For as long as banks have offered computer-based banking, they have agonized over how much — if anything — to charge for it.

Most banks do not charge for account access but do charge for bill payment. As more people adopt Internet bill payment, it is tempting to impose a monthly fee — most banks charge in the $5 range — to boost the Internet division’s bottom line. But fees, of course, turn customers off and probably stifle adoption of a service that is relatively cheap for the banks to offer, and which they therefore want to encourage. On Monday, Amsouth Bank of Birmingham, Ala., announced it had dropped fees for online banking and bill payment customers and will waive such fees for anyone who signs up by April 30.

Indeed, in a clever move, Amsouth is calling the offer “free Internet banking for life,” echoing the memorable “no fee for life” slogan that made the AT&T Universal credit card (now issued by Citibank) one of the most popular products in credit card history. Then again, Amsouth’s offer only applies to people who enroll by the deadline; AT&T was making a commitment to a product.

The fees that Amsouth is suspending are typical in size: $5 a month for online bill payment, $8 a month for Internet banking through Microsoft Money personal financial management software (the software and setup fee of $9.95 is also disappearing temporarily). Though none of these fees is very large, Amsouth estimates that people who take the offer will save $100 in the first year, which does seem like a substantial amount to spend just for the luxury of doing your banking through a channel that is less expensive for the bank.

A good argument can be made that banks should be giving away online banking and bill payment. Customers are smart, and they know the bank saves money each time they use the computer for a transaction instead of a teller — this is why online banking fees chafe them so much.

Moreover, since online banking is hardly ubiquitous, bankers should still be doing everything in their power to nurture its growth. Perhaps this means offering it free for the time being, then slapping on the fees once customers are hooked; perhaps it just means making up the fee income in areas that might seem more rational to consumers.

Citibank, perhaps the only major U.S. bank that charges nothing for online bill payment, no matter how many bills a customer pays, seems to have its price structure figured out logically. It was the biggest pioneer in online banking, and the DOS-based software it used to send out before it introduced Internet banking, Direct Access, may look primitive today but was trusty and efficient.

Direct Access, which was free, has been incorporated into Citi’s Internet banking offerings, which are also free — after all, it wouldn’t be fair to start charging for something that has been given away for years and years.

Sensibly enough, Citi charges customers nothing when they use completely self-service channels. When they use telephone bill payment by speaking with customer service representatives, the fee is $4.95 a month because of the human labor involved.

But in this pricing matter, Citi marches to a different drummer than banks in its peer group. Others seem to be using their pricing structures to try to reward certain types of customers or certain behaviors.

FleetBoston Financial Corp. charges $4.50 a month for online bill payment, except for various categories of private banking-type customers. Wells Fargo & Co. gives everyone two free months of Internet bill paying, and it waives the $5 monthly fee for everyone who always keeps at least $5,000 in his or her deposit accounts. Incidentally, the $5 fee is not for unlimited bill payment — pay more than 25 bills in any given month, and Wells will charge 40 cents per bill for the extras.

Bank One Corp. charges $4.95 a month for online bill payment, plus 30 cents per item after 20 bills have been paid in a month. Bank of America Corp. offers three free months of online bill payments, then charges $5.95 a month to people who do not have certain types of premier accounts.

Despite these fees, banks have had a good deal of success in getting people to use the services. Wells says that a staggering 25% of its customer base does online banking, though it does not say how many pay bills online and how many are paying for this service.

Yet it is tempting to wonder how many more people would migrate to these services if there were no fees. Is there a natural ceiling on the proportion of deposit account customers who will use online banking, and will that ceiling rise as younger and more computer-savvy people become a larger part of the customer base? How much higher would the ceiling be if everyone took Citibank’s approach?

Another consideration for banks is that the nonbank companies offering online bill payment and presentment, such as CheckFree Holdings Corp. and PayTrust Inc., tend to charge more.

PayTrust has two options: $4.95 a month plus 50 cents per transaction, or $12.95 a month with 30 free transactions (and 50 cents for every transaction over 30). CheckFree’s service is free for three months, then $12.95 a month for up to 35 payments (additional payments are $2.95 for each set of 10.) These nonbank companies — which are also big vendors to banks — can usually offer more than banks because they throw in bill presentment as part of the package.

In the early days of Internet banking, most banks were charging people monthly fees just for the privilege of looking at their accounts and moving money around; only a handful (notably Citibank) offered this for free. Today’s paradigm — free account access but monthly fees for bill payment — may be antiquated in an age when even edgier services, such as account aggregation and wireless account access, are being offered free. Banks may find it ever harder to justify these fees.

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