Spillover Effects in Irrigated Agriculture from the Groundwater Commons

Abstract

This study examines irrigation spillover effects within the groundwater commons of the San Luis Valley in Colorado. We investigate the common pool competition predicted by a theoretical model of crop production through water-use intensity, acreage size choices, and production intensity among irrigators. By specifying Spatial Probit and regular Spatial Durbin Models, we empirically measure not only the effects of these choices on neighbors, but also the effect of other factors that affect water use and cultivation choices at neighboring farming units. For all three response variables, the results show that irrigators consider neighbors’ responses, with the strength of spatial dependency being highest for production intensity. Additionally, there are significant spillover effects from changes in key covariates, demonstrating the inadequacy of estimating direct effects only. For example, a one-foot increase in depth-to-water has both direct and indirect positive effects on water-use intensity, but the indirect effect constitutes over 81% of the total effect.

Appendices

Appendix 1 Additional Derivations

1.1 Individual Problem

The first order conditions from irrigator j’s Nash equilibrium problem are as follows:

$$\begin{aligned} \begin{aligned} \frac{\partial {\varvec{\Uppi }^{j}}}{\partial w^j_k}&= \sum _{\ell =0}^K \frac{\partial \varvec{\pi }^{j*}_\ell \big (p_\ell , {\textbf {r}}, B^j(s^j(w^j, w^i),b), n^{j*}_\ell (\cdot ); {\textbf {x}}\big )}{\partial w^j_k} \\&= \sum _{\ell =0}^{K} \left[ \left( \frac{\partial \varvec{\pi }^{j*}_\ell (\cdot )}{\partial B^j(\cdot )} + {\frac{\partial \varvec{\pi }^{j*}_\ell (\cdot )}{\partial n^{j*}_\ell (\cdot )} \cdot } \frac{\partial n^{j*}_\ell (\cdot )}{\partial B^j(\cdot ) } \right) \left( \frac{\partial B^j(\cdot )}{\partial s^j(\cdot )} \cdot \frac{\partial s^j(\cdot )}{\partial w^j_k} \right) \right] \le 0 \\ \end{aligned} \end{aligned}$$

(9)

with equality if water withdrawal is positive, i.e. $w_k^j > 0$.

Using the Implicit Function Theorem, we see:

$$\begin{aligned} \frac{d w^i_k}{d w^j_m} = - \frac{ \displaystyle \frac{\partial }{\partial w^j_m}\left[ \frac{\partial {\varvec{\Uppi }^{i}}}{\partial w^i_k}\right] }{\displaystyle \frac{\partial }{\partial w^i_k}\left[ \frac{\partial {\varvec{\Uppi }^{i}}}{\partial w^i_k}\right] } = - \frac{\displaystyle \frac{\partial ^2 {\varvec{\Uppi }^{i}}}{\partial w^i_k \partial w^j_m }}{\displaystyle \frac{\partial ^2 {\varvec{\Uppi }^{i}}}{\partial {w^i_k}^2}}. \end{aligned}$$

(10)

The strategic effect of j’s total withdrawal will be made up of these individual effects, which are the cross-partial and the second order condition of own withdrawal on profits. Because of the external negative sign, the strategic effect will be positive if either numerator or denominator (but not both) is positive, while it would be negative if numerator and denominator match in sign. The effect will be small in absolute terms if the own second order condition is larger in magnitude than that due to the opponent, while the effect will be large if the other-directed second order condition is larger. The full second order conditions, with respect to $w^j_m$ and $w^i_k$ are as follows:

$$\begin{aligned}& \frac{\partial }{\partial w^j_m} \left[ \frac{\partial {\varvec{\Uppi }^{i}}}{\partial w^i_k}\right]= \sum _{\ell =0}^K \Bigg \{ \left( \frac{\partial ^2 \varvec{\pi }^{i*}_\ell (\cdot )}{ \partial B^i(\cdot )^2 } + \frac{\partial ^2 \varvec{\pi }^{i*}_\ell (\cdot )}{\partial n^{i*}_\ell (\cdot )^2} \cdot \frac{\partial n^{i*}_\ell (\cdot )}{\partial B^i(\cdot )} + \frac{\partial \varvec{\pi }^{i*}_\ell (\cdot )}{\partial n^{i*}_\ell (\cdot )}\cdot \frac{\partial ^2 n^{i*}_\ell (\cdot )}{\partial B^i(\cdot )^2 } \right) \nonumber \\ &\quad\quad\quad\quad\quad\quad\; \cdot \left( \frac{\partial B^i(\cdot )}{\partial s^i(\cdot )} \right) ^2 \left( \frac{\partial s^i(\cdot )}{\partial w^j_m} \cdot \frac{\partial s^i(\cdot )}{\partial w^i_k} \right) + \left( \frac{\partial \varvec{\pi }^{i*}_\ell (\cdot )}{\partial B^i(\cdot )} + \frac{\partial \varvec{\pi }^{i*}_\ell (\cdot )}{\partial n^{i*}_\ell (\cdot )} \cdot \frac{\partial n^{i*}_\ell (\cdot )}{\partial B^i(\cdot ) } \right) \\ &\quad\quad\quad\quad\quad\quad\; {\cdot} \left. {\left( {\frac{{\partial ^{2} B^{i} ( \cdot )}}{{\partial s^{i} ( \cdot )^{2} }} \cdot \frac{{\partial s^{i} ( \cdot )}}{{\partial w_{m}^{j} }} \cdot \frac{{\partial s^{i} ( \cdot )}}{{\partial w_{k}^{i} }} + \frac{{\partial B^{i} ( \cdot )}}{{\partial s^{i} ( \cdot )}} \cdot \frac{{\partial ^{2} s^{i} ( \cdot )}}{{\partial w_{m}^{j} \partial w_{k}^{i} }}} \right)} \right\} \end{aligned}$$

(11)

$$\begin{aligned}\frac{\partial }{\partial w^i_k} \left[ \frac{\partial {\varvec{\Uppi }^{i}}}{\partial w^i_k}\right]&= \sum _{\ell =0}^K \Bigg \{ \left( \frac{\partial ^2 \varvec{\pi }^{i*}_\ell (\cdot )}{ \partial B^i(\cdot )^2 } + \frac{\partial ^2 \varvec{\pi }^{i*}_\ell (\cdot )}{\partial n^{i*}_\ell (\cdot )^2} \cdot \frac{\partial n^{i*}_\ell (\cdot )}{\partial B^i(\cdot )} + \frac{\partial \varvec{\pi }^{i*}_\ell (\cdot )}{\partial n^{i*}_\ell (\cdot )}\cdot \frac{\partial ^2 n^{i*}_\ell (\cdot )}{\partial B^i(\cdot )^2 } \right) \nonumber \\& \quad\;\cdot \left( \frac{\partial B^i(\cdot )}{\partial s^i(\cdot )} \cdot \frac{\partial s^i(\cdot )}{\partial w^i_k} \right) ^2 + \left( \frac{\partial \varvec{\pi }^{i*}_\ell (\cdot )}{\partial B^i(\cdot )} + \frac{\partial \varvec{\pi }^{i*}_\ell (\cdot )}{\partial n^{i*}_\ell (\cdot )} \cdot \frac{\partial n^{i*}_\ell (\cdot )}{\partial B^i(\cdot ) } \right) \nonumber \\&\quad\;\cdot \left( \frac{\partial ^2 B^i(\cdot )}{\partial s^i(\cdot )^2} \left( \frac{\partial s^i(\cdot )}{\partial w^i_k} \right) ^2 + \frac{\partial B^i(\cdot )}{\partial s^i(\cdot )} \cdot \frac{\partial ^2 s^i(\cdot )}{\partial {w^i_k}^2} \right) \Bigg \} \end{aligned}$$

(12)

Using Eqs. (11) and (12), we can write $-\frac{d w^i_k}{d w^j_m}$ as:

$$\frac{{ {\sum\limits_{{l = 0}}^{K} \begin{gathered} \left\{ {\left( {\frac{{\partial ^{2} \pi _{\ell }^{{i*}} ( \cdot )}}{{\partial B^{i} ( \cdot )^{2} }} + \frac{{\partial ^{2} \pi _{\ell }^{{i*}} ( \cdot )}}{{\partial n_{\ell }^{{i*}} ( \cdot )^{2} }} \cdot \frac{{\partial n_{\ell }^{{i*}} ( \cdot )}}{{\partial B^{i} ( \cdot )}} + \frac{{\partial \pi _{\ell }^{{i*}} ( \cdot )}}{{\partial n_{\ell }^{{i*}} ( \cdot )}} \cdot \frac{{\partial ^{2} n_{\ell }^{{i*}} ( \cdot )}}{{\partial B^{i} ( \cdot )^{2} }}} \right)\left( {\frac{{\partial B^{i} ( \cdot )}}{{\partial s^{i} ( \cdot )}}} \right)^{2} \left( {\frac{{\partial s^{i} ( \cdot )}}{{\partial w_{m}^{j} }} \cdot \frac{{\partial s^{i} ( \cdot )}}{{\partial w_{k}^{i} }}} \right)} \right. \hfill \\ \left. {\quad \quad + \left( {\frac{{\partial \pi _{\ell }^{{i*}} ( \cdot )}}{{\partial B^{i} ( \cdot )}} + \frac{{\partial \pi _{\ell }^{{i*}} ( \cdot )}}{{\partial n_{\ell }^{{i*}} ( \cdot )}} \cdot \frac{{\partial n_{\ell }^{{i*}} ( \cdot )}}{{\partial B^{i} ( \cdot )}}} \right)\left( {\frac{{\partial ^{2} B^{i} ( \cdot )}}{{\partial s^{i} ( \cdot )^{2} }} \cdot \frac{{\partial s^{i} ( \cdot )}}{{\partial w_{m}^{j} }} \cdot \frac{{\partial s^{i} ( \cdot )}}{{\partial w_{k}^{i} }} + \frac{{\partial B^{i} ( \cdot )}}{{\partial s^{i} ( \cdot )}} \cdot \frac{{\partial ^{2} s^{i} ( \cdot )}}{{\partial w_{m}^{j} \partial w_{k}^{i} }}} \right)} \right\} \hfill \\ \end{gathered} } }}{{ {\sum\limits_{{l = 0}}^{K} \begin{gathered} \left\{ {\left( {\frac{{\partial ^{2} \pi _{\ell }^{{i*}} ( \cdot )}}{{\partial B^{i} ( \cdot )^{2} }} + \frac{{\partial ^{2} \pi _{\ell }^{{i*}} ( \cdot )}}{{\partial n_{\ell }^{{i*}} ( \cdot )^{2} }} \cdot \frac{{\partial n_{\ell }^{{i*}} ( \cdot )}}{{\partial B^{i} ( \cdot )}} + \frac{{\partial \pi _{\ell }^{{i*}} ( \cdot )}}{{\partial n_{\ell }^{{i*}} ( \cdot )}} \cdot \frac{{\partial ^{2} n_{\ell }^{{i*}} ( \cdot )}}{{\partial B^{i} ( \cdot )^{2} }}} \right)\left( {\frac{{\partial B^{i} ( \cdot )}}{{\partial s^{i} ( \cdot )}} \cdot \frac{{\partial s^{i} ( \cdot )}}{{\partial w_{k}^{i} }}} \right)^{2} } \right. \hfill \\ \left. {\quad \quad \left( {\frac{{\partial \pi _{\ell }^{{i*}} ( \cdot )}}{{\partial B^{i} ( \cdot )}} + \frac{{\partial \pi _{\ell }^{{i*}} ( \cdot )}}{{\partial n_{\ell }^{{i*}} ( \cdot )}} \cdot \frac{{\partial n_{\ell }^{{i*}} ( \cdot )}}{{\partial B^{i} ( \cdot )}}} \right)\left( {\frac{{\partial ^{2} B^{i} ( \cdot )}}{{\partial s^{i} ( \cdot )^{2} }}\left( {\frac{{\partial s^{i} ( \cdot )}}{{\partial w_{k}^{i} }}} \right)^{2} + \frac{{\partial B^{i} ( \cdot )}}{{\partial s^{i} ( \cdot )}} \cdot \frac{{\partial ^{2} s^{i} ( \cdot )}}{{\partial w_{k}^{{i2}} }}} \right)} \right\} \hfill \\ \end{gathered} } }}$$

(13)

With regard to the individual first derivative terms, we assumed for all crops k that an increase in the total marginal cost of water decreases profits, $\frac{\partial \varvec{\pi }^i_k(\cdot )}{\partial B^i(\cdot )} < 0$, while profits are non-decreasing in land allocation, $\frac{\partial \varvec{\pi }^i_k(\cdot )}{\partial n^{i*}_k(\cdot )} \ge 0$. Because of lift cost, increasing depth-to-water increases total marginal cost, $\frac{\partial B^i(\cdot )}{\partial s^i(\cdot )} > 0$. We assume that an agent’s own water use weakly increases their scarcity measure, $\frac{\partial s^i(\cdot )}{\partial w^i_k(\cdot )}\ge 0$; we also assume that another agent’s water use weakly increases¹⁶ agent i’s scarcity, $\frac{\partial s^i(\cdot )}{\partial w^j_m} \ge 0$. Less obvious, however, is how total marginal cost of water affects the land dedicated to various crops, $\frac{\partial n^{i*}_k (\cdot )}{\partial B^i(\cdot )}$: the farmer may decrease land for water-intensive crops and increase the allocation for fallowing or other crops, or there may be no reaction at all. In a particular growing season, it is unlikely the farmer will be able to purchase land quickly enough to grow additional crops that season. If we also assume that land is sold outside of the growing season (listed after a previous season), then we can see that in a particular game, total land is fixed, $\sum _{k=0}^K n^i_k = N^i$. Thus, when examining the derivatives, we know that their sum must add up to a constant, which is zero in the case of the one-stage game¹⁷ with no land sale: $\sum _{k=0}^K \frac{\partial n^i_k}{\partial B^i} = 0$.

With regard to the second derivative terms, we assumed convex profits, meaning $\frac{\partial ^2 \pi ^{i*}_k(\cdot )}{\partial B^i(\cdot )^2} \ge 0$. Resource cost functions are often convex, i.e. accelerating in cost (Schworm 1983; Kotani et al. 2008), and so we assume $\frac{\partial ^2 B^i(\cdot )}{\partial s^i(\cdot )^2} \ge 0$. Unfortunately, a priori, we cannot say much about the remaining second derivatives. There are two derivatives about the second order effect of water withdrawal—both i’s and j’s—on i’s depth-to-water. While we are comfortable assuming the first derivative is weakly positive (taking out water increases scarcity), we are unsure whether this effect is increasing, constant, or decreasing. The own second order effect, $\frac{\partial ^2\,s^i(\cdot )}{\partial w^i_k(\cdot )^2}$, reflects the curvature with regard to water withdrawal for only the farmer in question, while the cross-partial, $\frac{\partial ^2\,s^i(\cdot )}{\partial w^j_m \partial w^i_k(\cdot )}$, reflects how another farmer’s withdrawals change the marginal cost of withdrawal for farmer i. Additionally, we will remain agnostic on the second derivative of land allocation for crop k with respect to water cost, $\frac{\partial ^2 n^{i*}_k (\cdot )}{\partial B^i(\cdot )^2}$.

Examining the numerator of Eq. (13), an individual term in the summation is:

$$\begin{aligned} \begin{aligned}&\left( \underbrace{\frac{\partial ^2 \varvec{\pi }^{i*}_\ell (\cdot )}{\partial B^i(\cdot )^2 }}_{\ge 0} + \frac{\partial ^2 \varvec{\pi }^{i*}_\ell (\cdot )}{\partial n^{i*}_\ell (\cdot )^2} \cdot \frac{\partial n^{i*}_\ell (\cdot )}{\partial B^i(\cdot )} + \underbrace{\frac{\partial \varvec{\pi }^{i*}_\ell (\cdot )}{\partial n^{i*}_\ell (\cdot )}}_{\ge 0}\cdot \frac{\partial ^2 n^{i*}_\ell (\cdot )}{\partial B^i(\cdot )^2 } \right) \underbrace{\left( \frac{\partial B^i(\cdot )}{\partial s^i(\cdot )} \right) ^2}_{>0} \underbrace{\left( \frac{\partial s^i(\cdot )}{\partial w^j_m} \cdot \frac{\partial s^i(\cdot )}{\partial w^i_k} \right) }_{>0} \\&\quad+ \left( \underbrace{\frac{\partial \varvec{\pi }^{i*}_\ell (\cdot )}{\partial B^i(\cdot )}}_{<0} + \underbrace{\frac{\partial \varvec{\pi }^{i*}_\ell (\cdot )}{\partial n^{i*}_\ell (\cdot )}}_{\ge 0} \cdot \frac{\partial n^{i*}_\ell (\cdot )}{\partial B^i(\cdot ) } \right) \left( \underbrace{\frac{\partial ^2 B^i(\cdot )}{\partial s^i(\cdot )^2}}_{\ge 0} \cdot \underbrace{\frac{\partial s^i(\cdot )}{\partial w^j_m} \cdot \frac{\partial s^i(\cdot )}{\partial w^i_k}}_{>0} + \underbrace{\frac{\partial B^i(\cdot )}{\partial s^i(\cdot )}}_{>0} \cdot \frac{\partial ^2 s^i(\cdot )}{\partial w^j_m \partial w^i_k} \right) \end{aligned} \end{aligned}$$

(14)

Meanwhile, an individual term in the denominator is:

$$\begin{aligned} \begin{aligned}&\left( \underbrace{\frac{\partial ^2 \varvec{\pi }^{i*}_\ell (\cdot )}{\partial B^i(\cdot )^2 }}_{\ge 0} + \frac{\partial ^2 \varvec{\pi }^{i*}_\ell (\cdot )}{\partial n^{i*}_\ell (\cdot )^2} \cdot \frac{\partial n^{i*}_\ell (\cdot )}{\partial B^i(\cdot )} + \underbrace{\frac{\partial \varvec{\pi }^{i*}_\ell (\cdot )}{\partial n^{i*}_\ell (\cdot )}}_{\ge 0}\cdot \frac{\partial ^2 n^{i*}_\ell (\cdot )}{\partial B^i(\cdot )^2 } \right) \underbrace{\left( \frac{\partial B^i(\cdot )}{\partial s^i(\cdot )} \cdot \frac{\partial s^i(\cdot )}{\partial w^i_k} \right) ^2}_{>0} \\&\quad+ \left( \underbrace{\frac{\partial \varvec{\pi }^{i*}_\ell (\cdot )}{\partial B^i(\cdot )}}_{<0} + \underbrace{\frac{\partial \varvec{\pi }^{i*}_\ell (\cdot )}{\partial n^{i*}_\ell (\cdot )}}_{\ge 0} \cdot \frac{\partial n^{i*}_\ell (\cdot )}{\partial B^i(\cdot ) } \right) \left( \underbrace{\frac{\partial ^2 B^i(\cdot )}{\partial s^i(\cdot )^2}}_{\ge 0} \underbrace{\left( \frac{\partial s^i(\cdot )}{\partial w^i_k} \right) ^2}_{>0} + \underbrace{\frac{\partial B^i(\cdot )}{\partial s^i(\cdot )} }_{>0} \cdot \frac{\partial ^2 s^i(\cdot )}{\partial {w^i_k}^2} \right) \end{aligned} \end{aligned}$$

(15)

The first thing to note is that the sign of $\frac{d w^i_k}{d w^j_m}$ will depend on the signs of in Eqs. (14) and (15) added up over K. For each crop, these only differ according to curvature of profits with respect to land allocation and land reallocation with respect to water cost. The first terms of both Eqs. (14) and (15) will match in sign, so the second term that will determine the sign of the strategic effect. This second term is the change in profits per crop scaled by the adjustments in scarcity due to withdrawals and water’s cost due to scarcity.

The second thing to note is how these two expressions differ from each other: the denominator has a squared term $\left( \frac{\partial s^i(\cdot )}{\partial w^i_k}\right) ^2$, where the numerator has a product of matching signs but potentially not magnitudes, $\frac{\partial s^i(\cdot )}{\partial w^j_m} \cdot \frac{\partial s^i(\cdot )}{\partial w^i_k}$. Additionally, the final term in the numerator has the cross-partial of withdrawal, while that term in the denominator has the own second order effect. These terms are where potential differences in sign and magnitude arise.

1.2 Coordinated Problem

The first order conditions of the coordinated problem with respect to i’s water choices are as follows:

$$\begin{aligned}\begin{aligned} & \frac{\partial {\varvec{\Uppi }^{sp}}}{\partial w^i_k}= \sum _{\ell =0}^{K} \left( \frac{\partial \varvec{\pi }^i_\ell \big (p_\ell , {\textbf {r}}, B^i(s^i(w^i, w^j),b), n^{i*}_\ell (\cdot ); {\textbf {x}}\big )}{\partial w^i_k} + \frac{\partial \varvec{\pi }^j_\ell \big (p_\ell , {\textbf {r}}, B^j(s^j(w^i, w^j),b), n^{j*}_\ell (\cdot ); {\textbf {x}}\big )}{\partial w^i_k} \right) \\& \quad\quad\;= \sum _{\ell =0}^{K} \left( \frac{\partial \varvec{\pi }^i_\ell (\cdot )}{\partial B^i(\cdot )} \cdot \frac{\partial B^i(\cdot )}{\partial s^i(\cdot )} \cdot \frac{\partial s^i(\cdot )}{\partial w^i_k} + \frac{\partial \varvec{\pi }^i_\ell (\cdot )}{\partial n^{i*} _\ell (\cdot )} \cdot \frac{\partial n^{i*} _\ell (\cdot )}{\partial B^i(\cdot ) }\cdot \frac{\partial B^i(\cdot )}{\partial s^i(\cdot )} \cdot \frac{\partial s^i(\cdot )}{\partial w^i_k} \right) \\&\quad\quad\quad\;+\sum _{\ell =0}^{K} \left( \frac{\partial \varvec{\pi }^j_\ell (\cdot )}{\partial B^j(\cdot )} \cdot \frac{\partial B^j(\cdot )}{\partial s^j(\cdot )} \cdot \frac{\partial s^j(\cdot )}{\partial w^i_k} + \frac{\partial \varvec{\pi }^j_\ell (\cdot )}{\partial n^{j*} _\ell (\cdot )} \cdot \frac{\partial n^{j*} _\ell (\cdot )}{\partial B^j(\cdot ) }\cdot \frac{\partial B^j(\cdot )}{\partial s^j(\cdot )} \cdot \frac{\partial s^j(\cdot )}{\partial w^i_k} \right) \\&\quad\quad\;= \sum _{\ell =0}^{K} \left( \frac{\partial \varvec{\pi }^i_\ell (\cdot )}{\partial B^i(\cdot )} + \frac{\partial \varvec{\pi }^i_\ell (\cdot )}{\partial n^{i*} _\ell (\cdot )} \cdot \frac{\partial n^{i*}_\ell (\cdot )}{\partial B^i(\cdot ) } \right) \left( \frac{\partial B^i(\cdot )}{\partial s^i(\cdot )} \cdot \frac{\partial s^i(\cdot )}{\partial w^i_k} \right) \\&\quad\quad\quad\;+ \sum _{\ell =0}^{K} \hspace{-5.69054pt}\underbrace{ \left( \frac{\partial \varvec{\pi }^j_\ell (\cdot )}{\partial B^j(\cdot )} + \frac{\partial \varvec{\pi }^j_\ell (\cdot )}{\partial n^{j*} _\ell (\cdot )} \cdot \frac{\partial n^{j*}_\ell (\cdot )}{\partial B^j(\cdot ) } \right) \left( \frac{\partial B^j(\cdot )}{\partial s^j(\cdot )} \cdot \frac{\partial s^j(\cdot )}{\partial w^i_k} \right) }_{ E^i_k } \le 0 \\ \end{aligned} \end{aligned}$$

(16)

with equality if water withdrawal is positive, i.e. $w_k^i > 0$.

Again, assuming the irrigators are symmetric, we have for $w^j_k$

$$\begin{aligned} \begin{aligned} & \frac{\partial {\varvec{\Uppi }^{sp}}}{\partial w^j_k}= \sum _{\ell =0}^{K} \left( \frac{\partial \varvec{\pi }^j_\ell (\cdot )}{\partial B^j(\cdot )} + \frac{\partial \varvec{\pi }^j_\ell (\cdot )}{\partial n^{j*} _\ell (\cdot )} \cdot \frac{\partial n^{j*}_\ell (\cdot )}{\partial B^j(\cdot ) } \right) \left( \frac{\partial B^j(\cdot )}{\partial s^j(\cdot )} \cdot \frac{\partial s^j(\cdot )}{\partial w^j_k} \right) \\& \quad\quad\quad\;\;+ \sum _{\ell =0}^{K} \hspace{-5.69054pt}\underbrace{ \left( \frac{\partial \varvec{\pi }^i_\ell (\cdot )}{\partial B^i(\cdot )} + \frac{\partial \varvec{\pi }^i_\ell (\cdot )}{\partial n^{i*} _\ell (\cdot )} \cdot \frac{\partial n^{i*}_\ell (\cdot )}{\partial B^i(\cdot ) } \right) \left( \frac{\partial B^i(\cdot )}{\partial s^i(\cdot )} \cdot \frac{\partial s^i(\cdot )}{\partial w^j_k} \right) }_{ E^j_k } \le 0 \\ \end{aligned} \end{aligned}$$

(17)

with equality if water withdrawal is positive, i.e. $w_k^j > 0$.

Appendix 2 Additional Tables

We include tables for results for the larger 2- and 5-mile radii for all three regression specifications, as well as production intensity (share of acreage cropped) broken out by subdistrict. See Table 7, 8, 9, 10 and 11.

Table 7 Direct, Indirect and Total Effects for Water Use (acre-foot/acre) for 2- and 5-mile radii

Table 8 Direct, indirect and total effects for acreage irrigated for 2- and 5-mile radii

Table 9 Average partial direct and indirect effects for proportion of total available acres irrigated for 2- and 5-mile radii

Table 10 Production intensity for subdistrict with pumping fee—half-mile radius

Table 11 Production intensity for subdistrict with no pumping fee—half-mile radius