The biggest problem isn't really lowering overall consumption but lowering peak consumption. There's typically enough capacity during the night, but not on summer afternoons. Some power companies will pay you to install a device to cut off your AC for ~15 minutes during peaks (controlled by them, of course).

Nest has a program that gives the power company a small amount of control, in return for which you get a bill credit (and a free or discounted Nest).

https://nest.com/energy-partners/#rush-hour

And, of course, there's peak pricing, which is more what you're suggesting. But power companies are a regulated monopoly, so there are typically restrictions on how much they can increase the price. If they can lower peak consumption within their regulated prices, they can defer on infrastructure investments and thus save money.

There are various and somewhat complicated electricity regulatory systems in the US, it's not so simple as just raise the price.