I have sworned that the bulge warp did this in the negative setting. BUt no it does not but what you can do is do this manually using the Object>Envelope Distort>Make with Mesh.

Your settings will be one 1 row and 1 Column.

It is important when you adjustt the handles to do move them left or rigth just move them vertically straight up and straight down.

Als O suggest doing the whole group of text together.

You can edit the text witht he Object>Envelope Distort>Edit Content or turn on smart guides

You get this

If you do all the text together as one envelope then you should draw a no fill no stroke rectangle around the text so that there is a little more room at the top and bottom of the text or the letters might get clipped by the enevelope

If you want to but you will have to drag the mesh points as well. If you use the handles properly then it will work like I did it.

It is a good idea except for the editing part, but on the other had how hard is tio outline text and make it a brush?

I fhe OP keeps the original text live and outlines a copy of it then I say it is a preety good way o go usingg the width tool might in the long run be the simplest solution.

I would say that the above posters waay is bext.

But I found I had to group the color filled object behind the text with the text in order to get the proper result,

What the poster mentioned is that if you want to Rotate the art after the  Warp Effect you either turn the art with the bulge effect into a symbol then rotate it.

Or Apply a Effect>Distort and Transform>Trandform and only apply a rotation to the art .

If you wish to avoid that then you make the bulge this way

Object>Envelope Distort>Make with Warp>Bulge(with a negative setting group the text and the color filled rectangle behind it before apply the bulge warp.

Here is a video with the various ways describe by James and myself.

http://www.wadezimmerman.com/videos/PinchedType.mov

Edit: It is working properly.