For a low power, embedded controller, would a PIC be cheaper? It looks like the AVR chip would cost more and require more support ICs, like $5 compared with $2.
The answer is...'depends'.
ATtiny can be had down to under your target $2 mark for singles. I generally don't pay more than $1.50 for ATtiny85 and they can be programmed with Arduino IDE, if you only need a few IO pins. The only external support components are the pull-up resistor on the reset and a decoupling capacitor on power, and both of those are optional.
ATmega328 you can buy for $2-2.50 in singles, and the same rules for support circuitry apply if you're talking about keeping power consumption down and running it at 8MHz on the internal clock.
If you find the right eBay seller you can get a full Arduino pro micro clone (or mini, they don't follow naming conventions I've found) for very similar money which includes everything - fit headers and plug that in as the brains of your project and you're set to go.
The tiny x41 series are very capable, and not much more than an 85
The atmega328p is the low power variant. It might be more expensive. Pic might have some cheaper low power processors.
i'm only making a few, so I guess the power demands are more important than the cost. I was just curious. it would be annoying to change the batteries on a sensor more than once a year.
Andy, typically when people mention the 328 they are actually referring to the 328p. I've never used Pics but if they end up being the best for your project then cool.