Constrained Optimization with Equality Constraint problem

a. Consider the problem,

max x1 + x2 + x3 +x4 subject to :

(1)1/3 x1 + 1/3 x2 + 1/8 x3 + 1/8 x4 = 3

(2)x1x2x3x4 = 144

(3)x1, x2, x3, x4 are all greater or equal to 0

i. Write down the necessary condition for the solution of this problem.

ii. Show that the necessary conditions imply x1 = x2 and x3 = x4. Find the solution.

iii. Suppose that the Right Hand Side (RHS) value of the second constraint is changed to x1 x2

x3 x4 = 145. How the change in RHS value of the second constraint will affect the optimal

value of x1 + x2 + x3 + x4, without redoing the whole problem.

b. The farmer's available amount of land is 1000 acres, the unit net profit of crops 1 and 2 are

$10 and $8 respectively, and the production function are y1 = l1^0.6 and y2 = l2^0.8. What is

the optimal land allocation? Confirm that the solution for the optimal land-allocation is a

true maximum.