I think it's misleading to quote a day rate and then work less than a typically expected number of hours. The client will make an assumption that a day is approximately 8 hours and wouldn't be happy if he/she found out that you were only working 4 hours per "day."

I disagree. The client is much more interested in what you get done in a day than how much time you spend doing it.

The client is expecting a certain amount of work done in a day, and is measuring your 'days' against the 'days' offered by shittier programmers.