1. (a) Let f : R → R be a differentiable function.

Show that
there always exists a point c > 1 such that f (2) − 2f (1) =
cf � (c) − f (c). [4]
(b) Let f : (a, b) → R be a function such that there is some
c ∈ (a, b) satisfying f (c) = max{f (x) : x ∈ (a, b)}, where
(a, b) = {x ∈ R : a < x < b}. Assume that one-sided
derivatives f−� (c) and f+� (c) exist. Show that f−� (c) ≥ 0 and
f+� (c) ≤ 0. [8]
(c) Let f be a function which is continuous on [0, 1] and dif-
ferentiable on (0, 1), with f (0) = f (1) = 0. Assume that
there is some c ∈ (0, 1) such that f (c) = 1. Prove that
there exists some x0 ∈ (0, 1) such that |f � (x0 )| > 2. [12]

2. (a) A restaurant has a menu card containing three Chinese

items, six Indian items and five Continental items. Six
items are selected at random. Let X and Y denote respec-
tively the number of Indian and Continental items selected.
Compute the conditional probability mass function of X
given that Y = 3. Also compute E(X|Y = 3). [8+4=12]
(b) Let the joint probability mass function of X and Y be

p(1, 1) = 1/9, p(2, 1) = 1/3, p(3, 1) = 1/9,

p(1, 2) = 1/9, p(2, 2) = 0, p(3, 2) = 1/18,

p(1, 3) = 0, p(2, 3) = 1/6, p(3, 3) = 1/9,

where p(x, y) = P (X = x, Y = y) for all x = 1, 2, 3 and

y = 1, 2, 3. Find the correlation between X and Y . Also
find E(X|Y = 3). [8+4=12]

1
3. A consumer consumes two goods: visits to a nearby park (x) and
a composite consumption good (y), according to preferences u =
xy. The consumer’s income is R and the price of the composite
good is normalized to 1.
Initially, there is an entry fee p∗ per park visit. Suppose the
authorities are considering a proposal to reduce the per visit
entry fee from p∗ to p� .

(a) Find the increase in consumer’s surplus resulting from this

proposal. [3]
(b) Now suppose there is an alternative proposal to maintain
the per visit entry fee at the initial level p∗ , but hand out
a one-off cash voucher to the consumer (i.e., the consumer
would get the cash voucher only once, regardless of how
many times she visits the park). Find the minimum value
of the cash voucher that would make the consumer not
worse off under this alternative proposal, compared to the
earlier proposal to reduce the entry fee from p∗ to p� . [7]
(c) Next, suppose there is a third proposal which simultane-
ously reduces the per visit entry fee from p∗ to p� , and
charges a one-off lump-sum user fee (i.e., a fee which has
to be paid only once, regardless of the number of visits).
Find the maximum value of the lump-sum user fee that
would make the consumer not worse off under this third
proposal, compared to the initial situation (i.e., price p∗
per visit and no lump-sum payment). [7]
(d) Prove that C < ΔS < E, where ΔS is the solution to part
(a), E to part (b) and C to part (c) above. [7]

2
4. There is a worker who can purchase e units of education, e ∈
[0, ∞], at cost 2e2 /θ. The worker can be of high ability (θ = 2)
or low ability (θ = 1), with the worker knowing her own ability
(ability is exogenously given). Note that it is less costly for the
worker to achieve a particular level of education if she were of
high ability than if she were of low ability.
The worker can be hired by a firm paying wage w, and if hired,
her marginal product is θ, where θ is her ability. The firm does
not know the worker’s ability (it knows θ is either 1 or 2), but
might be able to infer it after observing her education choice.
In particular, if the firm believes the worker is of high ability,
it pays wage w = 2, while it pays wage w = 1 if it believes the
worker is of low ability.
The worker chooses e to maximize utility, and if she works for
the firm at wage w, then her utility, given e and θ, is u(w, e|θ) =
w − 2e2 /θ.

(a) Draw the indifference map of the worker in the education-

wage space (assume education to be measured along the
horizontal axis and wage to be measured along the vertical
axis) if she were of high ability. Also draw the indifference
map of the worker if she were of low ability. How many
times can an indifference curve of a worker who is of low
ability intersect an indifference curve of a worker who is of
high ability? [2+2+2]
(b) Suppose the firm perfectly infers the worker’s ability upon
observing her education choice. Suppose also the worker
chooses e = 0 if she is of low ability. Then how much
education would she purchase if she were of high ability?
[9]
(c) Continue to assume that the firm perfectly infers the
worker’s ability upon observing her education choice. Sup-

3
 pose, if the worker is of high ability, her education choice
matches what you found in part (b). Then show that she
would purchase no education if she were of low ability. [9]

5. An individual lives for two periods, 1 and 2, and has lifetime util-
ity function U (C1 ) + βU (C2 ), where C1 and C2 are the consump-
tions of this individual in period 1 and period 2 respectively,
and 0 < β < 1 is the subjective discount factor. The follow-
ing conditions are satisfied: U � (C) > 0, U �� (C) < 0, lim U � (C) =
C→0
∞, lim U � (C) = 0.
C→∞
The individual earns an exogenously given income w > 0 in pe-
riod 1 of his life and earns nothing in the second period of his
life. But he can lend or borrow freely at an exogenously given
rate of interest r > 0.
It follows that C1 = w − S and C2 = (1 + r)S, where S is the
savings made by the individual in period 1. Note that C1 (or S)
and C2 are endogenously determined and the exogenously given
parameters of the model are w, r and β.

(a) Suppose β(1 + r) = 1. Solve for C1 and C2 in terms of the

exogenously given parameters. [4]
(b) Now remove the restriction that β(1 + r) = 1. How would
savings S be affected if β goes up, ceteris paribus? [8]
(c) Continuing without the restriction that β(1 + r) = 1, show
that savings S goes up as w increases, ceteris paribus. Fur-
ther show that the increase in S is less than the increase in
w. [8+4]

4
 Group A (Microeconomics)
1. [30 marks: 4 + 20 + 6]

Consider a used car market with 600 buyers each willing to buy
exactly one used car, and 500 sellers each having exactly one
used car. Out of the 500 used cars, 400 are of good quality
(peaches) and 100 are of bad quality (lemons). The monetary
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.

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

Provide a clear explanation for your answer.

(b) (i) Derive, with a clear explanation, the supply of used cars
as a function of price. Draw this supply curve by plotting
number of used cars on x-axis and price on y-axis. [You
must label all the important points in the figure clearly.]
(ii) Derive, with a clear explanation, the demand for used cars
as a function of price. Draw this demand curve in the same
figure as in part (i). [You must label all the important
points in the figure clearly.]
(iii) Use the demand and supply functions above to find out
all possible competitive equilibria in the used car market
mentioning clearly which types of car, lemon or peach, are
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.

1
 2. [30 marks: 5 + 3 + 3 + 11 + 8]

Consider an industry with 2 firms – a private firm (indexed by

r) and a public firm (indexed by u) – producing a homogeneous
product and competing in quantities. The firms 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 firms respectively.
Each firm 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 firm’s
profit, (ii) public firm’s profit, (iii) consumer surplus, and (iv)
welfare (sum of consumer surplus and producer surplus).

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

(c) The public firm’s objective is to maximize welfare. For a given

qr , set up the public firm’s maximization problem and derive its
optimal choice of output qu . [This exercise gives you the reaction
function of the public firm.]

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

(i) Define the concept of equilibrium in this context and find

out the amounts of output, qr∗ and qu∗ , the two firms produce
in equilibrium. Find out the expressions of price, profits of
the two firms, 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

2
 and qr in y-axis). [You must label the important points in
the figure clearly.]

(e) Suppose that the marginal cost of the private firm falls to cr < c
while the marginal cost of the public firm remains the same at
c. Draw the new reaction functions and explain clearly how the
following outcomes change in the new equilibrium (as compared
to the old equilibrium): qr , qu , Q, price, profits of the two firms,
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

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
over interval [0, 1] so that the number of consumers with valua-
tion in between a and b, where 0 ≤ a < b ≤ 1, is b − a. There is
a monopoly firm with total cost of producing q units of the good
q
given by . The firm 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 firm.

(b) Derive, with a clear explanation, the monopoly price and profit
level.

(c) Suppose that the firm can, for a cost, get to know whether a
consumer belongs to the interval [0, 45 ], or to the interval ( 45 , 1].
What is the maximum amount the firm is willing to pay for this
information? Give a clear explanation for your answer.

3
 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 fixed 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 offer the nominal wage W > W .
A worker does not supply any labour if W < W . At W = W ,
a worker is indifferent between supplying and not supplying her
labour endowment. The number of workers available in the econ-
omy is fixed 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
W
rate, 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.

4
(c) Choose an arbitrary aggregate price level, P, and draw the ag-
gregate labour supply (LS ) and aggregate labour demand (LD )
curves, as functions of W , by plotting labour (L) on x-axis and
nominal wage (W ) on y-axis. Think about the labour market
equilibrium for the arbitrary aggregate price level P that you
have chosen.

Note that the equilibrium employment (L∗ ) in the economy de-

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
supply curve by plotting Y on x-axis and P on y-axis.

(e) Recall that the aggregate demand curve is given by a standard

downward-sloping curve. Explain the effectiveness of the stan-
dard monetary and fiscal 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:

Yt = (Kt )α (Lt )1−α , 0 < α < 1.

There is perfect competition in the factor market so that, in

equilibrium, each factor is paid its marginal product and the
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

5
 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 .
Kt
(ii) Define 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?

6
 (iv) How does k ∗∗ change with changes in τ ? Compare with the
response of k ∗ and explain the economic reason behind the
differential impact.

3. [30 marks: 9 + 5 + 6 + 4 + 6]

Consider an individual who lives for two periods. In the first

period, she earns an wage income W and takes her consumption-
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 first period (this can be considered as mandatory savings
of the individual) and returns the amount T in the form of a
lumpsum transfer in the second period. Suppose that the utility
derived by the individual who consumes c1 in the first period
and c2 in the second period is given by u (c1 ) + βu (c2 ) , where β
is a discount factor 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 first-order condition
of this utility maximization problem by showing your procedure
clearly. Provide a clear economic interpretation of the first-order
condition.

(b) Note that β < 1 implies that the individual is myopic (short-
sighted), she puts less weight on future period. Explain intu-
itively whether a more myopic individual will save more or less
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

7
 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 benefits 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.

8
 Group C (Mathematics)
1. [30 marks: 20 + 10]

Consider the following feasible region C in R2 for an optimization

program:

C := {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, y ≤ x}.

(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.

(b) Suppose a and b are independently drawn from [−1, 1] using

a probability distribution with cumulative distribution function
(cdf) F . What is the probability that the unique optimal solu-
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|.

Suppose B 2 − 4AC > 0.

(a) Argue that |A| ≥ |a|.

(b) Show that b2 − 4ac > 0.

(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:

9
 – 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.

10
 Group A

1. Answer the following questions.

(a) Find all maxima and minima of the function 𝑓(𝑥, 𝑦) = 𝑥𝑦,
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]

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
|𝑓(𝑥)| ≤ 𝑥 for all 𝑥 ∈ ℛ.
[20+10]

1
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
 (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,
𝑠, if 𝑟 and 𝜌 increase by the same amount (so that the condition
𝑟 = 𝜌 continues to hold)?
[20+10]

5. A profit maximizing monopolist produces a good with the cost

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

𝑇(𝑦) = 𝑝𝑦 + 𝑡;
where 𝑝 ≥ 0, 𝑡 ≥ 0 if 𝑦 > 0, and 𝑇(0) = 0. Thus, in order to
purchase any positive amount of the good, the consumer may have to
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
result.
(c) Calculate when social surplus is higher, explaining your result.
(d) Calculate when consumer’s surplus is higher, explaining your
result.
[10+10+5+5]

3
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)]

4
 Group C

7. Consider a world economy consisting of Home (H) and Foreign (F).

Each of these countries produces a single good that is both consumed
domestically and exported. Let Foreign output be the numeraire and
let p be the relative price of the H produced good. Assume full
employment in both countries, so that H produces a fixed output Y
and F produces a fixed output Y  . Let E be the Home expenditure
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
expenditure of Foreign consumers on the Home produced good.
Assume further 1      (i.e., the expenditure share of Home
consumers on the Home produced good is greater than the
expenditure share of Foreign consumers on the Home produced
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))?

Prove your claims.

[20+10]

5
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]

6
 Group A
1. Let f : ℜ → ℜ be a continuously differentiable function which has at
least three distinct zeros. (We say x is a zero of f if f (x) = 0). Let
g : ℜ → ℜ be defined as follows: g(x) = ex/2 f (x) for all x ∈ ℜ.

(i) Prove that g has at least three distinct zeros.

(ii) Prove that the function f + 2f ′ has at least two distinct zeros.

(10+20=30)

2. (i) Consider the following two variable optimization problem:

max (x2 + y 2 )
x,y

subject to
x+y ≤1
x, y ≥ 0

Find all solutions of this optimization problem.

(ii) In a kingdom far, far away, a King is in the habit of inviting 1000
senators to his annual party. As a tradition, each senator brings
the King a bottle of wine. One year, the Queen discovers that one
of the senators is trying to assassinate the King by giving him a
bottle of poisoned wine. Unfortunately, they do not know which
senator, nor which bottle of wine is poisoned, and the poison
is completely indiscernible. However, the King has 10 prisoners
he plans to execute. He decides to use them as taste testers to
determine which bottle of wine contains the poison. The poison
when taken has no effect on the prisoner until exactly 24 hours
later when the infected prisoner suddenly dies. The King needs
to determine which bottle of wine is poisoned by tomorrow so
that the festivities can continue as planned. Hence he only has
time for one round of testing. How can the King administer the
wine to the prisoners to ensure that 24 hours from now he is
guaranteed to have found the poisoned wine bottle?
(12+18=30)

1
3. (i) Let A and B be matrices for which the product AB is defined.
Show that if the columns of B are linearly dependent, then the
columns of AB are linearly dependent.
(ii) Let ei denote the column vector with three elements, each of
which is zero, except for the i-th element, which is 1. Consider
a linear transformation L : ℜ3 → ℜ3 with L(e1 ) = e1 , L(e2 ) =
e1 + e2 , and L(e3 ) = e2 + e3 . Does L map ℜ3 onto ℜ3 ? Prove
your answer.
(15+15=30)

2
 Group B

4. This question pertains to a situation in which a particular commodity,

like rice, is both available at a subsidised rate from a fair price shop
(ration shop) and at a higher price from the open market. Suppose
a consumer can buy a certain (fixed) quantity of rice at a lower price
from the ration shop (that is, there is a ration quota). In addition,
he can buy more of rice (assume a uniform quality of rice) from the
open market at a higher price. (You may assume that consumers
preferences are represented by standard downward sloping, smooth,
convex indifference curves.)

(i) Graphically depict the consumer’s equilibrium (assuming he ex-

hausts the ration quota and in addition buys from the open mar-
ket).
(ii) Suppose rice is a normal good. What will happen to the quan-
tity of rice purchased from the open market (over and above the
ration quota) in equilibrium if there is a cut in the ration quota?
Briefly explain.
(iii) Suppose rice is a normal good. What will happen to the quantity
purchased in the open market (over and above the ration quota)
if the subsidised price (price at which the ration quota rice could
be bought) is increased (but is still lower than the open market
price)? Will your conclusion change if rice is an inferior good?
Briefly explain.

(10+10+10=30)

3
5. There are plenty of fish in the Dull Lake. Boats can be hired by
fishermen to catch fish and sell it on the fish market. The revenue
earned each month from a total of x boats is given by the following
expression: Rupees 10, 000{4x − 12 x2 }. Each boat costs Rupees 20,000
each month.

(i) Derive the marginal and average revenue per boat

(ii) The Dull municipality is considering giving out permits for each
boat that fishes so they can track who is fishing from the lake. If
these permits are allocated freely, how many boats will fish every
month?
(iii) If total profit is to be the maximum possible, how many boats
should fish every month?
(iv) Dull municipality decides to charge for the permits instead of
giving them out for free. What should the per-boat charge for
the permit be if total profits are to be the maximum possible?

(5+10+5+10=30)

4
6. The Shoddy Theater screens movies every week and is located on a
university campus which has only students and faculty as residents.
It is the only source of watching movies for both faculty and students,
and is large enough to accomodate all faculty and students. Faculty
demand for movie tickets is given by 500 − 4PF = QF , where PF refers
to the price of the ticket paid by faculty and QF refers to the number
of tickets purchased by faculty. Demand by students is described by
100 − 2PS = QS , where PS refers to the price of the ticket paid by
students and QS refers to the number of tickets purchased by students.
The cost to service demand equals 500.

(i) If the price charged is to be the same for faculty and students,
what price would Shoddy Theater set in order to maximize its
profits?
(ii) Now imagine that Shoddy Theater decides to charge different
prices for faculty and students. What would these prices be, if
Shoddy Theater wants to maximize profits?

(15+15=30)

5
 Group C

7. Consider a Solow type economy, producing a single good, according

to the production function:

Y (t) = K(t)α L(t)(1−α)

Where, 0 < α < 1, and Y (t), K(t) and L(t) are the output of the
good, input of capital, and input of labour used in the production
of the good, respectively, at time t. Capital and labour are all fully
employed.

Labour force grows at the exogenous rate, η > 0, i.e.,

1 dL(t)
= η > 0.
L(t) dt

Part of the output is consumed and part saved. Let 0 < s < 1 be the
fraction of output that is saved and invested to build up the capital
stock. Also assume that there is no depreciation of capital stock.

Then it follows that:

dK(t)
sY (t) =
dt
d
Where dt
is the time derivative.

With this above given description of the economy, one can find out
the steady state growth rate of Y, for this economy. Growth rate of
1 dY (t)
output is given by: Y (t) dt
≡ gY .

Assume, the economy begins at date 0, from a per capita capital stock,
K(0)
k(0) ≡ L(0)
< k ∗ , where k ∗ denotes the steady state per capita capital
stock.

(i) Demonstrate formally, whether the growth rate of output (gY ),

at the beginning date 0, is greater, equal or less than the steady
state growth rate of output.

6
 (ii) For the same economy, consider, two alternative beginning date
scenarios: with per capita capital stock, given by:
Case 1. k(0); Case 2. k ′ (0). Where, k(0) < k ′ (0) < k ∗ Can you
compare the beginning date growth rates of output in the two
cases?
(iii) Next, consider two Solow type economies, namely, A and B. They
are isolated from each other and are working on their own. Both
the economies have absolutely the same description as given be-
fore, except for the fact that the fraction of income saved in
country A, denoted by sA is greater than the fraction of income
saved in country B, denoted by sB . Let k A (0) be the initial date
per capita capital stock in country A and k B (0), the initial date
per capita capital stock in country B, which are both less than
their respective steady state values. Assume, K A (0) < K B (0).
Can you figure out whether the initial date growth rate of out-
put in country A is greater than, equal or less than the initial
date growth rate of output in country B? In case you find the
data provided to you is insufficient to make any comment on this,
please point it out.

(20+5+5=30)

8. (i) What is the money multiplier? What determines its size? What
is the relation between the monetary base, the money multiplier,
and the money supply? Which of these variables can the central
bank change to change the money supply? What is the direction
of change in each case?
(ii) Why might the cash/deposit ratio and the reserve to asset ratio
be decreasing functions of the rate of interest? How does an
interest-sensitive money supply affect the LM curve? Illustrate
with a diagram, comparing this LM curve with the standard LM.
How does this change the effectiveness of counter-cyclical fiscal
policy (in a closed economy)? Explain.
(15+15=30)

7
9. (i) What is the difference between the real and the nominal exchange
rate? Give an example to explain this to someone who has not
studied economics. Is an increase in the real cost of imports an
improvement or a deterioration in the terms of trade?
(ii) A small open economy has a government budget surplus and a
trade deficit. Explain whether there is a private sector surplus,
deficit or balance. Examine the consequences in the short run for
output, the trade balance and the government budget balance of
a sudden fall in private consumption in this economy (due to an
epidemic in the small country) under (a) fixed exchange rates,
(b) flexible exchange rates. Use the Mundell-Fleming model with
perfect capital mobility. Explain the adjustment mechanisms.
(10+20=30)

8
 Group A
1. [30 marks: 5+10 +15].

Let X1 , ..., Xn be independent and identically distributed random variables

with a uniform distribution on (0, θ] where θ > 0.

(a) [5 marks] Write down the joint probability density function of X1 , ..., Xn .

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

2. [30 marks: 3+5+10+6+6]

A tournament consists of n 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)
2 .

(b) (5 marks) Suppose n > 3 and players 1, 2, 3 win every match against
players in {4, . . . , n}. Find the value of s4 + . . . + sn ?

(c) (10 marks) Suppose sn = s0 , sn−1 = s0 + 1, sn−2 = s0 + 2 for some

positive integer s0 and n ≥ 3. Show that
(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
 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.

2
 Group B
1. Consider an economy inhabited by identical agents of size 1. A represen-
tative agent’s preference over consumption (c) and labour supply (l) is
given by the utility function

u (c, l) = cα (24 − l)1−α , 0 < α < 1.

Production of the consumption good c is given by the production func-

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 is given by the allocation of consumption and

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

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

sumption demand, cD , to maximize her utility; a firm decides its labour
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.
(iii) [5 marks] Determine the competitive equilibrium allocation, cCE , lCE ,

and the relative price ratio, wp .

(b) [5 marks] Pareto efficient allocation:

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
 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.

4
 3. [30 marks = 6+7+7+10]
Consider a moneylender who faces two types of potential borrowers: the
safe type and the risky type. Each type of borrower needs a loan of the same
size L to invest in some project. The borrower can repay only if the investment
produces sufficient returns to cover the repayment. Suppose that the safe type
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
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.
(a) [6 marks] What is the highest interest rate, call it is , for which the
safe borrower wants the loan? What is the highest interest rate, ir , for
which the risky borrower wants the loan? Who is willing to pay a higher
interest rate, the risky borrower or the safe 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
given by p (1 + ir ) L − L. Derive, with a clear argument, the expression
of lender’s expected profit when he charges is .
(d) [10 marks] An equilibrium with credit rationing occurs when, at the
equilibrium interest rate, some borrowers who want to obtain loans are
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

5
 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 ).

6
(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
inefficiency of the financial sector θ?

2. [30 marks = 5+5+10+10]

Consider a Solow-Swan model with learning by doing. Assume that the

production function is of the form

Yt = Ktα (At Lt )1−α

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

Ż
(a) [5 marks] Derive an expression for 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

(c) [10 marks] Draw a diagram describing the dynamics of growth in the
model with learning by doing. Plot Z on the x − axis, and the appro-
priate functions on the y − axis

(d) [10 marks] In contrast to the model with no learning by doing, does an
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).

7
 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-
lution for the above problem yield constant consumption, C,
f in every
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-

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

(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
level of consumption, C.
e Compare this value derived with part c (i).

8
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 There are 6 questions: 2 in Group A, 2 in Group B, and 2 in Group C.

Answer 4 questions in total and at least 1 question from each Group.

Each question carries equal marks.

The maximum possible score is 100.

Group A
1. Suppose a government agency has a monopoly in the provision of internet connections.
The marginal cost of providing internet connections is 12 , whereas the inverse demand
function is given by: p = 1 q. The o¢ cial charge per connection is set at 0; thus,
1
the state provides a subsidy of 2
per connection. However, the state can only provide
budgetary support for the supply of 0:4 units, which it raises through taxes on con-
sumers. Bureaucrats in charge of sanctioning internet connections are in a position
to ask for bribes, and consumers are willing to pay them in order to get connections.
Bureaucrats cannot, however, increase supply beyond 0:4 units.

(a) Find the equilibrium bribe rate per connection and the social surplus.
(b) Now suppose the government agency is privatized and the market is deregulated;
however, due large …xed costs of entry relative to demand, the privatized company
continues to maintain its monopoly. Find the new equilibrium price, bribe rate
and social surplus, specifying whether privatization increases or reduces them.
(c) Suppose now a technological innovation becomes available to the privatized monopoly,
which reduces its marginal cost of providing an internet connection to c, 0 < c < 12 .
Find the range of values of c for which privatization increases consumers’surplus.

2. 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 minfX2 ; Y2 g, 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.

(a) What is the set of Pareto optimal outcomes in this economy? Justify your answer.
(b) What is the competitive equilibrium in this economy? Justify your answer.

1
 (c) Are the perfectly competitive equilibria Pareto optimal?
(d) Now consider another economy where everything is as before, apart from individ-
ual 2’s preferences, which are as follows: (a) among any two bundles consisting
of X and Y , individual 2 prefers the bundle which has a larger amount of com-
modity 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.

Group B
1. An economy comprises of a consolidated household sector, a …rm sector and the gov-
ernment. The household supplies labour (L) to the …rm. The …rm produces a single
good (Y ) by means of a production function Y = F (L) ; F 0 (L) > 0, F 00 (L) < 0; and
maximizes pro…ts = PY W L, where P is the price of Y and W is the wage rate.
The household, besides earning wages, is also entitled to the pro…ts of the …rm. The
household maximizes utility (U ), given by

U = 12 ln C + 21 ln M
P
d (L) ;

M
where C is consumption of the good and P
is real balance holding. The term d (L) de-
notes the disutility from supplying labour with d0 (L) > 0, d00 (L) > 0: The household’s
budget constraint is given by:

PC + M = WL + +M P T;

where M is the money holding the household begins with, M is the holding they end
up with and T is the real taxes levied by the government. The government’s demand
for the good is given by G. The government’s budget constraint is given by:

M M = PG P T:

Goods market clearing implies Y = C + G:

dY
(a) Prove that dG
2 (0; 1) ; and that government expenditure crowds out private
dC
consumption (i.e., dG
< 0).
(b) Show that everything else remaining the same, a rise in M leads to an equipro-
portionate rise in P .

2
2. Consider an IS-LM model where the sectoral demand functions are given by

C = 90 + 0:75Y;
G = 30;
I = 300 50r;
M
P d
= 0:25Y 62:5r;
M
P s
= 500:

Any disequilibrium in the international money market is corrected instantaneously

through a change in r. However, any disequilibrium in the goods market, which is
corrected through a change in Y , takes much longer to be eliminated.

(a) Consider an initial situation where Y = 2500, r = 15 . What is the change in the
level of I that must occur before there is any change in the level of Y ?
(b) Draw a graph to explain your answer.
(c) Calculate the value of (r; Y ) that puts both the money and goods market in
equilibrium. What is the value of investment at this point compared to (r =
0:2; Y = 2500)?

Group C
1. Answer the following questions.

(a) Let f : R ! R be a function de…ned as

jxj
f (x) = 8 x 2 R n f0g:
2x
Can f (0) be de…ned in a way such that f is continuous at 0? Justify your answer.
(b) Consider the following optimization problem:

max x(1 x);

x2[0; ]

where 2 [0; 1]. Let x be an optimal solution of the above optimization problem.
For what values of will we have x = ?
(c) A …rm is producing two products a and b. The market price (per unit) of a and b
are respectively 3 and 2. The …rm has resources to produce only 10 units of a and
b together. Also, the quantity of a produced cannot exceed double the quantity
of b produced. What is the revenue-maximizing production plan (i.e., how many
units of a and b) of the …rm?

3
2. Answer the following questions.

(a) A slip of paper is given to person A, who marks it with either (+) or ( ). The
probability of her writing (+) is 13 . Then, the slip is passed sequentially to B; C,
2
and D. Each of them either changes the sign on the slip with probability 3
or
1
leaves it as it is with probability 3
.

i. Compute the probability that the …nal sign is (+) if A wrote (+).
ii. Compute the probability that the …nal sign is (+) if A wrote ( ).
iii. Compute the probability that A wrote (+) if the …nal sign is (+).

(b) There are n houses on a street numbered h1 ; : : : ; hn . Each house can either be
painted blue or red.

i. How many ways can the houses h1 ; : : : ; hn be painted?

ii. Suppose n 4 and the houses are situated on n points on a circle. There
is an additional constraint on painting the houses: exactly two houses need
to be painted blue and they cannot be next to each other. How many ways
can the houses h1 ; : : : ; hn be painted under this new constraint?
iii. How will your answer to the previous question change if the houses are located
on n points on a line.

4
1. A researcher has 100 hours of work which have to be allocated
between two research assistants, Aditya and Gaurav. If Aditya is
allocated x hours of work, his utility is, −(x − 20)2 . If Gaurav is
allocated x hours of work, his utility is, −(x − 30)2 . The researcher
is considering two proposals: (I) Aditya does 60 hours and Gaurav
40 hours (II) Aditya does 90 hours and Gaurav 10 hours. Which
of the following statements is correct.
A. Proposal I is Pareto-efficient but Proposal II is not.
B. Proposal II is Pareto-efficient but Proposal I is not.
C. Both proposals are Pareto-efficient.
D. Neither proposal is Pareto-efficient.

2. The industry demand curve for tea is: Q = 1800 − 200P. The in-
dustry exhibits constant long run average cost (ATC) at all levels
of output at Rs 1.50 per unit of output. Which market form(s)
– perfect competition, pure monopoly and first-degree price dis-
crimination – has the highest total market (that is, producer +
consumer) surplus?
A. perfect competition
B. pure monopoly
C. first degree price discrimination
D. perfect competition and first degree price discrimination

3. The following information will be used in the next question also.

OIL Inc. is a monopoly in the local oil refinement market. The
demand for refined oil is

Q = 75 − P

where P is the price in rupees and Q is the quantity, while the

marginal cost of production is

M C = 0.5Q.

The fixed cost is zero. Pollution is emitted in the refinement of

oil which generates a marginal external cost (MEC) equal to 31
Rs/unit. What is the level of Q that maximizes social surplus?

1
 A. 50
B. 29 31
C. 17.6
D. 44

4. Refer to the previous question. Suppose the government decides to

impose a per unit pollution fee on OIL Inc. At what level should
the fee (in Rs/unit) be set to produce the level of output that
maximizes social surplus? You may use the fact that the marginal
revenue is given by: M R = 75 − 2Q.
A. 1/3
B. 2
C. 3/4
D. 5/3

5. Mr. X has an exogenous income W, and his utility from consump-

tion is given by U (c). With probability p, an accident can occur.
If it occurs, the monetary equivalent of the damage is T. Mr. X
can however affect the accident probability, p, by taking preven-
tion effort, e . In particular, e can take two values: 0, and a > 0.
Assume that p(0) > p(a). Let us also assume that the utility cost
of effort is Ae2 . Calculate the value of A below which effort will be
undertaken.
[p(a)−p(0)][u(W −T )−U (W )]
A. a2
p(a)−p(0)
B. u(W −T )−U (W )
p(a)p(0)a2
C. u(W −T )−u(W )
p(a)/p(0)
D. u(W −T )/u(W )
a2

6. Suppose Mr. X maximizes inter-temporal utility for 2 periods. His

total utility is given by

log(c1 ) + β log(c2 )

where β ∈ (0, 1) and c1 and c2 are his consumption in period 1

and period 2, respectively. Suppose he earns a wage only in period

2
 1 and it is given by W. He saves for the second period on which
he enjoys a gross return of (1 + r) where r > 0 is the net interest
rate. Suppose the government implements a scheme where T ≥ 0
is collected from agents (thus also from Mr. X) in the first year,
and gives the same amount, T , back in the second period. What
is the optimum T for which his total utility is maximized?
A. T = 0
W
B. T = 2β
βW
C. T = 2(1−β)
W
D. T = 2(1−β)

7. Suppose there is one company in an economy which has a fixed

supply of shares in the short run. Suppose there is new information
that causes expectations of lower future profits. How does this new
stock market equilibrium affect final output and the final price
level of the economy if you assume that autonomous consumption
spending and household wealth are positively related?
A. real GDP increases; price decreases
B. real GDP decreases; price increases
C. real GDP decreases; price decreases
D. real GDP increases; price stays constant.

8. A monopolist faces a demand function, p = 10 − q. It has two

plants at its disposal. The cost of producing q1 in the first plant is
300 + q12 , if q1 > 0, and 0 otherwise. The cost of producing q2 in
the second plant is 200 + q22 , if q2 > 0, and 0 otherwise. What are
the optimal production levels in the two plants?
A. 10 units in both plants,
B. 20 units in the first plant and 10 units in the second plant
C. 0 units in the first plant and 15 units in the second plant
D. None of the above.

3
 9. Consider a firm facing three consumers, 1, 2 and 3, with the follow-
ing valuations for two goods, X and Y (All consumers consume at
most 1 unit of X and 1 unit of Y .)

Consumers X Y
1 7 1
2 4 5
3 1 6
The firm can produce both goods at a cost of zero. Suppose the
firm can supply both goods at a constant per unit price of pX for
X, and py for Y. It can also supply the two goods as a bundle, for
a price of pXY . The optimal vector of prices (pX , py , pXY ) is given
by
A. (7,6,9).
B. (4,1,4).
C. (7,7,7).
D. None of the above.

10. Two individuals, Bishal (B) and Julie (J), discover a stream of
mountain spring water. They each separately decide to bottle some
of this water and sell it. For simplicity, presume that the cost of
production is zero. The market demand for bottled water is given
by P = 90−0.25Q, where P is price per bottle and Q is the number
of bottles. What would Bishal’s output QB , Julie’s output QJ ,
and the market price be if the two individuals behaved as Cournot
duopolists?
A. QB = 120; QJ = 120; P = 42
B. QB = 90; QJ = 90; P = 30
C. QB = 120; QJ = 120; P = 30
D. QB = 100; QJ = 120; P = 30

11. The next three questions (11, 12, 13) are to be answered together.
Consider the following model of a closed economy

4
 △Y = △C + △I + △G
△C = c△Yd
△Yd = △Y − △T
△T = t△Y + △T0

where △Y = change in GDP, △C = change in consumption, △I =

change in private investment, △G = change in government spend-
ing, △Yd = change in disposable income (i.e., after tax income),
△T = the change in total tax collections, t is the tax rate between
(0, 1), and △T0 = the change in that portion of tax collections that
can be altered by government fiscal policy measures. The value of
the balanced budget multiplier (in terms of G and T0 ) is given by:
1
A. 1−c(1−t)
−c
B. 1−c(1−t)
1−c
C. 1−c(1−t)
D. none of the above.

12. Refer to the previous question. Suppose the marginal propensity

to consume, c = .8, and t = .375. The value of the government
expenditure multiplier is
A. 2,
B. -1.6
C. .4
D. .5

13. Refer to the previous two questions. Suppose the marginal propen-
sity to consume, c = .8, and t = .375. The value of the tax multiplier
(with respect to T0 ) is
A. -1.6
B. 2
C. .4
D. .3

5
14. In the IS-LM model, a policy plan to increase national savings
(public and private) without changing the level of GDP, using any
combination of fiscal and monetary policy involves
A. contractionary fiscal policy, contractionary monetary pol-
icy
B. expansionary fiscal policy, contractionary monetary pol-
icy
C. contractionary fiscal policy, expansionary monetary pol-
icy
D. expansionary fiscal policy, expansionary monetary policy

15. Consider the IS-LM-BP model with flexible exchange rates but with
no capital mobility. Consider an increase in the money supply. At
the new equilibrium, the interest rate is , the exchange rate
is , and the level of GDP is , respectively.
A. higher, lower, higher
B. lower, higher, higher
C. lower, higher, lower
D. higher, lower, lower

16. Consider a Solow model of an economy that is characterized by the

following parameters: population growth, n; the depreciation rate,
δ; the level of technology, A; and the share of capital in output,
α. Per-capita consumption is given by c = (1 − s)y where s is the
exogenous savings rate, and y = Ak α , where y denotes output per-
capita, and k denotes the per-capita capital stock. The economy’s
golden-rule capital stock is determined by which of the following
conditions?
A. ∂c
∂k
= Ak α − (n + δ)k = 0
B. ∂c
∂k
= αAk α−1 − (n + δ) = 0
C. ∂c
∂k
= (n + δ)k − sAk α = 0
D. none of the above.

6
17. In the Ramsey model, also known as the optimal growth model,
with population growth, n, and an exogenous rate of growth of
technological progress, g, the steady-state growth rates of aggregate
output, Y, aggregate capital, K, and aggregate consumption, C, are
A. 0, 0, 0
B. n + g, n + g, n + g
C. g, n + g, n
D. n + g, n + g, g

18. Consider the standard formulation of the Phillips Curve,

πt − πte = −α(ut − un )

where πt is the current inflation rate, πte is the expected inflation

rate, α is a parameter, and un is the natural rate of unemployment.
Suppose the economy has two types of labour contracts: a propor-
tion, λ, that are indexed to actual inflation, πt , and a proportion,
1 − λ, that are not indexed and simply respond to last year’s infla-
tion, πt−1 . Wage indexation (relative to no indexation) will
the effect of unemployment on inflation.
A. strongly decrease
B. increase
C. not change
D. mildly decrease

19. Consider a Harrod-Domar style growth model with a (i) Leontieff

aggregate production function, (ii) no technological progress, and
(iii) a constant savings rate. Let K and L denote the level of
capital and labor employed in the economy. Output, Y, is produced
according to
Y = min{AK, BL}
where A and B are positive constants. Let L̄ be the full employment
level. Under what condition will there be positive unemployment?
A. AK > B L̄
B. AK < B L̄

7
 C. AK = B L̄
D. none of the above.

20. The next two questions (20 and 21) are to be answered together.
People in a certain city get utility from driving their cars but each
car releases k units of pollution per km driven. The net utility of
each person is his or her utility from driving, v, minus the total
pollution generated by everyone else. Person i’s net utility is given
by
∑
n
Ui (x1 , ..., xn ) = v(xi ) − kxj
j=1
j̸=i

where xj is km driven by person j, n is the city population, and

the utility of driving v has an inverted U-shape with v(0) = 0,
limx→0+ v ′ (x) = ∞, v ′′ (x) < 0, and v(x̄) = 0 for some x̄ > 0. In an
unregulated city, an increase in population will
A. increase the km driven per person
B. decrease the km driven per person
C. leave the km driven per person unchanged
D. may or may not increase the km driven per person.

21. Refer to the information given in the previous question. A city

planner decides to impose a tax per km driven and sets the tax
rate in order to maximize the total net utility of the residents.
Then, if the population increases, the optimal tax will
A. increase
B. decrease
C. stay unchanged
D. may or may not increase.

22. The production function

F (L, K) = (L + 10)1/2 K 1/2

has

8
 A. increasing returns to scale
B. constant returns to scale
C. decreasing returns to scale
D. none of the above.

23. Consider the production functions

F (L, K) = L1/2 K 2/3 and G(L, K) = LK.

where L denotes labour and K denotes capital.

A. F is consistent with the law of diminishing returns to
capital but G is not.
B. G is consistent with the law of diminishing returns to
capital but F is not.
C. Both F and G are consistent with the law of diminishing
returns to capital
D. Neither F nor G is consistent with the law of diminishing
returns to capital

24. A public good is one that is non-rivalrous and non-excludable. Con-

sider a cable TV channel and a congested city street.
A. A cable TV channel is a public good but a congested city
street is not
B. A congested city street is a public good but a cable TV
channel is not
C. Neither is a public good
D. Both are public goods.

25. Firm A’s cost of producing output level y > 0 is, cA (y) = 1 + y
while Firm B’s cost of producing output level y is, cB (y) = y(1−y)2

A. A can operate in a perfectly competitive industry but B

cannot
B. B can operate in a perfectly competitive industry but A
cannot

9
 C. Neither could operate in a perfectly competitive industry
D. Either could operate in a perfectly competitive industry.

26. Suppose we generically refer to a New Keynesian model as a model

with a non vertical aggregate supply (AS) curve. Under sticky
prices, the AS curve will be , and under sticky wages, the AS
curve will be , respectively.
A. horizontal, upward sloping
B. upward sloping, upward sloping
C. downward sloping, horizontal
D. upward sloping, horizontal

27. With perfect capital mobility, and , monetary policy is

at influencing output.
A. fixed exchange rates, effective
B. fixed exchange rates, ineffective
C. flexible exchange rates, ineffective
D. none of the above are correct

28. The next three questions (28, 29 and 30) use the following infor-
mation. Consider an economy with two goods, x and y, and two
consumers, A and B, with endowments (x, y) given by (1, 0) and
(0, 1) respectively. A’s utility is

UA (x, y) = x + 2y

while B’s utility is

UB (x, y) = 2x + y.
Using an Edgeworth box with x measured on the horizontal axis
and y measured on the vertical axis, with A’s origin in the bottom-
left corner and B’s origin in the top-right corner, the set of Pareto-
optimal allocations is

A. a straight line segment

B. the bottom and right edges of the box

10
 C. the left and top edges of the box
D. none of the above.

29. Referring to the information given in the previous question, the

following allocations are the ones that may be achieved in some
competitive equilibrium.
A. (0,1)
B. The line segment joining (0, 1/2) to (0, 1) and the line
segment joining (0, 1) to (1/2, 1)
C. The line segment joining (1/2, 0) to (1, 0) and the line
segment joining (1, 0) to 1, 1/2)
D. (1,0)

30. Referring to the information given in the previous two questions, if

the price of y is 1, then the price of x in a competitive equilibrium
A. must be 1/2
B. must be 1
C. must be 2
D. could be any of the above.

11
 SYLLABUS FOR MSQE
(Program Code: MQEK and MQED) 2016

Syllabus for PEA (Mathematics), 2016

Algebra: Binomial Theorem, AP, GP, HP, Exponential, Logarithmic Series,

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
optimization of functions of not more than two variables.

Elementary Statistics: Elementary probability theory, measures of central

tendency, dispersion, correlation and regression, probability distributions,
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
welfare economics.

Macroeconomics: National income accounting, simple Keynesian Model of

income determination and the multiplier, IS-LM Model, models of aggregate
demand and aggregate supply, Harrod-Domar and Solow models of growth,
money, banking and inflation.
 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 12 is priced at p = 2. Find the
optimal consumption of B and E when R = 1 and I = 3. [7]+[7]+[6]
1
2

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
differentiable in q.

(i) Derive the profit-maximising outcome.

(ii) Is the marginal revenue curve necessarily negatively sloped?

(iii) Can the monopolist fix the price of the magazine below the marginal cost of pro-
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
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
that inputs are essential and Inada conditions hold). Assume that capital evolves according
to
Kt+1 = (1 − δ)Kt + It
where It is the level of investment.

(i) Define yt = Ht .Show

Yt
that
yt = At f (kt )
where f (k) = F (k, 1).

(ii) Define kt = Kt
Ht and it = It
Ht . Show that
(1 − δ)kt + it
kt+1 =
1+n

(iii) Suppose the savings rate is given by st = σyt where σ ∈ [0, 1]. Derive the condition
that determines the steady state capital stock when α = 0. How many non-zero steady
states are there ?

(iv) Let γt = kt+1

kt be the gross growth rate. Suppose α = 0. Derive an expression for γt
dγt
and evaluate and discuss the sign for dk t
.
 3

kt+1
(v) Let f (kt ) = ktθ , A0 = 1, and α > 0. Along a balanced growth path show that 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.
4

(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]

8. Consider a cake of size 1 which can be divided between two individuals, A and B. Let
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.

(i) What is the set of Pareto optimal allocations in this economy?

(ii) Suppose A is asked to cut the cake in two parts, after which B can choose which of
the two segments to pick for herself, leaving the other segment for agent A. How should A
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 }.

(i) Suppose that 1 unit of output is to be produced and factor prices are 1, 2, 3 and 4 for
x1 , x2 , x3 and x4 respectively. Solve for the optimal factor demands.

(ii) Derive the cost function.

(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,
M M
( )d = 0.25Y − 62.5r, ( )s = 500.
P P
Any disequilibrium in the international money market is corrected instantaneously through
a change in r. However, any disequilibrium in the goods market, which is corrected through
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
the level of I that must occur before there is any change in the level of Y ?
(b) Draw a graph to explain your answer.
(c) Calculate the value of (r, Y ) that puts both the money and goods market in equilib-
rium. What is the value of investment at this point compared to (r = 2, Y = 2500)?
[10]+[5]+[5]