Question

In: Economics

A student must study for two exams: Class A and Class B. Happiness is determined by...

A student must study for two exams: Class A and Class B. Happiness is determined by the sum of the two grades. The student has 600 minutes to spend (as denoted by m) . If there is no studying, the student can get 50 grade points on each of the exams. In order to get one additional point on Class A, the student has to spend p1 = 30 minutes. On the other hand, in order to get one additional point on Class B, the student has to spend p2 = 20 minutes.

1) What would be the utility function that represents the preferences over grade points on Class A and Class B.

2) What would be the optimal amount of time for the student to choose to spend on Class A and Class B? How many grade points would the student get in each class?

3) Now, if it only takes 15 minutes to get an additional point in Class A, what would be the optimal amount of time for the student to choose to spend on Class A and Class B.

Expert Solution

1) Happiness is determined by the sum of two grades

i.e., Hapiniess = utility = marks in class A + marks in class B

= > utility = (xA +50) + (xB + 50)....{Note that even when there is no studying the student does secure 50 marks in each of the classes}

= > utility = xA + xB + 100

= > utility - 100 = xA + xB.(monotonic transformation)

= > U = xA + xB

Hence , the above utility function defines the preferences over grade points on class Aand class B.

2) Now the time constraint or the budget constraint here given that total time available to be devoted to studying =600:-

600 = 30xA + 20xB

= > 60 = 3xA + 2xB

[Since , here time spent is securing each additional marks in each class is the price paid for each additional marks.]

Now , note in the figure the black lines are the indifference curves and the red line is our budget constarint,

the highest utiltiy achieved by such indifference curves are generally at either of the corner points, In this the equilibrium lies on the vertical axis at xA=0, xB = 30

Mathematically, since xA and Xb are perfect subtitutes hence, in the equilbrium we are to obtain a corner slution that is either xA= 0 = > xB =30 = > utiltiy = 30

Or xB = 0 = > xA = 20 (from the budget equation) = > utility = 20

Since, the first case yields higher utility hence that is our solution here.

Thus, Note when we are solving for perfect substitutes utility function ; the good(class) for which the price (additional time spent )is lesser relatively , that good (class ) is allocated all the income(time) .

Thus optimal time in class A = 0 ; Class B = 600 minutes

3) hence when the time devoted to score additional marks in class A drops to 15 minutes now less than what is required for in class B = 20minutes;

The new budget line being :- 600 = 15xA + 20 xB

=> 60 = 1.5xA + 2xB.

Obviously the consumption point (equilbrium) shall shift to the other corner, and no amount of extra time shall be devoted to class B rather

600 minutes of additional studying shall be done for class A now. .

Rahul Sunny answered 1 month ago

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