metadata: authors: 0: author: Gine,Xavier
1: author: Jacoby,Hanan G.
count: India,South Asia
docna: 0: docna: Markets, contracts, and uncertainty in a
            groundwater economy
docty: Policy Research Working Paper
subtopic: Hydrology,Water and Food Supply,Climate Change and Agriculture,International Trade and Trade Rules,Regulatory Regimes,Social Policy,Common Property Resource Development,Legal Products,Judicial System Reform,Legislation,Legal Reform,Microfinance
teratopic: Social Development,Law and Development,Governance,Agriculture,Rural Development,Public Sector Development,Finance and Financial Sector Development,International Economics and Trade,Water Resources
abstracts: cdata!: Groundwater is a vital yet threatened
            resource in much of South Asia. This paper develops a model
            of groundwater transactions under payoff uncertainty arising
            from unpredictable fluctuations in groundwater availability
            during the agricultural dry season. The model highlights the
            trade-off between the ex post inefficiency of long-term
            contracts and the ex ante inefficiency of spot contracts.
            The structural parameters are estimated using detailed
            micro-data on the area irrigated under each contract type
            combined with subjective probability distributions of
            borewell discharge elicited from a large sample of
            well-owners in southern India. The findings show that, while
            the contracting distortion leads to an average welfare loss
            of less than 2 percent and accounts for less than 50 percent
            of all transactions costs in groundwater markets, it has a
            sizeable impact on irrigated area, especially for small
            farmers.  Uncertainty coupled with land fragmentation also
            attenuates the benefits of the water-saving technologies now
            being heavily promoted in India.
geo_reg_key: 517212,517191
repnme: repnme: Markets, contracts, and uncertainty in a
            groundwater economy
display_title: Markets, contracts, and uncertainty
            in a groundwater economy
listing_relative_url: /research/2016/06/26437790/markets-contracts-uncertainty-groundwater-economy
topicv3: Gravity Irrigation m1326421 1006,Water m1326522 137,Personal Property Rights m1326571 1907,Leasing m1326729 368,Trade m1326924 698
publishtoextweb_dt: 2016-06-06T00:00:00Z
historic_topic: Social Development,Law and Development,Governance,Agriculture,Rural Development,Public Sector Development,Finance and Financial Sector Development,International Economics and Trade,Water Resources
no_of_pages: 50
author_keys: 0: author_key: 658884
1: author_key: 84800
model_extractions: 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 South Asia. This paper develops a model of groundwater transactions under payoff uncertainty arising from unpredictable fluctuations in groundwater availability during the agricultural dry season. The model highlights the trade-off between the ex post inefficiency of long-term contracts and the ex ante inefficiency of spot contracts. The structural parameters are estimated using detailed micro-data on the area irrigated under each contract type combined with subjective probability distributions of borewell discharge elicited from a large sample of well-owners in southern India. The findings show that, while the contracting distortion leads to an average welfare loss of less than 2 percent and accounts for less than 50 percent of all transactions costs in groundwater markets, it has a sizeable impact on irrigated area, especially for small farmers. Uncertainty coupled with land fragmentation also attenuates the benefits of the watersaving 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._ Produced by the Research Support Team
Markets, Contracts, and Uncertainty in a Groundwater Economy Xavier Gin ́e and Hanan G. Jacoby<sup>_∗_</sup>

**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.<sup>1</sup> 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.<sup>2</sup> To explore the implications of water-saving technology in a setting where there are contracting distortions, we build and structurally estimate a model of south India’s groundwater economy incorporating three salient features: First, given high irrigation conveyance losses, groundwater transactions tend to be highly localized, typically involving bilateralmonopolistic contracts between a well-owner and a water-buyer on adjacent land (see Jacoby 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.<sup>3</sup> 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 contracts 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

> 1The 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.

> 2Due 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).

> 3During 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.

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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 referencepoint 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).<sup>4</sup> 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.

> 4Fehr et al. (2011), Hoppe and Schmitz (2011), and Bartling and Schmidt (2014) corroborate the reference-point idea experimentally.

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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 estimation 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, specifically 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 longterm 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.<sup>5</sup> 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

> 5Lafontaine 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.

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(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.<sup>6</sup> 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 property 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 available 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.

> 6Cary 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.<sup>7</sup>

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 simplifies 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 _lU_ = arg max _l {Ef_ ( _w/l_ ) _− c}_ and define the marginal return as

**Definition 1** _g_ ( _ω_ ) = _f_ ( _ω_ ) _− ωf_<sup>_′_</sup> ( _ω_ ) _._

The necessary condition for optimal planting 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 ∂lU /∂r <_ 0 _._<sup>8</sup>

> 7Constant 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
<u>Table 1: Model Decisions</u>

|||Timin|g|
|---|---|---|---|
|Contract type|Moniker|Ex-ante|Ex-post|
|Long-term|Seasonal|_p_, _τ_, _l_<br>surplus divided|-|
|Spot|Per-irrigation|_l_<br>surplus divided|_p_, _τ_|

_Notes: p_ is the price per unit of irrigation, _τ_ is the total transfer of groundwater, and _l_ 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_ = _lU E_ [ _f_ ( _w/lU_ ) _− c_ ] _._

In case _lU > 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.<sup>9</sup> 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 renegotiation of the terms, or hold-up, will therefore lead to deadweight losses due to aggrievement by one or both parties.<sup>10</sup> 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 _lb_ the buyer’s cultivated area such that _lU_ = _a_ + _lb_ . It is easy to see that _f_<sup>_′_</sup> ( _ωb_ ) = _f_<sup>_′_</sup> ( _ωs_ ) = _p_ which implies that _ωb_ = _ωs_ . Thus, _VU_ = _E_ [ _a_ ( _f_ ( _ωs_ ) _− c_ ) + _pωblb_ ] = _E_ [ _a_ ( _f_ ( _ωs_ ) _− c_ ) + _f_<sup>_′_</sup> ( _ωb_ ) _ωblb_ ] = _E_ [ _a_ ( _f_ ( _ωs_ ) _− c_ ) + _lb_ ( _f_ ( _ωb_ ) _− c_ )] = _E_ [( _a_ + _lb_ )( _f_ ( _ωs_ ) _− c_ )] _,_ where the penultimate expression follows from _Eg_ ( _ωb_ ) = _c_ , the necessary condition for the buyer’s optimal area cultivated.

> 10More 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

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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.<sup>11</sup> Summarizing, the seasonal contract has two salient features: First, by serving as reference 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 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;<sup>12</sup> the second term is his total revenue from the sale. The participation constraint ( _PC_ ) 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 _PC_ 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.

> 11For 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).

> 12Without 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_ � _<u>wl−s τ</u>_ � = _c_ . However, equation

(1) _⇒ Eg_ � _lwU_ � = _c ⇒ τ_ = _w_ (1 _− ls/lU_ ) _,_ which is a contradiction because _τ_ cannot be state-contingent.

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Given a binding _PC_ , the necessary conditions for the optimal contract are as follows: the solution to which is the water transfer-area pair ( _τC, lC_ ) _._ Divergence of supply and demand for irrigation ex-post creates a distortion. Since (3) implies _Ef_<sup>_′_�</sup><sup>_<u>w−</u>_</sup> _a_<sup>_<u>τC</u>_</sup> � = _f_<sup>_′_�</sup> _<u>τlCC</u>_ �, it is _not_ true, in general, that _f_<sup>_′_</sup> (<sup>_<u>w−</u>_</sup> _a_<sup>_<u>τC</u>_</sup> ) = _f_<sup>_′_</sup> (<sup>_<u>τ</u>_</sup> _lC_<sup>_<u>C</u>_)</sup><sup>_∀w_,whichwouldobtainif</sup><sup>_τ_werestate-</sup> 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_ (<sup>_<u>τ</u>_</sup> _lC_<sup>_<u>C</u>_)=</sup><sup>_g_(</sup> _lCw_ + _a_<sup>)=</sup><sup>_c_=</sup><sup>_g_(</sup> _l_<sup>_<u>w</u>_</sup> _U_<sup>)</sup> which implies that _lC_ = _lU − 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 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 perirrigation 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.<sup>13</sup> 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>u</u>_ = _lf_ ( _τ/l_ ) and _<u>v</u>_ = _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_<sup>_∗_</sup> = arg max( _u−_ _<u>u</u>_ )<sup>_η_</sup> ( _v −v_ <u>)</u><sup>1</sup><sup>_−η_</sup> , where _η_ is the buyer’s bargaining weight.<sup>14</sup> Therefore, _p_<sup>_∗_</sup> solves where the last line takes the limit of the second line as ∆ _→_ 0 _._ Thus, we obtain the standard surplus-splitting rule<sup>15</sup> Furthermore, once _f_<sup>_′_</sup> (<sup>_<u>τ</u>_</sup> _l_<sup>)</sup><sup>_−p∗_(</sup><sup>_τ_) =</sup><sup>_η_</sup> � _f_<sup>_′_</sup> (<sup>_<u>τ</u>_</sup> _l_<sup>)</sup><sup>_−f ′_(</sup><sup>_<u>w−</u>_</sup> _a_<sup>_<u>τ</u>_)</sup> <u>�</u> = 0, the buyer’s demand for irrigation is sated. Thus, the total transfer _τ_ must satisfy _f_<sup>_′_</sup> (<sup>_<u>τ</u>_</sup> _l_<sup>) =</sup><sup>_f ′_(</sup><sup>_<u>w−</u>_</sup> _a_<sup>_<u>τ</u>_)</sup><sup>_,_whichistheconditionfor</sup> 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 ex13We also abstract from any reallocation of property rights between the parties at this stage `a la Grossman 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.

> 15To 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_<sup>_∗_</sup> and the total transfer _τ_ . In particular, 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<sup>_p∗_(</sup><sup>_t_)</sup><sup>_dt_=(1</sup><sup>_−η_)</sup><sup>_lf_(</sup><sup>_<u>τ</u>_</sup> _l_<sup>) +</sup><sup>_ηa_</sup> � _f_ (<sup>_<u>w−</u>_</sup> _a_<sup>_<u>τ</u>_)</sup><sup>_−f_(</sup><sup>_<u>w</u>_</sup> _a_<sup>)</sup> � _,_ 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 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.<sup>16</sup> 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, _lP < lU − a_ ) in anticipation of ex-post rent appropriation.

# **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_<sup>_′_�</sup><sup>_<u>w−</u>_</sup> _a_<sup>_<u>τ</u>_</sup> �. Perhaps the complexity of this pricing scheme, the lack of observability (e.g., well flow may be manipulable by the seller),

> 16As before, the borewell owner fully cultivates his own plot before selling any groundwater (i.e., _ls_ = _a_ ). _Proof:_ Suppose not, then _Eg_ � _<u>w−lsτP</u>_ � = _c_ is necessary. However, equation (7) and _f_<sup>_′_</sup> (<sup>_<u>τ</u>_</sup> _lP_<sup>_<u>P</u>_)=</sup><sup>_f ′_(</sup><sup>_<u>w−</u>_</sup> _ls_<sup>_<u>τP</u>_</sup> ) _⇒ Eg_ � _<u>τlPP</u>_ � = _Eg_ � _<u>w−lsτP</u>_ � = _c/η,_ which is a contradiction.

10
or lack of third-party state verification, explains why we do not observe it in practice.<sup>17</sup> 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_ (<sup>_w−τC−_</sup> _a_<sup>_τP_</sup><sup><u>(</u></sup><sup>_τC_</sup><sup><u>)</u></sup> ) _− 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.<sup>18</sup>

# **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

> 17A 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.

> 18In 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_ ,<sup>19</sup> 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,_<sup>20</sup> _then (a) for some η, ∃ a unique r_<sup>_∗_</sup> ( _η_ ) _such that VC_ ( _r_<sup>_∗_</sup> ) = _VP_ ( _r_<sup>_∗_</sup> _, η_ ) _; (b)_ [ _VC_ ( _r_ ) _− VP_ ( _r, η_ )] ( _r_<sup>_∗_</sup> _− r_ ) _>_ 0 _._<sup>21</sup>

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 surplus 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, _lU_ = _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 arrangement. In this case, surplus _VP_ approaches _VU_ as _η_ approaches one. Moreover, at some low level of bargaining power _η_ = _<u>η</u>_ , _lP_ = 0 and _VP_ coincides with _VA_ . So, for some range of _η ∈_ <u>(</u> _<u>η,</u>_ 1) _, VP_ and _VC_ must cross. Given such a crossing (at _r_<sup>_∗_</sup> ), _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.

> 19For the seasonal contract, surplus is given by the private returns to the well-owner; since the _PC_ 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.

> 20In 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_ = _ε_ , the optimal transfer must be discretely less than _τC_ (0). In this case, _r_<sup>_∗_</sup> ( _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_<sup>_∗_</sup> .

> 21 _Proof:_ See Appendix A.

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Figure 1: Long-term versus spot contracts and uncertainty <!-- Start of picture text --> Surplus<br>VC<br>VP<br>VA<br>VU<br>C P A U<br>r ∗ rC rP rU Mean preserving spread<br><!-- End of picture text --> _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.<sup>22</sup> 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

> 22Our 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,<sup>23</sup> 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 overexploitation (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 Appendix 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 occurring 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 of the outlet pipe (i.e., full,<sup><u>3</u></sup> 4<sup>full,</sup><sup><u>1</u></sup> 2<sup>full,</sup><sup><u>1</u></sup> 4<sup>full,empty).24Next,usingthesameformat,</sup>

> 23Most 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.

> 24To 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.

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Figure 2: Distribution of groundwater supply uncertainty and pipe width <!-- Start of picture text --> Anantapur (holdout) Anantapur (holdout)<br>East Godavari (holdout) East Godavari (holdout)<br>Estimation sample Estimation sample<br>Total Total<br>0 .2 .4 .6 0 2 4 6 8<br>CV end-of-season flow Pipe width (inches)<br>30 3<br>20 2<br>10 1<br>0 0<br>30 3<br>20 2<br>10 1<br>Density 300 Density 03<br>20 2<br>10 1<br>0 0<br>30 3<br>20 2<br>10 1<br>0 0<br><!-- End of picture text --> _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 uncertainty (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
<u>Table 2: Precautionary Planting and Risk Aversion</u>

||(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 fow)||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_<sup>1</sup>||||-0.0146<br>(0.0231)||
|log(_CV_)_× RISK_<sup>1</sup>||||-0.00309||
|||||(0.0118)||
|_RISK_<sup>2</sup>|||||-0.144|
||||||(0.176)|
|log(_CV_)_× RISK_<sup>2</sup>|||||-0.0537<br>(0.0925)|
|_R_<sup>2</sup>|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_<sup>1</sup> is a self-assessed ranking of risk tolerance; _RISK_<sup>2</sup> 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 = <u>7</u><sup>IDtocomputetotalareairrigatedbyaborewell.25</sup> 3<sup>acre</sup> 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

> 25We carry this efficiency units assumption over to our structural estimation. To appreciate the simplification 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 transfer, 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 pipewidth (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, _RISKi_<sup>1</sup><sup>_,_isaself-assessedrankingofrisktolerance,with1</sup> 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), _RISKi_<sup>2isanindexofmarginalwillingness-</sup> to-pay for risk constructed from the characterstics of the preferred lottery. Ranging from 0 to 1, higher values of _RISKi_<sup>2indicategreaterriskaversion.Incolumns4and5ofTable</sup> 2, we report results of adding, respectively, _RISKi_<sup>1and</sup><sup>_RISK_</sup> _i_<sup>2,and,mostimportantly,</sup> 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 <!-- Start of picture text --> 0 1 2<br>log average plot size in adjacency<br>Borewell density<br>Groundwater transaction intensity<br>1<br>.8<br>.6<br>.4<br>.2<br><!-- End of picture text --> _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.

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# **3.5 Sub-samples and characteristics of adjacent plots**

Our subsequent analysis will rely on three subsamples of borewells: First, we have the _estimation_ 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/diferent 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.<sup>26</sup>

# **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.<sup>27</sup> 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. Groundwater 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

> 26As 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. 27In 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.

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Table 4: Descriptive Statistics by Subsample

|||||**Holdout **|**Samp**|**les**|
|---|---|---|---|---|---|---|
||**Estim**|**ation**|**Anan**|**tapur**|**E. G**|**odavari**|
|Choice _j_|_N_|_lj_|_N_|_lj_|_N_|_lj_|
|_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 <!-- Start of picture text --> 1st 99th<br>pctile pctile<br>0 .1 .2 .3 .4<br>CV end-of-season flow<br> 95% CI local fit<br>.8<br>.6<br>.4<br>Pr(P | P + C = 1)<br>.2<br>0<br><!-- End of picture text -->

_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 engaging 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).<sup>28</sup> 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 quantitative 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

# **4.3 Functional form**

We next assume that the intensive production function, _f_ , takes the form 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

> 28Based 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 (_lj_)<br>Surplus (_Vj_)|
|---|---|
|_U_ =unconstrained|�1_−α_<br>_c _<sup>_Ewα_�1</sup><sup>_/α_</sup><br>_αc_<br>1_−α_<sup>_lU_</sup>|
|_A_=autarky|_a_<br>_a_<sup>1</sup><sup>_−α_</sup>_Ew_<sup>_α _</sup>_−ca_|
|_L_=leasing|(1 +_γ_)<sup>_−_1</sup><sup>_/α_</sup>_lU −a_<br>(1 +_γ_)<sup>1</sup><sup>_−_1</sup><sup>_/α_</sup>_VU_ +_γca_|
|_C_ =seasonal contract|_lC_ solves _E_Ω<sup>_α−_1 </sup>= 1<br>_αc_<br>1_−α_ <sup>[</sup><sup>_E_Ω</sup><sup>_α_ +</sup><sup>_αlC/a −_(1</sup><sup>_−α_)]</sup>|
|_P_ =per-irrigation sale|_η_<sup>1</sup><sup>_/α_</sup>_lU −a_<br>_δVU_|
|_Notes:_ Ω= <sup>1</sup><br>_a_<br>�<br>_w_<br>�1_−α_<br>_c_<br>�1_/_|_α −lC_<br>�<br>, where_lC ≤_<br>�1_−α_<br>_c_<br>�1_/α wL_, and_δ_ = 1_−η_(1_−α_)<br>_α_<br>_η_<sup>1</sup><sup>_/α−_1</sup>_._|

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 _lU_ 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 Note that the parameter _λ_ in equation (10) and _ζ_ in equation (9) always enter the model together in the form _ζλ_<sup>_α_</sup> , which is why we normalize _ζ_ to one. Since expected groundwater supply is _w_ ̄ _i ≡ Ewi_ = _λRi_<sup>2</sup> � _k_<sup>_πkik_,wethinkof</sup><sup>_λ_asreflectingfactorsthatshiftthefirst</sup> 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 conditional on the transfer arrangement, we introduce a cost disturbance _εi ∼ N_ (0 _, σε_<sup>2) such</sup> that _ci_ = _<u>ce</u>_<sup>_εi_</sup> . 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 _lji_ = _lj_ ( _εji_ ) yields choice-specific residuals _εji_ for _j_ = _U, L, C, P_ , where _lji_ 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).<sup>29</sup> To allow for a discontinuity in irrigated area as one exits autarky, we introduce a transactions cost _κ_ .

We also allow _κ_ to vary across adjacencies to account for the availability of neighboring 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, _<u>ai</u>_ , which is plausibly unrelated to groundwater markets _except_ through borewell density. Thus, we incorporate heterogeneity in irrigable

> 29At 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 _<u>ai</u>_ into _κ_ as follows: 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 _<u>κj</u>_ , we allow for a separate _<u>κL</u>_ , but restrict _<u>κC</u>_ = _<u>κP</u> ≡_ _<u>κT</u>_ . Whereas we expect that _<u>κL</u> >_ _<u>κT</u>_ , 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, _lji_ , conditional on these choices. Choice probabilities are determined by the set of thresholds, ̃ _εjj′i_ , that solve the following crossing conditions for arrangement-specific value functions: The solution to the first of these equations yields _ε_ ̃ _AUi_ , 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 = ( _α, η, γ,_ _<u>κL, κT</u> , β, λ,_ _<u>c, σε</u>_ ) and data _Zi_ = ( _πki, Ri_<sup>2</sup><sup>_, ai,_</sup> _<u>ai</u>_ ) _,_ we have

25
where _φ_ is the standard normal pdf, and _R_<sup>_ε_</sup> _ji_<sup>istherelevantregionofintegrationforchoice</sup> _j_ . The simplest choice probability is for unconstrained cultivation, which is Pr( _U |_ Θ _, Zi_ ) = ̃ 1 _−_ Φ( ̃ _εAUi/σε_ ), where Φ is the standard normal cdf and the dependence of _εAUi_ on parameters and data is implicit. For autarky,

̃ ̃ where _εAi_ = max _{εLAi,_ ̃ _εCAi,_ ̃ _εPAi}_ represents the highest _εi_ that would induce any kind of 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 _R_<sup>_ε_</sup> _ji_<sup>hastwodisjointsegments.AppendixTableD1enumeratesall38</sup> 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 where _Hj_ = _|∂εji/∂lji|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_ ( _lji_ ) 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, _lUi < ai_ ,<sup>30</sup> 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 ̃ ̃ _lP >_ 0, not only must we have _εi < εAi,_ but also _εi < εCPi_ . To account for this additional right truncation, Pr( _P |_ Θ _, Zi_ ) multiplies the truncated density _HP_ ( _lPi_ ) _/_ Φ( ̃ _εAi_ ) in the third likelihood term.<sup>31</sup> 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.<sup>32</sup>

> 30This is, of course, why Pr( _U |_ Θ _, Zi_ ) drops out of the likelihood.

> 31The 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.

> 32With only a _single_ transaction type, say _P_ , the third term would collapse to _HP_ ( _lP i_ ) and the likelihood

26
Figure 5: Surplus and area irrigated under two possible model configurations <!-- Start of picture text --> VU VU<br>VA VA<br>VP −κT VC −κT VC −κT VP −κT<br>e ε e εPC ∗ e εCA ∗ e εUA ∗ Cultivation cost e εPC ∗ e ε e εPA ∗ e εUA ∗ Cultivation cost<br>lC + a<br>lP + a<br>lC + a lP + a<br>a a<br>lU lU<br>P C A U Cultivation cost C P A U Cultivation cost<br>Surplus<br>Surplus<br>irrigated<br>Area<br>Area irrigated<br><!-- End of picture text --> <!-- Start of picture text --> (a) PCAU Configuration (b) CPAU Configuration<br><!-- End of picture text --> _Notes:_ Solid line segments in bottom graphs show irrigated area schedule relevant for the optimal contractual arrangement given _ε_ .

# **4.8**

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 _<u>κj</u>_ 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 where _Iji_ = _lji_ + _ai_ for _j_ = _L, P_ and _Iji_ = _lji_ for _j_ = _U_ and the constant term _K_ = log(1 _− α_ ) _−_ log _<u>c</u>_ + _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 _<u>c</u>_ 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 of the fourth term of that equation (i.e., _α_<sup><u>1</u>log (�</sup> _k_<sup>_πkiw_</sup> _ki_<sup>_α_)=log</sup><sup>_wki_)andisthusidentified</sup> 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 landlease 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
<u>Table 6: Parameter Estimates</u>

|Parameter|Interpretation|Estimate|Std. Error|
|---|---|---|---|
|_α_|production function curvature|0.218|0.003|
|_η_|buyer’s bargaining weight|0.929|0.001|
|_γ_|leasing inefciency|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_|efective 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 <!-- Start of picture text --> 99th<br>pctile<br>1st<br>pctile<br>0 .2 .4 .6<br>CV end-of-season flow<br>95% CI Fit to data<br> Fit to model predictions<br>.5<br>0<br>-.5<br>-1<br>-1.5<br>log(irrigated area/borewell plot area)<br>-2<br><!-- End of picture text --> _Notes:_ Nonparametric regressions of actual and predicted log irrigated area on coefficient of variation of end-of-season 29
<u>Table 7: Model Predictions</u>

|||||**Holdout **|**Sampl**|**es**|**Ana**|**ntapur**_→_|
|---|---|---|---|---|---|---|---|---|
||**Est. **|**sample**|**Anan**|**tapur**|**E. Go**|**davari**|**E. G**|**odavari**|
||Data<br>(1)|Model<br>(2)|Data<br>(3)|Model<br>(4)|Data<br>(5)|Model<br>(6)|∆_π_<br>(7)|∆_π,_∆_R_<br>(8)|
|log(_I_)|1.67|1.68|1.06|1.01|2.32|2.60|1.47|2.56|
|_Pr_(_U_)|0.26|0.46|0.51|0.59|0.08|0.29|0.47|0.21|
|_Pr_(_A_)|0.23|0.15|0.47|0.16|0.26|0.10|0.15|0.10|
|_Pr_(_L_)|0.06|0.02|0.01|0.00|0.03|0.12|0.01|0.12|
|_Pr_(_P_)|0.21|0.15|0.01|0.20|0.05|0.19|0.13|0.25|
|_Pr_(_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, _Pr_ ( _j_ ), are simulated by drawing multiple values of _ε_ for each borewell and then averaging optimal choices. Column (7) reports mean predictions 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 Anantapur, 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 ( _π_ 0 _i, ..., π_ 4 _i_ ) 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 counterfactuals. We first simulate baseline expected surplus _E_ [ _Vj∗ − κj∗_ ] for each borewell owner, where _j_<sup>_∗_</sup> is the optimal water transfer arrangement chosen given a particular draw of _εi_ .

31
<!-- Start of picture text --> Figure 7: Welfare and irrigated area counterfactuals<br>0 median 10 20 0 median 10 20<br>borewell plot area (dry-equivalent acres) borewell plot area (dry-equivalent acres)<br>Remove uncertainty + consolidate Consolidate<br>Rebate transaction cost<br>15 40<br>30<br>10<br>20<br>% change in surplus 5<br>% change in area irrigated<br>10<br>0 0<br><!-- End of picture text --> _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 Paretoefficient allocation with each borewell owner earning (expected) first-best surplus _E_ [ _VU_ ]; (3) Consolidate land as in (2) while eliminating groundwater supply uncertainty, holding expected borewell discharge constant.<sup>33</sup> 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 33In this case, surplus is given by _E_ [ _VU_ 0<sup>] =</sup><sup>_α_ ̄</sup><sup>_wi_( ̄</sup><sup>_c/_(1</sup><sup>_−α_))1</sup><sup>_−_1</sup><sup>_/α Ee_(1</sup><sup>_−_1</sup><sup>_/α_)</sup><sup>_εi_(cf.Table5).</sup> 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.<sup>34</sup> 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 adoption<sup>35</sup> 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 postdrip, 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

> 34Indeed, since log( _VU_ 0<sup>_/VU_) = log(</sup><sup>_l_</sup> _U_<sup>0</sup><sup>_/lU_),thepercentagegainsaremathematicallyequivalent.</sup>

> 35Equation (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
<u>Table 8: Predicted Choice Transition Probabilites</u>

|||**P**|**ost-dri**|**p**|||
|---|---|---|---|---|---|---|
|**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.<sup>36</sup> 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_<sup>_∗_</sup> _/_ ∆ _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 _adjacency_ 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.

> 36One 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.

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