Expected Utility

This study uses a utility maximizing model to find the optimal irrigation strategy. This is because uncertainty in agriculture profitability is mainly because of random weather change and volatility in crop prices. My study examines the optimal irrigation strategy for a particular year given uncertain weather and simplifies the model by assuming price certainty. Granted, price uncertainty is likely to be a more important consideration for decisions made across growing seasons (e.g., irrigation technology and capital purchases) rather than for a single growing season (e.g., irrigation intensity and chemical purchases) ; however, farm output is a random variable when the agricultural inputs are chosen. Thus random agriculture output can be represented by a stochastic production function (Chavas 2004) shown as:

(3.1) y = f(x, μ)

where 𝑦 is output, 𝑥 is input, and 𝜇 is a random variable representing the uncertainty in production.

Next, the farmer’s subjective probability distribution of the uncertainty (𝜇) would be generated from information gathered before growing season begins. My study assumes that farmer has uniform probability for all conditions and that he would use his subjective probability to decide inputs and predict output. Thus, the farmer will choose inputs to maximize the expected utility of wealth (Chavas 2004) shown as:

(3.2) 𝑀𝑎𝑥_𝑥𝐸𝑈(𝜋, 𝑟) = 𝑀𝑎𝑥_𝑥{𝐸𝑈([𝑝𝑓(𝑥, 𝜇) − ∑^𝑁_𝑖=1𝑐_𝑖𝑥_𝑖], 𝑟)}

where 𝑝 is output price, 𝑟 is absolute risk aversion coefficient, 𝑐_𝑖 is the price of input i, and 𝜋 is wealth. Chavas (2004) states that maximizing the expected utility is equivalent to maximizing the certainty equivalent. Alternatively, the farmer’s decision on input can be shown (Chavas 2004) as:

(3.3) 𝑀𝑎𝑥_𝑥 𝐶𝐸 = 𝑀𝑎𝑥_𝑥{𝐸[𝑝𝑓(𝑥, 𝜇)] − ∑^𝑁_𝑖=1𝑐_𝑖𝑥_𝑖− 𝑅(𝑥, . )}

where 𝑅(𝑥, . ) is the risk premium. The risk premium (R) is the minimum amount of money that would make the producer feels indifferent about a risky asset versus a risk-free asset. The risk premium (R) also can be interpreted as the producer’s willingness to insure against risk and can be regarded as the implicit cost of private risk bearing. The necessary first order condition for maximizing the certainty equivalent (Chavas 2004) is:

(3.4) 𝜕𝐸[𝑝𝑦(𝑥,𝜇)]

𝜕𝑥_𝑖 = 𝑐_𝑖+^{𝜕𝑅(𝑥,.)}_𝜕𝑥

𝑖

The necessary first order condition of maximizing certainty equivalent in Equation 3.4 shows that the expected marginal value product of input (𝜕𝐸[𝑝𝑦(𝑥,𝜇)]

𝜕𝑥_𝑖 ) is equal to input cost (𝑐_𝑖) and marginal risk premium at optimal input chosen(^{𝜕𝑅(𝑥,.)}_𝜕𝑥

𝑖 ) (Chavas 2004). From maximizing expected utility of wealth and maximizing certainty equivalent, the marginal risk premium can be also defined (Chavas 2004) as:

(3.5) ^{𝜕𝑅(𝑥,.)}_𝜕𝑥

𝑖 = −𝐶𝑜𝑣{𝑈^′[^{𝑝𝜕𝑓(𝑥,𝜇)}_𝜕𝑥

𝑖 }/𝐸𝑈^′

Equation 3.5 shows that the marginal risk premium (^{𝜕𝑅(𝑥,.)}_𝜕𝑥

𝑖 ) represents the effect of input on the implicit cost of private risk bearing (Chavas 2004). The covariance term (−𝐶𝑜𝑣{𝑈^′[^{𝑝𝜕𝑓(𝑥,𝜇)}_𝜕𝑥

𝑖 }/

𝐸𝑈^′) in the right hand side of Equation 3.5 measures the marginal effect of input i on the implicit cost of private risk bearing (Chavas 2004). The covariance term also shows how input may reduce or increase the implicit cost of risk, providing an incentive for the farmer to reduce or increase input use. The sign of marginal risk premium (^{𝜕𝑅(𝑥,.)}_𝜕𝑥

𝑖 ) indicates whether input 𝑥_𝑖 is

risk increasing (^{𝜕𝑅(𝑥,.)}_𝜕𝑥

Based on this assumption, I expect that risk-averse behavior will induce the farmer to increase water use because water is an input that reduces the implicit cost of private risk bearing.

Also, I assume that the farmer has CARA risk behavior, which implies that the absolute risk aversion coefficient (r) is independent of initial wealth(𝜋). CARA also implies that initial wealth (𝜋) does not influence individual willingness to insure against risk (Chavas 2004). Thus the expected utility corresponding to negative exponential utility for a given risk absolute coefficient (r) is shown as:

(3.6) 𝐸𝑈(𝑟) = ∑ (1

𝑇) [1 − 𝑒^{−𝑟∗𝜋𝑡}]

𝑇

𝑖=1

where T is all possible conditions that may occur. The probability of (_𝑇¹) shows that we assume the farmer has uniform probability for all conditions. Chavas (2004) stated that under the CARA assumption and normality in wealth(𝜋_𝑡), maximizing expected utility [EU(.)] is globally valid with a maximizing certainty equivalent as shown below:

(3.7) max 𝐸𝑈(𝜋, 𝑟) = max 𝐶𝐸 = max 𝐸(𝜋) − 𝑅 𝑤ℎ𝑒𝑟𝑒 𝑅 = −0.5𝑈^′′

𝑈^′ 𝑉𝑎𝑟(𝜋) 𝑟 = −𝑈^′′

𝑈^′

Equation 3.7 represent the estimated expected utility or certainty equivalent using Mean-Variance analysis. The decision-maker who has risk averse (r>0) behavior always has the expected income higher than the certainty equivalent; meanwhile, the risk-loving decision maker

(r<0) always has the certainty equivalent higher than the expected income. Naturally, an alternative decision strategy with its distribution will be preferred if it offers a higher certainty equivalent or higher expected utility compared to another strategy for the set range of the absolute risk aversion coefficient (r). Accordingly, the Mean-Variance analysis will rank the most preferred irrigation strategy in Chapters 4 and 6.

Another method of estimating the certainty equivalent and expected utility is the Stochastic Efficiency Respect to Function (SERF) model. The SERF method ranks risky alternative strategy for welfare loss analysis in Chapter 5. Notably, the mean-variance analysis has a very restricted condition such that the certainty equivalent and risk premium estimation is only globally valid under normality assumption (Chavas 2004) whereas the SERF method relaxes the assumption of distribution (Hardaker et al. 2004). Therefore, Chapter 5 focuses on welfare loss analysis estimating loss using a utility-weighted risk premium between risky alternatives proposed by Hardaker et al. (2004) and Mjelde and Cochran (1988).

The SERF method identifies efficient utility sets by ordering alternative sets according to the certainty equivalent (CE) over a range of risk aversion coefficients (r) (Hardaker et al. 2004).

The certainty equivalent (CE) in my study corresponds to negative exponential utility (𝑈(𝑥) = 1 − 𝑒^{−𝑟∗𝜋}), and a given risk averse coefficient (r) is calculated as an inverse function of expected utility as follows:

(3.8) 𝐶𝐸(𝑟) = ln(1 − 𝐸𝑈(𝑟))^−1/𝑟

The CE is determined by expected utility and the degree of risk aversion.

This study identifies the irrigation strategy with the highest expected utility or highest certainty equivalent for several different values of the absolute risk aversion coefficient (r).

Accordingly, the study used the method proposed by Mccarl and Bessler (1989) to determine the

upper bound of absolute risk aversion coefficient (r) and then calculated the corresponding ARAC for particular risk averse behavior by using the method proposed by Babcock, Choi and Feinerman (1993). My study assumes that the lower bound for the absolute risk aversion coefficient (r) is 0 as a farmer is expected to have risk neutral or risk averse behavior but not risk loving behavior. The method of calculating the absolute risk aversion coefficient and upper bound of absolute risk aversion coefficient are discussed in Section 3.6.