WPS7694
Policy Research Working Paper 7694
Markets, Contracts, and Uncertainty
in a Groundwater Economy
Xavier Gine
Hanan G. Jacoby
Development Research Group
Agriculture and Rural Development Team
& Finance and Private Sector Development Team
June 2016
Policy Research Working Paper 7694
Abstract
Groundwater is a vital yet threatened resource in much of subjective probability distributions of borewell discharge
South Asia. This paper develops a model of groundwater elicited from a large sample of well-owners in southern
transactions under payoff uncertainty arising from unpre- India. The findings show that, while the contracting distor-
dictable fluctuations in groundwater availability during the tion leads to an average welfare loss of less than 2 percent
agricultural dry season. The model highlights the trade-off and accounts for less than 50 percent of all transactions costs
between the ex post inefficiency of long-term contracts and in groundwater markets, it has a sizeable impact on irrigated
the ex ante inefficiency of spot contracts. The structural area, especially for small farmers. Uncertainty coupled with
parameters are estimated using detailed micro-data on the land fragmentation also attenuates the benefits of the water-
area irrigated under each contract type combined with saving technologies now being heavily promoted in India.
This paper is a product of the Agriculture and Rural Development Team and Finance and Private Sector Development
Team, Development Research Group. It is part of a larger effort by the World Bank to provide open access to its research
and make a contribution to development policy discussions around the world. Policy Research Working Papers are also
posted on the Web at http://econ.worldbank.org. The authors may be contacted at hjacoby@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
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Markets, Contracts, and Uncertainty
in a Groundwater Economy
e and Hanan G. Jacoby∗
Xavier Gin´
Keywords: Irrigation, Water-saving technology, Groundwater markets, Contracts as
reference points
JEL codes: Q15, O13, L14
∗
Development Research Group, The World Bank, 1818 H St. NW, Washington DC, 20433. Gin´ e:
xgine@worldbank.org; Jacoby: hjacoby@worldbank.org. We owe a particular debt to K.P.C. Rao for his
eﬀorts in managing the ﬁeld work associated with this study, and to his survey team. We also thank
Jishnu Das, Quy-Toan Do, Greg Fischer, Guido Friebel, Oliver Hart, Ethan Ligon, Robert Townsend, Liam
Wren-Lewis and especially Mark Gersovitz for very useful comments, as well as seminar participants and
discussants for their ideas and suggestions.
1 Introduction
Water scarcity is one of the fundamental challenges facing developing country agriculture,
one that is only expected to be exacerbated by climate change. In India, by far the world’s
largest user of groundwater, millions of borewells have sprung up in recent decades (Shah,
2010). While groundwater exploitation has allowed increased agricultural intensiﬁcation, a
boon to the rural poor (Sekhri, 2014), unregulated drilling has also raised concern about
the sustainability of this vital resource (e.g., World Bank, 2005). India’s main policy related
to groundwater has been its eﬀort to promote water-saving technologies such as drip and
sprinkler irrigation, targeted particularly at small farmers.1 For such farmers, however,
the beneﬁt of water-saving technology depends on their ability to expand water sales to
neighboring cultivators, in other words on the the eﬃciency of groundwater markets.2
To explore the implications of water-saving technology in a setting where there are con-
tracting distortions, we build and structurally estimate a model of south India’s ground-
water economy incorporating three salient features: First, given high irrigation conveyance
losses, groundwater transactions tend to be highly localized, typically involving bilateral-
monopolistic contracts between a well-owner and a water-buyer on adjacent land (see Ja-
coby et al., 2004). Second, during the dry (rabi ) season, agricultural production relies almost
exclusively on borewell irrigation, the supply of which is, at least in part, unpredictable.3
Third, planting requires upfront and irreversible outlays. Insofar as a water-buyer has a
single borewell from which to purchase irrigation, his standing crop eﬀectively becomes an
investment speciﬁc to that trading relationship.
In our setting, bilateral transactions between well-owners and neighboring farmers take
one of two forms: spot contracts, in which groundwater is sold on a per-irrigation basis
throughout the season, and long-term (i.e., seasonal) contracts, which specify ex-ante the
price and area irrigated over the entire season. We develop a model in which spot con-
tracts are fully state contingent and thus ex-post eﬃcient, but, due to the classic hold-up
problem, ex-ante ineﬃcient. In particular, planting incentives of water-buyers are distorted.
Long-term contracts, by contrast, are assumed immune from hold-up, but lead to ex-post
1
The National Mission on Micro Irrigation, initiated in 2006, is perhaps the largest subsidy program
of its kind in the world. When combined with complementary subsidies recently oﬀered by several states,
including Andhra Pradesh, smallholders may be eligible to defray up to 90% of a system’s cost.
2
Due to the high degree of land fragmentation, groundwater markets are pervasive in India . The 2011-12
India Human Development Survey (Desai and Vanneman, 2011) indicates that, of the 83% of agricultural
households nationwide that do not own a borewell, 37% purchase irrigation (groundwater).
3
During the wet (kharif ) season, groundwater is typically used as a buﬀer against insuﬃcient rainfall or
shortfalls in surface water ﬂows rather than as the sole source of irrigation.
1
ineﬃciency inasmuch as a ﬁxed transfer of groundwater necessarily leads to a misallocation
across farms once the state of nature is revealed. Our model yields the sharp prediction
that as groundwater supply uncertainty increases, long-term contracts become unattractive
relative to spot arrangements.
The assumption that long-term (ex-ante) contracts deter holdup appeals to the reference-
point insight of Hart and Moore (2008) and Hart (2009) wherein a contract establishes what
each party in the transaction is entitled to; opportunism thus leads to deadweight losses
(see also Herweg and Schmidt, 2014, for a related model).4 In the earlier property rights
theory of the ﬁrm associated with Grossman and Hart (1986) and Hart and Moore (1990),
renegotiation is eﬃcient so that hold-up is virtually inevitable (Hart, 1995). Since there
is, consequently, no functional diﬀerence between contracts agreed upon ex-ante and those
agreed upon ex-post, payoﬀ uncertainty, in Hart’s (2009) terminology, can play no role.
A key contribution of this paper lies in quantifying the contracting distortion, as well as
its impact on the return to water-saving technology, using a structural econometric model.
A rather unique feature of a groundwater economy that allows us to do this is that buyers
and sellers are both agricultural producers, cultivating side-by-side with the same technology.
Our model of agricultural production under stochastic groundwater supply accounts for the
choice between seasonal contracts and per-irrigation sales, for water transfers through leasing,
as well as for the area irrigated under each such arrangement. We use data from a large
sample of borewell owners across six districts of Andhra Pradesh and Telangana states. The
specially-designed survey instrument takes particular care to elicit from each well owner a
subjective probability distribution of their borewell’s discharge near the end of the season
conditional on its initial discharge. The structural parameters of the model are identiﬁed
principally from the variation across borewells in this conditional probability distribution.
To assess external validity of the structural model, we retain two nonrandom holdout
samples corresponding to two of the six surveyed districts; borewells from the remaining
four districts constitute the estimation sample on which we ﬁt the model. Keane and Wolpin
(2007) argue for choosing “a [holdout] sample that diﬀers signiﬁcantly from the estimation
sample along the policy dimension that the model is meant to forecast (p. 1352).” The
analogue, in our setting, to a policy regime “well outside the support of the data” are the
large diﬀerences in ﬁrst and second moments of groundwater supply between estimation and
holdout districts.
4
Fehr et al. (2011), Hoppe and Schmitz (2011), and Bartling and Schmidt (2014) corroborate the
reference-point idea experimentally.
2
Structural estimation provides a threefold beneﬁt: First, it enables us to assess the
performance of actual contractual arrangements against the benchmark of Pareto-eﬃciency.
Relative to this ﬁrst-best counterfactual, we ﬁnd that observed contracts induce a substantial
reduction in irrigated area but only a modest welfare distortion. Second, structural estima-
tion allows us to compare the contracting distortion against another groundwater market
friction. In particular, once neighboring farmers all have borewells of their own, there is
limited scope for groundwater trade. A social planner, in this case, would want to drill
fewer wells (thus economizing on ﬁxed costs) and share more water among neighbors. To
capture the extent of this coordination failure, our empirical model incorporates a cost of
arranging groundwater transactions, which we ﬁnd to be of about the same magnitude as
the contracting distortion. Third, our structural estimates allow us to simulate irrigation
choices and returns to cultivation under a counterfactual water-saving technology, speciﬁ-
cally drip irrigation. Here we ﬁnd that switching from traditional to drip irrigation would
greatly stimulate groundwater market activity. We also show that, as a result, the farm-level
beneﬁts of drip adoption depend in a nuanced way on local patterns of land fragmentation.
This paper contributes to the empirical contracts literature in three ways. Early writings
in the transactions costs tradition (Williamson, 1971; Klein et al., 1978) recognize that long-
term contracts protect investments speciﬁc to a trading relationship but do so at a cost; in an
uncertain environment, contractual rigidity inevitably leads to resource misallocation, which
is obviated by ex-post or spot contracting. While the ensuing empirical literature investigates
the nature of long-term contracts, it has been largely silent on the choice of long-term over
spot contract, and, in particular, on how this choice is driven by the fundamental tradeoﬀ
between ex-ante and ex-post ineﬃciency.5 Second, structural estimation and quantitative
welfare analysis has been rare in the empirical contracts literature. Gagnepain et al. (2013) is
a notable exception. However, in their context of French public-sector contracts the tradeoﬀ
between ex-post renegotiation and ex-ante incentives is driven by asymmetric information
rather than, in our case, by payoﬀ uncertainty. Moreover, in the setting we consider, agents
have the option not to contract or trade at all (and many do not), which allows us to
investigate how payoﬀ uncertainty aﬀects overall market activity. Third, this paper is the
ﬁrst contract-theoretical application we are aware of incorporating subjective probabilities
5
Lafontaine and Slade (2012) review empirical studies of inter-ﬁrm contracting from various theoretical
perspectives. See Joskow (1987) on asset speciﬁcity and contract duration and Goldberg and Ericson (1987),
Masten and Crocker (1985), and Crocker and Masten (1988) on the structure of long-term contracts in
uncertain environments. Carlton (1979), Polinsky, (1987), and Hubbard and Weiner (1992) consider the
choice between long-term contracts and spot markets, but in these models there is no relationship-speciﬁc
investment; ﬁrms incur the transactions costs of long-term contracts to insure against cash-ﬂow variability.
3
(see Attanasio 2009, Delvande et al. 2011, and Mahajan et al. 2012 for reviews of related
work in other areas of economics).
Finally, there is burgeoning interest in the industrial organization of groundwater (e.g.,
Jacoby et al. 2004, Aggarwal 2007, Foster and Sekhri 2008, Anderson 2011, Banerji et al.
2012, Chakravorty and Somanathan 2014, and Michler and Wu 2014). None of this work,
however, focuses on groundwater supply uncertainty as a source of market failure nor on
how groundwater markets interact with water-saving technology adoption.6
The next section of the paper lays out the formal theoretical arguments. Section 3
describes our survey data and the groundwater economy of southern India in greater detail.
Section 4 adapts the theoretical model for the purposes of structural estimation and derives
the likelihood function. Estimation results and counterfactual simulations are reported in
Section 5. Section 6 concludes.
2 Theory
2.1 Preliminaries
We begin by brieﬂy enumerating the assumptions, leaving the more extended justiﬁcations
for Section 3.
A.1) Fragmentation: Agricultural production occurs on discrete plots of land of area a,
each owned by a distinct individual.
A.2) Borewells and groundwater: A reference plot has a borewell drawing a stochastic
quantity of groundwater w over the growing season, where w has p.d.f. ψ (w) on
support [wL , wH ].
As noted, groundwater is the sole (dry-season) irrigation source in our setting. Since prop-
erty rights to groundwater are not clearly delineated in India, there is no legal limit to
withdrawals. Upon striking an underground spring, farmers install the widest feasible pipe
consistent with the expected outﬂow. Likewise, because electricity is provided free at the
margin, farmers run their pumps for the maximum number of hours that power is avail-
able on any given day. Aside from pipe-width and electricity constraints, w depends on the
availability of groundwater in the aquifer at any given time and on the local hydro-geology.
6
Cary and Zilberman (2002) and Dridi and Khanna (2005) consider theoretically how spot markets in
surface water aﬀect water-saving technology adoption decisions in the developed country context. Pfeiﬀer
and Lin (2014) provide empirical evidence that the switch to water-eﬃcient center pivot irrigation in Kansas
led to an increase in groundwater use. Groundwater markets, however, are not relevant in this setting.
4
A.3) Agricultural technology: The common crop output production function, y = F (l, w, x),
depends on three inputs: land l, seed x, and water w, with land and seed used in ﬁxed
proportions. For any level of x, y/l = f (w/l) ≡ f (ω ), where ω is irrigation intensity
and the intensive production function, f , is increasing, concave, with f (0) = 0.7
Given A.3, we may write net revenue as l{f (ω ) − c}, where c is the cost of the required seed
per acre cultivated.
A.4) Risk preferences: Farmers are risk neutral.
Risk neutrality, a core assumption of transactions cost economics, is justiﬁed by evidence
presented below showing little, if any, role for risk preferences in irrigation decisions.
A.5) Land availability: A well-owner is not limited in the area of adjacent land that his
borewell can irrigate.
In invoking A.5, we abstract from any demand-side constraints that may arise when most
or all adjacent landowners also have their own borewells. While this assumption simpli-
ﬁes the theoretical analysis, it is unrealistic and will therefore be relaxed in the empirical
implementation.
Consider, ﬁrst, a well-owner’s choice of area cultivated (irrigated) when his own plot size
is not a constraint. Let U = arg max l {Ef (w/l) − c} and deﬁne the marginal return as
Deﬁnition 1 g (ω ) = f (ω ) − ωf (ω ).
The necessary condition for optimal planting
Eg (ω ) = c (1)
equates the expected marginal return to the marginal cost of cultivation.
Now, letting r index mean preserving increases in groundwater supply uncertainty, we
have
Proposition 1 (Precautionary planting) If g is strictly concave, then ∂ U /∂r < 0.8
7
Constant returns to scale is both technically convenient and empirically sensible. Diminishing returns
is unlikely to set in over the range of cultivated areas that we are considering. Moreover, under diminishing
returns, well-owners might simultaneously leave their own plot partially fallow while selling water to a
neighboring plot, a scenario virtually never observed in practice.
8
Proof: Follows directly from Theorem 1 of Diamond and Stiglitz (1974).
5
Table 1: Model Decisions
Timing
Contract type Moniker Ex-ante Ex-post
Long-term Seasonal p, τ , -
surplus divided
Spot Per-irrigation p, τ
surplus divided
Notes: p is the price per unit of irrigation, τ is the total transfer
of groundwater, and is area irrigated by the buyer.
In other words, well-owners may exhibit a precautionary motive analogous to that in the
savings literature (e.g., Kimball, 1990), in this case limiting their exposure to increases in
supply uncertainty by committing less area to irrigate.
The surplus generated by a borewell under unconstrained self-cultivation is
Deﬁnition 2 VU = UE [f (w/ U) − c] .
In case U > a, we may think of VU as the surplus derived by the well-owner if he could sell
an unlimited amount of groundwater in a competitive spot market.9 As mentioned, however,
groundwater transactions do not resemble this competitive, arm’s-length, ideal.
The next two subsections discuss each of the two observed forms of bilateral contracting
in turn, using Table 1 as an organizing framework.
2.2 Long-term contracts
The canonical long-term contract commits the well-owner to irrigate a buyer’s ﬁeld, or some
portion thereof, for the whole season at a pre-determined price. Following Hart and Moore
(2008), we think of such (ex-ante) contracts as establishing entitlements. Ex-post renegoti-
ation of the terms, or hold-up, will therefore lead to deadweight losses due to aggrievement
by one or both parties.10 To bring the tradeoﬀ between ex-ante and ex-post ineﬃciency into
9
To see why, let subscripts b and s denote water-buyer and seller, respectively. Further, let p be the
spot price and b the buyer’s cultivated area such that U = a + b . It is easy to see that f (ωb ) = f (ωs ) =
p which implies that ωb = ωs . Thus, VU = E [a(f (ωs ) − c) + pωb b ] = E [a(f (ωs ) − c) + f (ωb )ωb b ] =
E [a(f (ωs ) − c) + b (f (ωb ) − c)] = E [(a + b )(f (ωs ) − c)] , where the penultimate expression follows from
Eg (ωb ) = c, the necessary condition for the buyer’s optimal area cultivated.
10
More precisely, there are noncontractible actions that either party can take ex-post to add value to the
transaction. As long as a party feels he is getting what he is entitled to in the contract, he will undertake
such helpful actions, but if he feels shortchanged he will withhold them, generating a loss in surplus. In the
6
stark relief, we assume that these deadweight losses make hold-up prohibitively costly. Our
evidence, in fact, indicates that renegotiation of seasonal contracts is extremely rare.11
Summarizing, the seasonal contract has two salient features: First, by serving as ref-
erence point in, and hence as a deterrent to, renegotiation, it protects relationship-speciﬁc
investment (in our context, planting inputs); second, water allocations under the contract
are unresponsive to the state of the world.
Let τ denote the total transfer of groundwater at per unit price p to irrigate a ﬁeld of
size l. The optimal simple (i.e. single-price) contract solves
w−τ
max a Ef ( ) − c + pτ s.t.
p,l a
τ
P C : l f ( ) − c − pτ ≥ 0 (2)
l
τ
IC : τ = arg max l f ( ) − c − pτ
τ ∈[0,wL ) l
The ﬁrst term in the well-owner’s objective function (top line) is the expected revenue from
crop production on his own plot net of cultivation costs, which is diminished when he sells
water to a neighbor;12 the second term is his total revenue from the sale. The participation
constraint (P C ) stipulates that the crop revenue of the buyer net of both cultivation and
water costs cannot be negative. Finally, the incentive constraint (IC ) says that the transfer
is maximizing the buyer’s net revenue, subject to the constraint that the promised amount
cannot exceed the available supply of water in the lowest state of the world, wL . Note that
expectations are dropped in both the P C and IC because, under the contract, l and τ are
ﬁxed ex-ante. Thus, the seasonal contract oﬀers an assured supply of irrigation to the buyer;
the direct cost of production variability is borne fully by the seller on his plot.
words of Hart (2009): “Although our theory is static, it incorporates something akin to the notion of trust or
good will; this is what is destroyed if hold-up occurs.” (p. 270). Alternatively, Herweg and Schmidt (2014)
motivate the ineﬃciency of contract renegotiation using the notion of loss aversion.
11
For each of the 873 well-buyer-crop combinations in which a seasonal contract was undertaken in rabi
2011-12, our survey asks the borewell owner “Was the arrangement carried out as originally agreed?” with
possible responses: “(1) Yes; (2) No, price increased; (3) No, price decreased; (4) No, contract terminated.”
In all but one of these cases, the response was (1).
12
Without loss of generality, we assume that the constraint that the water-seller’s cultivated area ls
cannot exceed his plot area a is binding; i.e., the well-owner always fully cultivates his land before selling any
groundwater. Proof: Suppose not, then the optimal choice of ls requires Eg wl− s
τ
= c. However, equation
w
(1)⇒ Eg lU = c ⇒ τ = w(1 − ls /lU ), which is a contradiction because τ cannot be state-contingent.
7
Given a binding P C , the necessary conditions for the optimal contract are as follows:
w−τ
Ef =p
a
τ
g =c (3)
l
τ
f = p,
l
the solution to which is the water transfer-area pair (τC , C ). Divergence of supply and
demand for irrigation ex-post creates a distortion. Since (3) implies Ef w− a
τC
= f τC C
,
w − τC τC
it is not true, in general, that f ( a ) = f ( C ) ∀ w, which would obtain if τ were state-
contingent, as in a competitive spot market (see fn. 9). It follows as a corollary that the
distortion vanishes as uncertainty goes to zero, in which case g ( τCC
) = g ( Cw+a
) = c = g( wU
)
which implies that C = U − a. Thus, in the absence of uncertainty, the amount of land
irrigated and the economic surplus generated by the borewell would be the same under the
seasonal contract as under a competitive spot market; i.e., the long-term contract would
achieve the ﬁrst-best.
As usual, the roles of principal and agent here are entirely arbitrary; i.e., the constrained
Pareto eﬃcient allocation would be identical if the buyer were the monopsonist and it was
the seller whose PC was saturated. In other words, the division of ex-ante joint surplus is
both indeterminate and irrelevant for our purposes.
2.3 Spot contracts
Groundwater may also be sold on a per-irrigation basis. Once the season is underway,
however, commitments have been made. The potential seller has retained (i.e., refrained
from contracting out) the rights to some excess water from his well during the season whereas
the potential buyer has planted a crop in an adjacent plot. Since each party has some degree
of ex-post bargaining power, we use a Nash bargaining framework. To be clear, in a per-
irrigation arrangement there is a self-enforcing agreement to trade during the season, even
though the terms of these trades are not fully delineated ex-ante. Indeed, side-payments
may be made (or favors rendered) to secure an exclusive trading relationship. In other words,
as with the long-term contract, there is a division of ex-ante joint surplus (see Table 1) and,
just as in the long-term case, this division is irrelevant for allocations. We only assume that
8
any negotiations over this surplus are eﬃcient, leaving no money on the table.13
Turning to the ex-post stage, let τ be the amount of water already transferred to the
buyer and suppose that buyer and seller negotiate the price p of incremental transfer ∆.
The buyer’s net payoﬀ from consummating the trade is given by u = lf ((τ + ∆)/l) − p∆,
whereas that of the seller is v = af ((w − τ − ∆)/a) + p∆. The no-trade payoﬀs are given by
u = lf (τ /l) and v = af ((w − τ )/a), respectively. The absence of c in these payoﬀ functions
reﬂects the fact that all cultivation costs have already been incurred.
Given Nash bargaining, p∗ = arg max(u− u)η (v − v )1−η , where η is the buyer’s bargaining
weight.14 Therefore, p∗ solves
η (v − v ) − (1 − η )(u − u) = 0
f ( w−τ
a
−∆
) − f ( w−
a
τ
) f ( τ +∆
l
) − f(τ
l
)
ηa − (1 − η )l +p=0 (4)
∆ ∆
w−τ τ
−ηf − (1 − η )f +p=0
a l
where the last line takes the limit of the second line as ∆ → 0. Thus, we obtain the standard
surplus-splitting rule15
τ w−τ
p∗ (τ ) = (1 − η )f + ηf . (5)
l a
Furthermore, once f ( τ
l
) − p∗ (τ ) = η f ( τ
l
) − f ( w−
a
τ
) = 0, the buyer’s demand for irrigation
is sated. Thus, the total transfer τ must satisfy f ( τ l
) = f ( w−
a
τ
), which is the condition for
an ex-post eﬃcient allocation of groundwater conditional on area cultivated.
Now consider the buyer’s ex-ante problem of choosing area cultivated to maximize ex-
13
a la Grossman
We also abstract from any reallocation of property rights between the parties at this stage `
and Hart (1986), such as selling the borewell or the land itself. In the empirics, we allow for one form of
vertical integration: the well-owner can lease, at a cost, an adjacent plot without a well of its own.
14
While we take η as exogenous, the allocation of bargaining power may depend on local competition.
Given the spatial dispersion of borewells and high conveyance costs, we may think of groundwater markets
as trading networks. In Corominas-Bosch’s (2004) model of bilateral bargaining in such networks, a buyer’s
bargaining power depends, not only on the relative number of buyers and sellers, but also, critically, on the
link structure of the network, which may be extensive and thus diﬃcult to observe in practice.
15
To reiterate, this is ex-post surplus, which is distinct from ex-ante surplus in that it does not account
for (sunk) cultivation costs. That the water-buyer may have ex-post bargaining power (η > 0) is not in any
way inconsistent with the assumption that he is held to his PC in the long-term contract. As noted already,
the assignment of all ex-ante surplus to the well-owner is both an arbitrary and irrelevant assumption.
9
pected returns given price function p∗ and the total transfer τ . In particular,
τ
τ
P = arg max E lf − p∗ (t)dt − cl. (6)
l l 0
Observe that the per unit price of water is now state-dependent and, in particular, is no
longer constant as in the seasonal contract; each small increment of irrigation now has a
τ
diﬀerent cost. From (5), 0 p∗ (t)dt = (1 − η )lf ( τ
l
) + ηa f ( w−
a
τ
) − f(w
a
) , so only the ﬁrst
term on the right-hand side depends on l. The necessary condition for the buyer’s cultivation
choice is, therefore, simply
ηEg (τ /l) = c. (7)
Comparing equations (7) and (1), we see that they diﬀer by the factor η. Surplus extraction
on the part of the water seller eﬀectively taxes the marginal beneﬁts of cultivation, with the
tax rate decreasing in the buyer’s bargaining power.16
To summarize, spot contracts lead to an ex-post eﬃcient allocation but distort ex-ante
incentives. The latter ineﬃciency is due to the hold-up problem ﬁrst formalized by Grout
(1984); the buyer under-invests (indeed, P < U − a) in anticipation of ex-post rent appro-
priation.
2.4 Other contracts
Our approach, following in the tradition of the empirical contracts literature (e.g., Gagnepain
et al., 2013), has been to model only the principal arrangements observed in the data.
Nevertheless, it is worth a digression to discuss contracts that, although largely hypothetical,
are potentially more eﬃcient than those considered above.
2.4.1 First-best
It is clear from equation (7) that the ﬁrst-best contract has the seller commiting to η = 1.
This contract is tantamount to one in which the price of groundwater is indexed (cf. Hart,
2009) to the seller’s post-transfer marginal product f w− a
τ
. Perhaps the complexity of this
pricing scheme, the lack of observability (e.g., well ﬂow may be manipulable by the seller),
16
As before, the borewell owner fully cultivates his own plot before selling any groundwater (i.e., ls = a).
w −τ P w−τP
Proof: Suppose not, then Eg ls = c is necessary. However, equation (7) and f ( τ
lP ) = f ( ls ) ⇒
P
τP w−τP
Eg P
= Eg ls = c/η, which is a contradiction.
10
or lack of third-party state veriﬁcation, explains why we do not observe it in practice.17
Alternatively, a well-owner could achieve the ﬁrst-best allocation by subsidizing the
buyer’s planting cost at a rate of 1 − η ; obviously, allowing the planting investment to
be contractible obviates the hold-up problem. As a practical matter, however, it may be
quite diﬃcult for the seller to ensure the optimal ex-post demand for his water through such
an incentive scheme if the buyer is free to adjust the intensity of cultivation. While this
type of moral hazard problem, strictly speaking, lies outside of our model (because we have
assumed that land and inputs like seed are always used in ﬁxed proportions), it may explain
the absence of such planting subsidies in our setting.
2.4.2 Mixed
Thus far, we have analyzed each type of contract in isolation, not allowing borewell owners to
engage in both simultaneously. Our main reason for doing so is empirical; groundwater sales
to multiple buyers under diﬀerent contracts are rare in our data. The analysis of a mixed
contract, however, is straightforward. Given his residual water w − τC available ex-post,
the borewell owner sells an amount τP (τC ) on a per-irrigation basis to buyer B. Working
backwards, the amount sold to buyer A on a seasonal contract is the τC that maximizes
a Ef ( w−τC −a
τP (τC )
) − c + pτC , subject to the participation and incentive constraints.
Clearly, the mixed contract does not achieve the ﬁrst-best. While the allocation of water
between the seller’s plot and that of buyer B is ex-post eﬃcient, this is not the case for buyer
A. Indeed, neither the ex-post nor the ex-ante distortion is entirely eliminated.18
2.5 Characterizing the tradeoﬀ
Returning to the main argument, we have already seen that the distortion induced by the
long-term contract disappears when groundwater supply becomes perfectly certain, whereas
the distortion induced by the spot contract does not. Next, we establish a general result
about the dominance of long-term over spot contracts in our environment.
Recall that increases in r correspond to mean preserving increases in uncertainty, with
r = 0 indicating perfect certainty. Let Vj (r) be the surplus derived from contract of type
17
A share-contract by which groundwater is paid for out of the buyer’s crop partially mimics an indexed
price, though creates other incentive problems. Aggarwal (2007) ﬁnds that share-contracts for groundwater
are prevalent in parts of western India, but we have less than a handful of such cases in our data.
18
In the spirit of contracts as reference points, a buyer under a seasonal contract is precluded from selling
back water to the borewell owner, or to anyone else, on a per-irrigation basis as this would presumably
aggrieve the borewell owner.
11
j = C, P ,19 and note that VP (r, η ) also depends on the bargaining weight η .
Proposition 2 (Dominance) If g is strictly concave and τC (0) < wL ,20 then (a) for some
η , ∃ a unique r∗ (η ) such that VC (r∗ ) = VP (r∗ , η ); (b) [VC (r) − VP (r, η )] (r∗ − r) > 0.21
Simply put, under the conditions of proposition 2, there can be a level of uncertainty at
which the parties are indiﬀerent between seasonal and per irrigation arrangements. If so,
then the seasonal contract must dominate at low levels of uncertainty and per-irrigation sales
at high levels of uncertainty.
Figure 1 illustrates the intuition underlying proposition 2, showing how the economic sur-
plus generated by a borewell varies with uncertainty level r under alternative water transfer
arrangements. Regardless of arrangement, surplus always decreases with r (see Appendix).
In the case of autarky (A), in which the borewell irrigates exactly plot area a, surplus is
VA = aE [f (w/a) − c] . VA must lie strictly below ﬁrst-best surplus VU except at r = rU ; at
this level of uncertainty, U = a and autarky is the optimal unconstrained choice. When the
borewell owner sells water under a seasonal contract, surplus VC is also less than ﬁrst-best
(except under perfect certainty), coinciding with VA at some positive level of uncertainty
rC < rU . Note that VC declines relatively rapidly with r because higher uncertainty operates
upon two margins under a seasonal contract: It leads to greater ex-post misallocation of
groundwater across plots as well as to a contraction of overall area irrigated by the borewell
(precautionary planting). Only the latter eﬀect is operative under the per-irrigation arrange-
ment. In this case, surplus VP approaches VU as η approaches one. Moreover, at some low
level of bargaining power η = η , P = 0 and VP coincides with VA . So, for some range of
η ∈ (η, 1), VP and VC must cross. Given such a crossing (at r∗ ), VP coincides with VA at
a level of uncertainty rP between rC and rU . This shows that the spot contract can only
dominate the long-term contract at higher levels of uncertainty.
19
For the seasonal contract, surplus is given by the private returns to the well-owner; since the P C is
binding, the water-seller gets all the surplus. By contrast, in the per-irrigation case, we must consider
the joint surplus of well-owner and water-buyer. It might be argued that the choice of per-irrigation over
alternative arrangements should be governed by the water seller’s private returns as well. This, however,
runs counter to our assumption that all ex-ante negotiations are eﬃcient. In other words, situations in
which the per irrigation arrangement yields the highest joint surplus but fails to maximize the well-owner’s
private return would be resolved through side-payments.
20
In words, this latter condition states that the water transfer under perfect certainty is less than total
water available in the worst state of the world. Otherwise, VC has a discontinuity at r = 0; i.e., at r = , the
optimal transfer must be discretely less than τC (0). In this case, r∗ (η ) still exists for some η but it is not
necessarily unique. Part (b) of the proposition continues to hold, however, with respect to the largest r∗ .
21
Proof: See Appendix A.
12
Figure 1: Long-term versus spot contracts and uncertainty
Surplus
C P A U
∗ Mean preserving spread
Notes: C, P, A, and U denote regions where dominant arrangement is, respectively, long term contract, spot
contract, autarky, and unconstrained cultivation. Dashed portion of VU curve is unattainable given absence
of competitive spot markets.
3 Context
3.1 Groundwater markets survey
Our data come from a randomly selected survey of about 2300 borewell owners undertaken in
2012-13 in six districts of Andhra Pradesh (AP) and Telangana (until 2014, also part of AP).
The districts were selected to cover a broad range of groundwater availability, conditions for
which generally improve as one moves from the relatively arid interior of the state toward
the lusher coast.22 Drought-prone Anantapur and Mahbubnagar districts were originally
selected as part of a weather-index insurance experiment (Cole et al., 2013); all 710 borewell
owners were followed up from that study’s 2010 household survey. Guntur and Kadapa
districts, which fall in the intermediate range of rainfall scarcity, and the water-abundant
coastal districts of East and West Godavari, each contribute around 400 borewell owners.
All in all, our survey obtained information on 2,411 borewells in 144 villages (21-25 per
district). At the time of the survey, none of the plots on which these borewells were situated
22
Our sample is broadly representative of areas where groundwater is suﬃcient for rabi cultivation and
where it is the sole source of irrigation for that season (canal command areas were avoided).
13
was equipped with drip irrigation systems.
To capture transfers of groundwater, which typically occur between adjacent plots so
as to minimize conveyance losses,23 we departed from the usual household-based sampling
strategy. Instead, each respondent (borewell owner) was also asked to report on all the
plots adjacent to the one containing the reference borewell, including characteristics of the
landowner, details on how the plot was irrigated during the rabi, if not left fallow, and on the
transfer arrangement if one occurred. The number of adjacent plots varies from 1 to 7, with
a mode of 3. Not only does this adjacency approach provide information about transfers
that did happen but also about those that could have happened but did not.
3.2 Recharge and uncertainty
As in much of India, farmers in AP rely almost exclusively on groundwater during the rabi
(winter or dry) season, when rainfall is minimal and surface irrigation typically unavailable.
Indeed, recent years have seen an explosion of borewell investment as the costs of drilling
and of submersible electric pumpsets have fallen, raising concern about groundwater over-
exploitation (e.g., World Bank, 2005). Nonetheless, the time-series of depth to watertable
across AP in the last decade and a half is dominated by intra -annual variability (see Ap-
pendix Figure B.1). This is explained by the limited storage capacity of the shallow hard
rock aquifers that characterize the region. Most of the recharge from monsoon rains occur-
ring over the summer months is depleted through groundwater extraction in the ensuing
rabi season. In contrast to the deep alluvial aquifers of Northwest India, there are no deep
groundwater reserves to mine (see Fishman et al., 2011).
This annual cycle of aquifer replenishment and draw-down throughout AP is central to
our analysis of groundwater markets. Although farmers can observe monsoon rainfall along
with their own borewell’s discharge prior to rabi planting, they cannot perfectly forecast
groundwater availability over the entire season. To measure the degree of uncertainty, as
part of the borewell owner’s survey we ﬁelded a well-ﬂow expectations module, which was
structured as follows: First, we asked owners to assess the probability distribution of ﬂow
on a typical day at the start of (any) rabi season, the metric for discharge being fullness
3
of the outlet pipe (i.e., full, 4 full, 1
2
full, 1
4
full, empty).24 Next, using the same format,
23
Most irrigation water is transfered through unlined ﬁeld channels with high seepage rates. While our
survey also picked up a number of transfers to non-adjacent plots using PVC pipe, usually these cases
involved sharing of groundwater between well co-owners or between multiple plots of the same owner.
24
To appreciate how discharge can be fractional for an extended period, the metaphor for the aquifer to
keep in mind is that of a sponge rather than of a bathtub.
14
Figure 2: Distribution of groundwater supply uncertainty and pipe width
Anantapur (holdout) Anantapur (holdout)
10 20 30
3
2
1
0
0
10 20 30 East Godavari (holdout) East Godavari (holdout)
3
2
1
Density
Density
0
0
Estimation sample Estimation sample
10 20 30
3
2
1
0
0
Total Total
10 20 30
3
2
1
0
0
0 .2 .4 .6 0 2 4 6 8
CV end-of-season flow Pipe width (inches)
Notes: Dashed vertical lines represent medians for the full sample.
we asked about the probability distribution of end-of-season ﬂow conditional on the most
probable start-of-season ﬂow. Thus, the question was designed to elicit residual uncertainty
about groundwater availability.
The bottom left panel of Figure 2 shows substantial variability in groundwater uncer-
tainty (well-speciﬁc coeﬃcients of variation of end-of-season ﬂow) in the overall sample.
Notice that virtually no borewell owner (save ﬁve) report having a perfectly certain supply
of groundwater. Next, we consider the salience of groundwater supply uncertainty for dry
season production decisions.
3.3 Precautionary planting and risk aversion
Proposition 1 shows that uncertainty in groundwater availability can lead to precautionary
planting. This result, however, hinges on the properties of g , the marginal return to planting,
which is not directly observable. To motivate our theoretical assumptions, we now present
a reduced-form analysis of planting decisions. Ultimately, of course, it is only by estimating
the full structural model that we can distinguish precautionary planting per se from the
eﬀects of contracting distortions and other transactions costs.
Rabi season cultivation in southern India falls into two broad categories: wet crops (in
15
Table 2: Precautionary Planting and Risk Aversion
(1) (2) (3) (4) (5)
log(borewell plot area) 0.573*** 0.560*** 0.544*** 0.545*** 0.545***
(0.0155) (0.0149) (0.0146) (0.0144) (0.0145)
log(mean well ﬂow) 0.692*** 0.763*** 0.770*** 0.762***
(0.110) (0.110) (0.111) (0.110)
log(pipe width) 0.463*** 0.441*** 0.439*** 0.440***
(0.0366) (0.0405) (0.0406) (0.0405)
log(CV ) -0.709*** -0.311*** -0.249*** -0.226** -0.213***
(0.0256) (0.0459) (0.0468) (0.0882) (0.0749)
RISK 1 -0.0146
(0.0231)
log(CV ) × RISK 1 -0.00309
(0.0118)
RISK 2 -0.144
(0.176)
log(CV ) × RISK 2 -0.0537
(0.0925)
R2 0.599 0.634 0.645 0.645 0.645
Controls No No Yes Yes Yes
Notes: Robust standard errors in parentheses adjusted for clustering on borewell (*** p<0.01, **
p<0.05, * p<0.1). Sample size is 2,411 borewells. Dependent variable in all regressions is log(total
irrigated area by borewell); CV is the coeﬃcient of variation of end-of-season well ﬂow; RISK 1
is a self-assessed ranking of risk tolerance; RISK 2 is an index of marginal willingness-to-pay for
risk based on Binswanger lotteries. Unreported controls are as follows: pump horse-power, log of
well depth, number of other borewells within 100 meters, dummy for presence of recharge source.
our six districts, principally paddy, banana, sugarcane, and mulberry) and irrigated-dry or
ID crops (e.g., groundnut, maize, cotton, chillies), distinguished by the much greater water
requirements of the former. Since a ﬁeld that, planted to ID crops, would take 3 days to
irrigate would take a week to irrigate under wet crops, we use the equivalence 1 acre wet =
7
3
acre ID to compute total area irrigated by a borewell.25
We investigate the eﬀect of uncertainty on rabi area irrigated by a borewell conditional on
the plot area of that borewell. For 27% of the borewells, irrigated area is less than reference
plot area (also expressed in dry-equivalent acres), indicating that part of the borewell’s plot
25
We carry this eﬃciency units assumption over to our structural estimation. To appreciate the simpli-
ﬁcation thereby achieved, consider the implications of separate wet and ID crop technologies. In the ﬁrst
place, conditional on area choices of each crop type, farmers would presumably allocate groundwater ex-post
across crops in response to the realized w. This gives rise to one additional optimality condition for each
state of nature. Second, there would be two cultivated area choices, and farmers are observed opting for
mixed wet-ID cropping as well as for monoculture of either type. To rationalize the data, our empirical
model would need two structural error terms (instead of just the one ultimately assumed) and would have
to account for the two types of corner solutions in cropped area. Third, for any form of groundwater trans-
fer, each cell of the 3 × 3 matrix of wet-ID-mixed cropping decisions of borewell owner and groundwater
recipient would have to be compared to determine the optimal arrangement and the bivariate distribution
of the structural error terms partitioned accordingly. As the composite-crop model already captures the
fundamental trade-oﬀ between ex-ante and ex-post eﬃciency, we believe that a dual-crop model oﬀers little
in the way of additional insight relative to this enormous increase in computational burden.
16
was left fallow in the past rabi season, whereas for 46% of borewells, the opposite is true;
groundwater was either sold (irrigating the land of another farmer in the adjacency) or was
transferred to a leased plot. Thus, we regress log irrigated area on log of reference plot area
and the log(CVi ) of end-of-season ﬂow (see ﬁgure 2). The negative coeﬃcient on the latter
variable in column 1 of Table 2 means that that as groundwater uncertainty increases area
planted/irrigated declines. The impact of uncertainty is diminished, but not eliminated, after
controlling for mean end-of-season well ﬂow (based on the ﬁve-point scale) and outlet pipe-
width (col. 2). The result is also robust to controls for additional borewell characteristics
(pump horse-power, well depth, number of nearby wells, presence of groundwater recharge)
as seen in column 3.
While this evidence supports a precautionary planting motive, as well as the salience of
our uncertainty measure, we have not yet established the theoretical mechanism. To the
extent that variability in irrigation supply induces ﬂuctuations in household income, simple
risk aversion may explain why farmers limit their rabi planting in the face of uncertainty.
To assess this, we use two measures of preferences towards risk collected from well owners
in the survey. The ﬁrst measure, RISKi1 , is a self-assessed ranking of risk tolerance, with 1
indicating “I am fully prepared to take risks” and 10 indicating “I always try to avoid taking
risk.” The second measure is based on a Binswanger (1980) lottery played by each respondent
for real money. Following Cole et al. (2013), RISKi2 is an index of marginal willingness-
to-pay for risk constructed from the characterstics of the preferred lottery. Ranging from 0
to 1, higher values of RISKi2 indicate greater risk aversion. In columns 4 and 5 of Table
2, we report results of adding, respectively, RISKi1 and RISKi2 , and, most importantly,
their interactions with log(CVi ), in the corresponding baseline regression of column 3. The
estimated coeﬃcients on these interactions, and particularly their lack of signiﬁcance, betrays
no indication that highly risk averse borewell owners are especially responsive to groundwater
uncertainty. Precautionary planting, therefore, does not appear driven by risk preference.
For this reason, as noted in A.4, we assume risk neutrality throughout the paper.
3.4 Land fragmentation, ﬁxed costs, and groundwater markets
Aside from uncertainty, our environment is characterized by considerable land fragmentation
coupled with a high ﬁxed cost of borewell installation, on the order of US$1000 (excluding
the pump-set). Fragmentation is driven by the pervasive inheritance norm dictating equal
division of land among sons and the prohibitive transaction costs entailed in consolidating
spatially dispersed plots through the land market. In our data, nearly 80 percent of plots
17
Figure 3: Effects of average plot size in adjacency
1
.8
.6
.4
.2
0 1 2
log average plot size in adjacency
Borewell density
Groundwater transaction intensity
Notes: Nonparametric regressions of borewell density and groundwater transactions per plot, respectively,
on log average plot size for all 2306 reference borewell plot adjacencies.
were acquired through inheritance.
Land fragmentation would be irrelevant, of course, were groundwater markets frictionless.
In this case, Coasian reasoning suggests that allocations should be independent of asset
ownership. Thus, borewells would be just as likely on small plots as on large plots; the
owner of a small plot could simply sell any excess groundwater to a neighbor. Obversely,
small plots would be just as likely cultivated in the dry season as large plots; any plot
owner without a borewell of his own could purchase groundwater from that of a neighbor.
In Appendix B, however, we use plot-level data from our sample to strongly reject both
implications of frictionless groundwater markets. Thus, in practice, land fragmentation does
predict the distribution of borewells and fallowed area.
When we aggregate these data from around 9600 plots to the adjacency level in Figure 3,
two key facts emerge: First, borewell density–measured as the ratio of borewell irrigated area
to total area in the adjacency–is increasing in average adjacency plot area. Put another way,
in more fragmented adjacencies, borewell density is lower; this simply reﬂects the ﬁnding in
Appendix B that borewells are much less likely to be installed on small plots. Second, the
proportion of plots in the adjacency receiving any groundwater transfer from the reference
well (aside from transfers between its co-owners) is decreasing in average adjacency plot
area. So, borewell density and groundwater market activity are substitutes, both driven by
the degree of land fragmentation. We return to this feature in the econometric model.
18
3.5 Sub-samples and characteristics of adjacent plots
Our subsequent analysis will rely on three subsamples of borewells: First, we have the esti-
mation sample, consisting of 1646 observations from the districts of Mahbubnagar, Guntur,
Kadapa, and West Godavari. As the name implies, this sample will be used to estimate the
structural econometric model as described in the next section. Borewells from the remaining
districts shall comprise two distinct holdout samples reserved for model validation. Following
Keane and Wolpin (2007), the choice of holdout districts is dictated by their outlier status
with respect to the ﬁrst and second moments of groundwater supply. The top two panels of
Figure 2 show that the coeﬃcients of variation of end-of-season ﬂow (second moment) and
pipe-widths (ﬁrst moment) in the holdout districts are indeed well beyond the median values
for the sample as a whole. In the case of Anantapur, the most arid district, groundwater
uncertainty is extremely high and borewells have median pipe-width of only two inches. By
contrast, in the much wetter and groundwater abundant East Godavari, median pipe-width
is 4 inches, far above that of the other ﬁve districts, and CVs are also among the lowest.
Table 3: Characteristics of Adjacent Plots by Subsample
Holdout Samples
Estimation Anantapur E. Godavari Total
Mean number per adjacency 3.46 3.64 3.25 3.45
Mean area (acres) 3.16 2.59 3.20 3.04
% left fallow in rabi 11 40 2 14
% irrigated in rabi by
• reference borewell 34 16 46 33
of which, % irrigated under
-joint ownership 31 96 17 33
-land lease 7 2 4 6
-water sale 62 2 80 62
• own borewell 48 43 41 46
• other borewell 12 1 13 10
% owned by
• brother 12 24 5 12
• other relative 11 11 4 10
• unrelated/same caste 45 31 57 45
• unrelated/diﬀerent caste 32 34 34 33
Number of plots 4992 1106 1195 7293
Descriptive statistics on the nearly 7300 adjacent plots, shown in Table 3, provide insights
into the nature of groundwater transfers and how they vary across subsamples. Overall, a
19
third of adjacent plots are irrigated in whole or in part by the reference borewell. However,
this ﬁgure falls to just 16% of adjacent plots in Anantapur, even though the proportion owned
by brothers, who are more likely to be co-owners of the reference borewell, is comparatively
high in this holdout district. A similarly wide disparity exists in the percentages of plots
accessing other borewells, which may or may not be in the adjacency. Thus, transfers of
groundwater outside of joint ownership arrangements, and especially sales, are extremely
limited in Anantapur and commensurate with this is a relatively high fraction of adjacent
plots left fallow in rabi season. The situation is reversed for the holdout district of East
Godovari, which has more groundwater sales than average and much less fallow.26
4 Estimation framework
4.1 Preliminary analysis
Table 4 provides descriptive statistics for the estimation and holdout samples according
to the groundwater transfer choices made by the borewell owner.27 Half of the owners in
the estimation sample transferred groundwater to other plots in the adjacency during the
past rabi season, slightly favoring the seasonal contract over the per-irrigation sale, with
leasing trailing far behind in terms of popularity. Conditional on making a transfer, mean
area irrigated is highest for the seasonal contract, followed by leasing and the per-irrigation
arrangement.
As noted above, there are stark diﬀerences between the two holdout samples. Groundwa-
ter transactions are virtually nonexistent in Anantapur and more than half of the borewells
are unconstrained (U ) in the sense that they are irrigating less than their plot area. In East
Godavari, on the other hand, unconstrained cultivation is comparatively rare. And, while
the fraction of borewells with groundwater sales is certainly higher than in the estimation
sample, what is particularly striking is the large proportion of seasonal contracts among sales
in this holdout district.
Our theory suggests that spot contracts are more attractive than long-term contracts
26
As most sales transactions (86%) are between non-relatives, hold-up concerns are not prima facie
misplaced. Among these non-relative transactions, 60% are between members of the same caste. However,
57% of non-related adjacent plot owners are of the same caste as the owner of the reference borewell. Hence,
Anderson’s (2011) suggestion of caste-based barriers to groundwater trade does not ﬁnd support in our data.
27
In the 116 cases were the reference plot has multiple borewells, we allocate the total area of the adjacency
equally among wells, treating each well as an independent decision unit within its own (pro-rated) adjacency.
Also, in case of joint ownership of the reference borewell, we merge the plots of all co-owners found in the
adjacency.
20
Table 4: Descriptive Statistics by Subsample
Holdout Samples
Estimation Anantapur E. Godavari
Choice j N j N j N j
U =unconstrained 436 5.10 184 3.20 31 10.23
[0.26] (3.89) [0.51] (2.55) [0.08] (7.21)
A =autarky 381 6.29 170 3.82 104 13.00
[0.23] (4.11) [0.47] (2.40) [0.26] (26.94)
L =leasing 91 2.45 5 0.90 12 6.78
[0.06] (2.43) [0.01] (0.60) [0.03] (7.56)
C =seasonal contract 400 2.80 1 1 238 3.61
[0.24] (2.45) [0.00] − [0.59] (3.07)
P =per-irrigation sale 338 1.77 2 1.87 21 3.42
[0.21] (1.48) [0.01] (1.23) [0.05] (2.29)
Total 1646 − 362 − 406 −
Notes: Sample means (standard deviations). Proportions in square brackets.
Figure 4: Prevalence of spot contracts and uncertainty
.8
1st 99th
.6
pctile pctile
Pr(P | P + C = 1)
.4.2
0
0 .1 .2 .3 .4
CV end-of-season flow
95% CI local fit
Notes: Nonparametric regression of per-irrigation arrangement indicator (P ) on CV using sample of 1000
borewells engaged in any groundwater sale (either P = 1 or C = 1). Endpoints of conﬁdence interval
approximated by logit transformation.
21
when uncertainty is high. Figure 4 bears out this key implication. Conditional on engag-
ing in one of the two transactions, the probability of choosing the short-term per-irrigation
arrangement (P ) is increasing in the coeﬃcient of variation of well ﬂow (the arc-elasticity
of the probability between the 1st and 99th percentile of CV is 0.41).28 While this ﬁnding
supports the theoretical model, there are compelling reasons to press forward with structural
estimation. First, the structural model fully accounts for selection into who transacts in the
groundwater market. Second, and most importantly, the theoretical model has quantita-
tive implications for the contracting distortion that cannot be captured in a reduced-form
analysis.
4.2 Leasing
To account for leasing, we allow that this arrangement may entail an eﬃciency cost making it
less attractive than irrigating one’s own land. A rationalization for such costs, corroborated
by Jacoby and Mansuri (2009), is that underprovision of non-contractible investment (e.g.,
soil improvement) lowers the productivity of leased land. At any rate, without invoking some
sort of leasing cost, the existence of a market for groundwater and, indeed, the predominance
of groundwater sales over land leasing would be problematic (in the next subsection, we also
introduce a ﬁxed cost of leasing).
Thus, let γ > 0 be the proportional increment to cultivation costs that applies only to
leased land. Optimal leased area is then given by
L = arg max(a + l) {Ef (w/(a + l)) − c} − γcl. (8)
4.3 Functional form
We next assume that the intensive production function, f , takes the form
f (ω ) = ζω α , (9)
where 0 < α < 1. That is, F is Cobb-Douglas in land and water. For reasons that will
become clear shortly, we normalize the parameter ζ = 1. With these assumptions, the
implied g (marginal return to planting) is globally concave and, thus, by proposition 1
28
Based on a simple probit regression, the null hypothesis of no uncertainty eﬀect on contract choice
can be strongly rejected (p = 0.004); likewise (p = 0.003), if we condition on Ew/a, a measure of per acre
groundwater availability.
22
Table 5: Theoretical Solutions for Irrigated Area and Economic Surplus
Choice j Area ( j ) Surplus (Vj )
U =unconstrained 1−α α 1/α αc
c Ew 1−α U
A =autarky a a1−α Ewα − ca
L =leasing (1 + γ )−1/α U −a (1 + γ )1−1/α VU + γca
C =seasonal contract C solves E Ωα−1 = 1 αc
1−α [E Ωα + α C /a − (1 − α)]
P =per-irrigation sale η 1/α U −a δVU
1 1−α 1/α 1−α 1/α 1−η (1−α) 1/α−1
Notes: Ω = a w c − C , where C ≤ c wL , and δ = α η .
and consistent with the empirical evidence presented in Section 3, there is a precautionary
planting motive.
Combining equations (9) and (1) along with deﬁnition 2 yields expressions for U and VU
as reported in the ﬁrst row of Table 5. As seen in the remainder of the table, a closed form
for area irrigated is lacking only for the seasonal contract. In the case of per-irrigation sales,
we ﬁnd that the ratio of economic surplus to unconstrained or ﬁrst-best surplus, VP /VU , is
simply equal to a constant δ < 1 (see note to Table 5). Thus, 1 − δ represents the relative
distortion of the per-irrigation arrangement, which vanishes as the buyer’s bargaining weight
η approaches unity.
4.4 Borewell discharge
As noted in Section 3, well-owners report conditional probabilities for ﬁve water ﬂow states,
corresponding to “full”, 3/4, 1/2, 1/4, and zero ﬂow. For empirical purposes, therefore, the
groundwater distribution ψ (w) is discrete, consisting of ﬁve points of support k = 0, 1, 2, 3, 4
and corresponding borewell-speciﬁc probabilities, πki .
Since water discharge is proportional to the square of pipe radius Ri , we have
2
wki = λRi k. (10)
Note that the parameter λ in equation (10) and ζ in equation (9) always enter the model
together in the form ζλα , which is why we normalize ζ to one. Since expected groundwater
2
supply is w¯i ≡ Ewi = λRi k πki k , we think of λ as reﬂecting factors that shift the ﬁrst
23
moment of total eﬀective discharge, such as soil moisture retention capacity and, importantly
for later, the water-eﬃciency of the irrigation technology.
4.5 Cost disturbance
To explain why diﬀerent water transfer arrangements (including no transfers at all) are
chosen across observationally equivalent borewells, and also why diﬀerent areas are cultivated
2
conditional on the transfer arrangement, we introduce a cost disturbance εi ∼ N (0, σε ) such
εi
that ci = ce . We assume that cost heterogeneity reﬂects variation in local (adjacency-level)
conditions, such as soil texture and depth, or in the shadow price of inputs like seed and
fertilizer, rather than in cultivator characteristics.
Inserting ci into the expressions given in the ﬁrst column of Table 5 and inverting the
resulting functions ji = j (εji ) yields choice-speciﬁc residuals εji for j = U, L, C, P , where
ji is the area under that arrangement for borewell i.
4.6 Fixed transactions costs
Given the contracting distortions built into our model, a well-owner would want to remain
in autarky (choice A) over a range of ε. As it stands, however, the model implies that the
minimum groundwater market transaction would involve an inﬁnitesimal quantity of water
(area).29 To allow for a discontinuity in irrigated area as one exits autarky, we introduce a
ﬁxed transactions cost κ.
We also allow κ to vary across adjacencies to account for the availability of neighbor-
ing land to irrigate. A lease or water sale is diﬃcult, if not impossible, to arrange if all
nearby plots are already irrigated by their own borewells. In this sense, there is strategic
substitutability among borewells. Indeed, there may be greater incentives for farmers to
drill wells of their own in areas less conducive to groundwater markets in the ﬁrst place. In
other words, the density of borewells within an adjacency may be endogenous with respect
to groundwater market activity.
To capture heterogeneity in κ across adjacencies, therefore, we require, in eﬀect, an
instrument for borewell density. As Figure 3 suggests, we do have such an instrument:
average plot size in borewell i ’s adjacency, ai , which is plausibly unrelated to groundwater
markets except through borewell density. Thus, we incorporate heterogeneity in irrigable
29
At least for choices L and P . For the seasonal contract, choice C , the minimum transaction is the area
that can be irrigated by wL , the available groundwater supply in the lowest state.
24
land availability by introducing ai into κ as follows:
κji = κj + β log ai (11)
for water transfer type j = L, P, C . Note that β is assumed to be the same across transaction
types, which is consistent with its interpretation as the eﬀect of availability of adjacent land to
irrigate. As for the constant term κj , we allow for a separate κL , but restrict κC = κP ≡ κT .
Whereas we expect that κL > κT , since a land lease is likely to be more costly to arrange
than a water sale, our presumption is that the cost of arranging a water sale is independent
of its speciﬁc contractual terms.
4.7 Likelihood function
The mixed continuous/discrete choice likelihood function involves the probabilities of the
diﬀerent water arrangments (choices) j = U, A, L, C, P and the densities of irrigated areas,
ji , conditional on these choices. Choice probabilities are determined by the set of thresholds,
˜jj i , that solve the following crossing conditions for arrangement-speciﬁc value functions:
ε
VA (εAU i ) = VU (εAU i )
VA (εjAi ) = Vj (εjAi ) − κji j = L, C, P (12)
Vj (εjj i ) − κji = Vj (εjj i ) − κj i (j, j ) = {(C, L) , (P, L) , (C, P )} .
The solution to the ﬁrst of these equations yields ε ˜AU i , the upper limit of integration for
the autarky probability, which has a simple closed form. The second set of equalities in
(12) yield the cultivation costs at which the well-owner is just indiﬀerent between autarky
and transferring water under arrangement L, C and P , respectively. Given nonzero ﬁxed
costs, these thresholds do not have closed-form solutions and, hence, must be solved for
numerically given data and parameters. The third set of equalities give the cost thresholds
between alternative transfer arrangements and also do not have closed form solutions in
general.
Letting Pr(j |Θ, Zi ) denote the probability of choice j conditional on parameters Θ =
2
(α, η, γ, κL , κT , β, λ, c, σε ) and data Zi = (πki , Ri , ai , ai ), we have
1 ε
Pr(j |Θ, Zi ) = φ( )dε (13)
ε
ji
σε σε
25
where φ is the standard normal pdf, and ε ji is the relevant region of integration for choice
j . The simplest choice probability is for unconstrained cultivation, which is Pr(U |Θ, Zi ) =
1 − Φ(˜
εAU i /σε ), where Φ is the standard normal cdf and the dependence of ε ˜AU i on parameters
and data is implicit. For autarky,
εAU i /σε ) − Φ(˜
Pr(A|Θ, Zi ) = Φ(˜ εAi /σε ), (14)
where ε˜Ai = max {ε ˜LAi , ε ˜P Ai } represents the highest εi that would induce any kind of
˜CAi , ε
water transfer. For the probabilities of arrangements j = L, C, P , the regions of integration
are not easy to write out as there are many possible conﬁgurations of the relevant thresholds,
including cases where ε ji has two disjoint segments. Appendix Table D1 enumerates all 38
possible conﬁgurations and their associated integration limits.
Now, let dji take a value of 1 when well-owner i chooses water arrangement j = U, A, L, C, P
and zero otherwise. The likelihood contribution of borewell i is
dji
dU i Pr(j |Θ, Zi )
Li (Θ|dji , ji , Zi ) = HU ( U i) Pr(A|Θ, Zi )dAi Hj ( ji ) , (15)
j =L,P,C
Φ(˜ εAi )
where Hj = |∂εji /∂ ji |φ(εji /σε )/σε is the density of irrigated area under arrangement j ,
the product of the absolute value of the Jacobian of inverse transformation εji ( ji ) and the
density of the cost disturbance for the given j .
The ﬁrst two terms of the likelihood essentially form a tobit model, with HU accounting
for the uncensored observations, U i < ai ,30 and Pr(A|Θ, Zi ), as given by equation (14),
accounting for the censored observations. The third term of the likelihood can be understood
using Figure 5, which illustrates two possible conﬁgurations for a well-owner selling water
on a per-irrigation (P ) basis (VL is not relevant for these cases and hence does not appear
in the ﬁgure). In panel (a), P is optimal only for εi in the left tail. Thus, to observe
P > 0, not only must we have εi < ε ˜CP i . To account for this additional
˜Ai , but also εi < ε
right truncation, Pr(P |Θ, Zi ) multiplies the truncated density HP ( P i )/Φ(˜ εAi ) in the third
likelihood term.31 In panel (b), P is optimal for moderate εi with C dominating in the left
tail. As in the previous scenario, the third term of the likelihood must include Pr(P |Θ, Zi )
to account for this left truncation.32
30
This is, of course, why Pr(U |Θ, Zi ) drops out of the likelihood.
31
The presence of Pr(j |Θ, Zi )/Φ(˜ εAi ) is reminiscent of Cragg’s (1971) hurdle model. In our model,
however, there is one error term; i.e., the hurdle is not determined by a second independent error. Note as
well that the ﬁxed transactions cost (κji ) guarantees a discontinuity in the irrigated area schedule.
32
With only a single transaction type, say P , the third term would collapse to HP ( P i ) and the likelihood
26
Figure 5: Surplus and area irrigated under two possible model configurations
Surplus
Surplus
− − − −
∗ ∗ ∗ ∗
∗
Cultivation cost
∗
Cultivation cost
Area irrigated
Area irrigated
ℓ +
ℓ +
ℓ + ℓ +
ℓ ℓ
P C A U Cultivation cost C P A U Cultivation cost
(a) PCAU Conﬁguration (b) CPAU Conﬁguration
Notes: Solid line segments in bottom graphs show irrigated area schedule relevant for the optimal contractual
arrangement given ε.
4.8 Identiﬁcation
Although, in practice, identiﬁcation is secured through the full set of nonlinear cross-equation
restrictions embedded in the thresholds deﬁned by (12) and in the choice-speciﬁc residuals,
heuristically, it is helpful to think of particular moments of the data identifying particular
parameters. Thus, for example, the κj of equation (11) are identiﬁed from the fractions of
well-owners selling water and leasing in land whereas β is identiﬁed by the extent to which
these fractions vary with average plot size in the adjacency.
Note that, for j = L, P, U , log area irrigated is
1 α
log Iji = K + log η dP i − log(1 + γ ) dLi + log πki wki − εi (16)
α k
where Iji = ji + ai for j = L, P and Iji = ji for j = U and the constant term K =
log(1 − α) − log c + α log λ. It is evident from (16) that η and γ are identiﬁed oﬀ of mean
diﬀerences in irrigated area across arrangements (controlling for selection), α by the rate at
which irrigated area falls with variability in water supply, K by average area in unconstrained
cultivation, and σε by the residual variance of irrigated area. Although λ and c are conﬂated
would be identical to that of models with piecewise-linear budget constraints (see, e.g., Moﬃtt, 1986).
27
in the constant term K and hence are not identiﬁed from irrigated areas alone, λ enters
the choice probabilities distinctly, both through the value of autarky VA (see Table 5) and
through wL , water availability in the lowest state.
Equation (16) also makes clear that a nondegenerate groundwater supply distribution
ψ (w) is critical for model identiﬁcation even with substantial variation in wki (which is to
say, in pipe width) across borewells. In particular, if ∀i πki = 1 for some k , then α drops out
1 α
of the fourth term of that equation (i.e., α log ( k πki wki ) = log wki ) and is thus identiﬁed
solely oﬀ of nonlinearites. Finally, note that had we only had data on choices, rather than
on both choices and areas irrigated, the likelihood would involve only the Pr(j |Θ, Zi ). While
predicted choice probabilities based on estimates of such a likelihood would obviously match
the empirical choice probabilities extremely closely, identiﬁcation of the full set of model
parameters would be tenuous (e.g., γ and κ ¯ L could not be distinguished).
5 Results
5.1 Parameter estimates
Table 6 reports parameter estimates along with asymptotic standard errors. The estimates
appear reasonable. In particular, the curvature parameter α is considerably less than one,
whereas a value close to one would have implied little role for groundwater uncertainty. Our
estimate of buyer bargaining-power η translates (cf., deﬁnition of δ in Table 5) into a 3.7%
eﬃciency loss due to holdup in the per-irrigation arrangement, which is an upper bound
on the overall contracting distortion. The incremental cost of cultivating leased land versus
own land, γ , is precisely estimated at less than 1%. So, evidently, the paucity of land-
lease activity in the data is driven by the high ﬁxed transactions cost, κL , both in absolute
terms and relative to κT . Finally, as expected, we estimate β > 0, which says that ﬁxed
transactions costs are higher in adjacencies with greater borewell density (i.e., those with
larger plot sizes). Thus, owners of borewells surrounded by plots with borewells of their own
have greater diﬃculty arranging water sales.
5.2 Within-sample ﬁt
Table 7, column (2), reports mean predictions for the estimation sample, which can be
compared to the corresponding means of the data in column (1). As seen in the ﬁrst row, on
average, the model predicts (log) total area irrigated by a borewell extremely well. Figure
28
Table 6: Parameter Estimates
Parameter Interpretation Estimate Std. Error
α production function curvature 0.218 0.003
η buyer’s bargaining weight 0.929 0.001
γ leasing ineﬃciency 0.008 0.0004
κL leasing transaction cost - intercept 0.105 0.009
κT selling transaction cost - intercept 0.012 0.0095
β transaction cost - slope 0.043 0.006
λ eﬀective borewell discharge 0.367 0.034
¯
c mean cultivation cost 0.607 0.013
σε standard deviation cultivation cost 0.264 0.006
Notes: The maximized log-likelihood is -5599.7 on a sample of 1646 observations.
Figure 6: Irrigated area and groundwater supply uncertainty
.5
99th
pctile
log(irrigated area/borewell plot area)
-.5 0
1st
pctile
-1.5 -1 -2
0 .2 .4 .6
CV end-of-season flow
95% CI Fit to data
Fit to model predictions
Notes: Nonparametric regressions of actual and predicted log irrigated area on coeﬃcient of variation of
end-of-season ﬂow.
29
Table 7: Model Predictions
Holdout Samples Anantapur→
Est. sample Anantapur E. Godavari E. Godavari
Data Model Data Model Data Model ∆π ∆π, ∆R
(1) (2) (3) (4) (5) (6) (7) (8)
log(I ) 1.67 1.68 1.06 1.01 2.32 2.60 1.47 2.56
P r (U ) 0.26 0.46 0.51 0.59 0.08 0.29 0.47 0.21
P r ( A) 0.23 0.15 0.47 0.16 0.26 0.10 0.15 0.10
P r(L) 0.06 0.02 0.01 0.00 0.03 0.12 0.01 0.12
P r (P ) 0.21 0.15 0.01 0.20 0.05 0.19 0.13 0.25
P r (C ) 0.24 0.22 0.00 0.05 0.59 0.30 0.24 0.32
Notes: All ﬁgures are sample means. Log irrigated area, log(I ), and choice
probabilities, P r(j ), are simulated by drawing multiple values of ε for each
borewell and then averaging optimal choices. Column (7) reports mean predic-
tions when each borewell in the E. Godavari sample is assigned well-ﬂow state
probabilites (πki ) drawn at random from the Anantapur sample. Column (8)
is the same as column (7) except that the E. Godavari sample is also assigned
pipe widths (Ri ) drawn at random from the Anantapur sample.
6 shows that the model also reproduces, more or less faithfully, the bivariate relationship
between area irrigated (relative to borewell plot area) and the coeﬃcient of variation of
groundwater supply.
Comparing the remainder of columns (1) and (2) of Table 7 shows that the model is quite
successful at predicting choices of transfer arrangements (L, C , and P ), but far less so when
it comes to unconstrained self-cultivation (U ) versus autarky (A). While the latter result
may not seem encouraging, it must be emphasized that our model is extremely parsimonious.
Moreover, as alluded to earlier, we are not only estimating choice probabilities. Indeed, it
is remarkable that our likelihood, which recruits parameters such as c ¯, λ, and α to ﬁt the
mean (and variance) of irrigated area without the beneﬁt of any free constant terms (or ﬁxed
eﬀects), ends up ﬁtting choices as well as it does. That said, our model is, of course, not an
exact description of reality.
30
5.3 Out-of-sample ﬁt
Also in Table 7, we report predictions based on estimation sample parameters for borewells
in the holdout districts of Anantapur (column 4) and East Godavari (column 6). In Anan-
tapur, the gap between mean irrigated area predicted by the model and that found in the
data is an astonishingly narrow 5%, whereas the corresponding deviation in E. Godavari is
28%. Similarly, the model does somewhat better in capturing choices in Anantapur than
in E. Godavari, especially with respect to unconstrained cultivation (U ). In both districts,
however, the model greatly over-predicts the prevalence of the per-irrigation arrangement
and, in E. Godavari, the prevalence of leasing. Thus, as was the case within sample, the
model seems to predict irrigated area better than it predicts choices out of sample.
Importantly, the model nicely captures the contraction of area irrigated per borewell
and the concomitant collapse of the seasonal contract as we move from E. Godavari to
Anantapur and both groundwater scarcity and supply uncertainty increase. To disentangle
which of these changes–i.e., that of the ﬁrst or second moment–is driving the result, we
perform two comparative statics exercises reported, respectively, in columns (7) and (8) of
Table 7. In the ﬁrst, for each borewell in Anantapur, we replace the vector of subjective
probabilities (π0i , ..., π4i ) with a corresponding vector drawn at random (with replacement)
from the E. Godavari sample. This change alone raises the predicted prevalence of seasonal
contracts by 19 percentage points (from 0.05 to 0.24) and predicted area irrigated by 46%
(= 1.47 − 1.01). In the second exercise, we draw pipe-widths in addition to the πki vector
from the E. Godavari sample. Endowing borewells in Anantapur with the ﬁrst moments of E.
Godavari increases predicted area irrigated by an additional 109% (= 2.56 − 1.47), while the
prevalence of seasonal contracts rises by just 8 percentage points. In sum, regional variation
in contract type, as seen through the lens of the model, is driven primarily by variation in
uncertainty, which is to say by the second moment of groundwater supply, whereas variation
in irrigated area is largely due to variation in the ﬁrst moment.
6 Implications
6.1 Welfare
The structural model allows us to compare welfare and area irrigated under diﬀerent coun-
terfactuals. We ﬁrst simulate baseline expected surplus E [Vj ∗ − κj ∗ ] for each borewell owner,
where j ∗ is the optimal water transfer arrangement chosen given a particular draw of εi .
31
Figure 7: Welfare and irrigated area counterfactuals
15
40 30
% change in area irrigated
10
% change in surplus
20
5
10
0
0
0 median 10 20 0 median 10 20
borewell plot area (dry-equivalent acres) borewell plot area (dry-equivalent acres)
Remove uncertainty + consolidate Consolidate
Rebate transaction cost
Notes: Nonparametric regressions of predicted log changes in welfare (left panel) and irrigated area (right
panel) on borewell plot area.
Next, to assess the respective magnitudes of the contracting distortion, the ﬁxed transaction
cost, and the cost of precautionary planting, we perform three counterfactual experiments
in sequence: (1) Rebate the transaction cost associated with the optimal choice κj ∗ back
to the borewell owner; (2) Consolidate fragmented landholdings, which leads to the Pareto-
eﬃcient allocation with each borewell owner earning (expected) ﬁrst-best surplus E [VU ]; (3)
Consolidate land as in (2) while eliminating groundwater supply uncertainty, holding ex-
pected borewell discharge constant.33 We also simulate irrigated area under scenarios (2)
and (3); irrigated area under (1) is identical to that of the baseline scenario because we are
not allowing choices to adjust.
Figure 7 plots the percentage changes (relative to baseline) in surplus (left panel) and
in area irrigated (right panel) of each counterfactual against borewell plot area. Beginning
with scenario (1), borewell owners with smaller plots, ceteris paribus, are more likely to sell
groundwater and hence incur proportionally greater transaction cost than those with large
plots. The diﬀerence between the dashed consolidation line and the transaction cost rebate
line indicates the size of the uncertainty-induced contracting distortion; it averages about
1−1/α
33 0
In this case, surplus is given by E [VU ] = αw c/(1 − α))
¯i (¯ Ee(1−1/α)εi (cf. Table 5).
32
1.5% of the total rabi surplus generated by the borewell, similar in magnitude to the average
ﬁxed transaction cost of 1.8%. For the same reason as before, the contracting distortion
is greater for borewell owners with smaller plots. Interestingly, comparing the right and
left panels of the ﬁgure, the impact of groundwater market frictions on area irrigated is, in
percentage terms, roughly triple its impact on welfare.
Finally, shutting down uncertainty altogether by means of counterfactual scenario (3)
results in modest gains in both welfare (in this case, more so for larger plots) and in area
irrigated.34 Thus, in reference to the reduced form regressions in Table 2, our model suggests
that groundwater market distortions are far more important than the precautionary planting
motive in explaining the variation in irrigated area due to groundwater supply uncertainty.
6.2 Drip irrigation and land fragmentation
Based on agricultural experiments in India, water use eﬃciency is 2-3 times greater under
drip than under ﬂood irrigation (Narayanamoorthy 2006). Taking the conservative estimate
of a doubling of irrigation eﬃciency, we use the model, through a doubling of the eﬀective
discharge parameter λ, to simulate the gross return to drip adoption35
∆V ∗ = E [Vj ∗ − κj ∗ |2λ] − E [Vj ∗ − κj ∗ |λ]. (17)
Based on the four-district estimation sample, the average borewell owner would realize a 53%
surplus gain by converting his plot to drip irrigation, with an interquartile range of 44-65%.
The interesting question, however, is how these gains are achieved, since the gross return
reﬂects the full range of endogenous responses to increased irrigation eﬃciency, including
more intensive cultivation of the borewell plot, increased water sales, and switches between
alternative transfer arrangements. To gauge the nature and extent of these responses, Table
8 reports the predicted transition probabilites pre- to post-drip adoption. Thus, we see
that of those starting in unconstrained self cultivation (U ), 62% remain unconstrained post-
drip, 13% move to autarky (A), meaning that they now fully cultivate their own plot, and
25% start selling water, virtually all under the seasonal contract (C ). The most dramatic
transition is, of course, from autarky, where virtually no one remains post-drip, to P and
C (mostly the latter). Overall, the model predicts that the proportion of borewell owners
34 0
Indeed, since log(VU /VU ) = log( 0
U / U ), the percentage gains are mathematically equivalent.
35
Equation (17) is a simpliﬁcation for purposes of exposition. In practice, through suitable adjustments
to λ, we only allow for an increase in irrigation eﬃciency on the borewell plot area, not on the area of other
plots that may be (or could be) irrigated by the borewell.
33
Table 8: Predicted Choice Transition Probabilites
Post-drip
Pre-drip U A L P C All
U 0.622 0.126 0.000 0.003 0.249 0.462
A 0.000 0.011 0.004 0.234 0.751 0.153
L 0.000 0.000 1.000 0.000 0.000 0.023
P 0.000 0.000 0.031 0.968 0.000 0.146
C 0.000 0.000 0.014 0.166 0.820 0.216
All 0.287 0.060 0.031 0.215 0.407 1.000
Notes: Computed using estimation sample.
selling water would rise from 36% to 62% upon adoption of drip irrigation.36
Given the evident importance of groundwater markets in farmers’ exploitation of drip
irrigation, we next explore how land fragmentation, the driving force behind these markets,
aﬀects the beneﬁts of drip adoption. To isolate the role of groundwater market frictions, we
use the analog of equation (17) to compute ∆VU , the gross return to drip under the ﬁrst-best
(unconstrained) counterfactual. The ratio ρ = ∆V ∗ /∆VU thus represents the surplus gain
from drip adoption given frictions relative to the surplus gain absent frictions. The mean
of ρ in the estimation sample is 0.94; so, the distortions uncovered in this paper reduce the
gross return to drip adoption by an average of 6% relative to ﬁrst-best.
Looking beyond averages, Figure 8 shows that ρ increases (toward unity) with borewell
plot size. In other words, consistent with what we have already seen in Figure 7, groundwater
market frictions are more salient on smaller plots. However, ρ decreases with average adja-
cency plot size. Recall that adjacencies with larger plots have a higher borewell density (cf.
Figure 3) and, in our model, have a higher ﬁxed cost of groundwater transactions (because
β > 0). As a result, farmers surrounded by larger plots (i.e., more borewells) would proﬁt
less, ceteris paribus, by switching to drip irrigation, and this is especially true when they
themselves have a small plot. Paradoxically, fragmentation of neighboring land is conducive
to drip adoption even as fragmentation of own land is detrimental to it.
36
One caveat is that, as the eﬀective supply of groundwater expands, the ﬁxed cost of arranging a sale
(diﬃculty of ﬁnding a buyer) may increase. Since this type of general equilibrium consideration lies outside
of our model, we should think of the results in Table 8 as an upper bound on extensiﬁcation via the market.
34
Figure 8: Land fragmentation and returns to drip irrigation
0.98
0.96
0.94
0.92
ρ
0.9 5
0.88
Quintiles of ā
4
0.86
3
0.84
1 2
2
3
4 1
5
Quintiles of a
0.88-0.9 0.9-0.92 0.92-0.94 0.94-0.96 0.96-0.98
Notes: Average values of ρ (gross return to drip adoption normalized by ﬁrst-best gross return) by quintiles
a).
of plot area (a) and average adjacency plot area (¯
7 Conclusion
We have developed a model of contracting under Hartian payoﬀ uncertainty, which, in the
spirit of transactions cost economics, features a tradeoﬀ between ex-post and ex-ante ineﬃ-
ciency. Since long-term contracts are more protective of relationship-speciﬁc investment but
less ﬂexible than spot contracts, they are preferred in low uncertainty environments.
Structural estimation of the model against the backdrop of south India’s groundwater
economy allows us to quantitatively evaluate the contracting distortion in a real-world set-
ting. We ﬁnd that contractual form is sensitive to the extent of payoﬀ uncertainty in the
direction implied by the theory. Indeed, cross-sectional variation in uncertainty accounts
for most of the predicted (out-of-sample) regional diﬀerence in prevalence of long-term con-
tracts. However, the welfare cost of constrained-eﬃcient contracting turns out to be rather
modest, even though the impact on irrigated area is substantial.
Our analysis also points to another signiﬁcant source of ineﬃciency in groundwater mar-
kets. Due to uncoordinated borewell drilling, those with water to sell invariably have too
few willing buyers. This coordination failure combined with the contracting distortion at-
tenuates the return to drip irrigation. Thus, land fragmentation and groundwater supply
uncertainty, together, may constrain adoption of water-saving technology in India.
35
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40
Appendix (not for publication)
A Proof of Proposition 2
We begin by proving that under any water-transfer arrangement economic surplus, Vj (r)
j = U, A, C, P , strictly diminishes with mean-preserving increases in uncertainty r. But
ﬁrst, we paraphrase a result from Diamond and Stiglitz (1974, p. 340, fn. 8). For any twice
continuously diﬀerentiable function h(w)
h(w)ψr dw = hww (w)T (w, r)dw, (A.1)
where ψ is a p.d.f., ψr is its partial derivative with respect to r, and T (w, r) is a nonnegative
function deﬁned in equation (5) of Diamond and Stiglitz (1974).
Lemma 1 If g is strictly concave, then Vj (r) < 0 for j = U, A, C, P .
Proof. (i ) VU : Diﬀerentiating deﬁnition 2 and using the envelope theorem yields
VU (r) = U f (w/ U )ψr dw <0 (A.2)
by (A.1) and the concavity of f .
(ii ) VA : Follows from (A.2) with U replaced by a.
(iii ) VC : As noted in fn. 19, VC = aE f ( w−
a
τC
) − c + C f ( τC
C
) − c . Diﬀerentiating with
respect to r and using the envelope theorem leads to an expression analogous to (A.2).
(iv ) VP : Given the discussion in fn. 19,
w − τP τP
VP = aE f ( )−c + P f( )−c
a P
(A.3)
w
= (a + P )E f( )−c ,
a+ P
where the second line follows from the ex-post eﬃciency condition f ( τP
P
) = f ( w−
a
τP
). Dif-
41
ferentiating with respect to r in this case yields
w ∂ P w
VP (r) = E g −c + (a + p) f ψr dw
a+ P ∂r a+ P
(A.4)
1 ∂ P w
=c −1 + (a + p) f ψr dw,
η ∂r a+ P
where the second line uses equation (7). In the case of the per-irrigation arrangement, a
precise analog to Proposition 1 applies. Thus, given that g is concave, ∂ P /∂r < 0. The ﬁrst
term of (A.4) must, therefore, be negative and, using (A.1) again, the second term must also
be negative.
Proof of proposition 2: (a ) Proposition 1 implies that for suﬃciently large r, say
rU , U = a and, hence, VU (rU ) = VA (rU ). Recall that under perfect certainty, VC (0) =
VU (0) > VA (0). Given τC (0) < wL (see fn. 20) and lemma 1, VC is continuously decreasing
in r until it equals VA at some r = rC . Now use equation (7) under perfect certainty to
deﬁne η = c/g (w/a). For η ≤ η , P = 0 ∀r and, consequently, VP (r, η ) = VA (r) ∀r and, in
particular, for r = 0. Deﬁne rP (η ) as the solution to
VP (rP (η ), η ) = VA (rP (η )). (A.5)
Thus, clearly, rP (η ) = 0. Recall, also, that VP (r, 1) = VU (r) ∀r, so rP (1) = rU . To prove
part (a ), it is suﬃcient to show that rP (η ) ∈ (rC , rU ) for some η . This is so because
VP (0, η ) < VC (0) for all η < 1 and, therefore, by continuity, the functions VC (r) and VP (r, η )
must cross at r∗ < rC if indeed rP (η ) > rC (see ﬁgure 1).
Thus, it is suﬃcient to show that rP (η ) > 0 so that as η is increased from η to 1 rP (η )
eventually exceeds rC . Diﬀerentiating equation (A.5) with respect to η , substituting from
equation (7), and rearranging gives
1 ∂ P
c −1 + h (w) ψr dw rP (η ) = 0 (A.6)
η ∂η
w
where h(w) = (a + P )f ( a+ P
) − af ( w
a
). Since ∂ P /∂η < 0, and given (A.1), we have that
sign(rP (η )) = −sign(hww (w)). Diﬀerentiating h(w) twice, we get
1 w a+ P w
hww (w) = f − f (A.7)
a+ P a+ P a a
42
A Taylor expansion around P = 0 gives f w
a+ P
≈f w
a
−w P
a2
f w
a
. Substituting into
equation (A.7) and rearranging yields
− P w w w
hww (w) ≈ f + f . (A.8)
a(a + P ) a a a
Since the term in square brackets is just −g (w/a), the concavity of g ensures that hww < 0
and hence that rP (η ) > 0.
(b ) Having just established that rP (η ) > rC for some η ∈ (η, 1), it must be true that
VP (r, η ) > VC (r) over the interval (r∗ (η ), rP (η )).
B Groundwater in Andhra Pradesh
Figure B.1: Water table fluctuations: 1998-2014
Notes: Average depth to water table (meters below ground-level) from all state observation wells and
rainfall (millimeters) by month (Source: Government of Andhra Pradesh, Groundwater Department, 2014,
http://apsgwd.gov.in/swfFiles/reports/state/monitoring.pdf).
43
C Tests of frictionless groundwater markets
Here we test the two hypotheses implied by frictionless groundwater markets: that borewells
are just as likely to be sunk on small plots as on large plots and that small plots are just as
likely to be cultivated in the dry season as are large plots.
Our survey covers nearly 9600 plots, each of which either has a borewell itself or is adjacent
to a plot that does and, thus, could in principle receive a transfer of groundwater. Panel (a)
of Figure C.2 shows that borewells are actually much less likely on small plots than on large
ones. One might think that a random allocation of borewells across space could generate such
a pattern mechanically; larger plots would be more likely to have borewells insofar as they
constitute the majority of farmland area. But this ignores the fact that well placement is
determined by individual decision-makers at the plot-level. If we suppose, not unreasonably,
that the probability of ﬁnding a viable groundwater source (i.e., an underground spring) is
uniform across space and that each plot-owner makes the same number of drilling attempts,
then the likelihood of observing a borewell under the null hypothesis should be equal across
plots of diﬀerent size.
To be sure, owners of small plots may also be less wealthy and thus unable to aﬀord
multiple drilling attempts, or any attempts at all for that matter (see, e.g., Fafchamps and
Pender, 1997), so we need to control for wealth. To do so, we next focus only on the subset
of plots whose owners possess at least one other plot; otherwise, plot area and total owned
area are perfectly correlated. We then partial out the eﬀect of wealth (as proxied by total
landholdings) using dummies for each of the deciles of total landownership. The result,
shown in panel (b) of Figure C.2, conﬁrms that borewells are more likely on larger plots.
Returning to the full sample of plots, Figure C.3 also indicates that small plots are
much more likely to be left fallow in the dry season than large plots. Taken together with
the pattern of borewell placement, this evidence points strongly to frictions in groundwater
markets.
44
Figure C.2: Presence of a borewell and area of plot
1
proportion of plots with borewell
.2 .4 0.6 .8
0 2 4 6 8 10
plot area (acres)
95% CI local cubic polynomial fit
cum. distribution function
(a) Unconditional (all plots)
1
proportion of plots with borewell
.4 .6 .2 .8
0 2 4 6 8 10
plot area (acres)
95% CI unconditional
conditional plot area/tot. area owned.
(b) Partialling out wealth (multi-plot farmers)
Notes: Nonparametric regression of borewell indicator on plot area. Panel
(a) uses sample of 9584 plots that either have a borewell or are adjacent
to a borewell plot. Panel (b) uses a subsample of 4544 plots whose owners
possess at least one other plot. The solid curve replicates that of panel
(a) on the smaller sample, whereas the short-dashed curve partials out 9
dummies for the deciles of total land area owned.
45
Figure C.3: Dry-season fallow and area of plot
.4
proportion of plots left entirely fallow in rabi
.1 .2 0 .3
0 2 4 6 8 10
plot area (acres)
95% CI local cubic polynomial fit
Notes: Nonparametric regression of dry-season fallow indicator on plot area using sample of 9584 plots that
either have a borewell or are adjacent to a borewell plot.
D Choice conﬁgurations
46
Table D1: Choice probability integration limits by conﬁguration
Conﬁguration (1) (2) (3) (4) (5) (6) (7)
CAU (−∞, ε
˜CA ] [˜
εCA , ε
˜AU ] εAU , ∞)
[˜
CLAU ˜1
(−∞, ε CL ] ε1
[˜CL , ε
˜LA ] [˜
εLA , ε
˜AU ] εAU , ∞)
[˜
CLCAU (−∞, ˜1
ε CL ] ε1
[˜CL , ˜2
ε CL ] ε2
[˜CL , ε
˜CA ] [˜ ˜AU ]
εCA , ε εAU , ∞)
[˜
CLCPAU (−∞, ˜1
ε CL ] ε1
[˜CL , ˜2
ε CL ] ε2
[˜CL , ˜1
ε CP ] ε1
[˜CP , ε
˜P A ] [˜ ˜AU ]
εP A , ε εAU , ∞)
[˜
CLCPCAU (−∞, ˜1
ε CL ] ε1
[˜CL , ˜2
ε CL ] ε2
[˜CL , ˜1
ε CP ] ε1
[˜CP , ˜2
ε CP ] ε2
[˜CP , ε
˜CA ] [˜
εCA , ε
˜AU ] εAU , ∞)
[˜
CLPAU (−∞, ˜1
ε CL ] ε1
[˜CL , ˜1
ε P L] ε1
[˜P L, ε
˜P A ] [˜ ˜AU ]
εP A , ε εAU , ∞)
[˜
CLPCAU (−∞, ˜1
ε CL ] ε1
[˜CL , ˜P L ]
ε εP L , ε
[˜ ˜2CP ] ε2
[˜CP , ε
˜CA ] [˜ ˜AU ]
εCA , ε εAU , ∞)
[˜
CLPLAU (−∞, ˜1
ε CL ] ε1
[˜CL , ˜1
ε P L] ε1
[˜P L, ˜2
εP L] ε2
[˜P L, ε
˜LA ] [˜ ˜AU ]
εLA , ε εAU , ∞)
[˜
CLPLCAU (−∞, ˜1
ε CL ] ε1
[˜CL , ˜1
ε P L] ε1
[˜P L, ˜2
εP L] ε2
[˜P L, ˜2
εCL ] ε2
[˜CL , ε
˜CA ] [˜
εCA , ε
˜AU ] εAU , ∞)
[˜
CPAU (−∞, ˜1
ε CP ] ε1
[˜CP , ε
˜P A ] [˜
εP A , ε
˜AU ] εAU , ∞)
[˜
CPCAU (−∞, ˜1
ε CP ] ε1
[˜CP , ˜2
ε CP ] ε2
[˜CP , ε
˜CA ] [˜
εCA , ε
˜AU ] εAU , ∞)
[˜
CPCLAU (−∞, ˜1
ε CP ] ε1
[˜CP , ˜2
ε CP ] ε2
[˜CP , ˜1
εCL ] ε1
[˜CL , ε
˜LA ] [˜ ˜AU ]
εLA , ε εAU , ∞)
[˜
CPLAU (−∞, ˜1
ε CP ] ε1
[˜CP , ˜P L ]
ε [˜
εP L , ε
˜LA ] [˜
εLA , ε
˜AU ] εAU , ∞)
[˜
CPLAU (−∞, ˜1
ε CP ] ε1
[˜CP , ˜2
ε P L] ε2
[˜P L, ε
˜LA ] [˜
εLA , ε
˜AU ] εAU , ∞)
[˜
CPLCAU (−∞, ˜1
ε CP ] ε1
[˜CP , ˜P L ]
ε εP L , ε
[˜ ˜2
CL ] ε2
[˜CL , ε
˜CA ] [˜
εCA , ε
˜AU ] εAU , ∞)
[˜
LAU (−∞, ε
˜LA ] [˜
εLA , ε
˜AU ] εAU , ∞)
[˜
LCAU ˜1
(−∞, ε CL ] ε1
[˜CL , ε
˜CA ] [˜
εCA , ε
˜AU ] εAU , ∞)
[˜
LCLAU (−∞, ˜1
ε CL ] ε1
[˜CL , ˜2
εCL ] ε2
[˜CL , ε
˜LA ] [˜ ˜AU ]
εLA , ε εAU , ∞)
[˜
LCLPAU (−∞, ˜1
ε CL ] ε1
[˜CL , ˜2
εCL ] ε2
[˜CL , ε
˜P L ] [˜
εP L , ε
˜P A ] [˜ ˜AU ]
εP A , ε εAU , ∞)
[˜
LCPAU (−∞, ˜1
ε CL ] ε1
[˜CL , ˜2
εCP ] ε2
[˜CP , ε
˜P A ] [˜ ˜AU ]
εP A , ε εAU , ∞)
[˜
LCPCAU (−∞, ˜1
ε CL ] ε1
[˜CL , ˜1
εCP ] ε1
[˜CP , ˜2
εCP ] ε2
[˜CP , ε
˜CA ] [˜
εCA , ε
˜AU ] εAU , ∞)
[˜
LCPLAU (−∞, ˜1
ε CL ] ε1
[˜CL , ˜2
εCP ] ε2
[˜CP , ˜2
εP L] ε2
[˜P L, ε
˜LA ] [˜ ˜AU ]
εLA , ε εAU , ∞)
[˜
LPAU (−∞, ˜1
ε P L] ε1
[˜P L, ε
˜P A ] [˜
εP A , ε
˜AU ] εAU , ∞)
[˜
LPCAU (−∞, ˜1
ε P L] ε1
[˜P L, ˜1
εCP ] ε1
[˜CP , ε
˜CA ] [˜ ˜AU ]
εCA , ε εAU , ∞)
[˜
LPCLAU (−∞, ˜1
ε P L] ε1
[˜P L, ˜1
εCP ] ε1
[˜CP , ˜2
εCL ] ε2
[˜CL , ε
˜LA ] [˜
εLA , ε
˜AU ] εAU , ∞)
[˜
LPCPAU (−∞, ε
˜P L ] [˜
εP L , ˜1
εCP ] ε1
[˜CP , ˜2
εCP ] ε2
[˜CP , ε
˜P A ] [˜
εP A , ε
˜AU ] εAU , ∞)
[˜
LPCPLAU (−∞, ˜1
ε P L] ε1
[˜P L, ˜1
εCP ] ε1
[˜CP , ˜2
εCP ] ε2
[˜CP , ˜2
ε P L] ε2
[˜P L, ε
˜LA ] [˜
εLA , ε
˜AU ] εAU , ∞)
[˜
LPLAU (−∞, ˜1
ε P L] ε1
[˜P L, ˜2
εP L] ε2
[˜P L, ε
˜LA ] [˜ ˜AU ]
εLA , ε εAU , ∞)
[˜
LPLCAU (−∞, ˜1
ε P L] ε1
[˜P L, ˜2
εP L] ε2
[˜P L, ˜1
ε CL ] ε1
[˜CL , ε
˜CA ] [˜
εCA , ε
˜AU ] εAU , ∞)
[˜
PAU (−∞, ε
˜P A ] [˜
εP A , ε
˜AU ] εAU , ∞)
[˜
PCAU (−∞, ˜1
εCP ] ε1
[˜CP , ε
˜CA ] [˜
εCA , ε
˜AU ] εAU , ∞)
[˜
PCLAU (−∞, ˜1
εCP ] ε1
[˜CP , ˜1
ε CL ] ε1
[˜CL , ε
˜LA ] [˜ ˜AU ]
εLA , ε εAU , ∞)
[˜
PCLAU (−∞, ˜1
εCL ] ε1
[˜CL , ˜1
εCP ] ε1
[˜CP , ˜2
ε CL ] ε2
[˜CL , ε
˜LA ] [˜
εLA , ε
˜AU ] εAU , ∞)
[˜
PCPAU (−∞, ˜1
εCP ] ε1
[˜CP , ˜2
ε CP ] ε2
[˜CP , ε
˜P A ] [˜ ˜AU ]
εP A , ε εAU , ∞)
[˜
PCPLAU (−∞, ˜1
εCP ] ε1
[˜CP , ˜2
ε CP ] ε2
[˜CP , ε
˜P L ] [˜
εP L , ε
˜LA ] [˜ ˜AU ]
εLA , ε εAU , ∞)
[˜
PLAU (−∞, ε
˜P L ] ˜LA ]
[˜
εP L , ε [˜
εLA , ε
˜AU ] εAU , ∞)
[˜
PLCAU (−∞, ε
˜P L ] [˜ ˜1
εP L , ε CL ] ε1
[˜CL , ε
˜CA ] [˜
εCA , ε
˜AU ] εAU , ∞)
[˜
PLCLAU (−∞, ε
˜P L ] εP L , ε
[˜ ˜1CL ] ε1
[˜CL , ˜2
ε CL ] ε2
[˜CL , ε
˜LA ] [˜
εLA , ε
˜AU ] εAU , ∞)
[˜
˜jj (i subscript suppressed for convenience). Numerical superscripts
Notes: See text for deﬁnitions of the ε
refer to cases of a double value function crossing over the range of εi , with 1 being the threshold for the
leftmost crossing and 2 being the threshold for the rightmost.
47