Economics 214

Lecture 6

Polynomial Functions

. to1

from integers are polynomial in the exponents theand

numbers real are ,,2,1,0, parameters the

)(

form theakesfunction t polynomial univariateA 2

210

n

nia

xaxaxaaxfy

i

nn

Polynomial Functions The degree of the polynomial is

the value taken by the highest exponent.

A linear function is polynomial of degree 1.

A polynomial of degree 2 is called a quadratic function.

A polynomial of degree 3 is called a cubic function.

Roots of Polynomial Function

a

acbbxx

cbxaxy

bax

bxay

2

4, :roots

:function Quadratic

/ :root

:functionLinear

zero. equalfunction themakethat

argument its of values theare polynomial a of roots The

2

21

2

3 cases for roots of quadratic function

b2-4ac>0, two distinct roots. b2-4ac=0, two equal roots b2-4ac<0, two complex roots

Quadratic example

34

12

4

57

4

257

4

24497

2*2

2*3*477

2

4

2

1

4

2

4

57

4

257

4

24497

2*2

2*3*477

2

4

372

22

2

22

1

2

a

acbbx

a

acbbx

xxy

Plot of our Quadratic function

Roots of Quadratic Equation

-4

-2

0

2

4

6

8

10

12

14

-2 -1 0 1 2 3 4 5

x

y y

Exponential Functions

The argument of an exponential function appears as an exponent.

Y=f(x)=kbx

k is a constant and b, called the base, is a positive number.

f(0)=kb0=k

Exponential Functions

0

and with decreasesly montonical 0,b1When

axis.-y theacross function theof

reflection a is 1

ofgraph theThus1

b

1

exponents of rulesour Using

0

case In this . with increases then ,1When

x.of any valuefor parameter the

ofsign theas same theisfunction thisofsign The

lim

lim

x

x

x

x

x

xx

x

x

x

x

kb

xkb

b

bb

b

kb

xbb

k

Figure 2.16 Some Exponential Functions

Exponential functions with k<0

Exponential Functions with k<0, b=3/2 and 2/3

-8000

-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

1000

-25 -20 -15 -10 -5 0 5 10 15 20 25

x

y

y2

y1

Regional Growth in the U.S.

Region 1990.4 1995.4 2000.4 1991-1995 1996-2000 New England Region 303846 370457 510649 4.0% 6.6% Mideast Region 1000386 1209580 1600680 3.9% 5.8% Great Lakes Region 818083 1048335 1337536 5.1% 5.0% Plains Region 327328 417824 553255 5.0% 5.8% Southeast Region 1051354 1392657 1857465 5.8% 5.9% Southwest Region 443615 598893 850857 6.2% 7.3% Rocky Mountain Region 130333 184108 264158 7.2% 7.5% Far West Region 888273 1073627 1515873 3.9% 7.1%

Annual Grow RateYear and Quarter

New England Income

1264012154

Change

328640303846)(1.0816303846)()04.1(

)1()1)(1(

)1()1()1(

4:1992

316000303846)(04.1

)04.1()1(

4:1991

Income England New

.

)1(

relation thehave wegeneral,In

4:19914:19924:19904:1991

2

4:19902

4:1990

4:19904:19914:1992

4:19904:19904:1991

1

XXXX

XrXrr

XrrXrX

XXrX

formdecimalinrategrowthannualr

tyearinincomeXwhereXrX ttt

Growth Formula

form. decimalin expressed is

)1(

is period,each of end at the

rate by the growsit when periodin level its and ,

period,in variablea of valueebetween th iprelationsh The

gCompoundin

Period-of-End Discrete with FormulaGrowth

r

whereXrX

r

ntt, X

tn

nt

t

New England Personal Income Annual Growth

Rate = 4% Our bar Chart is

approximately a step function.

We assume growth doesn’t occur until end of the year.

050000

100000150000200000250000300000350000400000

1990:4

1991:4

1992:4

1993:4

1994:4

1995:4

PersonalIncome

Step Function

1990

19

90

19

91

19

92

19

93

19

94

19

95

Income

Year

Income Growth

We have depicted the growth of income over time as a step function.

It is usually more natural to think of continuous growth, which would be reflected in a smooth evolution over time of variables like income and population.

Multiple Compounding in one Period

t

k

t

1t

t

Xk

rX

kr

X

t, X

1

is period, theduring times compounded rate the

by growsit when , period,next theof beginning

at the valueits and , period of beginning at the

variablea of valueebetween th iprelationsh The

1

Effect of Number of compounds at 4%

Compounds 1,000.00$ 1 1,040.00$ 2 1,040.40$ 4 1,040.60$

12 1,040.74$ 365 1,040.81$

Continuous 1,040.81$

Effect of Number of Compounds at 4%

$1,000 Compounded over 1 year

$1,039.40

$1,039.60

$1,039.80

$1,040.00

$1,040.20

$1,040.40

$1,040.60

$1,040.80

$1,041.00

1 2 4 12 365 Continuous

Number of Compunds

$1,000

Figure 3.2 Compounding at Different Frequencies