ISI Admission Test 2024 Question Paper MS (QE) PEB

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ISI 2024
Question Paper · MS
(QE) PEB
EXAM YEAR TYPE SUBJECT

ISI 2024 Question Paper MS (QE) PEB

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1. (a) Let f : R → R be a differentiable function. Show that
there always exists a point c > 1 such that f (2) − 2f (1) =
m
.co
cf � (c) − f (c). [4]
m
c.c o∈ (a, b) satisfying f (c) = max{f (x) : x ∈ (a, b)}, where
(b) Let f : (a, b) → R be a function such that there is some
s e m
em l a
l as (a, b) = {x ∈ R : a < x < b}. Assume that one-sided
ag
ag derivatives f−� (c) and f+� (c) exist. Show that f−� (c) ≥ 0 and
f+� (c) ≤ 0. [8]
(c) Let f be a function which is continuous on [0, 1] and dif-
ferentiable on (0, 1), with f (0) = f (1) = 0. Assume that
there is some c ∈ (0, 1) such that f (c) = 1. Prove that
there exists some x0 ∈ (0, 1) such that |f � (x0 )| > 2. [12]

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c
2. (a) A restaurant has a menu .card containing three Chinese
s emand five Continental items. Six
a random. Let X and Y denote respec-
items, six Indian items
items are selected lat
g
a of Indian and Continental items selected.
tively the number
Compute the conditional probability mass function of X
given that Y = 3. Also compute E(X|Y = 3). [8+4=12]
(b) Let the joint probability mass function of X and Y be

p(1, 1) = 1/9, p(2, 1) = 1/3, p(3, 1) = 1/9,
m
m p(1, 2) = 1/9, p(2, 2) = 0, p(3, 2) = 1/18,
.co
m .co s em
p(1, 3) = 0, p(2, 3) = 1/6, p(3, 3) = 1/9,
s e a
where p(x, y) = P (X = x, Y = y) for all xgl= 1, 2, 3 and
g la a
a y = 1, 2, 3. Find the correlation between X and Y . Also
find E(X|Y = 3). [8+4=12]

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a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 1 of 4

3. A consumer consumes two goods: visits to a nearby park (x) and
a composite consumption good (y), according to preferences u =
xy. The consumer’s income is R and the price of the composite
good is normalized to 1.
Initially, there is an entry fee p∗ per park visit. Suppose the
authorities are considering a proposal to reduce the per visit
entry fee from p∗ to p� .

(a) Find the increase in consumer’s surplus resulting from this
proposal. [3]
(b) Now suppose there is an alternative proposal to maintain
the per visit entry fee at the initial level p∗ , but hand out
a one-off cash voucher to the consumer (i.e., the consumer
would get the cash voucher only once, regardless of how
many times she visits the park). Find the minimum value
of the cash voucher that would make the consumer not
worse off under this alternative proposal, compared to the
earlier proposal to reduce the entry fee from p∗ to p� . [7]
(c) Next, suppose there is a third proposal which simultane-
ously reduces the per visit entry fee from p∗ to p� , and
charges a one-off lump-sum user fee (i.e., a fee which has
to be paid only once, regardless of the number of visits).
Find the maximum value of the lump-sum user fee that
would make the consumer not worse off under this third
proposal, compared to the initial situation (i.e., price p∗
per visit and no lump-sum payment). [7]
(d) Prove that C < ΔS < E, where ΔS is the solution to part
(a), E to part (b) and C to part (c) above. [7]

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m
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4. There is a worker who can purchase e units of education, e ∈
[0, ∞], at cost 2e2 /θ. The worker can be of high ability (θ = 2)
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or low ability (θ = 1), with the worker knowing her own ability

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(ability is exogenously given). Note that it is less costly for the
. to achieve a particular level of education if she were of s e
m
worker
ehigh ability than if she were of low ability. l a
as
l The worker can be hired by a firm paying wage w, and if hired, ag
g
a her marginal product is θ, where θ is her ability. The firm does
not know the worker’s ability (it knows θ is either 1 or 2), but
might be able to infer it after observing her education choice.
In particular, if the firm believes the worker is of high ability,
it pays wage w = 2, while it pays wage w = 1 if it believes the
worker is of low ability.

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c
The worker chooses e to maximize utility, and if she works for

m . given e and θ, is u(w, e|θ) =
the firm at wage w, then her utility,
s e
a
2
w − 2e /θ.
g l
a
(a) Draw the indifference map of the worker in the education-
wage space (assume education to be measured along the
horizontal axis and wage to be measured along the vertical
axis) if she were of high ability. Also draw the indifference
map of the worker if she were of low ability. How many
times can an indifference curve of a worker who is of low

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ability intersect an indifference curve of a worker who is of
m . c
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high ability? [2+2+2]
m
e upon
e m s
lathe worker
(b) Suppose the firm perfectly infers the worker’s ability

las a g
observing her education choice. Suppose also

ag chooses e = 0 if she is of low ability. Then how much
education would she purchase if she were of high ability?
[9]
(c) Continue to assume that the firm perfectly infers the
worker’s ability upon observing her education choice. Sup-

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a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 3 of 4

pose, if the worker is of high ability, her education choice
matches what you found in part (b). Then show that she
would purchase no education if she were of low ability. [9]

5. An individual lives for two periods, 1 and 2, and has lifetime util-
ity function U (C1 ) + βU (C2 ), where C1 and C2 are the consump-
tions of this individual in period 1 and period 2 respectively,
and 0 < β < 1 is the subjective discount factor. The follow-
ing conditions are satisfied: U � (C) > 0, U �� (C) < 0, lim U � (C) =
C→0
∞, lim U � (C) = 0.
C→∞
The individual earns an exogenously given income w > 0 in pe-
riod 1 of his life and earns nothing in the second period of his
life. But he can lend or borrow freely at an exogenously given
rate of interest r > 0.
It follows that C1 = w − S and C2 = (1 + r)S, where S is the
savings made by the individual in period 1. Note that C1 (or S)
and C2 are endogenously determined and the exogenously given
parameters of the model are w, r and β.

(a) Suppose β(1 + r) = 1. Solve for C1 and C2 in terms of the
exogenously given parameters. [4]
(b) Now remove the restriction that β(1 + r) = 1. How would
savings S be affected if β goes up, ceteris paribus? [8]
(c) Continuing without the restriction that β(1 + r) = 1, show
that savings S goes up as w increases, ceteris paribus. Fur-
ther show that the increase in S is less than the increase in
w. [8+4]

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