EC 001: Micro Economic Theory

Department of Economics
University of Delhi
Internal AssessmeHt
1s Semester (2024-25)
Total Marks: 26
Time: 90 minutes

The exam has twvO sections. Answer both sections in separate sheets. The maximum
marks are assigned toeach question.
SectionA
1. Consider a two good X and Y economy for an individual to maximize its utility. In this
cconomy the govermment wants to set some norm of minimum utility to define poverty
level for each consumer. Now let's suppose each Individual face the utility function U=
X'y where X and Y are two goods available in the economy to consume. The price of X
is P1and the price of goodY is half the price of good X. Now the governments decides at
least Uo utility level (some real number) for subsistence of each individual. Any individual
do not have other source of incomc or initial wealth. Then

(a) What shouldbe the minimum expenditure, the government needs to ensure or transfer
to each individual toattain at least the targeted utility level Uo?
(b) Calculate the utility levcl that cach consumer will get if thc govcrnment ensures or
transfers that minimal level of income endowment to each consumer?
(6+7)
 Section B

1. Consider two firms operating in a small, price-taking econonuy witlh the

following production technologies:
n(Li, K) = L*K4
2(L2, Ky) = LY'K/4
2

The economy's total endowment of inputs is:(L, K) = (10,20).

(a) Derive the conditional cost functions for each firm to produce one
unit of output.
(b) State the Stolper-Samuelson Theorem in general terms
(c) Apply the Stolper-Samuelson Theorem to the context of this prob
lem.
(5+2+6)
 Delhi School of Economics
M.A. ECONOMICS, SUMMER SEMESTER 2024
COURSE ECO02. INTRODUCTORY MATHEMATICAL ECONOMICS
Mid-term Examination
5th November 2024

Instructions. Time: 70minutes. Macimum Marks 30. TWO PARTS. Use a separate booklet for
Attempt all questions from each part. Marks for each question given in the each part.
parenthesis.

Part-A: Calculus
Question 1: Is the statement
the atomic statements.
(p Ag) ’ (pVr) a tautological implication? Justify. Wherep,g & r are
[Marks 2])
Question 2: Consider the recursive sequence defined by:

1
an+1 zan t n.
Show that an converges.
(Marks 4)
Question 3: Let two non-empty sets, A and B, be grouns with
respect to a binary operation O.
1. Prove that if f: A’ B and g: B’ A are functions such
and g are inverses of each other. The symbol ida & ida that o.g =ida and g o f = ida, then j
f
A& B respectively. [The identity function represent the identity function on the set
ida:A’ Ais defined as: ida() = t for all zE A.
2. Does the theorem still hold if .both
existence of an identity element withsets
A and B satisfy all the
properties of a group except lor e
respect to the binary operation o? Justify your answer.
ONestion 4: Using e ð
definition, show that the following function is [Marks 2+2=4)
continuous at z = 1
if < 1
f(e) = 2- if 1< < 2
if c>2

[Marks 4)

1
 Part-B: Linear Algebra
Questlon 1:
a) Consider the subset {(,T2, T3)"eF:Z = 2r3}
(2+1+2
(i) Determine if it is a subspace in F, Check if the properties are satisfied.
(ii) Think of a linear transformation in F³ that generates the above subspace.
Is that transfornation "onto" or "one-one" or both? Give justification in support of your claim.
(ii) Calculate the "orthogonal perp" (i.e. the subspace orthogonal) to the subspace in (i) above.
D) Let UI,Un be a basis in V. Further let a = Sn, OUk and y
Under what comdition the inner product of z and u is defined? SBkUk, where ok,Be EF.
[Hint: think in terns of bass]
Question 2:
a) Let B denote the basis of R". Suppose that there exist alinear transformation T:R ’
R2, whose matrix with respect to B is What is the matrix of T with respect to the standard basis
of R2? Express your answer in terms of a, b, c, d.
(3|
b) Find the plane that gives best fit to the 4 values b = (2, 5,-1,3) at the corners (1, 0), (0, 1),
and (0, --1) of a square. The equations C 4 Dr +Ey =b at those 4 points are Az = b with th1ree (--1,0)
I =(C, D, E). At the centre of the square, show that C + Dz + Ey = nknowus
average of the b's.
c) Let Ube a2×2 orthogonal matrix with det U=1. Prove that Uis a rotation matrix. Docs work
for a 3 x 3 matrix as wel1? What additional condition might be required? [3

2
 Department of Economics
University of Delhi
M.A, Economics: Semester I

003: Basic Econometrics

Midterm Exam, November 6, 2024

Time: 90 minutes
Maximum MarkS: 26

Part B- 10 marks

Instructions: Do all questions. Each part should be answered on a separate amswer sheet. Start each
question on a fresh page. The marimum marks that can be awarded for each question are given in parentheses
belowthe question.
Part A- 10 marks

1. Provide the intuition and a proof for the law of total variance. (4 marks)

2. State which of the following statements are true. Provide examples to support your answers.
(a) Conditional independence of events implies unconditional independence of events
(b) Unconditional independence of events implies conditional independence of events
(c) Conditional independence of events does not imply unconditional independence of events
(a) Unconditional independence of events does not imply conditional independence of events
(4 marks)
3. Abox contains 100 balls, of which r are red. Suppose that the balls are drawn from the box one at a time,
at random, without replacement. Determine (a) the probability that the 1* ball drawm will NOT be red;
and (b)the probability that the 100h ball drawn will NOT be red. (2 marks)

Part B- 10 marks

1. State whether each of the following statements are true or false and provide a one-sentence proof if true or
counter-example if false (no more than 15 words).
(a) Consider a random sample (X1,...Xn) from a uniformdistribution on [0, 0] where 0 is unknown. The
Maximum Likelihood Estimator of the median of the distribution is the median sample value if n is
an odd-nunber.

(b) In any estimation problem, order statistics based on a random sample are always jointly sufficient.
(c) Suppose we have a randon sample (X1,...X,) from a normal distribution. The sample mean based
on the irst (n-1) ob8er vations is a biased estimator while the sample mean based on all n observations
is an unbiased estimator.

(6 marks)

1
Z. Let Jo(r) be the density function for a
uniform distribution ou (0, 1l, and fi(r)= cI is a densy
ol an alternative distribution, where e > Vou wut to
fipd a test which minimizes Type Ierror, gven a
level of signifcance a = 05 What is ho form of this optimal test
and how will the critical region tor ns
test and the level of Tvpe ll error vary
with the level of c.
(4 marks)

(Part C)

1. Does an increase in the error variance necessarily reduce the reliability of estimated
coefficients in a regression analysis? Justify your argument? 2marks
2. Consider

Food Expenditure = ao + a Householdsize + u,

Where u = Household size * e

Where e is a random variable with E(e) = 0 and var(e) = o . Assume that e is

independent of household size.
Provide a discussion that supports the assumption that the variance of food
expenditure increases with household size. Whether the çausal impact of household
size on food expenditure is captured by a, or not? (2+2=4marks)