# Examples of separable ordinary differential equations in economics

I'm currently teaching an integral calculus course for business students, and we're just about to discuss differential equations. They've worked hard, and I'd like to reward them with some economic applications of ODEs, but they can only handle simple separable equations.

I'm going to frame exponential growth in terms of economic growth(among other things,) and then I'm currently planning on looking at which demand functions have constant elasticity and looking at the logistic model of a population. I might be asking for too much, but I was wondering whether anyone could suggest a separable equation that arises from a simple model(they've all taken an introduction to economics, but no more.)

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You might want to take a look at "Further Mathematics for Economic Analysis" by Sydsaeter and Hammond. It is a textbook for economics students with a lengthy part on ODEs, so I'm sure you can find some examples there. – Michael Greinecker Oct 24 '12 at 8:10
Thanks! I'll look into that for the next time I teach the course. – Gordon Craig Nov 6 '13 at 20:06

Suppose you maintain a pond with fish (for profit, of course, this is economics!). When the food is abundant and there are not many fish, the population grows at a constant rate $k>1$ (reproduction rate minus death rate), so we have $y'=ky$. This is separable. Solve it. Give a numerical example. Conclude from the example that our assumptions are not realistic. So what is wrong with our assumtions? Abundant food!!! (Of course. This is economics after all:-) The next simple assumption is that the pond can support only some maximal population, say $A$. Which means that when the population approaches $A$ the death rate increases (starvation), so the net growth rate is not just $k$ but $k(1-y/A)$. When $y$ is small, (or $A$ is very large) we have almost $y'=ky$ as before. When $y$ is close to $A$, the net rate of change approaches $0$, as it should be. We obtain $y'=ky(1-y/A)$, another separable equation!
But this pond brings you no profit yet. To make a profit, you have to catch some fish, say at a constant rate. You obtain another separable equation $y'=ky(1-y/A)-c$. Discuss what happens for various values of parameters $k,A,c$.
You can go further and further with this model when time permits. Suppose that instead of harvesting a fixed amount $c$, you gauge the population somehow, and harvest $cy$, a fixed proportion of the population. This leads to another separable equation, as well as to a useful discussion, which strategy is better, $c$ or $cy$ in terms of long term profits and in terns the pond sustainability.