LHOMEWORK ASSIGNMENTS FOR COMPUTATIONAL ECONOMICS
SAN JOSÉ STATE UNIVERSITY
ECONOMICS DEPARTMENT
Thayer Watkins
Homework Assignment 1: The Geometry of Linear Programming Problems
Suppose a farm can produce either corn or soy beans.
Both corn and soy beans require land and water which the farm has in
limited amounts. Suppose the land available is 100 acres and the water
is 200 acre-feet. The resource requirements for the production of
one ton of corn and one ton of soy beans are as follows:
| RESOURCE | Corn
(One ton) | Soybeans(One ton) |
| Land | 2 acres | 1 acre |
| Water | 2 ac-ft | 4 ac-ft |
The prices of corn and soybeans are $200 and $300, respectively.
- 1. Draw a graph with production of corn on the horizontal axis and
of soy beans on the vertical axis. Plot up all the combinations of
corn and soybeans which do not use more land than is available. Shade
this area in in some color.
- 2. Do the same for the water supply.
- 3. Indicate the combinations of corn and soy bean production that
do not use more land or water than is available.
- 4. Plot the line that shows the combinations of corn and soy bean
production which have a value of $20,0000; i.e., the isovalue line. Do
the same for a value of $30,000. Are the lines parallel? Draw the
isovalue line for $10,000.
- 4. Which point in the set of feasible productions is on the
highest isovalue line? Read off from the graph the levels of production
at that point. What is the value of the production at that point?
This is the maximum value of output.
- 5. Now consider the effect of having more land. Suppose the amount
of land is 110 acres. Repeat the above steps and determine the
optimal combination of corn and soy beans and the maximum value of
output. Compute the increase in output over the original problem. What
is the increase in the value of output per unit increase in the
availabiility of land; i.e., divide the increase in output by 10 acres,
the increase in land availability. This is the marginal value of an
acre of land.
- 6. Set the land back to 100 acres and increase the availability of
water to 220 acre-feet. Determine the marginal value of another
acre-foot of water by finding the increase in the value of production
which can be achieved with the additional water. Now consider another
linear programming problem that involves minimizing V= 100r+200s subject
to the constraints that
2r + 2s >= 200 and r + 4s >= 300
- 7. Indicate the set of r and s such that
2r+2s>=200.
Indicate the set such that r+4s>=300. Plot the values of r and s such
that 100r+200s is equal to 20,000. Find the combination of r and s
which minimizes 100r+200s. How do their values compare to the marginal
values of land and water found in #5 above.
- 8. Consider the effect on the minimum of 100r+200s if the first
constraint 2r+2s>=200 is replaced by 2r+2s>=210. What is the change in
the minimum per unit change in the right hand constant of the constraint.
How does this value compare to the optimal value of x in the original
problem? Consider the change in the other constraint to r+4s>=330.
What is the change per unit change in the constant? How does the result
compare to the optimal value of y in the original problem?
- 9. Change the price of soybeans to $500 per ton and rework the
entire problem up to this point.
- 10. Set the price of soybeans back to $300 and introduce a constraint
of labor assuming there are 60 workers available and both corn and
soybeans require 1 worker per ton of product. Rework the entire problem
taking into account this additional constraint.
Homework Assignment 2: The Geometry of Integer and Mixed Linear
Programming Problems
- 1. Integer Variables:
Suppose a railroad can operate two types of trains on some route.
Type A trains require one engine and three crew members whereas
Type B requires two engines and a crew of five. The company has
ten engines and twenty two crew members. Plot a graph which shows
the numbers of Type A and Type B trains the railroad can provide.
First construct the set of points satisfying the constraints ignoring
the integer restrictions. Then indicate the points within this set
which involve integral numbers of the two types of trains.
- 2. Mixed Integer and Continuous Variables
Let x be a variable that can only take on nonnegative integral values.
The variable y, on the other hand, can take on any real nonnegative value.
If x and y must satisfy the constraints
2x + y <= 10, x + 3y <= 12.
Plot a graph showing the feasible combinations of x and y.
Homework Assignment 3: Exercises in Three Dimensional Geometry for
Linear Programming Problems
Herman Weyl, a mathematical physicist, once said something to the
effect that in any field of mathematics the angel of geometry is always
fighting the demon of algebra.