Consider the following constrained maximization problem:maximize y = x1 + 5 ln x2subject to k - x1 - x2 = 0,where k is a constant that can be assigned any specific value.a. Show that if k = 10, this problem can be solved as one involving only equality constraints.b. Show that solving this problem for k = 4 requires that x1 = 1.c. If the x’s in this problem must be nonnegative, what is the optimal solution when k = 4?d. What is the solution for this problem when k = 20? What do you conclude by comparing thissolution to the solution for part (a)?Note: This problem involves what is called a “quasi-linear function.” Such functions provide importantexamples of some types of behavior in consumer theory—as we shall see.
Consider the following constrained maximization problem:
maximize y = x1 + 5 ln x2
subject to k - x1 - x2 = 0,
where k is a constant that can be assigned any specific value.
a. Show that if k = 10, this problem can be solved as one involving only equality constraints.
b. Show that solving this problem for k = 4 requires that x1 = 1.
c. If the x’s in this problem must be nonnegative, what is the optimal solution when k = 4?
d. What is the solution for this problem when k = 20? What do you conclude by comparing this
solution to 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.
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