Lecture 4: Feasible Space and Analysis AGEC 352 Spring 2012 – January 25 R. Keeney
Feb 24, 2016
Lecture 4: Feasible Space and Analysis
AGEC 352Spring 2012 – January 25
R. Keeney
Linear Equations & SystemsRecall the following for y = mx +
b◦Linear equations have constant slope
Differentiate y = mx + b and the result is m
◦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
ZQQFtosubject
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?
ZQQFtosubject
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/
requiredCorn 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/
requiredCorn 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
050
100150
Premium Finest
Stan
dard
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
100200300
Corn constraint
Premium FinestStan
dard
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
50100150200250300350
Corn constraint Sugar constraintMach. constraint
Premium Finest
Stan
dard
Stu
ffQuestions1) 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?