What size should power wires be to run big appliances?
In the UK,

Copper                  Circuit
Wire size      Breaker  Type        Example
-------------  -------  ------      -------------------
1 or 1.5 mm²   6A       radial      lighting circuits.  
2.5 mm²        16A      radial      immersion heater.   
2 x 2.5 mm²    32A      ring-final  13A sockets.   
6 mm²          32A      radial      7 kW electric shower, cooker.  
10 mm²         40A      radial      9 kW electric shower.


A 2 mm² wire is insufficient for a 18A load in the UK. I believe some countries push the limits further - I'd check local regulations.

In the UK, you also have to downrate wire (i.e. reduce breaker size in amps) depending on how the cable is installed and on its length.

            Current Ratings in amps

                     In       Clipped to
Wire Size  In Wall   Conduit  Surface  
---------  -------   -------  -------
  1 mm²       11       13       15
  1.5 mm²     14       16.5     19.5
  2.5 mm²     18.5     23       27
  4 mm²       25       30       36
  6 mm²       32       38       46
  10 mm²      43       52       63