1. ECON 201 Indices
Week

2. Definition: An index number is a relative figure expressed as a percentage used to measure change in a variable between any two given time periods.
Definition: Base year is the year upon which comparison is made or a point of reference when looking at the relative change in the variable under consideration.

3. There are three major types of indices which are:
(a) Price Indices: They measure changes in prices of goods/services over a period of time. The most popular measure is the Consumer Price Index (CPI) which measures the changes in average prices of consumer goods and services. There is also the Producer Price Index (PPI) which measures the changes in average prices of goods at different stages of production.
(b) Quantity Indices: These measure the change or changes in the output or amount of goods consumed or demanded over a period of time.
(c) Value Indices: These are indices which measure change in monetary terms of goods and services over a given period of time. Some common value indices include the Export Index which measures the monetary value of exports of an economy over a given period of time; Import Index measures the monetary value of imports of an economy over a given period of time; and 𝒕𝒓𝒂𝒅𝒆 𝒅𝒆𝒇𝒊𝒄𝒊𝒕=𝒆𝒙𝒑𝒐𝒓𝒕 𝒊𝒏𝒅𝒆𝒙−𝒊𝒏𝒑𝒐𝒓𝒕 𝒊𝒏𝒅𝒆𝒙.

4. To compute index numbers we consider a single item or a basket of good and measure their percentage change over time.
There are unweighted index numbers and weighted index numbers. Unweighted index give the same or equal weighting to all the items in the basket. Weighted index numbers do not give same equal importance to all the items in the basket of goods.

5. Simple Index numbers
A simple index is a ratio of the price of a commodity at a time t divided by its value in the base year. They measure percentage change on a variable for an item after some time period. And are usually expressed as a percentage by multiplying it by 100.
𝑺𝒊𝒎𝒑𝒍𝒆 𝑷𝒓𝒊𝒄𝒆 𝑰𝒏𝒅𝒆𝒙 𝑺𝑷𝑰 = 𝑷 𝒏 𝑷 𝟎 ×𝟏𝟎𝟎%
where 𝑃 𝑛 is the price during period n and 𝑃 0 is the price in the base year.
𝑺𝒊𝒎𝒑𝒍𝒆 𝑸𝒖𝒂𝒏𝒕𝒊𝒕𝒚 𝑰𝒏𝒅𝒆𝒙 𝑺𝑸𝑰 = 𝑸 𝒏 𝑸 𝟎 ×𝟏𝟎𝟎%
where 𝑄 𝑛 is the price during period n and 𝑄 0 is the price in the base year.

6. Problems Consider the following data on some selected goods
Problems Consider the following data on some selected goods. Compute the simple price and simple quantity indices for all the goods and interpret your results.
Product
2000
2005
Price
Quantity
Bread
$1.50
53 000
$3.30
60 000
Cement
$9.85
14 000
$7.50
8 000
Shoes
$22.40
28 000
$25.30
27 000

7. Unweighted Aggregate Index Numbers
Simple aggregate price index is the ratio of the sum of several items in the new period over the sum of the prices of the same items in the base year, and is given by
𝑺𝒊𝒎𝒑𝒍𝒆 𝑼𝒏𝒘𝒆𝒊𝒈𝒉𝒕𝒆𝒅 𝑨𝒈𝒈𝒓𝒆𝒈𝒂𝒕𝒆 𝑷𝒓𝒊𝒄𝒆 𝑰𝒏𝒅𝒆𝒙 𝑺𝑼𝑨𝑷𝑰 = 𝑷 𝒏 𝒊 𝑷 𝟎 𝒊 ×𝟏𝟎𝟎%
where 𝑃 𝑛 𝑖 is the price of the item i during period n and 𝑃 0 𝑖 is the price of item i in the base year.
Simple aggregate quantity index is the ratio of the quantities demanded of several items in the new period over the sum of the quantities demanded of the same items in the base year, and is given by
𝑺𝒊𝒎𝒑𝒍𝒆 𝑼𝒏𝒘𝒆𝒊𝒈𝒉𝒕𝒆𝒅 𝑨𝒈𝒈𝒓𝒆𝒈𝒂𝒕𝒆 𝑸𝒖𝒂𝒏𝒕𝒊𝒕𝒚 𝑰𝒏𝒅𝒆𝒙 𝑺𝑼𝑨𝑸𝑰 = 𝑸 𝒏 𝒊 𝑸 𝟎 𝒊 ×𝟏𝟎𝟎%
where 𝑄 𝑛 𝑖 is the quantity of the item i demanded during period n and 𝑄 0 𝑖 is the quantity of item i demanded in the base year.
One disadvantage of the simple aggregate indices is that they give equal weighting or equal importance to all the items under consideration which is not always true.

8. Problems Consider the following prices and quantities that reflect the average weekly buying habits of a typical family in 1980 and Compute and interpret the following (a) a simple unweighted aggregate price index (b) a simple unweighted aggregate quantity index (c) a value index
Item
1980
1995
Unit Price ($)
Quantity
Unit Price($)
Apples
0.15 6
0.25 1
Milk
0.30 3
0.35 2
Bread
0.32
0.40
Eggs
0.50 5
0.65 4

9. Weighted Index numbers
Weighted index numbers take into account the differences in relative influence exerted by different products in a composite index.
The Value Index (VI) measures the percentage change in the value of items in the basket of goods considered and it is given by
𝑽𝑰= 𝑷 𝒏 𝑸 𝒏 𝑷 𝟎 𝑸 𝟎 ×𝟏𝟎𝟎%.
where 𝑄 𝑛 𝑖 is the quantity of the item i demanded during period n and 𝑄 0 𝑖 is the quantity of item i demanded in the base year. This index has the disadvantage that the resulting number shows the effect of both price and quantity changes. It is necessary to weigh the prices in period n by the same quantity used for the base period. The question becomes: Which quantity should be used 𝑄 𝑛

10. Laspeyres Price Index (LPI)
It uses the base year quantities as weights for the prices and is computed as:
𝑳𝑷𝑰= 𝑷 𝒏 𝑸 𝟎 𝑷 𝟎 𝑸 𝟎 ×𝟏𝟎𝟎%.
Laspeyres Quantity Index (LQI)
It uses the base year prices as weights for the prices and is computed as:
𝑳𝑸𝑰= 𝑷 𝟎 𝑸 𝒏 𝑷 𝟎 𝑸 𝟎 ×𝟏𝟎𝟎%.
Paasche Price Index (PPI)
It is computed in the same manner as the Laspeyres index, except that it uses the new prices as weights and is computed as
𝑷𝑷𝑰= 𝑷 𝒏 𝑸 𝒏 𝑷 𝟎 𝑸 𝒏 ×𝟏𝟎𝟎%.
Paasche Quantity Index (PQI)
𝑷𝑸𝑰= 𝑷 𝒏 𝑸 𝒏 𝑷 𝒏 𝑸 𝟎 ×𝟏𝟎𝟎%.

11. Fisher’s Price Index (FPI)
It is computed by taking the geometric square root of the LPI and the PPI, that is
𝐹𝑃𝐼= 𝐿𝑃𝐼×𝑃𝑃𝐼
𝐹𝑃𝐼= 𝑷 𝒏 𝑸 𝟎 𝑷 𝟎 𝑸 𝟎 𝑷 𝒏 𝑸 𝒏 𝑷 𝒏 𝑸 𝟎
Fisher’s Quantity Index (FQI)
It is computed by taking the geometric square root of the LQI and the PQI, that is
𝐹𝑃𝐼= 𝐿𝑄𝐼×𝑃𝑄𝐼
𝑭𝑸𝑰= 𝑷 𝟎 𝑸 𝒏 𝑷 𝟎 𝑸 𝟎 𝑷 𝒏 𝑸 𝒏 𝑷 𝒏 𝑸 𝟎
Marshall-Edgenorth Index (MEI)
This index tries to overcome problems of under or overstatement by using the arithmetic means of the quantities. It is given by
𝑀𝐸𝐼= 𝑃 𝑛 𝑄 𝑛 + 𝑄 𝑃 0 𝑄 𝑛 + 𝑄 = 𝑃 𝑛 𝑄 𝑛 + 𝑄 𝑃 0 𝑄 𝑛 + 𝑄 0

12. Index
Advantages
Disadvantages
Laspeyres
(1) It is easy to compare different indices because the base year is the same.
(2) It is cheaper in terms of collection of data that is used.
(3) It is less tedious to compute
(1) It can be outdated since the base year is static.
(2) It can overstate the magnitude of change, for example as prices rise; quantities demanded tend to diminish if there are no alternatives.
Paasche
(1) It is more reflective since current values are used as weights.
(1) It is time consuming to calculate where there is a series of years involved (i.e. the weights will be changing each time)
(2) It can only be calculated if up-to-date data is available (i.e. quantities are updated annually).

13. Consider the information of prices and quantities on five inputs of a company that manufactures fertilizer Compute and interpret the following (a) SPI (b) SPI (c) FPI (d) LQI (e) PQI (f) FQI (g) VI (h) MEI
Year 2009
Year 2010
Input
Price
Quantity A
3000
150
3500
180 B
8000 50
8500 C
5200
175
5800
190 D
4500
210
240 E
5600 80 65

14. Problems Associated With Index Number Construction
The following problems are often encountered
1. Base year selection: At times, it might be difficult to come up with a base year which is very reflective of the relative change in the variable under study.
2. Comparability: It is difficult to effectively compare index numbers computed using different base years, particularly in the case of the Paasche Index numbers.
3. Selection of Items: Selecting items which can constitute the basket can be a problem.
4. Inadequacy of data: The data may not be available completely or if available may not be complete.

15. Consumer Price Index (CPI)
A consumer price index (CPI) measures changes through time in the price level of consumer goods and services purchased by households. The CPI is defined as "a measure of the average change over time in the prices paid by urban consumers for a market basket of consumer goods and services."