Course Overview and Overview Course Overview and Overview of Optimization in Ag Economicsof Optimization in Ag Economics

Lecture 1Lecture 1

Course Outline

Static OptimizationStatic OptimizationOverview of OptimalityOverview of OptimalityReview of Linear AlgebraReview of Linear AlgebraOptimality ConditionsOptimality ConditionsAlgorithmsAlgorithmsOptimization on a ComputerOptimization on a Computer

Dynamic OptimizationDynamic OptimizationA Review of Dynamic MathematicsA Review of Dynamic MathematicsCalculus of VariationsCalculus of VariationsOptimal ControlOptimal ControlApplications of Optimal ControlApplications of Optimal Control

Overview of Optimization

The Basic Microeconomic ProblemThe Basic Microeconomic ProblemDefinition of EconomicsDefinition of EconomicsThe Consumer’s ProblemThe Consumer’s Problem

xMax ( )

. .

U x

s t p x Y

The Producer’s ProblemThe Producer’s Problem

Max

. . ( )

p y w x

s t y F x

Food and Diet Problem

Agricultural applications of the food and diet Agricultural applications of the food and diet problems include both human and animal diets.problems include both human and animal diets. The food and diet research can be characterized The food and diet research can be characterized

by two major focuses:by two major focuses:Least cost combination of foods to meet Least cost combination of foods to meet

dietary needs. Stigler’s “Cost of dietary needs. Stigler’s “Cost of subsistence”.subsistence”.

Least cost feed ration studies.Least cost feed ration studies.

The basic application would involve The basic application would involve minimizing the cost of a diet subject to minimizing the cost of a diet subject to some nutritional constraint:some nutritional constraint:

c c is a vector of prices for each food, is a vector of prices for each food,

min

. .x

c x

s t Ax b

xx is a vector of choice levels for each is a vector of choice levels for each food,food,

AA is a matrix of nutrients provided by is a matrix of nutrients provided by each food, and each food, and

bb is a vector of minimum nutritional is a vector of minimum nutritional requirements.requirements.

More advanced formulations of the diet More advanced formulations of the diet problem have been developed in the guise problem have been developed in the guise of the household production model.of the household production model.General form household production General form household production

problem:problem:

,

max

. .

y xU y

s t y F x

p x I

where where FF((xx) denotes the production ) denotes the production relationship between purchased relationship between purchased foodstuffs and consumable goods (foodstuffs and consumable goods (yy).).

pp is the price vector for purchased is the price vector for purchased foodstuffs, andfoodstuffs, and

II is income is income

A linear formulation of such a model can be A linear formulation of such a model can be expressed asexpressed as

In addition to foodstuffs, In addition to foodstuffs, xx can be can be augmented to include labor use.augmented to include labor use.

,

max

. .

y xU y

s t y Ax

p x I

Farm and Agribusiness Management Initially, linear programming was used to Initially, linear programming was used to

find optimal crop mix. This work has find optimal crop mix. This work has grown into large extension farm planning grown into large extension farm planning efforts such as OK farms. These models efforts such as OK farms. These models tend to be either general linear or integertend to be either general linear or integer

max

. .x

c x

s t Ax b

xx could be a vector of possible crop could be a vector of possible crop alternatives (wheat, cotton, and oats),alternatives (wheat, cotton, and oats),

cc was a conformable vector of net returns from was a conformable vector of net returns from each crop activity,each crop activity,

AA is a matrix of resource constraints is a matrix of resource constraints

bb is the vector of resource constraints. is the vector of resource constraints.

1 1 1

.2 .3 .1

25 100 10

Land

A Labor

Capital

One way that risk may enter the farm One way that risk may enter the farm management model is by complicating the management model is by complicating the objective function:objective function:

max

. .x

E U x

s t Ax b

where where EE[.] is the expected value operator, [.] is the expected value operator, UU(.) is the utility function,(.) is the utility function,

AA is the resource coefficients, is the resource coefficients,bb is the vector of resource constraints, is the vector of resource constraints,

andand xx is the level of each activity. is the level of each activity.

Freund shows that given that preferences Freund shows that given that preferences are negative exponential and returns are are negative exponential and returns are normally distributed, the expected utility normally distributed, the expected utility function becomes:function becomes:

exp

, 2

U x xE U x x x x

x N

Therefore, the maximization problem Therefore, the maximization problem becomes a nonlinear optimization problembecomes a nonlinear optimization problem

max2

. .

xx x x

s t Ax b

However, given that few closed form However, given that few closed form conjugates exist, technologies have evolved conjugates exist, technologies have evolved to allow direct optimization of more to allow direct optimization of more generalized problems:generalized problems:

,

yU y

y N x x x

Farm Firm Development

The typical farm firm development model is The typical farm firm development model is primarily interest in firm growth.primarily interest in firm growth.

1

2

1 2 3 3

11 1 1

12 22 2 2

23 33 3 3

max

0 0 0

0 0

. . 0 0

0 0 0

n

n

nn n n

x

x

c c c c x

x

A x b

T A x b

s t T A x b

A x b

Focusing on the first two constraintsFocusing on the first two constraints

So decisions made in year 1 could also So decisions made in year 1 could also affect the resources available in year 2. affect the resources available in year 2.

11 1 1

12 1 22 2 2 22 2 2 12 1

A x b

T x A x b A x b T x

Again generalizing the model, a decision in Again generalizing the model, a decision in year 1 may have multiple possible year 1 may have multiple possible outcomes:outcomes:

11

1211 1 12 2 22 3 32

22

32

11 11 1

12 12 12 2

22 22 22 3

32 32 32 4

max

0 0 0

0 0. .

0 0

0 0

x

xc p c p c p c

x

x

A x b

T A x bs t

T A x b

T A x b

pp11 denotes the probability of event 1 denotes the probability of event 1

occurring, occurring, pp22 is the probability of event 2 is the probability of event 2

occurring, and occurring, and pp33 is the probability of is the probability of

event 3 occurring.event 3 occurring. If event 1 occurs, the profit vector If event 1 occurs, the profit vector cc1212 and and

TT1212 resources transfer to period 2. resources transfer to period 2. 12 12 12 12 11

22 22 22 22 11

32 32 32 32 11

A x b T x

A x b T x

A x b T x

Production Response

Production response models have been used to Production response models have been used to study the impact of some policy or external shock study the impact of some policy or external shock to the sectorto the sector.. From the firm level, the effect of changing From the firm level, the effect of changing

fertilizer prices, labor availability, or support fertilizer prices, labor availability, or support prices on firm outputs, profits, and input prices on firm outputs, profits, and input demands can be mapped out, much like the demands can be mapped out, much like the duality approach to production.duality approach to production.

The firm level effects are then aggregated The firm level effects are then aggregated to the sector level.to the sector level.

An example of this type of model is An example of this type of model is CARD.CARD.