This is an interesting question. Some cities have, in fact, done this- for example, San Francisco, Seattle, and Washington D.C. have all passed laws requiring employers to pay a higher minimum wage than is in effect in the rest of the state. (Many local ordinances of this type will be phased in over a series of years.)
The federal minimum wage, set under the Fair Labor Standards Act, sets a “floor,” but states and some municipalities have gone beyond to provide a higher minimum wage for their residents. For example, the federal minimum wage is $7.25 per hour, Maine’s minimum wage is $7.50 per hour, and the city of Portland has passed an ordinance setting its minimum wage at $10.10 per hour starting in 2016.
However, a new law in Michigan bans municipalities from setting a higher minimum wage than the rest of the state. The reason that a state can prohibit a city from raising its minimum wage is that a state is a sovereign- capable of running itself, making its own laws, and subject only to the limited restrictions of the federal government- but a city is not. A municipality can make some laws of its own, but only over those matters that are allowed by the state.
The Michigan state government has promoted this bill as a way to protect state employers. Read this recent article: https://www.washingtonpost.com/