Transcript Econ 299 Chapter 01... - University of Alberta

Economic Data
Calculus and Economics
Basics of Economic Models
Advanced Calculus and Economics
Statistics and Economics
Econometric Introduction
Lorne Priemaza, M.A.
[email protected]
1.1 Data Types and Presentations
1.2 Real and Nominal Variables
1.3 Price Indexes
1.4 Growth Rates and Inflation
1.5 Interest Rates
1.6 Aggregating Data: Stocks and Flows
1.7 Seasonal Adjustment
Appendix 1.1 Exponentials and Logarithms
1) Describe Economy
Current and past data
Increases and
decreases
This information can
influence decisions
ie: GDP, interest rate,
unemployment, price,
debt, etc.
2) Test Theory
Does variable A affect
variable B?
ie: Smokers and the
cost to healthcare
ie: Married couples and
health
Data is essential for economists. Data can
be categorized by:
1) How it is collected:
 time series data
 cross-sectional data
 panel data
2) How it is measured:
 nominal data
 real data
-Collects data on one economic agent
(city/person/firm/etc.) over time
-Frequency can vary (yearly/monthly/
quarterly/weekly/daily/etc.)
-ie: Canadian GDP, GMC stock value,
your height, U of A tuition, world
population
University
Tuit 99/00
Tuit 00/01
Tuit 01/02
Tuit02/03
Alberta
3551.00
3770.00
3890.00
4032.00
British Columbia
2295.00
2295.00
2181.00
2661.00
Calgary
3650.00
3834.00
3975.00
4120.00
Concordia
1668.00
1668.00
1668.00
1668.00
Lethbridge
3360.00
3470.00
3470.00
3470.00
Manitoba
3005.00
2796.00
2807.00
2818.00
McGill
1668.00
1668.00
1668.00
1668.00
Ottawa
3760.00
3892.00
4009.00
4085.00
Year
GDP (In current US$, in trillions)
2007
3.52
2008
4.56
2009
5.06
2010
6.04
2011
7.49
2012
8.46
2013
9.49
2014
10.4
Source: World Development Indicators, The World Bank, www.worldbank.org
#
Year
Rating
#
Year
Rating
1
1987
7.5
1
1987
9
2
1988
6.5
2
1988
5
3
1990
7.3
3
1990
7
4
1991
8.3
4
1991
10/12
5
1992
7.1
5
1992
4
6
1994
8.7
6
1994
11
7
1997
9.4
7
1997
8
8
1999
9.2
8
1999
7
Source: www.thefinalfantasy.com
Time Series:
Source: the truth
One Agent
Many Time Periods
-Collects data on multiple economic
agents (locations/persons/firms/etc) at
one time
-Taken at one specific point in time
(September report, January report, etc.)
-ie: current stock portfolio, hockey
player stats, provincial GDP comparison,
last year’s grades
University
Tuit 99/00
Tuit 00/01
Tuit 01/02
Tuit02/03
Alberta
3551.00
3770.00
3890.00
4032.00
British Columbia
2295.00
2295.00
2181.00
2661.00
Calgary
3650.00
3834.00
3975.00
4120.00
Concordia
1668.00
1668.00
1668.00
1668.00
Lethbridge
3360.00
3470.00
3470.00
3470.00
Manitoba
3005.00
2796.00
2807.00
2818.00
McGill
1668.00
1668.00
1668.00
1668.00
Ottawa
3760.00
3892.00
4009.00
4085.00
Canadian Provincial Corporate Tax 2015
- Cross Sectional Data
BC
Alberta
Saskatchewan
Manitoba
Ontario
Nova Scotia
11%
12%
12%
12%
11.5%
16%
Source: Canada Revenue Agency (http://www.cra-arc.gc.ca/tx/bsnss/tpcs/crprtns/prv/menu-eng.html),
NDP platform (Alberta)
*Refers to the higher rate; not applicable to small business
Course
English 101
Philosophy
262
Llama
Studies 371
Economics
282
Economics
299
Hours
6
12
2
11
25
Cross Sectional:
Many Agents
One Time Period
-Combination of Time Series and
Cross-sectional Data
-Many economic agents
-Many time periods
-More difficult to use
-Often required due to data restrictions
-also referred to as pooled data
University
Tuit 99/00
Tuit 00/01
Tuit 01/02
Tuit02/03
Alberta
3551.00
3770.00
3890.00
4032.00
British Columbia
2295.00
2295.00
2181.00
2661.00
Calgary
3650.00
3834.00
3975.00
4120.00
Concordia
1668.00
1668.00
1668.00
1668.00
Lethbridge
3360.00
3470.00
3470.00
3470.00
Manitoba
3005.00
2796.00
2807.00
2818.00
McGill
1668.00
1668.00
1668.00
1668.00
Ottawa
3760.00
3892.00
4009.00
4085.00
Exercise: What kind of data is:
1) Election Predictions 10 days before an
election?
2) MacLean’s University Rankings?
3) Yearly bank account summary?
4) University Transcript after your 4th year?
1. Nominal variables
Measured using current prices
Provides a measure of current
value
Ie: a movie today costs $12.
2. Real variables
Measured using base year prices
Provides a measure of quantity
(removing the effects of price change
over time)
Ie: a movie today costs $4.00 in 1970
dollars
In 1970, a movie cost $0.50
BUT
$0.50 then was a lot more than $0.50 now.
Nominal Comparison:
Movie prices have increased by a
factor of 24 ($0.50 -> $12)
Real Comparison:
Movie prices have increased by a
factor of 8 ($0.50 -> $4)
Gross Domestic Product
-Monetary value of all goods and services
produced in an economy
How do nominal and real GDP differ?
-Current monetary value of all goods
produced:
∑ quantities X prices
-changes when prices change
-changes when quantities change
Assume:
prices quadruple (x4)
production is cut in half (x 1/2)
Nominal GDP (year 1) = 1 X 1 = 1
Nominal GDP (year 2) = 0.5 X 4 = 2
-although production has been devastated,
GDP reflects extreme growth
-Base year value of all goods currently
produced:
∑ quantities X prices
base year
-doesn’t change when prices change
-changes when quantities change
Assume:
prices quadruple (x4)
production is cut in half (x 1/2)
Real GDP (year 1) = 1 X 1 = 1
Real GDP (year 2) = 0.5 X 1 = 0.5
-real GDP accurately reflects the economy
-Used to convert between real and
nominal terms
-different indexes for different variables
or groups of variables
Ie: GDP Deflator
2002 = 100 (base year)
2010 = 125 (World Bank)
The “price” of GDP has risen 25%
between 2002 and 2010
Price Index
Nominal GDP 
x Real GDP
100
Nominal GDP
Real GDP 
x 100
Price Index
Price Index
Nominal 
x Real
100
Nominal
Real 
x 100
Price Index
University
Tuit 99/00
Tuit 00/01
3551
3770
3890
4032
100
103
106.1
109.273
Real Tuition (1999 dollars)
=(Nominal Tuition/Price Index)100
3551
3660
3667
3689.85
British Columbia (Nominal)
2295
2295
2181
2661
100
103
106.1
109.273
2295
2228
2056
2435.19
Alberta (Nominal)
Tuition Price Index*
Tuition Price Index*
Real Tuition (1999 dollars)
=(Nominal Tuition/Price Index)100
Tuit 01/02
*Based on 3% yearly inflation typical to years listed
Tuit02/03
X – variable you are adding
s – starting observation
n – last observation
n
X
is
i
6
In
this
class,
often
X

X

X

X

X

i
3
4
5
6
simplified to:
i 3
 X  add up ALL observations of X
 X  add up ALL observations of X at time t
t
 PQ
t 1
t
 P1t Q1t 1  P2t Q2t 1  ...  Pnt Qnt 1
Add up each good’s (1, 2…n) product
of its price today times its quantity
last term.
For example, if you were in charge of
food for a convention:
 PQ
t
t 1
 PBreakfastToday QBreakfastYesterday  PLunchTodayQLunchYesterday
 PDinnerToday QDinnerYesterday
Price indexes summarize how the
cost of baskets (collections of
goods or services) changes over
time.
The total cost of a basket is:
BasketCost   PQ
John is constructing a price
index to reflect his
entertainment spending
John values two activities
equally: seeing movies
and eating hot dogs
The prices of movies and hot
dogs have moved as
follows:
Year
Movies
Hot
Dogs
Price
Price
2000
$12.00
$1.00
2001
$20.00
$1.00
2002
$10.00
$2.00
Simple Price Index = ∑price X weight
Year
Movies
Price
Simple Price
Index
Hot Dogs
Weight
Price
Weight
2000
$12.00
0.5
$1.00
0.5
$6.50
2001
$20.00
0.5
$1.00
0.5
$10.50
2002
$10.00
0.5
$2.00
0.5
$6.00
Exercise: If John valued hot dogs three
times as much as movies, what would
the price indexes become?
-price indexes themselves are
meaningless
“The price of GDP was 78.9 this year”
-price indexes help us:
1) Compare between years
2) Convert between real and nominal
-to compare more easily, we normalize
to make the index equal 100 in the
base year
Raw Price Index t
Normalized Price Index t 
x 100
Raw Price Index base year
= 100 in base year
For example, if GDP was 310 in 1982,
dividing every year’s GDP by 310 and
then multiplying by 100 normalizes
GDP to be 100 in 1982.
Take 2000 as the base year:
Year
Movies
Price
Normalized
Price
Simple
Price Index
Index
Hot Dogs
Weight
Price
Weight
2000
$12.00
0.5
$1.00
0.5
$6.50
100
2001
$20.00
0.5
$1.00
0.5
$10.50
162
2002
$10.00
0.5
$2.00
0.5
$6.00
92.3
Does the base year chosen affect the
outcome?
Take 2002 as the base year:
Year
Movies
Price
Normalized
Price
Simple
Price Index
Index
Hot Dogs
Weight
Price
Weight
2000
$12.00
0.5
$1.00
0.5
$6.50
108
2001
$20.00
0.5
$1.00
0.5
$10.50
175
2002
$10.00
0.5
$2.00
0.5
$6.00
100
Note: Raw and normalized PI’s WORK the
same, normalized PI’s are just easier to
visually interpret
If instead of using inflation for our tuition
deflator, we use the education deflator,
we can first normalize it to 1999/2000:
Raw Education
Index*
Year
1999/2000
2000/2001
2001/2002
2002/2003
*Cansim series V735564, January Data
149.3
155.6
160.6
165.8
Calculation
Normalized Price
Index
149.3/149.3 X 100
100
155.6/149.3 X 100
104
160.6/149.3 X 100
108
165.8/149.3 X 100
111
University
Alberta
Tuition
1999/00
Tuition
00/01
Tuition
01/02
Tuition
02/03
3551
3770
3890
4032
100
104
108
111
Real Tuition (1999
dollars)
3551
3625
3602
3632.43
Calgary
3650
3834
3975
4120
100
104
108
111
3650
3687
3681
3711.71
Tuition Deflator
Tuition Deflator
Real Tuition (1999
dollars)
-base years can be changed using the
same formula learned earlier
-in the formula, always use the price
indexes from the SAME SERIES (same
base year)
-Up until this point, price index weights
have been arbitrary
-Arbitrary weights leads to bias, difficulty
in recreating data, and difficulty in
interpreting and comparing data
-One common price index (which the
Consumer Price Index uses) is the
Laspeyres Price Index
(The Paasche Price Index is the other
common index used)
-uses base year quantities as weights
-still = 100 in base year (automatically
normalized
LPIt = ∑ pricest X quantitiesbase year
---------------------------------- X 100
∑ pricesbase year X quantitiesbase year
-tracks cost of buying a fixed (base year)
basket of goods (ie: CPI)
Year
Movies
Karaoke
Price
Quantity
Price
Quantity
1
10
20
20
10
2
11
15
25
15
3
12
25
15
20
4
15
5
15
20
5
11
10
20
15
Laspeyres Price Index
Year
Cost in year t
Cost in Base Year
of base year basket
of base year basket
Laspeyres Price Index
1 (10*20)+(20*10)
400 (10*20) + (20*10)
400 400/400 X 100
100
2 (11*20)+(25*10)
470 (10*20) + (20*10)
400 470/400 X100
118
3 (12*20)+(15*10)
390 (10*20) + (20*10)
400 390/400 X 100
97.5
4 (15*20)+(15*10)
450 (10*20) + (20*10)
400 450/400 X 100
113
5 (11*20)+(20*10)
420 (10*20) + (20*10)
400 420/400 X 100
105
1) Using individual prices and quantities
-Same as before
2) Using basket costs
PaQb
Price of basket b in year a
P2012Q1997
Price in 2012 of what was bought in 1997
Laspeyres Price Index
Year
Cost in year t
Cost in Base Year
of base year basket
of base year basket
Laspeyres Price Index
1 (10*20)+(20*10)
(10*20) + (20*10)
400/400 X 100
100
2 (11*20)+(25*10)
(10*20) + (20*10)
470/400 X100
118
3 (12*20)+(15*10)
(10*20) + (20*10)
390/400 X 100
97.5
4 (15*20)+(15*10)
(10*20) + (20*10)
450/400 X 100
113
5 (11*20)+(20*10)
(10*20) + (20*10)
420/400 X 100
105
Laspeyres Price Index
Year
Cost in year t
Cost in Base Year
of base year basket
of base year basket
Laspeyres Price Index
1
400
400 400/400 X 100
100
2
470
400 470/400 X100
118
3
390
400 390/400 X 100
97.5
4
450
400 450/400 X 100
113
5
420
400 420/400 X 100
105
Every year, Lillian Pigeau
likes to travel.
The first year, she went to
Maraket,
the second year to Ohm,
and the third year to Moose
Jaw.
The costs of those trips are
as follows:
Year
Maraket
Ohm
Moose
Jaw
1
$800
$1,000
$650
2
$900
$1,100
$550
3
$600
$1,200
$700
LPI t
LPI1
PQ


(100)
PQ
PQ


(100)
 PQ
t
b
b
b
Year
Maraket
Ohm
Moose
Jaw
1
1
1
$800
$1,000
$650
1
1
2
$900
$1,100
$550
3
$600
$1,200
$700
$800
LPI1 
(100)
$800
LPI1  100
LPI t
LPI 2
PQ


(100)
PQ
PQ


(100)
 PQ
t
b
b
b
2
1
1
$800
$1,000
$650
1
1
2
$900
$1,100
$550
3
$600
$1,200
$700
$900
LPI 2 
(100)
$800
LPI 2  112.5
Year
Maraket
Ohm
Moose
Jaw
LPI t
LPI 3
PQ


PQ
PQ


 PQ
t
b
b
b
3
1
1
1
(100)
Year
(100)
$600
LPI 3 
(100)
$800
LPI 3  75
Maraket
Ohm
Moose
Jaw
1
$800
$1,000
$650
2
$900
$1,100
$550
3
$600
$1,200
$700
-An alternative to a normal price index is a
CHAINED PRICE INDEX
A chained price index gives a measure of
an aggregate good’s price from one
year/term to the next
-chained price indexes are less affected
by a base year
-chained price indexes can better
capture substitution away from goods
(You are not responsible for calculating
chained price indexes in class.)
-As time goes on, base years change
-Prices and quantities of horses and cars
in the 1960’s are a little different than
today
-This creates price indexes with different
base years, spanning different periods
-Sometimes these differing price indexes
need be spliced together
1) Find a year with price indexes from BOTH
series & calculate a conversion factor
Conversion factor = Price Index (new base)
--------------------------------------------------
Price Index (old base)
New = index you want to fill in
Old = index you want to convert
2) Multiply old index by conversion factor to
fill in new index
Year
1988
1989
1990
1991
1992
1993
1994
Price Index Price Index Calculations
(1989=100) (1992=100)
120
100
95
92
Price Index
(1992=100)
120 X (110/92)
100 X (110/92)
95 X (110/92)
110
100
95
95
Exercise: How would the full price index
look with 1989 as the base year?
143
120
114
Growth Rates are important concepts in
economics.
Inflation = growth rate of CPI (all items)
Growth = { (Xt – Xt-1)/ Xt-1 } X 100
= { ln(Xt) – ln(Xt-1) } X 100
Note: g = (Xt – Xt-1)/ Xt-1
UBC tuition in 2001/2002: $2181.
In 2002/2003 it was $2661
Growth = { (2661-2181)/2181 } X 100 = 22.01%
Growth = { ln(2661) – ln(2181) } X 100 = 19.89%
U of A tuition in 2001/2002: $3890.
In 2002/2003 it was $4032
Growth = { (4032-3890)/3890 } X 100 = 3.65%
Growth = { ln(4032) – ln(3890) } X 100 = 3.59%
Growth = {ln(Xt) – ln(Xt-1)} X 100
The log growth formula is only appropriate
when growth is small.
If the log growth formula reveals large
growth, use the normal growth formula
instead
If g is SMALL g ≈ ln (1+g)
ln(1+g) = ln [1+(Xt-Xt-1)/Xt-1]
= ln [(Xt-1+Xt-Xt-1)/Xt-1]
= ln [Xt/Xt-1]
= ln [Xt] – ln[Xt-1]
Therefore
g ≈ ln [Xt] – ln[Xt-1]
or (Xt-Xt-1)/Xt-1 ≈ {ln [Xt] – ln[Xt-1]}
1) Division Rule
ln(A/B) = ln(A) – ln(B)
2) Multiplication Rule
ln(AB) = ln(A) + ln (B)
3) Power Rule
ln(Ab) = b X ln (A)
Note:
ln (A+B) ≠ ln (A) + ln (B)
gA/B = [ln(At/Bt) – ln(At-1/Bt-1)] X 100
= [ln(At)-ln(Bt){ln(At-1)-ln(Bt-1)}] X 100
= [ln(At)-ln(At-1){ln(Bt)-ln(Bt-1)}] X 100
= [ln(At)-ln(At-1)] X 100 –
{ln(Bt)-ln(Bt-1)} X 100
gA/B = growth of A – growth of B
Recall that:
Real = nominal /(price index/100)
Ie: Real price=nominal price/(PI/100)
Therefore:
Real growth = nominal growth – PI growth
For example:
Real price change = nominal price change
-inflation
If tuition was $5000 last year and $5100 this
year, how much did real tuition change if
inflation is 3%?
Real growth = nominal growth – inflation
={(5100-5000)/5000}X100 -3
= (100/5000)X100 - 3
= 2-3
= -1%
gAB= [ln(AtBt) – ln(At-1Bt-1)] X 100
= [ln(At)+ln(Bt){ln(At-1)+ln(Bt-1)}] X 100
= [ln(At)-ln(At-1)+
{ln(Bt)-ln(Bt-1)}] X 100
= [ln(At)-ln(At-1)] X 100 +
{ln(Bt)-ln(Bt-1)} X 100
gAB = growth of A + growth of B
Recall that:
Per Capita GDP = GDP/Population
THEREFORE
GDP = Per Capita GDP X Population
THEREFORE
GDP growth = per capita GDP growth +
population growth
If each person produces 1% more, and population
grows by 2%, overall GDP growth is 3%
Interest rates are important in economics,
as they show the opportunity cost of a
project.
Different interest rates apply to different
situations.
Different interest rates are available to
different people.
Saving:
1 Year GIC: 0.85%
1 Year Cashable GIC: 0.4%
3 Year GIC: 1.05%
3 Year Cashable GIC: 0.5%
Bank Account: 0.0%
Borrowing
Bank of Canada Rate: 0.5%
1 year closed Mortgage: 2.89%
1 year open Mortgage: 6.3%
Bank of Canada rate for banks
Is less than
Chartered Banks’ rates for best customers
Is less than
Typical Bank Rate
Is less than
Risky Investor Bank Rate
More risk = higher rate
Super Savings Bank Account: 2% interest
Cash on hand: $100
2 DVD players:
Basic: $100
DVD Playback
Deluxe: $102
DVD/VCD/SVCD/AVI/DVD±R/CD/CD±R
3D Blu-Ray, Wi-Fi, Memory Card Slot,
Picture Viewer, Stop Memory, Shiny Red
Colour
You want the deluxe, so you invest for a
year, cash on hand in a year: $102
But, due to 3% inflation, the DVD players
now cost: $103 (basic) $105.06 (deluxe)
Now you can’t afford either
You’ve LOST buying power
rreal = (1+rnom)
--------- -1
(1+inf)
rreal= real interest rate
rnom= nominal interest rate
inf = inflation
rreal = (1+rnom-1-inf)
---------------(cross multiply to get…)
(1+inf)
rreal+ rreal*inf = rnom-inf
(rreal*inf is small)
rreal = rnom – inf
Last example: rreal = 2%-3%=-1%
Very few safe investments offer a return
greater than inflation.
You are losing buying power
Is buying today a better move?
WHY SAVE?
You can invest in Canada, the US, or
Mexico. Investment opportunities are
4%, 5%, and 15% respectively.
However, country currency inflation is 2%,
3% and 14%
Real interest rate then becomes:
Canada: 4%-2%=2%
US: 5%-3%=2%
Mexico: 15%-14% = 1%
Investment: $100
Interest rate: 2%
Derived Formula:
S = P (1+r)t
S = value after t years
P = principle amount
r = interest rate
t = years
Year
Calc.
1
Amount
100
100.00
2 100*1.02
102.00
3 100*1.022
104.04
4 100*1.023
106.12
5 100*1.024
108.24
If interest is compounded m times a year, 1/m
of the interest is paid each time
Modified Formula:
S = P (1+[r/m])mt
S = value after t years P = principle amount
r = interest rate
t = years
m = times compounded (monthly = 12, etc)
Infinite Compounding:
S = Pert
Year
Yearly
Biyearly
Monthly
Weekly
Daily
0
$100.00
$100.00
$100.00
$100.00
$100.00
1
$110.00
$110.25
$110.47
$110.51
$110.52
2
$121.00
$121.55
$122.04
$122.12
$122.14
3
$133.10
$134.01
$134.82
$134.95
$134.98
4
$146.41
$147.75
$148.94
$149.13
$149.17
5
$161.05
$162.89
$164.53
$164.79
$164.86
6
$177.16
$179.59
$181.76
$182.11
$182.20
7
$194.87
$197.99
$200.79
$201.24
$201.36
8
$214.36
$218.29
$221.82
$222.38
$222.53
9
$235.79
$240.66
$245.04
$245.75
$245.93
10
$259.37
$265.33
$270.70
$271.57
$271.79
More frequent compounding gives greater returns.
Which is the better investment: 25% compounded
annually or 24% compounded monthly?
rE = effective rate of interest if
compounded annually
P (1+rE)t = P (1+[r/m])mt
Solving for rE, we get:
rE = (1+[r/m])m-1
Which is the better investment: 25% compounded
annually or 24% compounded monthly?
rE
=
=
=
=
=
=
(1+[r/m])m-1
(1+[0.24/12])12-1
(1+0.02)12 -1
(1.02)12 -1
1.268-1
26.8%
infann = (1+infmon)12-1
In one month of 2005, gas prices rose from 98
to 112 cents a liter.
Infmon = [(112-98)]/98 X 100 = 14.3%
If this continued throughout the year, inflation
would reach:
infann = (1+0.143)12-1 = 397%
Some sketchy investments (some mutual
funds sold by a “friend”) use this misleading
calculation often.
infann = (1+infday)365-1
Cheezy loan inc. offers 0.1% daily interest on
payday loans. They advertise that a oneday payday loan of $1000 only costs $1!
However, yearly this becomes:
infann
=
=
=
=
(1+0.001)365-1
(1.001)365-1
(1.44-1)
44% interest!
If interest/return is expressed yearly,
but paid out multiple times per
year, effective interest/return is:
r m
rE  (1  )  1
m
If interest/return is expressed more
frequently (monthly, etc), effective
interest/return is:
m
rE  (1  r )  1
How much do I have to invest now to have a
given sum of money in the future?
PV = S/[(1+r)t]
PV = present value (money invested now)
S = sum needed in future
r = interest rate
t = years*
*Note: time can be in months (or any time period) if interest rate is also in
months (or any time period)
You and your spouse just got pregnant, and
will need to pay for university in 20 years. If
university will cost $30,000 in real terms in
20 years, how much should you invest now?
(long term GIC’s pay 5%)
PV = S/[(1+r)t]
= $30,000/[(1.05)20]
= $11,307
How does this change if it’s more than a onetime investment/payment?
(ie: $100 per year for 5 years, 7% interest)
PV= 100+100/1.07 + 100/1.072 + 100/1.073
+ 100/1.074
= 100 + 93.5 + 87.3 + 81.6 + 76.3 = $438.7
Or
PV = A[1-(1/{1+r})t] / [1- (1/{1+r})]
PV = A[1-xt] / [1-x]
x=1/{1+r}
PV = 100[1-(1/1.07)5]/[1-1/1.07] = $438.72
PV = A[1-(1/{1+r})t] / [1- (1/{1+r})]
PV = A[1-xt] / [1-x]
x=1/{1+r}
A = value of annual payment
r = annual interest rate
t = number of annual payments
Note: if specified that the first payment is
delayed until the end of the first year, the
formula becomes
PV = A[1-xt] / r
x=1/{1+r}
You won the lottery. Which is greater?
$800,000 now or $100,000 for the next 10
years at 5% real interest?
PV = A[1-(1/{1+r})t] / [1- (1/{1+r})]
PV = $100,000 {(1-[1/1.05]10)/(1-[1/1.05])}
= $100,000 {0.386/0.0476}
= $100,000 {8.11}
= $811,000
Take the money over 10 years
(Surprising how many take the lump sum)
Arithmetic Mean
-Averaging items that are added
together
(University grades, income, rent)
Ie: 3 numbers: 7, 15, and 20
Average= (7+15+20)/3 = 14
Geometric Mean
-Averaging items that are multiplied
together
(Interest rates, inflation)
Ie: 3 numbers: 7, 15, and 20
Geo Mean= (7x15x20)1/3 = 12.81
(Generally more useful than arithmetic)
Consider three investment opportunities: a
stable bank account with 3% interest, an
escalading GIC, or a risky investment, all
with the “same” return:
Year
Arithmetic Mean
Account
GIC
Investment
1
0.03
0.015
-0.500
2
0.03
0.020
-0.100
3
0.03
0.025
0.100
4
0.03
0.040
0.150
5
0.03
0.050
0.500
0.03
0.030
0.030
Although each investment has the same
arithmetic mean, the geometric means
clearly rank the investments.
Year
Account
GIC
Investment
1
0.03
0.015
-0.500
2
0.03
0.020
-0.100
3
0.03
0.025
0.100
4
0.03
0.040
0.150
5
0.03
0.050
0.500
Arithmetic Mean
0.03
0.030
0.030
Geometric Mean
0.03
0.027
-0.031
Assume an initial investment of $100:
Year
Account
GIC
Investment
1
103
101.50
50
2
106.09
103.53
45
3
109.273
106.12
49.5
4
112.551
110.36
56.925
5
115.927
115.88
85.3875
When investing with compound interest:
ALWAYS CONSIDER GEOMETRIC MEANS
As Arithmetic means are meaningless.
(Even though they’re sometimes reported.)
By definition:
(1+rgeo)T = (1+r1)(1+r2)(1+r3)…(1+rT)
(1+rgeo) = [(1+r1)(1+r2)(1+r3)…(1+rT)]1/T
rgeo= [(1+r1)(1+r2)(1+r3)…(1+rT)]1/T -1
It is EXTREMELY important to add 1 to each
interest rate.
If there is NO compounding…
the arithmetic mean will be an appropriate
measure of average returns
Ie) A person invests $1000 each year, takes
it all out, and then invests $1000 next
year.
Ie) A person invests in a poor GIC that
does not compound
Sometimes data needs to be AGGREGATED
– changed from one form (time period)
to another.
ie) monthly tuition payments => yearly tuition
payments
How to aggregate depends on whether the
variable is a STOCK or a FLOW
ie) I pay $500 a month in tuition. Therefore
yearly tuition is $500 (the average). -FALSE
Stock : a set, tangible value at a period of
time
Flow: a change to a stock variable
ie) Tuition:
Total tuition paid – stock variable
Monthly tuition payment – flow variable
Total tuition paid = ∑ Monthly tuition
payment
Stock : a set, tangible value at a time
Flow: a change to a stock variable
ie) Capital:
Kt = Kt-1 + It – Dt
K = Capital – stock
I = investment – flow
D = depreciation - flow
Stock : a set, tangible value at a time
Flow: a change to a stock variable
ie) Final Mark:
Final Markt = Final Markt-1 + Bribe effectt
+Scalingt
Final Mark = stock
Bribe Effect = flow
Scaling = flow
Stock : a set, tangible value at a time
Flow: a change to a stock variable
ie) Your mark
M=a1+a2+a3+a4+midterm+lab+final
M =end mark (stock)
A =mark gained by assignment
Midterm =mark gained by midterm
Lab =mark gained through lab component
final =mark gained through final (all flows)
Jedit = Jedit-1 – Darkt + Traint + Redeemt
– Aget – Battlet -66t
Jedi = number of Jedi (stock)
Dark = Jedi turning to dark side (flow)
Train = New Jedi’s trained (flow)
Redeem = Dark Jedi’s returning (flow)
Age = Jedi’s dying of old age (flow)
Battle = Jedi’s dying in battle (flow)
66 = Jedi’s killed by Emperor's order (flow)
Stock : a set, tangible value at a time
Flow: a change to a stock variable
What are the stocks and flows in:
1) Your Bank Account
2) Yearly Debt
3) Flirting with a girl/guy
Type of
Stock
Flow
Variable
Major
Measured at a point in Measured over a
Characteristic time
period (between
points in time)
Examples
Debts, wealth,
housing, stocks,
capital, tuition
Aggregation
Method
Average or
Use values from the
same time each year
Deficits, income,
building starts,
investment,
payments
Sum
(Average if
annualized)
Monthly Savings
Flow – ADD
Temperature
Stock – Average
Population
Stock – Average
Births
Flow - ADD
Vacancy Rate
We’ve had 10% vacancy a month.
a)That’s 120% vacancy a year (flow)
Or
b) That’s an average of 10% vacancy for
the year. (stock)
Building Starts
500 new buildings have started each
month
a)That’s 6000 new buildings a year (flow)
Or
b) That’s an average of 500 new buildings
this year (stock)
Money Supply
Canada’s money supply each month has
been $200 billion
a)That’s $2.4 trillion a year (flow)
Or
b) The money supply was $200 billion
that year (stock)
Investment
“Each month I invest $500 in elevators
inc. It’s bound to go up sometime!”
a)That’s an investment of $6,000 a year
(flow)
Or
b) That’s an average yearly investment of
$500 (stock)
Consumption
“My grocery bill is $300 a month
a)That’s an bill of $3,600 a year (flow)
Or
b) That’s average yearly groceries of
$300 (stock)
Job creation
“Our new evaporated water factory will
create 2,000 new jobs every month.
Now that’s the magic of government!”
a)24,000 jobs will be created this year
(flow)
Or
b) Government “magic” creates 2,000
jobs this year! (stock)
Two methods of determining costs of
durable goods (goods not consumed in 1
time period):
1) Purchase price
-actual sticker price paid for good
-one time price, ignores durability
2) User cost of Capital
-value of services received over time
-implicit rental rate
You buy a used printer (that only lasts one
year) for $20, to print 2,000 pages. Ink
and paper cost you $50, and photocopying
(“renting”) would cost $0.02 a sheet.
Buying = $20 + $50 = $70
Photocopying = 2,000 * $0.02 = $40
You would “rent” instead of buy…but most
printers last MORE than one year
Economist’s user cost of capital:
“How much would you be willing to pay per
term (ie: year) to rent capital that you
could buy for $X?”
-implicit rental rate
-BUYING the good is equivalent to
renting it for this amount each term
1) Depreciation – the more that an item
depreciates (more it costs to maintain),
the less likely one is to buy
-higher maintenance=>higher “implicit rent”
2) Opportunity cost of funds – the more
that a buyer can earn for his money, the
less likely he will be to buy
-higher interest rates =>higher “implicit rent”
3) Capital gains (loses) – a buyer is more
likely to purchase a product that keeps
its value over time
-gains value => lower “implicit rent”
-loses value =>higher “implicit rent”
User cost of capital = implicit rental rate
Pkt ( d + r - [Pkt+1 – Pkt]/Pkt )
d = depreciation
(more willing to rent a costly item)
r = return on alternate investments
(more willing to rent given high returns)
[Pkt+1 – Pkt]/Pkt = capital gains/losses
(less willing to rent an item that gains/holds value)
User cost of capital = implicit rental rate
=
Pkt ( d + r - [Pkt+1 – Pkt]/Pkt )
d = 1 (printer explodes)
r = 0 (no alternate investments)
[Pkt+1 – Pkt]/Pkt = 0 (no price change)
Implicit rental cost = $70(1+0-0)
= $70
Buy : $70 (implicit rental) > $40 (actual rental)
You decide to buy a tiny (almost
condemned) house for $200,000. The
house is so old and decrepit that
depreciation is 10%. You can invest
in a GIC at 5%, and expect the price
of the house to increase to $205,000
over the next year.
User cost of capital = implicit rental rate
=
Pkt ( d + r - [Pkt+1 – Pkt]/Pkt )
d = 0.10
r = 0.05
[Pkt+1 – Pkt]/Pkt = [205-200]/200 = 0.025
Implicit rental cost = $200,000(0.10+0.05-0.025)
= $200,000(0.125)
= $25,000
You decide to buy a new supercomputer.
The computer originally costs $2,000,
and depreciates 25% a year (since
you don’t have Norton Internet
Security). The purchase price
DECREASES 10% each year, and you
could alternately invest at 5%
User cost of capital = implicit rental rate
=
Pkt ( d + r - [Pkt+1 – Pkt]/Pkt )
d = 0.25
r = 0.05
[Pkt+1 – Pkt]/Pkt = -0.10 (decrease)
Implicit rental cost = $2000(0.25+0.05-[-0.10])
= $2000(0.4)
= $800
If you could rent the house for LESS
than $25k a year, you should rent
If you could rent the computer for MORE
than $800 a year, you should buy.
If Rent > User Cost of Capital, buy
If Rent < User Cost of Capital, rent
“Icon’s ice cream sales fell in November –
they should shut down.”
“The new federal budget has caused a
decrease in student unemployment this
May.”
“Apple CEO demands raise for increase in
sales in December.”
“Holes Greenhouse sales fall in March –
accountants perplexed.”
Many economics variables often have
PREDICTABLE seasonal movements.
Failure to appreciate these movements can
lead to wrong assumptions.
Is growth or loss:
1) A seasonal effect OR
2) A true change.
-Ice cream sales fall in winter
-Students get jobs in May
-Christmas boosts sales in December
-Flower sales rise for Valentines Day, then
fall afterwards
-Health Club memberships soar following
New Years’ resolutions
-Gas sales decrease in winter as certain
drivers chose not to drive.
Statistics Canada accounts for seasonal
adjustments by publishing two sets of
data:
1) Raw (not seasonally adjusted) data
2) Seasonally adjusted data
In order to make correct conclusions when
faced with seasonally adjusted data, one
should:
1) Use seasonally adjusted data
2) Compare between years (not between
months)
Note: Other factors other than seasons can
create variable movements:
a) long-term trends
b) Business cycle
c) Irregular shocks
These events are not factored out by
seasonal adjustments, but must be
identified in a decent study. (ie: plot the
trend on a graph and look for patterns)
Two key mathematical concepts used in
economics are exponentials and logarithms
(which are related concepts)
The features of exponentials are:
1 1
1
e  2.718  1  

...
1 1(2) 1(2)3
e x  exp( x)
e x  0,  1) If x  0, 0  e x  1
2) If x  0, e x  1
3) If x  0, e x  1
The key features of Logarithms are:
log e ( x)  ln( x) (economics always refers to base e, not base 10)
1) if ln( x)  z, e z  x
2) ln( x) is only defined for x  0 :
if 0  x  1  ln(x)  0
if x  1
 ln(x)  0
if x  1
 ln(x)  0
Note that exponentials and logarithms can be
interchanged to solve a problem:
if ln(x)  10
e x
x  22,026
10
1) Division Rule
ln(A/B) = ln(A) – ln(B)
2) Multiplication Rule
ln(AB) = ln(A) + ln (B)
3) Power Rule
ln(Ab) = b X ln (A)
Note
ln (A+B) ≠ ln (A) + ln (B)