L3 - Index Numbers Flashcards
What are index numbers?
- Data in the form of index numbers are ubiquitous throughout applied economics, appearing in many forms and in many areas.
- Obvious questions to ask are: what exactly is an index number and why are they used?
- An index number can be thought of as a type of summary statistic, so that it summarises a large amount of information on, say, prices and quantities, in a single value.
- It is thus used when presenting the underlying raw data is too time consuming or too complicated to comprehend easily.
How do you calculate Index numbers for a set of data?
- set one year as the base year and it takes a value of 100
- The values for the next years is then calculated in index form using the following formula:
- Index{t}= 100 x ((Observed{t})/(Observed{base}))
- Although any year can be picked it makes sense to pick a simpler value towards the end –> dont pick the first or last value
What is an issue when monitoring an index over time?
. One problem that often occurs, particularly when taking data from different ‘vintages’ of publications, is that the base changes every so often, usually every five or ten years, so that the data ‘jumps’ at these years.
- You can solve this problem by chaining the data
How can you Chain index numbers from different publications?
- This can only be done if the two indexes have an overlapping observation i.e. one date at which we have observation from both bases
- from the two sets of the data pick which set you are going to use as the base year period and divide that index number by the overlapping one of the other data set to get a conversion factor
- You can then time all the data in the other index by this to get them all in the desired base year
- if you want to switch base years to the other period you just use the inverse of the conversion factor
- If more than one base change occurs then chaining can be used repeatedly.
Why are Index Numbers Useful?
While simple, index numbers of this type have to be interpreted carefully.
- For an individual series, an index number in isolation is meaningless; what does a GDP index of 116.8 in 2004 actually tell you?
- It only becomes meaningful if it is compared to another value: in 2001 the index was 100, so that we can say that GDP has grown by 16.8% over the period 2002 to 2004.
- Comparing GDP indices across countries, for example, will be misleading unless the base year, base and underlying GDP values in the base year are identical, which is unlikely to say the least!
What do more sophisticated Indices do?
- More sophisticated index numbers attempt to combine information on a set of prices, say, by weighting the individual prices by some measure of their relative importance (perhaps given by the quantities sold of each of the products). - - This would define a price index, of which the Consumer Price and Retail Price Indices (CPI and RPI respectively) are the most familiar in the U.K, although the FTSE stock market indices are
another popular example. - for example Suppose we want to construct an overall energy price index using four energy price indices e.g. Coal, Gas, Electricity and Petroleum all taken in the base year of 2005
- An unweighted average would be wrong unless each type of energy is equally important i.e. consumed in equal
quantities - If we measure ‘importance’ by the quantities used of each fuel, then we can construct a weighted average using these quantities as weights. But what quantities do we choose? A traditional approach, similar to the previous example, is to choose a base year and to use the quantities prevailing in that year as a set of base-year weights.
What is Lasperyres price Index?
- for a basket of goods for each year but the prices in index form at a single base year
- figure out a base year for the quantities and set this as the base quantity year
- Then for each of the price index for each good multiple by the quantity at base year and total for all goods
- Do this for all years to give a set of cost for all the different years
- The index of the energy prices can then be calculating by taking the ratio of the costs for each year relative to 2000 (and conventionally multiplying by 100)
- This lead to what is known as a Laspeyres price index
What is the general Lasperyres Price Index formula?
P_t^L=100x((Σ(p{i,t}xq{i,0}))/((Σ(p{i,0}xq{i,0}))
- In this formula, the p{i,0} and q{i,0} are the base year prices and quantities, and p{i,t} are the ‘current’ year prices, this can be rewritten:
- P_t^L = 100 x ((Σ(p{i,t}/p{i,0) x p{i,0}q{i,0}))/(Σp{i,0}q{i,0})
= 100 x ((Σ(p{i,t}/p{i,0)) x w_i,0^L))/(Σw_i,0^L))
where w_i,0^L = p{i,0}q{i,0}, shows that P_t^L is a weighted average of the price relatives ,((p{i,t})/p({i,0})) with the weights given by the base year expenditures p{i,0}q{i,0} .
How do you calculate the expenditure share?
- s{i,0}=((p{i,0}q{i,0}))/(Σp{i,0}q{i,0})
= (w_i,0^L))/(Σw_i,0^L)
How can we link the expenditure share to the Lasperyres Price Index?
- s{i,0}= ((p{i,0}q{i,0}))/(Σp{i,0}q{i,0})
= (w_i,0^L))/(Σw_i,0^L) - if laperyres Price index equals P_t^L = 100 x ((Σ(p{i,t}/p{i,0) x p{i,0}q{i,0}))/(Σp{i,0}q{i,0})
= 100 x ((Σ(p{i,t}/p{i,0)) x w_i,0^L))/(Σw_i,0^L))
combining the two equations - P_t^L=100x((Σ(p{i,t}/p{i,0}))xs{i.0})
- A problem with the Laspeyres Price Index is that the choice
of 2000 for example as the base in our case was arbitrary. Adjusting the
base will change the base expenditures and the price relatives
which will cause the index values to change.
What is the Problem with the Laspeyres Price Index?
There is a related, probably more significant, defect with the Laspeyres index.
- As relative prices alter over time, one would expect quantities consumed to change.
- This is not allowed in the index, which is calculated using unchanging base year quantities. These may therefore soon become unrepresentative and the index then has to be rebased using quantities of a more recent vintage.
- We can see these changes occurring in the quantities of energy consumed (particularly coal) over successive years.
What is the Paasche Price Index?
-An alternative to using base year weights is to use current year weights which is the case with the Paasche Price Index
P_t^P=100x((Σ(p{i,t}xq{i,t}))/((Σ(p{i,0}xq{i,t}))
P_t^P = 100 x ((Σ(p{i,t}/p{i,0) x p{i,0}q{i,t}))/(Σp{i,0}q{i,t})
= 100 x ((Σ(p{i,t}/p{i,0)) x w_i,t^P))/(Σw_i,t^P))
How can the expenditure share formula be linked to the Paasche Price Index?
The expenditure share formula is:
- s{i,t}= ((p{i,0}q{i,t}))/(Σp{i,0}q{i,t})
= (w_i,t^P))/(Σw_i,t^P)
The Paasche Price Index is:
-P_t^P=100x((Σ(p{i,t}xq{i,t}))/((Σ(p{i,0}xq{i,t}))
P_t^P = 100 x ((Σ(p{i,t}/p{i,0) x p{i,0}q{i,t}))/(Σp{i,0}q{i,t})
= 100 x ((Σ(p{i,t}/p{i,0)) x w_i,t^P))/(Σw_i,t^P))
Combining the two equations:
P_t^P = 100 x ((Σ(p{i,t}/p{i,0) x s{i,t}))
= 100 x ((1/(Σ(p{i,0}/p{i,T} x s{i,t})
How do answers differ from the Laspeyres and Paasche Price Index?
Paasche’s Price Index tends to be slightly smaller than their Laspeyres counterparts, which is to be expected if consumption of energy is switched to those fuels that are becoming relatively cheaper, since the Paasche index, by using current weights, can capture this switch.
What are the advantages and drawbacks of the Price Indexes?
Both indices have advantages and drawbacks. The Laspeyres is simpler to calculate and to understand but loses legitimacy over time as its weights become unrepresentative.
- The Paasche, on the other hand, always has current weights, but is more difficult to calculate (although this is hardly a problem when computers do most of the calculations) and rather harder to interpret.
