So.

Minimum wage allows an economy to achieve the optimum allocation of labour, where demand for labour equals supply for labour. The supply for labour at any wage rate means the worker is willing and able to work at a given wage rate. In equilibrium, D for Labour = S for Labour meaning the optimal allocation of resources are achieved.

It is difficult to set a minimum wage because if the wage is set too low then demand would outweigh supply and there would be a deficit of labour. If it was set too high, more workers would be willing to work than there are jobs available, therefore creating involuntary unemployment.

Minimum wage means people receive a minimum amount to live on. The benefits of it are that businesses can keep costs fairly low, the amount of poverty is reduced as people theoretically earn enough to live on and for employees, being paid more than the minimum wage means that they have a feeling of being valued as labour more. If the minimum wage was to be raised, business costs would rise. Also,is likely that people will demand a higher wage from employers to keep their gap over the minimum wage, increasing business costs again. The increase in costs would be passed on in the form of higher prices on the goods they produce. Therefore would result in higher inflation and peoples spending power reduced. Give the wage price spiral a google, that explains it diagrammatically.

The eradication of absolute poverty through a minimum wage means crime rates will fall as there is less incentive to commit crimes and other behaviour that might be considered a negative social externality will fall, this making the economy overall more productive.

Hope this helps.