I have been playing a thought experiment with my problem solving students in relation to determining the impact on resources of constant growth. The scenario that I presented was:

*Martha is concerned about the rate at which natural resources are being used and wants some way of being able to calculate when a particular resource might run out. She wants to base her calculations on knowing what the current rate of usage is in terms of units of the resource, the available units of the resource, and the rate of growth for the resources usage. The rate of growth for the resources usage should be fixed at a percentage of usage of the current year's usage.*

*To improve her understanding, she would like to include a rate at which new resources are discovered and to allow this rate to decline over time. Like the growth in resource usage, the rate of resource discovery should be based on the current year's available resources. Her argument is that when the resources are plentiful, the resource discovery would correspond in some way but would decline as resource availability begins to shrink. The decline in the rate of discovery is supposed to indicate the increasing difficulty of discovering new resources.*

*As well as calculating how long a resource will last, she would like a to have a graph that shows each years resource usage, the available resource and the rate of discovery of new resources. She believes this will help her understand better the issues around resource usage.*

To try and solve this problem, I created a spreadsheet that would give me the amount of resource required for each year, the know available resource for that year, and the amount of new resource discovered during that year. I also calculated the declining rate to resource discovery for each year. By providing the initial seed values for the key variables, I can quickly plot what will happen for a resource over a set period (experiments reported here range over 100 to 500 years). That seems reasonable since we have managed to do most of the damage to the environment and diminish the resources in the last 100 years, in fact I would say in the last 60 years (i.e. in my life time).

Before I carried out this experiment, I was aware that a growth rate for resource usage of 1% required only 70 years for the resource requirement to double. That is if you start requiring 100 units of the resource in the first period, you will need at least 200 units by the 70th year, 400 units by the 140th year, and 800 units by the 210th year. With a 2% growth rate in resource usage the requires would double every 35 years. With a 10% growth rate the doubling of the resource requirements is approximately every seven years. Regardless of the rate of growth the result is an exponential growth curve. It is simply the slope of the curve that is the focus of the growth rate required.

It was this exponential growth curve that I hoped I would help my students discover. However, by trying to graph resource availability, I realised that the growth curve didn't tell the story as quickly as the diminishing resource curve. In a journal that I wrote for the students related to the development of my solution, I started with the simple case: a fixed amount of resource consumed at a constant rate. The time until the resource is fully utilised is easy to calculate (i.e. the initial quantity of the resource divided by the annual usage). Add in a simple growth in resource usage (an exponential curve) and it is now lasting less time. The decline in resource availability is dramatic compared to the growth curve. Resource consumption is cumulative while the resource requirement is simply increasing at the growth rate. After two years the available resource has reduced by double the requirement plus the additional requirement of the growth rate so the decline in resource availability is quite dramatic and becomes increasingly dramatic.

Lets use a simple table to illustrate this:

Resource Required | Available | Growth Rate | Time to extinction |
---|---|---|---|

100 units | 10,000 units | 0.00% | 100 years |

100 units | 10,000 units | 1.00% | < 70 years (doubling period) |

100 units | 10,000 units | 2.00% | < 56 years |

100 units | 100,000 units | 0.00% | 1,000 years |

100 units | 100,000 units | 1.00% | < 241 years |

100 units | 100,000 units | 2.00% | < 154 years |

100 units | 1,000,000 units | 0.00% | 10,000 years |

100 units | 1,000,000 units | 1.00% | < 465 years |

100 units | 1,000,000 units | 2.00% | < 268 years |

What this table seems to be showing is that even a very small growth in the resource usage causes quite a significant reduction in how long the resource lasts. The dramatic reduction is caused by the exponential nature of a percentage increase in resource usage. Below is the graph for the last line of this table. What is notable about this graph is the way that it drops away sharply as the resource approach extinction. This is something that people who talk about exponential growth comment on. They say that once 50% of the resource is gone, it only takes one more doubling period for the remainder of the resource to be used. As stated above, for a 2% growth rate in usage the doubling period is 35 years so in those last 35 years, 50% of the available resource is consumed. In the previous 35 years, 25% of the resource was used. So during the early years of a resources usage, there doesn't seem to be a problem. It is much later on that the problem reveals itself and by then it can be too late to find more resource or to find an alternative.

In reality, we often don't know how much of a resource actually exists hence in the original problem it specified the requirement to allow for the discovery of new resources in the model. If we assume that the discovery rate is based on the currently available quantity then what is the impact on resource depletion? What new resource discovery rate is required to ensure that the resource will never run out for the period that we seek to be able to use the resource?

With this approach to modelling, as long as the available resources for a each year are increasing so does the quantity being discovered (i.e. discovered is greater than used). As the required amount approaches the new discovery amount for any given year then the discovery amount begins to decline. In theory modeling a fixed discovery of new resources doesn't reflect either a limited supply but for discovery rates less than the growth in usage demonstrate the impact of a limited resource since once the known available resource begins to fall so does the quantity being discovered. This effect increases the decline in available resource leading to an earlier extinction of the resource.

Let's assume that we have 1,000,000 units of our resource available, we are using it at 100 units per year. and we think we only want it to last 300 years. What discovery rate do we need to ensure that the resource lasts for that period of time? We know from that without an increase in consumption it will last 10,000 years but we are going to have a modest growth of 2% per annum. The graph above showed that this would last for just short of 268 years. Experiments with the model in the spreadsheet show that a 0.1% growth in the available resource pushes the extinction point to 280 years, a 0.2% growth in the available resource pushes extinction to 291 years, and a 0.3% growth the available pushes the extinction point beyond 300 years. That seems achievable (see graph below).

What is dramatic about this graph and all of the graphs including a growth in finding resources where the resource runs out within 300 years, is a rapid decline once the available resource reaches its peak. For this example the peak availability was 1,524,425.95 units at 194 years. By 300 years, the available resource has dropped to 234,066.7 units and would be extinct by year 309.

We started with a healthy difference between the current usage and what was available (10,000 years supply if no growth). What happens if we start with a known 10 years of supply (i.e. a known availability of 1,000 units)? We will continue to use the modest 2% growth in usage so without a growth in supply, the resource will last just over 9 years. We have to find new resources to meet our requirements especially if we want this to last 100 years never mind that 300 years. At a find rate of 11.9992%, the resource will last 101 years as shown below.

Change the growth rate in discovering resource to 12% and the resource almost 385 years That doesn't seem a big difference but it is around this point that we seem to be discovering enough new resource to be able to make the supply sustainable but is it really? Look at how the graph dips as the available resources peak. There is little warning that we are no longer finding resources quick enough.

Reality differs from my simplistic model. Including the decline in the discovery rate does further highlights the problems. The desire to include a decline in the discovery rate is that as a resource becomes scarce, it becomes more difficult to find. The following graph is based on starting values of 10 units available and consuming 1 unit per year with a 2% growth rate in usage and an initial rate of 13.8% for find resources that is declining by 1% per year.

Up to year 279, everything seems fine but by year 349 all the resource is gone. Change the decline rate in finding new resources to 2% and the starting find rate percentage has to be 21% for the resources to last as long. The graph has an even more dramatic shape although the warning of decline in availability comes earlier.

How does reality match these models? These models are very simplistic where the rates of usage of any resource will vary while it is being used and so will the discovery rate. However, we already have evidence of the decline caused by over fishing (i.e. the catch rate exceeding the rate at which fish breed) or the decline in the discovery of gold once the easily accessible gold has been recovered. For renewable resources such as fish, there is potentially a sustainable level of use but other resources, such as fossil fuels, where there is a very low production rate, there may be no sustainable level of use. In fact with most resources, we are only guessing how long they will last.

The problem is that our growth mentality and desire for the latest, drives a growing need to obtain more of increasing difficult resources to find. What adds to the difficulty is that in order to satisfy the increase in the need for one resource (i.e. housing), we are reducing the ability to produce another resource (i.e. food). In some many ways we treat the planet as limitless when it really is a finite resource. What is more our awareness of our rate of consumption has become more obvious in the last 50 years when the period of growth in resource consumption has possibly been at its highest. The questions that we should be asking are whether continued growth is really possible and whether it is possible to build a community that is sustainable?

It isn't a natural disaster or the hand of an unseen god that will destroy us. It is our own greed, our lack of awareness of the limits of our planet, and our fear that others will take what we believe we need that will destroy humanity and the planet that we live on.

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