The solution to the allocation problem is a part of what is called “carbon accounting”.13 Carbon accounting can provide a system of product-level metrics for carbon content. It starts with a record of direct emissions. To compute indirect emissions, information on the carbon content of inputs is needed as well.
With this information at hand, it is possible to calculate the carbon content of inputs in just the same way as input prices are used to determine product costs. 14 13  See e.g. Stechemesser and Guenther (2012) for a review of the carbon accounting literature or, for a practical introduction, Eitelwein and Goretzki (2010). See also Carbon accounting - Wikipedia.