ECON 102: Mathematical Operations and Index Numbers Formulas
In this document, we present the most important formulas used in Module 2.
Date Convention
Flow Variables or Variables Representing Averages
The Date
The date corresponds to the center of the period:
• Annual: July 1st.
• Monthly: The 15th of each month.
• Quarterly: The dates of each quarter are respectively February 15th, May 15th, August 15th and
November 15th.
The Decimal Date
• Annual: Year + 0.5. For example, 1980.5 for the year 1980.
• Monthly: Year + month/12 -1/24. For example, the decimal date for March 2020 is 2020 +
3/12 − 1/24 = 2020.208.
• Quarterly: Year + Quarter/4 – 1/8. For example, the decimal date for the third quarter of 2019
is 2019 + 3/4 − 1/8 = 2019.625 Stock Variables
In this course, we rarely work with stock variables. Most macro-economic variables that we analyze are either flow variables (GDP, consumption, etc.) or averages (interest rate, price index, population etc.).
The Date
The date corresponds to the last day of the period:
• Annual: December 31st.
• Monthly: The last day of the month (28, 29, 30 or 31).
• Quarterly: The dates of each quarter are respectively March 31st, June 30th, September 30th
and December 31th.
Econ 102 Module 2: Formulas Page 1 of 4
The Decimal Date
• Annual: Year + 1 For example, 1981 for the year 1980.
• Monthly: Year + month/12. For example, the decimal date for March 2020 is 2020 + 3/12 =
2020.25.
• Quarterly: Year + Quarter/4. For example, the decimal date for the third quarter of 2019 is
2019 + 3/4 = 2019.75 Growth Rate
Notation: Xt is the value of X at time t, Xt−s is the value of X s periods before t and Xt+s is the value of X s periods after t.
Growth Rate Between Two Periods (gt) Not in percentage:
In percentage
We can also write:
Growth Rate of a Product
gt = Xt −Xt−1 Xt−1
gt = Xt −Xt−1 ×100 Xt−1
1+gt= Xt Xt−1
The growth rate of (XtYt) between t − 1 and t is: (1+gx)×(1+gy),
where gx and gy are the growth rates of Xt and Yt between t − 1 and t not expressed in percentage. You can then multiply by 100 to express the growth rate in percentage.
Growth Rate of a Product: An Approximation
The growth rate of (XtYt) between t − 1 and t is approximately equal to: gx + gy ,
where gx and gy are the growth rates of Xt and Yt between t − 1 and t expressed in percentage or not. The approximation is better when gx and gy are small.
Growth Rate of a Ratio
The growth rate of (Xt/Yt) between t − 1 and t is: (1+gx) −1,
(1+gy)
where gx and gy are the growth rates of Xt and Yt between t − 1 and t not expressed in percentage.
You can then multiply by 100 to express the growth rate in percentage.
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Growth Rate of a Ratio: An Approximation
The growth rate of (Xt/Yt) between t − 1 and t is approximately equal to: gx − gy ,
where gx and gy are the growth rates of Xt and Yt between t − 1 and t expressed in percentage or not. The approximation is better when gx and gy are small.
Annualized Growth Rate: Monthly Series
Let Xt be a monthly series. The annualized growth rate is: (1+gt)12 −1,
where gt is the growth rate of Xt between t − 1 and t not expressed in percentage. You can then multiply by 100 to express the growth rate in percentage.
Annualized Growth Rate: Quarterly Series
Let Xt be a quarterly series. The annualized growth rate is: (1+gt)4 −1,
where gt is the growth rate of Xt between t − 1 and t not expressed in percentage. You can then multiply by 100 to express the growth rate in percentage.
Annualized Growth Rate: An Approximation
For monthly series, the annualized growth rate is approximately equal gt × 12
and for quarterly series, if it approximately equal to gt ×4,
where gt is the growth rate of Xt between t−1 and t expressed in percentage or not. The approximations are better when the gt is small.
Average Annual Growth Rate
Let Xt and Xs, s > t, be two observations distant by (s − t) years, then the average annual growth rate over the period is
where
It is also approximately equal to
when g is small.
(1+g)1/(s−t) −1, 1 + g = Xs .
Xt g
s−t
Econ 102
Module 2: Formulas
Page 3 of 4
Log Approximation
The following only applies to strictly positive variables.
Log Properties
Growth rates
log(ab) = log(a) + log(b) log(a/b) = log(a) − log(b) log(ab) = b × log(a)
The growth rate between Xt−1 and Xt−1 is approximately equal to
log(Xt) − log(Xt−1) = log(Xt/Xt−1) = log(1 + gt) ,
where (1 + gt) = Xt/Xt−1. The approximation is better when gt is small. Index Numbers
An index number It/t0 is a ratio of a variable Xt over the same variable at a base period Xt0 multiplied by 100:
It/t0 = Xt ×100. Xt0
The index is called base 100 = t0.
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