Carbon footprinting: Building ESG metrics into your financial plans [Part 3]

Part three: Scope 3 planning using Input-Output Analysis (IOA)

Author: Steve Bows, Certified Master Anaplanner and Managing Director, Zooss Consulting.

In the first article of this series, we looked at the difference between Scope 1, Scope 2 and Scope 3 carbon emissions, and how you might incorporate Scope 1 (direct) and Scope 2 (electricity-based) emission planning in your Anaplan models. In the second article, we looked at how Life Cycle Assessment (LCA) data can be used to parameterise planning models that address Scope 3 (embodied carbon) as well.

The LCA approach reflects a bottom-up approach to emissions measurement, and is always the preferred approach where time and money permit. However, there will always be boundary problems inherent in attempting to measure emissions from what may be an infinite supply chain. Your direct suppliers and customers may permit you to perform measurements on their sites, but how about your supplier’s suppliers? Or their suppliers?

Introducing Input-Output Analysis (IOA)

So how do we complete the picture? The answer lies in some ingenious math developed by the Nobel Prize-winning economist Wassily Leontief in the 1940s. The technique is called Input-Output Analysis and uses a ‘wiring diagram’ of the myriad supply chains involved in producing economic output. Once all these supply chains have been mapped — by documenting the bilateral relationships between each industry and every other industry (the ‘intermediate demand’) — a very simple equation will give you a multiplier that can be applied to any change in input variable (e.g. consumer demand) to give you the change in economic output.

In practical terms, macroeconomists use this technique to derive, say, the change in GDP (output) that would result from an increase in government expenditure (part of ‘other final demand’ in the Yucatan example below). And because it is so widely used by treasury economists around the world for scenario planning, there is a good bet that your country’s statistical agency (e.g. the Australian Bureau of Statistics in Australia) goes to a great deal of trouble to collect all of the painstaking detail required to put together this ‘wiring diagram.’ It’s called an Input-Output Table, and they are normally available for the public to download [1].

Here’s the math: the Leontief equation (x=Ly) uses matrix algebra to provide a sum to infinity of all the supply chains represented by matrix T (intermediate demand) in an Input Output table and provides a way of calculating total output (x, aka GDP) from final demand (y). The table below shows a simple example using the economy of the Yucatan peninsula in Mexico, and how this can be divided up into the 5 matrices required for the equation.

Where do you use the matrix T? Well, L = (I-A)⁻¹ and A = Tx^⁻¹, so ultimately L is a function of T. Let's unpack that a bit:

  1. The matrix A is the matrix T (intermediate demand) multiplied by the inverse of x^, where x^ is the diagonalised version of x.
  2. The Leontief multiplier L is the inverse of the identity matrix I (=‘one’ in matrix algebra) minus the matrix A.

That’s a lot of very dense matrix algebra, but nothing that can’t be researched using math and economics textbooks elsewhere.

Environmentally extended Input-Output Analysis

This is all very interesting (if you’re a math nerd!), but what on earth does it have to do with carbon emissions? It turns out that you can ‘piggyback’ on the Leontief equation to transform a producer’s footprint (scope 1 emissions) into a consumption footprint (scope 3 emissions = embodied carbon). The environmentally extended Leontief equation has the consumer’s footprint f=qLy’ where q is the emissions per unit of output for a particular industry and y’ is the final demand from that specific consumer. If that consumer is the company you’re working for, then y’ can be thought of as the company expenditure (e.g. from the P&L forecast), mapped against the industries you have in your IO table. The larger your IO table, the more accurate your footprint will be.

So that’s y’, but what about qL — how do you calculate that? This number is effectively the embodied carbon (in kgCO2e or tCO2e) for each $ of consumption tagged against each industry. Unfortunately it needs matrix algebra again, and if you’ve been thinking about this as you’ve been reading, you’ll have realized this is the tricky bit. Anaplan is not really designed to handle matrices, despite its multidimensional nature, but I have read some great articles on how to get around this if you want to give it a go [2].

Practical uses of IOA in Anaplan

So in practice, you would probably not perform the calculation of embodied carbon per $ in Anaplan (qL=q(I-A)⁻¹= q(I-(Tx^⁻¹))⁻¹) but this is not really a problem as IO tables are published quite infrequently — once a year, or even every two years. You only need to refresh the calculations when you have a new IO table, so you could easily use a tool like MATLAB, or write the calculations in R [3] , and then import the resulting table using CloudWorks. It’s the dollar figures that will keep changing, as your forecast changes, not the emissions coefficient.

Integrating IOA and LCA calculations

We haven’t finished yet! Now we have a complete footprint using IOA, based on our company expenditure profile by industry. But it’s not particularly accurate, as it relies on summary data by industry, and an IO table which may be a little out of date. But what it is useful for is filling in the gaps where we don’t have LCA data. There are a variety of methods in the academic literature to achieve this hybridization, but I like Structural Path Exchange, mostly because it doesn’t require matrix algebra and, hence, we have a fighting chance of doing it in Anaplan. However, explaining Path Exchange would require me to first explain Production Layer Decomposition and then Structural Path Analysis, so I think I will wait until enough people ask me for it. But please do, if you’re interested, I’m very happy to explain it.


It's been challenging to write a condensed synopsis of Input-Output Analysis and its application for carbon footprinting in Anaplan, and I’m quite sure it has been just as challenging to absorb it all! There is a lot to learn and a fair bit of complex math to get your head around. The good news is that you don’t need to perform these calculations too often, so you can ask a mathematician to do them for you outside Anaplan if needs be.

But what you should take away is that there is a way of tagging each dollar of company expenditure with an estimate of the embodied carbon it represents, and that this can be used to supplement the more detailed BOM-based calculations that use LCA data. Combining the two methods gives you both completeness and accuracy in determining your Scope 3 footprint, which in turn completes the overall carbon footprint once you’ve added Scope 1 and Scope 2.

I hope you’ve enjoyed this series and found it thought-provoking at least. If you’d like any further information on research material, or if you’ve been doing similar analysis and would like to share your findings, please comment below. I’d love to hear what you all think in this newly emerging field of Anaplan use cases.


[1] For example, this is the latest set of Australian IO tables.

[2] For example: Matrix Inverse using Optimizer.

[3] For example: Input-Output Tables & the Leontief Inverse in R - Part I.


  • @sjbows Terrific blog, Steve, thank you for sharing your expertise! So appreciative of the effort it takes to do a multi-part series!