How Does Utility Vary with Tip $ for Servers?

In lecture last week, I mentioned my suspicions that tipping is an important part or our self-identity, that "I like thinking that I am someone who tips well." I demonstrated the idea with the hypothetical example of the difficulty one would face with the prospect of placing their tip on a different table, leaving the server no financially worse off but thinking you were a jerk (I know I picked this example up from somewhere else on the blogosphere, but where eludes my memory).

A student wrote me today with this:

While I do think that part of the reason people tip is to make ourselves feel good, I think there is something else going on too. If you have ever spent time with waitresses, a lot of their conversations surround tips and tippers. They feel insulted if someone doesn't tip or justify that the person is a bad individual, and they are happy if people leave them a 'big' tip. Anything in the middle is just routine and not newsworthy.

I don't know if these things things cause = amounts of happiness and anger, but if you are trying to reward a waitress by giving them a tip, although the financial outcome of the deal would be the same if you left your tip on another table she was waiting, would she get the same number of 'utils'? Would she get just as much happiness of getting a big tip and no tip vs. two medium sized tips? Would she wonder what she did wrong that you decided not to tip her? If part of your goal of tipping is signaling that she did a good job serving you, than this aspect of your tip would be lost if you left it on another groups' table.

Now, my wife has 3 years experience as a server, and she insists that for a regular meal order a tip of $1 or coinage actually creates negative utility (disutility) for servers. "The person wasn't just forgetful, absent-minded, or philosophically opposed to tipping, they just choose to tip little)" is the explanation I am given.

So, lets assume both my student and my wife are correct (pretty good assumption in my experience with these two, and my experience as a Bob Evans Bus Boy). If we normalize a $0 tip to 0 utils for a "regular meal order," then utility as a function of tip looks like the following:

where X > $1.

As signalers, what can we learn from this graph.

1. Terrible service warrants a small, but not zero, tip. Preferably coins.
   
2. Diminishing returns set in quickly and significantly. If we want to signal or "goodness" as human beings to servers, we do not get much bang-for-the-buck over the middle range of x. You can maximize your utility by staying on the lower end of the moderate tip and then use the tip-savings to spend on yourself.
3. To make an impression, you probably must go significantly above your regular tip, and yes the waitress will praise you when you are gone.