ISI Admission Test 2018 Question Paper MS (QE) PEB

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ISI 2018
Question Paper · MS
(QE) PEB
EXAM YEAR TYPE SUBJECT

ISI 2018 Question Paper MS (QE) PEB

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.
.co s e m

s em l a
a ag
There are 6 questions: 2 in Group A, 2 in Group B, and 2 in Group C.

Answer 4 questions in total and at least 1 question from each Group.

Each question carries equal marks.

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The maximum possible score is 100.

. c om e m .
em l as
l as Group A
ag
g
a a government agency has a monopoly in the provision of internet connections.
1. Suppose
The marginal cost of providing internet connections is 12 , whereas the inverse demand
function is given by: p = 1 q. The o¢cial charge per connection is set at 0; thus,
the state provides a subsidy of 21 per connection. However, the state can only provide
budgetary support for the supply of 0:4 units, which it raises through taxes on con-
sumers. Bureaucrats in charge of sanctioning internet connections are in a position

o m
to ask for bribes, and consumers are willing to pay them in order to get connections.
c
. beyond 0:4 units.
m
Bureaucrats cannot, however, increase supply
e
a s
l per connection and the social surplus.
g
(a) Find the equilibrium bribe rate
a
(b) Now suppose the government agency is privatized and the market is deregulated;
however, due large …xed costs of entry relative to demand, the privatized company
continues to maintain its monopoly. Find the new equilibrium price, bribe rate
and social surplus, specifying whether privatization increases or reduces them.
(c) Suppose now a technological innovation becomes available to the privatized monopoly,
which reduces its marginal cost of providing an internet connection to c, 0 < c < 21 .
m
m Find the range of values of c for which privatization increases consumers’ surplus.
c. o2. Consider an exchange economy consisting of two individuals 1 and m .co
m s e
e l a 2, and two goods,

las X and Y . The utility function of individual 1 is U = X + g
a Y , and that of individual
ag
1 1 1

2 is minfX2 ; Y2 g, where Xi (resp. Yi ) is the amount of X (resp. Y ) consumed by
individual i, where i = 1; 2. Individual 1 has 4 units of X and 8 units of Y , and
individual 2 has 6 units of X and 4 units of Y to begin with.

(a) What is the set of Pareto optimal outcomes in this economy? Justify your answer.
(b) What is the competitive equilibrium in this economy? Justify your answer.

m . c
c. o 1
s e m
s em
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(c) Are the perfectly competitive equilibria Pareto optimal?
(d) Now consider another economy where everything is as before, apart from individ-
ual 2’s preferences, which are as follows: (a) among any two bundles consisting
of X and Y , individual 2 prefers the bundle which has a larger amount of com-
modity X irrespective of the amount of commodity Y in the two bundles, and (b)
between any two bundles with the same amount of X, she prefers the one with a
larger amount of Y . Find the set of Pareto optimal outcomes in this economy.

Group B
1. An economy comprises of a consolidated household sector, a …rm sector and the gov-
ernment. The household supplies labour (L) to the …rm. The …rm produces a single
good (Y ) by means of a production function Y = F (L) ; F 0 (L) > 0, F 00 (L) < 0; and
maximizes pro…ts = PY W L, where P is the price of Y and W is the wage rate.
The household, besides earning wages, is also entitled to the pro…ts of the …rm. The
household maximizes utility (U ), given by

U = 21 ln C + 21 ln M
P
d (L) ;

where C is consumption of the good and M
P
is real balance holding. The term d (L) de-
notes the disutility from supplying labour with d0 (L) > 0, d00 (L) > 0: The household’s
budget constraint is given by:

PC + M = WL + +M P T;

where M is the money holding the household begins with, M is the holding they end
up with and T is the real taxes levied by the government. The government’s demand
for the good is given by G. The government’s budget constraint is given by:

M M = PG P T:

Goods market clearing implies Y = C + G:

(a) Prove that dY
dG
2 (0; 1) ; and that government expenditure crowds out private
consumption (i.e., dC
dG
< 0).
(b) Show that everything else remaining the same, a rise in M leads to an equipro-
portionate rise in P .

2

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.
.co s e m

s em l a
a ag
2. Consider an IS-LM model where the sectoral demand functions are given by

C = 90 + 0:75Y;
G = 30;
I = 300 50r;
m
co
M

om .
P d
= 0:25Y 62:5r;

. c M
= 500:
e m
em
P s

l as
as
Any disequilibrium
g
in the international money market is corrected instantaneously
a is
l a change in r. However, any disequilibrium in the goods market, which
a g
through
corrected through a change in Y , takes much longer to be eliminated.

(a) Consider an initial situation where Y = 2500, r = 15 . What is the change in the
level of I that must occur before there is any change in the level of Y ?
(b) Draw a graph to explain your answer.
(c) Calculate the value of (r; Y ) that puts both the money and goods market in
m
.co
equilibrium. What is the value of investment at this point compared to (r =
0:2; Y = 2500)?
e m
l as C
a g
Group
1. Answer the following questions.

(a) Let f : R ! R be a function de…ned as
jxj
8 x 2 R n f0g:
f (x) =
2x
Can f (0) be de…ned in a way such that f is continuous at 0? Justify your answer.
m
.co
(b) Consider the following optimization problem:
m
m .co max x(1
s em
x);
e laoptimization problem.
x2[0; ]

las ag
where 2 [0; 1]. Let x be an optimal solution of the above

ag For what values of will we have x = ?
(c) A …rm is producing two products a and b. The market price (per unit) of a and b
are respectively 3 and 2. The …rm has resources to produce only 10 units of a and
b together. Also, the quantity of a produced cannot exceed double the quantity
of b produced. What is the revenue-maximizing production plan (i.e., how many
units of a and b) of the …rm?

m . c
c. o 3
s e m
s em
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2. Answer the following questions.

(a) A slip of paper is given to person A, who marks it with either (+) or ( ). The
probability of her writing (+) is 13 . Then, the slip is passed sequentially to B; C,
and D. Each of them either changes the sign on the slip with probability 32 or
leaves it as it is with probability 13 .

i. Compute the probability that the …nal sign is (+) if A wrote (+).
ii. Compute the probability that the …nal sign is (+) if A wrote ( ).
iii. Compute the probability that A wrote (+) if the …nal sign is (+).

(b) There are n houses on a street numbered h1 ; : : : ; hn . Each house can either be
painted blue or red.

i. How many ways can the houses h1 ; : : : ; hn be painted?
ii. Suppose n 4 and the houses are situated on n points on a circle. There
is an additional constraint on painting the houses: exactly two houses need
to be painted blue and they cannot be next to each other. How many ways
can the houses h1 ; : : : ; hn be painted under this new constraint?
iii. How will your answer to the previous question change if the houses are located
on n points on a line.

4

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