What graphics card are you getting and what 450W PSU are you considering? Not all power supplies are created equal so links to the GPU and the PSUs you are planning to get would be helpful, as would a budget for the PSU.
Radeon 5570 and the cheapest power supply i can get.
BAD! VERY BAD!
never ever ever ever go for the cheapest unit you can get. I would go for either the Corsair 430CX that twoboxer linked to or the antec EA380D, both of them would have plenty of power for that system and arent likely to explode spectacularly. http://www.newegg.com/Product/Product.aspx?Item=N82E168...
I WILL get the cheapest power supply that will work, I'm low budget, what I'm asking for is a cheap power supply that will work with my computer that is around 450W. I'm not one of those people who has lots of money to dump into their computer, all i care about is getting a power supply that works and is cheap. I'm not going to upgrade again.
The units we are suggesting arent very expensive, $35 after MIR isnt much for a power supply, especially a decent one that isnt going to blow up. If you get that logisys PSU, i doubt your system will survive another year. A super cheap PSU is never a good value, especially one that cheap that shouldnt be more than a 300W unit, its a fire hazard and can kill your system if it goes.
You can put a 1200W unit into a tiny office PC, nothing bad will happen.
PSUs output a constant voltage, the components draw as much power as they need from the PSU, just because you put a 1200W PSU in doesnt mean that the system will be using 1200W, it justs means it would be able to, the system only draws as much power as it needs exactly at that time.
There is no such thing as too much power(within reason, ^1200w example^) but you can have too little or poor power. Just because its cheap is not a good way to go as others are trying to explain, it is best to spend the extra couple of dollors than to have your PSU or entire PC go up in smoke and flames, maybe even harming you or damaging your home. Cheap PSU's normally do not meet there advertised rated power nor do they put out clean power signals like the models that were suggested to you because they use cheap/inferior parts, and both of these factors can affect the longevity of your PC's parts and in turn cost you much more in the long run.
I have to agree with the suggestions hunter315 and Twoboxer gave you, really if you want to any cheaper, your going to get less quality parts and are taking quite a risk, you have been warned, cant say we didnt try.
The Corsair Builder series is a step backwards. Their old CX line was, but the new ones aren't even 80+ certified. In that range, about the best you'll do is the 380W Antec Earthwatts, which is 80+ bronze and runs very cool and quiet.
Edit: and, as they've said, the PSU is NOT the place to go cheap. You risk your system with a chokemax brand like Logisys. At a minimum, look for full range active PFC (no little voltage switch) and 80+ certification.