Question 6:

Consider the following optimization problem:

max3 2/5 lnx1+lnx2

(x1,x2,x3)R+

subject to 4x1+x3100

3x2= 2x3.

a. Fill in the blanks (if multiple items are provided in a parenthesis after a blank, fill in

the blank with one of them):

i. There is ___________(some, no, uncertain) loss of generality to replace the constraint 4x1+x3100 by the equality 4x1+x3= 100 because, if 4x1+x3<100 then we can ________(raise, reduce)x1slightly___________ (with, without) violation of any constraint and, with this altered value ofx1, the objective ______(increases, decreases, remains unchanged).

ii. For any (x1, x2, x3)R3+that satisfies the two constraints whilex3= 0, whenx3is increased slightly,x2________(increases, does not increase) and the objective_______ (increases, does not increase because it is independent ofx3). Thus,

there is________ (some, no, uncertain) loss to assumex3>0 at any optimum.

b. Rewrite the optimization problem into an equivalent form to which the Lagrange method with equality constraints can be applied directly:

c. Write down the Lagrangian of this problem:

d. Write down all the first-order conditions for the Lagrangian:

e. The solution for the optimization problem is:x1=____________ ,x2=____________ ,x3=_______ . The Lagrange multiplier for the constraint 4x1+x3= 100 is equal to_________ , and that for the constraint 3x2= 2x3is equal to___________ .