Extra LP Notes

With Vegetable Farm Example

Whole Farm Planning

Whole-farm planning is largely a matterof enterprise selection. What crops andlivestock enterprises will be produced onthis farm in the next year?

Background: Enterprise Combinations

Economic theory behind whole-farmplanning.

Production Possibility Curve

Definition: A Production Possibility Curve(PPC) is the geometric representation ofthe combination of products that can beproduced with a given set of inputs. Itcan be defined for an entire economy orfor a single firm.

Graph of PPC

enterprise 1

enterprise 2

Types of Enterprise Relationships Competitive with constant substitution Competitive with increasing substitution Supplementary Complementary

Competitive with Constant Substitution

enterprise 1

enterprise 2These enterprises usethe same inputs, in thesame ratios.

Competitive with Increasing Substitution

enterprise 1

enterprise 2The enterprises use differentratios of inputs and inputsexperience diminishing marginal returns in each case.

Supplementary

enterprise 1

enterprise 2

supplementary range

enterprise 1 makes use ofsome inputs that are not needed for enterprise 2

Complementary

enterprise 1

enterprise 2 as we produce more ofenterprise 1, we can also produce more of enterprise 2

Examples

Competitive Constant Sub:

Competitive Increasing Sub:

Supplementary:

Complementary

corn and milo

corn and cotton

soybeans and winter stockers

broilers and cattle

Terms

Physical substitution ratio:

Profit Ratio

Quantity of Output LostQuantity of Output Gained

Profit per unit of gained outputProfit per unit of lost output

Physical Substitution Ratio

The physical substitution ratio is the slopeof the Production Possibility Curve.

Profit Ratio

Profit Ratio is the slope of the isoprofit line:

= 1* Y1 + 2 *Y2

where 1 is profit per unit of enterprise 1,Y1 is the number of units (e.g. acres)

produced, 2 is the profit per unit of enterprise 2 and Y2 is the number of unitsproduced.

Decision Rule

Physical Substitution Ratio =Price Ratio

Graph: Point of Tangency

enterprise 1

enterprise 2

isoprofit lines and PPC

In real life

We don't know the PPC. We are goingto approximate this process using atechnique called "Linear Programming."

Linear Programming

Linear programming maximizes or minimizes a particular linear objective function, subject to linear restrictions.

Here our objective function is to maximizethe returns over variable costs. This is aone-year or short-run plan.

Returns over variable costs

The returns over variable costs come fromthe enterprise budgets.

Farm Planning Process

Inventory available resources Select enterprises to be considered. Obtain appropriate Enterprise Budgets. Figure out the "technical coefficients"

and "RHS" (limits) Develop linear programming tableau. Find optimal enterprise combination.

Resource Inventory

The resource inventory tells you howmuch of each resource (e.g. land, labor,other inputs) you have on the farm.

Labor resources is usually calculated forseveral periods of the year. Land may beof several different types.

Technical Coefficients

Technical Coefficients tell you how muchof each resource you need to produce oneunit of a given enterprise.

For example, it takes one acre of row-cropland to produce one acre of cotton.

Restrictions in LP

Each limited resource requires one linear restriction in the LP model. Theyare normally "inequality constraints."

Consider a simple example:

Vegetable production in Zaire.

Possible enterprises: Lettuce and tomatoes.

Each bed of lettuce makes a profit of30 "Zaires" (local currency). Each bedof tomatoes makes a profit of 40 Zaires.

Marketing Restrictions

Marketing: The local market will absorbno more than the output of: 16 beds of tomatoes8 beds of lettuce

Labor Restriction

The student who wants to grow vegetablescan work up to 24 hours per week on hisgarden.

Tomatoes require 1 hour per week. Lettuce requires 2 hours per week.

Setting up the LP: Objective Function

= 1 Y1 + 2 Y2

Y1 is the number of tomato bedsY2 is the number of lettuce beds

= 40Y1 + 30 Y2

RestrictionsY1 ≤ 16 (marketing restriction for tomatoes)Y2 ≤ 8 (marketing restriction for lettuce)

Y1 + 2Y2 ≤ 24 (labor)

So we can produce no more than 16 beds oftomatoes and 8 beds of lettuce. And we mustlimit our labor so that the amount expendedis less than 24 hours per week.

All Together in Equation Form

Objective max 40Y1 + 30 Y2 =

Subject to: Y1 ≤ 16

Y2 ≤ 8 Y1 + 2Y2 ≤ 24

Graphing the constraints

(mktg 2)

(mktg 1)

16

8

(labor)

24

12

lettuce

tomatoes

Creating The "PPC"

(mktg 2)

(mktg 1)

16

8

(labor)

24

12

lettuce

tomatoes

Feasible Region

Feasible Region

16

8

lettuce

tomatoes

Feasible Region

(8,8)

(16,4)

Optimizing: Max profit $760

16

8

lettuce

tomatoes

Feasible Region

isoprofit linesslope = -30/40

Profit-Max Combination

(16,4)

With more enterprises

With more than two enterprises, we can'tgraph the solution. We will use some software to find our answer.

First we must put the problem in properform.

Equation Form Again

Objective max 40Y1 + 30 Y2 =

Subject to: Y1 ≤ 16

Y2 ≤ 8 Y1 + 2Y2 ≤ 24

The LP "tableau"

Y1 Y2 Type RHS OBJ 40 30 MT1 1 0 LE 16 MT2 0 1 LE 8 LBR 1 2 LE 24

Where LE means less than or equal to and RHS stands for "right hand side"

The RHS

The RHS (right-hand side) contains theamount of the constrained resource youhave available.

Technical Coefficients

The numerical values in the constraint rows,other than the RHS entries, are the technical coefficients.

Objective Function

The values in the OBJ row are the amount of profit per unit of enterpriseproduced. In your farm plan, you willget these values from the Enterprise Budgets.

For your OBJ values:Use Returns above Variable Costs.

Using Excel to Solve the LP

RHS USEDObj 40 30 MAX 760mkt1 1 0 LE 16 16mkt2 0 1 LE 8 4labor 1 2 Le 24 24answers 16 4

I took the tableau for the vegetable example, and solved it using Excel Solver (a tool in Excel). I get the answers 16 beds of tomatoes, 4 beds of lettuce, and profit of $760. If you pop this page open, you can see the formulas I used. You'll learn how to use Solver in the following slides.