A question about utility maximization. Nancy Lerner is trying to decide how to a

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A question about utility maximization.

Nancy Lerner is trying to decide how to allocate her time in studying for her Economics course.There are two examinations in this course. Her overall score for the course will be the minimum of her scores on the two examinations.She has decided to devote a total of 1,200 minutes to studying for these two exams, and she wants to get as high an overall score as possible. She knows that on the first examination if she doesn’t study at all, she will get a score of zero on it. For every 10 minutes that she spends studying for the first examination, she will increase her score by one point.If she doesn’t study at all for the second examination she will get a zero on it. For every 20 minutes she spends studying for the second examination, she will increase her score by one point.

What is Nancy's utility maximizing scores for exam 1 and exam 2? What is her maximum utility? Find how much time she must spend on each exam to maximize her overal score?

I am able to determine that Nancy's budget line here would be 10X + 20Y = 1200, where X = exam 1 score, Y = exam 2 score, however, I am having trouble determining the utility maximization since I can't figure out how to determine her utility function, U(X, Y), from there I can use the Lagrangian method to determine how to maximize her utility. It could be that I am going about this the wrong way though. Any help would be appreciated.

Thank you.

Explanation / Answer

This problem is a little tricky since her score will be determined by the minimum of her two scores. In order to maximize her overall score she needs to maximize the minimum score of the two exams.

overall score = min [exam 1, exam 2]

score exam 1= 0.1x; where x is the number of minutes spent studying on exam 1

score exam 2= 0.5y; where y is the number of minutes spent studying on exam 2

Therefore, it is likely that the optimal strategy would be to get the same score on both exams, because diverting extra time to one exam lowers the score on the other exam and thus the overall score.

Taking the first derivative reveals that the marginal benefit of time spent on exam 1 is twice that of exam two, she will need to spend twice as much time studying for exam 2 as for exam 1 to get the same score on both.

The budget equation : 1200 = x + y where x is the number of minutes spent studying for exam 1 and y is the number of minutes spent on exam 2. You can substitute 2x in for y which yields:

1200= x + 2x

1200= 3x

x=400

So she should spend 400 minutes studying for exam one and 800 minutes studying for exam 2. This yields scores of 40 on both exams and thus a 40 in the class. Diverting any extra time to either exam will lower the score of the other exam and thus lower her overall score.

I hope this helped.

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