From a pure COST analysis, it's likely proper plates are probably cheaper in the long run... depending of course on the cost of the plates, how often your kids break them and how you wash them.
From an ENVIRONMENTAL analysis, it's a bit more complicated. In the People's Republic of California... mostly because of our STUPID government... water and electricity are in short supply. It takes a lot of water to make paper (and thus paper plates) and typically involves cutting down forests (which adds to global warming, if you believe in that) and they eventurally end up in the landfill which are very near their capacity in CA (we're already trucking trash to Arizona and Mexico, burning fossil fuels to do so). BUT it takes energy and water to wash plates and there are other issues of those plates are plastic.
Me? I don't worry about it. Step one: kick the wackos out of our state government. Maybe we can send our governor to the White House and get rid of him that way?
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