DSE 2009 Q. 26 and 27

[Nag27]

Hi, Can anyone explain the solution of Q. No 26 and 27 in the DSE 2009 paper? The question and the options underneath are as follows:


Suppose that a typical graduate student at the Delhi School of
Economics lives in a two good world, books ( x ) and movies ( y ), with utility
function u(x, y) = x^(1/5) y^(4/5) . Prices of books and movies are 50 and 10 respectively.
Suppose the University is considering the following schemes.
Scheme 1: 750 is paid as fellowship and additional 250 as book grant. Naturally, book
grant can only be spent on books.
Scheme 2: 1000 as scholarship and gets one movie free on each book they purchase.
Believing that books and movies are perfectly divisible, compute the optimal
consumption bundle under each scheme.

26. Optimal consumption bundle under scheme 1 is
a) (4 books, 80 movies)
b) (5 books, 75 movies)
c) (6.5 books, 57.5 movies)
d) (10 books, 50 movies)


27. Optimal consumption bundle under scheme 2 is
a) (4 books, 80 movies)
b) (4 books, 84 movies)
c) (5 books, 75 movies)
d) (5 books, 80 movies)

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I don't understand how to include the 250 books grant into the optimisation problem.

[maahi]

part 1 : assume his total  money income to be 1000 which he can optimally divide as 80 movies  and 4 books but he has to spend 250 on books so  he must buy 1 more book in total 5 books and movies will be then 750/10 = 75 .

[duck]

Hi.. :)


26) budget constraint is : 50x +10y =750
by solving optimisation problem, u will get x=3 , however he is given 250 as book grant so he has to purchase 5 books(x) and spend all of his income on movies.
So, optimum: x=5 and y=750/10= 75

27) budget constraint is: 50x +10y =1000 +10y
                              ⟹ 40x + 10y=1000
(since, he gets one movie free on each book he purchase)

Now, solve like all other problems,
u will get x=5 and y=80

[Nag27]

Thank you so much! Are there other types of optimisation problems with a modification in the budget constraint? Could you share some?