Article

# Using mathematics to understand and manage climate change

By Jaime Lezama | September 21, 2020

The Kaya identity, which combines physics, economics and demographics in a mathematical equation, exposes the gap between climate-change ambition and reality.

Created almost 30 years ago by Japanese energy economist Yoichi Kaya, the Kaya identity combines in an elementary mathematical equation physics, economics and demographics. The equation can be used to understand climate change from different angles, offering an important source for interdisciplinary research.

The identity equation starts with the apparently meaningless equality:

CO2 = CO2

In other words, we are simply stating that world CO2 emissions are equal to themselves.

We will leave the left side of the equation untouched and proceed to multiply and divide the right side by a series of wisely chosen factors:

CO2 = CO2 x
 E E
x
 GDP GDP
x
 P P
, where:

E is the world consumed energy, GDP is the world gross domestic product, and P is the world population.

By reordering the terms of the right side, we obtain:

CO2 =
 CO2 E
x
 E GDP
x
 GDP P
x
 P

We succeeded to force the appearance of three factors:

•  CO2 E
is the carbon intensity of energy, the quantity of CO2 released for producing the energy we use.
•  E GDP
is the energy intensity, the energy we use for generating a dollar of GDP.
•  GDP P
is the world gross domestic product per capita.

### A reality check on ambitious targets

In order to keep the global temperature rise to below 2°C by 2100, net greenhouse gas emissions in 2050 would need to be reduced by 40% to 70% compared to 2010 levels. With this in mind, the following exercise considers the feasibility of a 66% emissions reduction by 2050 using the Kaya identity:

 CO2 3
= [
 CO2 E
x
 E GDP
x
 GDP P
x P ] / 3

The first reality check is provided by the United Nations’ prediction that the world population (P) will rise to 9.8 billion by 2050. This represents a 1.3-fold increase compared to the current population.

As a result, the three remaining terms of our equation now need to be reduced by a factor of four (3 x 1.3) in order to achieve the goal of reducing emissions by 66% by 2050.

However, the GDP per capita term is also expected to increase. If we assume 2% growth per year until 2050 (a conservative estimate), in 30 years we will end up with a GDP per capita higher by approximately a factor of two (due to compounding of increases). Keep in mind, though, that growth assumptions may have to be reviewed in light of the COVID-19 health crisis. It is too early at this stage to assess the impact of the pandemic on long-term growth projections for both population and GDP.

The two outstanding terms of the equation – carbon intensity of energy and energy intensity – now need to be reduced by a factor of eight (4 x 2).

It has taken 30 years for energy intensity in Europe to decrease by just 37%. If we assume a similar world pace for the next 30 years, we would have a fall in energy intensity of 30% by 2050. The factor of eight is now reduced to a factor of five (70% of 8).

This leaves us with one remaining lever: carbon intensity of energy. For global CO2 emissions to be 66% lower in 2050, the Kaya identity tells us carbon intensity by this time needs to have reduced by a factor of five, or 80%. Is it possible to cut 80% of our carbon footprint of energy by 2050, when in the last 30 years we were only able to achieve a reduction of less than 10%? Will the decarbonization be rapid enough to counterbalance the economic and demographic growth trajectories?

The transition to a carbon-neutral society is both an urgent challenge and an opportunity to build a better future for all. Yes, the situation is difficult and uncertain. Yes, the situation is only going to get more challenging. But we have choices and there are options. And we have a clear timeframe of the next 30 years in which we need to act. We have to act.

Jaime Lezama, Associate Director, Climate and Resilience Hub, Willis Towers Watson.

##### Author

Associate Director, Climate and Resilience Hub, Willis Towers Watson