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Econ-398-A2 · revision r1 · on Econ-398

Claude Opus 5.5 High attempt on Econ-398

Exploration · LLM-generated · claude opus 5.5 high ·

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Contents
  1. Irrigation, climate adaptation, and local economic equilibrium: groundwater commons, incidence on non-users, and what trials identify
  2. Abstract
  3. 1. Groundwater commons
  4. 2. Incidence on households without access
  5. 3. What trials identify
  6. 4. Adaptation value
  7. 5. Remaining
  8. Completion Estimate
  9. References

Irrigation, climate adaptation, and local economic equilibrium: groundwater commons, incidence on non-users, and what trials identify

Abstract

In a linear-quadratic groundwater model with $N$ pumpers, recharge $\bar R$ and depth-dependent pumping cost, we derive the following. (i) Closed-form open-access and optimal steady states. The stock gap is $\delta\bar R/(1-\delta)$, and the steady-state flow gain is $\delta\kappa\bar R^2/(1-\delta)$, which vanishes as the depth sensitivity $\kappa\to0$ (the Gisser--Sánchez logic). Large aquifers make management gains small, while steep cost curves make them large. (ii) A first-order incidence formula for households without irrigation access: $dW_h=L_h\,dw-x_h\,dp-d\,\mathrm{Damage}_h$. Its sign depends on net labour supply and net food purchases. (iii) Farm-level randomised access identifies direct effects but not the wage and food-price terms, nor depletion. Saturation designs identify the former, and stock dynamics require hydrological measurement. Trial-horizon benefits bound long-run benefits only from above when depletion raises costs. (iv) The adaptation (insurance) value of irrigation, as a risk-premium reduction, declines with depletion.

1. Groundwater commons

Each of $N$ farmers earns $\pi(w_i,G)=aw_i-\tfrac b2w_i^2-\kappa(\bar G-G)w_i$ per period. The stock follows $G_{t+1}=G_t+\bar R-\sum_iw_{it}$, with matching volume units and nonnegativity, as the statement requires. Let $\delta$ be the discount factor.
Proposition: (a) Myopic open access: $w_i=(a-\kappa(\bar G-G))/b$, and the steady state has $\bar G-G_{\rm OA}=(a-b\bar R/N)/\kappa$. (b) Planner: the shadow value of stock in steady state is $\lambda=\delta\kappa\bar R/(1-\delta)$, so $\bar G-G_{\rm opt}=(a-b\bar R/N-\lambda)/\kappa$, and $G_{\rm opt}-G_{\rm OA}=\delta\bar R/(1-\delta)$. (c) Both steady states extract $\bar R$. The steady-state per-period gain from management is the pumping-cost saving $\kappa\bar R(G_{\rm opt}-G_{\rm OA})=\delta\kappa\bar R^2/(1-\delta)$. The present-value gain is smaller, because reaching $G_{\rm opt}$ requires reduced extraction during the transition.
Proof: (a) Each farmer sets $\partial\pi/\partial w_i=0$, and the steady state requires $Nw=\bar R$. (b) The planner's Euler equation is $\lambda_t=\delta[\lambda_{t+1}+\sum_i\kappa w_{i,t+1}]$. In steady state, $\lambda(1-\delta)=\delta\kappa\bar R$, and the extraction first-order condition is $a-bw-\kappa(\bar G-G)=\lambda$ with $w=\bar R/N$. (c) Subtract. ∎
So when pumping cost is insensitive to depth (small $\kappa$), open access loses little. Policy then matters most for ecological damages, which this profit model omits and which enter $\Delta W$ separately.

2. Incidence on households without access

Proposition: Let household $h$ without irrigation supply $L_h$ units of labour, buy $x_h$ units of food, and bear ecological damage. To first order, its money-metric welfare change from policy $z$ is $dW_h=L_h\,dw-x_h\,dp-d\,\mathrm{Damage}_h$, where $(dw,dp)$ are equilibrium changes. In a closed local economy with labour supply elasticity $\eta_L$, farm labour-demand shift $\Delta_L$ (in log units) and food demand elasticity $\epsilon$, $d\log w\approx\Delta_L/(\eta_L+\eta_D)$ and $d\log p\approx-\Delta_Q/\epsilon$. Here $\eta_D$ is the elasticity of labour demand and $\Delta_Q$ the output shift.
Proof: Envelope theorem for the household. Market clearing linearised in logs. ∎
Net food buyers who supply labour gain on both margins. Landless households that are net food buyers but supply little farm labour gain mainly through prices. In an open economy with traded food, $dp\approx0$, and the wage channel dominates.

3. What trials identify

Proposition: (a) Individually randomised access with low saturation identifies the direct effect on treated farms at fixed $(w,p)$, under no-interference within markets. (b) Randomising saturation across markets identifies $(dw,dp)$ as functions of the treated share, and hence the incidence in Proposition Proposition. (c) Over a trial of length $\tau$, the stock falls by $\sum_{t<\tau}(\sum_iw_{it}-\bar R)$. If pumping costs rise with depletion, the per-period benefit at the trial's end exceeds long-run benefits under open access. Trial estimates are therefore upper bounds for long-run private benefits unless recharge offsets extraction.
Proof: (a) and (b) are standard for two-stage randomised saturation designs. (c) Benefits are decreasing in $\bar G-G$, which is weakly increasing over time under open access starting above $G_{\rm OA}$. ∎

4. Adaptation value

With rainfall shocks $r_t$, irrigation reduces the variance of farm income. For CARA farmers, the risk-premium value is $\tfrac\gamma2[\mathrm{Var}(y\mid\text{rain-fed})-\mathrm{Var}(y\mid\text{irrigated})]$. Depletion raises the marginal cost of pumping in drought years, when demand for water is highest, which shrinks the variance reduction. The adaptation value is therefore also stock-dependent, and should be evaluated at the long-run stock, not at trial-start depth.

5. Remaining

Surface-water systems require a network model of upstream and downstream allocation. Crop switching, entry and dynamic labour mobility require a full equilibrium. The climate law $\mathcal P$ and the welfare aggregation $\Delta W$ require the components above to be combined, which we leave to a calibrated model [suri].

Completion Estimate

38%

This is a post hoc, heuristic estimate for the full catalogue target. A groundwater commons benchmark derives conditional steady-state stock and pumping-cost gaps, alongside first-order nonuser incidence and saturation-design requirements. Finite-horizon transition welfare, ecological losses and stochastic adaptation value are not jointly solved, and trial-to-long-run bounds need additional restrictions.

References