How can differential equations be used to maximize profit?

[ojsimon]

Hi

I am doing a research project on maximizing profit using mathematics, and found some high level phd theisis on maximizing profit using differential equations. I was wondering if anyone could explain in a simpler form how differential equations can be used to maximise profit.

Thanks

Olie

 

[John Creighto]

Differential equations are great for modeling things. For instance compounded interest follows the simple differential equation:

$$\frac{dy}{dt}=ky$$

If an economic system was modeled with differential equations then optimization routines could be applied to try and maximize profits.

 

[ojsimon]

wait so that formula is just the formula for comound interest differentiated? so y=xk^n where n= time invested. k=intrest rate and x=initial amound. dy/dx=nx^n-1 ? How does this work? Thanks

 

[pbandjay]

It works like this...

$$\frac{dy}{dt} = ky$$

$$\displaystyle\int_{y(0)}^{y(t)}\frac{dy}{y}=\displaystyle\int_{0}^{t}kdt$$

$$\ln y(t) - \ln y(0) = kt - k*0$$

$$\ln \frac{y(t)}{y(0)} = kt$$

let y(0) = y_0

$$y(t) = y_0e^{kt}$$

 

[Feldoh]

Well in a lot of cases certain business could be seen to be dependent on each other. So you might model these as a system of differential equations and figure out how to maximize profits.

Really there are infinite possibilities only bounded on how you make your economic model.

 

[John Creighto]

wait so that formula is just the formula for comound interest differentiated? so y=xk^n where n= time invested. k=intrest rate and x=initial amound. dy/dx=nx^n-1 ? How does this work? Thanks

The formula for compound interest is:

$$a(t) = \left(1 + \frac {r} {n}\right) ^ {nt}$$

which approaches in the limit as N approach infinity:


$$A(t) = A_0e^{rt}$$
(see continuously compounded interest)

The previous post showed how to solve the differential equation I gave in an earlier post, you can verify this by differentiating the results. This is relevant with regards to investing because investments are compared on the basis of rate of return. If you pick up a book on engineering economics it will show how to treat non geometric profit returns in terms of an equivalent rate of return based on the time value of money.

Generally, I think company's give a fixed rate of return based on their current capital, I believe their is a quantity called return on capital that gives an idea of the expected growth rate. I'll give more details later.