Nope! Although perfect competition does exist in labor markets (sub contracting) where compensation is entirely production based, it is not in the lower income demographic.
Perfect competition does not exist in any capitalistic market, especially in the higher wage markets. This is because we arn't perfect people. Greed and ignorance and stupidity and vainaty and and a plethora of other human traits prevent pefect markets. If we had perfect markets, then stock prices wouldn't fluxuate so wildly, neither would the price of commidaties or unemployment rates.
In many cases people who get big paying jobs get those jobs not due to their skills or the value of their labor, they get those jobs due to cronyism, nepatisim, percieved and actual power, and of course corruption.
But that doesn't mean that the free market doesn't do a better job of allocating jobs and resources than government does. There is plenty cronyism, nepatisim and corruption in government also, maybe more. This is in addition to the desire of our government officials to create a welfare state in which those who are less in social and economic standing are kept at their low levels so that those who are of higher standing can continue to keep their relative economic wealth and power.
There are a lot of jobs that simply aren't worth minimum wage. There are lots of people who would be happy to take below minimum wage jobs. When I was in college I had a friend who had a minimum wage job. Her job was answer the phone in an office that rarely got phone calls. She did all of her studying and homework while getting paid minimum wage. I would have taken her job in a heart beat for a wage less than minimum wage - and I would have been very happy to do so. Now some of her co-workers had much more difficult jobs, but they also got paid minimum wage. Their salaries HAD to be lower than what they otherwise would have been - because due to minimum wage laws, my friend HAD to be paid minimum wage which had the effect of leaving less money in the budget to pay her co-workers who had more difficult jobs.
And this effect works in reverse also. Cronyism/nepatisim and corruption at higher paying jobs tends to reduce the money that is left over in the lower paying wage pool for people that are not as socially fortunate. Let's say that a company has 10 million dollars that economically they are able to budget for employment compensation, and it takes 201 employees to run the company. If that company pays $5,000,000 to one employee, then the rest of the employees can only average $25,000 each. Now a lot of people would suggest that the $5 million dollar employee is "worth" what he makes (assuming that there is perfect competition created by our quazi-capitalistic society). But does he really bring as much value to the company as the other 200 employees all put together? Maybe, but highly unlikely. I would seriously doubt if that $5 million dollar employee is capable of putting in 80,000 hours of work per week. I would seriously doubt if that top employee has as much capability and skill and knowledge as the other 200 employees all put together. Now if that top employee only made $400,000 (which still puts him in the top 1% of all income earners) then the other 199 workers can have an average wage in excess of $47,000 each. Some people may suggest that the top paying employee is worth much more than a half million - but there are 200 other employees who on average are only making 10% of his wage, and many of them are likely just as smart and capable and motivated as he is - and a few of them would likely literally kill to have his salary.
Our world is full of talented and capable and motivated individuals. For every CEO of a huge company making mega-bucks, there are tens of thousands of people who are just as talented and capable and educated and motivated - they just don't happen to have the right contacts. One of the services that my company offers is a resume service. I can tell you from personal experiance that Ivey League graduates are a dime a dozen. It always amazes me when we have someone come in who has a Ph.D and they are looking to take whatever job they can find, at whatever salary they can find, and the most money that they have ever made is less than what I happen to make.
So why do top CEO's get paid so much if they truely arn't worth it to the company? Because their salaries are determined by a Board of Directors who tend to be part timers. Members of the Board naturally want to make as much as they personally can, and generally it is not accepted by the share holder to pay part time board members more than the full time CEO. So for the board members to increase their salary for political reasons they have to increase the salary of the CEO - ultimately resulting in inflated salaries for all of the top executives and the board. And naturally, when the board is looking for a new CEO, they select a personal friend, who they then justify paying a larger salary to than the last CEO because this guy is "better" than the last guy and thus "deserves" the big salary (the board members would get run-off if they admitted to hiring someone who was addmittedly worse than the last CEO). Mega-salaries tend to be the results of cronyism and corruption much more frequently than a measure of the talent or worth of an individual.
For every famous rock band, there are thousands of garage bands that are just as good. For every "leading man/lady" actor, there are thousands who are just as good trying to scratch their way to the top. Rare tallent is not nearly as rare as we tend to believe it is, nor is it as valuable as our society tends to accept. Just yesterday I was on Bourban Street in New Orleans, on that one street there was probably 50 different performers (excluding the multitude of sex artists of varying skills and talents). Out of those 50 performers, probably 10 were as good as it is humanly possible to be. Out of those 10, probably only one has the desire and is willing to put in the effort to make it big - and even with his/her tallent and desire and willingness, his chances of ever becoming "big time" are next to none unless he has the right contacts. And since the pool of money for paying entertainers is limited, and since we pay the famous entertainers mega-salaries, the people who are stuck on Bourban Street are pretty much economically screwed.
If we really wanted to have government to control wages (which I am in no way recommending) - setting a MAXIMUM WAGE standard would do a lot more for the poor than having a minimim wage.