Econ 299 Quantitative Methods in Economics

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Econ 299Quantitative Methods in Economics

• Economic Data
• Calculus and Economics
• Basics of Economic Models
• Advanced Calculus and Economics
• Statistics and Economics
• Econometric Introduction

Lorne Priemaza, M.A. Lorne.priemaza_at_ualberta.ca

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1. Data Description, Presentation, and
Manipulation

• 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

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Why do economists need data?

• 1) Describe Economy
• Current and past data, as well as increases and
  decreases
• This information can influence decisions
• ie GDP, interest rate, unemployment, price,
  debt, etc.

• 2) Test Theory
• Data is needed to test a hypothesis that one
  aspect of the economy impacts another
• ie Smokers and the cost to healthcare
• ie Married couples and health

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1 Data Types

• 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

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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

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Albertas 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
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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
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 www.thefinalfantasy.com
Source the truth
Time Series One Agent Many Time Periods
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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 years
  grades

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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
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Timothy A. Students 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
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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

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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
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1.1 Data Types

• Exercise What kind of data is
• 1) Election Predictions 10 days before an
  election?
• 2) MacLeans University Rankings?
• 3) Yearly bank account summary?
• 4) University Transcript after your 4th year?

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1.2 Real and Nominal Variables

• 1. Nominal variables
• Measured using current prices
• Provides a measure of current value
• Ie a movie today costs 12.

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1.2 Real and Nominal Variables

• 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

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A Movie in 1970

• 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 -gt 12) Real
Comparison Movie prices have increased by a
factor of 8 (0.50 -gt 4)
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GDP example

• Gross Domestic Product
• -Monetary value of all goods and services
  produced in an economy
• How do nominal and real GDP differ?

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Nominal GDP

• -Current monetary value of all goods produced
• ? quantities X prices
• -changes when prices change
• -changes when quantities change

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The Problem with Nominal GDP

• 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

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Real GDP

• -Base year value of all goods currently produced
• ? quantities X prices base year
• -doesnt change when prices change
• -changes when quantities change

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The Solution of Real GDP

• 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

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Price Indexes (Indices)

• -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

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GDP Converting Between Real and Nominal
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General Conversion Equations
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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
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1.3 How to Calculate Price Indexes

• -simple price index
• -weighted sum of individual prices of a good or
  group of goods
• Price index ?price X weight

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Example 1
Year Movies Hot Dogs
Price Price
2000 12.00 1.00
2001 20.00 1.00
2002 10.00 2.00

• 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

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Example 1

• To construct a price index, simply sum the
  products of the prices and weights
• Price index ?price X weight

Year Movies Hot Dogs 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?
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1.3.1 Normalizing Price Indexes

• -price indexes themselves are meaningless
• The price of GDP was 78.9 this year
• -price indexes help us
• Compare between years
• Convert between real and nominal
• -to compare more easily, we normalize to make the
  index equal 100 in the base year

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Normalizing the price index

• 100 in base year

For example, if GDP was 310 in 1982, dividing
every years GDP by 310 and then multiplying by
100 normalizes GDP to be 100 in 1982.
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Example 1a - Normalized

• Take 2000 as the base year

Year Movies Hot Dogs Price 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?
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Example 1b - Normalized

• Take 2002 as the base year

Year Movies Hot Dogs Price 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 PIs WORK the same,
normalized PIs are just easier to visually
interpret
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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

Year Raw Education Index Calculation Normalized 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
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Example Tuition
University Tuit 99/00 Tuit 00/01 Tuit 01/02 Tuit02/03
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
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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)

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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
• -Two common universal price indexes are the
  Paasche and the Laspeyres Price Indexes

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1.3.2 Laspeyres Price Index

• -uses base year quantities as weights
• -still 100 in base year
• 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)

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1.3.2 Paasche Price Index

• -uses current year quantities as weights
• -still 100 in base year
• PPIt ? pricest X quantitiest
• ---------------------------------- X 100
• ? pricesbase year X quantitiest
• -compares cost of current basket now to cost of
  current basket in base year

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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
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Example Laspeyres (Base year 1)
Laspeyres Price Index Laspeyres Price Index
Cost in year t Cost in Base Year Cost in Base Year
Year of base year basket of base year basket of base year basket of base year basket Laspeyres Price Index Laspeyres Price Index
1 (1020)(2010) 400 (1020) (2010) 400 400/400 X 100 100
2 (1120)(2510) 470 (1020) (2010) 400 470/400 X100 118
3 (1220)(1510) 390 (1020) (2010) 400 390/400 X 100 97.5
4 (1520)(1510) 450 (1020) (2010) 400 450/400 X 100 113
5 (1120)(2010) 420 (1020) (2010) 400 420/400 X 100 105
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Example Paasche (Base year 1)
Paasche Price Index Paasche Price Index
Cost in year t Cost in Base Year Cost in Base Year
Year of year t basket of year t basket Paasche Price Index Paasche Price Index
1 (1020)(2010) 400 (1020) (2010) 400 400/400 X 100 100
2 (1115)(2515) 540 (1015) (2015) 450 540/450 X 100 120
3 (1225)(1520) 600 (1025) (2020) 650 600/650 X 100 92.3
4 (155)(1520) 375 (105) (2020) 450 375/450 X 100 83.3
5 (1110)(2015) 410 (1010) (2015) 400 410/400 X 100 103
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Comparing Paasche and Laspeyres
Year Laspeyres Price Index Laspeyres Price Index Paasche Price Index Paasche Price Index
1 400/400 X 100 100 400/400 X 100 100
2 470/400 X100 118 540/450 X 100 120
3 390/400 X 100 98 600/650 X 100 92
4 450/400 X 100 113 375/450 X 100 83
5 420/400 X 100 105 410/400 X 100 103
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Choosing Paasche or Laspeyres

• Current year weights Paasche
• Base year weights Laspeyres

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2 Price Index Calculation Methods

• 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

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Method 1 Individual Prices and Quantities
Laspeyres Price Index Laspeyres Price Index
Cost in year t Cost in Base Year Cost in Base Year
Year of base year basket of base year basket of base year basket of base year basket Laspeyres Price Index Laspeyres Price Index
1 (1020)(2010) (1020) (2010) 400/400 X 100 100
2 (1120)(2510) (1020) (2010) 470/400 X100 118
3 (1220)(1510) (1020) (2010) 390/400 X 100 97.5
4 (1520)(1510) (1020) (2010) 450/400 X 100 113
5 (1120)(2010) (1020) (2010) 420/400 X 100 105
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Method 2 Basket Costs
Laspeyres Price Index Laspeyres Price Index
Cost in year t Cost in Base Year Cost in Base Year
Year of base year basket of base year basket of base year basket of base year basket Laspeyres Price Index 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
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Method 2 Example
Year Maraket Ohm Moose Jaw
1 800 1,000 650
2 900 1,100 550
3 600 1,200 700

• 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

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Method 2 Paasche (Year 1 Base Year)
Year Maraket Ohm Moose Jaw
1 800 1,000 650
2 900 1,100 550
3 600 1,200 700
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Method 2 Paasche (Year 1 Base Year)
Year Maraket Ohm Moose Jaw
1 800 1,000 650
2 900 1,100 550
3 600 1,200 700
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Method 2 Paasche (Year 1 Base Year)
Year Maraket Ohm Moose Jaw
1 800 1,000 650
2 900 1,100 550
3 600 1,200 700
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1.3.2.1 Chained Price Indexes

• -A chained price index gives a measure of an
  aggregate goods 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 years link, then multiply links
  together

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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 terms quantities

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1.3.2.1 Laspeyres Chained Price Index (LCPI)

• LCPI1100
• LCPI2LCPI1 x LCL1, 2
• LCPI3LCPI2 x LCL2, 3
• LCPI3LCPI1 x LCL1, 2 x LCL2, 3
• and so on

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Example Laspeyres Chained Index
Laspeyres Price Index Laspeyres Price Index
Cost in year t Cost in t-1 Cost in t-1
Year of t-1 basket of t-1 basket of t-1 basket of t-1 basket Link Index Link Index Link Index
1 N/A N/A N/A N/A N/A 100 100
2 (1120)(2510) 470 (1020) (2010) 400 1.175 117.5 117.5
3 (1215)(1515) 405 (1115) (2515) 540 0.75 88 88
4 (1525)(1520) 675 (1225) (1520) 600 1.125 99 99
5 (115)(2020) 455 (155) (1520) 375 1.21 120 120
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1.3.2.1 Paasche Chain Link

• -uses current term quantities as weights
• -still 100 in base year
• PCLt-1,t ? pricest X quantitiest
• ----------------------------------
• ? pricest-1X quantitiest
• -compares cost of current basket now to cost of
  current basket last term

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1.3.2.1 Paasche Chained Price Index

• PCPI1100
• PCPI2PCPI1 x PCL1, 2
• PCPI3PCPI2 x PCL2, 3
• PCPI3PCPI1 x PCL1, 2 x PCL2, 3
• and so on
• -similar to Laspeyres
• -notice how chained an non-chained PIs still
  tell similar stories

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1.3.3 Splicing Price Indexes

• -As time goes on, base years change
• -Prices and quantities of horses and cars in the
  1960s 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

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1.3.3 Splicing Price Indexes

• 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

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Ie Price Index (Computers)
Year Price Index Price Index Calculations Price Index
(1989100) (1992100) (1992100)
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?
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1.3.4 Nominal, Relative, and Real Price Indexes

• Nominal Price Index
• -price index for a good or service
• -describes movement of prices over time
• ie education, gas, coffee
• Note CPI (consumer price index) for all goods is
  used to measure inflation

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1.3.4 Nominal, Relative, and Real Price Indexes

• Relative Price Index
• -price index for a good or service relative to
  (compared to) another
• -describes movement of prices over time compared
  to another good or service
• Relative Price Index Price Index A
• 
  ----------------- X 100
• Price Index B

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1.3.4 Nominal, Relative, and Real Price Indexes

• Real Price Index
• -price index for a good or service relative to
  all others
• -describes movement of prices over time compared
  to all other goods
• Real Price Index Price Index A
• 
  ----------------- X 100
• CPI (all goods)

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Example
Year Education Price Index Recreation Price Index CPI (all goods) Education relative to Recreation Education relative to all goods (Real)
1999/2000 149 110 111 (149/110)x100136 (149/111)x100134
2000/2001 156 112 115 (156/112)x100139 (156/115)x100136
2001/2002 161 112 116 (161/112)x100144 (161/116)x100138
2002/2003 166 114 121 (166/114)x100145 (166/121)x100137
Increase faster price growth
(relatively) Decrease slower price growth
(relatively)
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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

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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
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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

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Why two growth formulas? (proof)

• If g is SMALL g ln (1g)
• ln(1g) ln 1(Xt-Xt-1)/Xt-1
• ln (Xt-1Xt-Xt-1)/Xt-1
• ln Xt/Xt-1
• ln Xt lnXt-1
• Therefore g ln Xt lnXt-1
• or (Xt-Xt-1)/Xt-1 ln Xt lnXt-1

68
Log Review

• 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 (AB) ? ln (A) ln (B)

69
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

70
Example Relative Growth Rate

• Recall that
• Real nominal /(price index/100)
• Ie Real pricenominal price/(PI/100)
• Therefore
• Real growth nominal growth PI growth
• For example
• Real price change nominal price change
• -inflation

71
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)/5000X100 -3
• (100/5000)X100 - 3
• 2-3
• -1

72
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

73
Example Multiplicative Growth Rate

• 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

74
Example Growth of education relative to
recreation (Reference)
Year Education Price Index Recreation Price Index CPI (all goods) Education relative to Recreation Education relative to all goods
1999/2000 149 110 111 136 134
2000/2001 156 112 115 139 136
2001/2002 161 112 116 144 138
2002/2003 166 114 121 145 137
75
Example Growth of education relative to
recreation

• Education growth (2002-2003)
• ln(166)-ln(161) X 100 3.06
• Recreation growth (2002-2003)
• ln(114)-ln(112) X 100 1.77
• Relative growth (2002-2003)
• ln(145)-ln(144) X 100 1.29
• 3.06-1.771.29
• Numbers in table above were rounded, actual
  numbers were not.

76
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.

77
1.5 Interest Rate Examples (Sept 2011)

• Saving
• 1 Year GIC 1
• 1 Year Cashable GIC 0.75
• 3 Year GIC 1.35
• 3 Year Cashable GIC 1.2
• Bank Account 0.0
• Borrowing
• Bank of Canada Rate 1
• 1 year closed Mortgage 3.5
• 1 year open Mortgage 6.3

78
1.5 Interest Rate Rules

• Bank of Canada rate for banks
• Is less than
• Chartered Banks rates for best customers
• Is less than
• Typical Bank Rate
• More risk higher rate

79
1.5.2 Real Vrs. Nominal Rates

• Super Savings Bank Account 2 interest
• Cash on hand 100
• 2 DVD players
• Basic 100
• DVD Playback
• Deluxe 102
• DVD/VCD/SVCD/AVI/DVDR/CD/CDR
• 3D Blu-Ray, Wi-Fi, Memory Card Slot, Picture
  Viewer, Stop Memory, Shiny Red Colour

80
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 cant afford either
• Youve LOST buying power

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1.5.2.1 Calculating real interest

• rreal (1rnom)
• --------- -1
• (1inf)
• rreal real interest rate
• rnom nominal interest rate
• inf inflation

82
1.5.2.1 Easy Interest Formula

• rreal (1rnom-1-inf)
• ---------------- (cross multiply to
  get)
• (1inf)
• rreal rrealinf rnom-inf
• (rrealinf is small)
• rreal rnom inf
• Last example rreal 2-3-1

83
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?

84
Example Calculating currency interest

• 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 4-22
• US 5-32
• Mexico 15-14 1

85
1.5.3.1 Annual Compounding

• Investment 100
• Interest rate 2

Year Calc. Amount
1 100 100.00
2 1001.02 102.00
3 1001.022 104.04
4 1001.023 106.12
5 1001.024 108.24
Derived Formula S P (1r)t S value after
t years P principle amount r interest rate t
years
86
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 (1r/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
87
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
At a 10 year return of 10, daily compounding
gives an EXTRA 12 return. Thats more than the
first years interest.
88
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 (1rE)t P
(1r/m)mt Solving for rE, we get rE
(1r/m)m-1
89
1.5.3.3 Effective Rate of Interest

• Which is the better investment 25 compounded
  annually or 24 compounded monthly?

rE (1r/m)m-1 (10.24/12)12-1
(10.02)12 -1 (1.02)12 -1 1.268-1
26.8
90
1.5.3.3 Annualizing Monthly Inflation
infann (1infmon)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 (10.143)12-1 397 Some
sketchy investments (some mutual funds sold by a
friend) use this misleading calculation
often.
91
1.5.3.3 Short Term Loans
infann (1infday)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 (10.001)365-1
(1.001)365-1 (1.44-1) 44
interest!
92
Effective Interest Rate Formulas
If interest/return is expressed yearly, but paid
out multiple times per year, effective
interest/return is
If interest/return is expressed more frequently
(monthly, etc), effective interest/return is
93
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/(1r)t PV
present value (money invested now) S sum needed
in future r real, compound interest rate t
years
94
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
GICs pay 5) PV S/(1r)t
30,000/(1.05)20 11,307
95
1.5.3.4 Continued Deposits
How does this change if its more than a one-time
investment/payment? (ie 100 per year for 5
years, 7 interest) PV 100100/1.07 100/1.072
100/1.073 100/1.074 100 93.5 87.3
81.6 76.3 438.7 Or PV A1-(1/1r)t /
1- (1/1r) PV A1-xt / 1-x
x1/1r PV 1001-(1/1.07)5/1-1/1.07
438.72
96
1.5.3.4 Annuity Formula
PV A1-(1/1r)t / 1- (1/1r) PV
A1-xt / 1-x x1/1r 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
A1-xt / r x1/1r
97
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 A1-(1/1r)t / 1-
(1/1r) PV 100,000 (1-1/1.0510)/(1-1/1.0
5) 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)
98
1.5.4 Calculating average returns
Arithmetic Mean -Averaging items that are added
together (University grades, income, rent) Ie
3 numbers 7, 15, and 20 Average (71520)/3 14
99
1.5.4 Calculating average returns
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)
100
1.5.4 Calculating average returns
Consider three investment opportunities a stable
bank account with 3 interest, an escalading GIC,
or a risky investment, all with the same return
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
101
1.5.4 Calculating average returns
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
102
1.5.4 Investment Results
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
103
1.5.4 Investments and means
When investing with compound interest ALWAYS
CONSIDER GEOMETRIC MEANS As Arithmetic means are
meaningless. (Even though theyre sometimes
reported.)
104
1.5.4 An Easy Method For Solving for the Geo.
Mean
By definition (1rgeo)T (1r1)(1r2)(1r3)(1r
T) (1rgeo) (1r1)(1r2)(1r3)(1rT)1/T rgeo
(1r1)(1r2)(1r3)(1rT)1/T -1 It is
EXTREMELY important to add 1 to each interest
rate.
105
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
106
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 gt 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
107
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
variable Monthly tuition payment flow
variable Total tuition paid ? Monthly tuition
payment
108
1.6.1 -Stocks and Flows
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
109
1.6.1 -Stocks and Flows
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
110
1.6.1 -Stocks and Flows
Stock a set, tangible value at a time Flow a
change to a stock variable Ie) Your
mark Ma1a2a3a4midtermlabfinal 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)
111
1.6.1 -Stocks and Flows
Jedit Jedit-1 Darkt Traint Redeemt
Aget Battlet -666t Jedi number of Jedi
(stock) Dark Jedi turning to dark side
(flow) Train New Jedis trained (flow) Redeem
Dark Jedis returning (flow) Age Jedis dying
of old age (flow) Battle Jedis dying in battle
(flow) 666 Jedis killed by Emperor's order
(flow)
112
1.6.1 -Stocks and Flows Exercises

• Stock a set, tangible value at a time
• Flow a change to a stock variable
• What are the stocks and flows in
• Your Bank Account
• Yearly Debt
• Flirting with a girl/guy

113
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 or Use values from the same time each year Sum (Average if annualized)
114
Stock or Flow?

Monthly Savings Flow ADD Temperature
Stock Average Population Stock
Average Births Flow - ADD
115
Stock or Flow?

• Vacancy Rate
• Weve had 10 vacancy a month.
• Thats 120 vacancy a year (flow)
• Or
• b) Thats an average of 10 vacancy for the year.
  (stock)

116
Stock or Flow?

• Building Starts
• 500 new buildings have started each month
• Thats 6000 new buildings a year (flow)
• Or
• b) Thats an average of 500 new buildings this
  year (stock)

117
Stock or Flow?

• Money Supply
• Canadas money supply each month has been 200
  billion
• Thats 2.4 trillion a year (flow)
• Or
• b) The money supply was 200 billion that year
  (stock)

118
Stock or Flow?

• Investment
• Each month I invest 500 in elevators inc. Its
  bound to go up sometime!
• Thats an investment of 6,000 a year (flow)
• Or
• b) Thats an average yearly investment of 500
  (stock)

119
Stock or Flow?

• Consumption
• My grocery bill is 300 a month
• Thats an bill of 3,600 a year (flow)
• Or
• b) Thats average yearly groceries of 300 (stock)

120
Stock or Flow?

• Job creation
• Our new evaporated water factory will create
  2,000 new jobs every month. Now thats the magic
  of government!
• 24,000 jobs will be created this year (flow)
• Or
• b) Government magic creates 2,000 jobs this
  year! (stock)

121
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
122
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
70 Photocopying 2,000 0.02 40 You would
rent instead of buybut most printers last MORE
than one year
123
1.6.2 The User Cost of Capital
Economists 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
124
1.6.2 Factors Affecting User Cost

• Depreciation the more that an item depreciates
  (more it costs to maintain), the less likely one
  is to buy
• -higher maintenancegthigher 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 gthigher implicit rent

125
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 gt lower implicit
rent -loses value gthigher implicit rent
126
1.6.2 The User Cost of Capital
User cost of capital implicit rental rate Pkt (
d r - Pkt1 Pkt/Pkt ) d
depreciation (more willing to rent a costly
item) r return on alternate investments (more
willing to rent given high returns) Pkt1
Pkt/Pkt capital gains/losses (less willing to
rent an item that gains/holds value)
127
1.6.2 Simple Choice Example
User cost of capital implicit rental rate Pkt
( d r - Pkt1 Pkt/Pkt ) d 1 (printer
explodes) r 0 (no alternate investments) Pkt1
Pkt/Pkt 0 (no price change) Implicit rental
cost 70(10-0) 70 Buy 70 (implicit
rental) gt 40 (actual rental)
128
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.
129
1.6.2 House Example
User cost of capital implicit rental
rate Pkt ( d r - Pkt1 Pkt/Pkt ) d 0.10
r 0.05 Pkt1 Pkt/Pkt 205-200/200
0.025 Implicit rental cost 200,000(0.100.05-0.
025) 200,000(0.125) 25,000
130
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 dont have Norton Internet
Security). The purchase price DECREASES 10 each
year, and you could alternately invest at 5
131
1.6.2 Computer Example
User cost of capital implicit rental rate Pkt
( d r - Pkt1 Pkt/Pkt ) d 0.25 r 0.05
Pkt1 Pkt/Pkt -0.10 (decrease) Implicit
rental cost 2000(0.250.05--0.10)
2000(0.4) 800
132
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 gt User Cost of Capital, buy If Rent
lt User Cost of Capital, rent
133
1.7 Seasonal Adjustment
Icons 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.
134
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.
135
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.
136
1.7 Seasonal Adjustment

• Statistics Canada accounts for seasonal
  adjustments by publishing two sets of data
• Raw (not seasonally adjusted) data
• Seasonally adjusted data

137
1.7 Dealing with Seasons

• In order to make correct conclusions when faced
  with seasonally adjusted data, one should
• Use seasonally adjusted data
• Compare between years (not between months)

138
1.7 Dealing with Seasons

• Note Other factors other than seasons can create
  variable movements
• long-term trends
• Business cycle
• 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)

139
APPENDIX 1.1 EXPONENTIALS AND LOGARITHMS
Two key mathematical concepts used in economics
are exponentials and logarithms (which are
related concepts) The features of exponentials
are
140
APPENDIX 1.1 EXPONENTIALS AND LOGARITHMS
The key features of Logarithms are
141
APPENDIX 1.1 EXPONENTIALS AND LOGARITHMS
Note that exponentials and logarithms can be
interchanged to solve a problem
142
From Section 1.4, Log Review

• 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 (AB) ? ln (A) ln (B)