Quantitative Business Analysis Assignment: ECO2IQA, Semester 2, 2018

This document presents a complete solution to a Quantitative Business Analysis assignment, likely for an economics course. The assignment is divided into two sections: Section A comprises multiple-choice questions addressing core concepts such as price elasticity of demand and equilibrium price and quantity determination. Section B delves into more complex problems, including supply and demand analysis, migration and matrix analysis, and a national income model. The solutions in Section B demonstrate step-by-step calculations and explanations, including the derivation of equilibrium prices and quantities, matrix operations, and the application of economic models to real-world scenarios. The assignment covers various economic principles and analytical techniques relevant to business decision-making and economic modeling, providing a comprehensive overview of the subject matter. The assignment is designed to test the students' understanding of quantitative methods and their ability to apply them to economic problems.

Running Head: QUANTIATIVE BUSINESS ANALYSIS
Quantiative Business Analysis
Name of the Student
Name of the University
Course ID

1QUANTIATIVE BUSINESS ANALYSIS
Table of Contents
Section A ......................................................................................................................................... 2
Answer 1 ..................................................................................................................................... 2
Answer 2 ..................................................................................................................................... 2
Answer 3 ..................................................................................................................................... 2
Answer 4 ..................................................................................................................................... 2
Answer 5 ..................................................................................................................................... 2
Answer 6 ..................................................................................................................................... 2
Answer 7 ..................................................................................................................................... 2
Answer 8 ..................................................................................................................................... 2
Answer 9 ..................................................................................................................................... 2
Answer 10 ................................................................................................................................... 3
Section B ......................................................................................................................................... 3
Answer 1: Supply and Demand ...................................................................................................... 3
Answer 2: Migration and Matrices ................................................................................................. 8
Part 1 ........................................................................................................................................... 8
Part 2 ........................................................................................................................................... 8
Part 3 ........................................................................................................................................... 8
Part 4 ........................................................................................................................................... 9
Part 5 ........................................................................................................................................... 9
Answer 3: National Income Model ............................................................................................... 10
Part 1 ......................................................................................................................................... 10
Part 2 ......................................................................................................................................... 10
Part 3 ......................................................................................................................................... 12

2QUANTIATIVE BUSINESS ANALYSIS
Section A
Answer 1
d. -0.25
Answer 2
c. (P∗, Q∗) = (40,140)
Answer 3
d. 2500
Answer 4
c. 14.3
Answer 5
d. Upper Triangular matrix
Answer 6
b. is the product of elements in the principal diagonal
Answer 7
a. F is a 2x6 matrix
Answer 8
b. Q > 20
Answer 9

3QUANTIATIVE BUSINESS ANALYSIS
a. 0.5
Answer 10
b. the curve will shift parallel to the right
Section B
Answer 1: Supply and Demand
The demand function of wine is given as
𝑄𝑑 = 𝑘 − 4𝑃
The supply function of wine is given as
𝑄𝑠 = 1 + 3𝑃
At equilibrium,
𝑄𝑑 = 𝑄𝑠
𝑂𝑟, 𝑘 − 4𝑃 = 1 + 3𝑃
𝑂𝑟, 7𝑃 = 𝑘 − 1
𝑂𝑟, 𝑃 = 𝑘 − 1
7
𝑄𝑑 = 𝑘 − 4𝑃
= 𝑘 − 4 × ( 𝑘 − 1
7 )
= 𝑘 − 4𝑘 − 4
7

4QUANTIATIVE BUSINESS ANALYSIS
= 7𝑘 − 4𝑘 + 4
7
= 3𝑘 + 4
7
𝑬𝒒𝒖𝒊𝒍𝒊𝒃𝒓𝒊𝒖𝒎 𝒑𝒓𝒊𝒄𝒆(𝑷 ∗) = 𝒌 − 𝟏
𝟕
𝑬𝒒𝒖𝒊𝒍𝒊𝒃𝒓𝒊𝒖𝒎 𝑸𝒖𝒂𝒏𝒕𝒊𝒕𝒚 (𝑸 ∗) = 𝟑𝒌 + 𝟒
𝟕
a.
At, k =3
𝐷𝑒𝑚𝑎𝑛𝑑 (𝑄𝑑 ) = 3 − 4𝑃
𝑆𝑢𝑝𝑝𝑙𝑦 (𝑄𝑠) = 1 + 3𝑃
𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑝𝑟𝑖𝑐𝑒 = 𝑘 − 1
7
= 3 − 1
7
= 2
7
= 𝟎. 𝟐𝟗
𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 = 3𝑘 + 4
7
= (3 × 3) + 4
7

5QUANTIATIVE BUSINESS ANALYSIS
= 9 + 4
7
= 13
7
= 𝟏. 𝟖𝟔
b. At k =4
𝐷𝑒𝑚𝑎𝑛𝑑 (𝑄𝑑 ) = 4 − 4𝑃
𝑆𝑢𝑝𝑝𝑙𝑦 (𝑄𝑠) = 1 + 3𝑃
𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑝𝑟𝑖𝑐𝑒 = 𝑘 − 1
7
= 4 − 1
7
= 3
7
= 𝟎. 𝟒𝟑

6QUANTIATIVE BUSINESS ANALYSIS
𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 = 3𝑘 + 4
7
= (3 × 4) + 4
7
= 12 + 4
7
= 16
7
= 𝟐. 𝟐𝟗
c.
at k = 2
𝐷𝑒𝑚𝑎𝑛𝑑 (𝑄𝑑 ) = 2 − 4𝑃
𝑆𝑢𝑝𝑝𝑙𝑦 (𝑄𝑠) = 1 + 3𝑃
𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑝𝑟𝑖𝑐𝑒 = 𝑘 − 1
7

7QUANTIATIVE BUSINESS ANALYSIS
= 2 − 1
7
= 1
7
= 𝟎. 𝟏𝟒
𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 = 3𝑘 + 4
7
= (3 × 2) + 4
7
= 6 + 4
7
= 10
7
= 𝟏. 𝟒𝟑
K =1/2

8QUANTIATIVE BUSINESS ANALYSIS
For feasible solution to exists, k must be greater than 1. At any value of k which is less than 1,
price would be negative which is not possible in real world.
Answer 2: Migration and Matrices
𝑃 = (
𝑝11 𝑝12 𝑝13
𝑝21 𝑝22 𝑝23
𝑝31 𝑝32 𝑝33
) , 𝑥𝑡 = (
𝑥𝑡1
𝑥𝑡2
𝑥𝑡3
), 𝑥 0 = (
𝑥01
𝑥02
𝑥03
)
Part 1
We know that, 𝑥𝑡 = 𝑃𝑥𝑡−1
Therefore, 𝑥1 = 𝑃𝑥0
And, 𝑥2 = 𝑃𝑥1
= 𝑃. 𝑃𝑥0
= 𝑃2𝑥0
Part 2
𝑥1 = 𝑃𝑥0
= (
𝑝11 𝑝12 𝑝13
𝑝21 𝑝22 𝑝23
𝑝31 𝑝32 𝑝33
) × (
𝑥01
𝑥02
𝑥03
)
= (
𝑝11 𝑥01 𝑝12 𝑥02 𝑝13 𝑥03
𝑝21 𝑥01 𝑝22 𝑥02 𝑝23 𝑥03
𝑝31 𝑥01 𝑝32 𝑥02 𝑝33 𝑥03
)
Part 3
𝑃 = (
0.75 0.25 0.01
0.1 0.65 0.54
0.15 0.1 0.45
)

9QUANTIATIVE BUSINESS ANALYSIS
𝑥0 = (
8
12
20
)
Therefore, 𝑥1 = 𝑃𝑥0
= (
0.75 0.25 0.01
0.1 0.65 0.54
0.15 0.1 0.45
) × (
8
12
20
)
= (
9.2
19.4
11.4
)
Therefore, after migration in period 1, the number of workers in region 1 = 9.2 million, the number
of workers in region 2 = 19.4 million and the number of workers in region 3 = 11.4 million
Part 4
𝑥2 = 𝑃2𝑥0
= (
0.75 0.25 0.01
0.1 0.65 0.54
0.15 0.1 0.45
) × (
0.75 0.25 0.01
0.1 0.65 0.54
0.15 0.1 0.45
) × (
8
12
20
)
= (
0.589 0.351 0.147
0.221 0.5015 0.595
0.19 0.1475 0.258
) × (
8
12
20
)
= (
11.864
19.686
8.45
)
Therefore, after migration in period 2, the number of workers in region 1 = 11.86 million, the
number of workers in region 2 = 19.69 million and the number of workers in region 3 = 8.45
million
Part 5

10QUANTIATIVE BUSINESS ANALYSIS
The number of workers in each region after an influx of international migrants to each region in
Looneyland by 5,000,000 after period 1 is:
Region 1 = (9.2 + 5) million = 14.2 million
Region 2 = (19.4 + 5) million = 24.4 million
Region 3 = (11.4 + 5) million = 16.4 million
Answer 3: National Income Model
𝐶 = 20 + 0.85 (𝑌 − 𝑇) … … … … … … … … . (𝑖)
𝑇 = 25 + 0.25 𝑌 … … … … … … … … … … … … (𝑖𝑖)
𝐼 = 155 … … … … … … … … … … … … … … … (𝑖𝑖𝑖)
𝐺 = 100 … … … … … … … … … … … … … (𝑖𝑣)
Part 1
In the equations given, Y is the endogenous variable and I and G are the exogenous variables.
Part 2
We know that,
𝑌 = 𝐶 + 𝐼 + 𝐺
= 20 + 0.85 (𝑌 − 𝑇) + 155 + 100
= 20 + 0.85 𝑌 − 0.85 𝑇 + 225
= 275 + 0.85 𝑌 − 0.85 𝑇

11QUANTIATIVE BUSINESS ANALYSIS
→ 0.15 𝑌 + 0.85 𝑇 = 275 … … … … … … … … … . . (𝑣)
− 0.25 𝑌 + 𝑇 = 25 … … … … … … … … … … … … . . (𝑣𝑖)
Let the equations (v) and (vi) be expressed by the following matrix notation, AX = B.
Where, 𝐴 = ( 0.15 0.85
−0.25 1 ) , 𝑋 = (𝑌
𝑇) , 𝐵 = (275
25 )
𝐴𝑋 = 𝐵
→ 𝐴−1𝐴𝑋 = 𝐴−1𝐵
→ 𝑋 = 𝐴−1𝐵
Now,
|𝐴| = | 0.15 0.85
−0.25 1 |
= [(0.15 × 1) − (−0.25 × 0.85)]
= 0.3625
𝐴𝑑𝑗. (𝐴) = ( 1 −0.85
0.25 0.15 )
∴ 𝐴−1 = 𝐴𝑑𝑗. (𝐴)
|𝐴|
= 1
0.3625 ( 1 −0.85
0.25 0.15 )
= (2.76 −2.34
0.69 0.41 )
∴ 𝑋 = 𝐴−1𝐵 = (2.76 −2.34
0.69 0.41 ) × ( 275
25 ) = (700.5
200 )

12QUANTIATIVE BUSINESS ANALYSIS
∴ The required value of Y, as obtained from the matrix method is 700.5
Part 3
𝑌 = 𝐶 + 𝐼 + 𝐺
→ 𝑌 = 20 + 0.85 (𝑌 − 𝑇) + 155 + 100
→ 𝑌 = 20 + 0.85 (𝑌 − 25 − 0.25 𝑌) + 155 + 100 … … … … 𝐹𝑟𝑜𝑚 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 (𝑖𝑖)
→ 𝑌 = 275 + 0.85 (0.75 𝑌 − 25)
→ 𝑌 = 275 + 0.6375 𝑌 − 21.25
→ 𝑌 = 253.75 + 0.6375 𝑌
→ 0.3625 𝑌 = 253.75
→ 𝒀 = 𝟕𝟎𝟎
Thus, the required value of Y as obtained by solving the simultaneous equations (i) and (ii) is 700.