UPDATE: Crusoe pdf link fixed

Tyler Cowen posted Saturday arguing that a consistent theory of human welfare would mean that either

  • Rich people really like being rich and so higher taxes will make them much worse off but won’t cause them to loose their ambitions


  • Rich people don’t really care that much about being rich in which case higher taxes aren’t that bad but will really discourage work effort.

It seemed to me the standard cardinal approach to human welfare could easily produce someone who didn’t care much about being rich and who wasn’t that impacted by taxes.

However,  I wanted to build a model to see how that panned out. So I modeled a single person, Crusoe, who pays taxes and receives government services. He decides how much to work and because I created his utility function we can see how damaged he is from taxation.

Here is a snapshot of the results



The take away relative to Cowen’s post is that Crusoe is relatively insensitive both to the tax rate when it comes to labor supply and utility. In fact a 35% tax rate hardly makes a ding in his utility.  So we know that such people can easily exist. Whether they do exist is another matter.

A few things that I found out from playing with the model (but perhaps should have already known).

  • Without government services, taxes or wages make no difference in Crusoe’s labor supply choice. That is, if Crusoe has to work for everything he consumes then the substitution and income effects exactly cancel out. This extends from the log utility function.
  • Government services, however, throw a wrench in things. Now Crusoe can kick back and still have some guaranteed income. He takes advantage of that.
  • The tax and government services effects interact. The more government services the more taxes effect labor supply. The higher taxes, the more government services effect labor supply.

Of course, Cursoe is about as simple of a guy as you can get. I’d like to expand him out with a more general utility function and perhaps more complex tax code as well as an additional person to look at specific efforts to counteract inequality.

One cute thing to note is that Crusoe does indeed have a Laffer Curve and with the default settings it tops out at tax rates somewhere in the high 60s. The Laffer Curve peak will move as the ratio of government services to the wage changes.

I embedded Crusoe into a pdf which allows me to share him on the blog. You can play around with the model here.

Now for the nerdier among you here are some details on Crusoe

  • He has log utility which of course satisfies all of the derivative conditions
  • His utility is additively separate in consumption and leisure
  • He takes the real wage and government services as given
  • His utility is scaled up by a factor 15 to make the scale on par with hours


Note: Leisure preference is how is much weight leisure has in Crusoe’s utility function. Without a weighting he will spend much more time working than people really do. This is impart because he doesn’t need any down at all. In the model the only reason he stops working is because he enjoys leisure. He could work 24 hours a day if he wanted and not suffer any loss in productivity. This, of course is unrealistic.

You can adjust this to see how people who tend to work more would respond to taxes.