Chiang/Wainwright: Fundamental Methods of Mathematical Economics
Instructor’s Manual
CHAPTER 13 Exercise 13.1 1. For the minimization problem, the necessary conditions become f 0 (x1 ) = 0
and
x1 > 0
f 0 (x1 ) = 0
and
x1 = 0
f 0 (x1 ) > 0
and
x1 = 0
These can be condensed into the single statement f 0 (x1 ) > 0 x1 ≥ 0
and
x1 f 0 (x1 ) = 0
2. (a) Since λi and ∂Z/∂λi are both nonnegative, each of the m component terms in the summation expression must be nonnegative, and there is no possibility for any term to be cancelled out by another, the way (−3) cancels out (+3). Consequently, the summation expression can be zero if and only if every component term is zero. This is why the one-equation condition is equivalent to the m separate conditions taken together as a set. ∂Z ∂Z = 0. This is because, for each j, xj ∂xj ∂xj must be nonpositive, so that no “cancellation” is possible.
(b) We can do the same for the conditions xj
3. The condition xj λi
∂Z = 0 (j = 1, 2, · · · , m) can be condensed, and so can the conditions ∂xj
∂Z = 0 (i = 1, 2, · · · , m). ∂λi
4. The expanded version of (13.19) is: m X ∂Z = fj − λi gji ≥ 0 xj ≥ 0 and ∂xj i=1 ∂Z = ri − g i (x1 , · · · , xn ) ≤ 0, λi ≥ 0 ∂λi
xj (fj − and
m X i=1
λi [ri − g i (x1 , · · · , xn )] = 0
(i = 1, 2, · · · , m; j = 1, 2, · · · , n)
87
λi gji ) = 0