Utility Curves For Multiple Individuals

Previous post in the "Mathematics of Utilitarianism" series: Utility Curves

A central principle in utilitarianism is that all individuals' interests are considered equally- even those that are not human. My utility is worth just as much as your utility or some frog's or even that stop sign on the corner, although the stop sign probably doesn't have any utility since it's, well, a stop sign, which means we don't have to factor it into our models. But we do have to factor in me, you, and some frog. Here's how:


Figure 1 shows the superposition of two individuals' utility curves:


I couldn't resist putting a :) into an equation, but this notation is going to be easier to work with:


This is a mathematical representation of the idea that all individual's interests are considered equally. We could, however, treat the individuals differently by weighting, i.e. by multiplying one (or both) of the individual's utilities by some number (other than 1). For example, we could model the Three-fifths compromise with the equation:


Here, individual 1 is white and individual 2 is black.

We can easily extend this model to the general case where all individuals' utility is considered by using a summation:


Here, n is the number of all individuals. i is each individual's index number. So I could be individual number 6543, you could be indivudal number 2345, that frog could be individual number 7339, etc.

We can include a weighting by writing the summation as follows:

Mathematically, Equation 3 is a special case of Equation 4: where all a_i are equal to 1.

Next post in the "Mathematics of Utilitarianism" series: Total Lifetime Utility