Notes

Outline

Farm Portfolio Problem: Part III Lecture VII

Target MOTAD The target MOTAD model is a two-attribute risk and return model. Return is measured as the sum of the expected return of each activity multiplied by the activity level. Risk is measured as the expected sum of the negative deviations of the solution results from a target-return level. Risk is then varied parametrically so that a risk-return frontier can be traced out.

"Mathematically," Mathematically, the model is stated as

Discrete Sequential Stochastic Programming Target MOTAD, direct expected utility, and even MOTAD begin to develop the concept of constraints being stochastic or met with some level of probability. In target MOTAD, income under a certain state exceeds the target level of income with some probability.

"In direct expected utility maximization..." In direct expected utility maximization the level of wealth transferred to the objective function was represented by a constraint which had some level of probability. In MOTAD, we minimized the expected negative deviations which implied stochastic constraints.

"However," However, in each of these cases, the primary impact of stochastic constraints was on the objective function or some threshold level of risk (as was the case in target MOTAD).

"The variant of model that..." The variant of model that we want to develop is referred to as Discrete Sequential Stochastic Programming (DSSP), although other names have been attributed to it.  This work grows out of work by Cocks, and focuses on decision processes which are strung out over a discrete number of decision periods.

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"At a discrete point in..." At a discrete point in the future, the farmer has to make a decision, for example a stocking rate on cattle.  Given this first round decision and a random outcome, such as rainfall, there is then a subsequent decision to be made, for example whether to sell cattle or buy feed. Each state occurs with a given level of probability and each “node” can contribute to the objective function.

"A mathematical formulation" A mathematical formulation

"In this model x1 represents..." In this model x1 represents the acres of wheat planted, x2 is the number of stockers purchased, x3 the tons purchased under outcome 1, and x4 the tons of feed under outcome 2. The first two equations, then, simply balance the feed requirements under each state of nature.  For example, if there is good rainfall in state 1, then  more grazing will be produced by the wheat ,x2, and less feed will have to be purchased than in state 2.  C1 and c2 are then the cost of feed in each state weighted by the probability of that state.

"The third equation then transfers..." The third equation then transfers the cattle purchased into the next decision period.  X5 is a variable modeling the number of stockers sold, while x6 models any additional stockers purchased.  The total number of stockers in the next production period is x7.  Given the number of cattle transferred into the next period the feed balance relationships determine the level of feed that must be purchased.

"Chance Constrained Programming." Chance Constrained Programming. The DSSP problem above assumes that the possible outcomes can be represented in a finite number of states, although several pieces of applied research have examined the efficiency of approximating the moments of a continuous distribution with a finite number of points.

"An alternative would be to..." An alternative would be to constrain the probability.  For example, assume that you want to constrain the probability that profit will be less than a fixed level T (to borrow the target MOTAD concept).  Mathematically, this constraint becomes:

"Under normality," Under normality, we can transform this constraint via the confidence interval:

Generalized Mean-Variance A Reformulation of the EV Problem The typical mean-variance crop selection model is expressed as

"An extension of this model..." An extension of this model involves appending a term on the constraints which accounts for risk in the constraints.  Specifically, rephrasing the profit function as

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"This specification gives rise to..." This specification gives rise to a related pair of mathematical programming models. The primal

"The dual" The dual

"This specification is consistent with..." This specification is consistent with chance constrained programming.  Specifically, maximizing the primal above can be viewed as maximizing the certainty equivalent of a risky revenue subject to the constraint that the probability of the marginal value of the constraints is less than a given critical level with some probability.  Mathematically,

"Portions of the relationship between..." Portions of the relationship between the chance constrained probolem and the generalized mean-variance programming formulation are dependent on the Kuhn-Tucker conditions.

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