At a premium

a prize, bonus, or award given as an inducement, as to purchase products, enter competitions initiated by business interests, etc.
a bonus, gift, or sum additional to price, wages, interest, or the like.
Insurance. the amount paid or to be paid by the policyholder for coverage under the contract, usually in periodic installments.
Economics. the excess value of one form of money over another of the same nominal value.
a sum above the nominal or par value of a thing.
the amount paid to the lender of stock by the borrower, typically a short seller.
the amount the buyer of a call or put option pays to the seller, quoted in dollars per share of stock.
a fee paid for instruction in a trade or profession.
a sum additional to the interest paid for the loan of money.
of exceptional quality or greater value than others of its kind; superior:
a wine made of premium grapes.
of higher price or cost.
of or relating to premiums:
to work in premium sales.
at a premium,

at an unusually high price.
in short supply; in demand:
Housing in that area is at a premium.

noun
an amount paid in addition to a standard rate, price, wage, etc; bonus
the amount paid or payable, usually in regular instalments, for an insurance policy
the amount above nominal or par value at which something sells

an offer of something free or at a specially reduced price as an inducement to buy a commodity or service
(as modifier): a premium offer

a prize given to the winner of a competition; award
(US) an amount sometimes charged for a loan of money in addition to the interest
great value or regard: to put a premium on someone’s services
a fee, now rarely required, for instruction or apprenticeship in a profession or trade
at a premium

in great demand or of high value, usually because of scarcity
above par

n.

c.1600, “reward given for a specific act,” from Latin praemium “reward, profit derived from booty,” from prae- “before” (see pre-) + emere “to buy,” originally “to take” (see exempt (adj.)). Insurance sense is 1660s, from Italian premio. Adjectival sense of “superior in quality” is first attested 1925, originally in reference to butter.
At a higher price than usual owing to scarcity; also, considered more valuable, held in high esteem. For example, Since that article came out, the firm’s stock has been selling at a premium and Space is at a premium in most stores. This idiom uses premium in the sense of “bounty” or “bonus.” [ Mid-1800s ]
Also see: put a premium on
see:

at a premium
put a premium on