b. The chance is .5 that an individual buyer’s value is less than $330
thousand. The chance that both values are less than the reserve is
(.5)(.5) .25. The chance that one bidder will meet the reserve is .5.
The chance that both values exceed the reserve is .25. If both values
are above $330 thousand, the expected auction price is: (2/3)(330)
(1/3)(360) $340 thousand.
c. With Pmin$330 thousand, the seller’s expected revenue is
(.25)(300) (.5)(330) (.25)(340) $325 thousand. This is $5
thousand more than the expected revenue in part a (with Pmin
$300 thousand).Chapter 17
- a. Increasing or decreasing returns to scale implies that either the
objective function or some constraint is nonlinear. Thus, the LP
formulation cannot be used.
b. The LP method can handle any number of decision variables. The
earlier problem of producing a maximum level of output contained
more variables (3) than constraints (2).
c. A downward-sloping demand curve implies a nonlinear revenue
function. (The revenue function is linear only if the demand curve is
horizontal, that is, the price is constant.) Thus, the LP formulation
cannot be used.
d. Here, the constraints are Q 1 /Q 2 .4 and Q 1 /Q 2 .6. These can be
rewritten as Q 1 .4Q 2 0 and Q 1 .6Q 2 0, respectively. Since
these are both linear, the LP formulation applies. - a. The slope of the objective function (10/15) lies between the slopes
of the two constraints (2/5 and 6/3). Therefore, the optimal
solution has both constraints binding: 2x 5y 40 and 6x 3y 48.
The solution is x 5 and y 6. The value of the objective function
is 140.
b. The slope of the objective function (.75) lies outside the slopes of the
two constraints (1/.5 and 1/1). Therefore, the optimal solution has
y 0 and only the second constraint is binding: x y 16. Thus, x 16
and the minimum value of the objective function is 12. - a. The formulation is
Minimize:
Subject to:
6M2C66 (calories),2M6C90 (protein)2M2C50 (calcium).1M.15C
34 Answers to Odd-Numbered ProblemsBMAns.qxd 9/26/11 11:18 AM Page 34