In this essay we will discuss about the theory of crop water production function.
Relationship between crop production and water received is called the crop-water production function. In other words, the functional relationship between crop yield and water use is defined as crop-water production function. Crop water production functions describe the relationship of crop yield (Y) response to varying levels of water input and can be useful for various water management applications.
Knowledge of the relationship between crop yield and water use would greatly contribute in:
1. Prediction of yield response to water
2. Planning of strategies for water supply at farm and project level
3. Evaluation of alternate cropping patterns in relation to water availability, irrigation scheduling to crops and cropping systems
4. Allocation of water among crops and area within the project
5. Planning of strategies for use of limited water supply
6. Economic analysis of irrigation project plans, operations and impacts on income as opposed to investment costs.
Research aimed at determining this function can be categorised into three groups, according to different considerations of what constitutes a desirable level of water use:
1. Agronomists and other production oriented scientists often aim for the level of water inputs necessary to achieve maximum yield per unit land area
2. Irrigation engineers, at least in theory, desire to maximise the efficiency of irrigation water use (irrigation efficiency)
3. Economists argue that water, to be used efficiently, should be applied up to the point where the price of the last unit of water applied is just equal to the revenue obtained as a result of its application.
A simple model of production can be used to demonstrate these three different goals, as presented in Fig 7.3.
A production function in which crop yield (Y) is a function of the amount of water received by the crop in terms of rainfall (P) and irrigation (I) can be defined as follows:
Y = f (P + I) …(1)
Average yield Y̅ which is output divided by input can be written as:
Y̅ = Y / (P + I) …(2)
Marginal yield (Ŷ) is defined as the change in production associated with the addition of one unit input.
It can be written as:
Ŷ = ∂Y/∂(P + I) …(3)
Maximum yield is achieved when the marginal yield is equal to zero. Maximum water use efficiency requires that the derivative of the average yield is equal to zero,
(P + I)-1 [∂Y/∂(P + I) – (Y/(P + I)] = 0 …(4)
Equation (4) shows that, as long as some quantity of water is applied, water use efficiency is maximal where it is equal to the marginal production.