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Advanced MathQ&A LibraryConsider 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.Question

Asked Jan 20, 2020

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Consider the following constrained maximization problem:

maximize y = x_{1} + 5 ln x_{2}

subject to k - x_{1} - x_{2} = 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 x_{1} = 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.

Step 1

Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and specify the other subparts (up to 3) you’d like answered.

Given maximization problem is,

Step 2

(a)Given constraint can be re-written as,

Step 3

First order condition for maximization is given by,

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