Bargaining Over Environmental Budgets: A Political Economy Model with Application to French Water Policy

In decentralized water management with earmarked budgets ﬁnanced by user taxes and distributed back in the form of subsidies, net gains are often heterogeneous across user categories. This paper explores the role of negotiation over budget allocation and coalition formation in water boards, to provide an explanation for such user-speciﬁc gaps between tax payments and subsidies. We propose a bargaining model to represent the sequential nature of the negotiation process in water districts, in which stakeholder representatives may bargain upon a fraction of the budget only. The structural model of budget shares estimated from the data on French Water Agencies performs well as compared with reduced-form estimation. Empirical results conﬁrm the two-stage bargaining process and provide evidence for systematic net gains from the system for agricultural water users.


Basic Setting
We assume there are n committee members (also referred to as "players"), from which k 137 are representatives of water user categories, i = 1, . . . , k, with k ≤ n. Non-water users, 138 j = k + 1, . . . , n, are stakeholders not directly impacted by committee decisions on budget. 139 We denote by γ i the fraction of total taxes paid by the ith category of users: where t i is the amount of taxes paid by the ith category of users. We assume, without 141 loss of generality, that γ 1 ≤ γ 2 ≤ · · · ≤ γ k .

142
The committee members decide on the distribution of the budget, normalized to 1 without 143 loss of generality, among the k users. The policy space is X ≡ can possibly concern the welfare of all k categories of users. We assume that each player 148 j = k + 1, . . . , n assigns a weight β ij to user category i, i = 1...k, such that for any j = 149 k + 1, . . . , n and i = 1, . . . , k, β ij ∈ [0, 1] and k i=1 β ij = 1. Then, given the vector of shares 150 x = (x 1 , ..., x k ) the utility of player j, j = k + 1, . . . , n, is where u j is a twice continuously differentiable function such that u j > 0 and u j < 0. We  Players act both as voters and as proposers. The voting activity is described by a weighted 155 majority game. Let q i denote the voting weight (the number of representatives) of category i. 156 We assume that all other voters have weight equal to 1. The quota Q of the game could be any number between ( k i=1 q i) +(n−k) 2 2 and k i=1 q i +(n−k). Therefore, our framework allows 158 for a wide range of voting mechanisms. When Q = ( k i=1 q i) +(n−k) 2 , to pass the proposal 159 the approval of the majority of members is necessary, while when Q = k i=1 q i + (n − k), 160 unanimity is required. Unless otherwise specified, we assume that Q is the majority quota. 161 We denote by W (W m ) the set of winning (minimal winning) coalitions. 3

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The distribution of proposal powers is represented by the vector p = (p 1 , p 2 , ..., p n ) such 163 that p i ≥ 0 for all i = 1, ..., n, and n i=1 p i = 1, where p i denotes the probability that 164 player i is in charge of making a proposal. The probability to act as a proposer, p i , is likely 165 to reflect the proportion of representatives for category i and therefore the relative voting 166 weight w i = q i /n. We consider however a more general case and we do not impose p i and w i 167 to be equal, as other factors than q i may influence voting weights. 168 The game has two stages. The first stage is a BD bargaining game on the fraction, 169 denoted α, of the budget distributed proportionally to tax payments. This is a sequential 170 game with a possibly infinite number of rounds and each player is informed of all options.

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At each round t, a proposer i(t) is selected and makes a proposal α(i, t), which members of 172 the committee may approve or reject. If the subset of members approving the proposal is a 173 winning coalition, the proposal is adopted, and if not, the game moves to round t + 1 and 174 the procedure is repeated. If no proposal is ever accepted, the players receive γ. 175 If α = 1 is selected, the game ends after the first stage and the whole budget is distributed to make a proposal x(i) ∈ X which can be accepted or not by other players. If the subset of 181 players approving the proposal is a winning coalition, it is adopted, otherwise, the vector γ 182 is adopted for the residual budget. 183 2 For any real number x, x denotes the smallest integer greater than x. 3 A coalition S is a winning one if and only if i∈S q i ≥ Q. If, moreover, by dropping any player j we reverse the inequality, i.e., i∈S\{j} q i < Q for any j ∈ S, then such a coalition S is called minimal winning.
with the description of the second stage of the game. The outcome of the second stage is the allocation of residual budget (1 − α) among categories 187 of users. Nature draws proposer j with probability p j ≥ 0, where n j=1 p j = 1.

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Proposer j selects vector We assume that ties are broken in favor of the proposer.

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For the proposal to be accepted, the proposer should consider the cost of "buying" a 195 minimal winning coalition. Letting S be any such coalition, the problem of proposer j can 196 be written as: Let us denote by C(α, S, j) the value of this problem and We also denote by x * j (α) for j = 1...n the optimal solution to problem (4) and we proceed 199 as if this solution were unique. The component x ij is the share of the budget offered to player i by proposer j.

8
In such a case, players voting on behalf of category i have weight equal to w i = q i + m i .

205
Further, the set of supporters of category i votes in favor of the proposal if and only if 206 Things are as if proposer j representing category i makes a proposal to win the votes of a 207 winning coalition in a weighted majority game with {1, 2, ..., k} as the set of players and w i 208 being the weight of player i. The probability of player i to be selected as a proposer is now 209 equal to: The set of (minimal) winning coalitions of this simple game is denoted by ( W m ) W. 211 We have and therefore, Equivalently in this case: To obtain a closed-form expression for x * ij (voter i's equilibrium share when j is the The first stage is a one-dimensional BD bargaining game , once we account for the backward 220 solution of the second stage on the residual budget allocation, vector x * j (α) for all j, j = 221 9 1, ..., n. In the first stage of the game, each player i views the choice of α as the choice in a The expected utility V i (α) of player i is equal to: Note that when j = i, player i's equilibrium share x * ij is either equal to 0 or to (1 − α) γ i . The player's expected utility is therefore based on two numbers: first, the probability denoted by P i that i is considered in the continuation game when i is not the proposer himself, and second, the coalition S i of players who receive a positive share in his proposal. Without loss of generality, we assume that S i does not contain player i. Player i's share x i can be expressed as: We obtain that: From our assumptions on u i , it follows that V i (α) < 0 for all α ∈ [0, 1], i.e., func-226 tion V i (α) is strictly concave on the unit interval. We denote by α * i the (unique) peak for The following proposition summarizes the properties displayed by the preferred peaks of 232 the different groups.
has a unique maximum on the interval (0, 1) and it is defined The thresholds γ i and γ i are calculated as: and Proof: see Appendix 1. i.e., γ i . Being a "cheap" coalitional partner, he is included in any coalition when he is not a 249 proposer receiving an offer equal to γ i . When he is a proposer he gets strictly more than γ i .

250
On the contrary, part (iii) of the proposition states that any player i with relatively high 251 γ i , i.e., with γ i above γ i , prefers to share the whole budget according to the mechanical 252 rule as his preferred α * i = 1. The reason is that in the bargaining game, player i being an 253 "expensive" coalitional partner receives no offer when he is not a proposer.   First, it is reasonable to assume that members of RBCs prefer to avoid sharp changes in 320 budget allocation one year to the next. 8 321 Second, risk aversion is likely to play a role in the objective of water users to bargain 322 over a residual budget, once a non-random part of the latter is decided upon.

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There are various ways to define the stable part of budget: stationary over time, or as a 324 function of observed and predetermined variables such as tax payments. This is this second 325 possibility that we consider here; water users collectively determine the share of budget 326 allocated according to their relative tax burden, and then they bargain over deviations from 327 this simple and "equitable" rule.

328
Interviews with Water Agency executives and RBC members (see footnote 7) reveal that 329 their policy is to maintain a reasonable stability in subsidies granted to water user categories  An interesting feature of our model is that it includes as special cases the absence of bar-341 gaining (equivalent to α = 1) and bargaining over the full budget (when α = 0). Therefore, 342 these two polar cases can be considered equivalent to a situation in which the outcome of 343 the bargaining game corresponds to a one-stage game. ing water sources for ecosystems. Symmetrically to the fact that residential water users do not pay taxes directly, they do not receive direct subsidies, which are granted to local 364 communities instead.

365
The proportion of subsidies received by each user category (agriculture, industry, resi-366 dential users) is computed for each river basin and each multi-year intervention programme. 9

367
Although emission and water-use tax rates as well as subsidy rates are defined over the represented by specific professional members in the RBCs, but whose interests may also be 381 represented by representatives of rural communities. We assume that this is the case for 382 farmers, but it is not possible to single out industry representatives for agrofood and food 383 processing on the one hand, and for other industries on the other. We therefore assume that 384 industry representatives do not represent farmers' interests.

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Regarding projects subsidized by Water Agencies and which concern ecosystem conser- than one user category. However, local communities and therefore residential water users 388 are the most important beneficiaries of these projects (CGDD, 2012). We therefore affect 389 natural resource and ecosystem conservation projects to local communities. Table 1  There is a clear ranking of tax contributors with agriculture, industry and residential users 393 in increasing order, which is also observed for subsidies from the Water Agency. However, 394 ratios of subsidy over tax are fairly heterogeneous on average across user categories.

396
We consider here the special case of three categories of water users: as discussed above, in 397 most water boards or agencies, water users paying taxes and receiving subsidies are residential 398 users, industry and farmers. From the discussion above, it follows that a more detailed 399 description of the equilibrium peaks α * i and the median α * requires more detailed information 400 on the parameters of the game. We illustrate Proposition 1 through a special case which 401 will prove useful in the application to French Water Agency policy, where water users can 402 be grouped into three major categories (local communities, industry, agriculture).

403
Consider then the case k = 3 and the simple majority game. From data presentation above, we let γ 1 < γ 2 < γ 3 , with player 1 corresponding to agriculture, player 2 to industry, and player 3 to residential users. As before, we denote by x i share of group i from the bargaining game. Since player 1 is the "cheapest", he is always in the winning coalition, therefore his share is: Consider then player 2. It is included in the winning coalition by group 1 but not by 17 group 3: , with probability p 2 , γ 2 , with probability p 1 , αγ 2 , with probability p 3 .
Since player 3 is the "most expensive", it is invited as a coalition partner by neither group 1 nor group 2: From the assumption on u 1 it follows that V 1 (α) < 0 and therefore, α * 1 = 0.

404
Results are summarized in Figure 1, and details are provided in Appendix 2.
first-order conditions (25) and (26) (see Appendix 2) can be solved explicitly for α * 2 and α * 3 : and Interestingly, since for CRRA utility functions u i (0) = ∞, the two extreme cases with 411 α * = 0 (see Figure 1) disappear, i.e., at equilibrium a positive part of the budget is al-412 ways shared according to the mechanical rule. We assume from now on that risk aversion 413 parameters ρ are constant over time.

414
Omitting river basin and time indexes for the sake of clarity, the system of budget share 415 equations can be written, for water user category i, 18 where From Equation (18), it can be seen that for any value of α * , user category 1 (agriculture) 419 always gains from bargaining because The structural model of bargaining consists of the system of non linear equations for subsidy shares, with probabilities p i , tax shares γ i and risk-aversion parameter ρ i on the right-hand side. Because probabilities (that a representative of category j is a proposer) are not observed and correspond to the subsidy internal committee, we assume that they are related to observed political representation of water users in the RBCs. More precisely, we specify a logit probability: p ijt = P rob(user i from river basin j at time t is the proposer) where w i = q i /n is the observed voting weight and, without loss of generality, category 1 is 422 chosen as the reference.

423
The optimal parameter α * in river basin j at time t equals α * 2 ifp 3 /p 2 < γ 3 /γ 2 , equals α *  implying that ifp 3 /p 2 is not too far from w 3 /w 2 , the number of observations such that 459 α * = α * 2 would be far greater than the two other cases. We check during estimation that this 460 is the case, which implies that parameter ρ 3 is not identified because α * is almost always 461 equal to α * 2 . Therefore, we consider only the case α * = α * 2 .

462
To avoid possible small-sample bias because of excessive over-identification, we consider  these test statistics are well below 0.05, so that the assumption of a single-stage game with full or no bargaining is strongly rejected, when α * is evaluated at the sample mean.
[ We compare our GMM structural parameter estimates with reduced-form estimates. To Agencies, and also the fact that subsidies aim at helping water users reduce their tax burden 515 paid to water Agencies. We therefore consider only w and γ as explanatory variables. This 516 also has the advantage of matching exactly variables used in the structural model. 517 We perform a regression analysis of the relative subsidies received by two out of the three 518 main water users (agriculture and industry, because these shares sum to 1), as a function of 519 relative representation in RBC and/or tax shares of each user category.

520
The system of reduced-form equations is the following: i, k = 1, 2 (agriculture, industry), j = 1, 2, . . . , 6; t = 1, 2, . . . , T, where x ijt is the share of total subsidies received by the user category i (agriculture,  Table 3 indicate that models are observationally equivalent at the 5 percent level, and that 558 the structural model would be preferred to reduced-form Model III at the 10 percent level. 559 We therefore conclude that our structural model performs well in predicting relative subsidy  ities that a representative of a particular category is chosen as a proposer (p). From Table 4 565 reporting average proportions of representatives (w) together with estimated probabilities, 566 one can see that averagep and w are close for industry. However, the probability estimate 567 24 is about twice the average proportion w for farmers, while it is lower by about one-third 568 for local communities. We therefore identify an additional factor for the systematic excess 569 ratio of subsidy over tax for farmers, due to the nature of the bargaining process. Farmers 570 receive a larger share of subsidies than their relative contribution to total taxes, not only be-571 cause they are often well represented in RBSs (as reflected by a relatively large w i ), but also 572 because the probability that a farmer representative is chosen as a proposer (p i ) is higher.

573
[ The bargaining model presented in this paper draws upon Baron and Ferejohn (1989) and Since u i (·) < 0 it follows from (24) that V i (α) ≤ 0.

628
The derivative of V at α = 0 is: One can check that: (14).
Therefore, the behavior of player 2 can be described as follows:

644
In a similar way, the thresholds on the tax share for player 3 can be expressed as follows: The behavior of player 3 can be summarized as: , function V 3 (α) has an inferior maximum α * 3 on 649 [0, 1] and it is defined from V 3 (α) = 0:  Estimation method: nonlinear two-step GMM. Standard errors in parentheses are estimated from a heteroskedasticity-consistent robust variance-covariance matrix.*, ** and *** respectively denote parameter significance at 10, 5 and 1 percent level. Parameter α is estimated in a second stage from GMM estimates, and at the sample mean.  Standard errors (in parentheses) are estimated from a heteroskedasticity-consistent robust variance-covariance matrix.*, ** and *** respectively denote parameter significance at 10, 5 and 1 percent level. Instruments for Model I and Model II equations: (1, γ 1 , γ 2 , w 2 ).