# How can I model immigration and xenophobia economically using these equations?

I'm trying to put together a toy model for immigration ($I$), labor prices ($w$), and a measure of xenophobic/nationalistic sentiment ($A$). What I've come up with so far is the following:

1. $$\frac{dI}{dt} = -\alpha \cdot A + \beta \cdot \frac{\frac{dw}{dt}}{w}$$ The rate of change of immigration is negatively proportional to the level of xenophobia/nationalism ($A$) and positively correlated to the percentage change in price $w$ (as companies will push for higher levels of immigration as wage prices rise, e.g. more H1Bs for expensive tech labor, etc.).

2. $$\frac{\frac{dw}{dt}}{w} = \gamma \cdot g - \epsilon \cdot n - \eta \cdot \frac{\frac{dI}{dt}}{I},\;\; n\equiv \frac{\frac{dN}{dt}}{N}$$ The percent change in price is proportional to the growth rate of the economy ($g$) and the rates of increase in the domestic population $n$ and immigration $I$ with $\eta > \epsilon$ since foreign labor is cheaper.

3. $\frac{dA}{dt}$ = some function that increases linearly with $I$ up to some level $I_0$ then increases much faster before plateauing. This represents how xenophobia increases much faster after some threshold.

My questions for you that I was hoping you could help me with:

1. Do these simple equations make sense to you intuitively/economically? Are there any simple changes I can make to improve them?

2. Should I be using percent changes (like $\frac{dw}{w}$ in these models as shown above) or is that just going to be painful when I solve for values of $I$ and $w$ and $A$?

3. Do you have any thoughts on what function might be suitable for $A$? I was thinking of something along the lines of an augmented sigmoid but need something thats less symmetric than that. Would a piece-wise function be difficult to work with?

Thanks!

• To facilitate communication, I would suggest to use $w$ for "labor price", $g$ for the growth rate of the economy, $n$ for the growth rate of domestic population and in general do not use $r,h,g$ to respresent some constant - $r$ is almost always used to represent the real interest rate, $g$ is used for growth rates and $h$ usually stands for some function. Use Greek letters (lower cap usually) to represent constants. – Alecos Papadopoulos Apr 16 '17 at 12:54
• Thanks for the suggestion. I've now made those changes to my quesiton. – Essam Apr 16 '17 at 16:11
• I made a small correction. Another thing: when you write $dw/w$, do you mean $(dw/dt)/w$,? – Alecos Papadopoulos Apr 16 '17 at 16:15
• Yep, changed that too. – Essam Apr 16 '17 at 16:24
• What considering (in your immigration function) host country economic conditions relative to some global economic performance measure. If you can get more specific (looking at immigration from A to B) then you can look at changes in relative economic growth trends. There are tons of things to consider but this seems like a big one that is omitted here. – 123 Apr 17 '17 at 19:48