ISI Admission Test 2022 Question Paper MS (QE) PEB

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ISI 2022
Question Paper · MS
(QE) PEB
EXAM YEAR TYPE SUBJECT

ISI 2022 Question Paper MS (QE) PEB

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Group A

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1. Answer the following questions.

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g la(a) Find all maxima and minima of the function 𝑓(𝑥, 𝑦) = 𝑥𝑦, ag
a subject to the constraints 𝑥 + 4𝑦 = 120 and 𝑥, 𝑦 ≥ 0.
(b) Find the points on the circle 𝑥 + 𝑦 = 50 which are closest to
and farthest from the point (1,1).
(c) For what values of 𝛼 are the vectors (0,1, 𝛼), (𝛼, 1,0) and
(1, 𝛼, 1) in ℛ linearly independent?
[10+15+5]
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2. Let 𝑓: ℛ → ℛ be a continuous function.

(a) Let 𝑄 denote the set of rational numbers. Prove that, if
𝑓(𝑄) ⊆ {1,2,3, … }, then 𝑓 is a constant function.
(b) Calculate the value of 𝑓′(0) when 𝑓 is differentiable and
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[20+10]

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3. Suppose a set of 𝑁 = {1,2, … , 𝑛} political parties participated in an
election; 𝑛 ≥ 2. Suppose further that there were a total of 𝑉
voters, each of whom voted for exactly one party. Each party
𝑖 ∈ 𝑁 received a total of 𝑉 votes, so that 𝑉 = ∑ 𝑉 . Given the
vector (𝑉 , 𝑉 , … , 𝑉 ), whose elements are the total number of votes
received by the 𝑛 different parties, define 𝑃 (𝑉 , 𝑉 , … , 𝑉 ) as the
probability that two voters drawn at random with replacement
voted for different parties and define 𝑃 (𝑉 , 𝑉 , … , 𝑉 ) as the
probability that two voters drawn at random without replacement
voted for different parties. Answer the following questions.

(a) Derive the ratio as a function of 𝑉 alone.

(b) Consider the special case where 𝑉 = for all 𝑖 ∈ 𝑁. For this
case, find the probabilities 𝑃 and 𝑃 .
[25+5]

Group B
4. Consider an agent living for two periods, 1 and 2. The agent
maximizes lifetime utility, given by:
1
𝑈 (𝐶 ) + 𝑈 (𝐶 ),
(1 + 𝜌)
where 𝜌 > 0 captures the time preference, while 𝐶 and 𝐶 are the
agent’s consumption in period 1 and period 2, respectively. The
agent supplies one unit of labor inelastically in period 1, earning a
wage 𝑤. A portion of this wage is consumed in period 1 and rest is
saved (denoted 𝑠). In period 2 the agent does not work, but receives
interest income on the savings. Principal plus the interest income on
savings goes to finance period 2 consumption. Thus, 𝐶 + 𝑠 = 𝑤 and
𝐶 = (1 + 𝑟)𝑠, where 𝑟 is the rate of interest. Assume that the per
period utility function can be represented by (and only by) any
positive linear transformation of the form 𝑈(𝐶) = , where
0 < 𝜃 < 1.
2

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(a) Demonstrate, deriving your claim, how optimal savings, 𝑠, would
respond to changes in 𝑟 .
(b) Now suppose, initially, 𝑟 = 𝜌. What happens to optimal savings,
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𝑠, if 𝑟 and 𝜌 increase by the same amount (so that the condition
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𝑟 = 𝜌 continues to hold)?
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[20+10]
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5.l A profit maximizing monopolist produces a good with the cost
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a function 𝐶(𝑥) = 𝑐𝑥, 𝑐 > 0, where 𝑥 is the level of output, 𝑥 ≥ 0. It
sells its entire output to a single consumer with the following utility
function:
𝑢(𝑦) = 𝜃 𝑦 − 𝑇(𝑦);

where 𝑦 is the amount of the good purchased by the consumer and T
is the payment made by the consumer to the monopolist to purchase
the output; 0 ≤ 𝑦 ≤ 𝑥; 𝜃 > 0. Suppose
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𝑇(𝑦) m
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s e = 𝑝𝑦 + 𝑡;
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where 𝑝 ≥ 0, 𝑡 ≥ 0 if 𝑦 l> 0, and 𝑇(0) = 0. Thus, in order to
ag of the good, the consumer may have to
purchase any positive amount
pay a lump-sum amount 𝑡, or a per unit price 𝑝, or both.
(a) Find the profit of the monopolist when it can choose any
non-negative combination of 𝑡 and 𝑝.
(b) Find the profit of the monopolist when it can choose any
non-negative p, but is forced to set 𝑡 = 0. Calculate how this
profit relates to the profit derived in part (a) and explain your
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(c) Calculate when social surplus is higher, explaining your result.

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(d) Calculate when consumer’s surplus is higher, explaining

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6. Answer the following questions.
(a) Let the input demand functions of a profit-maximizing
competitive firm operating at unit level of output be given by:
( / ) ( / )
𝑥 = 1 + 3𝑤 𝑤 and 𝑥 = 1 + 𝑏𝑤 𝑤 ;

where 𝑤 and 𝑤 are input prices. Find the values of the
parameters 𝑎, 𝑏 and c.

(b) Check whether the following data, summarizing the observed
input-output choices of a competitive firm under three different
output-input price situations, are consistent with the hypothesis of
profit maximization by that firm.

P w q x
Observation 1 50 20 20 25
Observation 2 45 15 24 36
Observation 3 40 20 16 16

Here 𝑝 and 𝑤 denote output price and input price, respectively,
while 𝑞 and 𝑥 denote, respectively, the units of output supplied
and input demanded by the firm. Each of the three rows specifies
an observation of the output-input price configuration and the
output-input choice of the firm under that particular price
configuration.

(c) Suppose, for the production function 𝑓(𝑥 , 𝑥 ), the cost function
of a competitive firm is 𝑐(𝑞; 𝑤) = 𝑤 𝑤 𝑞, where 𝑤 =
(𝑤 , 𝑤 ) is the input price vector and 𝑞 is the level of output;
𝛼 ∈ (0,1). Derive the conditional input demand functions and the
production function of the firm.
[10+10+(5 +5)]

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Group C

7. Consider a world economy consisting of Home (H) and Foreign (F).
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Each of these countries produces a single good that is both consumed

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domestically and exported. Let Foreign output be the numeraire and

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let p be the relative price of the H produced good. Assume full
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employment in both countries, so that H produces a fixed output Y
g l and F produces a fixed output Y  . Let E be the Home expenditure
a in terms of its own good and let E  be the Foreign expenditure
measured in terms of the foreign good. We will treat E as a
parameter of the model, while 𝐸 ∗ is endogenous. Assume that
consumers have Cobb-Douglas utility functions with fixed
expenditure shares. Let  be the share of expenditure of Home
consumers on the Foreign produced good and let   be the share of

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expenditure of Foreign consumers on the Home produced good.

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Assume further 1      (i.e., the expenditure share of Home
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consumers on the Home produced good is greater than the
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expenditure share of Foreign consumers on the Home produced

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good). World income equals world expenditure, and goods markets
clear.

Now, suppose E falls.

(a) What will happen to 𝑝?

(b) What will happen to the trade balance of Home, denominated in
units of the Foreign good (i.e., to 𝑝(𝑌 − 𝐸))?
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Prove your claims.

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8. Consider the Solow growth model with constant average propensity to
save 𝑠, labor supply growth rate 𝑛, no technological progress and
zero rate of depreciation. Let 𝑣 denote the capital-output ratio.

(a) Prove that, at the steady state, = 𝑛.

(b) Now suppose that, in some initial situation, > 𝑛. Explain how
market forces will operate to restore, over time, the equality
= 𝑛.

(c) In the process of adjustment in (b), in which direction will the real
wage and real rental on capital change? Explain.
[8+16+6]

9. Answer the following questions.
(a) Using a simple Keynesian model of income determination, derive
and explain the conditions under which a rise in the marginal
propensity to save will reduce aggregate savings in the economy.
(b) Using a model of aggregate demand and aggregate supply, explain
how an increase in fuel prices would impact aggregate output,
employment and the price level.
[15+15]

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