To take on this issue first we must make some assumptions about economics (These are not moral assumptions):

1.) If a business uses the labor of a citizen, they must pay them the market value of that labor or minimum wage (whichever one is higher) in order for the employee to agree to work for them.

2.) That we are talking about "at will" employees. This means that they can quit or be let go. (In other words their labor is not part of a contract, unlike soldiers.)

3.) That a business must be solvent in order to operate (Making a profit or breaking even).

4.) That a business must incorporate the price of the employees' wages into the cost they charge for their goods or services.

5.) That were talking about legitimate businesses (Not black markets or "under the table" wages).

6.) That we have a progressive tax systems. (That the more money you make the higher percentage of that money you have to pay to the government).

7.) That competition exists, if one person can offer the exact same goods or services cheaper those are goods and services more likely to be used.

8.) That markets are free, people are not forced to use a particular product over the other (I can buy Coke and/or Pepsi).

9.) That supply and demand are dynamic, not static (people will be less likely to buy something the more expensive it is and vice-versa).

10.) That we use goods a services in our lives, that effect the cost of living.

11.) That were in a economy (there is several avenues for work that grant compensation.

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Moral assumption:

1.) That some employees are more skilled than others (due to aptitude, experience, training, etc).

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Now under these assumptions we can see that the company hiring unskilled labor can hire a candidate with more experience or training if they apply, but if you have two applicants for a position who are fresh out of HS with no work experience, it is a toss up. But if we take away minimum wage, the business can hire both for low wage like $2/hour, and simple give them raises for the more productive they are. If there happens to be a employee who never receives a higher wage, they can always apply to another employer, as can their more productive counterpart. The business is able to reward productivity with higher compensation to retain that productive employee. But under minimum wage the company may not be able to hire both, leaving one person unemployed, and the business suffers a opportunity cost if they hired the less productive employee.

Also keep in mind that a lot of goods a services we use are produced by business that employ unskilled employees (Think the mall and grocery store). So if minimum wage is raised so is cost of living for everyone.

Another thing to keep in mind is that business make a higher amount of revenue and hence a higher percentage of their profits will go to taxes, which makes it harder for them to stay in business.

The higher wage a business has to pay an employee of the bat is a higher risk every time the hire a candidate whose market value may be below minimum wage to them. This could lead a company to discriminate against persons of a certain race, gender, creed, origin, etc. In a attempt to reduce risk.

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PS. Read animal farm if you're for keeping minimum wage and pretend you're the horse.