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
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
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 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
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
Average return from stocks
• return r
• Dow Jones $742.12 in February 1978
• Dow Jones $12650.36 in February 2008
30742.12 12650.36
ln(742.12) 30 ln(12650.36)
ln(12650.36) ln(742.12)9.45%
30
re
r
r
Average return from stocks
• return r
• Dow Jones $742.12 in February 1978
• Dow Jones $9908.39 in February 2010
32742.12 9908.39
ln(742.12) 32 ln(9908.39)
ln(9908.39) ln(742.12)8.10%
32
re
r
r
Average return from stocks
• return r
• Dow Jones $742.12 in February 1978
• 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
Average return from stocks
• 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
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
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