Economical Calculus: Price inflation and the value of money

Economical Calculus: Price inflation and the value of money

One of the applications of calculus is in economics, calculus can be used to determine price inflation and the value of money.

Inflation rates are of two types namely continuous rate and annual rates.
 In order to determine price change $\Delta{y}$ over a year ,we use the annual rate:
$$\begin{equation} \Delta{y}=(annual\ rate)\times{y}\times{\Delta{t}} \end{equation}$$
We can apply the continuous rate to each instant $dt$, the price change is $dy$:
$$\begin{equation} dy=(contiuous\ rate)\times{y}\times{dt} \end{equation}$$
dividing by $dt$ yields a differential equation for the price:
$$\begin{equation} \frac{dy}{dt}=(continuous\ rate)\times{y}=0.05y. \end{equation}$$
The solution is thus $y_0{e}^{0.05t}$,if we set t=1 then $e^{0.05}\thickapprox{1.0513}$ and the annual rate is 5.13 percent.Normally, when you ask a bank what interest they pay,they give both rates of 8 percent and 8.33 percent .
The higher one they call the "effective rate" it comes from compounding(and depends on how often they do it). if the compounding is continuous , every $dt$ brings an increase of $dy$ and $e^{0.08}$ is near 1.0833.
Now the interval drops from a month to a day to a second.That leads to $(1+\frac{1}{n})^n$ and in the limit to e.here we compute the effects of 5 percent continuous interest.

Future Value:A dollar now has the same value as $e^{0.05T}$ dollars in T years.

Present Value:A dollar in T years has the same value as $e^0.05T$ dollars now.

Doubling Time:Prices double $(e^{0.05T}=2)$ in $T=\ln(\frac{2}{0.05})\thickapprox{14 years}$.
With no compounding , the doubling time is 20 years. Simple interest adds on 20 times 5 percent=100 percent.
With continuous compounding the time is reduced by the factor $\ln{2}=0.7$, regardless of the interest rate.

Example: In 1626 the Indians sold Manhattan for 24 dollars.Our calculations indicate that they knew what they were doing.Assuming 8 percent compound interest, the original 24 dollars is multiplied by $e^{0.08t}$. After t=365 years the multiplier is $e^{29.2}$ and the 24 dollars has grown to 115 trillion dollars.With that much money they could buy back the land and pay off the national debt.
this seems farfetched.Possibly there is a big flow in the model, it is absolutely true that Benjamin franklin left money to Boston and Philadelphia,to be invested for 200 years. In 1990 it yielded millions(not trillions,that takes longer).