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Econ 299Quantitative Methods in Economics
Economic DataCalculus and EconomicsBasics of Economic ModelsAdvanced Calculus and Economics Statistics and EconomicsEconometric Introduction
Lorne Priemaza, M.A.
Lorne.priemaza@ualberta.ca
1. Data Description, Presentation, and Manipulation
1.1 Data Types and Presentations1.2 Real and Nominal Variables1.3 Price Indexes1.4 Growth Rates and Inflation1.5 Interest Rates1.6 Aggregating Data: Stocks and Flows1.7 Seasonal AdjustmentAppendix 1.1 Exponentials and Logarithms
Why do economists need data?
1) Describe EconomyCurrent and past data Increases and decreasesThis information can influence decisions
ie: GDP, interest rate, unemployment, price, debt, etc.
2) Test TheoryDoes variable A affect variable B?
ie: Smokers and the cost to healthcare
ie: Married couples and health
1 Data Types
Data is essential for economists. Data can be categorized by:
1) How it is collected:time series data cross-sectional datapanel data
2) How it is measured:nominal datareal data
Time Series 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
Alberta’s Tuition – Time Series
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
China GDP – Time Series
Source: World Development Indicators, The World Bank, www.worldbank.org
Final Fantasy Quality - Time Series Data# Year Rating
1 1987 7.5
2 1988 6.5
3 1990 7.3
4 1991 8.3
5 1992 7.1
6 1994 8.7
7 1997 9.4
8 1999 9.2
Source: www.thefinalfantasy.com
# Year Rating
1 1987 9
2 1988 5
3 1990 7
4 1991 10/12
5 1992 4
6 1994 11
7 1997 8
8 1999 7
Source: the truth
Time Series: One AgentMany Time
Periods
Cross Sectional Data
-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
99/00 Tuition – Cross Sectional
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
Timothy A. Student’s Weekly Time Spent Studying for Midterms
- Cross Sectional Data
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
Panel Data
-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
Pooled Tuition
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
1.1 Data Types
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?
1.2 Real and Nominal Variables
1. Nominal variables
Measured using current pricesProvides a measure of current value
Ie: a movie today costs $12.
1.2 Real and Nominal Variables
2. Real variables
Measured using base year pricesProvides a measure of quantity (removing the effects of price change over time)
Ie: a movie today costs $4.00 in 1970 dollars
A Movie in 1970
In 1970, a movie cost $0.50BUT
$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)
GDP example
Gross Domestic Product -Monetary value of all goods and services produced in an economy
How do nominal and real GDP differ?
Nominal GDP
-Current monetary value of all goods produced:
∑ quantities X prices
-changes when prices change-changes when quantities change
The Problem with Nominal GDP
Assume: prices quadruple (x4)production is cut in half (x
1/2)
Nominal GDP (year 1) = 1 X 1 = 1Nominal GDP (year 2) = 0.5 X 4 = 2
-although production has been devastated, GDP reflects extreme growth
Real GDP
-Base year value of all goods currently produced:
∑ quantities X prices base year
-doesn’t change when prices change
-changes when quantities change
The Solution of Real GDP
Assume: prices quadruple (x4)production is cut in half (x
1/2)
Real GDP (year 1) = 1 X 1 = 1Real GDP (year 2) = 0.5 X 1 = 0.5
-real GDP accurately reflects the economy
Price Indexes (Indices)
-Used to convert between real and nominal terms
-different indexes for different variables or groups of variables
Ie: GDP Deflator2002 = 100 (base
year)2010 = 125 (World
Bank)The “price” of GDP has risen
25% between 2002 and 2010
GDP – Converting Between Real and Nominal
GDP Real x 100
Index PriceGDP Nominal
100 x Index Price
GDP NominalGDP Real
General Conversion Equations
Real x 100
Index Price Nominal
100 x Index Price
Nominal Real
Example: Tuition
University Tuit 99/00 Tuit 00/01 Tuit 01/02 Tuit02/03
Alberta (Nominal) 3551 3770 3890 4032
Tuition Price Index* 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
Tuition Price Index* 100 103 106.1 109.273
Real Tuition (1999 dollars)=(Nominal Tuition/Price Index)100 2295 2228 2056 2435.19
*Based on 3% yearly inflation typical to years listed
1.3 How to Calculate Price Indexes-simple price index
-weighted sum of individual prices of a good or group of goods
Simple Price Index = ∑price X weight
Example #1
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 MoviesHotDogs
Price Price
2000 $12.00 $1.00
2001 $20.00 $1.00
2002 $10.00 $2.00
Example #1
Simple Price Index = ∑price X weight
Year Movies Hot DogsSimple Price
Index
Price 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?
1.3.1 Normalizing Price Indexes
-price indexes themselves are meaningless
“The price of GDP was 78.9 this year”
-price indexes help us:1) Compare between years2) Convert between real and
nominal-to compare more easily, we
normalize to make the index equal 100 in the base year
Normalizing the price index:
= 100 in base year
100 x Index Price Raw
Index Price RawIndex Price Normalized
year baset
t
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.
Example #1a - Normalized
Take 2000 as the base year:
Year Movies Hot DogsSimplePrice Index
Normalized Price Index
Price 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?
Example #1b - Normalized
Take 2002 as the base year:
Year Movies Hot DogsSimplePrice Index
Normalized Price Index
Price 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
Example #2 – Tuition
If instead of using inflation for our tuition deflator, we use the education deflator, we can first normalize it to 1999/2000:
YearRaw Education
Index* CalculationNormalized Price
Index
1999/2000 149.3 149.3/149.3 X 100 100
2000/2001 155.6 155.6/149.3 X 100 104
2001/2002 160.6 160.6/149.3 X 100 108
2002/2003 165.8 165.8/149.3 X 100 111
*Cansim series V735564, January Data
Example: Tuition – Converting from real to nominal
UniversityTuition 1999/00
Tuition00/01
Tuition 01/02
Tuition02/03
Alberta 3551 3770 3890 4032
Tuition Deflator 100 104 108 111
Real Tuition (1999 dollars) 3551 3625 3602 3632.43
Calgary 3650 3834 3975 4120
Tuition Deflator 100 104 108 111
Real Tuition (1999 dollars) 3650 3687 3681 3711.71
1.3.1.1 Changing Base Years
-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)
1.3.2 Common Price Indexes
-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)
1.3.2 Laspeyres Price Index
-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)
Example: Movies and Karaoke
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
Example: Laspeyres (Base year 1)
Laspeyres Price Index
Cost in year t Cost in Base Year
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
2 Price Index Calculation Methods1) Using individual prices and
quantities-Same as before
2) Using basket costsPaQb
Price of basket b in year aP2012Q1997
Price in 2012 of what was bought in 1997
Method 1 – Individual Prices and Quantities
Laspeyres Price Index
Cost in year t Cost in Base Year
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
Method 2 – Basket Costs
Laspeyres Price Index
Cost in year t Cost in Base Year
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
Method 2 Example
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 OhmMooseJaw
1 $800 $1,000 $650
2 $900 $1,100 $550
3 $600 $1,200 $700
Method 2 – Laspeyres (Year 1 Base Year)
100
)100(800$
800$
)100(
)100(
1
1
11
111
LPI
LPI
QP
QPLPI
QP
QPLPI
bb
btt
Year Maraket OhmMooseJaw
1 $800 $1,000 $650
2 $900 $1,100 $550
3 $600 $1,200 $700
Method 2 – Laspeyres (Year 1 Base Year)
5.112
)100(800$
900$
)100(
)100(
2
2
11
122
LPI
LPI
QP
QPLPI
QP
QPLPI
bb
btt
Year Maraket OhmMooseJaw
1 $800 $1,000 $650
2 $900 $1,100 $550
3 $600 $1,200 $700
Method 2 – Laspeyres (Year 1 Base Year)
75
)100(800$
600$
)100(
)100(
3
3
11
133
LPI
LPI
QP
QPLPI
QP
QPLPI
bb
btt
Year Maraket OhmMooseJaw
1 $800 $1,000 $650
2 $900 $1,100 $550
3 $600 $1,200 $700
1.3.2.1 Chained Price Indexes
-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
-to form a chained price index, one must first form each year’s “link”, then multiply links together
1.3.2.1 Laspeyres Chain Link (LCL)
-uses last term quantities as weights-still = 100 in base year
LCLt-1,t = ∑ pricest X quantitiest-1
---------------------------------- ∑ pricest-1 X quantitiest-1
-tracks cost of buying a last term’s quantities
1.3.2.1 Laspeyres Chained Price Index (LCPI)
LCPI1=100LCPI2=LCPI1 x LCL1, 2
LCPI3=LCPI2 x LCL2, 3
LCPI3=LCPI1 x LCL1, 2 x LCL2, 3
and so on…
Example: Laspeyres Chained Index
Laspeyres Price Index
Cost in year t Cost in t-1
Year of t-1 basket of t-1 basketLink Index
1 N/A N/A N/A N/A N/A 100
2 (11*20)+(25*10) 470 (10*20) + (20*10) 400 1.175 117.5
3 (12*15)+(15*15) 405 (11*15) + (25*15) 540 0.75 88
4 (15*25)+(15*20) 675 (12*25) + (15*20) 600 1.125 99
5 (11*5)+(20*20) 455 (15*5) + (15*20) 375 1.21 120
1.3.3 Splicing Price Indexes
-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
1.3.3 Splicing Price Indexes
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 inOld = index you want to convert2) Multiply old index by conversion
factor to fill in new index
Ie: Price Index (Computers)
Year Price Index Price Index Calculations Price Index
(1989=100) (1992=100) (1992=100)
1988 120 120 X (110/92) 143
1989 100 100 X (110/92) 120
1990 95 95 X (110/92) 114
1991 92 110
1992 100
1993 95
1994 95
Exercise: How would the full price index look with 1989 as the base year?
1.4 Growth Rates and Inflation
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
1.4 Growth Rates Example
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%
1.4 Log Growth Restrictions
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
Why two growth formulas? (proof)
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]}
Log Review
1) Division Ruleln(A/B) = ln(A) – ln(B)
2) Multiplication Ruleln(AB) = ln(A) + ln (B)
3) Power Ruleln(Ab) = b X ln (A)
Note:ln (A+B) ≠ ln (A) + ln (B)
Example: Relative Growth Rate
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
Example: Relative Growth Rate
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
Example: Relative Growth Rate
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%
Example: Multiplicative Growth Rate
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
Example: Multiplicative Growth Rate
Recall that:Per Capita GDP = GDP/Population
THEREFOREGDP = Per Capita GDP X Population
THEREFOREGDP growth = per capita GDP growth +
population growth
If each person produces 1% more, and population grows by 2%, overall GDP growth is 3%
1.5 Interest Rates
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.
1.5 Interest Rate Examples (Aug 2015)
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%BorrowingBank of Canada Rate: 0.5%1 year closed Mortgage: 2.89%1 year open Mortgage: 6.3%
1.5 Different Interest Rates
Bank of Canada rate for banksIs less than
Chartered Banks’ rates for best customers
Is less thanTypical Bank Rate
Is less thanRisky Investor Bank Rate
More risk = higher rate
1.5.2 Real Vrs. Nominal Rates
Super Savings Bank Account: 2% interest
Cash on hand: $1002 DVD players: Basic: $100
DVD PlaybackDeluxe: $102DVD/VCD/SVCD/AVI/DVD±R/CD/CD±R3D Blu-Ray, Wi-Fi, Memory Card Slot,
Picture Viewer, Stop Memory, Shiny Red Colour
1.5.2 Real Vrs. Nominal Rates
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 eitherYou’ve LOST buying power
1.5.2.1 Calculating real interest
rreal = (1+rnom) --------- -1
(1+inf)
rreal= real interest raternom= nominal interest rateinf = inflation
1.5.2.1 Easy Interest Formula
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%
1.5.2.1 Depressing Interest Facts
Very few safe investments offer a return greater than inflation.
You are losing buying power
Is buying today a better move?
WHY SAVE?
Example: Calculating currency interestYou 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%
1.5.3.1 Annual Compounding
Investment: $100 Interest rate: 2%
Year Calc. Amount
1 100 100.00
2 100*1.02 102.00
3 100*1.022 104.04
4 100*1.023 106.12
5 100*1.024 108.24
Derived Formula:
S = P (1+r)t
S = value after t years
P = principle amount
r = interest ratet = years
1.5.3.2 More Frequent Compounding
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 = yearsm = times compounded (monthly = 12,
etc)
Infinite Compounding: S = Pert
Compounding Comparison
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.
1.5.3.3 Effective Rate of Interest
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
1.5.3.3 Effective Rate of Interest
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%
1.5.3.3 Annualizing Monthly Inflation
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.
1.5.3.3 Short Term Loans
infann = (1+infday)365-1
Cheezy loan inc. offers 0.1% daily interest on payday loans. They advertise that a one-day 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!
Effective Interest Rate Formulas
If interest/return is expressed yearly, but paid out multiple times per year, effective interest/return is:
1)1( mE m
rr
If interest/return is expressed more frequently (monthly, etc), effective interest/return is:
1)1( mE rr
1.5.3.4 Present Value
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 futurer = interest ratet = years**Note: time can be in months (or any time period) if interest rate
is also in months (or any time period)
1.5.3.4 Tuition Example
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 one-time 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
OrPV = 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
1.5.3.4 Continued Deposits
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 paymentr = annual interest ratet = 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}
1.5.3.4 Annuity Formula
1.5.3.4 Example
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)
1.5.4 Calculating average returns
Arithmetic Mean-Averaging items that are added together(University grades, income, rent)
Ie: 3 numbers: 7, 15, and 20Average= (7+15+20)/3 = 14
1.5.4 Calculating average returns
Geometric Mean-Averaging items that are multiplied together(Interest rates, inflation)
Ie: 3 numbers: 7, 15, and 20Geo Mean= (7x15x20)1/3 = 12.81
(Generally more useful than arithmetic)
1.5.4 Calculating average returns
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
Consider three investment opportunities: a stable bank account with 3% interest, an escalading GIC, or a risky investment, all with the “same” return:
1.5.4 Calculating average returns
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
Although each investment has the same arithmetic mean, the geometric means clearly rank the investments.
1.5.4 Investment Results
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
Assume an initial investment of $100:
1.5.4 Investments and means
When investing with compound interest:
ALWAYS CONSIDER GEOMETRIC MEANS
As Arithmetic means are meaningless.
(Even though they’re sometimes reported.)
1.5.4 An Easy Method For Solving for the Geo. Mean:
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.
1.5.4 Geometric Note
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
1.6 Aggregating Data – Stocks and Flows
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
1.6.1Stocks and Flows
Stock : a set, tangible value at a period of time
Flow: a change to a stock variable
ie) Tuition:Total tuition paid – stock variableMonthly tuition payment – flow
variableTotal tuition paid = ∑ Monthly tuition
payment
1.6.1 -Stocks and Flows
Stock : a set, tangible value at a timeFlow: a change to a stock variable
ie) Capital:Kt = Kt-1 + It – Dt
K = Capital – stockI = investment – flowD = depreciation - flow
1.6.1 -Stocks and Flows
Stock : a set, tangible value at a timeFlow: a change to a stock variable
ie) Final Mark:Final Markt = Final Markt-1 + Bribe
effectt +Scalingt
Final Mark = stockBribe Effect = flowScaling = flow
1.6.1 -Stocks and Flows
Stock : a set, tangible value at a timeFlow: a change to a stock variable
ie) Your markM=a1+a2+a3+a4+midterm+lab+fin
alM =end mark (stock)A =mark gained by assignmentMidterm =mark gained by midtermLab =mark gained through lab
componentfinal =mark gained through final (all
flows)
1.6.1 -Stocks and 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)
1.6.1 -Stocks and Flows Exercises
Stock : a set, tangible value at a timeFlow: a change to a stock variable
What are the stocks and flows in:1) Your Bank Account2) Yearly Debt3) Flirting with a girl/guy
1.6.1 -Stocks and Flows Summary
Type of Variable
Stock Flow
Major Characteristic
Measured at a point in time
Measured over a period (between points in time)
Examples Debts, wealth, housing, stocks, capital, tuition
Deficits, income, building starts, investment, payments
Aggregation Method
Average orUse values from the same time each year
Sum(Average if annualized)
Stock or Flow?
Monthly Savings
Flow – ADD
Temperature
Stock – Average
Population
Stock – Average
Births
Flow - ADD
Stock or Flow?
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)
Stock or Flow?
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)
Stock or Flow?
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)
Stock or Flow?
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)
Stock or Flow?
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)
Stock or Flow?
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)
1.6.2 – The User Cost of Capital
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
1.6.2 – Simple Choice Example
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 = $70Photocopying = 2,000 * $0.02 = $40
You would “rent” instead of buy…but most printers last MORE than one year
1.6.2 – The User Cost of Capital
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
1.6.2 – Factors Affecting User Cost
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”
1.6.2 – Factors Affecting User Cost
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”
1.6.2 – The User Cost of Capital
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)
1.6.2 – Simple Choice Example
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)
1.6.2 – House Example
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.
1.6.2 – House Example
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
1.6.2 – Computer Example
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%
1.6.2 – Computer Example
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
1.6.2 – User Cost of Capital
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, buyIf Rent < User Cost of Capital, rent
1.7 Seasonal Adjustment
“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.”
1.7 Seasonal Adjustment
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.
1.7 Seasonal Adjustment
-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.
1.7 Seasonal Adjustment
Statistics Canada accounts for seasonal adjustments by publishing two sets of data:
1)Raw (not seasonally adjusted) data2)Seasonally adjusted data
1.7 Dealing with Seasons
In order to make correct conclusions when faced with seasonally adjusted data, one should:
1)Use seasonally adjusted data2)Compare between years (not
between months)
1.7 Dealing with Seasons
Note: Other factors other than seasons can create variable movements:
a) long-term trendsb)Business cyclec) Irregular shocksThese 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)
APPENDIX 1.1 – EXPONENTIALS AND LOGARITHMS
1 0, xIf 3)
1 0, xIf 2)
10 0, xIf 1) ,0
)exp(
...3)2(1
1
)2(1
1
1
11718.2
x
x
xx
x
e
e
ee
xe
e
Two key mathematical concepts used in economics are exponentials and logarithms (which are related concepts)
The features of exponentials are:
APPENDIX 1.1 – EXPONENTIALS AND LOGARITHMS
0ln(x) 1 xif
0ln(x) 1 xif
0ln(x) 1x0 if
:0for x definedonly is )ln( )2
z,)ln( if 1)
10) basenot e, base torefers always (economics )ln()(log
x
xex
xxz
e
The key features of Logarithms are:
APPENDIX 1.1 – EXPONENTIALS AND LOGARITHMS
026,22
10 ln(x) 10
x
xe
if
Note that exponentials and logarithms can be interchanged to solve a problem:
From Section 1.4, Log Review:
1) Division Ruleln(A/B) = ln(A) – ln(B)
2) Multiplication Ruleln(AB) = ln(A) + ln (B)
3) Power Ruleln(Ab) = b X ln (A)
Noteln (A+B) ≠ ln (A) + ln (B)
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