I estimate that the world currently has 62% of the resources that it had in 0 A.D., a fraction that is decreasing by more than 1% annually. By the time it falls below 20%, less than 30 years from now, we will be forced to consume less each year, which will likely result in a decrease in population. If we are unable to increase the world’s supply of non-renewable resources to compensate for what we consume, then to avoid a loss of population we must increase our use of renewable resources.
This can be done by first deciding how much consumption we want to have (proportional to the square root of the population size we want). We could choose, for example, to freeze world consumption at what it will be in 2010, or we could choose to let it grow to twice the 2010 level. Once we are able to supply that amount by renewable resources, we won't be able to increase how much we use without increasing the supply.
Next, we need to decide when we will start increasing the amount of renewable resources that we use; keeping in mind that waiting longer will force us to work faster. If we start at the end of 2010, we will need to add about 1% each year to the fraction of consumption supplied from renewable resources (the “renewable fraction”) while keeping consumption constant at the 2010 level. If we wait until the end of 2020, holding consumption constant at that level, the renewable fraction will be 1.5%. Waiting until 2030 increases the renewable fraction to more than 6%. By the population peak (2037), the renewable fraction will be 9%.
The renewable fraction is proportional to the total amount of consumption that we choose as a maximum. With consumption limited to what it will be in 2010 and assuming we are not using any renewable resources, we will need to get 1% of what we consume from renewable resources in 2011. In 2012, we add 1% to get 2%; the following year, we add the renewable fraction to get 3%, and so on. If we choose instead to double the amount of consumption, the renewable fraction doubles to 2% per year; we are still starting to increase our amount of renewable consumption at the end of 2010, but we are allowing consumption to grow as it has until it reaches the limit (around 2031). Note that for any given year, using the previous year's consumption to determine how much renewable resources we will use will reduce the actual renewable fraction somewhat when we allow consumption to grow to a larger target value.