How many people have ever lived?  Keyfitz's calculation updated
(done June 18, 1999)
This work was done at the request of Prof. Glen Paige
for his new book, "Non-killing Political Science"

A simple model of population growth is the exponential model where the population at time t, P(t), is given by Cert for some appropriate constants C and r.  This model is broadly accurate, especially when an organism invades a new ecological niche and the environment seems to have infinite capacity for growth.  This model accurately describes the initial growth of a bacterial culture in a fresh dish of growth medium, or the human population as it discovered agriculture.  More refined models are needed as environmental and predatory constraints impinge, but this model works well with most human populations because many human populations rapidly shift from one environmental disequilibrium to another (and thus present a moving target that the environmental constraints fail to effectively constrain).  It is of course never that simple:  Black Death in Europe caused populations to drop dramatically for a few centuries.

Regardless of how one models P(t), the population at time t, P(t) may be used to estimate the total number of humans that have lived in a given time period.  From a time A to a later time B, the integral of P(t) on [A,B] gives the total number of person-years for that time interval.   If one has an estimate of the average lifespan, say 25 years as in Keyfitz's work, the number of people who lived from time A to time B is approximately (1/25) of the integral of P(t) on the [A,B].

Keyfitz assumed that exponential growth occured in various historical time intervals [A, B], but with possibly different constants C and r for each time interval.  He chose C and r to match given values of P(t) at A and B.  This requires solving two equations in 2 unknowns:

• P(A)=C erA
• P(B)=C erB
By standard algebra, P(B)/P(A)=er(B-A) and hence r={Ln[P(B)]-Ln[P(A)]}/(B-A).  Once r is known, C is obtained immediately from C=P(A)e-rA.  Note that Ln is the natural logarithm, the logarithm to the base e.

Keyfitz then integrates Cert on the interval [A,B].  An anti-derivative is (C/r)ert and the definite integral is the anti-derivative at B minus the anti-derivative at A.  The total person-years from A to B is (C/r)[erB-erA].  By algebra, this is equal to (1/r)[P(B)-P(A)].  By substituting the value of r described just above,

the total person-years from A to B is [P(B)-P(A)](B-A)/{Ln[P(B)]-Ln[P(A)]}.

This formula is now applied to a system of time intervals from 1,000,000 BC to the present.  The sum of the total person-years for all these intervals gives the total person-years for all human life.  By dividing by the average lifespan, one obtains an estimate of how many people have ever lived.  The data at ends of intervals are taken from a recent textbook on population, except for the first data point of 2 people at -1,000,000.  That data point was proposed by Keyfitz.

 Year People People-Years Since Previous Data Point -1000,000 2 0 -9000 7,500,000 4.91* 1011 0 300,000,000 7.14 * 1011 1650 507,500,000 6.51 * 1011 1750 795,000,000 6.41 * 1010 1800 969,000,000 4.40 * 1010 1850 1,265,000,000 5.55 * 1010 1900 1,656,000,000 7.26 * 1010 1950 2,513,000,000 1.027 * 1011 1960 3,027,000,000 2.76 * 1010 1970 3,678,000,000 3.34 * 1010 1980 4,415,000,000 4.04 * 1010 1990 5,275,000,000 4.83 * 1010 2000 6,199,000,000 5.72 * 1010
The total of the entries of the last column is about 2,402 billion person-years (2,402,000,000,000).  If one divides by 25 as an estimate of average lifespan, one estimates that 96,100,000,000 people have lived on the earth.