If I had a pound for every time someone assumed I studied maths because I wanted to be an economist without writing essays, I’d have enough to make it worth following the stock market. However, once the indignation fades, I can see the attraction—there are a lot of interesting uses of mathematics in economics. One of the most basic, yet most important, is modelling unemployment.
Unemployment might be caused by too few jobs in an area. Or, it may also be due to a lack of information being provided to employers or potential workers: there may be perfectly good jobs available that qualified workers simply don’t know about. This sort of unemployment is called frictional unemployment.
We split the labour force $L$ into two separate populations: employed ($N$) and unemployed ($U$). We then define $s$ and $f$ to be the rates at which people gain and lose employment:
The rate of change in unemployment is:
\begin{align*}
\frac{\text{d} U}{\text{d}t}&=\text{number becoming unemployed} -\text{number entering work}\\
&=sN(t)-fU(t)
\end{align*}
If we assume that the total size of the labour force is constant, then this leads us to:
$$\frac{\text{d}u}{\text{d}t}+(s+f)u=s,$$
where $u$ is the proportion of the labour force that is unemployed. A lovely first order ODE, which can be solved using the integrating factor method (an exercise left for the reader). Simple enough that even an economist would understand!
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