Optimizing Adjustments to Transboundary Water Sharing Plans: A Multi-Basin Approach

Abstract

Afghanistan contributes water supplies to Iran, Pakistan, Tajikistan, Turkmenistan, and Uzbekistan. However, with the exception of the Helmand Basin, Afghanistan has negotiated transboundary water sharing agreements with no downstream country. This paper describes a constrained optimization framework to minimize economic costs within each of nine Afghan transboundary basins of adapting to potential water sharing agreements. Model results show impacts of water agreements on farm income and food security for each Afghan basin. Our results show that unrestricted trading reduces the economic costs of adapting to water sharing treaties by two to 6 % compared to the conventional water sharing system. A higher scale of reservoir storage capacity as well as market trading of water among regions moderates costs of water shortages, both with and without water agreements in place.

Appendix

Overview.

The appendix summarizes relevant mathematical documentation of the multi-basin hydro-economic model design to inform water policy decisions in Afghanistan. It outlines the mathematical framework of the optimization model built using General Algebraic Modeling Systems to include sets, parameters, variables, equations and the objective function specification. The complete model show hydrological, economic and institutional constraints. The code for the hydro-economic model and spreadsheet results are available at the New Mexico State University web page at https://water-research.nmsu.edu/

Sets.Sets characterize the foundation of the multi-basin framework. Each set constitute elements as described below. In our model, sets and corresponding elements are defined for all studied transboundary river basins in Afghanistan.

sets	set name	description	set elements
k	crop	Crop selection is an important choice variable for farm income, food security, and policy assessment that influences both.	/ Alfalfa, Cotton, Melon, Potato, Pulses, Rice, Tomato, Wheat/
t	Year	Year reflect a 20-year horizon	/year1 * year 20/
i	Region by basin	Each basin divided into 3 regions	/upper, middle, lower/
j	Water right priority	Priority among regions inside each basin	/j1, j2, j3/
r	Water allocation rule	Water right systems for sharing water shortages when they occur	/Upstream priority, free market/
s	Storage reservoir capacity scale	Scale at which new reservoirs could be built. None reflects base condition and has no storage.	/none, small, medium, large/
b	River Basin	Lists of all river basins studied in the country	/Balkh, Patika, Helmand, Farah, Kabul, Kokcha Kunduz, Murghab, Upper_Harirud
y	Treaty	Treaty agreement on transboundary river basins.	/Without treaty, With treaty/
rji(r,j,i)	Mapping set	Assigns priorities to regions in a basin to vary water right regime	us_priority.j1.upper,us_priority.j2.middle,us_priority.j3.lower free_mkt.j1.upper,free_mkt.j1.middle,free_mkt.j1.lower
rwa(r)	Upstream priority	Upstream users receive priority when shortages occur	upstream priority
rfm(r)	free_market	No restrictions on water reallocations from baseline	free market

Parameters.

Terms ending in _p refer to parameters, data read by the model. They are:

Parameter	sets	Label	units
Nat_supp_p	(b)	Natural runoff inflow to basin	MCM / year
Nat_suppl_p	(b,t)	Stochastic water supply by year with set mean and standard deviation	MCM / Year
Nat_supply_p	(b,t,y)	Natural water supply adjusted by downstream delivery treaty requirement	MCM / Year
Pop_p	(b,t)	Forecast population by basin and years	thousands
Popratio_p	(b,t)	Ratio of given year population to base year	1 and above
Rho_p		Discount rate (1%)	unitless
Capacity_p	(b,s)	Maximum reservoir storage capacity	MCM
Com_cost_mcm	(b,s)	Construction cost per million cubic meters storage capacity	$US/MCM
Prop_right_p	(b,i)	Proportion of right to water compared by region within basin	0–1
Right_p	(b,i)	Customary absolute right to water supply by region	MCM / year
Shortage_p	(b,r,i,t,s,y)	Water storage relative to full supply right	0–1
Basin_use_p	(b,r, t,s,y)	Total water use in the basin	MCM / year
Change_store_p	(b,r, t,s,y)	Total change in storage	MCM / year
Bc_p	(b,i,k)	Water use per unit land (irrigation depth)	M / year
Land_p	(b,i,k)	Observed land in production	Ha / year
Price_p	(k,b)	Observed crop prices	$US per metric ton
Price_elast_p	(k,b)	Price elasticity of demand	unitless
Yield_p	(b,i,k)	Observed crop yield	Metric tons/ha
Cost_p	(b,i,k)	Production costs	$US per ha
A0_p	(b,k,t)	Intercept in price-dependent demand function	$US per metric ton
A1_p	(b,k,t)	Slope of price-dependent demand function	$US per metric ton
B0_p	(b,k,t)	Intercept in crop water production function (PMP)	Metric tons/ha
B1_p	(b,k,t)	Slope in crop water production function (PMP)	Metric tons/ha

Variables.

As characterized in the multi-basin model, variables are endogenous (unknown) and end in _v to distinguish the unknown data from parameters. These variables are optimized by the model, while respecting all important bounds. The most important variables are:

Variables	sets	label	units
Hectares_v	(b,r,i,k,t,s,y)	Land use	1000 ha
T_hectares_v	(b,r,i, t,s,y)	Total land use	1000 ha
Uses_v	(b,r,i, t,s,y)	Water use	MCM/year
Sum_uses_v	(b,r, t,s,y)	Summed water use	MCM/year
Production_v	(b,r,i,k,t,s,y)	Land in production details	1000 ha
Yield_v	(b,r,i,k,t,s,y)	Crop yields	Tons/ha
Crop_price_v	(b,r, k,t,s,y)	Crop price	$US/ton
Netrev_v	(b,r,i,k,t,s,y)	Net revenue per unit land	$US/ha
Con_surp_v	(b,r, k,t,s,y)	Consumer surplus (willingness to pay in excess of price)	$US 1000 / year
Tgrossrev_v	(b,r,i,k,t,s,y)	Total gross revenue over all lands	$US 1000 / year
Ag_ben_j_v	(b,r,i,k,t,s,y)	Discounted ag benefits with details	$US 1000 / year
Tot_wel_v	(b,r, t,s,y)	Discounted total welfare	$US 1000 / year
Wel_wo_cd_v	(b,r, s,y)	Discounted total welfare without reservoir capacity expansion	$US 1000 / year
Ttot_wel_v	(b,r, s,y)	Discounted total welfare with reservoir capacity expansion	$US 1000 / year

Equations

Mathematical expressions connecting the relationship between variables and parameters used in the optimization frame to characterize the most important indicators such as hydrology, land in production, crop production, and economic policy choices.

Hydrology Block

$$ \mathrm{X}\left(\mathrm{t}\right)\sim \mathrm{N}\left(\upmu, {\upsigma}^2\right). $$

(1)

Basin headwater supply for the year t, X(t)

$$ \mathrm{X}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)\kern0.5em =\kern1em \mathrm{Bc}\_\mathrm{p}{\left(\mathrm{b},\mathrm{i},\mathrm{k}\right)}^{\ast }\ \mathrm{hectares}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right) $$

(2)

Crop water use equation calculated based on land in production

$$ \mathrm{Uses}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{t},\mathrm{s},\mathrm{y}\right)\kern0.5em =\mathrm{sum}\left(\mathrm{k},\mathrm{X}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)\right) $$

(3)

Water use summed over crops

$$ \mathrm{sum}\_\mathrm{uses}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{t},\mathrm{s},\mathrm{y}\right)=\mathrm{sum}\left(\mathrm{i},\mathrm{Uses}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{t},\mathrm{s},\mathrm{y}\right)\right) $$

(4)

It calculates water use summed over canals

Agronomy Block

$$ \mathrm{T}\_\mathrm{hectares}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{t},\mathrm{s},\mathrm{y}\right)=\mathrm{sum}\left(\mathrm{k},\mathrm{hectares}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)\right) $$

(5)

Land in production summed over crops

$$ \mathrm{Yield}\_\mathrm{v}\ \left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)=\mathrm{B}0\_\mathrm{p}\left(\mathrm{b},\mathrm{i},\mathrm{k}\right)+\kern0.5em \mathrm{B}1\_\mathrm{p}{\left(\mathrm{b},\mathrm{i},\mathrm{k}\right)}^{\ast }\ \mathrm{hectares}\_\mathrm{v}\ \left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right) $$

(6)

Crop yields declines as acreage in production increases

$$ \mathrm{Production}\_\mathrm{v}\ \left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)=\mathrm{Yield}\_\mathrm{v}\ {\left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)}^{\ast }\ \mathrm{hectares}\_\mathrm{v}\ \left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right) $$

(7)

It calculates crop production based on crop yield and acreage in production.

$$ \mathrm{Tot}\_\mathrm{prod}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)=\mathrm{sum}\left(\mathrm{i},\mathrm{production}\_\mathrm{v}\ \left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)\right) $$

(8)

Total crop production summed over regions.

Economics Block

$$ \mathrm{crop}\_\mathrm{p}\mathrm{rice}\_\mathrm{v}\ \left(\mathrm{b},\mathrm{r},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)=\mathrm{BB}0\_\mathrm{p}\ \left(\mathrm{b},\mathrm{k},\mathrm{t}\right)+\left[\mathrm{BB}1\_\mathrm{p}{\left(\mathrm{b},\mathrm{k},\mathrm{t}\right)}^{\ast }\ \mathrm{Tot}\_\mathrm{p}\mathrm{rod}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)\right] $$

(9)

Linear demand function of crop price for each basin as influence by water right systems, region, time, storage capacity and treaty

$$ \mathrm{con}\_\mathrm{surp}\_\mathrm{v}\ \left(\mathrm{b},\mathrm{r},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)={0.5}^{\ast }\ \left[{\left[\mathrm{BB}0\_\mathrm{p}\left(\mathrm{b}\mathrm{b},\mathrm{k},\mathrm{t}\right)-\mathrm{crop}\_\mathrm{p}\mathrm{rice}\_\mathrm{v}\ \left(\mathrm{b},\mathrm{r},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)\right]}^{\ast }\ \mathrm{Tot}\_\mathrm{p}\mathrm{rod}\_\mathrm{v}\ \left(\mathrm{b},\mathrm{r},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)\right] $$

(10)

Consumer surplus representing gains in economic benefits to the Afghan people as water right systems, treaty agreement and storage capacities are varied.

$$ \mathrm{Netrev}\_\mathrm{v}\ \left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)=\mathrm{crop}\_\mathrm{p}\mathrm{rice}\_\mathrm{v}{\left(\mathrm{b},\mathrm{r},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)}^{\ast }\ \mathrm{Yield}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)-\mathrm{Cost}\_\mathrm{p}\left(\mathrm{b},\mathrm{i},\mathrm{k}\right) $$

(11)

Net revenue per unit land is price times yield minus costs per unit land

$$ \mathrm{Ag}\_\mathrm{Ben}\_\mathrm{j}\_\mathrm{v}\ \left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)=\mathrm{Netrev}\_\mathrm{v}{\left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)}^{\ast }\ \mathrm{hectares}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right) $$

(12)

Agricultural benefits by river basin, water right systems, agricultural region, crop, time period, storage capacity and treaty.

$$ \mathrm{To}\_\mathrm{wel}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)=\mathrm{sum}\ \left[\mathrm{i},\mathrm{ag}\_\mathrm{ben}\_\mathrm{j}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)\right]+\mathrm{con}\_\mathrm{surp}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right) $$

(13)

Discounted net present value of total economic welfare

$$ \mathrm{Tot}\_\mathrm{wel}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{t},\mathrm{s},\mathrm{y}\right)=\mathrm{sum}\ \left(\mathrm{k},\mathrm{To}\_\mathrm{wel}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)\right) $$

(14)

Summed total welfare over crops

$$ \mathrm{Wel}\_\mathrm{wo}\_\mathrm{cd}\_\mathrm{v}\ \left(\mathrm{b},\mathrm{r},\mathrm{s},\mathrm{y}\right)=\mathrm{sum}\ \left(\mathrm{t},\mathrm{DF}{\left(\mathrm{t}\right)}^{\ast }\ \mathrm{Tot}\_\mathrm{wel}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{t},\mathrm{s},\mathrm{y}\right)\right) $$

(15)

Net present value of farm income plus consumer surplus

$$ \mathrm{Ttot}\_\mathrm{wel}\_\mathrm{v}\ \left(\mathrm{b},\mathrm{r},\mathrm{s},\mathrm{y}\right)=\mathrm{Wel}\_\mathrm{wo}\_\mathrm{cd}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{s},\mathrm{y}\right)-\mathrm{Tot}\_\mathrm{COM}\_\mathrm{cost}\_\mathrm{p}\left(\mathrm{b},\mathrm{s}\right) $$

(16)

Total net benefits after subtracting dam construction cost

$$ \mathrm{Tot}\_\mathrm{b}\_\mathrm{v}=\mathrm{sum}\ \left(\left(\mathrm{b},\mathrm{r},\mathrm{s},\mathrm{y}\right),\mathrm{Ttot}\_\mathrm{wel}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{s},\mathrm{y}\right)\right) $$

(17)

Total benefits looped over all changeable indices

$$ \mathrm{shad}\_\mathrm{p}\mathrm{rice}\_\mathrm{p}\left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{t},\mathrm{s},\mathrm{y}\right)=\mathrm{uses}\_\mathrm{v}.\mathrm{m}\ \left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{t},\mathrm{s},\mathrm{y}\right) $$

(18)

Shadow prices for all water allocation rules except unrestricted trading

$$ \mathrm{shad}\_\mathrm{p}\mathrm{rice}\_\mathrm{p}\left(\mathrm{b},\mathrm{rfm},\mathrm{i},\mathrm{t},\mathrm{s},\mathrm{y}\right)=\mathrm{sum}\_\mathrm{uses}\_\mathrm{v}.\mathrm{m}\left(\mathrm{b},\mathrm{rfm},\mathrm{t},\mathrm{s},\mathrm{y}\right) $$

(19)

Shadow prices for unrestricted trading

$$ \mathrm{Storage}\_\mathrm{Capacity}\_\mathrm{p}=\mathrm{scale}\_{\mathrm{p}}^{\ast }\ \mathrm{natural}\_\mathrm{supply}\_\mathrm{p}. $$

(20)

Water storage capacity is defined by natural supply at various values of the scale parameter (scale_p)

$$ \mathrm{Tot}\_\mathrm{Com}\_\mathrm{cost}\_\mathrm{p}\left(\mathrm{b},\mathrm{s}\right)=\mathrm{Cost}\_\mathrm{Per}\ \mathrm{Unit}\ \mathrm{Storage}\_{\mathrm{p}}^{\ast }\ \mathrm{Storage}\_\mathrm{Capacity}\_\mathrm{p}. $$

(21)

$$ \mathrm{Ttot}\_\mathrm{wel}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{s},\mathrm{y}\right)=\mathrm{Wel}\_\mathrm{wo}\_\mathrm{cd}\_\mathrm{v}\left(\mathrm{b},\mathrm{r},\mathrm{s},\mathrm{y}\right)\hbox{--} \mathrm{Tot}\_\mathrm{Com}\_\mathrm{cost}\_\mathrm{p}\left(\mathrm{b},\mathrm{s}\right). $$

(22)

Total net benefits obtained after dam cost.

Bounds.

Bounds characterize constraints on institutions, technology, and water user behavior, while also producing realistic responses to future climate or policy adaptations to climate. A lower bound (.lo) on each water flows and land used assures no crop production or water supplies have negative values. An upper bound (.up) guards against a variable exceeding that set level.

$$ \mathrm{hectares}\_\mathrm{v}.\mathrm{lo}\left(\mathrm{b},\mathrm{r},\mathrm{i},\mathrm{k},\mathrm{t},\mathrm{s},\mathrm{y}\right)=0. $$

(23)

Agricultural land in production cannot be negative.

$$ \mathrm{uses}\_\mathrm{v}.\mathrm{up}\ \left(\mathrm{b},\mathrm{rwa},\mathrm{i},\mathrm{t},\mathrm{s},{}^{\hbox{'}}\mathrm{Without}\_{\mathrm{treaty}}^{'}\right)={1.0}^{\ast }\ \mathrm{wet}\_\mathrm{wat}\_\mathrm{use}\left(\mathrm{b},\mathrm{rwa},\mathrm{i},\mathrm{t},\mathrm{s},{}^{\hbox{'}}\mathrm{Without}\_{\mathrm{treaty}}^{'}\right) $$

(24)

uses_v . up (b, rwa, i, t, s, ^'With_treaty^’) = 0.8^* wet_wat_use(b, rwa, i, t, s, ^'With_treaty^’) (25)

Bounds water use in production, with use in the face of a treaty set to 80% of without treaty level

$$ \mathrm{sum}\_\mathrm{use}\mathrm{s}\_\mathrm{v}.\mathrm{up}\left(\mathrm{b},\mathrm{rfm},\mathrm{t},\mathrm{s},{}^{\hbox{'}}\mathrm{Without}\_{\mathrm{treaty}}^{'}\right)={1.0}^{\ast }\ \mathrm{basin}\_\mathrm{use}\_\mathrm{p}\left(\mathrm{b},\mathrm{rfm},\mathrm{t},\mathrm{s},{}^{\hbox{'}}\mathrm{Without}\_{\mathrm{treaty}}^{'}\right) $$

(26)

$$ \mathrm{sum}\_\mathrm{use}\mathrm{s}\_\mathrm{v}.\mathrm{up}\left(\mathrm{b},\mathrm{rfm},\mathrm{t},\mathrm{s},{}^{\hbox{'}}\mathrm{With}\_{\mathrm{treaty}}^{'}\right)={0.8}^{\ast }\ \mathrm{basin}\_\mathrm{use}\_\mathrm{p}\left(\mathrm{b},\mathrm{rfm},\mathrm{t},\mathrm{s},{}^{\hbox{'}}\mathrm{With}\_{\mathrm{treaty}}^{'}\right) $$

(27)

When water trading is allowed among regions in each basin, only total water use is bounded.

Objective.

The objective of the model is to maximize discounted net present value subject to the hydrologic and institutional constraints described above.

The objective function Ttot_wel_v(b,r,s,y) specified above is maximized for each basin, water shortage sharing rule, treaty agreement and storage capacity level. A total of 144 model runs are optimized.