# Equation of (percentage) grow [closed]

To find a particular change, the equation $$\dfrac{x_1-x_0}{x_0}$$ is used,

but in Economics, instead, they used $$\dfrac{x_1-x_0}{\frac{x_1+x_0}{2}}$$, where in denominator is the average.

Which equation is correct, and precise? And what “intuition” is behind that?

• Both are correct in that they have different meanings. The first is appropriate if you are thinking of an abrupt change from $x_0$ to $x_1$. The second is appropriate if you are thinking of a gradual change, and you are estimating the rate of change by letting the denominator represent the average value upon which the change is based during the gradual period of change. Jun 28, 2021 at 15:48
• Thank you for your answer. Here when they use different equations, they get different results, and it is difficult to set relativity in gradual and abrupt change.
– Nick
Jun 30, 2021 at 13:19

Their problem here is that for example they see going from $$400$$ to $$500$$ as $$+25\%$$ but going from $$500$$ to $$400$$ as $$-20\%$$, and they do not like this as it disrupts their elasticity calculations.
A solution which could work for them would be to use logarithms, and here $$\log_e(500)-\log_e(400) =\log_e(1.25)\approx 0.22314$$ and in the opposite direction $$\log_e(400)-\log_e(500) =\log_e(0.8) \approx -0.22314$$.
But that is a bit complicated to explain and is not really a percentage change. So instead they redefine percentage change (why not - they already plot price and quantity a strange way round on the axes, mix superscripts with exponents in equations, and see no need in general to be consistent with mathematicians or statisticians), and here get $$\frac{500-400}{450}\approx 0.22222$$ and $$\frac{400-500}{450}\approx -0.22222$$ which they think is close enough and they are willing to call a percentage change.
• @Nick then you will have something else again. The problem comes when you get multiple changes, for example $400$ to $500$ to $600$ and then back to $400$. Logarithms can handle this, while (possibly redefined) percentages are messier Jun 30, 2021 at 13:45