Contrary to popular belief, Capitalism isn't simply "buy low, sell high". Capitalism is founded upon several principles, including the free-market theory. One of the core principles of free-market theory is that you pay the full, unsubsidized cost of the goods and services you use. Subsidies create an artificial and unsustainable market. If you don't meet that condition, you don't, and can't, have a "market economy" that is central to Capitalism.
The labor a business uses requires a certain amount of money to live and maintain the health and well being of the person supplying that labor. That "minimal living wage", that is, a wage that pays the full cost of the “necessities” of life in that society, varies by location, but it is indeed a real, measurable amount that is NOT dependent upon the lifestyle of any individual, it's calculable based upon what is necessary for a typical family to live and remain as a healthy, productive part of that society, and it is already calculated for all parts of the USA (see related links). If you are paying less than that minimum living wage, then someone else must subsidize your labor. For purposes of this discussion, it doesn't matter who is providing the subsidy, the fact is that your business is being subsidized, and therefore violates one of the core principles of capitalism.
The "X job isn't worth that much" argument is nonsense.
If that job is necessary to the operation of your business, and it can't be automated (or costs too much to automate), then you must pay some person to do that job. Without that labor, you have no business. Therefore, it is inherently worth at least the cost of a "minimum living wage". If the cost of labor (or other costs) forces you to price your product above what people will pay or keeps you from making a profit, then you don't have a viable capitalist business model. That is how capitalism works.
Therefore, paying at least such "minimum living wage", is actually necessary for capitalism to be sustained. Any business that cannot make a profit while paying a living wage is unsustainable as a for-profit capitalist business. For-profit businesses with unsustainable business models should be allowed to fail, to be replaced by better business models. That's part of the fundamental principles of capitalism. Since greed and the "profit motive" have repeatedly proven they will be used to distort and manipulate the market for personal gain when possible, having laws requiring businesses/employ to pay at least that wage, are prudent, if not required, for capitalism to be sustainable.
Shouldn't minimum wage be left up to the market to determine?
A person does not have a right (legally or ethically) to enter into a contract that obligates/requires anyone else to subsidize his life, unless he has explicit permission from the person/entity paying the subsidy. Nor, does an employer have a right to request or expect any employee to be subsidized by anyone else (taxpayers, parents, charities, etc.). Ergo, no employer has a right to offer, and no employee has a right to accept, a wage that requires subsidies from ANY OTHER PARTY, without the consent of the person/entity providing the subsidy. So the market can and will determine how high wages are for any given position, except that the rate must not be less than the "minimum living wage".
Notes for clarification/completeness:
In capitalism, there are times when products are sold below cost, "going out of business", "clearance", "liquidation", "bankruptcy", "loss leaders", etc. but you'll notice that it's the "things" that are sold below cost, not the labor being paid less than a living wage. It's also not the normal course of business, all of those things are temporary in term, and all but "loss leader" items are one-time events when the business is failing/failed. No entity that relies upon subsidies should be operated as a for-profit entity.
Exceptions to the "minimum living wage" rule:
Some percentage of the population is not "able bodied" and/or has another form of disability. As a civilized society, we have said that government and/or charity programs will subsidize the living costs of these people. Similarly, teens/students living with their parents and/or seniors living with their family, may not need a full "living wage". In such cases, IF AND ONLY IF the person/entity providing the subsidy has agreed, then the person receiving the subsidy might be paid a lower wage consistent with their abilities AND the amount of subsidy provided.
Non-profit making activities:
The above is not to say that only profitable activities are valuable or worth pursuing, even in a capitalist society. For example, military, law enforcement, courts, and penal systems are necessary to society, yet they should never be organized to be profitable, as the profit motive calls for expansion and growth, and that is exactly what you don't want to encourage in those systems. It's not even appropriate to design those systems to attempt to recover their operating costs. Indeed, it's vital that such services be available to all, not just those who can afford to pay for them at any given moment.
There are also many types of activities that "serve the public good" which might not be economically viable as "for-profit" entities. Most government services, charities, and activities that serve the interests of society as a whole, are valuable and often necessary, and these activities are best operated as not-for-profit entities that rely upon taxes, fees, donations and/or other forms of subsidies for their continued operation. Not-for-profit entities, because they rely upon external subsidies, should be subject to caps on the maximum amount they can pay their labor, even/especially their most senior executives. That should be part of the terms or "costs" of operating on subsidies.
Expanded discussion and examples:
Labor is not a "thing", it's value isn't dependent solely upon how much you think it contributes to the company. Labor is a person, who must earn enough money to live on, and those costs rise over time, even if the contribution of their labor does not increase. Wages MUST increase with inflation.
Most "things", e.g. equipment, needs maintenance, and those maintenance costs increase over time. The same is true of labor. That the contribution of the equipment to your business does not increase over time, doesn't mean your maintenance costs will remain constant. The same is true of labor.
"Labor is prior to and independent of capital. Capital is only the fruit of labor, and could never have existed if labor had not first existed. Labor is the superior of capital, and deserves much the higher consideration. Capital has its rights, which are as worthy of protection as any other rights. Nor is it denied that there is, and probably always will be, a relation between labor and capital producing mutual benefits. The error is in assuming that the whole labor of community exists within that relation. A few men own capital, and that few avoid labor themselves, and with their capital hire or buy another few to labor for them."
- Abraham Lincoln, 1861 in his first annual address to Congress.
It does not matter that you think the labor is only worth $5/hr. If it costs a minimum of $10/hr x40 hours/wk for "labor" to live, then you have to pay at least $10/hr, no matter how much you think the job is worth. In this regard, it's no different than buying a piece of equipment. You don't get to say "I'm only going to pay $5 for this $10 machine because I think it's only worth $5". You either pay $10, or you don't get the machine to use.
Minimum wage is intended to be a "living wage"
It has been from it's inception. The claim that it's "not supposed to be" a permanent, wage that can support a family is a relatively modern one invented by minimum wage opponents. But even a cursory look at history proves that is has always been intended as a sustainable income for working adults.
"It seems to me to be equally plain that no business which depends for existence on paying less than living wages to its workers has any right to continue in this country. By "business" I mean the whole of commerce as well as the whole of industry; by workers I mean all workers, the white collar class as well as the men in overalls; and by living wages I mean more than a bare subsistence level-I mean the wages of decent living."
- Franklin D Roosevelt, 1933 in his statement on the National Industrial Recovery Act, which established the federal minimum wage in 1933