For example, between 1990 and 2005, in the United States, the greenhouse gas intensity of the economy declined at a rate of -1.9% per year, but total emissions increased at the rate of 1.0% per year. Average income is measured as per capita Gross Domestic Product (GDP), in international dollars of purchasing power parity ($PPP). (Note that population times per capita GDP equals GDP.) Greenhouse gas intensity is measured as tons of emissions in carbon equivalents per million dollars of GDP. As exponents of multiplicands are added, relationship among the variables can be simply expressed: the growth rates of the three variable on the left side of the equation sum to the growth rate of the variable (emissions) on the right side. As the table shows, global growth rates for population and per capita income outpaced the rate of decline in intensity—so GHG emissions rose; this is also true of the majority of nations, including the United States. For Brazil, Mexico, Indonesia, Iran, Australia, Spain, South Africa, and Turkey, any improvements in intensity were annulled by increases in population alone. For these countries, population growth combined with per capita GDP growth overwhelmed whatever intensity improvements they achieved. For the Russian Federation and Ukraine, economic contraction following the dissolution of the Soviet Union contributed to decreases in their emissions. Stabilizing emissions would require an accelerated decline in intensity. For the United States, the situation was similar: for emissions in 2000 to have remained at 1990 levels, intensity would have had to decline at the rate of -3.2% per year, rather than the actual -1.9%. Because natural gas produces less total CO 2 per kilowatt hour than coal (at a ratio of about 0.6 to 1 on a Btu basis ), CO 2 intensity in the United Kingdom declined between 1990 and 2000 at a rate of -2.8% per year, and CO 2 emissions declined at a rate of -0.5% per year. Thus equation 3 yields a negative growth in emissions (numbers do not add precisely, due to rounding):
The examples of France and the United Kingdom show that for a period of time, at least, greenhouse gas intensity improvements can be sufficient to absorb growth in population and economic activity, so that actual emissions decline. From 1980-1986, U.S. CO 2 intensity declined at a rate of -3.6% per year, and emissions declined at a rate of -0.5% per year. For most nations most of the time, the combination of population growth and per capita GDP growth has more than offset forces tending to depress emissions, so emissions have increased. For the purpose of thinking about the United States slowing and then reversing its increase in greenhouse gas emissions, the historic trends in population, income growth, and greenhouse gas intensity indicate the magnitude of the challenge: 
To simply stop the growth in GHG emissions, the three factors on the left side of the equation must sum to zero. Assuming that population continues to grow at +0.9% through 2020, and that per capita GDP grows at the rate of +1.8% of 1990-2005, then GHG intensity would have to decline at the rate of -2.7% per year to stabilize emissions. But the U.S. target for the Copenhagen Accord is to reduce GHG emissions for 2020 to 17% below 2005 emissions. The answer is, it would take a rate of intensity decline of about -4.6% per year beginning in 2010, to reach the level of 1,570 in 2020. Conclusion

In the end, the interactions of the variables, population, income, and intensity of emissions ( equation 1) , together with the inexorable force of compounding growth rates over time ( e quation 2 ) are inescapable conditions determining both the risks of climate change and the costs, benefits, and tradeoffs of options for responding.
In the context of climate change and possible responses to the risk associated with it, three variables strongly influence the levels and growth of greenhouse gas (GHG) emissions: population, income (measured as per capita gross domestic product [GDP]), and intensity of emissions (measured as tons of greenhouse gas emissions per million dollars of GDP).
(Population) × (per capita GDP) × (Intensityghg) = Emissionsghg
This is the relationship for a given point in time; over time, any effort to change emissions alters the exponential rates of change of these variables. This means that the rates of change of the three left-hand variables, measured in percentage of annual change, sum to the rate of change of the right-hand variable, emissions.
For most countries, and for the world as a whole, population and per capita GDP are rising faster than intensity is declining, so emissions are rising. Globally, for the variables above over the period 1990-2005, the rates of change (∆) in annual percent sum as follows (numbers do not add precisely because of rounding):
Population ∆ + per capita GDP ∆ + Intensityghg ∆ = Emissionsghg ∆
(+1.4) + (+1.7) + ( -1.6) = (+1.6)
As can be seen, global emissions have been rising at a rate of about 1.6% per year, driven by the growth of population and of economic activity.
Within this generalization, countries vary widely. (Unless otherwise noted, comments about countries refer to the top-20 emitters as of 2005, who accounted for about 75% of world emissions that year.) Between 1990 and 2005, in some countries, including Brazil, Mexico, Indonesia, and South Africa, population growth alone exceeded the decline in intensity. For most countries, and for the world as a whole, per capita GDP growth exceeded the intensity improvement each achieved. Countries for whom intensity improvements were greater than their per capita GDP increases included Germany, the United Kingdom, the United States, France, and South Africa. And both the Russian Federation and the Ukraine, following their economic contractions in the 1990s, posted negative numbers for population, per capita income, intensity, and GHG emissions between 1990 and 2005. Besides the Russian Federation and the Ukraine, only the United Kingdom and Germany reduced their GHG emissions for the period (Germany being helped by reductions in the former East Germany).
Stabilizing greenhouse gas emissions would mean the rate of change equals zero. Globally, with a population growth rate of 1.4% per year and an income growth rate of 1.7% per year, intensity would have to decline at a rate of -3.1% per year to hold emissions at the level of the year that rate of decline went into effect. Within the United States, at the 1990-2005 population growth rate of 1.1% per year and income growth rate of 1.8% per year, intensity would have had to decline at a rate of -2.9% per year to hold emissions level; however, U.S. intensity declined at a rate of -1.9%, leaving emissions to grow at 1.0% per year.
Looking to the future, under auspices of the Copenhagen Accord, the United States has submitted a target of reducing emissions from the 2005 level by 17% in 2020. This would require the United States to reduce the intensity of its emissions by some -4.6% per year during the 2010-2020 decade. This implies that the rate of intensity decline needs to better than double.