Here's an interesting article. Two economists, Ferreira and Gyourko, looked at a large sample of cities in which the Democratic candidate for mayor had either won or lost by a narrow margin. Then they looked to see if there were important differences in the policies those cities produced. They found none, suggesting that parties really don't matter all that much locally. Electoral competition fosters convergence, running counter to evidence gathered at the state and national level.
This is a pretty interesting and surprising finding, and it might actually be true, but there are a few rather glaring omissions:
1) They don't differentiate between cities with partisan elections and those with nonpartisan elections. (I'm guessing there'd be a much bigger party effect on policy in the partisan election cities.) Shockingly, they were aware of this distinction and just tried to find out the mayor's party affiliation anyway, sometimes using party registration records. Footnote 14 bizarrely suggests that L.A. functionally has partisan elections. (Hahn v. Villaraigosa? And just how many voters knew Riordan was a Republican?)
2) They don't incorporate mayoral strength into their calculations. A weak Republican mayor working with a strong Democratic city council just isn't going to cut city services all that much.