ISI Admission Test 2023 Question Paper MS (QE) PEB

FOR ISI EXAM PREPARATION

ISI 2023
Question Paper · MS
(QE) PEB
EXAM YEAR TYPE SUBJECT

ISI 2023 Question Paper MS (QE) PEB

Notes · Sample Papers · Previous Year Papers · Mock Tests

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Group A (Microeconomics)

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1. [30 marks: 4 + 20 + 6]

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Consider a used car market with 600 buyers each willing to buy
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m car. Out of the 500 used cars, 400 are of good quality
exactly one used car, and 500 sellers each having exactly one
eused l as
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la (peaches) and 100 are of bad quality (lemons). The monetary ag
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a valuation of owning a peach is Rs. 100 for a buyer and Rs. 50 for
a seller. On the other hand, the monetary valuation of owning
a lemon is Rs. 10 for both a buyer and a seller. A seller knows
whether the car she owns is a peach or a lemon, whereas a buyer
only knows that there are 400 peaches and 100 lemons. Both
the buyers and the sellers know the various valuations.

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.your answer.
(a) What outcome maximizes the aggregate surplus of the economy?

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Provide a clear explanation for

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l explanation, the supply of used cars
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(b) (i) Derive, with a clear
a price. Draw this supply curve by plotting
as a function of
number of used cars on x-axis and price on y-axis. [You
must label all the important points in the gure clearly.]
(ii) Derive, with a clear explanation, the demand for used cars
as a function of price. Draw this demand curve in the same
gure as in part (i). [You must label all the important
points in the gure clearly.]
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m (iii) Use the demand and supply functions above to nd .out c
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emarket
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all possible competitive equilibria in the used car

las mentioning clearly which types of car, lemongor
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ag bought and sold in each equilibrium.

(c) Now suppose that buyers also know the identity of all cars, that
is, whether any given car is a peach or a lemon. Use a similar
demand-supply analysis as above to solve for all possible com-
petitive equilibria in the used car market in this scenario.

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2. [30 marks: 5 + 3 + 3 + 11 + 8]

Consider an industry with 2 rms – a private rm (indexed by
r) and a public rm (indexed by u) – producing a homogeneous
product and competing in quantities. The rms face an inverse
demand function p = a − bQ, a > 0, b > 0, where Q = qr + qu
denotes aggregate output, and qr and qu denote the amounts of
output produced by the private and public rms respectively.
Each rm i faces the total cost of production cqi , i = r, u, 0 <
c < a.

(a) For any qr and qu , derive the expressions for (i) private rm’s
pro t, (ii) public rm’s pro t, (iii) consumer surplus, and (iv)
welfare (sum of consumer surplus and producer surplus).

(b) The private rm’s objective is to maximize its own pro t. For
a given qu , set up the private rm’s maximization problem and
derive its optimal choice of output qr . [This exercise gives you
the reaction function of the private rm.]

(c) The public rm’s objective is to maximize welfare. For a given
qr , set up the public rm’s maximization problem and derive its
optimal choice of output qu . [This exercise gives you the reaction
function of the public rm.]

(d) Recall that the two rms compete in quantities.

(i) De ne the concept of equilibrium in this context and nd
out the amounts of output, qr∗ and qu∗ , the two rms produce
in equilibrium. Find out the expressions of price, pro ts of
the two rms, consumer surplus and welfare in the equilib-
rium.

(ii) Illustrate this equilibrium by drawing the two reaction func-
tions you have derived in parts (b) and (c) (plot qu in x-axis

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and qr in y-axis). [You must label the important points in
the gure clearly.]
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(e) Suppose that the marginal cost of the private rm falls to cr < c
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while the marginal cost of the public rm remains the same at
ec.mDraw the new reaction functions and explain clearly how the l as
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la following outcomes change in the new equilibrium (as compared ag
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a to the old equilibrium): q , q , Q, price, pro ts of the two rms,
r u

consumer surplus and welfare. [There is no need to derive the
exact expressions; just qualitative answers are enough.]

3. [30 marks: 6 + 4 + 20]

There is a unit mass of consumers all of whom want to purchase

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at most 1 unit of a good. Consumer v has a valuation v for this
good, where v ∈ [0, 1]. Assume that v is uniformly distributed

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over interval [0, 1] so that the number of consumers with valua-
tion in between a and b,lawhere 0 ≤ a < b ≤ 1, is b − a. There is
a monopoly rm with a gtotal cost of producing q units of the good
q
given by . The rm does not know the identity of any consumer
3
and hence must charge a uniform price to all the consumers.

(a) For any price p, derive, with a clear explanation, the demand
facing the monopoly rm.

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(b) Derive, with a clear explanation, the monopoly price and pro t
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level.
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(c) Suppose that the rm can, for a cost, get to knowawhether a
l ( , 1].
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consumer belongs to the interval [0, ], or to the g4
interval 4

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What is the maximum amount the rm is willing to pay for this
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information? Give a clear explanation for your answer.

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Group B (Macroeconomics)
1. [30 marks: 4 + 6 + 10 + 5 + 5]

Consider an aggregate demand and aggregate supply model
where, in the short run, aggregate capital is xed at the level K.
The aggregate demand curve, aggregate output (Y ) demanded
as a function of aggregate price level (P ), is given by a standard
downward-sloping curve. The aggregate supply curve, aggregate
output (Y ) supplied as a function of aggregate price level (P ), is
not standard, and the question leads you to derive the aggregate
supply curve.

The aggregate production function is linear in capital and labour
(L): Y = AL+K, A > 0. The labour union is very powerful and
dictates the minimum aggregate nominal wage rate as W . Each
worker is endowed with one unit of labour which they supply
inelastically if the producers o er the nominal wage W > W .
A worker does not supply any labour if W < W . At W = W ,
a worker is indi erent between supplying and not supplying her
labour endowment. The number of workers available in the econ-
omy is xed at L.

(a) Derive, with a clear explanation, the aggregate labour supply
(LS ) in this economy as a function of the aggregate nominal
wage rate, W.

(b) Note that the marginal product of labour is constant, A > 0.

(i) Derive, with a clear explanation, the aggregate labour de-
mand (LD ) in this economy as a function of the real wage
rate, W
P
.
(ii) Using your answer to part (i) above, derive the aggregate
labour demand (LD ) in this economy as a function of the
aggregate nominal wage rate, W.

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(c) Choose an arbitrary aggregate price level, P, and draw the ag-
gregate labour supply (LS ) and aggregate labour demand (LD )
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curves, as functions of W , by plotting labour (L) on x-axis and

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nominal wage (W ) on y-axis. Think about the labour market
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equilibrium
ehave chosen.
for the arbitrary aggregate price level P that you
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as
l Note that the equilibrium employment (L ) in the economy de- ag
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pends on the arbitrary price level P that you choose. Derive,
with a clear explanation, the equilibrium employment (L∗ ) as a
function of aggregate price level P.

(d) Derive, with a clear explanation, aggregate output (Y ) supplied
as a function of aggregate price level (P ). Draw this aggregate
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supply curve by plotting Y on x-axis and P on y-axis.

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la Explain the e ectiveness of the stan-
(e) Recall that the aggregate demand curve is given by a standard

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downward-sloping curve.
dard monetary and a scal policies in this set up.

2. [30 marks: 14 + 16]

Consider the following version of the Solow growth model. The
aggregate output at time t, Yt , depends on the aggregate capital
stock (Kt ) and aggregate labour force (Lt ) in the following way:
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Yt = (Kt )α (Lt )1−α , 0 < α < 1.
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There is perfect competition in the factor market aso that, in
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ag equilibrium, each factor is paid its marginal product
total output is distributed to all the households in the form of
wage earnings and interest earnings. Households save a propor-
tion 0 < s < 1 of their disposable income in every period. All
household savings are invested which augment the capital stock
over time. There is no depreciation of capital. Population and

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therefore the aggregate labour force grows at a constant rate
n > 0.

(a) The government taxes the interest earnings at the rate 0 < τ <
1. Wage earnings are not taxed. The government uses the col-
lected taxes to fund government consumption; in particular, the
tax collection is not used for investment at all.

(i) Derive, with clear explanations, the expressions for aggre-
gate wage earning, aggregate interest earning and aggregate
savings (St ) of the economy in terms of Yt .
(ii) De ne kt ≡ Kt
Lt
, the capital-labour ratio in period t. Derive,
with a clear explanation, the law of motion of capital-labour
ratio, that is, the equation with kt+1 on the left-hand side
and kt on the right-hand side.
(iii) Derive, with a clear explanation, the steady-state level of
capital-labour ratio in this economy, k ∗ , and examine how
k ∗ changes with changes in the tax rate τ.

(b) As in part (a) above, the government continues taxing interest
earnings at the rate τ and wage earnings are not taxed. But
consider now that the tax revenue collected is used to fund in-
vestment by the government so that the capital stock is further
augmented by this public investment.

(i) Derive, with a clear explanation, the expression for aggre-
gate investment in this economy.
(ii) Derive, with a clear explanation, the new law of motion of
capital-labour ratio.
(iii) Derive the new steady-state level of capital-labour ratio
in this economy, k ∗∗ , and compare it with k ∗ . Does the
comparison make economic sense?

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(iv) How does k ∗∗ change with changes in τ ? Compare with the
response of k ∗ and explain the economic reason behind the
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di erential impact.
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3. [30 marks: 9 + 5 + 6 + 4 + 6]
m an individual who lives for two periods. In the rst s e m
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a savings decision once income is realized. In the second period,
she has no wage income but receives the return along with her
principal amount of savings, s. The gross rate of interest is
R > 1. Suppose that there is a government which collects an
amount T in the form of lumpsum tax from the wage income
in the rst period (this can be considered as mandatory savings

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of the individual) and returns the amount T in the form of a
c
lumpsum transfer in the second .period. Suppose that the utility
em consumes c in the rst period
derived by the individual swho
la is given by u (c ) + βu (c ) , where β
1

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is a discount factor a
and c in the second period
2 1 2

with 0 < β < 1 and the utility function u is
strictly increasing and strictly concave.

(a) Set up the individual’s utility maximization problem by specify-
ing her budget constraint clearly. Derive the rst-order condition
of this utility maximization problem by showing your procedure

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clearly. Provide a clear economic interpretation of the rst-order

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(b) Note that β < 1 implies that the individual is myopic (short-
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sighted), she puts less weight on future period. Explain
g a more or less
a
itively whether a more myopic individual will save
than a less myopic individual. Verify your intuition by deter-
ds
mining the sign of .
dβ
(c) Note also that the individual’s personal savings, s, depends
on the government mandated savings T . Explain intuitively

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whether the government mandated savings increases or decreases
personal savings. Verify your intuition by determining the sign
ds
of .
dT
(d) One rupee received in bene ts in period 2 would require an in-
1
dividual to save an amount (< 1 since R > 1) in period
R
1. Explain intuitively whether the government mandated sav-
ings make the individual cut back her personal savings at a rate
1
higher or lower than . Verify your intuition.
R
(e) In the light of your answer to part (b) and from the expression
ds
of you expect that the rate of change in personal savings
dT
in response to a change in T depends on the discount factor β.
Prove that more myopic individuals reduce their personal savings
at a higher rate.

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Group C (Mathematics)
1. [30 marks: 20 + 10]
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Consider the following feasible region C in R for an optimization
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program:
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g l a C := {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, y ≤ x}. ag
a
(a) Suppose a ̸= 0, b ̸= 0. Show that an optimal solution to

max ax + by
(x,y)∈C

is either (0, 0), (1, 0), (1, 1). Describe all possible values of a and
b for which each of {(0, 0), (1, 0), (1, 1)} is an optimal solution.

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(b) Suppose a and b are independently drawn from [−1, 1] using
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a probability distribution with cumulative distribution function
e that the unique optimal solu-
as
(cdf) F . What is the probability
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tion to the above optimization problem is (1, 0)?

2. [30 marks: 5 + 10 + 15]

Suppose A, B, C, a, b, c are real numbers and A ̸= 0, a ̸= 0.
Suppose for all real values of x, the following holds:

|ax2 + bx + c| ≤ |Ax2 + Bx + C|.

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Suppose B 2 − 4AC > 0.
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a
a (c) Show that B 2 − 4AC ≥ b2 − 4ac.

3. [30 marks: 5 + 5 + 15 + 5]

Let F be a class of functions from [0, ∞) to [0, ∞) with the
following properties:

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– The functions f1 (x) = ex − 1 and f2 (x) = ln(x + 1) are in
F.
– If f (x) and g(x) (not necessarily distinct) are in F, then
the functions f (x) + g(x) and f (g(x)) are in F.
– If f (x) and g(x) are in F and f (x) ≥ g(x) for all x ∈ [0, ∞),
then f (x) − g(x) is in F.

(a) For any positive integer n, show that h(x) = nx is in F.

(b) If f (x) and g(x) are in F , show that the functions ln(f (x) + 1)
and ln(g(x) + 1) are in F.

(c) If f (x) and g(x) are in F, show that the function f (x)g(x) +
f (x) + g(x) is in F.

(d) If f (x) and g(x) are in F, show that the function f (x)g(x) is in
F.

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