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Given that gold has no end-of-life cycle and cannot be considered waste due to its significant economic value (with one ounce worth around $2,300 USD), it is not something people discard or that becomes part of the waste stream. What would be an appropriate method to measure the GHG emissions for gold sold to a pawn shop, where its value is exchanged for currency?

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This is an extreme case of the classic "more than one life cycle" question in LCA, you have this for all products which are at least partially made from "recyclates", be it paper, plastic, or metals. It is a somewhat philosophical question how much burden should be given to the recyclate at the input of a life cycle, and how much credit to the output. There are different approaches, cut-off (ignore both sides), the famous/infamous Circular Footprint Formula used in Ennvironmental Footprint, and others. The choice of this approach largely influences the result for gold, and you should thus discuss and specify it in goal and scope of your study.
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