Quantitative Methods for Economics, Tutorial 1

7/30/2019 Quantitative Methods for Economics, Tutorial 1

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Quantitative Methods for EconomicsKatherine EyalTutorial 1


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TUTORIAL 1

26th-30th July 2010

ECO3021S

PART 1

1. In the following paired statements, let p be the first statement and q the second.

Indicate for each case whether p q, p q or p q.

(a) It is a holiday; it is Christmas Day.

(b) The geometric figure has four sides; it is a rectangle.

(c) Two ordered pairs (a, b) and (b, a) are equal; a is equal to b.

(d) x Z; x R.(e) The petrol tank in my car is empty; I cannot start my car.

2. Simplify the following expressions:

(a) x2 x0

(b) a3b2a5b(c)

x1/5y6/5

z2/5

5(d) ln eab + ln e3a

(e) elnx2

3. A firms output Y is related to capital input K, labour input L and natural resourceinput R by the production function


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Find

(a) AB

(b) CA

6. A firm with 5 retails stores stocks TVs (t), hi-fis (h), DVD players (d) and computers(c).

Shop 1 has 10t, 15h, 9d and 12c.





Express the current inventory in the form of a matrix.

7. Solve the following system of equations Ax = b.

4 1 4 65 5 0 34 4 10 60 0 0 0

x1x2x3x4

=

3214

8. Show in the 3x3 case that there exists only one matrix I that satisfies AI= IA = A,for any matrix A, and that is the (3x3) identity matrix.


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(g) ln

x

(x + 1)2 (x + 2)3

(h) log2 (2x) log2 (x + 1)

10. The components of the 3-vector a are Vuyos weekly expenditures on food, clothingand housing. The components of the 3-vector b are Siyandas weekly expenditureson food, clothing and housing. Interpret the vectors a + b and 52a.

11. If

a =

25

,b =

71

find

(a) a + b

(b) 4a

(c) 6b(d) 2a

3b

12. Let

A =

1 0 53 2 1

2 1 0

,B =

3 1 20 1 1

0 1 0

(a) Find 2B.(b) Find AB.


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15. Solve the following systems of equationsAx

=b.

(a)

1 2 4 10 0 2 3

x1x2x3x4

=

56

(b) 1 3 23 5 62 4 3

x1x2x3

=57

8

ADDITIONAL QUESTIONS

16. Prove the following

(a) xm xn = xm+n

(b) (xm)n = xmn

(c) logayb

= b loga y


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TUTORIAL 1 SOLUTIONS

2010

ECO3021S

1. In the following paired statements, let p be the first statement and q the second.Indicate for each case whether p q, p q or p q.

(a) It is a holiday; it is Christmas Day.

In order for it to be a holiday, it is not necessary that it is Christmas day, sop q.

If it is Christmas Day, it is sufficent to conclude that it is a holiday, so p q.Thus q is sufficient, but not necessary, for p, i.e. p q.

(b) The geometric figure has four sides; it is a rectangle.

In order for the geometric figure to have four sides, it is not necessary that it isa rectangle, so p q.

If it is a rectangle, it is sufficent to conclude that the geometric figure has foursides, so p q.Thus q is sufficient, but not necessary, for p, i.e. p q.

(c) Two ordered pairs (a, b) and (b, a) are equal; a is equal to b.

In order for two ordered pairs (a, b) and (b, a) to be equal, it is necessary that ais equal to b, so p q.


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(a) x2 x0 = x2

(b) a3b2a5b = a8b3

(c)

x1/5y6/5

z2/5

5=

xy6

z2

(d) ln eab + ln e3a = ab + 3a

(e) eln x2 = x2

3. A firms output Y is related to capital input K, labour input L and natural resourceinput by the production function

Y = 2K1/2L1/3R1/6

Write down a linear relationship between the logarithms of Y,K,L,R.

ln Y = ln2 +1

2ln K +

1

3ln L +

1

6ln R

4. Let

A =

1 0 53 2 1

2 1 0

,B =

3 1 20 1 1

0 1 0


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(b)

CA =

24 2014 2

6. A firm with 5 retails stores stocks TVs (t), hi-fis (h), DVD players (d) and computers(c).






Express the current inventory in the form of a matrix.

Shop number

t h d c10 15 9 1220 14 8 516 8 15 625 15 7 165 12 20 18

7. Solve the following system of equations Ax = b.


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Part 2


(a)x3

x3= x6

(b)x2y7

x3y5=

y2

x

(c)

x1 y12 = x

2

y2

(y x)2=

1x 1

y

2=

y xxy

2etc.

(d) ln Ae4 + ln e7 = ln A + 11

(e) ln e + log1

10= 1 1 = 0

(f) lnx2

(x + 1)3= 2ln x 3 l n (x + 1)

(g) lnx

(x + 1)2 (x + 2)3= 1

2ln x 2 l n (x + 1) 3 l n (x + 2)

(h) log2 (2x) log2 (x + 1) = log22x

x + 1

10. The components of the 3-vector a are Vuyos weekly expenditures on food, clothingand housing. The components of the 3-vector b are Siyandas weekly expenditureson food, clothing and housing. Interpret the vectors a+ b and 52a.


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(c)

6b =426

(d)

2a 3b =

410

213

=

17

7

12. Let

A =1 0 53 2 1

2 1 0

,B = 3 1 20 1 10 1 0

(a) Find 2B.

2B =

6 2 40 2 20

2 0

(b) Find AB.

AB =

3 6 29 6 8

6 3 5

13. Let


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14. A clothing store discounts all its jeans, jackets and suits by 20% at the end of theseason. IfV1 is the value of the stock in its three branches prior to the discount, findthe value ofV2 after the discount where

V1 =

5000 4500 600010000 12000 7500

8000 9000 11000

V2 = 0.8 5000 4500 600010000 12000 7500

8000 9000 11000

=

4000 3600 48008000 9600 6000

6400 7200 8800

15. Solve the following systems of equations Ax = b.

(a)

1 2 4 10 0 2 3

x1x2x3x4

=

56

This is in row echelon form. Columns 1 and 3 are basic, so we assign arbitraryvalues to x2 and x4. Let x2 = and x4 = .Then


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(b) 1 3 23 5 6

2 4 3

x1x2

x3

=

57

8

The augmented matrix

1 3 23 5 62 4 3

5

78

1 3 20 4 12

0 2 7

582

(2) 3 (1)

(3) 2 (1)

1 3 20 1 30

2 7

52

2

(2)1

4

1 0 70 1 3

0 0 1

122

(1) 3 (2)

(3) + 2 (2)

1 0 00 1 0

0 0 1

1582

(1) 7 (3)

(2) + 3 (3)


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Quantitative Methods for Economics, Tutorial 1