A simple deterministic profit function was presented in Equation 15.2. The introduction of risk creates a more general expression of the farmer’s decision problem. Instead of all profits, prices and quantities being represented by single values, some are represented as probability distributions. In particular, quantity produced and price of product may be expressed as probability distributions with a known or predicted mean value and, at least, a known or predicted variance. In other words, some of the constraints to Equation 15.2 become stochastic, including the production function and some of the price constraints. The presence of stochastic variables on the right-hand side of the equation means that profit (the left-hand side variable in Equation 15.2) is also stochastic. Rather than maximizing profit, the farmer is now said to maximize expected profit. That is, the farmer chooses inputs to maximize the mean or expected value of profit rather than actual profit.

Consideration of risk caused economists to consider even more general representations of the farmer’s decision problem. Rather than maximizing expected profit, the objective function can include terms for variance of profit or the probability of losses or gains relative to expected profit. Decision makers who are risk averse may prefer to forego some expected profit in exchange for reduced probability of losses below the expected value. A simple objective function that incorporates risk has the left hand side of the equation include expected profit minus some coefficient multiplied by variance of profit. A larger coefficient on the variance of profit term indicates greater risk aversion by the decision maker and a greater willingness to exchange lower expected profit for reduced variance of profit. A zero coefficient on the variance term means the decision maker is risk neutral and will maximize expected profit regardless of variance of profit. In general, this farmer objective function in the presence of risk suggests that two farmers faced with the same yield and price distributions may choose different input quantities solely due to their differing preferences for risk. Risk management comes into focus as an important aspect of farm management.