Now I am not an economist, and I am not qualified to speak to the relative merits of each of these models.
However, I am a scientist, and I have do have experience dealing with conditional models. By conditional models, I mean that there are circumstances under which the model applies and explains a lot and circumstances under which it doesn't. I can give you an example. If you work in animal behavior, you learn very quickly that animals -- like people -- respond to incentives. Give the rat a treat, and you can get the rat to do what you want. However, incentive learning does not apply under all circumstances. One example is habitual learning. If you teach a rat that every time it presses a lever that it gets a treat, it will press that lever a lot. However, after the animal has repeatedly done this over and over for a long time, you find that if you remove the reward the animal will just keep on pressing that lever -- even though it gets nothing for it. We say then that the response has become habitual and resistant to reward devaluation. So I could say that incentives matter in learning generally, but there are circumstances where they don't.
This is what I see when I read this debate about the minimum wage. It could be that the perfect competition model functions rather nicely except in the particular circumstance of a small increase in minimum wage. The question then becomes: what is small? Is it 10 cents? Is it a dollar? When scientists create models, we also have to spend a lot of time defining when and where they apply. Under 99% of situations humans deal with, Newtonian mechanics predicts outcomes just fine. It is just situations near the speed of light where relativity becomes important.
I'd say that within a dollar or so, increases in the minimum wage produce employment effects that are generally too small to measure. They also don't seem to produce much of a measurable reduction in the poverty rate, which is not surprising, because so few workers actually earn minimum wage.
We do have a lot of good literature on price controls, though, and the results are generally pretty much what you'd expect. Artificially raise the price, and you get too much supply and not enough demand; artificially lower it, and you get too much demand and not enough supply. Lowering product quality may compensate for some of these problems (employers turn off the air conditioning and end overtime; landlords stop painting and become profoundly disinterested in pest control.) But there is no such thing as a free lunch.
These effects mount over time. Rent control was a great deal in New York in 1955. By 1990, rent controlled, or rent stabilized apartments were generally pretty squalid--do not believe television shows showcasing palatial apartments available for a song. Any apartment that great is occupied by a 90 year old lady who moved in during World War II, and you can bet the landlord has a camera on the door so he can prove that whoever tries to inherit it on joint tenancy did not actually live there. More typical was my ex-boyfriend's rent stabilized deal on the Upper East Side--$1400 a month for two small bedrooms overlooking an airshaft, a stove that hadn't worked since about 1970, and don't forget to step over the dead roaches as you come through the door. The better the deal, the more likely it comes with a "key fee", aka a very large bribe that returns much of the consumer surplus to the landlord. And of course, the owner of such a desireable property can afford to be choosy about tenants, so they tend to go to affluent people.
Rent control/stabilization is also the reason that developers refuse to build housing for anyone poor enough to attract sympathetic politicians to their plight--and thus, in part, for New York's sub-2% vacancy rate. When I was apartment hunting in DC, I surprised the hell out of potential landlords by showing up with notarized copies of my credit report, wads of cash, and checks ready for deposit and first/last rent--the standard procedure for procuring an inexpensive apartment in New York. Apparently, the standard procedure here is to think about it for a few days and then mosey over and sign a lease sometime that month.
But over the short term, price controls can look like they are working (for some values of the word "working"), because price elasticity is almost always much greater over the long run than short term. The real reason that minimum wages haven't been particularly worrisome in the United States is that there hasn't been much of a long run over the past few decades; inflation and GDP growth have eroded its value before it got too onerous. Of course, supporters of the minimum wage are working very hard to change that.
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