Fundamental Welfare Theorems of Economics:

First Fundamental Welfare Theorem:

Every competitive equilibrium is Pareto efficient (given that preferences are not (locally) satiated).
 
That is  to  say  a'free market  in  equilibrium is Pareto efficient, provided  the following conditions  true:

1) No externalities
2)  Perfect competition
3)  No transaction costs
4)  Full information

Second Fundamental Welfare Theorem

Under  some (fairly weak) assumptions, every Pareto efficient allocation can be decentralised as a competitive equilibrium (i.e., a price vector can be found such  that  if  the endowments are  set  equal  to  the (initial) Pareto  eficient allocation,  all  agents will demand  exactly these  endowments (which is  the Pareto allocation.) given the prices.