If one were to look at silver/gold proof coins that are minted by/for/on behalf of the government, one could see that some coins would bear a face value, such as the 1/2 oz $25 American Eagle gold coins.
How do the mint set the value? Surely it cannot be a value plucked from the sky. Furthermore, what does the value imply to the bearer? Is it redeemable for $25 at any time?
How do the mint set the value? Surely it cannot be a value plucked from the sky.
Actually, that's exactly how it is set - plucked from the sky. Not by the mint, though, by Congress (in the case of the US, similarly in other countries). The US mint merely executes orders given by the US Congress and signed into law by the US President. For example, the wildly popular Silver Eagle bullion coin is authorized by the Statue of Liberty - Ellis Island Commemorative Coin Act. The act specifies the denomination ($1 in this case), the metal composition (99.9% pure silver, in this case) and the visual appearance. See description here.
ALL the US coins that are minted - are minted exactly as described and prescribed in the relevant legislation, and that includes the denomination.
Precious metal coins are not intended for circulation, but rather for collectors. Thus, the denomination doesn't correspond to the coins' real (intrinsic) value - the actual value of the metal is significantly higher than the denomination of the coin (with the notable exception of the Canadian $20 for $20 series).
Furthermore, what does the value imply to the bearer? Is it redeemable for $25 at any time?
Yes, they're legal tender. Obviously it would be ridiculous to do that since the metal content worth much more than the denominated value. It has also been set as a legal precedent that payment with these coins cannot be taken at face value, but rather at their real (metal content) value, for the purposes of income tax assessment.
The coins are produced as bullion, not intended for circulation. The fact that 1 ounce is stamped with a 'face value' means the coin is counted as issued money as would other coinage, but with its value over $1100, the $50 is meaningless, in my opinion.
I'd imagine those in favor of a gold standard would encourage the mint to use a higher value, say, $1000/oz, which would put a minimum price on that coin, and in effect, start a shift toward their goal.
In the early 1900's [or even earlier in some countries] Proofs were intended to be struck as concept and served multiple purposes, they helped mints finalize the issues with design; put a new process, etc. These were initially presented to Dignitaries [Kings & Ministers] in those days. To that extent they were carefully prepared, the dies were polished and coins individually struck often more than one time to give the nice look.
Over a period of time, Proofs have become a coin collector’s delight. The value put on them is just a value, so that it's still a coin and not a medal. These are typically called NCLT [Non-Circulating legal Tender].
The mints make a good amount of premium from minting these coins and selling these directly to the collectors. As there is very little profit in striking coins on behalf of Govt for circulation , the mints make a good amount of profits via such collectibles.
Most of the times the value is either traditional or just plucked out of the air or to match the commemorative theme under which it’s released ... say 100 years of something would have a value of 100 on the coin. However the intrinsic value of such coins is much higher.
