Lecture 4: Feasible Space and Analysis

AGEC 352Fall 2012 – September 5

R. Keeney

Linear Equations & Systems

Recall the following for y = mx + b◦Linear equations have constant slope

Differentiate y = mx + b and the result is b

◦If restrict y and x to be non-negative we are only dealing with the 1st quadrant of the Cartesian plane

Linear Equations & SystemsThe solution to two linear

equations is an (x,y) pair that defines the intersection◦Two linear equations also

May have no solution Identical slope, different intercepts

May have no non-negative solution Different slope and intercept, intersect outside

the 1st quadrant May have many solutions

Same slope and intercept

Production Possibilities FrontierA producer has a

given amount of inputs

Must choose the best quantity of different outputs

Assumptions◦Costs are sunk on

inputs◦Profit will be maximized

where revenue is maximized

ZQQF

tosubject

QPQPR

),(

:

max

21

2211

Graph of the PPF

Q1

Q2 1) How do we interpret the PPF?

2) What does feasible mean in terms of the PPF?

3) How do we solve the economic problem (Revenue maximization) that goes with the PPF?

What does the PPF have to do with linear equation systems?The shape of the PPF is not

known in generalIntro economics draws it smooth

and bowed out◦Because we need to teach 2 things

1) Elasticity of supply Requires a smooth curve with a derivative

2) Declining marginal transformation Requires that additional units of 1st output given

up produce smaller yields of the 2nd output

Linear functions may be a good approximation to a PPFNo elasticities but the problem is

easier to solveCan still represent bowed out PPF

to a degree

Q1

Q2

Constraints and inequalitiesThe PPF is a constraint representing◦1) available technology (ways to turn

inputs into outputs)◦2) available quantities of inputs (Z)

It is more appropriately represented as a boundary of the entire feasible set it defines◦Why might this be?

ZQQF

tosubject

QPQPR

),(

:

max

21

2211

An example with two outputsA manufacturer makes two

brands of beverages◦PF = Premium Finest◦SS = Standard Stuff

The manufacturer has three resources available for making the beverages◦C = corn (600 bushels)◦S = sugar (600 pounds)◦M = machinery (200 hours)

Technical information How do the inputs become

output?Resource

PF SS

Corn 5 bu/gallon

3 bu/gallon

Sugar 4 lbs/gallon

2 lbs/gallon

Machinery

1 hr/gallon

2 hr/gallon

Some analysis of this informationIdentify the most limiting

resource for each beverage.◦How would we do that?

Most limiting resourceThe goal is to see what resource is

“most scarce” for each product◦The “most scarce” resource will drive the

economics of the productStep 1: For each input, divide the

total available quantity by the requirement per gallon of the beverage

Step 2: Identify the most limiting as the lowest number (i.e. it limits the beverage quantity to X)

Most Limiting cont.Resource Required for

PFTotal Total/

required

Corn 5 bu/gallon 600 120

Sugar 4 lbs/gallon 600 150

Machinery

1 hr/gallon 200 200

Most Limiting cont.Resource Required for

SSTotal Total/

required

Corn 3 bu/gallon 600 200

Sugar 2 lbs/gallon 600 300

Machinery

2 hr/gallon 200 100

Most limiting summaryWe will never make more than

120 gallons of PFWe will never make more than

100 gallons of SSIf we make 120 gallons of PF, we

make no SSIf we make 100 gallons of SS, we

make no PF

Feasible Space

Is this the right feasible space?

0 20 40 60 80 100

120

140

0

50

100

150

Premium Finest

Sta

nd

ard

Stu

ff

Feasible SpaceIdentifying the most limiting

resource for each output tells us…◦1) the correct endpoints

(intersections with the axes) for the feasible space but nothing about the points in between

◦2) which inputs are most likely to determine the economics of the optimal output mix

To visualize the feasible space, we need to graph a set of inequalities

A ConstraintCorn available is 600 bushels

◦PF uses 5 bushels per gallon◦SS uses 3 bushels per gallon

Need to write a total corn usage inequality

5*PF + 3*SS ≤ 600To graph this we need to

◦1) convert it to an equality/equation◦2) identify two points for the

equation

Corn constraint cont.5*PF + 3*SS = 600The easiest two points to get from

this equation are◦1) when PF = 0◦2) when SS = 0

Plug zero in for one of the outputs, solve for the other that solve the equation

We already did that in the most limiting factors so we have

(PF, SS) = {(0,200), (120,0)}

Corn constraint cont.

All combinations of PF and SS that can be produced considering only the corn limit◦Remember the inequality, everything inside of

the line can be produced as well

0 30 60 90 120 1500

100

200

300

Corn constraint

Premium FinestSta

nd

ard

Stu

ff

Other constraintsReturning to the most limiting factor

analysis we can find two points for each of the other resources as well

Sugar: (PF, SS) = {(0,300), (150,0)}Machinery: (PF, SS) = {(0,100),

(200,0)}

Graph those in the same space as the corn constraint…

Feasible Space

0 30 60 90 120 150 180 2100

50

100

150

200

250

300

350

Corn constraint Sugar constraintMach. constraint

Premium Finest

Sta

nd

ard

Stu

ff

Questions1) Which side of the line is

feasible and which is infeasible?

2) Which constraints ‘define’ the feasible space?

3) How would we go from feasible analysis to ‘best’ analysis?

4) What combo uses all corn and machinery time?