ISI Admission Test 2021 Question Paper MS (QE) PEB

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ISI 2021
Question Paper · MS
(QE) PEB
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ISI 2021 Question Paper MS (QE) PEB

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Group A
1. Let f : ℜ → ℜ be a continuously differentiable function which has at
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least three distinct zeros. (We say x is a zero of f if f (x) = 0). Let
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x/2
g : ℜ → ℜ be defined as follows: g(x) = e f (x) for all x ∈ ℜ.

(i) m s e
s e Prove that g has at least three distinct zeros. l a
g la(ii) Prove that the function f + 2f has at least two distinct zeros.
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a (10+20=30)

2. (i) Consider the following two variable optimization problem:

max (x2 + y 2 )
x,y

subject to
x+y ≤1
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g a a King is in the habit of inviting 1000
Find all solutions of this optimization problem.
laway,
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(ii) In a kingdom far, far
senators to his annual party. As a tradition, each senator brings
the King a bottle of wine. One year, the Queen discovers that one
of the senators is trying to assassinate the King by giving him a
bottle of poisoned wine. Unfortunately, they do not know which
senator, nor which bottle of wine is poisoned, and the poison
is completely indiscernible. However, the King has 10 prisoners
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he plans to execute. He decides to use them as taste testers to
determine which bottle of wine contains the poison. The .poison
m .co s em24 hours
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when taken has no effect on the prisoner until exactly

las later when the infected prisoner suddenly dies.gThe
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ag to determine which bottle of wine is poisoned by tomorrow so
that the festivities can continue as planned. Hence he only has
time for one round of testing. How can the King administer the
wine to the prisoners to ensure that 24 hours from now he is
guaranteed to have found the poisoned wine bottle?
(12+18=30)

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3. (i) Let A and B be matrices for which the product AB is defined.
Show that if the columns of B are linearly dependent, then the
columns of AB are linearly dependent.
(ii) Let ei denote the column vector with three elements, each of
which is zero, except for the i-th element, which is 1. Consider
a linear transformation L : ℜ3 → ℜ3 with L(e1 ) = e1 , L(e2 ) =
e1 + e2 , and L(e3 ) = e2 + e3 . Does L map ℜ3 onto ℜ3 ? Prove
your answer.
(15+15=30)

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Group B

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4. This question pertains to a situation in which a particular commodity,

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like rice, is both available at a subsidised rate from a fair price shop
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(ration shop) and at a higher price from the open market. Suppose
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a consumer
from the ration shop (that is, there is a ration quota). In addition, ag
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ag he can buy more of rice (assume a uniform quality of rice) from the
open market at a higher price. (You may assume that consumers
preferences are represented by standard downward sloping, smooth,
convex indifference curves.)

(i) Graphically depict the consumer’s equilibrium (assuming he ex-
hausts the ration quota and in addition buys from the open mar-

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ket).

m . What will happen to the quan-
e the open market (over and above the
(ii) Suppose rice is a normal good.
s
a
gl if there is a cut in the ration quota?
tity of rice purchased from

a
ration quota) in equilibrium
Briefly explain.
(iii) Suppose rice is a normal good. What will happen to the quantity
purchased in the open market (over and above the ration quota)
if the subsidised price (price at which the ration quota rice could
be bought) is increased (but is still lower than the open market
price)? Will your conclusion change if rice is an inferior good?
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(10+10+10=30)

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5. There are plenty of fish in the Dull Lake. Boats can be hired by
fishermen to catch fish and sell it on the fish market. The revenue
earned each month from a total of x boats is given by the following
expression: Rupees 10, 000{4x − 12 x2 }. Each boat costs Rupees 20,000
each month.

(i) Derive the marginal and average revenue per boat
(ii) The Dull municipality is considering giving out permits for each
boat that fishes so they can track who is fishing from the lake. If
these permits are allocated freely, how many boats will fish every
month?
(iii) If total profit is to be the maximum possible, how many boats
should fish every month?
(iv) Dull municipality decides to charge for the permits instead of
giving them out for free. What should the per-boat charge for
the permit be if total profits are to be the maximum possible?

(5+10+5+10=30)

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6. The Shoddy Theater screens movies every week and is located on a
university campus which has only students and faculty as residents.
It is the only source of watching movies for both faculty and students,
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and is large enough to accomodate all faculty and students. Faculty
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mprice of the ticket paid by faculty and Q refers to the number las
demand for movie tickets is given by 500 − 4PF = QF , where PF refers
to e
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of tickets purchased by faculty. Demand by students is described by a
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ag 100 − 2P = Q , where P refers to the price of the ticket paid by
S S S

students and QS refers to the number of tickets purchased by students.
The cost to service demand equals 500.

(i) If the price charged is to be the same for faculty and students,
what price would Shoddy Theater set in order to maximize its
profits?
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(ii) Now imagine that Shoddy Theater decides to charge different

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prices for faculty and students. What would these prices be, if

g lato maximize profits?
Shoddy Theater wants
a
(15+15=30)

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Group C

7. Consider a Solow type economy, producing a single good, according
to the production function:

Y (t) = K(t)α L(t)(1−α)

Where, 0 < α < 1, and Y (t), K(t) and L(t) are the output of the
good, input of capital, and input of labour used in the production
of the good, respectively, at time t. Capital and labour are all fully
employed.

Labour force grows at the exogenous rate, η > 0, i.e.,

1 dL(t)
= η > 0.
L(t) dt

Part of the output is consumed and part saved. Let 0 < s < 1 be the
fraction of output that is saved and invested to build up the capital
stock. Also assume that there is no depreciation of capital stock.

Then it follows that:
dK(t)
sY (t) =
dt

Where dtd is the time derivative.

With this above given description of the economy, one can find out
the steady state growth rate of Y, for this economy. Growth rate of
output is given by: Y 1(t) dYdt(t) ≡ gY .

Assume, the economy begins at date 0, from a per capita capital stock,
k(0) ≡ K(0)
L(0)
< k ∗ , where k ∗ denotes the steady state per capita capital
stock.

(i) Demonstrate formally, whether the growth rate of output (gY ),
at the beginning date 0, is greater, equal or less than the steady
state growth rate of output.

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(ii) For the same economy, consider, two alternative beginning date
scenarios: with per capita capital stock, given by:
Case 1. k(0); Case 2. k ′ (0). Where, k(0) < k ′ (0) < k ∗ Can you
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compare the beginning date growth rates of output in the two
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cases?

s
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(iii)

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the economies have absolutely the same description as given be-
fore, except for the fact that the fraction of income saved in
country A, denoted by sA is greater than the fraction of income
saved in country B, denoted by sB . Let k A (0) be the initial date
per capita capital stock in country A and k B (0), the initial date
per capita capital stock in country B, which are both less than

o m
their respective steady state values. Assume, K A (0) < K B (0).
c
. initial date growth rate of out-
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Can you figure out whether the
e than, equal or less than the initial
a s
put in country A is greater
date growth rate of loutput in country B? In case you find the
ag is insufficient to make any comment on this,
data provided to you
please point it out.

(20+5+5=30)

8. (i) What is the money multiplier? What determines its size? What
is the relation between the monetary base, the money multiplier,
and the money supply? Which of these variables can the central
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bank change to change the money supply? What is the direction

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of change in each case?
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(ii) Why might the cash/deposit ratio and the reserve
g a How does an
a
be decreasing functions of the rate of interest?
interest-sensitive money supply affect the LM curve? Illustrate
with a diagram, comparing this LM curve with the standard LM.
How does this change the effectiveness of counter-cyclical fiscal
policy (in a closed economy)? Explain.
(15+15=30)

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9. (i) What is the difference between the real and the nominal exchange
rate? Give an example to explain this to someone who has not
studied economics. Is an increase in the real cost of imports an
improvement or a deterioration in the terms of trade?
(ii) A small open economy has a government budget surplus and a
trade deficit. Explain whether there is a private sector surplus,
deficit or balance. Examine the consequences in the short run for
output, the trade balance and the government budget balance of
a sudden fall in private consumption in this economy (due to an
epidemic in the small country) under (a) fixed exchange rates,
(b) flexible exchange rates. Use the Mundell-Fleming model with
perfect capital mobility. Explain the adjustment mechanisms.
(10+20=30)

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