2.7 Consider the following constrained maximization problem: maximize y x1 5 lnx2 subject to k % x1 % x2 0, where k

2.7 Consider the following constrained maximization problem:

maximize y ¼ x1 þ 5 lnx2 subject to k % x1 % x2 ¼ 0, where k is a constant that can be assigned any specific value.

a. Show that if k ¼ 10, this problem can be solved as one involving only equality constraints.

b. Show that solving this problem for k ¼ 4 requires that x1 ¼ %1.

c. If the x’s in this problem must be non-negative, what is the optimal solution when k ¼ 4? (This problem may be solved either intuitively or using the methods outlined in the chapter.)

d. What is the solution for this problem when k ¼ 20? What do you conclude by comparing this solution with the solution for part (a)?

Note: This problem involves what is called a quasi-linear function. Such functions provide important examples of some types of behavior in consumer theory—as we shall see.