ISI Admission Test 2026 Sample Paper MS (QE) PEB

FOR ISI EXAM PREPARATION

ISI 2026
Sample Paper · MS
(QE) PEB
EXAM YEAR TYPE SUBJECT

ISI 2026 Sample Paper MS (QE) PEB

Notes · Sample Papers · Previous Year Papers · Mock Tests

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1. Consider a monopoly firm facing a market demand function p =
12 → q, where p is price, and q is quantity. The monopolist has
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a single plant that can produce an output of q at a cost of C(q),
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2
where C(q) = 2 + q2 , if q > 0, and C(q) = 0 otherwise.
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(a) Find the optimal monopoly output. What is the
la deadweight loss, and social welfare at the optimal ag
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a monopoly output?
(b) Suppose the monopoly firm can price discriminate perfectly,
and can also sell in the world market for a constant price
of 8. Solve for the optimal monopoly outcome.
(c) Next suppose the monopoly firm has access to two plants,
1 and 2. Plant i, i = 1, 2, can produce an output of qi at a
cost of qi2 /2.
o m
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. cost function, i.e. the
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e to produce a total output
i. Solve for the firm’s aggregate

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glone, or both the plants. Use the
total cost in case it decides

a
of q, using either
aggregate cost function to solve for the optimal
monopoly output.
ii. What is the optimal monopoly outcome if the
monopoly firm can price discriminate perfectly?

[8 + 6 + 10 = 24]
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m 2. An economy consists of two agents 1 and 2 and an initial
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e building of a bridge: d = 1 if the bridge is built and d =a0 if it
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is not. The bridge costs Rs 60 to build. If the bridge
a the remaining Rs 40 is distributed among the two agents; if it
is not built, Rs 100 is distributed among the agents. An
allocation in the economy consists of a triple (d, x1 , x2 ) where
x1 + x2 = 40 if d = 1 and x1 + x2 = 100 if d = 0. In each case
x1 , x2 ↑ 0. Agent i, i = 1, 2 has a valuation vi ↑ 0 for the

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bridge. Agent i’s utility from the allocation (d, x1 , x2 ) is
vi d + xi .

(a) Suppose v1 = 25, v2 = 45. Is the allocation (d = 0, 40, 60)
Pareto-e!cient? Justify your answer.

(b) Suppose v1 = 25, v2 = 45. Is the allocation (d = 0, 70, 30)
Pareto-e!cient? Justify your answer.

(c) Show that if the allocation (d = 1, x1 , x2 ) is Pareto-e!cient,
then v1 + v2 ↑ 60.

[5 + 5 + 14 = 24]

3. (a) An economist collects data from an experiment. Each data
point is one of the two types: (i) male or (ii) female. The
date is collected and sent in three boxes. One box contains
data of only male type, another box contains data of only
female type, and a third box contains data of both male
and female type. When the three boxes reach the o!ce of
the economist, there are labels on each box (see Figure 1).
The economist is told that the labels in every box is wrong.

Box 1 Box 2 Box 3

male female male & female

Figure 1: Each box is incorrectly labeled

The economist asks an MSQE student to sample data from
the boxes and figure out the correct labels. What is the
minimum number of samples that the MSQE student needs
to draw to figure out the correct labels of all the boxes?

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Describe your answer logically by showing which box(es)
need to be sampled and how many times.
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(b) Three students are standing in a straight line. Student 1 at
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the front, student 2 next, and student 3 at the last position.
m are 3 red hats and 2 blue hats. A teacher comes s e m
s eand
There
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a the colour of the hats of students in front of her, but not
her own hat or the hats of students behind her. So, student
3 sees the colour of the hats of student 1 and student 2;
student 2 sees the colour of the hat of student 1; and student
1 does not see the colour of anyone’s hat. Starting with
student 3, followed by student 2, and finally student 1,

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each student is asked if she knows the color of her own hat.

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The students can either answer yes or no. Assume students
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answer truthfully and answer of each student is revealed to
all the students.
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i. What is the probability that student 3 says no?
ii. Suppose student 3 says no. What is the probability
that student 2 says no? Note that student 2 knows the
answer of student 3.
iii. Suppose student 3 and student 2 both say no. What
is the probability that student 1 says no? Note that
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student 1 knows the answers of student 3 and student
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2.

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la= 23]
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[5 + 18
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4. (a) How many real solutions does the following equation have?
2
(x2 → 5x + 6)(x →7x+12) = 1

(b) A function f : ↓ ↔ ↓ is called quasi-convex if for every

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x, y ↗ ↓ and every ω ↗ [0, 1], we have

f (ωx + (1 → ω)y) ↘ max{f (x), f (y)}

i. Show that a convex function is quasi-convex.
ii. Show that a non-decreasing function is quasi-convex.
Note that f is non-decreasing if x > y implies f (x) ↑
f (y).

[10 + 5 + 4 = 19]

1→ω
5. (a) Suppose that utility function is given by u = ln c + b (1→l)
1→ω
where c and l represent consumption and labour supply, and
parameters b and ε are positive. Further, the wage rate per
unit of labour supply is given by w. Agents consume out of
their labour income which forms their budget constraint. If
agents maximise utility subject to their budget constraint,
how does labor supply depend on the real wage rate w.
Clearly show all the derivations and explain the result.

(b) Now consider that agents live for two periods and they
have the following utility function:
! !
u = ln c1 + 2b (1 → l1 ) + ϑ[ln c2 + 2b (1 → l2 )] where ci ,
li represent consumption and labour supply in period
i = 1, 2 respectively, the parameter ϑ is positive while the
rest of the notation is the same as in part (a) above.
Further, any consumption made or income earned in the
second period is discounted at the rate, 1 + r. Agents
maximise their utility subject to the life-time budget
constraint. Further, leisure in period i = 1, 2 is denoted by
qi = 1 → l i .

i. Find out the expression for the ratio of the optimal qq12
w2
in terms of the wage ratio, w 1
, and other parameters

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of the model. How does l1 vary with w
w1
2
? Explain the
result.
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1→l1
ii. Let q ↑ be the ratio of optimal q1 to q2 , that is, 1→l = q↑
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c.toow . Interpret your result in a couple of sentences.
2

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w2
and w = w↑ . Calculate the elasticity of q ↑ with respect

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1

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↑

l as [8 + 15 + 7 = 30] ag
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