Indexes
Values, Quantities and Prices
First, we note that values – as measured in money terms – consist of two components: quantities and prices. A price is value per unit of quantity, and when multiplied with the amount of quantity we obtain the total value. In symbolic form we can write this as follows:
v=q.p (1)
Here v stands for value, p for price and q for quantity. The following table gives an example of a single good, valued at two moments in time, labeled 1 and 2 (e.g. referring to the years 2010 and 2011).
Quantities make sense for homogeneous goods that comes in bulk, measured by units of mass (kilogram, ton) or volume (liter, gallon). In the example above the price would then be 3 money units (Euro, Dollar, etc.) per kilogram. Multiplying such prices with the quantity in kilograms (10) yields the monetary value of the good at that particular period, say 2010 (30). For manufactured goods quantities are typically given in countable units, e.g. one piece of soap. The price for such a good is then given as money value per unit. In the above example we would have 10 pieces of soap, each valued 3 money units. For services we can also make the above breakdown. Here quantities can be measured in time units, with prices the monetary values per time unit, such as the hourly or daily fee for the provision of the service. In the above example, for 10 hours of work, valued at the fee of 3 money units per hour, the total value is 30.
For both goods and services we speak of quantities when the good or service is “elementary” for the purpose of analysis. Flour (measured in kilograms), milk (liters), bars of soaps (units), cleaning services (hourly fee); all these goods and services are typically treated as single, elementary units, even though they may be composites in terms of production processes, with many inputs being combined to yield the good or service. Such elementary units can be precisely described and have real prices and quantities that can be measured in price and production statistics. In contrast, the analysis as done in National Accounts usually pertains to aggregated values, where many goods and services are combined. An example is total production in the food industry. Here the total obviously contains many diverse products, each with its own specifications and units of quantities. In order to use the identity (1) given above, we must give q a somewhat different meaning, denoting an abstract (and in some way weighted) total consisting of many elementary units. Such totals are called volumes. So for National Accounts the identity (1) is typically interpreted as representing the breakdown of values into volumes and prices. Exactly how volumes are calculated from elementary goods and services, and what the units of such volumes are, if any, are two of the core questions that we will investigate in this paper. But before delving into these questions we must prepare some groundwork. So for the time being we will forget about the complexities involved with aggregation into volumes and go back to elementary units with real prices and quantities.
Changes in Values, Quantities and Prices
Since both quantities and prices are measured for a particular period it obviously makes sense to look at their changes from one period to the next. In the following table we calculate the changes in value, quantity and price of the item presented as follows:
Note that the symbol for change is ∆, the Greek delta letter. The changes themselves are easy to calculate as differences between the values, quantities or prices in both periods. Note that the units of these changes are identical to the units of the items for which the change is calculated. E.g. the unit of ∆p is the same as for p, money per unit of quantity.
Is there a relation like (1) relating these changes to one another? Let us investigate this question. Using the multiplication rule d(xy)=(dx)y+x(dy) from differential calculus we obtain from (1) the following relation:
∆v=∆q.p+q.∆p (2)
However, we have a problem in this relation: for what period do we give p and q? Say we use the first period:
∆q.p(1)+q(1).∆p=2.3+10.1=16
But this is not equal to the change in value as given in table 2 (∆v=18). Let us try the second period:
∆q.p(2)+q(2).∆p=2.4+12.1=20
Again, this is not the true change. It turns out that we have to mix the two periods in the sense of either taking the first period for p and the second for q or vice versa.
∆v=∆q.p(1)+q(2).∆p=2.3+12.1=18
∆v=∆q.p(2)+q(1).∆p=2.4+10.1=18
It is actually easy to understand why this must be the case. Look at the figure below, which is a plot of prices (measured horizontally) and quantities (measured vertically). Our good is represented by point (10,3) for period 1 and (12,4) for period 2.
The area of rectangle A connecting the points (0,0), (10,0), (10,3) and (0,3) is equal to 30 and represents the value of the good in period 1. Similarly, the value of the good in period 2 is given by the rectangle connecting the points (0,0), (12,0), (12,4) and (0,4) and is equal to 40. As can be seen from the diagram this rectangle consists of the sum of four smaller rectangles A, B, C and D, with A giving the original value in period 1. So when going from the value in period 1 (A) to the value in period 2 (A+B+C+D) we can do two things:
Keep the price fixed at 3 and let the quantity go from 10 to 12, this contribution is given by rectangle B and is equal to ∆q.p(1)=2.3=6. Next we take the new quantity in period 2 and apply the price change. This contribution is given by the sum of rectangles C and D and is equal to q(2).∆p=12.1=12.
Keep the quantity fixed at 10 and let the price go from 3 to 4, this contribution is given by rectangle D and is equal to q(1).∆p=10.1=10. Next we let the quantity change at the new price. This contribution is given by the sum of rectangles B and C and is equal to ∆q.p(2)=2.4=8.
It turns out that the first method is of fundamental importance for National Accounts, so we will paraphrase it to consist of two steps:
Change the quantity from period 1 to period 2 at the fixed price of period 1; this will give the new quantity valued at the old price or, as National Accountants say, the new quantity at previous years’ prices or – more commonly, though not strictly correct – at constant price;
Apply the price change to this quantity change at constant price; since the price goes from period 1 to 2 (old to new) this is often called inflating the quantity change at constant price.
Before we can go into more details we have to address one problem: our changes, such as ∆p, have units, and so will depend on these units. This is clearly undesirable, since choosing different units of measurement will therefore affect the calculated changes. We want to measure changes in a way that does not depend on units.
Percentage changes in values, quantities and prices
The obvious solution to the above problem is to introduce percentage changes instead of our nominal changes. To do this, simply divide the change ∆p by p(1), the original price. We will denote this percentage change by %∆p. Optionally, you can multiply the result with 100, but we will not do so here. The percentage changes for the example are given in the following table.
Again we ask the question: is there an easy way to relate the percentage changes for values, quantities and prices? Let us find out. First, take relation (2) and divide by p(1).q(1):
∆v⁄(p(1).q(1) )=(∆q.p(1))⁄(p(1).q(1) )+(q(2).∆p)⁄(p(1).q(1) )
On the left-hand side we can substitute v(1) for p(1).q(1), this is identity (1). On the right-hand side we can cross out p(1) from the first term (since it appears in both the numerator and the denominator). We then get:
%∆v=%∆q+q(2) ⁄q(1) .%∆p (3)
So to get the percentage change in values you can add the percentage changes in quantity and price, but you have to multiply the percentage price change with a factor q(2) ⁄q(1) . We can check relation (3) by inserting the numbers from the above tables (where we write %∆p as 1⁄3):
0.6=0.2+12⁄10.1⁄3
Although we now have a nice additive relationship between the percentage changes in v, q and p, we do have the inconvenient factor q(2) ⁄q(1) . Can we do something about this?
Indexes of values, quantities and prices
To come to better solution, let us write the percentage change in values as follows:
%∆v=((v(2)-v(1)))⁄v(1) =v(2) ⁄v(1) -1 (4)
We call a factor like v(2) ⁄v(1) a relative (in this case a value relative). Let us now substitute this into relation (3):
v(2) ⁄v(1) -1=q(2) ⁄q(1) -1+q(2) ⁄q(1) .(p(2) ⁄p(1) -1)= q(2) ⁄q(1) .p(2) ⁄p(1) -1
To get from the second to the third term, multiply away the parentheses and cross out the common factor q(2) ⁄q(1). If we also cross out the common -1 we get:
v(2) ⁄v(1) =q(2) ⁄q(1) .p(2) ⁄p(1) (5)
Now this is very nice. To get the value relative v(2) ⁄v(1) we multiply the quantity relative q(2) ⁄q(1) with the price relative p(2) ⁄p(1) . Of course we could have saved ourselves a lot of trouble by simply rewriting the value relative using identity (1):
v(2) ⁄v(1) =(q(2).p(2))⁄(q(1).p(1) )=q(2) ⁄q(1) .p(2) ⁄p(1)
Whatever way we look at it, the choice of relatives instead of percentage changes makes it possible to go from absolute values in relation (1) to changes without units in (5). Both are multiplicative relations. The next table calculates the value, quantity and price relatives for our example.
Note that we have included a factor 100 here, which is standard practice. Note also that we give these relatives a subscript consisting of the two periods (separated by a comma), with the second coming first (i.e. period 2 as related to period 1). Using these special subscripts we have:
v(2,1)=q(2,1).p(2,1) (6)
Relatives as introduced here are better known as indexes and we will henceforth follow suit. However, please note that although all relatives are indexes, the reverse is not true: not all indexes are simple relatives as introduced here. With this renaming we can now conclude with the paraphrasing of (6) in words:
Value index = Volume index multiplied with price index
This will be the fundamental relationship that underlies the derivation of national accounts aggregates in constant prices, as we will see below. But before continuing with this, let us investigate some properties of indexes in general.
Properties of indexes
Take the quantity relative q(2) ⁄q(1) relating period 2 to period 1. We can write a similar relative, relating period 3 to period 2: q(3,2)=q(3) ⁄q(2). Multiplying these two relatives, and crossing out the common q(2) gives q(3,1), relating period 3 to period 1:
q(3,1)=q(3) ⁄q(2) .q(2) ⁄q(1)
Hence, one can always relate any period to any other period using elementary relatives, going back only one period, which we will henceforth call link relatives.. Obviously the q series must have a beginning; let us call this first period: 0. We can relate this period to itself as follows:
q(0,0)=q(0) ⁄q(0) .100=100
Using link relatives q(1,0), q(2,1) etc. we can now construct a “chain” of link relatives relating any period back to the first period as follows:
q(0,0)=100
q(1,0)=(q(1,0).q(0,0))/100
q(2,0)=(q(2,1).q(1,0))/100
The year t for which q(t,t) is set to 100 is commonly called the reference year. In the example above t=0, but it is easy to see that any year can be taken as reference year, simply by dividing the index series by the value for this year. This is called re-referencing the index series. An example:
We get series 2 by dividing all terms of series 1 by 103 (and multiplying with 100).
One more property of an index: because of relation (4), you can always get the percentage change (%∆ now multiplied with 100 ) by subtracting 100 from the index (e.g. index 103 gives a 3% change). Vice versa, adding 100 to the percentage change will give you the index (5% change gives an index of 105). Note that an index can be smaller than 100, e.g. 97.1 in the above table. In this case we get as percentage change is 97.1 – 100 = -2.9%, a negative change.