Yes – sort of.
Having the price of gold fixed is what the gold standard is – one unit of currency is equal to and backed by one unit of gold. For example, gold was fixed at $20.67 per ounce until 1933 – that was the price of gold. Assuming that you could redeem currency for gold, the price would stay fixed at that – no buyer would buy for more than that (since that was the price the government would charge) and no seller would sell for less than that (for the same reason).
Of course, real economies rarely work so cleanly. Usually you _couldn’t_ freely transact gold with the government (at least in quantities that would matter) so whatever price gold was fixed at was somewhat meaningless as the government wouldn’t actually give you that price. This meant that the true market price of gold would fluctuate.
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