UNCWilmington
ECN 321
Department of Economics and Finance
Dr. Chris Dumas
Examples of Consumer Choice Problems
that can be solved with Linear Programming
Example (1):
Alicia Silverstone's Problem
Suppose Cher (played by Alicia Silverstone in the movie, Clueless) is trying to decide
how to allocate her leisure time on a particular vacation weekday (you've got to see the
movie, but, suffice it to say that, for Cher, every weekday is a vacation day).
Specifically, she is trying to decide how many hours to spend at the beach and how many
hours to spend at the mall.
All else equal, she figures that an hour spent at the beach
gives her about onethird more satisfaction than an hour spent at the mall.
At most,
suppose she has 8 hours of leisure time to divide between the beach and the mall.
However, due to the weather, suppose there will be only 6 good beach hours this day.
Also, due to hours of operation, suppose the mall is open only 10 hours.
In addition,
suppose Cher spends, on average, $5 per hour at the beach and $10 per hour at the mall.
Finally, suppose Cher has a total of $60 to spend on leisure this day.
What will Cher
choose to do?
Solution:
Identify the choice variables:
Let x1 = beach hours and
x2 = mall hours.
Set up the optimization problem:
max
U = 1.33
⋅
x1 + 1
⋅
x2
x1, x2
Subject to: (1) x1
≥
0
(2) x2
≥
0
(3) x1 + x2
≤
8
(4) x1
≤
6
(5) x2
≤
10
(6) 5
⋅
x1 + 10
⋅
x2
≤
60
To facilitate graphing, constraint (3) above can be rewritten as : x2
≤
8  x1
Similarly, constraint (6) above can be rewritten as : 10
⋅
x2
≤
60  5
⋅
x1
x2
≤
(60  5
⋅
x1)/10
x2
≤
6  1/2
⋅
x1
1
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View Full DocumentUNCWilmington
ECN 321
Department of Economics and Finance
Dr. Chris Dumas
To find the Feasible Region, graph the constraints:
Finding the coordinates for Extreme Point C:
The x2's of constraints (3) and (6) are equal at Point C; hence,
8  x1 = 6  1/2
⋅
x1
2 = 1/2 x1
x1 = 4
Plugging x1 = 4 back into either constraint, we find x2 = 4
Thus, Extreme Point C is (4,4).
Finding the coordinates for Extreme Point D:
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 Fall '08
 Dumas
 Economics, Microeconomics, Optimization, Department of Economics and Finance, extreme point, Dr. Chris Dumas

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