint are represented by c0. Individual farmers are assumed to take costs as given.


of the allocation of permits between types.The benefits from trade are strongly influ-enced by the initial allocation. Given the sig-nificant legal, political, and informationbarriers preventing a discriminatory alloca-tion of pumping rights that would lead toan efficient result, this figure shows theimportance of allowing trade in this situation.In our Coachella application an efficient allo-cation would put most of the permits in thehands of H-type growers, so an initial alloca-tion that skews in the opposite direction pro-duces even greater gains to trade relative tothe baseline solution.Finally, panel E shows the percentage

change in surplus from trade as a functionof the groundwater supply elasticity. We con-sider a range for ε that encompasses thepotential storativity values for the Coachellagroundwater basin estimated by Tyley(1974). The percentage change in surplus isincreasing in ε, but at a very small rate,showing that the gains result is extremelyinsensitive to our storativity assumption ofs = 0.11. The pumping supply elasticity is arelatively unimportant parameter becausepumping costs are fixed by the allowedpumping volume, X0. The main role of ε inthe trading model is to calibrate reducedpumping costs relative to open access. Thevalue of ε, however, assumes increasedimportance in the presence of buyer powerbecause, analogous to the seller power case,the distortion from a given value of θ andthe resulting surplus loss is magnified themore inelastic is supply.Overall, the efficiency gains of groundwater

markets relative to the no-trade scenario arelarge on a percentage basis for a broad rangeof the model parameters.20 Because theseparameter ranges represent a wide spectrumof market conditions, they suggest that ourresults for Coachella are likely to hold moregenerally, that is, the gains from groundwatertrade can be quite large for manygroundwater-dependent agricultural regions.

Conclusion

Regulation of groundwater is on the near-termhorizon for California and likely for many otherjurisdictions as well, as climate change makesrainfall and surface-water supplies more vari-able and in many cases less bountiful, therebyplacing increased demands on the availablegroundwater supplies. Due to informational,legal, and political impediments, regulators willhave little opportunity to assign property rightsto groundwater in an economically efficientmanner, opening the door to groundwater mar-kets as a device to achieve allocative efficiencyand increase returns to agricultural stakeholdersoperating on a restricted basin.The existing literature on surface water trad-

ing (e.g. Sunding et al. 2002; Hagerty 2019) pro-vides little guidance to regulators andstakeholders in understanding how groundwa-ter markets may operate, as most surface watertrades have been through bilateral negotiationsbetween water-supply organizations, whereasgroundwater rights are likely to be in the handsof individual landowners and limited by thehydrologic connectivity of the basin over whichthey operate. This article has been devoted tounderstanding the essential economic factorsthat will impact emergent groundwatermarkets.Our theoretical model, when expressed in its lin-ear form, described a groundwater trading equi-librium in terms of six market parameters thatcanbe expressed as pure numbers: the heteroge-neity of demand for groundwater across users(α), the price elasticity of groundwater demand(η), the total allowable extraction defined rela-tive to the open-access equilibrium (X0), the irri-gation efficiency (δ), the price elasticity ofgroundwater supply (ϵ), and the degree of buyer(θ) or seller (ξ) market power.We argued that buyer or seller market

power could be a key consideration in manygroundwater trading markets due to theirrestricted geographic coverage and barriersto entry, high and increasing concentrationamong producers and processor-shippers formany industries, and relative lack of impedi-ments to formation of buyer or seller coali-tions. Results from applying a flexibleoligopoly-oligopsony model to groundwatertrading showed that either buyer or sellerpower had limited impacts on the overall gainsto trade but that even relatively modest buyeror seller power could tilt the gains from tradesignificantly in the direction of the entitiesexercising the power.

20 If we adjust α above its base value in Coachella to evaluatesensitivity of model parameters when the trading equilibria occurin the upward-sloping portion of the excess supply function(instead of the vertical portion as in the base Coachella case), weobserve that higher α values shift the curves downward, reflectinglower gains to trade with more similar demands but with negligibleimpact on their shape. These results are available upon requestfrom the authors.


We applied the model to the Coachella Valleyin California where we were able to obtainCoachella-specific estimates for each of themodel’s parameters except for market power.Given the basin-wide reduction in groundwaterpumping of 20% needed to achieve sustainabilityof the basin, we estimated that the economic ben-efits with perfectly competitive trade are 36%greater than that under a “command-and-con-trol” scenario where pumping is restricted buttrade is not allowed. Simulations that varied mar-ket conditions showed that the gains from traderemained large over a reasonable rangeof param-eter values, meaning results are likely to general-ize to other basins where trading might occur.

Given evidence that the presence of buyeror seller power has only a minor impact onthe overall gains from trade, concerns overmarket power should not constitute a compel-ling argument to avoid trading. However, dis-tributional impacts, which may beconsiderable, may impact some stakeholders’incentives to support or oppose a tradingregime. The majority of the gains from tradeaccrue to the players with market power; thesewill tend to be large operations that may alsowield considerable political influence. None-theless, both buyers and sellers will benefitoverall from trade even with severe marketpower. Concerns about market power maybe better directed at the initial allocation ofpermits among players, because the closerthe initial allocation is to the efficient outcome,the less are the impacts of market power.

Our framework quantified the gains fromtrade in the short-run, given fixed croppingpatterns and irrigation technologies. One com-pelling area for future research is the identifi-cation of market impacts over a long-runhorizon, whereby all potential margins ofadjustment, including cropping pattern andirrigation efficiency, are considered. A secondand highly challenging potential area for fur-ther research is to consider trading in an envi-ronment of bilateral oligopoly. A fullerunderstanding of the potential of groundwatermarkets will be important as many agriculturalregions continue to grapple with changes togroundwater and climate in the years to come.

Supplementary Material

Supplementary material are available atAmerican Journal of Agricultural Economicsonline.

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Appendix

Simulation Details

Herein we provide detailed explanations ofthe methods for estimating model parameters,the total gains from trade, and the distribu-tional impacts of market power.

Model Parameters

The demand-shift index, α, captures the degreeof heterogeneity of water demands amonggroundwater users. This parameter was calcu-lated with data from Riverside County’s 2016Crop Report for Coachella Valley andUniversity of California Cooperative Exten-sion (UCCE) Cost and Return Studies.21 Wefocus on the ten leading crops, which are listedby total acreage in table 2, along with informa-tion on total production value and averageapplied water per acre. To estimate α, we firstobtained a point on the average product curveof irrigation water for each crop by dividingper acre production by applied water per acre.Given the assumption of linear demands, thispoint can then be related to a point on irriga-tion water’s marginal value product (MVP)curve with information on the per-unit outputprice as shown in Bruno (2018).

We extrapolated this point to the MVPintercept or “choke point” using the demandelasticity estimate described below and com-pared these estimates across crops. In thisfashion, the demand shift index can be com-puted for any pair of crops. For consistencywith the conceptual model, we conducted thesimulation with H and L demand types and asingle value for α. To this end, based on table 2,

we bundled dates, sweet corn, carrots, water-melon, and broccoli as the L crops and gener-ated an H crop bundle consisting of tablegrapes, lemons, bell peppers, romaine lettuce,and mandarins. We calculated an acreage-weighted average of the intercepts for eachcrop’s water demand to arrive at an interceptvalue for the bundle. The ratio of the MVPintercepts between theH and L bundles servesas the estimate of α. The sensitivity analysisexplores different combinations for the Hand L bundles.The Coachella Valley Water District

(CVWD), the largest water agency in the val-ley, meters extraction at each groundwaterwell within its service area that pumps morethan 25 acre feet (AF) per year and chargesvolumetric prices for groundwater pumpingknown as “replenishment assessment charges”or RAC. TheRAC ranged from $0 to $129/AFacross three subregions from 2000 to 2016,with an average over this time period of $65/AF. These plausibly exogenous prices wereutilized by Bruno and Jessoe (2018) to esti-mate a price elasticity of groundwater demandfor agricultural users in the Coachella regionbased on monthly panel data on well-levelgroundwater extraction and prices.22 Theirresearch design exploits the deployment ofthree location-based pricing regimes withinthe Coachella Valley. They find that demandis inelastic, with a preferred point estimate of−0.17 that is statistically significant and robustto alternative specifications.23

The total endowment of groundwater pump-ing rights, X0

H +X0L, was estimated to be 80%

of Coachella’s open-access groundwaterextraction, implying that a 20% reduction inwater use is needed to correct for basin over-draft. This figure was calculated by comparingthe average annual groundwater extractionwithin the service area of the CVWD to thatwhich would be allowed if it were to eliminateits reported 70,000 AF/year of groundwater

21 UCCECost andReturn Studies for all available commoditiescan be accessed here: https://coststudies.ucdavis.edu/en/current.

22 The well pumping data do not indicate the crop(s) being fedfrom the well, precluding crop-specific elasticities from being esti-mated. Bruno and Jessoe (2018) estimate an average waterdemand elasticity across all Coachella crops.

23 A handful of other recent papers have estimated a price elas-ticity of groundwater demand including Gonzalez-Alvarez, Kee-ler, and Mullen (2006) for Georgia, Hendricks and Peterson(2012) for Kansas, and Smith et al. (2017) for Colorado. The esti-mates are difficult to compare directly due to differences in cropsproduced, sources of price variation, and rainfall. Bearing thiscaveat in mind, the Bruno-Jessoe estimate lies about midwaybetween the 0.1 base estimate of Hendricks and Peterson andthe 0.27 base estimate of Gonzalez-Alvarez et al. The estimatesof Smith et al. are somewhat more elastic, ranging from 0.5 to 0.77.


https://coststudies.ucdavis.edu/en/current

overdraft (CVWD 2016). Proportional toacreage held by each type, 53% of this capwas allocated to the H types in the basesimulation.Drip technology is widely used in the Coa-

chella Valley. An irrigation efficiency of 85%was chosen for δ because it is the reportedaverage distribution efficiency for drip tech-nology (Rogers et al. 1997) and is also the effi-ciency rate used in UCCE Cost and ReturnStudies for drip irrigation systems(O’Connell et al. 2015).Estimating ε requires specifying the func-

tion for marginal groundwater extractioncosts, c(X), for Coachella. These costs con-sist of the sum of the incremental energyextraction costs and the volumetric pump-ing charge (i.e. the RAC) imposed byCVWD. Energy extraction costs are anincreasing function of total pumping, X,and, thus, a reduction in basin-wide ground-water use under SGMA will decrease themarginal energy pumping costs faced byusers on the basin.24

To estimate marginal energy extractioncosts, let h(X) represent the depth to the watertable (in feet) from the surface, which dependson total basin pumping X (in acre feet). Let pe

represent the electricity price ($ per kwh) andthe constant ϕ denote the kwh requirement toraise an acre foot of water one foot (kwh perAF per foot). The marginal energy cost perAF of groundwater extracted can then beexpressed as ϕpeh(X) (Rogers and Alam2006), thereby yielding the following expres-sion for c(X) for the Coachella Valley: c(X) = ϕpeh(X) +RAC. Imperial Irrigation Dis-trict, the local energy provider, reportspe = $0.0952 as the 2016 per kwh electricityprice faced by irrigated agriculture in the Coa-chella area andRogers andAlam (2006) reportϕ = 1.551 to be the kwh requirement to lift oneAF of water one foot.25 Assuming an averagedepth to thewater table of 108 feet and average2016 RAC rates of $84.60, the imputed price of

groundwater for 2016 would be $100.50 perAFon average (DWR 2019).26

In order to capture how a cap on pumpingwould affect pumping costs, we need to specifya relationship for h(X), that is, how the depthto the water table changes with basin-widepumping. Informed by Fetter (2001), weassume h Xð Þ= X

s , so that h0(X) > 0, which

implies c0(X) > 0. Here s represents the stora-

tivity of the aquifer, defined as the volume ofwater released from groundwater storage perunit decline in the depth to water in a well(Fetter 2001). Thus, the inverse storativitycan be interpreted as the change in the depthto the water table due to a change in ground-water extraction. Storativity is a pure numberthat is equal to the specific yield of an aquiferif groundwater is unconfined (Fetter 2001).Although both confined and unconfinedgroundwater conditions are present in theIndio subbasin beneath the Coachella Valley(DWR 2015), we focus on the storativity foran unconfined aquifer for simplicity. Tyley(1974) estimated specific yields ranging from0.06 to 0.15 for the unconfined parts of theIndio basin. We take the simple average ofthese values for a baseline storativity valueof 0.11.The inverse of the supply elasticity in terms

of applied water (x = δX) evaluated at theopen-access equilibrium quantity and price,normalized to (x*, c*) = (1, 1), is thus1ε =

∂c xð Þ∂x

x*c* =

ϕpe

δs = 1:58, given s = 0.11,ϕ = 1.551, pe = $0.095, and δ = 0.85.We use thisinverse supply elasticity to parameterize themarginal cost function defined in equation (4)to estimate themarginal pumping costs given a20% reduction in basin-wide groundwaterextraction, c X = :8ð Þ= 1− 1

ε

� �+ 1

ε :8ð Þ= :68, thatis, marginal pumping costs under the con-strained allocation are estimated to be about68% of their normalized value under openaccess.The final parameter needed to estimate

gains to groundwater trade in the CoachellaValley is the level of either buyer or sellermarket power. Because groundwater tradingis only on the horizon, there exist no data fromactual trades to estimate values for θ or ξ

24 We assume for simplicity that marginal extraction costs arethe same across all users despite users in different regions of Coa-chella facing location-based volumetric groundwater charges. Thisassumption impacts the gains from trade only if H or L types areconcentrated in one of the three pricing regions. However, exam-ination of the cropping patterns indicates no concentration by typein any single region.

25 The constant ϕ = 1.551 accounts for pump efficiency byassuming an electric pump output of 0.885 water horsepower-hourper kwh (Rogers and Alam 2006).

26 Mean depth to the water table was calculated by averagingthe depths in 2016 across the ten active irrigation wells in the Coa-chella Valley Indio sub-basin that are monitored and reported tothe California Statewide Groundwater Elevation Monitoring(CASGEM) program. At pumping quantities, depths, and pricesobserved in Coachella in 2016, c* = (1.551) * ($0.0952) *108 ft + $84.60 = $100.50.


using, for example, methods of the new empirical industrial organization, as discussed in Kaiserand Suzuki (2006), Perloff, Karp, andGolan (2007), and elsewhere. Our approach is to first solvefor the market equilibrium price, trade volumes, and surplus measures under perfect competi-tion and then show their sensitivity to alternative magnitudes of seller market power, as mea-sured by ξ > 0, recognizing that buyer power (θ > 0) will yield a similar impact based on thetheoretical model.

Gains from Trade

Recall the excess supply relationship defined by the set of all (X, ρ) ∈IR2 and X ≥ 0 such that:

ρ=ES−1 Xð Þ= 0,X ≤ �X

ρ=ES−1 Xð Þ= 2ηδσL−

2ηδX0

L +2ηδX , �X ≤X ≤X0

L

X ρð Þ=X0L,ρ>

2η

α 1+ ηð Þ1 + α

� �,

8>>>>><>>>>>:

where �X =X0L−σL and σL = 1

δα 1 + ηð Þ1 + α − η

2δc0. Additionally, recall ED−1 Xð Þ= 2δ

η σH −X0H −X

� �where σH = 1

δ1 + η1 + α −

η2δc

0 and inverse demand functions for H and L types:

ðB:1Þ D−1H Xð Þ= 2

η

1 + η

1+ α

� �−2ηδX ,

ðB:2Þ D−1L Xð Þ= 2

η

α 1 + ηð Þ1 + α

� �−2ηδX :

We solved the following expression in the main text for gains from trade as a percentage ofsurplus under command and control:

ðB:3Þ %4 =

ÐXT

0 ED−1 τð Þ−ES−1 τð Þ� �dτÐX0

H0 D−1

H τð Þ−c0� �dτ +

ÐX*L

0 D−1L τð Þ−c0� �

dτ*100

where c0 = c X0H +X0

L

� �is the marginal pumping cost under the constrained equilibrium and

X*L = 1

δα 1 + ηð Þ1 + α − η

2c0

� �<X0

L is defined as the quantity pumped by unconstrained L types,

D−1L X*

L

� �= c0. Figure 6 illustrates the surplus under command and control that is captured in

the denominator of equation (B.3) and shows how basin-wide pumping costs change with arestriction on pumping at the basin level.

We know that for Coachella, the constraint is nonbinding for the L types for a range of prices,and we know that in equilibrium the L types sell all their water (on vertical portion of excess sup-ply) as shown in figure 4. Plugging in the above functional forms for ES−1, ED−1, andD−1

i , i=H,L, and simplifying yields the following expression:

ðB:4Þ %4 =

Ð �X0

2η

1 + η1 +α

� �− 2

ηδX0H − 2

ηδτ� �

dτ +ÐXT

�X2η

1−αð Þ 1 + ηð Þ1 + α

� �+ 2

η δX0L−δX

0H

� �− 4

ηδτh i

dτÐX0H

02η

1 + η1 + α

� �− 2

ηδτ−c0

� �dτ +

ÐX*L

02η

α 1 + ηð Þ1 + α

� �− 2

ηδτ−c0

� �dτ

*100:

Performing the integration and simplifying yields:


ðB:5Þ %4 =2η

1 + η1 + α

� �τ− 2

ηδX0Hτ−

1ηδτ

2� �

�X0 + 2

η1−αð Þ 1 + ηð Þ

1 + α

� �τ + 2

η δX0L−δX

0H

� �τ− 2

ηδτ2

� � XT

0

2η

1 + η1 + α

� �τ− 1

ηδτ2−c0τ

� �X0

H0 + 2

ηα 1 + ηð Þ1 + α

� �τ− 1

ηδτ2−c0τ

� � X*L

0

*100:

ðB:6Þ %

4 =2η

1 + η1 + α

� ��X − 2

ηδX0H�X − 1

ηδ�X� �2 + 2

η1−αð Þ 1 + ηð Þ

1 + α

� �XT + 2

η δX0L−δX

0H

� �XT − 2

ηδ XT� �22η

1 + η1 + α

� �X0

H − 1ηδ X0

H

� �2−c0X0

H + 2η

α 1 + ηð Þ1 + α

� �X*

L−1ηδ X*

L

� �2−c0X*

L

*100:

Where

XT =Ω2δ

XT <X0L

X0L otherwise

8>>><>>>:

is the equilibrium quantity traded. Plugging parameter values for Coachella (table 1)yields %4 = 35.8%.The numerator in equation (B.6) is the normalized monetary gains, G, from trade, that is,

ðB:7Þ G=2η

1 + η

1 + α

� ��X −

2ηδX0

H�X −

1ηδ �X� �2 + 2

η

1−αð Þ 1 + ηð Þ1 + α

� �XT +

2η

δX0L−δX

0H

� �XT −

2ηδ XT� �2

= 1:32

We undo the normalization by multiplying G by the nominal values for (x*, c*) (229,867 AF,$100.50/AF) in the open-access equilibrium to obtain the estimated monetary gains, G*, togroundwater trade in the Coachella Valley in 2016 dollars, given the baseline parameter values:

ðB:8Þ G* = 1:37*$100:50*229,867 = $30:42million:

In equilibrium, L types sell their entire allocation, meaning 37.6% of the open-access 229,867AF quantity is traded, which is 86,430 AF. The range of market-clearing prices, ρ to �ρ, is deter-mined by the range between the point in which the excess supply curve becomes vertical andwhere ED−1 intersects the vertical ES−1, as shown in figure 4:

ðB:9Þ ρ=ES−1 X0L

� �=2δη

X0L−X

0L + σL

� �= 2:16

ðB:10Þ �ρ=ED−1 X0L

� �=2δη

σH −X0H −X0

L

� �= 2:14

Multiplying by baselinemarginal extraction costs of $100.5AF, we see ρ=$217 and �ρ =$225 peracre-foot. Thus, total price (P = ρ + c0) paid for groundwater in this context would range between$317 and $325. Dividing the total gains by the quantity traded reveals an annual average value ofabout $352 per AF traded.

A.1.1. Estimating Distributional Impacts

Expressions for changes in consumer and producer surplus shown in figure 3 are similarlyexpressed here as a function of parameters from the linear model. Recall the expressions forthe percentage change in consumer and producer surplus as a function of the market powerparameter, ξ, equations (20) and (21), which characterize the case where the constraint is alwaysbinding for the L types and in equilibrium the L types apply some portion of their water to their


land (on the upward-sloping portion of excess supply). We can express these as functions of theparameters for the linear model:

ðB:11Þ %4CS=

ÐXSP ξð Þ0

2δη σH −X0

H −τ� �

−ρSP� ��

dτ−ÐXT

02δη σH −X0

H −τ� �

−ρT� �

dτÐXT

02δη σH −X0

H −τ� �

−ρT� �

dτ*100,

ðB:12Þ %4PS=

ÐXSP ξð Þ0 ρSP ξð Þ− 2δ

η τ−X0L + σL

� �� dτ−

ÐXT

0 ρT − 2δη τ−X0

L + σL� ��

dτÐXT

0 ρT − 2δη τ−X0

L + σL� ��

dτ*100,

where ρSP = 2δη σH −X0

H −σH −σL +X0

L−X0H

2 + ξ

h i� �, ρT = δ

η σH + σL− X0H +X0

L

� �� , XSP = 1

δΩ

2 + ξ, and

XT = Ω2δ. Plugging baseline parameter values from table 1 with an adjustment to α as explained

in the main text, and a choice for ξ, yields changes in market surplus for buyers and sellers. Theseexpressions are calculated for every value of ξ to obtain figure 3.


Documents

The Gains from Agricultural Groundwater Trade and the