Convergence formula

The convergence formula defines a gradual transition from the current distribution of emissions, to an equal per-capita distribution in the "convergence year".

Note this is not meant to be a prediction, but a possible framework for allocating tradeable emissions quotas under a global climate agreement. See emissions distribution

Adjusting the convergence year

Select "convergence to equal-per-capita from the distribution menu. You can then adjust the convergence year by dragging the blue arrow on the per-capita emissions plot. You should see point where the lines converge move to the left or right accordingly.

Mathematical formula

The allocation for each country (or region) is simply calculated by

(allocation for country "c" in year "y") =
(global emissions budget in year "y") x (share for country "c" in year "y").

The "share" for each country (i.e. the fraction of the global emissions budget) is calculated every year as follows:

For country "c" in year "y" the share "s" is given by:

s_{c, y+1} = s_{c, y} + f_y. ( p_{c, y} / p_{w, y} - s_c,y )

where "p_{c, y}" is the population of that country, "p_{w, y}" is the world population in that year.

f_y is a factor determining the rate of convergence.

A simple linear formula gives

f_y = 1 / ( y_conv - y )
where "y_conv" is the convergence year (which must be agreed in advance).

Alternatively, if you choose the "expert" version, you can experiment with an exponential formula:

f_y = e ^{{q ( t -1 ) }}
where t = ( y - y_start ) / ( y_conv - y_start )
and q is an arbitrary "convergence factor"

Some key features of this convergence formula (with either definition of f_y) are:

The sum of the shares in any year must always be equal to one. To stay within a fixed global emissions budget, you can only increase the share of one country, by decreasing the share of another.
In the convergence year f=1, so shares are then exactly proportional to populations
The formula for allocating shares between countries is independent of the formula for defining the global emissions budget
The share in any year is only dependent on the population in the previous year, and on the share in the start year. A policy agreement based on this formula, would not need to specify any projections of future population changes.
The same formula applies to all countries.

Other options for convergence criteria may also be considered, for example "energy demand" perhaps mixing population, economic activity, natural resources, local climate (feedback?), etc.

Expert options

Only available, if you choose expert version from the top menu

Convergence factor

The linear convergence formula is used by default. This is simpler, but results in a sharp kink for some countries in the start year. The exponential formula allows a smoother transition, although the penalty for higher emissions in earlier years, is a more rapid drop later.

To choose the exponential convergence formula, click the button labelled "EC" at the top right of the plot. You should now see that the lines become more curved, and a new double-arrow (cyan) control appears. Dragging this control to the left or right changes the "convergence factor" as defined above. You can see the exact value of this factor in the pop-up info, the default value is 6.0.

The population cut-off year

It may be argued that this formula provides no incentive for countries to restrain future population growth. Therefore the option is provided the option to freeze the population used for calculating convergence, in a certain year which we call the "population cutoff year". Specifically, p_{c, y} and p_{w, y} in the formula above are fixed in and after the cutoff year (but not before that).

This cutoff year would have to be agreed in advance. On the plot it is shown by the position of the green upward-pointing arrow, which appears if you click the button labelled PC at the top right of the plot.

You may have observed that, when the population cutoff option is enabled, the per-capita emissions do not converge exactly in the convergence year, or appear to diverge thereafter. This is to be expected, since after the cutoff year, countries with growing populations have to divide their share among more people, and so get less for each person, whereas those with falling populations will have more for each person. Note that the per-capita emissions shown on the plot are calculated by taking the shares calculated using the "cutoff year" population as defined above, multiplying these by the global emissions budget, and then dividing by the actual (projected) population (not by the "cutoff" population).