I recently came across this paper about minimum wage effects. The paper presents a far more complicated model than the one I used in my previous post, but I did notice the opportunity to use some real data for the methods shown in that post.
The paper presents a graph of the distribution of wages, with data taken from the US CPS (Current Population Survey). See the figure on the right below:
So the curves represent the distribution of the hourly wages of workers in the US. This is somewhat similar to the supply curves discussed in my previous post.
Note that there’s some big differences. For the hourly wage curve, the CPS is asking employed workers “what is your current hourly wage?”. For a firm’s labor supply curve, they are asking all possible employees (including currently unemployed workers) “What is the lowest wage you will be willing to take to work for us?”.
The difference between these two could be shrunk by imagining a hypothetical magic firm which:
- Hired from the population of employed, without regard to what the workers actually do.
- Being hired means that literally nothing changes for the employee on their end (there are many intangibles at play beyond just the wage).
Anyway, we don’t need to get too bogged down on differences. The point is that we want to play with real data and that surveying for a firm’s labor supply curve is hard whereas surveying the current wages is easy.
I went ahead and collected the CPS data and produced my version of the wage figure above:
It may not be exactly the same as the paper’s figure due to some clarity issues about their method, but it matches pretty closely.
Next step is to model the distribution of the wages. In my previous post I used the Half-Normal distribution, but since this is data from all wages, then you’d expect to use the full Normal distribution. Furthermore, you can see that the low end is squeezed whereas there’s a long tail for the higher wages. This means a Log-Normal distribution would be a good fit, though you could argue that some other distribution may be a better fit.
Below are the inverse CDFs of the Log-Normal distributions fit to the wage data:
And here are the fitted distributions overlayed with the real wage data (dotted lines):
It’s a pretty close fit. And now that we have these distributions, we can run the rest of the machinery described in the previous post. We can pretend that this wage curve corresponds to our magic firm’s supply curve. Since we have a differentiable function representing the supply curve, we can next generate a marginal factor cost curve from this supply curve.
Focusing on only the “All Workers” (Black) curve and show it’s corresponding MFC curve, this is what it looks like:
Here’s what it looks like once you zoom into the left half:
And just like in the MW post, we can obtain the distribution resulting from a $15 minimum wage and see how it compares (shown here in the dotted lines):
You could go on further and imagine the firm’s demand curve and infer further from there.
Anyway, I hope this post did a good job of grounding the math in real data (with all the caveats mentioned above).
