Non Linear Essay

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Lecture 4: Non-Linear Functions

ECO-10023

Quantitative Methods

M. Jami Husain
ROOM CBA 2.028, CHANCELLOR BUILDING e-mail: m.j.husain@mngt.keele.ac.uk

Lecture 4

Non-Linear Functions

ECO10023: Quantitative Methods

M. Jami Husain 2011/12

LECTURE CONTENT

Lecture Outline: • Powers functions • Solving Quadratic equations • Logs and Exponential functions • Applications

Reading:

• Bradley 4.1-4.4 and Jacques 2.1, 2.3, 2.4

ECO10023: Quantitative Methods

M. Jami Husain 2011/12

Motivation for non-linear functions

• Better representation of relationship • Derived functions: Linear demand gives non linear total revenue function Q = 5 – P/3 demand function P = 15 – 3Q inverse demand function

then total revenue is: TR = P.Q = 15Q – 3Q2 which is quadratic is Q We can plot this …..

ECO10023: Quantitative Methods

M. Jami Husain 2011/12

Demand and Total Revenue
20 18 16 15 18 18

Q 0 1 2 3 4 5

P 15 12 9 6 3 0

TR 0 12 18 18 12 0

14 12 10 9 8 6 4 3 2 0 0 0 6 12 12 12

0

1

2
Quantity

3

4

5

Demand and Total Revenue
20 18 18 18

Points to note: a) is symmetrical b) slope non – constant

16 15 14 12 10 9 8 6 4 3 2 0 0 0 6 12 12 12

0
ECO10023: Quantitative Methods

1

2
Quantity

3

4

5

M. Jami Husain 2011/12

That was one General Case….

e.g.

(2 x − 3) 2 = (2 x − 3)(2 x − 3)
M. Jami Husain 2011/12

ECO10023: Quantitative Methods

Rules of Powers
o There are a set of rules that we should revise – o Exponents are used lots in Econ/Finance

I.

x .x = x

m

n

m+n

e.g.

x2 times x3 = x5 72.73=75 xn. x=xn+1

II.

xm = x m−n xn

e.g.

x5 = x 5− 2 = x 3 2 x
M. Jami Husain 2011/12

ECO10023: Quantitative Methods

1 x2 x.x = = 2 = x −2 x 4 x.x.x.x x
Hence: III.

x

−n

1 = n x
= n , we have x0=1

IV. Using (2) with m

xm m−n 0 1= n = x = x x
V.

(xm)n = xmn
M. Jami Husain 2011/12

m=3 and n=2 then (x3)2 =...

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