Groundwater Market Model – John Robin Inston

1 1 Groundwater Market Model

In Cialenco & Ludkovski [[cialencogroundwatermarketmodel2025]] the authors describe a groundwater market model defined on filtered probability space $(\Omega, \mathcal{F}, \mathbb{P}, \mathbb{F}:=(\mathcal{F}_{t})_{t=0}^T)$ assuming that $J$ agents trade groundwater among themselves within one basin. Throughout, all processes are assumed $\mathbb{F}$-adapted and we use boldface to denote vectors $\boldsymbol{\varphi}=(\varphi_{1}, \dots, \varphi_{J})$ and Fraktur letter to denote sums of quantities across agents $\mathfrak{W}=\sum_{j=1}^J W_{j}$.

The profit and loss of the $j$-th agent is given by \[ L_{j}(t):=\sum_{k=1}^K \overline{f}_{j}^k(t, \varphi_{j}^k)\cdot \varphi_{j}^k(t)+\psi_{j}(t)\cdot p(t), \] where: 1. $K$ denotes the types of goods that can be produced; 2. $\varphi_{j}^k$ denotes the units of good $k$ for individual $j$; 3. $\overline{f}_{j}^k(t, \varphi_{j}^k)$ is a function (assumed continuous) denoting net profit in dollars ($) per $\varphi_{j}^k$ units of good $k$ at time $t$; 4. $\psi_{j}(t)$ denotes the amount of water traded in acre-feet (ac-ft) at time $t$ with the convention that $\psi_{j}>0$ means selling and $\psi_{j}<0$ means buying; 5. $p(t)$ denotes the price of traded water in $/ac-ft at time $t$.

The agents control $\pi_{j}(t)$ consists of the type of goods to produce, the quantity of goods to produce, and the amount of water to trade, hence \[ \pi_{j}(t):=(\boldsymbol{\varphi}_{j}(t), \psi_{j}(t)), \] subject to the additional production constraints \[ 0 \leq n_{j}^k(t)\leq \varphi_{j}^k(t)\leq N_{j}^k(t)\quad(\forall k,t), \] where the lower bound $n_{j}^k(t)$ represents agricultural constraints. The processes $\boldsymbol{n}(t)$ and $\boldsymbol{N}(t)$ are taken as inputs to the model and can be either deterministic or stochastic.

The production of one unit of good $k$ requires $a^k(t)$ ac-ft of water and so we denote the $j$-th agent’s water needs for producing goods at time $t$ by \[ C_{j}(t):=\sum_{k=1}^K\varphi_{j}^k(t)a^k(t). \] The water availability for agent $j$ $\{W_{j}(t)\}$ has dynamics described by \[ W_{j}(t+1)=W_{j}(t)+\theta_{j}R(t+1)-C_{j}(t)-\psi_{j}(t)\quad (t=0, \dots, T-1), \] with $W_{j}(0) = \theta_{j}H(0)$ where: 1. $R(t)$ denotes the water recharge process (primarily from surface precipitation) from $t$ to $t+1$. 2. $\theta_{j}$ denotes the fixed proportion of recharge for agent $j$ allocated by regulators where we assume $\theta_{j}>0$ such that $\sum_{j}\theta_{j}=1$.

The recharge process $R(t)$ is given exogenously and follows a Markovian structure. The dynamics of the water table height $H(t)$ is given by \[ H(t+1)=H(t)+R(t+1)-\mathfrak{C}(t)\quad (t=0,1\dots,T-1), \] and for simplicity we equate this with available water for all $J$ agents \[ H(t)=\mathfrak{W}(t). \] We assume that agents cannot use more water than total available to them for that period, namely, imposing the water budget constraints \[ -\sum_{i \neq j}W_{i}(t)\leq C_{j}(t)+\psi_{j}(t)\leq W_{j}(t). \] The agents have the opportunity for groundwater banking, that is \[ C_{j}(t)+\psi_{j}(t)<W_{j}(t), \] yielding an inter-temporal shift in water consumption. Moreover, the traded water amounts must balance out via the market clearing condition \[ \sum_{j}\psi_{j}(t)=0. \] Denote by $\mathscr{S}_{j}$ the set of all feasible controls $\pi_{j}$ of agent $j$ i.e. stochastic processes satisfying (1.3) and (1.4). Considering only Markovian policies $\pi_{j}\in\mathscr{S}_{j}$, the process $(\boldsymbol{W}(t), R(t))$ is Markov and the agents solve \[ \sup_{\pi_{j}\in\mathscr{S}_{j}}\mathbb{E}\left[ \sum_{s=t}^T U_{j}(L_{j}(s))\middle|\boldsymbol{W}(t)=\boldsymbol{w}, R(t)=r \right], \] where the functions $U_{j}$ account for each agent’s idiosyncratic utility, as well as the temporal discount factor, and are required to be monotone increasing and concave.

2 2 One-Period Fixed Parameter Market Model

We consider the one-period model consisting of two agents $j=1,2$ each able to plant two crop types $k=1,2$. Each agent is looking to maximize profit \[ A_{j}(\pi_{j}, \boldsymbol{\pi}_{-j}, p)=\mathbb{E}\left[ U_{j}(L_{j}( \pi_{j}, p) \right]=L_{j}(\pi_{j}, p) \] with respect to $\pi_{j}\in\mathscr{S}_{j}$ where we are considering the linear utility function $U_{j}(x)=x$. We note that in the one-period case we are taking everything as fixed, hence nothing is stochastic and we can drop the expectation. Each agent’s profit is given by \[ L_{j}(\pi_{j}, p):=\sum_{k}^{}\overline{f}_{j}^k(\varphi_{j}^k)\cdot \varphi_{j}^k + \varphi_{j}\cdot p \] which we rewrite as \[ L_{j}(\pi_{j},p)=G_{j}(C_{j})+\varphi_{j}p=G_{j}(C_{j})+(W_{j}-C_{j})p, \] where we have some assumed agents good production given her constraints $G_{j}$ and that self-sufficient farmers will treat trading as a bonus or a necessity (i.e. $\varphi_{j}=W_{j}-C_{j}$). We consider the net profit function \[ \overline{f}_{j}^k(\varphi_{j}^k)=f_{j}^k(\varphi_{j}^k)^{\alpha_{j}^k-1}-q_{j}^k\varphi_{j}^k, \] for constants $f_{j}^k, q_{j}^k\geq 0$ and $\alpha_{j}^k \in (0,1)$. The Lagrangian for agent $j=1,\dots,J$ assuming no production constraints and for fixed price $p$ becomes \[ \mathcal{L}(\boldsymbol{\varphi}_{j}; \lambda_{j}) = \sum_{k}^{}(f_{j}^k(\varphi_{j}^k)^{\alpha_{j}^k}-q_{j}\varphi_{j}^k)+\left( W_{j}-\sum_{k}^{}a^k\varphi_{j}^k \right)p+\lambda_{j}\left( R-\sum_{i,k}^{}a^k \varphi_{i}^k \right), \] and the combined first-order conditions for agents $j=1, \dots, J$ are \[ \begin{cases} f_{j}^k\alpha_{j}^k(\varphi_{j}^k)^{\alpha_{j}^k-1}-q_{j}^k-a^kp\lambda_{j}=0 \\ R-\sum_{i,k}^{} a^k \varphi_{i}^k = 0,\quad \varphi_{j}^k \geq 0 \,\quad k = 1, \dots, K. \end{cases} \] Letting \[ d_{j}^k:=\left( \frac{a^k}{\alpha_{j}^k f_{j}^k} \right)^{\frac{1}{\alpha_{j}^k-1}} ~~\&~~e_{j}^k:=\frac{q_{j}^k}{a_{k}}, \] then for any $\lambda_{j}\geq -p - e_{j}^k$ \[ \begin{align} \varphi_{j}^k & = (p + \lambda_{j}+e_{j}^k)^{\frac{1}{\alpha_{j}^k - 1}}d_{j}^k,~~k=1, \dots, K \\ R & = \sum_{k,j}^{}(p+\lambda_{j}+e_{j}^k)^{\frac{1}{\alpha_{j}^k-1}}a^k d_{j}^k \end{align} \] Clearly, for any $p\geq 0$ there exists $\lambda_{j}\in \mathbb{R}$ for $j = 1, \dots, J$ such that this expression for $R$ holds and so our expression for $\varphi_{j}^k$ is satisfied. Moreover, since the optimization problem at hand is convex with affine constraints, these conditions are sufficient for $\boldsymbol{\varphi}$ to be the joint maximizer.

For a fixed price $p>0$ announced by the price setter, assuming that agent $j$ has no constraints on available water to buy/sell, her optimal production is given by \[ \varphi_{j}^k=(p+e_{j}^k)^{\frac{1}{\alpha_{j}^k-1}}d_{j}^k, \] which decreases as a function of $p$, i.e. lower price will yield larger production and hence larger consumption. To meet these optimal production rates, agent $j$ would buy water if \[ W_{j}\leq \sum_{k}^{}a^k \varphi_{j}^k=\sum_{k}^{}a^k(p+e_{j}^k)^{\frac{1}{\alpha_{j}^k - 1}}d_{j}^k. \] Due to the monotonicity of the power function, there exists unique price $\tilde{p}_{j}$ such that the above inequality becomes an equality. Thus, for any $p \leq \underline{p}=\min_{j} \tilde{p}_{j}$ no agent would be willing to sell the water. The agent will consume all the allocated water $W_{j}$ and by direct calculations we obtain the optimal production rates \[ \varphi_{j}^k=(\tilde{p}_{j}+e_{j}^k)^{\frac{1}{\alpha_{j}^k-1}}d_{j}^k. \] This implies that the agent $k$ chooses the Lagrange multiplier $\lambda_{j}$ corresponding to the NE such that $p+\lambda_{j}=\tilde{p}_{j}$, hence $\lambda_{j}>0$ making the agent $j$ artificially boost profit from producing the goods. Similarly, if $p>\overline{p}=\max_{j}\tilde{p}_{j}$ each agent prefers to sell water, thus no water is traded, and the multipliers are chosen so that $\lambda_{j}<0$ which can be interpreted as the agent $j$ produces goods pretending that the water price is the indifferent price.

From here, to reach the maximum profit for a fixed price $p$, each agent $j$ should choose $\lambda_{j}=0$. Taking $\lambda_{j}=0$ for all $j$, we get the unique price $p^\circ$ that clears the market \[ \sum_{j,k}^{}(p^\circ+e_{j}^k)^{\frac{1}{\alpha_{j}^k-1}}a^kd_{j}^k = R. \] Clearly, this price $p^\circ$ is Pareto optimal. We also note that if there were a central planner who would implement the cooperative solution of maximizing the total production revenue of all farmers $\sum_{j}^{}G_{j}(C_{j})$ subject to $\sum_{j}^{}C_{j}=R$ this yields the same f.o.c. \[ \varphi_{j}^k = (\lambda^{SO}+e_{j}^k)^{\frac{1}{\alpha_{j}^k-1}}d_{j}^k,\quad \sum_{j,k}^{}a^k\varphi_{j}^k=R, \] and hence the same optimal production rates $\varphi_{j}^k$ corresponding to $p^\circ$. Moreover, we can interpret $p^\circ$ as the shadow cost of water (i.e. the Lagrange multiplier for the budget constraint) from the central planners perspective. Note that the social optimum is driven by the total available water $\mathfrak{W}$ and agent specific allocations $W_{j}$ play no role beyond cash transfers within the subsidiary farmers in the community.

Similar computations and arguments can be extended to the case with nontrivial production bound $n_{j}^k, N_{j}^k$. For example, assuming that $\sum_{j}^{}\underline{c}_{j}<R<\sum_{j}^{}\overline{c}_{j}$ the social optimum exists and is computed from KKT conditions \[ \varphi_{j}^k=n_{j}^k \vee (\lambda^{SO}+e_{j}^k)^{\frac{1}{\alpha_{j}^k-1}}d_{j}^k \wedge N_{j}^k,\quad \sum_{j,k}^{}a^k\varphi_{j}^k=R. \] Correspondingly, the max profit function $C_{j}(C)$ are computed through $\varphi_{j}^k=n_{j}^k \vee (\lambda_{j}(C)+e_{j}^k)^{\frac{1}{\alpha_{j}^k-1}}d_{j}^k\wedge N_{j}^k$ where $\lambda_{j}(C)$ are the Lagrange multipliers such that $\sum_{j,k}^{}a^k\varphi_{j}^k=R.$

3 3 One Period Agent Continuum Model

We now look to modify our one period finite agent model to instead consider a continuum of infinitesimally small agents. We assume that agents in the continuum are characterized by some selection of model parameters, for example $f^k$, and that all remaining parameters are constant across the continuum for each crop $k$. On our abstract player space $\mathcal{J}$ there is some measure $\mu$ describing the distribution of aggregate agent behavior. We denote the distribution induced by $\mu$, of for example by $f^k$, by \[ G_{f}^k(x)=\mu(f^k < x), \] with the corresponding density given by $g_{f}^k$ assuming that $G_{f}^k$ is differentiable. We are particularly interested in resulting continuum distribution of optimal agent production $G_{\varphi}^k$ and resulting water consumption $G_{C}{}^k$ given by \[ \begin{cases} \varphi^k:= n^k \vee \Phi^k \wedge N^k \\ \Phi^k := \left( \frac{{pa^k+q^k}}{\alpha^kf^k} \right)^{\frac{1}{\alpha^k-1}}. \end{cases} \] Intuitively we can think of the continuum case as the target that the finite $J$ player model should tend towards asymptotically as we take $J\to \infty$. Note that our existing model has no parameter scaling and relies on the input parameters to determine the convergence of the final productivity. We attempt to construct a scalable model requiring an in-built dependence on the total number of players $J$ scaling production $\varphi_{j}\to 0$ as $J \to \infty$.

To clarify notation we write $^*\varphi_{j}^k$ for our scalable model parameters and we note that our aim is to have \[ \lim_{ J \to \infty } ~\sum_{j=1}^{J}~^*\varphi_{j}^k = m^k \] where $m^k$ is the population optimal average production for the continuum \[ m^k=\int_{\mathcal{J}}^{}{x}~d{G_{\varphi}^k(x)}. \] To construct this scalable model we need to consider the following points: (1) the agent characterization model components; (2) the distribution of agents across each characterizing components; (3) the discretization of continuous parameter spaces.

3.1 3.1 One Period Agent Continuum - Productivity

As a starting point assume that agents are characterized by productivity $f^k$ across the continuum and that all other model parameters are constant across the continuum for each crop $k$. To simplify our analysis we our expression for $\Phi^k$ as \[ \Phi^k=\left( \frac{\alpha^k}{pa^k+q^k} \cdot f^k \right)^{\frac{1}{1-\alpha^k}}=(\xi^{k}\cdot f^k)^{\frac{1}{\beta^k}}=h(f^k), \] where we have defined \[ h(x) = (\xi^k\cdot x)^{\frac{1}{\beta^k}},\quad\xi(p) = \frac{\alpha}{pa + q}\quad\&\quad \beta={1-\alpha}. \] From the non-negativity of all terms and since $\beta^k<1$ we have that $h$ is one-to-one, hence we can write the density of $\Phi^k$ as \[ g_{\Phi}^k(y)=g_{f}^k(h^{-1}(y))\cdot \left| \frac{d}{dy} h^{-1}(y) \right| =g_{f}^k\left( \frac{y^{\beta^k}}{\xi^k} \right)\cdot \frac{\beta^k}{\xi^k} x^{\beta^k-1}. \] From this the density of $g_{\varphi}^k$ includes a weighting and point masses due to the production bounds $n^k, N^k$ and so \[ g_{\varphi}^k(x)=\begin{cases} \mu(\Phi^k \leq n^k) & x=n^k \\ g_{\Phi}^k(x) & x \in [n^k, N^k] \\ \mu(\Phi^k \geq N^k) & x = N^k. \end{cases} \] Thus the population average optimal production can be computed from the integral \[ m^k = \int_\mathcal{J} x g_\varphi^k(x)dx = n^k \mu(\Phi^k\leq n^k) + \int_{n_k}^{N^k}x\Phi^k(x)dx+N^k\mu(\Phi^k\geq N^k). \] Note: It should be possible to compute the spread of the distribution of optimal agent production. It should also be simple to verify the result numerically.

We proceed to consider several cases each of which considers $f^k$ to follow a standard non-negative distribution. family such as the continuous uniform, the two-parameter gamma and the scaled two-parameter beta.
#### 3.1.1 Case 1 - Uniform Productivity

Assume that $\mu$ induces a uniform distribution $f^k\sim\mathcal{U}([A^k, B^k])$ for some $0\leq A^k<B^k$ with \[ g_f^k(x)=\frac{1}{B^k-A^k}\mathbb{1}_{\{x\in[A^k, B^k]\}}. \] The density of $\Phi^k=h(f^k)$ is given by \[ g_\Phi^k(x) = \frac{1}{B^k-A^k}\mathbb{1}_{h^{-1}(x)\in[A^k, B^k]}\cdot \left|\frac{\beta^k x^{\beta^k-1}}{\xi^k}\right| = \frac{\beta^k x^{\beta^k-1}}{\xi^k(B^k-A^k)}\mathbb{1}_{\{x\in[h(A^k), h(B^k)]\}}. \] Noting that in this case productivity is bounded on $[A^k, B^k]$ we define $\lambda_1^k := (n^k\vee h(A^k))\wedge h(B^k)$ and $\lambda_2^k := (N^k\wedge h(B^k))\vee h(A^k)$ so that we can write the player average of $\varphi^k$ as \[ \begin{aligned} m^k & = \int x dG_\varphi(x) = n^k\cdot\mathbb{P}(\Phi^k\leq \lambda_1^k) + \int x\mathbb{1}_{\Phi^k\in[\lambda_1^k, \lambda_2^k]}g_\Phi^k(x)dx + N^k\cdot\mathbb{P}(\Phi^k\geq \lambda_2^k) \\ & = \frac{n^k(h^{-1}(\lambda_1^k)-A^k)}{B^k-A^k} + \frac{\beta^k[(\lambda_2^k)^{\beta^k+1}-(\lambda_1^k)^{\beta^k+1}]}{\xi^k(B^k-A^k)(\beta^k+1)} + \frac{N^k[B^k-h^{-1}(\lambda_2^k)]}{B^k-A^k}. \end{aligned} \] From this we can look to construct a finite model for $^*\varphi^k_{j}$. First considering the case for unbounded production we have that \[ \lim_{ J \to \infty } \sum_{j=1}^{J}~^*\varphi_{j}^k= \]