## Tuesday, May 29, 2012

### A tale of exponential interest

I have a savings account at ING bank which gives me 4% interest monthly. Each month I receive payment of both the principal I deposited plus the compound interest they gave me over the past months.

Notice that the interest, or the present increment or my balance, $\frac{d x(t)}{dt}$, is equal to the current balance itself times time interest $k$.
$$\frac{d x(t)}{dt} = kx(t)$$
We write each function as a variable for convenience.
$$\frac{d x}{x} = kdt$$
And solve by integration
$$\int \frac{d x}{x} = k \int dt$$
$$\log x = kt + C_0$$
(we abuse notation and name the upper limit of integration with the same name than the integrating variable) yielding
$$x(t) = C_1e^{kt}$$

What does it mean? It means that everybody would see their savings accounts grow exponentially. Now, would it correspond to exponential wealth? Would the currency be debased at the same speed for the interest to be payable or can we extract oil fast enough to cover our land with things and thus effectively increasing our wealth?

$C_1=e^{C_0}$

Here is an interesting discussion at Do the Math.