Why is a finite world a problem? I can think of many answers:

1. A finite world is a problem because we and all of the other creatures living in this world share the same piece of "real estate." If humans use increasingly more resources, other species necessarily use less. Even "renewable" resources are shared with other species. If humans use more, other species must use less. Solar panels covering the desert floor interfere with normal wildlife; the use of plants for biofuels means less area is available for planting food and for vegetation preferred by desirable insects, such as bees.

2. A finite world is governed by cycles. We like to project in straight lines or as constant percentage increases, but the real world doesn't follow such patterns. Each day has 24 hours. Water moves in waves. Humans are born, mature, and die. A resource is extracted from an area, and the area suddenly becomes much poorer once the income from those exports is removed. Once a country becomes poorer, fighting is likely to break out. A recent example of this is Egypt's loss of oil exports, about the time of the Arab Spring uprisings in 2011 (Figure 1). The fighting has not yet stopped. 

Figure 1. Egypt's oil production and consumption, based on BP's 2013 Statistical Review of World Energy data.

Figure 1. Egypt's oil production and consumption, based on BP's 2013 Statistical Review of World Energy data.

The interconnectedness of resources with the way economies work, and the problems that occur when those resources are not present, make the future much less predictable than most models would suggest.

3.  A finite world means that we eventually run short of easy-to-extract resources of many types, including fossil fuels, uranium, and metals.  This doesn't mean that we will "run out" of these resources. Instead, it means that the extraction process will become more expensive for these fuels and metals, unless technology somehow acts to hold costs down. If extraction costs rise, anything made using these fuels and metals becomes more expensive, assuming businesses selling these products are able to recover their costs. (If they don't, they go out of business, quickly!) Figure 2 shows that a recent turning point toward higher costs came in 2002, for both energy products and base metals.

Figure 2. World Bank Energy (oil, natural gas, and coal) and Base Metals price indices, using 2005 US dollars, indexed to 2010 = 100. Data source: World Bank.

Figure 2. World Bank Energy (oil, natural gas, and coal) and Base Metals price indices, using 2005 US dollars, indexed to 2010 = 100. Base metals exclude iron. Data source: World Bank.

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