The Illiquidity of Water Markets2026

This paper is incredible. It examines markets vs. quotas and finds that when there exists liquidity constraints quotas are more efficient. A natural question to ask is for generalizations of this idea, insofar as the structure of the mode of production itself is such that liquidity constraints are prevalent and exist almost by construction (competition, placing beings in relation, etc.) for household-firms. In this case, poor apricot farmers did worse that wealthy ones. Efficiency is computed in a clever fashion, as output data is not available. I write up their model in the following paragraphs. Farmers are heterogeneous in their willingness and ability to pay for water. The quotas are efficients because of the marginal returns to water. They state that in the presence of frictions, appropriate institutions need to be designed when it comes to allocating resources. With poor farmers and non-enforcable contracts, endogeneous borrowing constraints emerge.

The setting is the historic water auction in the city of Mula. It used to be, following the Reconquista of Murcial in the 1200s. Shareholders owned a cartel corporation in which an English (sequential outcry ascending price) auction sold water usage rights and dividends were paid at the end of the year. In 1966 the W syndicate helped establish quotas proportional to the size of the farmers' plots. Wealthy and poor farmers purchased similarly throughout the year, but there is a critical season for the trees where price increased dramatically. Poor farmers could not match what rich farmers were buying, and thus had to reduce their usage, which reduced output.

The model is as follows. There are \(N\) rational and forward looking farmers \(i\). There is moisture \(M\), money \(m\), and time \(t\). Seasons \({\omega}_{t} \in \{1,2,\cdots, 53\}\) matter since demand is seasonal. The model runs forever with discount between periods denoted by \(\beta \in (0,1)\), so today mattters more than the future. Water \(j\) only brings utility during the critical season. Its supply is exogeneous to the farmer, and farmers can store it, although there is a probability of evaporation. Water is contained in the production function \(h(j_{it},M_{it},\omega_{it};\gamma)\) of apricots that is increasing in moisture level \(M_{it}\). Utility is profits and preferences are given by \[\begin{split} u_{it}(\omega_{it}, M_{it}, w_t, p_t, \varepsilon_{it} \gamma, \zeta) & = h(\omega_{it}, M_{it}, w_t \gamma) \\ & - \zeta \mathbb{1}\{\omega_{it} > 0\} - p_t j_{it} + \varepsilon_{it} \end{split}\] with \(j_{it} \in \{0,1,\dots,J\}\) being water bought by farmer \(i\), \(p_t\) is the price of water, and \(\varepsilon_{it}\) is a random shock for farmer \(i\), all at time \(t\). \(\mathbf{1}(\cdot)\) is an indicator function. \(\zeta\) is a parameter for irigation costs, and \(\gamma\) is a parameter for the transformation rate of water into apricots during the fruit’s growth season. Every week the farmer decides how much water to buy to maximize apricot revenue minus water fixed and variable costs. Rainfall \(r_t\) and the distribution \(\sigma_{\varepsilon}\) both implicitly affect the profit equation. It is assumed that \[\forall j_{it} > 0 \Rightarrow (m_{it} - p_t j_{it}) \geqslant 0\] which means that farmers cannot buy water if they do not have the liquid money for it: there is a missing credit market. Farmers are heterogeneous in \(m_{it}\) and \(\varepsilon_{it}\), both private information. Thus, farmers who are wealthy can always buy water, but poor farmers might have to buy water to store before price surges in the critical season, when they will be unable to do so. This dynamic system has the following state variables: how wet the soil is, what week of the year it is, the price of water/rainfall, cash holdings, and productivity shocks.

Suppose both wealthy and poor farmers do not differ in measurement error. Denote \(\text{PW}\) as the point in which apricot trees die from low moisture (\(\forall t \Rightarrow M_{it} \geqslant \text{PW}\)), and \(\text{FC}\) as full capacity to hold water. Moisture evolves weekly as follows: \[M_{it} = \min \Big\{ M_{i,t-1} + r_{t-1} + \frac{j_{i,t-1} \cdot 432{,}000}{\text{area}_i} - \text{ET}(M_{i,t-1}, \omega_{t-1}), \text{FC} \Big\}\] with 432,000 the number of liters contained in each \(j\), are measured in square meters, and \(\text{ET}(M_{i,t-1}, \omega_{t-1})\) the evapotranspiration function. As previously mentuoned, water has diminishing returns for a pair of reasons. One, the wetter the soil, the faster the water evaporates. Two, we can reach \(\text{FC}\). Next, they describe farming as a deterministic clock that peaks at the critical season. They also link rainfall with the price of water. Droughts increase the price of water. The computer model draws \((p_t, r_{t-1})\) from previous years in the current iteration of simulacra. Empirical bootstrapping. Productivity shocks are drawn for the computer simulacrum i.i.d. from a Gumbel distribution with CDF \(F(\varepsilon_{it}; \sigma_\varepsilon) = e^{-e^{-(\varepsilon_{it} / \sigma_\varepsilon)}}\). The variance is \(\frac{\sigma_{\varepsilon}^2 \pi^2}{6}\). They then use Nested Logit to model the fact that irrigating 0 units of \(j\) and irrigating \(>1\) units of \(j\) are the real choices, and not that buyig 0,1,2,3, and so on units of water with equal distinction. The nested tree separates between a branch that decides to buy water or not, and a branch that if buying, decides how many units to buy. The asusmption is made that \(\varepsilon\) affects the buying water choices \((1,2,\cdots,J)\) together. The parameter to control how correlated the group choices are is \(\lambda \in (0,1]\). Finally, the cash holdings cash balance is as follows: \[m_{it} = m_{i,t-1} - p_{t-1} j_{i,t-1} + \phi_{i0} + R_{it} + \nu_{it}\] with \(m_{i,t-1}\) being the starting balance, \(p_{t-1} j_{i,t-1}\) the cost of water, \(\phi_{i0}\) a baseline endowment, \(R_{it}\) apricot revenue, and \(\nu_{it}\) some random shock, an unexpected inheritance or burden and so on. Revenue \(R_{it}\) is only collected on week 24, so \[R_{it} = \begin{cases} 0 & \text{if } \omega_t \neq 24 \\ \text{Rev}_{it} & \text{if } \omega_t = 24 \end{cases}\] this means farmers need to make revenue last for a whole year. Revenue is given by the production that happened during the critical season (weeks 18-32): \[\text{Rev}_{it} = \sum_{\omega_t=18}^{32} h(j_{it}, M_{it}, \omega_t; \gamma)\] We can write the production function as \[h(j_{it}, M_{it}, \omega_t; \gamma) = \big[ \gamma \cdot (M_{it} - \text{PW}) \cdot \text{KS}(M_{it}) \cdot \text{Z}(\omega_t) \big]\] with \(\text{KS}(M_{it}\) being a hydric stress coefficient, and \( \text{Z}(\omega_t)\) dummies for the critical season. Thus we can write the revenue function as \[\text{Rev}_{it} = \sum_{\omega_t=18}^{32} \gamma \cdot (M_{it} - \text{PW}) \cdot \text{KS}(M_{it})\] The value function is given by \[\begin{split} V(&M_{it}, \omega_t, p_t, r_t, m_{it}, \varepsilon_{it}) \equiv \max_{j_{it} \in \{0,1,\dots,J\}} \Big\{ h(j_{it}, M_{it}, \omega_t; \gamma) \\ & - \zeta \mathbb{1}\{j_{it} > 0\} - p_t j_{it} + \varepsilon_{it} \\ &+ \beta \, \mathbb{E} \big[ V(M_{i,t+1}, \omega_{t+1}, p_{t+1}, r_{t+1}, m_{i,t+1}, \varepsilon_{i,t+1}) \\ & \mid M_{it}, \omega_t, p_t, r_t, m_{it}, \varepsilon_{it}, j_{it} \big] \Big\}, \\ & \text{s.t. } M_{it} \geqslant \text{PW}, \quad j_{it} p_t \leq m_{it}, \quad \forall j_{it} > 0 \end{split}\] which is the monstrous Bellman equation. They calculate the parameters \(\Theta \equiv (\gamma, \sigma_{\epsilon}, \zeta, \lambda)\) using two steps. First, they compute transition probability matrices for the following observable state variables: moisture, week, price, and rain., and then use transition matrices to forward simulate the value function. The simulacrum behaves like historical data. The authors exclude poor farmers to avoid estimating that they are choosing not to buy because the do not value the water, which would be a biased estimated demand elasticity.

They then do a welfare analysis. There are two ways of wasting water, due to heterogeneity (having water when they are unproductive), and due to decreasing marginal returns (DMR), recieving water when soil moisture is high. Then, they compare Quotas (allocate water units uniformly) with markets. This solves the DMR problem, no one has a super wet plot while the neighbor has a dry one, but it does not solve the heterogeneity problem. In a perfect world, markets are flawless, solve both problems: farmers only buy when soil is dry and they are javing a productive week. With liquidity constraints, markets fail and create both types of waste. From the text: quotas are more efficient than markets when there is large heterogeneity in wealth and small heterogeneity in productivity. Markets are more efficient in the opposite case. In the general case where there is heterogeneity in both wealth and productivity, the efficiency of markets relative to quotas is ambiguous.

They compare four different scenarios, a free market, naive quotas, smart quotas that randomizes water among the bottom X% of farmers (the post 1966 system), and a centrally planned scenario that they call highest valuation. The smart quota is better than the market and the naive quota, but worse than the high valuation. The authors also respond to possible counter-argumentation: running tests on if rich farmers are better than the poor, if poor farmers are risk averse, and so on. In reality, for winter crops there is no difference, high correlation between wealth and productivity does not change the ranking of allocations, and the data shows that even if the poor farmers are risk averse, that alone cannot explain the observed behavior, it HAS TO BE that there are liquidity constraints. They compare this scenario to Coase's ideas on vertical integration, and the quota system is perfectly rational.

The authors say: Markets might not be efficient when liquidity constraints are present. This is true. They also remind us that this change was not done for egalitarian purposes, it was a purely efficiency based measure. Thus, the usual planning (equality but on average people might be worse off) vs. markets (efficient but unequal) is not necessarily true, insofar as even within consequentialist ontology that through normative commitments derives markets as the best there is, there are still scenarios in which they do not achive this purpose, and even inside of this methodology we can encounter contradictions and counterargumentation. This paper was incredible. I am happy I read it.