[SDGT]

I don't think they're quite competing with directly with Walmart, but only due to the targeted demographic.

Walmart targets everyone, which usually results in more of a sway towards the lowest common denominator. Target's marketing (Target's target marketing?) is seemingly always geared towards upper middle class, or whatever makes up that income level now. It even shows in their branding. Take a look at how Target specific brands are expressed (Archer Farms, Market Pantry, etc) and compare them to their counterparts over at wally world.

Here's an excellent example of this: Starbucks embedded cafes in Target stores under the Starbucks brand. They did the exact same thing in Walmart, but ran under the Seattle's Best marquee in order to provide the lower priced option.