I am studying the basics of statistical mechanics and Boltzmann distribution.

I tried to use the idea to find natural income distributions, through the method of maximization of probability using Lagrange multipliers, where:

average energy => income per capita
number of particles => number of people
Energy levels => income ranges

The total population, income per capita and income ranges are fixed. There is a minimum and a maximum income. The interval between ranges are all equal.

When the income per capita is below the average between max and min income the curve seems realistic.

When it is equal to that average, all ranges have the same number of people, and the curve is linear.

But when it is above the average, the distribution is odd because there are many more people in the high income levels, but I don't think it is impossible. At least it is not contradictory. The problem is that in this case, one of the Lagrange multipliers (usually called $\beta$), is negative.

In statistical mechanics $\beta$ is associated to the inverse of absolute temperature. So it can not be negative. But on the other hand, there is nothing in the method of Lagrange multipliers that obliges it (as I can see).

The graph is shown below.

So, is the fact that $\beta$ must be positive a specific feature of energy configuration of particles, or has it a broader interpretation? .negative beta - income distribution