FREE IGNOU MEC 103 QUANTITATIVE METHODS SOLVED ASSIGNMENT 2024-25
Section
A
Answer
the following questions in about 700 words each.
The
word limits do not apply in case of numerical questions. Each question carries
20 marks.
1. a. Consider the following utility maximization problem, in which
the utility function of the consumer is given by U(x,y)=x0.5y 0.5. The consumer
income is I=100 and px=2 and py=3. Find the utility maximizing x and y when the
budget constraint of the consumer may or may not bind. (Apply Kuhn-Tucker
method)
 |
| FREE IGNOU MEC 103 QUANTITATIVE METHODS SOLVED ASSIGNMENT 2024-25 |
1. Utility Maximization
Problem
a. Finding the Utility-Maximizing
Quantities of xxx and yyy
To solve the utility maximization problem, we
will use the Kuhn-Tucker method. The consumer's utility function is given by:
U(x,y)=x0.5y0.5U(x, y)
= x^{0.5} y^{0.5}U(x,y)=x0.5y0.5
The consumer's budget constraint is:
pxx+pyy=Ip_x x + p_y y
= Ipxx+pyy=I
where:
- px=2p_x = 2px=2
(price of xxx)
- py=3p_y = 3py=3
(price of yyy)
- I=100I = 100I=100
(income)
Thus
, the budget constraint becomes:
2x+3y=1002x + 3y = 1002x+3y=100
Step 1: Formulate the Lagrangian Function
The Lagrangian function L\mathcal{L}L for this problem is:
L=x0.5y0.5+Ξ»(100−2x−3y)\mathcal{L}
= x^{0.5} y^{0.5} + \lambda (100 - 2x - 3y)L=x0.5y0.5+Ξ»(100−2x−3y)
where Ξ»\lambdaΞ» is the Lagrange
multiplier.
Step 2: Compute the Partial Derivatives
b. Explain the theorem of second order optimum.
The Second-Order Condition in optimization helps determine
whether a given critical point is a maximum or minimum.
Theorem of Second-Order Optimum (for unconstrained problems):
Second-Derivative Test: For a function f(x)f(x)f(x) with
critical point x0x_0x0, if f′′(x0)<0f''(x_0) < 0f′′(x0)<0, then
x0x_0x0 is a local maximum; if f′′(x0)>0f''(x_0) > 0f′′(x0)>0, then
x0x_0x0 is a local minimum.
Hessian Matrix (Multivariable Case): For a function
f(x1,x2,…,xn)f(x_1, x_2, \ldots, x_n)f(x1,x2,…,xn), the Hessian matrix HHH
is composed of the second-order partial derivatives:
2.
Given the input matrix and the final demand vector: π΄
= [ 0.05 0.25 0.34 0.33 0.10
0.12 0.19 0.38 0 ] π = [ 1800 200 900 ]
2. Input-Output Analysis
Given:
Input-output matrix AAA:
A=[0.050.250.340.330.100.120.190.380]A
= \begin{bmatrix} 0.05 & 0.25 & 0.34 \\ 0.33 & 0.10 & 0.12 \\
0.19 & 0.38 & 0 \end{bmatrix} A=0.050.330.190.250.100.380.340.120
Final demand vector ddd:
d=[1800200900]d =
\begin{bmatrix} 1800 \\ 200 \\ 900 \end{bmatrix} d=1800200900
a)Explain the economic meaning of the elements 0.33,0 and 200
a) Economic Meaning of Specific
Elements
- 0.33 (in position (2,1)): This represents the fraction of output from the first
industry that is used as input by the second industry. In other words, 33%
of the output produced by the first industry is used as an input in the
second industry.
- 0 (in position (3,3)): This indicates that the output of the third industry
is not used as input in the third industry. There is no interdependence or
self-supply within the third industry.
- 200 (in vector ddd): This is the final demand for the output of the second
industry. It means that the total demand for the output of the second
industry is 200 units.
b) Explain the economic meaning of (if any) of the third column sum
b) Economic Meaning of the Third Column
Sum
To
find the sum of the third column in matrix AAA:
Third column sum=0.34+0.12+0=0.46\text{Third
column sum} = 0.34 + 0.12 + 0 = 0.46Third column sum=0.34+0.12+0=0.46
The
sum of the third column represents the total proportion of the output from the
third industry that is used as input by all industries. In this case, 46% of
the third industry's output is used as input across the industries.
c) Explain the economic meaning of (if any) of the third row sum
To
find the sum of the third row in matrix AAA:
Third row sum=0.19+0.38+0=0.57\text{Third
row sum} = 0.19 + 0.38 + 0 = 0.57Third row sum=0.19+0.38+0=0.57
The
sum of the third row represents the total proportion of the output from the
third industry that is used as input by the third industry and all other
industries. In this case, 57% of the output from the third industry is used as
input in the other industries.
d) Write out the specific input-output matrix equation for this
model
d) Input-Output Matrix Equation
The
specific input-output matrix equation can be written as:
x=(I−A)−1dx = (I - A)^{-1} d
x=(I−A)−1d
where
xxx is the vector of outputs from each industry, III is the identity matrix,
and AAA is the input-output matrix.
Let's
write the matrix equation explicitly:
·
Find the determinant of I−AI - AI−A (let's
denote it as det(I−A)\text{det}(I - A)det(I−A)).
·
Find determinants of matrices obtained by
replacing each column of I−AI - AI−A with the final demand vector ddd in turn.
·
Compute each variable xix_ixi using:
e) Find the solution output levels of the three industries using Cramer’s
rule.
Section
B
Answer
the following questions in about 400 words each. The word limits do not apply
in case of numerical questions. Each question carries 12marks
3. Using Simplex method solve the problem: Maximise π = 6π₯1 + 2π₯2 + 5π₯3
Subject to ( 2 3 1 1 0 2 1 2 5 ) ( π₯1 π₯2 π₯3 ) ≤ ( 10 8 19 ) x1, x2,
x3 ≥ 0
4. a.) A salesman has 50% chance of making a sale. If two customers
enter the shop, what is the probability that the salesman will make a sale?
b.) If the events A and B are independent, then show that Ac , Bc ,
A and B are pair wise independent.
5. a) Discuss the nature of the following time path: π¦π‘ = 3 π‘ + 1
b) Solve equation: π¦π‘+1 − 1 3 π¦π‘ = 6 πππ π¦0 = 1
6. Examine the following functions for maxima and minima:
a) π§ = 2π₯ 2 + π₯π¦ + 4π¦ 2 + π₯π§ + π§ 2 + 2
b) π§ = π 2π₯ − π π¦ + π 2π§ − 2(π₯ + π) + π¦
7.
Write short notes on following:
e) Euler’s theorem
f) Riemann Sum
g) Point of inflexion
h) Chi square test of goodness of fit
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Submission Date :
· 30
April 2025 (if enrolled in the July 2025 Session)
· 30th Sept, 2025 (if enrolled in the January
2025 session).
IGNOU Instructions for the MEC 103 QUANTITATIVE METHODS Assignments
MEC 103 QUANTITATIVE METHODS
Assignment 2024-25 Before attempting the assignment,
please read the following instructions carefully.
1. Read the detailed
instructions about the assignment given in the Handbook and Programme Guide.
2. Write your enrolment
number, name, full address and date on the top right corner of the first page
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3. Write the course title,
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4. Use only foolscap
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5. Write the relevant question
number with each answer.
6. You should write in your
own handwriting.
GUIDELINES FOR IGNOU Assignments 2024-25
MEG 02 QUANTITATIVE METHODS
Solved Assignment 2024-25 You will find it useful to keep the following points in
mind:
1. Planning: Read the questions carefully. Go through the units on which
they are based. Make some points regarding each question and then rearrange
these in a logical order. And please write the answers in your own words. Do
not reproduce passages from the units.
2. Organisation: Be a little more selective and analytic before drawing up a
rough outline of your answer. In an essay-type question, give adequate
attention to your introduction and conclusion. The introduction must offer your
brief interpretation of the question and how you propose to develop it. The
conclusion must summarise your response to the question. In the course of your
answer, you may like to make references to other texts or critics as this will
add some depth to your analysis.
3. Presentation: Once you are satisfied with your answers, you can write down
the final version for submission, writing each answer neatly and underlining
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MEC 103 QUANTITATIVE METHODS Handwritten Assignment
2022-23
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IGNOU and other university students. There are several types of handwritten
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FREE IGNOU MEC 104 ECONOMICS OF GROWTH AND DEVELOPMENT SOLVED ASSIGNMENT 2024-25
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