Editing Real and nominal value (section)

==Real growth rate==

The real growth rate <math>r_t</math> is the change in a nominal quantity <math>Q_t</math> in real terms since the previous date <math>t-1</math>. It measures by how much the buying power of the quantity has changed over a single period.

:<math>r_t = \frac{P_0 \cdot Q_t}{P_t} \Bigg/ \frac{P_0 \cdot Q_{t-1}}{P_{t-1}} - 1</math>

::<math>= \frac{P_{t-1} \cdot Q_t}{P_t \cdot Q_{t-1}} - 1</math>

::<math>= \frac{Q_t}{Q_{t-1}} (\frac{P_t}{P_{t-1}})^{-1} - 1</math>

::<math>= \frac{1 + g_t}{1 + i_t} - 1</math>

where <math>g_t</math> is the nominal growth rate of <math>Q_t</math>, and <math>i_t</math> is the inflation rate.

:<math>1 + r_t = \frac{1 + g_t}{1 + i_t}</math>

For values of <math>i_t</math> between −1 and 1 (i.e. ±100 percent), we have the [[Taylor series]]

:<math>(1 + i_t)^{-1} = 1 - i_t + i_t^2 - i_t^3 + ...</math>

so

:<math>1 + r_t = (1 + g_t)(1 - i_t + i_t^2 - i_t^3 + ...)</math>

:::<math>= 1 + g_t - i_t - g_t i_t + i_t^2 + \text {higher order terms.}</math>

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

:<math>r_t = g_t - i_t</math>