Agricultural Groundwater Markets:Understanding the Gains from Trade and the Role

of Market Power∗

JOB MARKET PAPER

The latest version is available for download at ellen-bruno.com/research.

Ellen M. Bruno†

December 20, 2017

AbstractThis paper models and quantifies the efficiency gains from using market-basedinstruments relative to command and control to manage groundwater. A the-oretical model of a groundwater market is developed to show how the mag-nitude and distribution of the gains from trade change as market structurevaries. Market structure is a key consideration because the presence of largegrower-shippers and/or competition among a few water agencies on a sharedbasin may be defining components of future groundwater markets. Using datafrom a groundwater basin in southern California, this paper demonstrates theexistence of large gains from trade, despite the potential for market power.Importantly, the price elasticity parameter used to calibrate the model is gen-erated with a fixed-effects econometric model that utilizes well-level panel dataon groundwater extraction and exploits temporal and cross-sectional variationin extraction prices. Simulations that vary market conditions show that re-sults generalize to other groundwater basins.

∗I am grateful to Richard Sexton and Katrina Jessoe for their guidance and support. I wouldalso like to thank Pierre Mérel, Michael Hanemann, Richard Howitt, James Wilen, Daniel Sumner,Jeffrey Williams, and Kevin Novan for helpful comments and suggestions. This paper benefitedfrom participant comments at presentations at the Western Agricultural Economics Associationand the Agricultural and Applied Economics Association annual meetings, the University of West-ern Australia, and the Australian Agricultural and Resource Economics Society. Funding for thisresearch came from: Giannini Foundation of Agricultural Economics, UC Water Security andSustainability Research Initiative funded by the UC Office of the President (Grant No. MR-15-328473), and the National Science Foundation Climate Change, Water, and Society IGERT (DGENo. 1069333).†Ph.D. Candidate, Department of Agricultural and Resource Economics, University of Califor-

nia, Davis. Email: embruno@ucdavis.edu.

1 Introduction

Improved management of water resources is becoming increasingly important in

the face of climate change. We expect to see more variability in precipitation,

with droughts, floods, and other extreme climate events occurring more frequently

(Kunkel et al. 2013). Because groundwater serves as a buffer to fluctuations in

surface water supplies, it can significantly reduce the cost of climate change for

agriculture. However, many groundwater basins worldwide have seen declines in

groundwater storage over time because groundwater is extracted at a rate faster

than it can be replenished (Rodell, Velicogna, and Famiglietti, 2009). This is largely

due to the lack of regulation of this common property, open-access resource. Since

groundwater management will have impacts on agriculture, attention must be given

to the choice of policy instrument.

Nowhere are concerns about groundwater management more pronounced than in

California. Groundwater accounts for 40% of the agricultural water supply on aver-

age (California Department of Water Resources, 2016) and several areas throughout

the state have seen significant declines in groundwater storage (Faunt, Belitz, and

Hanson, 2009). In recognition of the value in maintaining a reliable groundwater

supply, California passed legislation in 2014 that provides a statewide framework for

local agencies to manage groundwater. The 2014 Sustainable Groundwater Manage-

ment Act requires overdrafted basins throughout California to reach and maintain

long-term stable groundwater levels. However, California’s new groundwater law

is silent as to how groundwater agencies should achieve sustainability, even though

the cost effectiveness of different policy instruments may vary substantially. With

minimal assumptions on permit market structure, this paper evaluates the efficiency

gains of groundwater markets relative to a command-and-control regime for manag-

ing groundwater.

1

Economists have espoused the merits of market-based instruments to manage the

environment (e.g., Goulder and Parry 2008) and growing empirical work points to

substantial cost savings from the deployment of market-based instruments relative

to command and control policies for air pollution (e.g., Fowlie, Holland, Mansur

2012). Less is known, however, about the performance of economic instruments for

managing water. The existing literature focuses on the gains from surface water

transfers, suggesting that there are large benefits from the reallocation of water

from low-value to high-value users (e.g., Vaux and Howitt 1984; Hearne and Easter

1997). However, these arguments often rest on questionable assumptions regarding

competition, information, and the availability of substitutes, and have yet to be

empirically corroborated.

The performance of market-based instruments is further complicated by the pres-

ence of other market failures. Market structure will be an important feature to

consider when evaluating the merits of markets to manage groundwater. Driven by

the presence of large, vertically integrated farming-shipping operations, formation of

coalitions among buyers or sellers, and/or competition among a few water agencies

on a shared basin, market power in trading may occur due to the concentration of

permits among a handful of buyers or sellers; this will decrease the gains from allow-

ing groundwater trade and could have significant implications for the distribution

of benefits from groundwater markets. This paper addresses the question of how

either seller or buyer market power affects the efficiency gains and distributional

consequences from groundwater trade. An evaluation of the merits of groundwater

trading in the presence of imperfect competition contributes to the policy discussion

regarding the potential role of markets to manage groundwater.

In this paper, I first develop a theoretical framework to study agricultural ground-

water use and trading. After deriving unconstrained groundwater demand functions,

property rights for pumping are allocated such that the aggregate use is restricted.

2

I then derive permit demand and supply curves, which are used to quantify the

gains from trade. Using a flexible model framework that can reflect any degree of

buyer or seller market power in the permit market for groundwater, I identify the

relationship between market power and the efficiency of water trading. Simulations

show how the total efficiency gains and the distribution of benefits between buyers

and sellers changes with market structure. Results based on linear groundwater

demand curves show that the total impacts of market power are relatively small,

however, the distributional impacts are considerably larger. Distributional impacts

are important from a policy perspective because they may influence the political

feasibility of implementing groundwater markets. I show that those with market

power (whether as buyers or sellers) may be able to capture large shares of the gains

from trade.

The contribution of the theoretical model is twofold. First, this paper extends a

branch of literature that evaluates the impacts of market power in permit markets

to include groundwater. Stemming from the seminal paper by Hahn (1984), this

literature considers the initial distribution of property rights, strategic behavior of

competitors, and impacts in the final product market, with applications to fisheries

and pollution (e.g., Misiolek and Elder 1989; and Montero 2009). Second, this paper

relaxes assumptions regarding market structure to more broadly characterize the

impacts of imperfect competition. Whereas previous literature makes assumptions

on market structure, e.g., Cournot competition, or one or two dominant firms with

a competitive fringe (Westskog 1996; and Montero 2009), this paper uses a flexible

framework for imperfect competition in the permit market, allowing the model to

depict the entire range of possible market power settings for either buyer or seller

power. A key feature of this analysis is the demonstration of market power impacts

via simulations that vary a flexible market power parameter.

The model is applied to a groundwater basin in southern California that under-

3

lies the Coachella Valley, a major production region for citrus, dates, grapes, and

other vegetable row crops. I contribute to the literature on water markets by esti-

mating the gains from groundwater trade for the water district serving this area. A

novel feature of this application is that all model parameters are either constructed

or estimated using observational data from Coachella Valley. Importantly, the price

elasticity of demand for groundwater in the Valley is estimated with econometric

methods that utilize the panel structure of monthly, well-level groundwater extrac-

tion and price data. Results show that the economic surplus with trade is over twice

as large as that under command and control.

The estimation of the price elasticity of groundwater demand represents another

key contribution of the paper. A reliable estimate of this elasticity is essential to

assessing the potential of price as a tool to manage groundwater. Previous estimates

of agricultural water demand elasticities have varied widely, ranging from −0.001 to

−1.97 (Scheierling, Loomis, Young 2006). Of these previous estimates, those that

used groundwater extraction data have (1) relied on estimates of pumping costs for

price because groundwater itself is typically free of charge and (2) focused on areas

without surface water access (e.g. Hendricks and Peterson 2012; Gonzalez-Alvarez,

Keeler, and Mullen 2006). The Coachella Valley Water District is unique because it

deploys a location-based volumetric pricing scheme for groundwater. This ground-

water extraction price allows me to estimate a price response without attenuation

and amplification bias due to incomplete information on energy costs to extract

groundwater (Mieno and Brozovic 2017). Furthermore, data on surface water use

enables the model to control for substitution and isolate the price response in ground-

water use. Exploiting temporal and cross-sectional variation in prices, I provide an

estimate of the price elasticity of groundwater demand for the Coachella Valley that

is the first of its kind to not rely on estimates of pumping cost for price.

In the following section, I provide background on California’s 2014 Sustainable

4

Groundwater Management Act and the avenues through which market power may

arise in groundwater markets. In Section 3, a theoretical model of agricultural

groundwater use is developed. I derive unconstrained groundwater demand func-

tions and then assign property rights for pumping. Section 4 introduces trade;

permit demand and supply curves are derived. Within this framework, I compare

the perfectly competitive and imperfectly competitive solutions and determine the

effects of market power on market outcomes and distribution of welfare. Section

5 contains an application to California’s Coachella Valley. This is followed by a

simulation analysis in Section 6 that tests the sensitivity of these results. The final

section concludes.

2 Background

Groundwater is a significant component of California’s water supply that has his-

torically been unmeasured and unmanaged. Prompted by years of severe drought,

the California state legislature passed a landmark groundwater law in 2014 that

provides a statewide framework for local groundwater agencies to coordinate data

management and organize basin management plans to eliminate overdraft. The law

applies to 127 of 515 basins in California, which account for 96% of the groundwater

pumping in the state. The Sustainable Groundwater Management Act (SGMA)

gives local agencies the authority and flexibility to correct overdraft conditions in a

variety of ways.

Water agencies governing overdrafted basins will likely need to reduce basin-wide

pumping to achieve groundwater sustainability targets. Regulators may choose to

do this by restricting individual pumping volumes, after establishing property rights

for groundwater. However, the assignment of property rights for groundwater use

will cause efficiency loss in the absence of water trading, if the regulating agency

5

lacks perfect information and/or faces legal restrictions to setting allocations.1 Thus,

groundwater trading may emerge as one possible avenue for reaching basin sustain-

ability targets. Groundwater markets have the added benefit of eliminating uncer-

tainty in reaching a target, as compared to other economic instruments like taxes.

Prior to implementation of a trading regime, it is important to understand the mag-

nitude and distribution of benefits of groundwater markets, particularly given the

likely presence of market power. As groundwater markets become more prevalent

in California and elsewhere, determining the competitive outlook for these markets

and understanding the consequences resulting from any market power abuses will

be one of the fundamental challenges facing policymakers.

Due to the nature of water institutions and the market structure of agricul-

tural production in California, market power may be a defining component of future

groundwater trading. Several structural elements may give rise to imperfectly com-

petitive groundwater markets. For many groundwater basins in California, multiple

groundwater management agencies are emerging to jointly manage the groundwater

on a shared basin. If trading then occurs among a handful of groundwater agencies,

it will be characterized by a small-numbers problem. Also, a significant amount of

current surface water trading and groundwater banking (artificial aquifer recharge)

in California is organized at the district or county level among districts on behalf of

the farmers within their service areas. Water districts may be able to act as implicit,

legal cartels that are able to influence price in purchases or sales of groundwater.

Even under the authority of a single agency for a given groundwater basin,

market power will be a concern if only a few players are active on one side of the

market. The agricultural sector has seen significant structural change over the last

several decades, leading to fewer, larger, and vertically integrated farming-shipping1California’s correlative rights doctrine, which gives landowners the right to use the ground-

water underneath his or her land, may determine how property rights are allocated under theSGMA.

6

operations. Most groundwater rights in California are overlying rights, which are

based on ownership of the land above the aquifer and give the landowner the right

to pump. It is likely that permits for pumping, when properly defined to facilitate

transfers, will be allocated based off land size, causing the concentration of permits

in the hands of the large players. Furthermore, the absence of legal restrictions

regarding the formation of coalitions among buyers or sellers may lead to groups

of players coordinating interests and dominating the market. Given that trading

will be restricted to the boundaries of a given groundwater basin, the structure

of California agricultural production and the legal environment regarding coalition

formation may give rise to consolidation of groundwater rights when they become

properly defined under the SGMA.

In this paper, I focus specifically on the Coachella Valley Groundwater Basin

in southern California, which is an ideal setting for many reasons. Multiple water

agencies in the Coachella Valley have been approved by the California Department

of Water Resources as “Groundwater Sustainability Agencies” to jointly manage the

Coachella Valley Groundwater Basin over the coming years. They are required to

work together to reach sustainability targets for the entire basin, e.g., by trading

groundwater pumping rights among agencies. Additionally, since the Valley is home

to large grower-shippers like Grimway Farms for vegetables and Sun World for table

grapes, the concentrated structure of Coachella’s agricultural production may en-

able the exercise of market power in an emergent permit market. Lastly, access to

the Coachella Valley Water District’s data on monthly groundwater extraction and

price by well enables me to estimate the groundwater demand elasticity, a necessary

component to quantifying the gains from trade.

7

3 Modeling Framework

I develop a theoretical model for studying agricultural groundwater use and trading.

The model is framed as if a single authority governs an entire groundwater basin,

the groundwater basin defines the market, and permit trading occurs among farm-

ers. However, this framework may be adapted to represent an aquifer with several

governing agencies where trade occurs among agencies. Within this framework, I

investigate the magnitude of the gains from trade, the distribution of benefits among

traders, and how both are affected by market power.

We build up from an unmanaged, open-access groundwater setting to tradable

property rights for pumping. Throughout, we assume farmers draw groundwater

from a common aquifer. For simplicity, we assume there are two types of farmers

pulling from the aquifer, low (L) and high (H), that are homogeneous within their

type. Each produces a single output. Farmers of type L grow a low-value crop, such

as rice or cotton, with individual production functions denoted qLj = gj(xLj, y).

Farmers of type H grow a high-value crop, e.g., a produce commodity or tree nut,

with individual production functions denoted qHj = fj(xHj, z).2 Production func-

tions are assumed to exhibit diminishing marginal productivity to variable inputs.

The variable x represents applied groundwater and y and z are other inputs to

production.

There are Ni identical individuals j = 1, ...Ni within each type i ∈ H,L. We

distinguish the two types such that growers of the high-value crop have a higher will-

ingness to pay for water at any given quantity of groundwater. To account for system

distributional losses, pumped groundwater is distinguished from the amount of wa-

ter applied to the crop. Aggregate water quantities are denoted by X and x with-2This formulation assumes farmers have already preselected into producing certain crops, e.g.,

based on heterogeneous ability levels or land quality. This is a short-run analysis. Changes incropping patterns that might occur over time due to alternative groundwater management regimesare not considered.

8

out individual-specific or type-specific subscripts, where uppercase is for pumped

groundwater and lowercase is for applied groundwater. That is, X = XH +XL and

x = xH +xL, where total H-type pumped water is XH =∑NH

j=1XHj (uppercase) and

total H-type applied water xH =∑NH

j=1 xHj (lowercase). The relationship between

pumped and applied water is given by the efficiency parameter δ with 0 < δ < 1 such

that xi = δXi. We must distinguish pumped groundwater from applied groundwater

because applied water is what determines production outcomes, whereas pumped

groundwater is the good to be traded.

The marginal pumping cost of water is denoted by c(X) and is an increasing

function of the total amount pumped out of the aquifer due to reducing the water

table as more water is pumped. Marginal pumping costs are positive, increasing,

and differentiable, i.e. c(X) > 0 and c′(X) > 0. When individual pumping is

small relative to the basin total, farmers face the same costs and take the marginal

pumping cost as given, but collectively their decisions determine basin-wide pumping

costs.3

3.1 Open-Access Groundwater Use

I begin with the profit maximization problem for farmers in the unmanaged, open-

access case. Total costs of production come from groundwater pumping costs and

the costs of other inputs, such as fertilizers and pesticides. The price of groundwater

is denoted wx and the prices of the other inputs are denoted wy and wz. When the

aquifer is an unmanaged common-pool resource, the price of groundwater equals the

marginal pumping cost, which individual users regard as constant, and is denoted

by wx = c. Firms are assumed to choose inputs (xH , z) for H types or (xL, y) for L3One possible extension of this work is to expand on this farm-level groundwater optimization

problem by allowing individual firms to believe they pump enough to influence their own pumpingcosts. This may be because they are big enough to impact the water table with their consumptionor they pump enough to create a cone of depression at the site of a well.

9

types to maximize farm profits, where Pi is the output price for the crop produced

by type i, i ∈ L,H. An individual of type H faces the following profit maximization

problem:

maxxHj ,z

πHj = PHfj(xHj, zj)− cxHjδ− wzzj. (1)

The first-order conditions for each of the H types are:

PH∂fj(xHj, zj)

∂xHj=c

δ, PH

∂fj(xHj, zj)

∂zj= wz (2)

and are similar for each of the L types. Solving for xHj and xLj reveals the

groundwater demand curves for each farmer j, which are functions of crop output

price, input prices, and the marginal pumping cost of groundwater: xHj(PH , wz, c),

xLj(PL, wy, c). When optimizing, farmers equate the marginal value product of an

additional unit of groundwater to its price, which here is the marginal pumping

cost adjusted by the efficiency parameter. These expressions recognize that a unit

pumped yields less than a unit applied due to inefficiencies in delivery.

3.2 Competitive Equilibrium

The market equilibrium comes from equating the aggregate water demand relation-

ships across both types with the aggregate water supply relationship. Aggregate ap-

plied water demands are given by xH+xL =∑NH

j=1 xHj+∑NL

j=1 xLj. The aggregate wa-

ter supply relationship is just marginal pumping costs as a function of total pumping,

adjusted by system distributional losses. If we define equilibrium pumping quantities

and costs by writing x∗H = xH(PH , wz, w∗x), x∗L = xL(PL, wy, w

∗x), and w∗x, respec-

tively, then the equilibrium condition is given by c(X∗) = c(x∗H(w∗

x)

δ+

x∗L(w∗x)

δ) = w∗x.

To obtain analytical solutions, I assume aggregate demands for applied water

are linear and parallel such that the H type demand curve is higher than that of the

10

L type for all quantities of applied water.4 As shown in (3), we have intercepts of γ

and αγ for the H and L types, respectively, with 0 < α < 1, and slope coefficients

assumed to be the same, given by β2:

xH = γ − β

2wx and xL = αγ − β

2wx. (3)

Throughout, I assume marginal pumping costs are increasing and linear with positive

intercept and slope coefficients, c(X) = θ + µX with θ, µ > 0. Aggregate applied

water demand is the sum of the total water demands from each type: x = xH +xL =

(α + 1)γ − βwx. The intersection of this function with the supply relationship for

applied water, wx = θ + µδx, reveals the competitive, open-access equilibrium price

(w∗x) and quantity (x∗):

x∗ =δγ(α + 1)− δβθ

δ + βµ,w∗x = θ + µ(

γ(α + 1)− βθδ + βµ

). (4)

In what follows, I invoke normalizations (units of measurement for price and wa-

ter quantity) such that the competitive market equilibrium price (i.e., pumping cost)

and quantity are each equal to one (x∗, w∗x) = (1, 1). Evaluated at the competitive

equilibrium, the absolute value of the demand elasticity is given by η = β and the

supply elasticity is ε = δµ.5 Given the normalizations, we can rewrite the demands

for L and H types and the aggregate supply relationship as functions of readily

interpretable parameters: the supply and demand elasticities evaluated at the com-

petitive equilibrium, ε and η respectively, the demand shift parameter, α ∈ (0, 1),

reflecting differences in water demands between H and L types, and the water distri-4I choose to define functional forms for water demands at the aggregate level for analytical

simplicity. I divide aggregate demands by the number of agents in each type to reveal individualwater demands.

5The elasticity of demand is given by η = |∂xD

∂wx

wx

xD | = β evaluated at the equilibrium (1, 1)

with xD(wx) = γ(α + 1) − βwx. The elasticity of supply is given by ε = ∂xS

∂wx

wx

xS = δµ at (1, 1)

where xS(wx) = δµwx−

δµθ. These imply the following substitutions, which are used to rewrite the

original expressions: β = η, θ = 1− 1ε , γ = 1+η

1+α , µ = δε .

11

bution efficiency parameter δ ∈ (0, 1). This substitution eliminates the dependence

of the model on units of measurement, as all parameters defining the market are

pure numbers. Restating all solutions with respect to this normalization, we have

the following:

Equilibrium groundwater quantity and price: (x∗, w∗x) = (1, 1)

Aggregate high-type demand: xH = 1+η1+α− η

2wxH

Aggregate low-type demand: xL = α( 1+η1+α

)− η2wxL

Individual farm marginal value products:

MV PHj = 2η( 1+η

1+α−NHxHj)

MV PLj = 2η(α( 1+η

1+α)−NLxLj)

Aggregate inverse groundwater supply: wx = (1− 1ε) + 1

εx.

3.3 Establishing Property Rights for Groundwater

Let us now assume that a regulatory agency intervenes by establishing non-tradable

property rights for pumping. To induce conservation, the agency sets an aggre-

gate endowment that is less than the amount pumped in the open-access scenario.

Without the ability to trade, both types must reduce pumping to their assigned

allocation. Although it is possible to arrive at the socially optimal solution through

a discriminatory set of water allocations where each type is allocated the amount

it would pump in the socially optimal setting, we logically assume the regulator

lacks such information or the political ability or legal right to implement such an

allocation.6 Instead we assume that allocations are based on some simple rule, such

as equal amounts to each farmer or on a pro rata basis by land size, rendering it

highly unlikely that the allocation equates marginal value products across types.

Suppose each farmer receives an initial groundwater allocation, denotedX0ij, that

6Recall that most groundwater rights in California are overlying rights, meaning the legal rightsto groundwater are generally tied to landholdings.

12

is assumed to be the same across farmers within each farmer type. In the absence of

markets, each farmer is constrained to choose xij(·) ≤ δX0ij. Assuming farmers take

pumping costs as given, we have the following constrained optimization problem for

all individual H types, where λH,j is the Lagrange multiplier associated with the

constraint for individual j:

maxxHj ,zj ,λHj

πHj = PHfj(xHj, zj)− cxHjδ− wzzj − λHj(xHj − δX0

Hj). (5)

The Kuhn-Tucker conditions are:

PH∂fj(xHj, zj)

∂xHj− c

δ− λHj ≤ 0 and xHj(PH

∂fj(xHj, zj)

∂xHj− c

δ− λHj) = 0 (6)

PH∂fj(xHj, zj)

∂zj− wz ≤ 0 and zj(PH

∂fj(xHj, zj)

∂zj− wz) = 0 (7)

xHj − δX0Hj ≤ 0 and λH(xHj − δX0

Hj) = 0 (8)

λHj ≥ 0, xHj ≥ 0, zj ≥ 0. (9)

Any meaningful allocation of property rights in this setting must cause a binding

constraint on pumping, at least for the high types. Therefore, in equilibrium we

must have a strictly positive shadow price λ∗Hj > 0; at the constrained optimum,

the H types utilize the full allocation, i.e. x∗Hj = δX0Hj. The constrained maximiza-

tion problem and associated Kuhn-Tucker conditions are similar for the L types.

However, the constraint does not necessarily bind for the L types, even though it

binds for the H types, because L types have a lower willingness to pay for ground-

water at any quantity than the H types. In what follows, however, I focus on the

case where the allocation also binds for the L types.7

If we assume the constraint binds for both types, we get the following expressions7I do this for simplicity, to avoid the discontinuity associated with the kinked supply curve

that would result if the allocation was only sometimes binding for the L types.

13

for the shadow prices:

λ∗Hj = PH∂fj(δX

0Hj, zj)

∂xH− c

δ(10)

λ∗Lj = PL∂gj(δX

0Lj, yj)

∂xL− c

δ. (11)

This framework is important for establishing the baseline for a water market to

emerge; we derive the excess demand/excess supply functions for permits based on

this constrained equilibrium. A necessary condition for water markets to emerge is

that the shadow price for the H types must be strictly greater than that of the L

types at the constrained equilibrium. Applying functional forms to equations (10)

and (11), I characterize the necessary condition in terms of aggregate demands for

each type. Since farmers are identical within each type, we drop the individual-

specific subscript and define X0i as the aggregate endowment for one type, such that

X0i = NiX

0ij. The shadow prices become:

λ∗H =2

η(1 + η

1 + α− δX0

H)− c

δ(12)

λ∗L =2

η(α(1 + η)

1 + α− δX0

L)− c

δ. (13)

Equation (14), which is equivalently the difference in marginal value products

between types evaluated at the optimum x∗ij = δX0ij, gives the necessary condition

for trading to occur:

λ∗H − λ∗L = MV PH(δX0H)−MV PL(δX0

L)

=2

η[(1− α)(1 + η)

1 + α+ δX0

L − δX0H ] > 0.

(14)

When equation (14) holds, we know there exists some set of positive permit

prices where trading will occur. Thus, in the subsequent analysis we assume (14) is

satisfied. I define Ω = (1−α)(1+η)1+α

+ δX0L − δX0

H for simplicity. Rearranging equation

14

(14),(1− α)(1 + η)

1 + α> δ(X0

H −X0L), (15)

shows that when the difference in open-access quantity demanded between the two

types exceeds the difference in their effective allocation, there will be trade. Since

the term, (1−α)(1+η)1+α

, is positive for all values of α and η, an initial allocation which

distributes permits equally between types is sufficient for trading to occur.

4 Tradable Property Rights

I now introduce trade by using the groundwater demand functions and the ex-

ogenous allocation of pumping rights to create excess demand and excess supply

functions for pumping permits. Let us consider a homogeneous product market for

groundwater permits where two types buy and sell water pumping permits. Selling

or supplying groundwater in this context is simply being paid not to pump up to

one’s allocation of groundwater. Therefore, we define trade in terms of pumped

groundwater, as opposed to applied groundwater. Let ρ represent permit price and

XTi be the quantity of permits traded by type. Each farmer receives an allocation

appropriate for his type and relates this to his input demand for water to formulate

excess demand/excess supply functions. We define the excess function for farmer j

in type i, denoted Eij(ρ), as the difference between his water demand (for pumped

water) at price ρ and water allowance, X0ij:

Eij(ρ) ≡ xij(ρ)

δ−X0

ij. (16)

If Eij(ρ) > 0, individual j has excess demand at groundwater price ρ because his

water demand exceeds his endowment of water rights. If Eij(ρ) < 0 at groundwater

price ρ, then individual j has excess supply because his water demand is less than

15

Figure 1: Excess Supply and Demand for Groundwater

his allocation. As before, we drop the individual-specific subscript and define X0i as

the aggregate endowment for one type since farmers are identical within each type.

Given that (14) holds, the H types will be net demanders and the L types will be

net suppliers in the water market. Applying functional forms to the excess equation

(16) obtains the following excess demand and excess supply curves:

Excess Demand: XTH =

1

δ(1 + η

1 + α)−X0

H −η

2δρ (17)

Excess Supply: XTL = X0

L −1

δ

α(1 + η)

1 + α+

η

2δρ (18)

The excess demand/supply curves are functions of the demand elasticity and

other parameters from the profit maximization problems. Fixing these exogenous

parameters at particular levels fixes the intercepts and slopes for the excess functions.

Figure 1 shows a graphical derivation of excess supply and excess demand for permits

from linear aggregate groundwater demands for the two types, which are labeled

16

Xi. The initial endowment of pumping rights is equal between types (X0L = X0

H), as

shown by the vertical line. ED(XTH) denotes inverse excess demand of permits for

the H types and ES(XTL ) denotes inverse excess supply of permits for the L types.

In equilibrium, the L-type farmers sell permits to the H-type farmers. In Figure

1, (ρ∗, XT ∗) denote the market-clearing permit price and quantity pumped under

perfect competition.

4.1 Perfectly Competitive Solution

Equating excess demand of equation (17) and excess supply of equation (18) yields

the following market-clearing price (ρ∗) and quantity (XT ∗) under perfect competi-

tion:

(XT ∗, ρ∗) = (

Ω

2δ,

1

η(1 + η − δ(X0

H +X0L))). (19)

The presence of the permit market alters the farmers’ optimization problems.

Marginal pumping costs now consist of the permit equilibrium price plus the physical

marginal pumping costs, adjusted by the efficiency parameter. Additional benefits

are captured by farmers when we allow agents to buy and sell excess pumping rights.

The gains from trade are given by the following expression, denoted G, which is

calculated as the sum of consumer and producer surplus in the permit market:

G =

∫ XT∗

0

ED(τ)− ES(τ)dτ =Ω2

2δη. (20)

Welfare in the permit market depends on the demand elasticity, the demand shift

and efficiency parameters, and the original endowment of property rights to both

types. It is strictly positive by (14), demonstrating the existence of net benefits

from voluntary trading. This area is shown in Figure 1 by the shaded triangle.

For use in the sensitivity analysis in Section 6, we also express the gains as a

percent increase from the surplus under non-tradable property rights for pumping,

17

i.e. command and control. Surplus under the command and control scenario is given

by (δX0H)2+(δX0

L)2

ηand calculated from the area under the demand curves and above

price when types are restricted to pumping X0i . The percentage increase in gains

due to trading relative to the no-trade scenario is given by following expression:

%4 =Ω2

2δ

(δX0H)2 + (δX0

L)2∗ 100. (21)

4.2 Imperfectly Competitive Solutions

Next, we evaluate how groundwater markets are impacted by market power. We

focus on within-basin market power with two cases: (1) a small number of sellers

exercise oligopoly power over many buyers (NL small, NH large) and (2) a small

number of buyers exercise oligopsony power over many sellers (NH small, NL large).

In the first case, the H types are assumed to behave competitively. In the second

case, sellers are competitive while permit buyers are not.

A convenient way to introduce either buyer or seller market power into our model

is to introduce market power parameters ξ and θ, which lie on the unit interval and

are interpreted as measures of market competitiveness. Various papers have used

this approach (e.g. Suzuki et al. 1994; Zhang and Sexton 2002). This framework

allows for the complete range of competitive outcomes among buyers and sellers

to be represented. For example, ξ, θ = 0 gives the perfectly competitive solution

in (19), while ξ = 1, θ = 0 gives the seller monopoly solution, and θ = 1, ξ = 0

depicts buyer monopsony. Various degrees of oligopoly power can be depicted by

0 < ξ < 1, θ = 0 and various degrees of oligopsony power by 0 < θ < 1, ξ = 0.8

Perloff, Karp, and Golan (2007) among others derive the market power param-

eters ξ and θ using a conjectural variations framework. They show that the market8We do not consider simultaneous oligopoly and oligopsony power (0 < ξ < 1, 0 < θ < 1). This

falls in the realm of multilateral bargaining and the problem is fundamentally intractable withoutstrong assumptions on the bargaining environment.

18

power parameters can be related to the concepts of perceived marginal revenue and

perceived marginal cost curves, which are then used to predict market outcomes.

The perceived marginal revenue curve is a linear combination of the monopoly

marginal revenue curve and the market demand curve (perfect competition marginal

revenue curve), with weights given by the seller market power parameter ξ. Similarly

for buyer market power, the perceived marginal cost curve is a weighted average of

the supply curve and the monopsonist’s marginal factor cost curve.9

When depicting various degrees of oligopoly power (0 < ξ < 1, θ = 0), the

intersection of the perceived marginal revenue curve with the sellers’ excess sup-

ply function determines the permit market volume. Figure 2 shows the perceived

marginal revenue curve and the equilibrium price and quantity that result in this

flexible framework when sellers are assumed to be imperfectly competitive. In the

figure, (ρSP , XTSP ) denote the permit price and quantity with seller market power.

Relative to the competitive outcome, fewer permits are traded and at a higher price.

This creates a welfare loss, shown on Figure 2 by the shaded area.

4.2.1 Seller Market Power

We introduce seller market power (ξ > 0, θ = 0) into the existing modeling frame-

work. Starting from the excess demand and excess supply curves for permits from

equations 16 and 17, we first derive the perceived marginal revenue curve, as shown

in Figure 2. As noted, the perceived marginal revenue curve (PMR(XTH)) is derived

as a linear combination of the monopoly marginal revenue curve (MR(XTH)) and

the market inverse demand curve (ED(XTH)) with weights determined by the seller

9The market power parameters used in this modeling framework are sometimes interpretedas conjectural elasticities, which capture a firm’s expectation about how its rivals will react toits actions (Kaiser and Suzuki, 2006). For this paper, we simply interpret the market powerparameters as summarized measures of market competitiveness; the parameters ξ and θ representrealizations at any point in time of an unobserved game being played, thus we do not rely on firmsforming conjectures.

19

Figure 2: Groundwater Permit Market with Seller Market Power

market power parameter, ξ.

PMR(XTH) = ξMR(XT

H)+(1−ξ)ED(XTH) =

2

η(1 + η

1 + α−δX0

H)−(1+ξ)2

ηδXT

H (22)

By equating the perceived marginal revenue curve with the sellers’ excess inverse

supply, we derive the equilibrium quantity of permits and characterize this as a

function of the equilibrium quantity under perfect competition. Plugging that result

back into excess demand reveals the equilibrium price, ρSP , which can be written as

a function of the perfectly competitive permit price.

XTSP =1

δ

Ω

2 + ξ= (

2

2 + ξ)XT ∗

, (23)

ρSP =2

η(1 + η

1 + α− δX0

H −Ω

2 + ξ) = ρ∗ + (1− 2

2 + ξ)Ω

η. (24)

20

These results are completely summarized in terms of the demand elasticity pa-

rameter, the demand shift and efficiency parameters, the initial assignment of prop-

erty rights, plus the degree of competition on the seller side (ξ). If ξ = 0, the equilib-

rium outcome reverts to the perfect competition solution. By taking the derivative

with respect to the market power parameter, we can see how seller market power

affects market outcomes:

∂XTSP

∂ξ= −1

δ

Ω

(2 + ξ)2< 0

∂ρSP

∂ξ=

2

η(

Ω

(2 + ξ)2) > 0. (25)

The greater the market power exercised by the sellers, the fewer the permits that

are traded and the higher the permit price.10 This creates an inefficiency relative

to a competitive permit market. The deadweight loss (DWL) due to the exercise of

market power is given by the following area:

DWL =

∫ XT∗

XTSP (ξ)

ED(τ)− ES(τ)dτ =Ω2

2δη(

ξ

2 + ξ)2 > 0. (26)

Since 12δη

> 0, the deadweight loss area is strictly positive. This deadweight loss is

equal to the total welfare in the perfectly competitive permit market calculated in

(20), multiplied by the term ( ξ2+ξ

)2, which depends on the degree of market power

in the permit market. We can use this to characterize the gains from trading under

seller market power, denoted and given by (27):

GSP = (1− (ξ

2 + ξ)2)

Ω2

2δη. (27)

As shown by (28), gains to trading under seller power are a decreasing function of10These inequalities always hold because of the assumption on (14).

21

market power:∂GSP

∂ξ= −2Ω2

δη

ξ

(2 + ξ)3< 0. (28)

For a better understanding of magnitude, we express (27) as a percentage de-

crease in welfare relative to the perfectly competitive solution. Equation (29) gives

the percent change in total surplus (TS), which depends on the degree of market

power. When gains from trade under imperfect competition are expressed as a per-

centage change from surplus under perfect competition, they become an expression

of ξ only, and are thus robust to assumptions on parameters α, δ, η,X0.

%4 TS = −(ξ

2 + ξ)2 ∗ 100 (29)

We look at how (29) varies over the range of possible market power settings with

a simulation that varies market structure. Simulations that vary the market power

parameter allow us to see how the potential gains to trade are impacted by market

power.

From Figure 3, we can see that the additional gains from trade decline mono-

tonically and at an increasing rate with increasing seller power. Gains from trade in

Figure 3 are expressed as a percentage change relative to the gains from trade un-

der perfect competition. As the market power parameter converges to 1 (monopoly

case), the additional surplus from trading is 11% smaller than that under perfect

competition. Although the potential for market power is a concern for developing

groundwater markets, that concern should not be used as an argument against al-

lowing trade, because even in the presence of pure monopoly, gains from trade are

reduced only by 11%, a result that is robust to other parameters in the model.

22

Figure 3: The Effect of Seller Market Power on Total Gains from Trade

4.2.2 The Effect of Seller Power on the Distribution of Welfare

To assess the distributional impacts of market power, we compare how the gains

from trade accrue differently to buyers or sellers. Equation (30) depicts the percent

change in consumer, i.e. buyer, surplus due to seller market power relative to that

under perfect competition. This percent change in consumer surplus is decreasing

in ξ because fewer permits are traded and at a higher price with increasing seller

power, meaning buyers are increasingly worse off with increasing seller power. Also

expressed as a percent change from surplus under perfect competition, equation

(31) shows the percent change in seller or producer surplus due to seller power.

This expression is increasing in the market power parameter, meaning producers

are capturing a larger share of the surplus as their market power increases.

%4 CS =

∫ XT∗

XTSP (ξ)ED(τ)dτ − ρSPXTSP + ρ∗XT ∗∫ XT∗

0ED(τ)dτ − ρ∗XT ∗

∗ 100 (30)

%4 PS =ρSPXTSP − ρ∗XT ∗

+∫ XT∗

XTSP (ξ)ES(τ)dτ

ρ∗XT ∗ −∫ XT∗

0ES(τ)dτ

∗ 100 (31)

23

Figure 4: The Effect of Seller Power on Consumer Surplus

Figures 4 and 5 show graphically the percent change in buyer/consumer and

seller/producer surplus relative to that under perfect competition as market power

increases. Consumer surplus declines more rapidly than the overall gains to trade

which are shown in Figure 3. At ξ = 1, consumer surplus is over 50% smaller than

it would be under perfect competition, with most of the loss to buyers captured by

sellers, given the relatively small deadweight loss. On the other hand, producer or

seller surplus increases quickly as the market power parameter increases: the gains

from trade accrue rapidly to those with market power. At ξ = 1, seller surplus is

43% greater than that under perfect competition.11

These distributional impacts are important for several reasons. Relative winners

and losers from a trading environment will help determine the political feasibility

of implementing groundwater markets and may also influence the initial allocation

of permits, which can exacerbate the efficiency losses. Additionally, these impacts

may be undesirable from a regulator’s perspective because a large share of the gains

are accruing to those with market power, which by definition is only a few and are11These simulations rely on parameter estimates that are explained in Section 5 and summarized

in Table 5.

24

Figure 5: The Effect of Seller Power on Producer Surplus

likely to be large agribusiness enterprises with significant landholdings.

4.2.3 Buyer Market Power

In the same way, we can alternatively introduce buyer market power (ξ = 0, θ > 0)

into the framework. In this case, the buyers act as if they are facing a perceived

marginal factor cost curve, which results in an equilibrium with less trade and a

lower price. The perceived marginal factor cost curve is a linear combination of the

monopsony marginal factor cost curve (MFC(XTL )) and the market inverse supply

curve (ES(XTL )) with weights given by the oligopsony power parameter, θ:

PMFC(XTL ) = θMFC(XT

L ) + (1−θ)ES(XTL ) =

2

η(2δXT

L − δX0L−

α(1 + η)

1 + α), (32)

where the marginal factor cost is given by MFC(XTL ) = 2

η(2δXT

L − δX0L −

α(1+η)1+α

).

Equilibrium quantity, denoted XTBP , is determined by the intersection of the per-

ceived marginal factor cost curve and the buyers’ excess demand. Permit price, ρBP ,

25

is determined where XTBP intersects the excess supply curve:

XTBP =1

δ

Ω

2 + θ= (

2

2 + θ)XT ∗

, (33)

ρBP =2

η(α(1 + η)

1 + α− δX0

L +Ω

2 + θ) = ρ∗ + (

2

2 + θ− 1)

Ω

η. (34)

Equations (33) and (34) show that buyer power will depress trade in the water

market and reduce price relative to the competitive equilibrium. By taking the

derivative with respect to the market power parameter, we can see how seller market

power affects market outcomes:

∂XTBP

∂θ= −1

δ

Ω

(2 + θ)2< 0

∂ρBP

∂θ= −2

η

Ω

(2 + θ)2< 0 (35)

The solutions to the buyer market power case are symmetric to the seller power

scenario. In either case, a larger market power parameter implies fewer permits

traded and thus efficiency loss relative to the competitive equilibrium. As one would

expect, the price is higher than the competitive counterpart when there is seller

power, and lower in the presence of buyer power. Both sets of solutions depend on

the demand elasticity and other parameters from the profit maximization problems.

5 Application to Coachella Valley, California

We now turn to estimating the magnitude of the gains to trade relative to command

and control for a particular irrigation district. The Coachella Valley, CA is an

ideal setting for an application of the model because it is both subject to mandates

under SGMA and home to large grower-shippers that may dominate the market for

permits if trading occurs in the future. A novel feature of this application is that all

26

the model parameters are either constructed or estimated using observational data

from this area. Micro-level panel data on groundwater extraction and price from the

Coachella Valley Water District allow me to generate a well-identified price elasticity

estimate. With estimates of the groundwater demand elasticity, the demand shift

parameter, and the efficiency parameter that are reflective of a real-world setting,

I quantify the gains to trade and express the benefits relative to surplus under

command and control.

5.1 Background

The Coachella Valley is a productive agricultural region in southern California that

depends partly on groundwater for irrigation and is subject to restrictions under

SGMA. It has roughly 65,000 acres in crop production with a total production value

of over half a billion dollars a year. Producing 95% of the nation’s dates, the area is

also well-known for the production of table grapes, citrus fruits, bell peppers, and

other vegetables. Identified in Figure 6, it is located in Riverside County just north

of the Salton Sea and southeast of the San Bernardino Mountains.

Receiving only about 4 inches of precipitation in an average year, agriculture in

the area depends on both groundwater and imported Colorado River water for irri-

gation. The Coachella Valley groundwater basin has suffered at times from ground-

water overdraft, the condition wherein pumping exceeds groundwater recharge over

a period of years. As a result, three of the Valley’s four groundwater subbasins

have been classified as “medium priority” by the California Department of Water

Resources, which means the local agencies are subject to a timeline and set of goals

mandated by SGMA for reaching groundwater sustainability. However, relative to

the “high priority” basins in the state, this area does not face overdraft conditions

as severe as some others.

The Coachella Valley Water District (CVWD), the largest water agency in the

27

Figure 6: California Groundwater Basin Map

Source: California Department of Water Resources

28

Figure 7: Replenishment Assessment Charge (RAC) by Region

Coachella Valley, is unique in that it charges location-specific volumetric prices to

groundwater users for pumping. CVWD imports Colorado River water to artificially

replenish the aquifer in three locations throughout its service area, and charges sep-

arate prices to groundwater users within those three areas to fund the replenishment

program. The three geographic regions, East Whitewater, West Whitewater, and

Mission Creek, share some common features. Within each region, all customers face

a uniform price per acre-foot for groundwater extraction, called the Replenishment

Assessment Charge (RAC). When rate changes occur, they happen in the same

month of the year for all regions facing a rate increase. The regions differ in some

important ways as well; both the groundwater extraction fee and the level of price

increases vary substantially across regions. We exploit this plausibly exogenous vari-

ation in prices across space and time to estimate the price elasticity of groundwater

demand for the Coachella Valley.

Figure 7 shows the variation in extraction prices across time and region. The

RAC for the West Whitewater region began prior to 2000, so users in that area have

29

a nonzero groundwater price throughout our sample set. All other rate increases

occurred during July of each year. The East Whitewater region faces the lowest

RAC, remaining below $50/AF through June 2014. Prices vary only at the region-

year level. Figure 7 showcases the variability in the price changes across regions,

e.g., periods of time when rates changed for the East Whitewater region, but not

the others.

Most of the groundwater users in the CVWD service area have access to surface

water in addition to groundwater. Surface water from the Colorado River is deliv-

ered to CVWD customers and to the groundwater replenishment facilities via the

Coachella Canal. Unlike the RAC, canal rates vary by usage category (agricultural,

municipal, etc.) and not by region. Surface water is delivered periodically to the

groundwater replenishment facilities for recharge at the sites and these delivered

water quantities vary due to drought conditions and other factors.

Groundwater use in the Coachella Valley spans agricultural, urban, and recre-

ational uses. RAC program data are not collected by usage category, so CVWD is

unable to segregate agricultural wells from municipal wells, golf wells, etc. As of

2005, 45% of water use came from the agricultural sector, 33% of water use came

from urban (residential and industrial) users, and 17% of water was used by golf

courses (Coachella Valley Water District, 2012). Data are not available on how this

make-up has changed from 2000 to 2016, or how it varies among regions.

5.2 Price Elasticity of Demand for Groundwater

The price elasticity of demand for groundwater is not only an important parameter in

estimating the gains from trade, but it is also a necessary statistic in understanding

the potential of price as a lever to manage groundwater demand. The deployment

of three different pricing regimes for groundwater extraction within the Coachella

Valley provides an opportunity to estimate the effect of prices on groundwater use. I

30

take advantage of monthly, well-level data spanning 17 years to estimate the effect of

a uniform, volumetric pumping fee on well-level groundwater extraction, controlling

for well fixed effects, month aggregate shocks, and other potentially confounding

factors. I find the groundwater demand to be highly inelastic, with an estimate of

-0.19. To my knowledge, this provides the first micro-level estimate of the price

elasticity of groundwater demand that does not rely on estimates of pumping cost

for price. In what follows, I introduce the data, explain the empirical strategy, and

present estimation results.

5.2.1 Data

The dataset is an unbalanced panel that spans 17 years and over 900 individual

wells overlying the Coachella Valley Groundwater Basin. Received directly from

the Coachella Valley Water district, the dataset includes groundwater extraction

per month of every well in the RAC program from 2000 to 2016.12 Although the

RAC program for the West Whitewater region began prior to 2000, this is not the

case for the other two regions. The Mission Creek and East Whitewater regions

began facing groundwater fees in July 2004 and January 2005, respectively. The

unbalanced panel reflects the addition of new wells to the program over time.

Monthly groundwater extraction varies across seasons and regions. Average ex-

traction may vary by region because of differences in land use, the RAC, availability

of surface water, and other factors. The West Whitewater region has both the high-

est average extraction and the highest average RAC across the three regions whereas

the East Whitewater region has the lowest average extraction and lowest average

price. Figure 8 displays average monthly groundwater extraction by region over

time. The vertical line depicts when the RAC was introduced. Figure 8 shows the12The RAC program includes all groundwater wells that extract more than 25 acre feet in a

year. Extraction must be reported to CVWD on a monthly basis and is subject to the RAC of theregion in which the well is located.

31

Figure 8: Average Monthly Groundwater Extraction by Region

32

clear seasonal fluctuations associated with groundwater extraction.

Table 1 provides descriptive statistics of all the variables in the dataset, which in-

clude other factors that impact groundwater use and may be correlated with ground-

water prices. In addition to well-level groundwater extraction, historical price (RAC)

data were received directly from the Coachella Valley Water District.

Table 1: Descriptive Statistics

Mean Std Dev Min MaxMonthly Groundwater Extraction (AF) 58.12 67.93 0.00 537.10Replenishment Assesment Charge ($/AF) 64.40 34.22 0.00 128.80Monthly Cummulative Precip (Inches) 0.19 0.26 0.00 1.76Monthly Average of Max Daily Temp (F) 89.21 13.29 67.84 111.29Monthly Average of Min Daily Temp (F) 61.24 13.13 40.68 84.71Growing Degree Days (8-32C) 6.23 1.26 3.38 7.44Harmful Degree Days (over 34C) 4.15 4.46 0.00 11.18Annual Recharge to Facilities (Hund. Thous. AF) 0.53 0.70 0.00 2.57Monthly Surface Water Use (Hund. Thous. AF) 0.27 0.07 0.08 0.40Annual Recycled Water Use (Thous. AF) 0.35 0.42 0.00 1.72Annual State Water Project Allocation (%) 53.93 26.64 5.00 100Drought Index: None (%) 25.87 38.44 0.00 100.00Drought Index: D0 (%) 17.59 22.65 0.00 97.94Drought Index: D1 (%) 23.40 25.32 0.00 100.00Drought Index: D2 (%) 22.88 27.35 0.00 100.00Drought Index: D3 (%) 10.18 21.41 0.00 100.00Drought Index: D4 (%) 0.07 0.13 0.00 0.41Proportion Citrus (%) 15.51 2.53 12.56 21.64Proportion Tree, Vine (%) 27.45 2.18 24.84 33.89Proportion Vegetable, Melon, Misc. (%) 46.56 4.07 34.59 50.81Proportion Field, Seed (%) 4.06 0.91 1.30 5.01Proportion Nursery (%) 6.42 1.32 4.53 9.63Area Irrigated (Acres) 54807 3733 43613 59626Observations 65108Note: Drought indices give the percentage of land in Riverside County experiencing differentdegrees of dryness from D0 representing "abnormally dry" conditions up to D4 representing "ex-ceptional drought" conditions.

Data on drought and aggregate weather shocks were obtained to account for

ways in which these factors may affect groundwater extraction and prices over time.

Daily precipitation and temperature data were collected from a weather station at

33

the Indio Fire Station (National Climatic Data Center #4259) in Riverside county.

Daily precipitation data were summed to aggregate to the monthly level. Growing

degree day and harmful degree day variables were constructed from daily average

temperatures and used in place of temperature.13 The California Department of

Water Resource’s State Water Project allocation announcements, which range from

0-100% of quantities requested in State Water Project surface water contracts, may

represent aggregate shocks that influence the amount of water available to ground-

water replenishment facilities within CVWD’s service area. Lastly, monthly values

of the U.S. Drought Monitor Index for Riverside County were collected over the

relevant time period (U.S. Department of Agriculture, 2017). D0 represents the

percentage of land in Riverside County facing "abnormally dry" conditions in a

given year. D4 is the most extreme degree of drought, representing conditions of

"exceptional drought". All are expressed as a percentage of land coverage in that

category.

Data on the availability of alternative water supplies were also obtained to ac-

count for the possibility of these factors affecting both prices and groundwater ex-

traction. Surface water is used for both direct consumption at the farm level and for

groundwater recharge across the regional groundwater replenishment facilities. The

amount of water delivered to groundwater replenishment facilities for the sake of

recharging the aquifer was reported in documents produced by the Coachella Valley

Water District (CVWD, 2016). Monthly consumptive use of surface water diverted

from the Colorado River for direct use is reported by the Bureau of Reclamation and

represents surface water use on behalf of all users in the service area of the Coachella13Agronomists model temperature by converting daily temperatures to growing degree days,

a nonlinear transformation of temperature that more accurately reflects the way crops respondto heat. Growing degree days are derived by summing the degrees above a lower baseline andbelow an upper threshold during the growing season. Following Richie and NeSmith (1991) andSchlenker, Hanemann, and Fisher (2007), we use the range from 8C to 32C, within which plantgrowth is assumed to be linear. Days with temperatures above 34C, assumed to be harmful toplant growth, are used to construct harmful degree days.

34

Valley Water District (U.S. Department of Interior, 2017). Furthermore, users in

two regions receive limited recycled water to be used in-lieu of groundwater. Data

on recycled water deliveries by region were collected from documents published by

the Coachella Valley Water District (CVWD, 2016).

5.2.2 Empirical Strategy

To estimate the price elasticity of demand for groundwater in the Coachella Valley, I

take advantage of substantial variation in groundwater prices across regions and time

within the water district’s service area. I use panel data on groundwater extraction

and replenishment charges (RAC) spanning 17 years to estimate a fixed effects model

using ordinary least squares.

The basic estimating equation is

ln(wit) = αi + βln(RACit) +D′mδ +X ′itγ + εit, (36)

where the dependent variable, ln(wit), is the natural log of groundwater extraction

for well i in month t. The parameter of interest β is the coefficient on the natural

log of groundwater prices, denoted ln(RACit). Equation (36) includes a vector

of control variables, represented by Xit, which includes precipitation, degree days,

State Water Project (SWP) allocations, and surface water quantities for both direct

consumption and recharge at the replenishment facilities. Well-specific fixed effects,

αi, allow flexibility in capturing differences across wells; this controls for all well

characteristics possibly correlated with RAC that are unchanged over time, i.e.,

differences in soils or other hydrogeologic features. The variable Dm is a set of

monthly dummy variables that allows us to net out seasonal fluctuations in pumping.

Lastly, εit is an idiosyncratic error term. Standard errors are clustered at the well

level to allow for serial correlation within a well over time.

35

Because the RAC changes annually and prices have increased over time, annual

fixed effects soak up substantial variation in price changes. For this reason, I in-

tentionally omit year and month-year fixed effects in this empirical approach, and

instead explicitly account for relevant aggregate (basin-wide) shocks and region-

varying shocks by conditioning on time-varying observables. To account for the fact

that drought and weather may affect both groundwater use and prices, I control for

precipitation, degree days, and the percentage of land in Riverside county facing

different levels of drought. Since some of CVWD’s water for recharge at the ground-

water replenishment facilities comes from the State Water Project, I also condition

on annual SWP water project allocations. Furthermore, since regional differences

in prices are partially driven by the amount of water recharged at replenishment

facilities over time, I include regional groundwater recharge as a control variable,

thus capturing relevant time-varying, regional shocks.

For the coefficient of interest β to capture the causal effect of price changes

on groundwater use, no region-level time-varying unobservables that affect extrac-

tion can be systematically correlated with prices. While it cannot be demonstrated

that this identifying assumption holds, I empirically examine its plausibility in what

follows. Threats to identification of β in equation (36) come from region-level time-

varying omitted variables that are both correlated with RAC and influence ground-

water extraction. Identification of the impact of price on extraction comes from

variation in the RAC across regions and time, while controlling for time-invariant

well characteristics, aggregate shocks, and other confounding factors.

The inclusion of surface water variables, one for direct consumption and the

other for groundwater recharge at regional replenishment facilities, is motivated

by features of our institutional setting. Surface water availability and use may be

correlated with both groundwater prices and groundwater extraction. Given a fixed

annual allocation of Colorado River water, the amount of groundwater extracted, the

36

quantity surface water applied, and the water remaining for groundwater recharge

within the basin are likely correlated. The RAC is largely determined by replenish-

ment program costs, which is a function of the amount of water used for recharge at

the various groundwater replenishment facilities. By including both surface water for

direct consumption and surface water for recharge at the facilities (which varies by

region) as control variables in the specification, we eliminate these potential threats

to identification.

Changes in energy prices may confound the estimation of β since energy prices

may be correlated with the RAC and likely affect groundwater extraction; pumping

costs are partially determined by the energy requirements to lift the groundwater

from below. In our empirical setting, a single utility supplies energy and all users

face the same rates and rate changes. The region faced virtually no rate changes over

the relevant time period. Imperial Irrigation District, the main energy provider for

Coachella Valley groundwater pumpers, did not change electricity rates from 2000 to

2014 and only increased them by $0.03 per kWh for 2015 and 2016. Furthermore, we

need not worry about differences in energy costs across individuals that are driven

by differences in pressure or depth to the water table because this is not likely

correlated with the agency-determined RAC.

Differences in land use across region and time may also pose a threat to iden-

tification because land use affects groundwater extraction and may be correlated

with the RAC. Groundwater wells in this dataset span agricultural and municipal

uses, and a portion of wells irrigate grasses on golf courses. The East Whitewater

region is predominantly agricultural, and this could potentially affect the replenish-

ment activities of the facility in the area, correlating land use with the RAC. These

characteristics are only problematic if they influence the replenishment charge in

that region. Coachella Valley Water District Engineering Reports, which justify

RAC rate changes, show no evidence of correlation between the RAC and land use

37

(CVWD, 2016).

Differences in hydrogeology that coincide with the rate zones may affect both

pumping decisions and the costs at respective replenishment facilities, possibly con-

founding the estimation of β. The three areas differ in proximity to the Salton Sea,

and thus may vary in groundwater quality. Distance to the Salton Sea, soil type, and

other natural features are time-invariant and thus captured with well fixed effects.

5.2.3 Results

Results from the estimation of equation (36) are reported in Table 2. Column (1)

reports results from an OLS regression of the log of groundwater extraction on

the log of prices controlling for well fixed effects. Column (2) further controls for

month fixed effects and column (3) controls for aggregate shocks by conditioning on

basin-wide surface water consumption, drought, precipitation, squared precipitation,

degree days, and SWP allocations. Column (4) further conditions on region-level,

time-varying observables, namely, delivered water for recharge at the regional re-

plenishment facilities.

With a price elasticity point estimate of -.19 in the preferred specification of col-

umn (4), results suggest that prices impact extraction in a modest, yet statistically

significant way. If the price of groundwater is increased by 1%, we expect this to

cause a decrease in extraction of 0.19%. Results from Table 2 show the robust-

ness of results to the inclusion of controls for seasonal effects, aggregate shocks, and

groundwater recharge.

Table 3 presents the robustness of the baseline results to the inclusion of ad-

ditional control variables for other potentially relevant factors. Also, to check the

sensitivity to assumptions on functional form, Table 3 includes results for the OLS

regression of groundwater extraction on prices in levels, controlling for well and

month fixed effects, aggregate shocks, and groundwater recharge. As a basis for com-

38

Table 2: Double-Log Fixed-Effects Regression

(1) (2) (3) (4)ln(Pump) ln(Pump) ln(Pump) ln(Pump)

ln(RAC) -0.166∗∗∗ -0.210∗∗∗ -0.203∗∗∗ -0.188∗∗∗(0.041) (0.042) (0.049) (0.053)[0.071] [0.083] [0.064] [0.046]

ln(Recharge) -0.007(0.007)[0.005]

N 54500 54500 54500 54500Well FE Yes Yes Yes YesMonth FE No Yes Yes YesAggregate Shocks No No Yes YesNotes: In parentheses are standard errors clustered at the well level, with *, **, *** denotingsignificance at the 10%, 5%, and 1% level. In brackets below are standard errors clustered at theregion level. Aggregate shocks include basin-wide consumptive use of surface water, SWP allocationannouncements, and weather controls (drought indices, precipitation, squared precipitation, anddegree days).

parison, column (1) presents results without additional controls, i.e., the preferred

specification from Table 2. Overall, results are robust to alternative specifications

and alternative assumptions on functional form.

Since land use likely affects groundwater extraction and is not otherwise captured

in a model without year fixed effects, a concern may be that baseline estimates are

confounded by the exclusion of acreage and crop choice variables. Column (2) reports

results from a specification that controls for aggregate land use measures, i.e., the

total irrigated acreage in Coachella Valley and the percentage of land in five crop

categories: citrus, tree/vine, vegetable/melon, field/seed, and nursery. Column (3)

further accounts for unexplained aggregate shocks by including a linear time trend.

Columns (4) and (5) explore the robustness of results to additional region-level,

time-varying observables.14 Users in the West and East Whitewater regions receive

limited amounts of recycled water from the Coachella Valley Water District to be

used in-lieu of groundwater. The amount of recycled water that is processed and14The sample size is restricted in (2) and (4) due to data limitations; land use data were only

available for 2002-2015 and recycled water use data were only available from 2006-2015.

39

Table3:

Rob

ustnessto

Alterna

tive

Specification

s

Add

itiona

lCon

trolsfor

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Baseline

Land

Use

Tim

eTr

end

RecycledWater

Lagged

Prices

AgRegionOnly

Levels

ln(RAC

)-0.188∗∗∗

-0.136∗∗

-0.118∗

-0.173∗∗∗

-0.175∗∗∗

-0.201∗∗∗

(0.053

)(0.069

)(0.062

)(0.064

)(0.061

)(0.078

)[0.046

][0.025]

[0.014

][0.007

][0.038

]RAC

-0.255∗∗∗

(0.064

)[0.065

]N

5450

050

533

5450

04173

853

924

2266

354

933

WellF

EYes

Yes

Yes

Yes

Yes

Yes

Yes

Mon

thFE

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Agg

rega

teSh

ocks

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Notes:The

baselin

especification

correspo

ndsto

thepreferredspecification

ofcolumn(4)in

Tab

le2.

Eachsubsequent

columnrepo

rtsan

alternate

specification

.W

iththeexceptionof

column(7),

which

provides

estimates

oftheOLS

regression

inlevels,thedepe

ndentvariab

leis

thelogof

grou

ndwater

extraction

.Stan

dard

errors

clusteredat

thewelllevel

arerepo

rted

inpa

rentheses,with*,

**,*

**deno

ting

sign

ificanceat

the10%,5

%,

and1%

level.Stan

dard

errors

clusteredat

theregion

levela

rerepo

rted

belowin

brackets.Aggregate

shocks

includ

eba

sin-wide,mon

thly

consum

ptive

useof

surfacewater,a

nnua

lSW

Pallocation

anno

uncements,a

ndweather

controls(droug

htindices,precipitation,

squa

redprecipitation,

anddegree

days).

40

delivered within a year affects the agency’s costs, and thus may influence the RAC,

and may also impact well-level groundwater extraction. Column (5) includes a one

year lag of RAC prices to account for the possibility that groundwater pumpers may

be slow to adjust to price changes. The insensitivity of the elasticity estimate to the

inclusion of additional controls demonstrates the robustness of the estimate.

Since the East Whitewater region is predominantly agricultural, I report results

in column (6) from a restricted sample of wells in the East Whitewater region only.

Although we do not know which individual wells within this dataset are pumping

groundwater for agricultural purposes, I proxy for this by focusing on the region

that is primarily agricultural. Results show a slightly more elastic estimate when

focusing on this region alone.

Column (6) presents results from an OLS regression of groundwater extraction

on RAC in levels, controlling for well and month fixed effects, aggregate shocks, and

groundwater recharge. The coefficient estimate of -0.25 in column (7) translates to

a demand elasticity point estimate of -.28, when evaluating at the mean values for

extraction and RAC. Since the gains from trade are increasing in the price elasticity

of demand, my use of the -.19 estimate from the double-log specification provides a

conservative, lower bound in the following analysis.

5.3 Other Simulation Model Parameters

The remaining parameters for the simulation model were also estimated with data

from the Coachella Valley. First, the demand shift parameter α, which is the ratio

of marginal value products between low- and high-value crops at any quantity of

groundwater, was calculated with data from Riverside County’s 2015 Crop Report

for Coachella Valley and University of California Cooperative Extension (UCCE)

Cost and Return Studies. For simplicity, we focus on the four leading crops (grapes,

lemons, bell peppers, and dates), which are listed with production value and acreage

41

Table 4: Top Four Crops Grown in the Coachella Valley, CA

Crop Revenue Acreage Revenue ValuePer Acre Type

Grapes $131,852,825 7,802 $16,900 HighLemon/lime $93,824,406 3,887 $24,138 HighBell Peppers $87,891,750 4,490 $19,575 HighDates $36,184,900 7,765 $4,660 Low

in Table 4.

To estimate α, I first estimated a point on the value marginal product curve of

water for each crop. UCCE Cost and Return Studies report applied water per acre

by crop and Riverside County’s Crop Report discloses production and value per acre

by crop. Used together, a point on the average product curve of water by crop was

identified by dividing average production by applied water per acre. As shown in

Appendix A, economic theory can then be applied to relate this point to a point on

the value marginal product curve.

With the per-acre value reported in Riverside County’s 2015 Crop Report for

Coachella Valley, I derived a point on the marginal revenue product curve, extrap-

olated to the intercept with an elasticity estimate, and compared across crops. In

this fashion, the demand shift index can be computed for any pair of crops. For the

purposes of this simulation, I needed to arrive at a single alpha; the estimate of α is

given by the ratio of the intercepts between H- and L-type groundwater demands. I

used dates as the low-value crop and generated a high-value crop bundle consisting

of table grapes, lemons, and bell peppers.

Another model parameter, the total endowment of pumping rights, X0H + X0

L,

was estimated to be 80% of Coachella’s “business as usual” groundwater extraction,

meaning a 20% reduction in water use to correct for basin overdraft. This was

calculated by comparing the average annual groundwater extraction in Coachella to

that which would be allowed on the basin if it were to eliminate its reported 70,000

42

Table 5: Parameter Estimates

Parameter Estimate SourceDemand Shift Index .1 Estimated by comparing intercepts of H(0 < α < 1) tand L type water demands, which were

proxied with estimates of water demandsfor Coachella’s four leading crops.

Demand Elasticity (η) -.19 Estimated with fixed effects regressionsusing well-level pumping and price datafrom CVWD.

Total Allowable .8 Calculated by comparing average annualExtraction (X0

L +X0H) basin total pumping with CVWD’s

annual overdraft estimates.Irrigation Efficiency (δ) .85 Rogers et al. (1997)

AF/year of groundwater overdraft (Coachella Valley Water District, 2016). Lastly,

an irrigation efficiency of 85% was chosen for δ because this is the reported average

distribution efficiency for drip technology. Drip technology is used by growers of

grapes, lemons, and bell peppers in the Coachella Valley (Rogers et al., 1997). An

irrigation efficiency of 85% is also the distribution efficiency used in UCCE Cost

and Return Studies for drip irrigation systems (O’Connell et al., 2015). All model

parameters for Coachella Valley are summarized in Table 5.

5.4 Results for Coachella Valley, California

Using the parameter estimates summarized in Table 5, I estimate the magnitude of

the gains from groundwater trade for the Coachella Valley. Recall that the expres-

sions derived in Section 3 reflect only the case when permit allocations are binding

for both types. At the parameter values estimated for the Coachella Valley, how-

ever, the constraint is not binding for the L types. Results reported here reflect the

discontinuity associated with a kinked supply curve that results from an allocation

that binds for the H types only.15

15In this case, the inverse excess supply curve ES(XT ) = 0 up until the point where the Ltypes are willing to sell at a positive price; this point is denoted x = X0

L − 1δ (α(1+η)1+α ) in equation

43

Equation (37) reports an expression for the percent change in surplus from al-

lowing trade relative to a command-and-control scenario when the L types are not

constrained. Using this expression and our estimates of the model parameters, I show

that the gains from trade are large for two different allocation scenarios. The two sce-

narios considered are: (1) an equal allocation between types (X0L = X0

H = .4) and (2)

an unequal allocation that is proportional to acreage shares (X0L = .26, X0

H = .54).16

For an equal allocation between types, the percent change in surplus relative to

command and control is

%4 = ∫ x0ED(τ)dτ +

∫ Ω2δ

x(ED(τ)− ES(τ))dτ∫ X0

H

0xH(τ)dτ −X0

H [2δη

(1δ( 1+η

1+α)−X0

H)] +∫ x

0xL(τ)dτ − x

∗ 100 = 427% (37)

where x = X0L− 1

δ(α(1+η)

1+α) and x = α( 1+η

1+α)− η

2. Thus, surplus with trade is over five

times greater than the benefits under a no-trade scenario.

The value of the gains from trade for the Coachella Valley in this case can be

calculated, given estimates of both aggregate annual groundwater extraction and the

full per-unit pumping price. An irrigator’s full price per acre-foot of groundwater is

equal to the RAC plus the energy cost to pump an acre-foot of water to the surface.

Using an average depth to water from the land surface of 152 feet and the local

energy provider’s $.0618/kWh energy price for agricultural uses in 2016, I apply an

engineering formula from Rogers and Alam (2006) to calculate the average per-unit

energy cost of extraction. With an approximate (full) groundwater price of $99/AF

and an aggregate annual groundwater extraction of 464,400 AF, the gains from trade

are approximately $119 million annually.

(37). Thus, we have a kinked supply curve and a quadrilateral-shaped gains area. Furthermore, anon-binding constraint for the L types affects the surplus calculation under the baseline command-and-control scenario, which alters the denominator of equation (37).

16This allocation was calculated with acreage values from Table 4, scaled to reflect the totalallowable extraction of 80% of “business as usual” pumping.

44

Given California’s legal history, an allocation of permits that is proportional

to land holdings is a more likely scenario. For an allocation of permits based on

acreage, the percent change in surplus relative to command and control is

%4 = 144%. (38)

Surplus with trade is over twice as large in this case. With an approximate ground-

water price of $99/AF and an aggregate annual groundwater extraction of 464,400

AF, the annual gains from trade for the Coachella Valley in this case are approx-

imately $73 million. Since the L types receive fewer permits in this scenario, the

gains from trade are smaller than if permits were allocated equally between types.

Magnitudes generated with base Coachella values represent possible gains to

Coachella Valley groundwater users from water trading, if the water district imposed

pumping restrictions of a reasonable magnitude and distributed permits for pumping

in one of these ways. Given the model assumptions and parameter estimates of

Table 5, the gains from trade are large. These estimates suggest that, given a 20%

reduction in pumping from the status quo, surplus is at least twice as large with

trade than without.

Since we have no information on the magnitude of potential market power in this

area, these gains are calculated assuming a perfectly competitive permit market.

However, from analysis in Section 4, we know the gains from trade can be reduced

by at most 11% when ξ = 1, θ = 0 or θ = 1, ξ = 0, regardless of the initial allocation

of permits. The efficiency gains of groundwater markets relative to command and

control remain large even in the presence of market power. Thus, market power

should not be used as a possible argument against the formation of groundwater

markets. Even in the presence of monopoly power or monopsony power, efficiency

gains of markets relative to command and control are substantial.

45

6 Sensitivity Analysis

I now explore how these results generalize to other settings, which may differ from

Coachella Valley due to a variety of reasons. Other agricultural regions grow different

crops and vary in degrees of groundwater overdraft. Among other things, the crops

being grown in an area impact water demands and the sensitivity of users to changes

in water price. With simulations that vary market conditions, I perform a sensitivity

analysis to see how reasonable alternative values for the model parameters affect the

results.

Figure 9 shows how the gains from groundwater trade change as market condi-

tions vary, where the surplus with trade is expressed as a percentage change from

the surplus under a command-and-control scenario. Both the equal and acreage-

based allocation scenarios are shown. Panel A shows how the gains change as we

vary the demand shift parameter, followed by the demand elasticity, and the total

allowable extraction. Table 5 summarizes the base parameter values chosen for the

simulations; these values are shown with a vertical line in each panel of Figure 9.

The first panel of Figure 9 shows the percentage change in surplus from allowing

trade (relative to the no-trade scenario) as the alpha parameter varies. The demand

shift parameter α ∈ (0, 1), defined as the ratio of marginal value products between

types, reflects the differences in water demands between groundwater users. The

larger the value of α, the smaller is the difference in water demands between the two

types. When α is small and there is a large difference in demands between the H and

L types, we expect there to be greater potential gains from trade. We see this to be

true; the gains from trade decrease as alpha increases. The gains from trade relative

to command and control converge to zero as α → 1. The degree of heterogeneity

among groundwater users is an important determinant of the magnitude of the

benefits from trade.

46

Figure 9: Gains from Trade Sensitivity Analysis

47

The second panel of Figure 9 shows the percentage increase in surplus with trade

(relative to the no-trade scenario) as a function of the demand elasticity. More

elastic values lead to a greater percentage increase in the gains from trade. More

trading occurs with more elastic demand; both the quantity of permits traded and

the permit price are increasing in the demand elasticity. We see from panel B that

the percent increase in benefits is large for a wide range of elasticity values.

The bottom panel of Figure 9 depicts the gains with trade as a function of the

total endowment of property rights for pumping, i.e., the total allowable extraction

on the basin. Expressed again as a percentage change from benefits under command

and control, the gains with trade are decreasing at a decreasing rate as the endow-

ment increases. Even as the total allowable extraction approaches the aggregate

quantity pumped in open-access (where X0L + X0

H = 1), there are large gains from

trade. Thus, even if the total restriction on pumping is small relative to the open-

access consumption, an endowment of property rights that does not equate marginal

value products across types will hurt farmers relative to a trading scenario.

Overall, the efficiency gains of groundwater markets relative to command-and-

control instruments remain large and positive for a broad range of the model param-

eters. Since these parameter ranges represent a wide spectrum of market conditions,

we have reason to believe these results hold more generally; this range of parame-

terizations would likely encompass the conditions characterizing many basins where

trading might occur. Assuming a reasonable degree of heterogeneity among ground-

water users, the gains from trade can be quite large for many groundwater-dependent

agricultural regions.

48

7 Conclusion

As regulators around the world strive for improved groundwater management, they

may consider market-based instruments to improve the allocation of limited water

resources. Prior to implementing a voluntary groundwater trading program, it is

important to understand how groundwater markets will function, how they might

be impacted by market power, and how market power influences other types of

behavior. This paper evaluates the potential for groundwater trading and sheds light

on the role for markets to manage water in the presence of imperfect competition.

I estimate the efficiency gains of groundwater markets relative to command-and-

control instruments in the presence of this market failure.

Imperfect competition may be an important characteristic of future groundwater

markets. When rights for use are allocated by land size or historical pumping in an

area characterized by large, vertically integrated farming operations, permits may

be concentrated in the hands of a few large players. Additionally, in areas where

trading occurs on the level of the water districts and agencies jointly governing a

basin, there is a small-numbers problem that will influence trading. To understand

the nature and magnitude of these impacts, the decisions of heterogeneous farmers to

buy or sell tradable water permits were modeled such that one side of the market has

a flexible degree of market power. I estimated the gains from trade in a real-world

setting and see how large of an issue market power can be.

Overall, I show that regardless of the degree of competition, there are significant

net benefits from a groundwater trading regime. With parameters estimated with

data from the Coachella Valley, I find that surplus with trade is over twice as large

as the surplus under command and control, assuming perfect competition. Given

that the gains can be reduced by at most 11% with market power on one side of

the market, we conclude that the gains remain large regardless of permit market

49

structure. Furthermore, I show that the gains remain large over a broad range of

parameterizations by showing the sensitivity to changes in the demand elasticity,

the endowment of property rights, and the demand shift index, which captures the

degree to which marginal value products differ between buyers and sellers. Thus,

although other groundwater basins will exhibit features that differ from Coachella

Valley, these results are likely to hold more broadly.

To evaluate the role of market power, I show both how the total gains from

trade decrease and how the distribution of benefits changes relative to the perfectly

competitive baseline. The reduction in the total gains from trade from any degree of

market imperfection is small. Considering the magnitude of the potential gains from

trade, this welfare loss should not be used as an argument against the formation of

groundwater markets. Groundwater users as a whole are still significantly better off

than under command and control. However, the distributional impacts of market

power can be large, socially undesirable, and a factor in the political economy of

implementing trading regimes. The majority of the gains from trade accrue to the

few with market power, which are likely large agribusiness enterprises.

The distributional impacts of market power provide insight into how groundwater

markets may develop. First, a groundwater trading regime will require considerable

political support among groundwater users. The distributional dimension of market

power may dissipate support from the group being subjected to the market power,

since much of this group’s potential gains from trading will be captured by those

with market power. However, the converse is true in that a group that thinks

it has market power in trading would be strongly supportive. Second, a trading

regime must be preceded by the allocation of property rights for pumping. Big

players may exert influence on the method of permit allocation prior to trading,

exacerbating the distributional impacts of market power. Although our results imply

that permits should be allocated to avoid their concentration in the hands of a few

50

buyers or sellers, this may be infeasible due to the political environment and/or

limited information on behalf of the regulator. There is room for additional work on

the interaction between different permit allocations and the exercise of market power,

with the goal of determining the best permit allocations given limited information.

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A Appendix: Deriving the Demand Shift Index

Assuming linear and declining value marginal products of water throughout, the

value marginal product curve will have twice the slope of the average revenue product

curve. Aggregate H-type and L-type demand curves both take the following form:

VMPx = a− bx. (A.1)

Thus, average revenue product curves are:

ARPx = a− b

2x (A.2)

53

By Equation A.1, we know the demand elasticity is:

|η| = 1

b

V MPxx

. (A.3)

Since we have an estimate of η and a point on the average revenue product curve, I

solve a system of three equations and three unknowns (a, b, VMPx) to find a point

on the value marginal product curve. Eliminating a and b, we can isolate VMPx,

VMPx =ARPx1 + 1

2η

(A.4)

The parameter of interest is the difference in marginal value products between types

at any x,

α = VMPH − VMPL =ARPH,x − ARPL,x

1 + 12η

(A.5)

The demand shift index can be computed for any pair of crops. For the purposes of

this simulation, I needed to arrive at a single alpha. Focusing on the top four crops,

I grouped them into high- and low-value crop bundles to calculate one α value.

54