ISI Admission Test 2020 Question Paper MS (QE) PEB

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ISI 2020
Question Paper · MS
(QE) PEB
EXAM YEAR TYPE SUBJECT

ISI 2020 Question Paper MS (QE) PEB

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.
.co s e m

s em l a
a ag

Group A
1. [30 marks: 5+10 +15].
o m
m . c
coX be independent and identically distributed random variables
Let X ,....,
s e m
em distribution on (0, θ] where θ > 0.
1 n

s
with a uniform
g l a
g la[5 marks] Write down the joint probability density function of X , ..., X . a
(a)
a
1 n

(b) [10 marks] Suppose xi is a realization of Xi , for each i = 1, ..., n. And
suppose the value of θ is unknown. Find the value of θ that maximizes
the joint p.d.f. in part (a) given that x1 , ..., xn have been observed. (This
is called the maximum likelihood estimate of θ.)

(c) [15 marks] Consider the function: f (x, y) = x2 + y 2 − 2x

(i) Find the maximum value of f over the region {(x, y) | 2x2 + 3y 2 −
2x ≤ 100}
m
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(ii) Find the minimum value of f over the region {(x, y) | 2x2 + 3y 2 −
m
e
2x ≥ 100}

2. [30 marks: 3+5+10+6+6] a
l s
g
A tournament consists ofan players and all possible C(n, 2) = n(n−1)
2 pair-
wise matches between them. There are no ties in a match: in any match, one
of the two players wins. The score of a player is the number of matches she
wins out of all her (n − 1) matches in the tournament. Denote the score vector
of the tournament as s ≡ (s1 , . . . , sn ) and assume without loss of generality
s1 ≥ s2 ≥ . . . ≥ sn .

(a) (3 marks) For any 2 ≤ k ≤ n, show that s1 + . . . + sk ≥ C(k, 2), where
C(k, 2) = k(k−1) .
o m
c
2

m (b) (5 marks) Suppose n > 3 and players 1, 2, 3 win every matchmagainst .
c. o s e
e m players in {4, . . . , n}. Find the value of s + . . . + s ?4

=gl
a
n

las (c) (10 marks) Suppose s = s , s n = s + 1, s
0 n−1
a 0s + 2 for some
n−2 0

ag positive integer s and n ≥ 3. Show that
0

(n − 2)(n − 3)
s0 ≤ .
2n

(d) (6 marks) A tournament generates a score vector s such that

sj − sj+1 = 1 for all j ∈ {1, . . . , n − 1}.

1
m . c
c. o s e m
s em
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Page 1 of 8

What is the score vector of this tournament? For every Player j, who does
Player j beat in this tournament?

(e) (6 marks) Suppose there are six players, i.e., n = 6. There is a tourna-
ment such that each player has a score of at least two and difference in
scores of any two players is not more than one. What is the score vec-
tor of this tournament? Construct a tournament (describing who beats
who) which generates this score vector.

3. [30 marks: 6+6+6+4+8]

Consider the following equation in x:

(x − 1)(x − 2) · · · (x − n) = k, (1)

where n > 1 is a positive integer and k is a real number. Argue whether
the following statements are true or false by providing a proof or a counter
example.

(a) (6 marks) Suppose n = 2. There is a real solution to Equation (1) for
every value of k.

(b) (6 marks) Suppose n = 3. There is a real solution to Equation (1) for
every value of k.

(c) (6 marks) For all k ≥ 0 and for every positive integer n > 1, there is a
real solution to Equation (1).

(d) (4 marks) For all k < 0 and for every odd positive integer n > 1, there
is a real solution to Equation (1)

(e) (8 marks) For all k < 0, there is some even positive integer n such that
a real solution to Equation (1) exists.

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Group B
m
co
1. Consider an economy inhabited by identical agents of size 1. A represen-

om .
tative agent’s preference over consumption (c) and labour supply (l) is

. c
given by the utility function
e m
e m l as
as u (c, l) = cα (24 − l)1−α , 0 < α < 1.
l Production of the consumption good c is given by the production func- ag
g
a tion c = Al, where A > 0 is the productivity of labour. Both the com-
modity market and labour market are perfectly competitive: the buyers
and sellers take the price as given while taking demand and supply de-
cisions. Let us denote the hourly wage rate by w > 0 and price of the
consumption good by p > 0.

(a) [25 marks: 13 + 7 + 5] Competitive Equilibrium:

A competitive equilibrium
m
is given by the allocation of consumption and

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labour, cCE , lCE , and the relative price ratio, wp , such that, given w

e m
and p, a representative agent decides her labour supply, lS , and con-

as
sumption demand, cD , to maximize her utility; a firm decides its labour
l
ag
demand, lD , and supply of consumption good, cS , to maximize its profit;
and, finally, both the commodity market and labour market clear, that
is, lD = lS and cD = cS .

(i) [13 marks] Set up the representative agent’s utility maximization
problem. Write down the first order conditions for this maximiza-
tion problem and determine lS and cD as functions of w and p.
(ii) [7 marks] Set up a firm’s profit maximization problem. Determine
lD and cS as functions of w and p.

m



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(iii) [5 marks] Determine the competitive equilibrium allocation, cCE , lCE ,
m and the relative price ratio, wp .

m .co s em
e la
(b) [5 marks] Pareto efficient allocation:

las a g
ag For this economy define the concept of a Pareto efficient allocation of
consumption and labour. Find out a Pareto efficient allocation of consumption
and labour in this economy. Provide a clear explanation.

3
m . c
c. o s e m
s em
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2. [30 marks = 12 + 18]

(a) [12 marks] Ms. A’s income consists of Rs.1,00,000 per year from pension
plus the earnings from whatever she sells of the 2,000 kilograms of rice
she harvests annually from her farm. She spends this income on rice (x)
and on all other expenses (y). All other expenses (y) are measured in
rupees, so that the price of y is Rs. 1. Last year rice was sold for Rs. 20
per kilogram, and Ms. A’s rice consumption was 2,000 kilograms, just
the amount produced on her farm. This year the price of rice is Rs. 30
per kilogram. Ms. A has standard convex preferences over rice and all
other expenses. Answer the following two questions without referring to
any utility function or indifference curves.

(i) [7 marks] What will happen to her rice consumption this year –
increase, decrease, or remain the same? Give a clear explanation
for your answer.
(ii) [5 marks] Will she be better or worse off this year compared to last
year? Explain clearly.

(b) [18 marks] There are two goods x and y. Mr. B has standard convex
preferences over the two goods. He has endowments of ex > 0 units of
good x and ey > 0 units of good y. He does not have any other source
of income. When the price of good y is Rs. 1 and the price of good x is
Rs. px , he decides neither to buy nor to sell good x.

(i) [8 marks] Suppose that, for good x, the prices have become Rs.
pL < px if an individual is a seller and Rs. pH > px if an individual
is a buyer. The price of good y remains Rs. 1 no matter whether
an individual buys or sells good y. Write down the equation of the
new budget constraint and draw it labelling the important points
clearly.
(ii) [10 marks] Will Mr. B buy or sell good x? By how much? Give
a clear explanation for your answer without referring to any utility
function or indifference curves.

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3. [30 marks = 6+7+7+10]
Consider a moneylender who faces two types of potential borrowers: the
m
om
safe type and the risky type. Each type of borrower needs a loan of the same
. co
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size L to invest in some project. The borrower can repay only if the investment
em
em
produces sufficient returns to cover the repayment. Suppose that the safe type
l as
l as
is always able to obtain a secure return of R from the investment, where R > L.
On the other hand, the risky type is an uncertain prospect; he can obtain a
ag
ag
higher return R0 (where R0 > R), but only with probability p. With probability
1 − p, his investment backfires and he gets a return of 0. The money lender has
enough funds to lend to just one applicant, and there are two of them – one
risky, one safe. Each borrower knows his own type, but the moneylender does
not know the borrower’s type. He just knows that one borrower is a safe type
and the other one is a risky type. Since the moneylender has enough funds
to lend to just one applicant, when both the borrowers apply for the loan, he
gives the loan randomly to one of them, say by tossing a coin. Assume that
the lender supplies the loan from his own resources and his opportunity cost
is zero.
o m
c
. rate, call it i , for which the
m
safe borrower wants the loan? eWhat is the highest interest rate, i , for
(a) [6 marks] What is the highest interest s

l as the loan? Who is willing to pay a higher r

ag or the safe borrower?
which the risky borrower wants
interest rate, the risky borrower
(b) [7 marks] The lender’s objective is to maximize his expected profit.
Argue clearly that the lender’s effective choice is between two interest
rates, is and ir . (That is, argue that the lender will not choose any
interest rate strictly lower than min {ir , is } , any interest rate strictly
higher than max {ir , is } , or any interest rate strictly in between ir and
is .)
(c) [7 marks] Argue that when the lender charges ir , his expected profit is
m
m .co
given by p (1 + ir ) L − L. Derive, with a clear argument, the expression

.co m
of lender’s expected profit when he charges is .

m s e
e la
(d) [10 marks] An equilibrium with credit rationing occurs when, at the

las equilibrium interest rate, some borrowers who want to obtain loans are
ag
ag unable to do so; however, lenders do not raise the interest rate to elimi-
nate the excess demand.

Explain clearly that we have an equilibrium with credit rationing when
R
p< .
2R0 − R

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m . c
c. o s e m
s em
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Group C
1. [30 marks=3+8+2+10+4+3]

Consider an economy where identical agents (of mass 1) live for two peri-
ods: youth (period 1) and old age (period 2). The utility function of a repre-
sentative agent born at time t is given by

u (c1,t , c2,t+1 ) = log (c1,t ) + β log (c2,t+1 ) ,

where c1 denotes consumption in youth, c2 denotes consumption in old age,
and 0 < β < 1 is the discount factor reflecting her time preference. In her
youth the representative agent supplies her endowment of 1 unit of labour
inelastically and receives the market-determined wage rate wt . So in her youth
the agent faces the budget constraint c1,t + st = wt , where st denotes her
savings. When old, she just consumes her savings from youth plus the interest
earning on her savings, st rt+1 , where rt+1 is the market-determined interest
rate in period t + 1. That is, when old, her budget constraint is c2,t+1 =
(1 + rt+1 ) st .

(a) [3 marks] Set up the agent’s utility maximization problem by showing
her choice variables clearly.
(b) [8 marks] Write down the first order conditions for this maximization
problem and derive the savings function. Explain how savings, st , if it
does, depends on the interest rate rt+1 .

The production function of the economy is given by Yt = AKtα Lt1−α ,
0 < α < 1, where K and L denote the amounts of capital and labour in
the economy, respectively. Capital depreciates fully after use, that is, the
rate of depreciation of capital is one. Factor markets being competitive, the
equilibrium factor prices are given by their respective marginal products.

(c) [2 marks] Derive the equilibrium wage rate (wt ) of the economy in terms
of Kt . [Keep in mind that the mass of agents is 1 and each agent supplies
her endowment of 1 unit of labour inelastically.]

The role of the financial sector (banks, stock market, and so on) is to mo-
bilize the savings of households to bring it for effective use by the production
sector. But the financial sector does not work well and a fraction 0 < θ < 1
of aggregate savings gets lost (vanishes in thin air) in the process of interme-
diation.

(d) [10 marks] Derive the law of motion of capital (that is, express capital
in period t + 1, Kt+1 , in terms of capital in period t, Kt ).

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(e) [4 marks] Derive the steady state amount of capital of the economy.

(f ) [3 marks] How does the steady state amount of capital depend on the
m
om
inefficiency of the financial sector θ?
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. c e m
m
2. [30 marks = 5+5+10+10]
e a Solow-Swan model with learning by doing. Assume that the gl as
l as
Consider
a
agproduction function is of the form
α 1−α
Y = K (A L )
t t t t

where A is the level of technological progress and grows at the rate g > 0, L
is the population with grows at the rate n > 0, K is the capital stock, Y is
GDP, and α ∈ [0, 1]. Assume that

K̇ = sY − δK

m
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K
Define Z = AL as the capital labor ratio in efficiency units. Let output per
worker be given by Q = AZ α . The parameter s ∈ [0, 1] denotes the savings

e m
rate. The parameter δ ∈ [0, 1] denotes the depreciation on capital.

l as for
g
Ż
(a) [5 marks] Derive an expression
a
Z

(b) [5 marks] Instead of assuming that the rate of technological progress is
constant (g), now assume that the instantaneous increase in A is pro-
portional to output per worker, i.e., there is learning by doing

Ȧ = γQ.

Show that the law of motion of capital is given by

Ż = (s − γZ)Z α − (δ + n)Z
o m
m (c) [10 marks] Draw a diagram describing the dynamics of growth c
.in the
c. o m
model with learning by doing. Plot Z on the x − axis, andethe appro-
s
e m l a
l as priate functions on the y − axis
agby doing, does an
ag (d) [10 marks] In contrast to the model with no learning
increase in the investment rate raise the balanced-growth rate ? What
does this tell you about the change in policy having level effects versus
growth effects in the model with learning by doing in contrast to the
model when there is no learning by doing ? Show your answer using the
diagram in part (c).

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m . c
c. o s e m
s em
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3. [30 marks =10+10+3+3+4 ]
PT n−t ln(c )
Suppose households who live till T periods maximize n=t β n
where cn represents their income in period n = t, t + 1, t + 2, ....T and β is
a parameter with 0 < β < 1. Suppose per period income and the saving
of households are yn and sn respectively and the activity starts from the
beginning of their life t. Further, the net interest rate on saving in between
any two periods is exogenously fixed at r and so the gross rate of return is
1 + r. Households have only two activities in every period - consuming and
saving.

(a) [10 marks] Write down the sequence of budget constraints (one for each
period) and the aggregate budget constraint derived from these periodic
budget constraints where on the left hand side, consumption levels for
all the periods appear, and on the right hand side, income in all periods
appears.

(b) [10 marks] Under what condition between β and r, is the optimal so-
f in every
lution for the above problem yield constant consumption, C,
period ?

(c) Suppose the condition that you derive in (b) holds. Then answer the
following questions in (c) and (d)

(i) [3 marks] For a transitory change in income in period t only, calcu-
e
late the change in the constant level of consumption, C.
(ii) [3 marks] For a transitory change in income in period t + k only,
e
calculate the change in the constant level of consumption, C.

(d) [4 marks] For a permanent change in income (assume the same amount
of income change in all periods), calculate the change in the constant
e Compare this value derived with part c (i).
level of consumption, C.

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