$15 is probably too much, but the idea of having no minimum wage is crazy. I'm sure other people will talk about the welfare of workers/availability of jobs. Welfare is adequately ensured by in-work benefits, and the State is perfectly happy to create the illusion of employment in order to please voters. The minimum wage is necessary to protect all taxpayers from the inevitable desire of corporations to get something for nothing.

The simple fiscal argument for minimum wage legislation goes like this. Both the UK and the US have systems of in-work benefits that top up wages to a level sufficient to live on. So from firms’ perspective, when there is slack in the labour market (unemployment) they have little incentive to pay wages high enough to live on. And from workers’ perspective, they have little incentive to demand higher wages, especially if the consequence might be unemployment. If we scrap the minimum wage the co-existence of unemployment with in-work benefits drives down wages to below subsistence level, which is obviously societally untenable. 

As the majority of government tax income comes from households, not corporations/businesses, over time this becomes unsustainable: all unskilled workers become in effect employees of the state, and the higher skilled are forced to subsidise the wages of the unskilled through rising taxes. There would inevitably be calls for in-work benefits to be cut and significantly more stigmatisation of the poor. Unskilled workers would effectively be incorporated into that weird category of "government handout scroungers". So in-work benefits without a legislated minimum wage are fiscally unsustainable and socially divisive when there is persistent unemployment, and unemployment is a necessary feature of the system as it is.