At the expense of

cost or charge:
the expense of a good meal.
a cause or occasion of spending:
A car can be a great expense.
the act of expending; expenditure.
expenses.

charges incurred during a business assignment or trip.
money paid as reimbursement for such charges:
to receive a salary and expenses.

to charge or write off as an expense.
to be expensed.
at the expense of, at the sacrifice of; to the detriment of:
quantity at the expense of quality.
noun
a particular payment of money; expenditure
money needed for individual purchases; cost; charge
(pl) incidental money spent in the performance of a job, commission, etc, usually reimbursed by an employer or allowable against tax
something requiring money for its purchase or upkeep: the car was more of an expense than he had expected
at the expense of, to the detriment of: he succeeded at the expense of his health
verb
(transitive) (US & Canadian) to treat as an expense for book-keeping or tax purposes
n.

late 14c., from Anglo-French expense, Old French espense “money provided for expenses,” from Late Latin expensa “disbursement, outlay, expense,” noun use of neuter plural past participle of Latin expendere “to weigh out money, to pay down” (see expend).

Latin spensa also yielded Medieval Latin spe(n)sa, whose sense specialized to “outlay for provisions,” then “provisions, food,” which was borrowed into Old High German as spisa and is the root of German Speise “food,” now mostly meaning prepared food, and speisen “to eat.”
v.

1909, from expense (n.). Related: Expensed; expensing.
Also, at one’s expense.
Paid for by someone, as in The hotel bill for the sales force is at the expense of the company. [ Mid-1600s ]
To the detriment or injury of a person or thing, as in We can’t speed up production at the expense of quality, or The laughter was all at Tom’s expense. [ Late 1600s ]
see:

at the expense of
go to the trouble (expense)
money (expense) is no object