Question

Rental costs for office space have been going up at 9.3% per year compounded annually for the past 6 years of office space rent is now $21 per square foot per month, what were the rental rates 6 years ago?The ratos 6 years ago were about sper square foot(Round to two decimal places as needed)

Answer 1

We have rent prices that increase by 9.3% each year.

We know that the price now is $21 per square foot per month.

We have to calculate the price 6 years ago.

As we know that prices increase by 9.3% we can relate the prices of two years as:

$\begin{gathered} P(t)=(1+0.093)\cdot P(t-1) \\ P(t)=1.093\cdot P(t-1) \end{gathered}$

We can find a function that only depends on t, the year from now, as:

$\begin{gathered} P(t)=1.093\cdot P(t-1) \\ P(t)=1.093\cdot(1.093\cdot P(t-2))=1.093^2\cdot P(t-2) \\ P(t)=1.093^t\cdot P(t-t)=1.093^t\cdot P(0) \end{gathered}$

As P(0) is the price now, we can replace it and we have a formula for the price of any given year:

$\begin{gathered} P(t)=P(0)\cdot1.093^t \\ P(t)=21\cdot1.093^t \end{gathered}$

The price 6 years ago correspond to t = -6, so we can calculate it as:

$\begin{gathered} P(-6)=21\cdot1.093^{-6} \\ P(-6)=\frac{21}{1.093^6} \\ P(-6)\approx\frac{21}{1.705} \\ P(-6)\approx12.32 \end{gathered}$

Answer: the prices 6 years ago were $12.32 per square foot per month.