To: The Honorable Vanessa Frito Minister of Income Policy Kingdom of Mediocria

From: Math Iz Us, Inc.

Subject: An income policy for Mediocria

As requested, we present below an analysis of some aspects of an income policy for Mediocria. The basic task is to investigate the way in which income distribution will affect the fraction of their earnings that Mediocrian citizens spend. To this end we establish some preliminary results. In what follows, we shall use data provided by the Mediocrian Census Commission and the Mediocrian Bureau of Statistics. For the moment we denote the relevant quantities by letters.

Let P denote the population of Mediocria. The total amount of money available for the Mediocrian government to pay out in salaries each year is I Mediocrian dollars. Consequently the average annual per capita income A of Mediocrian citizens (measured in Mediocrian dollars) is given by: A = I/P.

Denote by f(x) the fraction of the Mediocrian population whose salary is no more than x dollars per year. As P and I are very large, it is an acceptable approximation to assume that the domain of f is the closed interval [0, I]; clearly its range is the closed unit interval [0, 1]. Let mA denote the largest permissible salary that can be earned in Mediocria (thus m denotes what multiple of the average salary the highest salary is).