You're making some false assumptions to begin with. First of all, the economy is not the government's to own or tinker with. Economics itself is the study of the use of alternative resources which have alternative uses (*Thomas Sowell, Basic Economics p1). The things within the bounds of this economy all have alternative uses and the amounts that people set for their labor are based on their abilities to be useful to others.
Basically people's wages are established based on their worth to others. As a medical professional I have a worth to others in this society that commands so much money for minutes of my time. When someone graduates from high school with few skills, their worth to an employer is small. They can demand high wages but without a valid reason they will only get what value they can bring to an employer. If for instance the person is a young healthy strong male, he may only be worth $5.50 an hour to McDonald's because his skills are limited to grilling a hamburger, running the French fryer or filling a bag with the order. However at this time he may be worth $10.50 an hour to a roofer or some other construction company because he is young and strong. To a company looking for a manager to oversee their payroll department he is worth $0.00 because he has no background in that area.
You ask is there no degree of income inequality which would bother you? and I would answer NO! Payment for services is voluntary. People decide the worth of what someone does based on the scarcity of their skills and the demand for those same skills. If their skills are in demand but are also plentiful their value is less than if those skills are limited and in great demand. So the young strong guy will find more for his skills to move objects or do physical labor than would someone with a bad back and congestive heart failure. CEOs, professional athletes, actors, doctors and etc. have the ability to do things that few others can do. CEOs add wealth to companies which are owned by the wealthy and ordinary individuals. Professional athletes have the ability to perform at a physical level that surpasses others and provides many thousands of others with entertainment. Same for actors. Doctors can usually improve someone's health status.
This economy is made up of the millions of transactions that take place daily between individuals seeking to satisfy their needs, whether they are perceived or real needs is irrelevant. To alter what I decide is the worth of something or someone creates a weakness within the economy. It is no longer a free economy. Also to assume that you can wipe out poverty by artificially raising the bottom level (increasing the minimum wage) you only move the bottom up temporarily and once the economy readjusts those persons who you've claimed to help are no better off. And quite often many of them become unemployed in the interim.
Another fallacy is that there are a tremendous number of people out there making minimum wage trying to raise a family. The reality (read what Dr Thomas Sowell has written on the subject of minimum wages and statistics of who earns it -books and short essays) is that of those making hourly wages that would classify them as "living below the poverty level" 70% of them are kids, of the remaining 30%, about 80% of them are adults with second jobs picking up spare money. Of those remaining 6% most move out of that income level within a year as they acquire skills that make them more valuable to employers. Another thing that Dr Sowell has demonstrated using census bureau data is that poverty is not something that is lasting for most individuals within it. Statistics show that it appears we have a consistent percentage of the population living in poverty. However, it has been shown that those in poverty today will not likely be in poverty within 5 years and usually by the time they are ready to retire they have acquired financial security. Dr Sowell has shown that about 20% of those classified as living below the poverty line remain there for more than one decade. And that percentage decreases when viewed for two decades or more.
A final thought. Since president Johnson's war on poverty (initiated in 1964), the number of poor people has remained almost unchanged since 1959 (before the "war" began) through 2002 (39.5 million in 1959 and 31.1 million in 2002 *John Stossel, Give Me A Break p219). The percentage of the population in poverty was decreasing before the "war" began and has increased several times between 1964 and 2002. The greatest amount of wealth in the history of this world has been transferred from the middle and upper income classes to these "poor" and yet we see no end to this war in sight. If tomorrow by government fiat the minimum wage was set at $500 an hour so every person that worked a 2,000 hour year would now be a millionaire, within 12 months when the economy began to finally settle down, a loaf of bread, a gallon of milk and an automobile would be the same proportion of cost greater than a million dollar income then as they are above an $11,000 a year income today ($5.50 x 2,000 hours). And I would venture that you would hear the same people out there whining that it's impossible to raise a family on a million dollars a year, we need to raise the minimum wage.
I wonder how much medical professionals would make without socialized health care programs like Medicaid and Medicare.