Lecture 5:
Objective Equations
AGEC 352
Fall 2012 – September 10
R. Keeney
Linear Programming

Linear
◦ All of the functions are linear in the variables
 Constraint inequalities (feasible space)
 Objective equation

Programming
◦ Nothing to do with computer code
◦ Comes from a term in planning
◦ First application was in military procurement
and distribution
Review Feasibility

In lecture 4 we developed a graphical
approach to the feasible space
◦ Boundary defined by the linear constraints
 Inequalities create a region on either side of a
constraint that is feasible (half-space)
 Combine the inequalities using the most restrictive
in any area of the graph

The shape of the feasible space is the key
in linear programming
◦ Linear segments create kinks or corner points
Level Curves

An equation in 3 variables is impossible to
graph in 2 dimensions
◦ In economics, we use the level curve
Y
Z
X
Level Curve Example
Y
Production Function
Y = f(X1 | X2)
X1
Level Curve Example
Y
For any ‘level curve’ below, the X2 value is fixed but it is a variable.
Y when X2 = 3
Y when X2 = 2
Y when X2 = 1
X1
Level curves in economics

Typically start with Iso◦ Prefix meaning equal
Isorevenue
 Budget line (isoexpenditure)
 Isoquant
 Indifference curve (isoutility)

Isorevenue

The assumptions of the PPF (outputoutput) model
◦ Price taking = decision maker has no impact
on prices (exogenous)
◦ Chooses the output mix that maximizes
revenue

3 variables: Revenue, quantity of a,
quantity of b
Isorevenue and Units
R = PaQa + PbQb
Qb = R/Pb – (Pa/Pb)Qa
Qb units = bushels of b
R units = dollars
Pb units = dollars per bushel of b
Pa units = dollars per bushel of a
Qa units = bushels of a
Cancel out and we see the isorevenue is in
units of the Y axis variable.
Isorevenue graph
Qb
Intercept = R/Pb
R = PaQa + PbQb
Isorevenue
Qb = R/Pb – (Pa/Pb)Qa
PPF
Slope = -Pa/Pb
Qa
Isorevenue graph
Qb
Intercepts
R1/Pb
R2/Pb
R3/Pb
Since Pb is fixed, the intercept
measures revenue.
R2 is optimal because it is the
highest intercept that is still
feasible given the PPF.
Slope = -Pa/Pb
Qa
Linear programming of the outputoutput model
Works exactly the same as graphically
solving the model with the non-linear PPF
 There is no tangency result but we are
still looking for the isorevenue line with
the greatest intercept that is still feasible

LP PPF model
Qa
For given prices, the solution is found by
identifying the objective equation level
curve that is 1) feasible and 2) has the
highest intercept.
Qa
Implications

Linear programming solutions will always
occur at a corner point of the feasible
space
◦ If the slope of the objective equation is
exactly equal to the slope of a boundary
constraint, multiple solutions (including 2
corner points) exist

We can solve linear programs just by
solving for all possible corner points and
evaluating the revenue at each one
LP Algebraic Form
max R  8 PF  6 SS
subject to :
corn : 5 PF  3SS  600
sugar : 4 PF  2 SS  600
m achinery: PF  2 SS  200
non  negativity: PF  0, SS  0
Corn and sugar intersection
5 PF  3SS  600
4 PF  2 SS  600
5 PF  3SS  4 PF  2 SS
PF   SS
 5SS  3SS  600
SS  200
PF  200
These constraints do not
intersect in the 1st
quadrant.
They do not generate a
relevant corner point
since we have nonnegativity constraints.
Only one of these
constraints can bind in a
solution.
Standard Stuff
Feasible space graph
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Premium Finest
Corn constraint
Mach. constraint
Sugar constraint
Corn and machinery
corn : 5 PF  3SS  600
m achinery: PF  2 SS  200
5 PF  3SS  3PF  6 SS
PF  1.5SS
5(1.5SS )  3SS  600
10.5SS  600
SS  600/ 10.5
PF  900/ 10.5
If this is better than
producing only PF
or only SS, then it is
the optimal
solution.
Lab for Wednesday

Posted after class
◦ Use a spreadsheet to graph and solve for a
two variable linear program
 Discussion questions at end