ISI Admission Test 2016 Syllabus and Sample Paper MS (QE) PEB

FOR ISI EXAM PREPARATION

ISI 2016
Syllabus and Sample
Paper · MS (QE) PEB
EXAM YEAR TYPE SUBJECT

ISI 2016 Syllabus and Sample Paper MS (QE) PEB

Notes · Sample Papers · Previous Year Papers · Mock Tests

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SYLLABUS FOR MSQE
(Program Code: MQEK and MQED) 2016

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Syllabus for PEA (Mathematics), 2016

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Algebra: Binomial Theorem, AP, GP, HP, Exponential, Logarithmic Series,
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Sequence, Permutations and Combinations, Theory of Polynomial Equations;
(up to third degree).

Matrix Algebra: Vectors and Matrices, Matrix Operations, Determinants.

Calculus: Functions, Limits, Continuity, Differentiation of functions of one
or more variables. Unconstrained Optimization, Definite and Indefinite
Integrals: Integration by parts and integration by substitution, Constrained
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optimization of functions of not more than two variables.

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Elementary Statistics: Elementary probability theory, measures of central

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tendency, dispersion, correlation and regression, probability distributions,

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standard distributions-Binomial and Normal.

Syllabus for PEB (Economics), 2016

Microeconomics: Theory of consumer behaviour, theory of production,
market structure under perfect competition, monopoly, price discrimination,
duopoly with Cournot and Bertrand competition (elementary problems) and
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Macroeconomics: National income accounting, simple Keynesian Model of
s e income determination and the multiplier, IS-LM Model, models of aggregate
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PEB (2016)
Answer any 6 questions. All questions carry equal marks.

1. Consider an exchange economy consisting of two individuals 1 and 2, and two goods,
X and Y. The utility function of individual 1 is U1 = X1 + Y1 , and that of individual 2 is
min{X2 , Y2 }, where Xi (resp. Yi ) is the amount of X (resp. Y ) consumed by individual i,
where i = 1, 2. Individual 1 has 4 units of X and 8 units of Y, and individual 2 has 6 units
of X and 4 units of Y to begin with.

(i) What is the set of Pareto optimal outcomes in this economy? Justify your answer.

(ii) What is the competitive equilibrium in this economy? Justify your answer.

(iii) Are the perfectly competitive equilibria Pareto optimal?

(iv) Now consider another economy where everything is as before, apart from individual
2’s preferences, which are as follows: (a) among any two any bundles consisting of X and
Y, individual 2 prefers the bundle which has a larger amount of commodity X irrespective
of the amount of commodity Y in the two bundles, and (b) between any two bundles with
the same amount of X, she prefers the one with a larger amount of Y . Find the set of
Pareto optimal outcomes in this economy. [6]+[6]+[2]+[6]

2. Consider a monopolist who can sell in the domestic market, as well as in the export
market. In the domestic market she faces a demand pd = 10 − qd , where pd and qd are
domestic price and demand respectively. In the export market she can sell unlimited
2
quantities at a price of 4. Suppose the monopolist has a single plant with cost function q4 .

(a) Solve for total output, domestic sale and exports of the monopolist.

(b) Solve for the domestic and world welfare at this equilibrium. [10]+[10]

3. A consumer consumes electricity, denoted by E, and butter, denoted by B. The per
unit price of B is 1. To consume electricity the consumer has to pay a fixed charge R, and
a per unit price of p. If consumption of E ≤ 12 then p = 1; otherwise p = 2. The utility
function of the consumer is 3E + B, and her income is I > R.

(i) Draw the consumer’s budget line.

(ii) If R = 0 and I = 1, find the consumer’s optimal consumption of E and B.

(iii) Consider a different pricing scheme where there is a rental charge of R and the price
of E is 1 for any X ≤ 1/2, and every additional unit beyond 21 is priced at p = 2. Find the
optimal consumption of B and E when R = 1 and I = 3. [7]+[7]+[6]
1

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4. A monopoly publishing house publishes a magazine, earning revenue from selling the
magazine, as well as by publishing advertisements. Thus R = q.p(q) + A(q), where R is
total revenue, q denotes quantity, p(q) is the inverse demand function, and A(q) is the
advertising revenue. Assume that p(q) is decreasing and A(q) is increasing in q. The cost
of production c(q) is also increasing in the quantity sold. Assume all functions are twice
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differentiable in q.

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(i) Derive the profit-maximising outcome.
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emmarginal revenue curve necessarily negatively sloped?
(ii) Issthe
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duction?[7]+[7]+[6]

5. Consider a Solow style growth model where the production function is given by
Yt = At F (Kt , Ht )
where Yt = output of the final good, Kt is the capital stock, At = the level of technology,
and Ht = the quantity of labor used in production (the labor force). Assume technology
is equal to At = A0 (1 + α)t where α > 0 is the growth rate of technology, A0 is the time 0
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level of technology, and Ht+1 = (1+n)Ht , where n > 0 is the labour force growth rate. The
production function is homogenous of degree 1 and satisfies the usual properties. (Assume

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that inputs are essential and Inada conditions hold). Assume that capital evolves according
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where It is the level of investment.ag
Kt+1 = (1 − δ)Kt + It

Yt
(i) Define yt = H t
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yt = At f (kt )
where f (k) = F (k, 1).

(ii) Define kt = K It
Ht and it = Ht . Show that
t

(1 − δ)kt + it

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kt+1 =

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1+n

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(iii) Suppose the savings rate is given by st = σyt where σ ∈ [0, 1]. Derive the condition

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that determines the steady state capital stock when α = 0. How many non-zero steady
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states are there ?

ag (iv) Let γt = kt+1
kt be the gross growth rate. Suppose α = 0. Derive an expression for γ t
dγt
and evaluate and discuss the sign for dk t
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(v) Let f (kt ) = ktθ , A0 = 1, and α > 0. Along a balanced growth path show that kt+1
kt
and yt+1
yt grow at the same rate. [2]+[3]+[5]+[5]+[5]

6. Consider the aggregate supply curve for an economy given by
Pt = Pte (1 + μ)F (ut , z)
where Pt = actual price level at time period t, Pte = expected prices at time t, and the
function, F, given by,
F (ut , z) = 1 − αut + z
captures the effects of the unemployment rate (ut ) at time t and the level of unemploment
benefits (z) on the price level (through their effects on wages). Assume μ > 0 denotes the
monopoly markup. Assume μ and z are constant.

(i) Show that the aggregate supply curve can be transformed to be written in terms of
πt (the inflation rate) and the expected inflation rate, πte , i.e. πt = πte + (μ + z) − αut ,
P e −Pt
where πt = Pt−1Pt−Pt and πte = t+1Pt . What is this equation called ? Briefly interpret it.

(ii) Now assume that πte = θπt−1 where θ > 0. What is this equation called ? Re-write
the equation in the above bullet and interpret when θ = 1 and θ = 1.

(iii) Let πte = πt−1 Derive the natura rate of unemployment, and express the change in
the inflation rate in terms of the natural rate. Briefly interpret this equation.

(iv) How would you think about wage indexation in this model ? Does wage indexation
increase the effect of unemployment on inflation? Assume πte = πt−1 . [8]+[3]+[6]+[6]

7. Consider an inter-temporal choice problem in which a consumer maximises utility,
u(c2 )
U (c1 , c2 ) = u(c1 ) + ,
1+δ
where ci is the consumption in period i, i = 1, 2, and δ is the discount factor (measure of
the consumer’s impatience), subject to
c2 Y2
c1 + = Y1 + ≡ W,
1+δ 1+r
where Yi is the consumer’s income in period i = 1, 2, and r is the rate of interest. Assume
ci > 1 ∀i.

(i) Let u(ci ) = log(ci ). Find a condition such that there is consumption smoothing.

(ii) Plot the two cases where (a) the consumer biases its consumption towards the future,
and (b) where the consumer biases it consumption towards the present. Put c2 on the
vertical axis and c1 on the horizontal axis.

(iii) Suppose there is consumption smoothing. Solve for c∗1 = c1 (r, Y1 , Y2 ). Interpret this
equation.

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(iv) Define YP , the permanent income, as that constant stream of income (YP , YP ) which
gives the same lifetime income as does the fluctuating income stream (Y1 , Y2 ). What does
this imply about the optimal choice of c1 , c2 , and YP ? Interpret your result graphically.
[5]+[6]+[4]+[5]

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8. Consider a cake of size 1 which can be divided between two individuals, A and B. Let
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α (resp. β) be the amount allocated to A (resp. B), where α + β = 1 and 0 ≤ α, β ≤ 1.
Agents A’s utility function is uA (α) = α and that of agent B is uB (β) = β.
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(i) Whatemis the set of Pareto optimal allocations in this economy? l
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agtwo segments to pick for herself, leaving the other segment for agent A. How should A
(ii) Suppose A is asked to cut the cake in two parts, after which B can choose which
the
cut the cake?

(iii) Suppose A is altruistic, and his utility function puts weight on what B obtains, i.e.
uA (α, β) = α + μβ, where μ is the weight on agent B’s utility. (a) If 0 < μ < 1, does the
answer to either 8(i) or 8(ii) change? (b) What if μ > 1? [5]+[5]+[10]

9. A firm uses four inputs to produce an output with a production function
f (x1 , x2 , x3 , x4 ) = min{x1 , x2 } + min{x3 , x4 }.

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x , x , x and x respectively. Solve for e m
(i) Suppose that 1 unit of output is to be produced and factor prices are 1, 2, 3 and 4 for

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the optimal factor demands.

(ii) Derive the cost function. gl
1 2 3 4

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(iii) What kind of returns to scale does this technology exhibit? [6]+[8]+[6]

10. Consider an IS-LM model where the sectoral demand functions are given by
C = 90 + 0.75Y,
G = 30, I = 300 − 50r,
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( )d = 0.25Y − 62.5r, ( )s = 500.
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a change in r. However, any disequilibrium in the goods market, which is corrected through
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a change in Y , takes much longer to be eliminated.
(a) Now consider an initial situation where Y = 2500, r = 1/5. What is the change in
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rium. What is the value of investment at this point compared to (r = 2, Y = 2500)?
[10]+[5]+[5]

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