IPAT  A mnemonic that crops up from time to time in discussing ecological impact  is an attempt to remind people that pollution or extraction are complex things and, in doing so, simplifies things. The mnemonic describes the equation:
I = P * A * T
Where I
is an ecological impact of some sort; P
is the relevant
population; A
is a measure of the affluence of the individuals in
the population; and T
is the contribution that relevant technology
makes to the impact. The terms are not defined, so they can be, and
often are, interpreted in many ways. To some extent, the connection
is obvious, people do something, that something invokes some
technology, or maybe group of technologies, the technology emits or
extracts according to some rule relating to whatever the 'something'
is, and the more people there are the more impact there is. Simple.
And misleading.
Tim Jackson, in the book Prosperity without Growth^{1}, has a page describing IPAT where he takes figures from 2007 and 1990 and works out total CO2 emissions. His equation is:
CO2 = P * average income * average CO2 over the economy
This is reasonable given that he is using published averages, and he
calculates a percentage rise in CO2 emissions between 1990 and 2007
and is pleased to see that his result agrees with other published
results. So far, so good. However, if we write the equation so that
the calculation of the averages is made clear (with $
representing
the total value of the economy) we get:
CO2 = P * $/P * CO2/$
The P
and $
terms cancel out and we are left with the identity
CO2 = CO2
. Suddenly we no longer have the population or the
economy  all we have to do is measure impact.
This is not new. In the early 1970s, Paul Ehrlich and John Holdren put up the IPAT equation, partly to emphasise the involvement of population growth. At that time, Barry Commoner had been working on the contribution of technology to various pollutants and his response was to emphasis, in a number of properly worked scenarios, that the cancellation of terms did indeed work, and all that was necessary was to measure the pollutant. There followed an academic dog fight and the supposed clarity of IPAT became distinctly murky^{2}.
The problem is that the terms of the equation are interdependent, and the equation itself gives no room for identifying these dependencies. This means that a change in one may have an unexpected change in another, and that there is no help in analysing a problem.
Let's look at some possible examples:
 The
P * average income
focuses on population, though by itself it tells us nothing about pollution levels. A population increases in poor countries will not generate demand for technology based goodies and the impact will not change. A population increase in a rich country will have the opposite effect.
 The
average CO2 over the economy
term focuses on relative decoupling, where efficiency improvements result in less CO2 per GDP $. However, a technology change that uses renewable energy, or a process change that uses less energy overall, may result, through the Jevons Paradox, in an increase in production and demand, leading to increases in supply from polluting companies as they try to match the apparent opportunity.
 We don't know if the economy is demand or supply driven, so we don't know what buttons to push if we want to reduce output.
 The equation structured around a static view of economic indicators. It does not provide a basis for a model that deals with dynamic aspects of markets or interpretations of GDP. There is nothing that references societal structures  society as a medium of change.
Above all, the global view that this encourages tells us nothing about which parts of the world we have to focus on. To be fair to Tim Jackson, having described the IPAT equation and given us a helpful example, he then proceeds to ignore it. Life is far too complicated to do otherwise.

This was the first edition. In the second edition Jackson has removed his explanatory sidebar, but retains a position that the IPAT identity can provide a rule of thumb in some circumstances. ↩

This history, and the related multiple interpretations of the equation, is described by Martin Chertow in The IPAT Equation and its Variants, MIT 2001 ↩