Plastic or Paper Bag? — The Problem of Estimating the Ecological Footprint

Paper bag or plastic bag for life — which one is more environmentally friendly?

Many environmentally aware people may ask themselves questions such as: Should I be using a paper or a plastic bag, and if I do choose the plastic option, which kind of plastic bag should I be using, single use bag or bag for life? Similarly, should I buy milk in a glass bottle or in a compound material carton? And is an electric vehicle really “cleaner” than one with an internal combustion engine when the entire supply chain is included? How “green” are products that purport to help protecting the environment really, and by what standard and metric? How can I assess the “greenness” of products in order to make an informed decision? Perhaps I am not alone in my constant struggle with making the correct decisions in my purchasing habits with regards to what is best for the environment or what has the smallest ecological footprint.

Apart from economical constraints, there is a certain degree of freedom in our choices. In all questions and choices, environment conscious consumers are trying to find the correct behaviour that minimises the ecological (or carbon or energy) footprint of their actions. Most of these actions are necessary to maintain our lives such as buying food or using means of transport. However, it is not quite easy to find a simple answer to these questions. This article tries to raise awareness of the complexity in the decision process, what could help to mitigate the situation and offer some help for the individual.

Some time during my university years (late 80ies), when Johann Brait, a good friend of my father and member of the Upper Austrian state government, visited our home, I tried to make a case for the government issuing a law in favour or reusable and recyclable glass bottles. The government member challenged my position by asking if I had taken all into account that influences the environmental footprint of said glass bottles, such as but not limited to the increased weight for transport, the washing process consuming a lot of energy and using strong detergents, etc. Well, I defended my position, glass bottles just “felt” greener as the material, albeit man-made, appears to be more “natural” than plastic, ergo glass bottles must be better. In the end I had to admit that the matter needs a bit more thought.

The lesson for me was that I have to recognise and consider the importance of system boundaries in time and space for any kind of assessment and its result, similar to modelling any system in my professional career. The result of any optimisation problem is a function of the selected contributing factors and the metric applied. What is and what is not included is a design choice; it can even be a political choice such that the calculations yield the desired outcome.

The choice of factors included and components taken into account, the system boundary, matters. In the example with the glass bottles, a bill of materials on a wide scale can be established to see what goes into the life cycle of the bottles and their usage. More in general, having made a choice on the system boundary and a list of what is included, I reckoned at the time that the ecological (or energy or carbon) footprint of an item x_j could obviously be expressed as a linear combination of the footprints (or carbon or energy) of all constituents by their contributions x_i weighted by their fractional contribution w_ij as well as an external quantity b_j:

x_j = sum_ij ( w_ij * x_i ) + b_j , with i != j

or using matrix notation with the item footprint vector X, the weight matrix W and the external vector B:

X = W * X + B

In principle, it appeared obvious (from first year Engineering mathematics) that a solution for all items X can be found by solving the set of linear equations using simple linear algebra:

X = inverse(I-W) * B

The vector X would then contain the ecological (or energy or carbon) footprint of the item of interest as well as all other items involved in the system. The only practical problem is that a linear combination of all constituents and contributing factors or weights has to be found for all items involved, and recursively for all their constituents, too. The scope could become quite wide and escalate, from glass, its ingredients and processes, washing detergent and its impact, etc. Where do you stop? To limit the scope one needs perhaps a minimum threshold for what is included in the assessment.

The choice of the threshold or cut-off line poses another problem: the issue of convergence. As a thought experiment, let’s start with a minimum set of constituents and a high threshold. The system of equations would easily yield a result for the ecological (or energy or carbon) footprint of all components. Let’s lower the threshold and see what the result looks like. It is likely to change but becomes more accurate. As the threshold is lowered even more the solution is expected to converges toward the true values; at some point, the point of diminishing returns is reached when more complexity does not yield any better results with a very small quantifiable error margin.

The above reasoning assumes systems to be in a steady state. There is no notion of time. However, system dynamics needs to be included for one simple reason: the ecological (or energy or carbon) footprint of any constituent, component or production factor varies over time following to certain time constants. Some factors may decrease due to economy of scale, some may increase because of economy of scale, i.e. the sheer quantity of material needed and the impact in the far future, e.g. think of mining Lithium ore in Chile for the global supply of batteries for green-looking electric vehicles. So what sometimes looks innocuous and eco-friendly may turn out to be more damaging at a system level as not all processes scale well.

It is quite difficult to assess the contributions of all components that make up a product or service over time. Some constituents may not contribute much today and fall under the threshold, but they may do in years to come, i.e. they have very large time constants or in other words, the effect becomes visible after a very long delay or lag. Since business have their own planning horizon, the time constants of these cost factors may be much larger than the period taking into account. Consequently, they are considered externalities which have to be picked up by future generations while the profits will go to the current business owners or shareholders.

It does sound like the quadrature of the circle when it comes to estimating the externalities beyond a certain planning or accounting horizon, the residual externalities. Perhaps some better modelling could help, e.g. dynamic input-output modelling that augments the input-output analysis (1) with dynamic behaviour for cost factors and component’s ecological footprint, both as a function of time and scale. While the latter may decrease ecological contribution in short term, it may well increase the impact in the long run. The equations above could be extended with a set of ordinary differential equations with the rate of change \dot(X(t)) being a non-linear function F(X(t)) of the state X(t) and an external excitation function G(t):

\dot(X(t)) = F(X(t)) + G(t)

The result of the integration of these equations would yield a better estimate of the total ecological (or energy or carbon) footprint of any item. This knowledge can be used to factor in estimated residual externalities in the cost of all products and services as contingency, possibly through a taxation system that takes the estimated externalities residuals per type of product into account. For example, if simulations of Lithium production and battery life cycles show future system costs, they have to be factored in now. As such this approach goes beyond market forces which are proportional control mechanisms only (with their tendency to instability). The estimated residual externalities approach incorporates estimates of the future costs based on model predictions (like model predictive control).

Coming back to the original question: paper or plastic bag, which one is better for the environment? The answer is perhaps that it is difficult to say if a product is “greener” than another one. It is the context and the quantity that matters, the system view and the choice of boundaries in the analysis. Nevertheless, it would be nice to get some advice.

Finding a solution to the problem of choice is intractable for an individual; but it could be offered as a service by a global, accredited and trustworthy international organisation based on global and dynamic input-output modelling; these models and processes ought to be peer-reviewed and published. I could imagine some kind of natural language processing being employed at the user interface level: ask a question and state the context. The system works out the influencing factors, uses the comprehensive mathematical models in the background and produce advice using language synthesis. So the answer to the question “Should I take a paper or plastic bag?” could be: “Well, it depends…, 42 perhaps?” or “Why do you need a bag?”.

On top of an advice service, I could imagine some kind of Carbon badge being issued to individuals, households and businesses: The yearly expenditure, e.g. the aggregated shopping list could be used with known model data to calculate a) the total ecological (or energy or carbon) footprint and b), the predicted biggest gains in behaviour. That could be very helpful for all of us.

Recently I learned that the subject of my reasoning on linear combinations has already been developed by Wassily Leontief in the so-called input-output analysis (which is quite obvious for an engineer).

Contributes to the technological foundations for the self-driving revolution at Five, UK. Interested in sustainable economies and renewable energy.

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