Real and nominal value

Real value

The nominal value of a commodity bundle tends to change over time. In contrast, by definition, the real value of the commodity bundle in aggregate remains the same over time. The real values of individual goods or commodities may rise or fall against each other, in relative terms, but a representative commodity bundle as a whole retains its real value as a constant from one period to the next. Real values can for example be expressed in constant 1992 dollars, with the price level fixed 100 at the base date.

The price index is applied to adjust the nominal value ${\displaystyle Q}$ of a quantity, such as wages or total production, to obtain its real value. The real value is the value expressed in terms of purchasing power in the base year. The index price divided by its base-year value ${\displaystyle P_t/P_0}$ gives the growth factor of the price index. Real values can be found by dividing the nominal value by the growth factor of a price index. Using the price index growth factor as a divisor for converting a nominal value into a real value, the real value at time t relative to the base date is:

${\displaystyle \frac{P_0\cdot Q_t}{P_t}}$

Real growth rate

The real growth rate ${\displaystyle r_t}$ is the change in a nominal quantity ${\displaystyle Q_t}$ in real terms since the previous date ${\displaystyle t-1}$. It measures by how much the buying power of the quantity has changed over a single period.

${\displaystyle r_t=\frac{P_0\cdot Q_t}{P_t}/\frac{P_0\cdot Q_{t-1}}{P_{t-1}}-1}$

${\displaystyle =\frac{P_{t-1}\cdot Q_t}{P_t\cdot Q_{t-1}}-1}$

${\displaystyle =\frac{Q_t}{Q_{t-1}}(\frac{P_t}{P_{t-1}}{)}^{-1}-1}$

${\displaystyle =\frac{1+g_t}{1+i_t}-1}$

where ${\displaystyle g_t}$ is the nominal growth rate of ${\displaystyle Q_t}$, and ${\displaystyle i_t}$ is the inflation rate.

${\displaystyle 1+r_t=\frac{1+g_t}{1+i_t}}$

For values of ${\displaystyle i_t}$ between −1 and 1 (i.e. ±100 percent), we have the Taylor series

${\displaystyle (1+i_t{)}^{-1}=1-i_t+i_t^2-i_t^3+...}$

so

${\displaystyle 1+r_t=(1+g_t)(1-i_t+i_t^2-i_t^3+...)}$

${\displaystyle =1+g_t-i_t-g_t{i}_t+i_t^2+\text{higher order terms.}}$

Hence as a first-order (i.e. linear) approximation,

${\displaystyle r_t=g_t-i_t}$

Real wages and real gross domestic products

The bundle of goods used to measure the consumer price index (CPI) is applicable to consumers. So for wage earners as consumers, an appropriate way to measure real wages (the buying power of wages) is to divide the nominal wage (after-tax) by the growth factor in the CPI. Gross domestic product (GDP) is a measure of aggregate output. Nominal GDP in a particular period reflects prices that were current at the time, whereas real GDP compensates for inflation. Price indices and the U.S. National Income and Product Accounts are constructed from bundles of commodities and their respective prices. In the case of GDP, a suitable price index is the GDP price index. In the U.S. National Income and Product Accounts, nominal GDP is called GDP in current dollars (that is, in prices current for each designated year), and real GDP is called GDP in [base-year] dollars (that is, in dollars that can purchase the same quantity of commodities as in the base year).

Example

Real interest rates

As was shown in the section above on the real growth rate,

${\displaystyle 1+r_t=\frac{1+g_t}{1+i_t}}$

where

${\displaystyle r_t}$ is the rate of increase of a quantity in real terms,

${\displaystyle g_t}$ is the rate of increase of the same quantity in nominal terms, and

${\displaystyle i_t}$ is the rate of inflation,

and as a first-order approximation,

${\displaystyle r_t=g_t-i_t.}$

In the case where the growing quantity is a financial asset, ${\displaystyle g_t}$ is a nominal interest rate and ${\displaystyle r_t}$ is the corresponding real interest rate; the first-order approximation ${\displaystyle r_t=g_t-i_t}$ is known as the Fisher equation. Looking back into the past, the ex post real interest rate is approximately the historical nominal interest rate minus inflation. Looking forward into the future, the expected real interest rate is approximately the nominal interest rate minus the expected inflation rate.

Cross-sectional comparison

Not only time-series data, as above, but also cross-sectional data which depends on prices which may vary geographically for example, can be adjusted in a similar way. For example, the total value of a good produced in a region of a country depends on both the amount and the price. To compare the output of different regions, the nominal output in a region can be adjusted by repricing the goods at common or average prices.