Who; and on what basis, determines those figures?
I'd prefer precise rules if they exist, over general principles like "US has higher GDP so they pay more".
These are decided by the UN Committee on Contributions.
UN resolution 70/245 of 23 December 2015, determines the contributions of Member States to the regular budget of the United Nations for 2016, 2017 and 2018.
- Decides that the scale of assessments for the period from 2016 to 2018 shall be based on the following elements and criteria:
(a) Estimates of gross national income;
(b) Average statistical base periods of three and six years;
(c) Conversion rates based on market exchange rates, except where that would cause excessive fluctuations and distortions in the income of some Member States, when price-adjusted rates of exchange or other appropriate conversion rates should be employed, taking due account of its resolution 46/221 B of 20 December 1991;
(d) The debt-burden approach employed in the scale of assessments for the period from 2013 to 2015;
(e) A low per capita income adjustment of 80 per cent, with a threshold per capita income limit of the average per capita gross national income of all Member States for the statistical base periods;
(f) A minimum assessment rate of 0.001 per cent;
(g) A maximum assessment rate for the least developed countries of 0.01 per cent;
(h) A maximum assessment rate of 22 per cent;
And UN resolution 55/235 from July 2001 determines the assessment for peacekeeping operations:
“the rates of assessment for peacekeeping operations should be based on the scale of assessments for the regular budget of the United Nations, with an appropriate and transparent system of adjustments based on levels of Member States”.
UN Committee on Contributions is in charge of assessments
The Committee on Contributions advises the General Assembly on the apportionment , under Article 17, of the expenses of the Organization among Members broadly according to capacity to pay. The Committee also advises the General Assembly on the assessments to be fixed for new Members, and on appeals by Members for a change of assessments.
The precise calculations are outlined in annual reports, e.g. 2016 report ("Annex I - Outline of the methodology used for the preparation of the United Nations scale of assessments for the period 2016-2018").
... current scale of assessments was based on the arithmetic average of results obtained using national income data for base periods of three and six years for the periods 2011-2013 and 2008-2013.
The methodology used in the preparation of each set of results took as its starting point the gross national income (GNI) of the States Members of the United Nations during the corresponding base periods as a first approximation of the capacity to pay, and applied conversion factors, relief measures and limits to the scale in order to arrive at the final scale.
The adjustments (once the GNI is normalized to single currency using exchange rates) are mainly affected by debt:
- The next step in the scale methodology was the application of the debt burden adjustment in each machine scale. ... Under this approach, the debt burden adjustment is the average of 12.5 per cent of total external debt for each year of the period (what has become known as the debt stock method), based on an assumed repayment of external debt within eight y ears...
Further adjustment was based on per capita income:
- The next step was the application of the low per capita income adjustment in each machine scale. ... The overall average figures for the current scale were $10,511 for the three year base period and $9,861 for the six year base period, and these were fixed as the starting points, or thresholds, for the corresponding adjustments. The share in GNI of each Member State whose average per capita GNI was below the threshold was reduced by 80 per cent of the percentage by which its average per capita GNI was below the threshold.
Further adjustments were floor rate, max rate and ceiling rate.