9781118041581

Answers to Odd-Numbered Problems 35

where M and C are the nonnegative quantities of milk and cereal. A

graph shows that the lowest contour touches the feasible region at the

corner formed by the protein and calcium constraints. (The slopes of

these constraints are 1/3 and 1, respectively; the slope of the

typical cost contour is .1/.15 2/3.) Solving 2M 2C 50 and

2M 6C 90, we find C 10 and M 15. The minimum cost of a

healthy diet is $3.

b. If we increase the calcium requirement by a small amount (say, by 4

units to 54), the new solution becomes C 9 and M 18. The cost of

meeting this higher health requirement is $3.15. Therefore, the

shadow price of an extra unit of calcium is .15/4 $.0375.

7. a. The formulation is

Maximize:

Subject to:

Since bonds have better returns, the investor would like to make

T as large as possible. Clearly, the first two constraints never are

binding. However, the last two constraints do bind the proportion

of bonds. Solving .4B 4T 2.5 and B T 1, we find B  .417

and T  .583. The expected return of this portfolio is 5.17

percent.

b. The formulation is

Maximize:

Subject to:

Notice that treasury bonds dominate (are more profitable and safer)

than treasury bills, corporate bonds, and municipal bonds.

Eliminating these three securities reduces the binding constraints to

5T J 3.5 and T J 1. The solution is T .625 and J .375. The

portfolio’s expected return is 6.75 percent.

c. If risk is not an issue, the manager should invest 100 percent of the

portfolio in junk bonds (J 1), earning a maximum rate of return

and just meeting the maturity constraint.

9. a. Let x 1 and x 2 denote the levels of the two processes. At a unit level,
   process 1 produces 2 units of H and 1 unit of P for a total

BTCMJ1.0.

5B5T3.5C3M1J3.5

4B6T4.4C5.6M8J

BT1.0.

.4B4T 2.5

.4B4T1.5

5B5T3.5

4B6T

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