Illustration, Notation, and Generalization
The simplest case of a bundle of commodities (goods) is one that has only one commodity. In that case, output or consumption may be measured either in terms of money value (nominal) or physical quantity (real). Let i designate that commodity and let:
- Pi = the unit price of i, say, $5
- Qi = the quantity of i, say, 10 units.
The nominal value of the bundle would then be price times quantity:
- nominal value of i = Pi x Qi = $5 x 10 = $50.
Given only the nominal value and price, derivation of a real value is immediate:
- real value of bundle i = Pi x Qi/Pi = Qi = 50/5 = 10.
The price "deflates" (divides) the nominal value to derive a real value, the quantity itself.
Similarly for a series of years, say five, given only nominal values of the good and prices in each year t, a real value can be derived for each of the five years:
- real value of bundle i in year t = nominal value of Qit/Pit = Qit.
This example generalizes for nominal values relative to real values across different years for which P, a price index comparing the general price level across years, is available. Consider a nominal value (say of an hourly wage rate) in each different year t. To derive a real-value series from a series of nominal values in different years, divide nominal value in each year by Pt, the price index in that year. By definition then:
- real value in year t = nominal value in year t/Pt.
| Numerical example:
If for years 1 and 2 (say 20 years apart) the nominal wage and P are respectively
real wages are respectively:
The real wage so constructed in each different year indexes the amount of commodities in that year that could be purchased relative to other years. Thus, in the example the price level increased by 33 percent, but the real wage rate still increased by 20 percent, permitting a 20 percent increase in the quantity of commodities the nominal wage could purchase. |
The generalization to a commodity bundle from the single-good illustration above is to a bundle of quantities of different commodities and different years. This has practical use, because price indexes and the National Income and Product Accounts are constructed from such bundles of commodities and their respective prices.
A sum of nominal values for each of the different commodities in the bundle is also called a nominal value. A bundle of n different commodities with corresponding prices and quantities for each year t defines:
- nominal value of that bundle in year t = P1t x Q1t + . . . + Pnt x Qnt.
From the above:
- Pt = the value of a price index in year t.
The nominal value of the bundle over a series of years and corresponding Pt define:
- real value of the bundle in year t = Qt = nominal value of the bundle in year t/Pt.
Alternatively, multiplying both sides by Pt:
- nominal value of the bundle in year t = Pt x Qt.
So, every nominal value can be dichotomized into a price-level part and a real part. The real part Qt is an index of the quantities in the bundle.
An illustration of a nominal-value sum is nominal GDP. An illustration of a real-value sum (or quotient) is real GDP.
Read more about this topic: Real Versus Nominal Value (economics)