exponential and logarithmic functions - peter cramton

Exponential and Logarithmic

Functions

Professor Peter Cramton

Economics 300

Modeling growth

• Exponential functions

– Constant percentage growth per unit time

• Logarithmic functions

– Inverse of exponential functions

Growth of money

• Interest rate r

• Value of Xt after 1 time period: Xt+1 = (1 + r)Xt

– r = 10%; $10 today is worth (1.1)10 = $11 next year

• Value of Xt after 2 time periods:

Xt+2 = (1 + r)Xt+1 = (1 + r)(1 + r)Xt = (1 + r)2Xt

• Value of Xt after n time periods

Xt+n = (1 + r)Xt+n-1 = (1 + r)nXt

• $1 earning 5% for 50 years = $11.47

• $1 earning 10% for 50 years = $117.39

– Doubling interest rate has a huge impact

Exponential growth

10 20 30 40 50

10

20

30

40

50

r=5%

r=10%

n (years)

(1 )nr

More frequent compounding

• Once per year:

• Twice per year:

• k times per year:

• times per year:

(1 )r2

2(1 )r

1 1(1 ) (1 ) (1 )kr

kr

rk mrrk m

1lim(1 )r

m r

mm

e

1 2

10 2.593742

100 2.704814

10000 2.718146

100000000 2.718282

1(1 )k

kk

Most important constant in economics

2.718281828459045235360287471352662497757247093699

959574966967627724076630353547594571382178525166

427427466391932003059921817413596629043572900334

295260595630738132328627943490763233829880753195

251019011573834187930702154089149934884167509244

761460668082264800168477411853742345442437107539

077744992069551702761838606261331384583000752044

933826560297606737113200709328709127443747047230

696977209310141692836819025515108657463772111252

389784425056953696770785449969967946864454905987

9316368892300987931

1lim(1 )k

kk

e

10 20 30 40 50

10

20

30

40

50

Exponential growth

r=5%

r=10%

n (years)

(1 )nrrne

Continuous

compounding

Annual

compounding

Effective rate vs. annual rate

• Annual rate of rA = 10% with continuous

compounding

• What is effective rate rE over the year?

• rA = 10% then rE = 10.52% (continuous compounding)

• rA = 10% then rE = 10.50% (daily compounding)

1

1

A

A

r

E

r

E

r e

r e

3651365

(1 ) 1Ar

Er

The Mating Game

A surprising application of e

[See mating-game.nb]

Present value (discrete)

• What is $1 next year worth today

• With r = 10%, than $1 today is worth $1.10 next

year

1

1

(1 )

1

(1 )(1 )

t t

tt

nt nt t nn

X r X

XX

r

XX X r

r

Present value (continuous)

• What is $1 next year worth today

• With r = 10%, than $1 today is worth $1.10 next

year

1

11

r

t t

rtt tr

rnt nt t nrn

X e X

XX X e

e

XX X e

e

Net present value

• Investments generate costs and revenues over

time

• What is the value today of the sequence of cash

flows from an investment?

10

0

0

Revenue Cost

...1 (1 ) (1 )

1, where = 1

1

t t t

nn t

n tt

nt

t

t

CF

CF CFCFNPV CF

r r r

NPV CFr

Examples

• Perpetuity: value of $1 each period forever

• Annuity: value of $1 each period for n periods

Discount rate 0

1Discount factor 1

1

r

r

1

1d

1

1

n

nd

Example

• Perpetuity: value of $1 each period forever

• r = 10%; = .909; perpetuity = $11.00

Discount rate 0

1Discount factor 1

1

r

r

2

2

1 ...

... , subtracting yields

(1 ) 1

1

1

d

d

d

d

Example

• Annuity: value of $1 each period for n periods

• r = 10%; = .909; 20-year annuity = $9.36

Discount rate 0

1Discount factor 1

1

r

r

2 1

2

1 ...

... , subtracting yields

(1 ) 1

1

1

n

n

n

n

n

n

n

n

d

d

d

d

Logarithms

• Inverse of exponential function

• b is the base

• Most commonly b = 10 or b = e

• log base e is called natural logarithm:

log ( ) finds exponent y such that y

by x b x

ln( )

log( )

y x

y x

Log is inverse of exponential function

Log base 10

• Example of log base 10

• x 1 10 100 1000 10000

• y 0 1 2 3 4

• Examples of log scales

– Shock waves (Richter scale for earthquakes)

(2011: Virginia 5.8, Japan 9.0; 1585 times larger)

– Sound waves (decibels for sound)

– Radio waves (Hz, kHz, MHz, GHz)

10

10

log ( )

yx

y x

Log base 10 and base 2

1 2 3 4 5

10

100

1000

104

105

Log plot of exponential growth

10xy

xy e

2xy

Properties of logarithmic functions

log 1

log log log

log log log

log log

loglog

log

b

b b b

b b b

y

b b

ab

a

b

xy x y

xx y

y

x y x

xx

b

Natural logs (base e)

• Continuous growth models

• Same properties hold

• Example: Yahoo Finance (plotting stock history)

ln 1

ln ln ln

ln ln ln

ln lny

e

xy x y

xx y

y

x y x

DJ vs. S&P vs. Nasdaq (linear scale)

DJ vs. S&P vs. Nasdaq (log scale)

Average return from stocks

• return r

• Dow Jones $742.12 in February 1978


30742.12 12650.36

ln(742.12) 30 ln(12650.36)

ln(12650.36) ln(742.12)9.45%

30

re

r

r


• return r



32742.12 9908.39

ln(742.12) 32 ln(9908.39)

ln(9908.39) ln(742.12)8.10%

32

re

r

r


• return r


• Dow Jones $15,801.79 in February 2014

36742.12 15801.79

ln(742.12) 36 ln(15801.79)

ln(15801.79) ln(742.12)8.50%

36

re

r

r


• return r, accounting for inflation

• Dow Jones $742.12 in Feb 1978; CPI 62.5

• Dow Jones $15,801.79 in Feb 2014; CPI 234.1

36(742.12 / 62.5) 15801.79 / 234.1

ln(742.12 / 62.5) 36 ln(15801.79 / 234.1)

ln(15801.79 / 234.1) ln(742.12 / 62.5)4.83%

36

re

r

r

Average growth rate

• Value at time 0: V0

• Value at time T: VT

• Assume constant percentage growth per unit time

0

0

0

ln( ) ln( )

ln( ) ln( )

rT

T

T

T

V e V

V rT V

V Vr

T

Inflation in Mexico, 1957-1990

Cobb-Douglas production

• One special case:

• In general:

• Q = real GDP

• L = labor

• K = capital

• are parameters

1 12 2

1 23y x x

Q AL K

and

Cobb-Douglas production

• We measure Q, L, and K at each time:

• Taking logs:

• Nice linear model!

• Can estimate parameters with econometrics

• Using subtraction

• What is ?

ln ln ln ln

t t t t

t t t t

Q A L K

Q A L K

1 1 1 1ln ln (ln ln ) (ln ln ) (ln ln )t t t t t t t tQ Q A A L L K K

1ln lnt tQ Q

How long does it take for something to

double?

• With r = 10% it takes 7 years for value to double

• With r = 5% it takes 14 years for value to double

• Moore’s Law of electronics: a doubling every 18 months – r = .6931/1.5 = 46%

0

0/ 2

ln( ) ln(2)

ln(2) .6931

rn

n

rn

n

rn

V e V

e V V

e

nr r

Properties of logarithmic functions

log 1

log log log

log log log

log log

loglog

log

b

b b b

b b b

y

b b

ab

a

b

xy x y

xx y

y

x y x

xx

b

Simplify

10log (100)

ln

10 5

2

5

10

ln

1log

1ln

x xe e

x

x ye

100

0x x

5

10 10log 5logx x

5 2( ln ln )x y

Problem

• $10,000 invested at 5% with continuous

compounding

• When do you have $15,000?

• Use formula for future value: Xt+n = Xt ern

• 15,000 = 10,000 e.05n

• Solve for n

e.05n = 15,000/10,000 = 1.5

.05 n = ln(1.5) n = ln(1.5)/.05 = 8.1093

exponential and logarithmic functions - peter cramton

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