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How do I negotiate a lower APR with a credit card company when the credit card issuer will not budge?

-- Leverage and persistence! If your balance is a maximum of 50% of your credit line and you make at least minimum payments each month, then you can remind the issuer that you are shopping the market for a new credit card unless they lower your rate. If the card is maxxed out then consider transferring the balance to another credit card, even if the other card has a similar rate. Believe me the issuer will come calling (or crawling back) with a lower rate to regain your business.