Shadow Price of Water Under Varying Risk Behavior
Sankalp Sharma (University of Nebraska β Lincoln)
Tajamul Haque (Council for Social Development/LANDESA)
Rifat Mansur (North Carolina State University)
Anil Giri (University of Central Missouri)
Presentation prepared for IWREC, 2016 1
Presentation prepared for IWREC, 2016 2
What is the problem?
β’ Acute groundwater shortage in India being worsened by disproportionate
irrigation use.
β’ Water rights tied to land acquisition.
β’ Owner of a land parcel can freely access groundwater.
β’ Rice, the second-most prominent crop is inefficiently produced in several states.
β’ For example, Chhattisgarh, Odisha, Haryana, U.P and Punjab use 35% or more
water than West Bengal.
β’ Punjab is the most inefficient at 51.2 percent. 5337 litres/ kg rice production.
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Causes of inefficiency
β’ More irrigation increases land productivity.
β’ Punjab produces 58.5 qtl/ha, whereas West Bengal produces 41.2 qtl/ha.
β’ Power subsidies provided to farmers in most states.
β’ Incentivizes them to pump more groundwater.
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Government scheme
β’ Water ceiling proposed for non-domestic uses.
β’ If a producer uses less water, then cash-subsidy is provided.
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Better solution required
β’ Cash incentive scheme creates burden on tax-payers.
β’ Water ceiling restricts a producerβs optimal choices.
β’ They may require more water.
β’ The government scheme will be better supported by a water market.
β’ Producers must be given choice of buying/selling water from the government
and/or neighboring producers.
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Question & Objectives
βIs there an amount that producers are willing to pay above the
government mandated cap for irrigation, under varying producer
risk aversion?β
Objectives:
β’ We determine the willingness-to-pay for water in the agricultural context
(India).
β’ Our model incorporates producer behavior under risk aversion and yield
uncertainty.
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Theoretical model
Let π be the profit function of a producer.
π =
π=1
π
πππ¦π π₯π , πΌ β π€πΌ β
π=1
πΎ
πππ₯π
Where,
ππ is the output price, π¦π is the uncertain yield, π is the number of crops
πΌ is irrigation use, π€ is the energy price required to pump water.
π₯π are the non-irrigation inputs (fertilizer, chemical, etc.), πΎ is the total number of
inputs, ππ are the input prices.
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Theoretical model: continued
Re-writing the profit function as:
π = π β π€πΌ β
π=1
πΎ
πππ₯π
Where,
π = π=1π π¦π, is the random aggregate yield per-acre.
π = π π + β π π, is the Just-Pope production function.
π π = π΄Ξ π=1πΎ π₯ππ½ππΌπ½πΌ, h π = π΅Ξ π=1
πΎ π₯ππ½πβ²
πΌπ½πΌβ²π (both π and β are Cobb-Douglas)
π βΌ π΅ π 1, π 2 , where π 1 and π 2 are the shape parameters.
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Producers are Expected Utility maximizers
Two types of producers: Risk Neutral and Risk Averse
Let each producer have a concave utility function π(π).
Standard assumptions apply: πβ² π β₯ 0 and πβ²β² π β€ 0
Functional form, if producer is risk neutral:
π π = πΌπ
Functional form, if producer is risk averse:
π π = βexp(βππ)
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Optimization problem
Producerβs optimization problem can be written as:
πππ₯πΌ, π₯ππ(π)
Subject to,
πΌ β€ πΌWhere, πΌ is the government mandated cap on irrigation.
Lagrangian under risk neutrality:
πΏ π₯π , πΌ, π = β« πΌπ(π, π, π½, π½β² )πΎ π β π€πΌ β
π=1
πΎ
πππ₯π β π πΌ β πΌ
Lagrangian under risk aversion:
πΏ π₯π , πΌ, π = β« βexp(βππ(π, π, π½, π½β² )πΎ π ) β π€πΌ β
π=1
πΎ
πππ₯π β π πΌ β πΌ
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Solutions
Optimal π for risk neutrality comes from the first-order condition w.r.t. πΌ.
πβ = π΄π½πΌπ₯1π½1π₯2π½2πΌπ½πΌβ1 + π΅
12
π½πΌβ²
2π₯1
π½1β²
2 π₯2
π½2β²
2 πΌπ½πΌβ12
π 1 + π 2βπ€
For a producer to have positive willingness-to-pay for irrigation above the
endowment, we need:
1. πΌβ = πΌ (from the Kuhn tucker conditions: if πΌβ < πΌ then π = 0, has to be true.
2. π΄π½πΌπ₯1π½1π₯2π½2πΌπ½πΌβ1 + π΅
1
2
π½πΌβ²
2π₯1
π½1β²
2 π₯2
π½2β²
2 πΌπ½πΌβ12
π 1+π 2> π€
Optimal π for risk aversion, follows similarly from the second Lagrangian
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Simulation Analysis
We require the following parameters to determine π for both risk neutrality and
aversion.
1. Production parameters: π΄, π½π , π½πΌ (mean elasticities) and π΅, π½πβ², π½πΌβ² (risk
elasticities)
2. Optimal inputs: π₯πβ (π = 1,2) and πΌβ = πΌ .
3. Mean (π) and variance (π) of yield.
4. Shape parameters of the yield distribution: π 1 and π 2.
5. Cost of energy: π€
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Simulation Analysis: Input use
β’ We use Nitrogen (N) and Phosphorous Pentoxide (π2π5) as our two non-
irrigation inputs.
β’ According to the World Bank database, Indian producers used 150 kg/ha
fertilizers in 2015.
β’ Endowment limit for πΌ = 2616 litres/kg, stress level πΌ = 2000 litres /kg.
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Simulation Analysis: cost of energy
β’ Indian producers pay 13% of the total electricity cost of pumping due to the high
subsidy rates.
β’ Energy required to lift a feet of groundwater: weight of water Γ feet of lift.
β’ Lifting 1233420 ft-kgs of water approximately requires 1.02 kWh of power.
β’ Producer requires 8.26-e7 kWh of energy to pump an additional liter of water.
β’ Cost of electricity for non-domestic use in Punjab is stated as Rs. 6.75 per kWh.
β’ Marginal cost of groundwater is 8.26e-7 Γ 6.75 = 5.57-e6 per-kg (liter).
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Simulation Analysis: Results
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Simulation Analysis: Results
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Summary & Further Work
Further work:
β’ Precipitation deficit unaccounted for in current paper.
β’ Use real-world data to compute elasticities.
β’ Incorporate water table values in the optimization problem.
β’ If yieldβs response to irrigation is low, then producer willingness to pay for
water is 0.
β’ If risk elasticity of irrigation increases, producerβs are willing to pay a higher
price.
β’ Under stress irrigation levels, producers are clearly willing to pay more.
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Thank You!
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Questions