Nominal salary inflation adjustment problem

Question

Given that average inflation from 2014 until 2024 is 4%, how big increase in nominal salary you would need in order for your real salary to increase 30% from 2014 until 2024?

Answer 1

Assume that your salary is $x$ dollars per year in 2014 (that's both nominal and real). Now, let's suppose that in 2015, all prices inflate by a factor of $r$. That means that the buying power of your salary declines by a factor of $r$ from 2014 to 2015 - you're still earning $x$ nominal dollars, but only $x/r$ real dollars. At this rate, you'll find you're only earning $x/r^{10}$ real dollars in 2024.

You want to be earning $1.3x$ real dollars in 2024. Hence, we need to figure out how large a nominal raise $\delta$ you need to get in order to achieve your desired salary. In other words, we must solve:

$$1.3x = \frac{x}{r^{10}} + \delta$$

for $\delta$. This is straightforward:

$$\delta = \left(1.3 + \frac{1}{r^{10}}\right)x$$

Hence, you need a nominal raise in the amount of $1.3 + \frac{1}{r^{10}}$ times your 2014 salary ($x$) in order to get a $30\%$ increase in your real salary by 2014. Plugging in $r = 1.04$, we find that your nominal salary must increase by a factor of $\boxed{1.98}$ over 10 years.

If, instead of having the raise as a one-time bulk increase, you want to have a percentage raise each year, we can compute the needed yearly raise $d$ as follows:

$$d = 1.98^{1/10} = 1.07$$

Hence, you must have a nominal yearly raise of $\boxed{7\%}$ in each of 2015, 2016, ..., 2024 in order to achieve a $30\%$ increase in real wages by 2024.