Water productivity: Not a helpful indicator of farm-level optimization

November 11th, 2014

Dr. Dennis Wichelns, Bloomington, Indiana, US

Many authors in recent years have suggested that increases in water productivity are needed to ensure food security in 2050 and beyond.1,2,3,4 Some authors call for increasing the ‘crop per drop’ or ‘value per drop,’ generated with water in agriculture in rainfed and irrigated settings.5,6 Those phrases are essentially analogous to the notion of increasing water productivity, which often is defined as the ratio of some measure of output (either crop mass or value) to some measure of water input (either water applied or transpired).

At first glance, the call to increase water productivity seems appropriate and compelling, given the obvious need to increase crop production to meet increasing food demands. Yet, water productivity, as defined in the literature, is simply a measure of total output or total value, divided by the amount of a single input used in production. The resulting ratio describes the average amount of output or value associated with the water applied or consumed. The ratio does not describe the incremental productivity of water, and it does not account for the contributions of other inputs in crop production. For these reasons, water productivity cannot serve as an indicator of economic efficiency, which requires consideration of incremental gains and costs, including opportunity costs.

My goal in this note is to demonstrate that farmers in most settings would not choose to maximize water productivity, as that term has been defined in the literature. Such a strategy would result in less output and smaller revenues, while placing smallholders at greater risk of food insecurity. I depict this perspective by examining four plausible water production functions. Two of these are defined in terms of water applied, while two are defined in terms of evapotranspiration. In three of the four cases, a strategy of maximizing water productivity would not result in maximizing crop yield or net revenue. In the fourth case, the point of maximum water productivity cannot be identified, such that an effort to maximize the value of water productivity would not be helpful in determining the optimal level of evapotranspiration.

Conceptual framework

Water productivity is defined most often as the average amount of output per unit of water applied on a field (Equation 1) or per unit of water evapotranspired (Equation 2).

Equation 1: WP AW (kg per m3) = Output (kg per ha) / Water Applied (m3 per ha)

Equation 2: WP ET (kg per mm) = Output (kg per ha) / Water Evapotranspired (mm)

One can assess the ordinal measure of water productivity when viewing a crop water production function, by considering the slope of a ray from the origin through any point on the function. For example, suppose a farmer applying 6,000 m3 per ha of irrigation water obtains a grain yield of 7,500 kg per ha, as depicted in Figure 1. The water productivity at Point A, which describes that combination of applied water and grain yield is equivalent to the slope of a ray from the origin, through that point. In this example, the water productivity is 1.25 kg per m3 of water applied.

All points to the right of Point A will be characterized by smaller values of water productivity, as the slope of a ray from the origin through those points will be smaller. Similarly, all points on the production function to the left of Point A will be characterized by larger values of water productivity. Indeed, for any quadratic crop water production function with a positive vertical intercept, the value of water productivity will decline monotonically as the volume of applied water increases. Given this characteristic, the strategy of maximizing water productivity would require the farmer to apply as little water as possible.

The quadratic crop production function depicted in Figure 2 has a negative vertical intercept. For such a production function, the value of water productivity will increase initially, with larger volumes of applied water, reach a maximum value, and then diminish for all larger volumes of applied water. The volume that maximizes water productivity will be found where a ray from the origin is just tangent to the crop production function. That unique point might maximize water productivity, but in most irrigation settings, it will not maximize net revenue, nor achieve any other farm-level optimizing goal, such as maximizing crop production for household consumption or minimizing the risk of production alternatives.

Rays from the origin also are helpful in characterizing the water productivity status of linear relationships describing crop yield as a function of ET. For example, if the yield:ET relationship has a positive vertical intercept, water productivity will decline throughout the relevant range of ET values (Figure 3). Given such a relationship, a strategy of maximizing water productivity would require one to minimize ET. Yet, this would likely not be an optimal strategy, as it would also result in minimizing crop yield.

If the yield:ET relationship has a negative vertical intercept, water productivity will increase throughout the relevant range of ET values. In that case, a strategy of maximizing water productivity would not be helpful in determining the optimal amount of ET, as the calculated values of water productivity would continue to increase with higher levels of ET, without an upper bound.

Empirical examples

Many empirical examples of crop water production functions appear in the literature. Any of these can be used to demonstrate the implications of considering a strategy of maximizing water productivity at the farm level. I choose two empirical relationships describing yield as a function of water applied, and two describing yield as a function of ET. The four examples correspond to the four cases described above, in which the vertical axis intercepts are either positive or negative.

Quadratic crop water production functions

Zhang and Oweis (1999)7 present empirical estimates of crop water production functions for durum wheat produced with supplemental irrigation in Syria. The estimated function for the sum of applied water (rainfall plus irrigation) is:

Equation 3: Yield = -5.8556 + 0.0329 (R + IR) – 0.00002164 (R + IR)2 (R2 = 0.87)

where yield is expressed in tons per ha, and the sum of rainfall and irrigation is expressed in mm. Given the negative vertical axis intercept, water productivity increases through the initial range of applied water values, reaches a maximum, and then declines, as depicted conceptually in Figure 2. The unique point at which water productivity is maximized likely is unrelated to the amounts of rainfall and irrigation that would maximize net revenue.

If one assumes a seasonal rainfall of 250 mm, and inserts this amount into Equation 3, one obtains the following crop water production function for supplemental irrigation:

Equation 4: Yield = 1.0169 + 0.02208 (IR) – 0.00002164 (IR)2

where IR represents supplemental irrigation, as rainfall already is embedded the function.

This function has a positive vertical axis intercept, such that it resembles the conceptual model presented in Figure 1, in which water productivity declines with increasing amounts of supplemental irrigation, throughout the full range of irrigation values. Thus, a criterion of maximizing water productivity would not be helpful in determining the optimal amount of supplemental irrigation.

Zhang and Oweis (1999)7 determine the profit maximizing amount of supplemental irrigation, by considering the estimated input costs, the incremental productivity of water along the production function, and the expected price of durum wheat. Given a seasonal rainfall of 250 mm, the profit maximizing amount of supplemental irrigation is 454 mm. That optimizing strategy could not have been determined through an effort to maximize water productivity.

Linear crop water relationships

Karam et al. (2009)8 examine durum wheat production in the Bekaa Valley of Lebanon, where they conduct experiment station trials involving irrigation and fertilizer treatments. Using data from three years of results, they estimate linear yield:ET relationships for the Waha and Haurani cultivars (Figure 4). The authors note correctly that for each 10 mm increase in ET, the grain yield of Waha increases by 50 kg, while the grain yield of Haurani increases by 38 kg. Indeed, the incremental increase in grain, per unit of ET, is constant along a linear yield:ET relationship. Yet, the calculated value of water productivity is different at every point. In the estimated yield:ET relationships for the Waha and Haurani cultivars, water productivity becomes smaller as ET increases, given that the vertical axis intercepts of the estimated relationships are positive.

Igbadun et al. (2009)9 estimate crop water relationships for maize in the Mkoji sub-catchment of the Great Ruaha River basin of Tanzania. Using data collected during field experiments in 2005, they estimate linear functions pertaining to both seasonal ET and seasonal water applied (Figure 5). The incremental gain in maize yield per unit of ET (15.4 kg per mm) is about twice as large as the incremental gain per unit of water applied (7.4 kg per mm), as might be expected. Yet, the calculated values of water productivity become larger as either ET or water applied increases, throughout the relevant range depicted in Figure 5. Thus, water productivity would not be a helpful indicator of the optimal amount of ET or water applied.

Summing up

Estimates of water productivity provide limited insight regarding farm-level crop production in rainfed and irrigated settings. Higher estimates of water productivity are not necessarily associated with higher yields or larger amounts of production for sale or home consumption. There is no economic rationale for maximizing water productivity.10 Ratios of total output to the amount of a single input used in production depict the average productivity of that input and, thus, are not helpful indicators of economic efficiency. Farmers wishing to maximize net revenue or the amount of output they generate for household consumption would not choose the irrigation depths that maximize water productivity. In some cases, implementing such a criterion would result in minimal irrigation and very little crop production.

References:

Molden, D., Oweis, T., Steduto, P., Bindraban, P., Hanjra, M.A., Kijne, J. 2010. Improving agricultural water productivity: Between optimism and caution. Agricultural Water Management 97(4): 528-535. Cai, X., Molden, D., Mainuddin, M., Sharma, B., Ahmad, M.-U.-D., Karimi, P. 2011. Producing more food with less water in a changing world: Assessment of water productivity in 10 major river basins. Water International 36(1): 42-62. Spiertz, H. 2012. Avenues to meet food security: The role of agronomy on solving complexity in food production and resource use. European Journal of Agronomy 43, 1-8. Brauman, K.A., Siebert, S., Foley, J.A. 2013. Improvements in crop water productivity increase water sustainability and food security: A global analysis. Environmental Research Letters 8(2): 24-30. Rockström, J., Barron, J., Fox, P. 2003. Water productivity in rainfed agriculture: Challenges and opportunities for smallholder farmers in drought-prone tropical agroecosystems, In: J.W. Kijne, R. Barker, D. Molden (Eds.) Water Productivity in Agriculture: Limits and Opportunities for Improvement. Wallingford, UK: CAB International. Stroosnijder, L., Moore, D., Alharbi, A., Argaman, E., Biazin, B., van den Elsen, E. 2012. Improving water use efficiency in drylands. Current Opinion in Environmental Sustainability 4(5): 497-506. Zhang, H., Oweis, T. 1999. Water-yield relations and optimal irrigation scheduling of wheat in the Mediterranean region. Agricultural Water Management 38: 195-2011. Karam, F., Kabalan, R., Breidi, J., Rouphael, Y., Oweis, T. 2009. Yield and water-production functions of two durum wheat cultivars grown under different irrigation and nitrogen regimes. Agricultural Water Management 96: 603-615. Igbadun, H.E., Tarimo, A.K.P.R., Salim, B.A., Mahoo, H.F. 2007. Evaluation of selected crop water production functions for an irrigated maize crop. Agricultural Water Management 94: 1-10. Wichelns, D. 2014. Do estimates of water productivity enhance understanding of farm-level water management? Water 6(4): 778-795.

Dr. Wichelns is an Agricultural and Natural Resource Economist, with experience in Academia, Research Institutes, Production Agriculture, and International Consulting. He has served on the faculty of several colleges and universities, and he has conducted research in several countries in Asia and Africa. Dr. Wichelns has directed two research centers and he has served as Principal Economist with the International Water Management Institute. He is co-Editor-in-Chief of Agricultural Water Management and the Founding Editor-in-Chief of Water Resources & Rural Development.

The views expressed in this article belong to the individual authors and do not represent the views of the Global Water Forum, the UNESCO Chair in Water Economics and Transboundary Water Governance, UNESCO, the Australian National University, or any of the institutions to which the authors are associated. Please see the Global Water Forum terms and conditions here.