Is 22/2 Electrical Wire Too Thin To Carry About 50 Watts For A Distance Of About 50 Feet?

Assuming you are hooking into a 120 volt outlet, there is a problem.  And especially if you are hiding the wire in the wall.  You need the same size wire as the wiring that feeds the existing outlet, probably 14 awg or maybe 12 awg.
You may be able to treat it as an extension cord (not in the wall).  I don't know the code on that.
Anything hooked to a standard outlet can draw as much amperage as the circuit breaker for that circuit allows.  If the wire is not of sufficient size to carry the 15 or 20 amps the breaker will allow, the wire and its insulation can catch fire and set surrounding objects on fire.  The fact that it a small 50 watt bulb may be irrelevant.  

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Short Answer .... Yes it will work (but see NOTE below)
 
Long Answer.....
 
Ohms Law
R = Resistance (ohms)     I = Current (amps)
E = Voltage (volts)           P = Work (watts)
 
R*I=E      I*E=P    P/E=I    E/I=R
 
If using a 50 watt bulb and 120 volts, you are pulling .416 amps.    22/2 has a maximum load of .92 amps (slightly more than double of your load) for 1000 feet.  Though, as freedom says, it might not meet code since there is no safety ground.. Codes vary from place to place. While codes might require it... a safety ground is not really needed for this particular application, and you will find that many incandescent fixtures do not even have a connector for a safety ground.  It would be a very good idea to add a .75 amp/120vt inline fuse to the live (hot) lead, because if it should short out, no breaker is going to be small enough to trip.. Thus the wire could start a fire if it shorts out without a fuse.  Resistance over that distance, within the wire, is a mere .8 ohms, and is negligable.. The light bulb resistance is 288 ohms.
 
If you are using 220/240vt 50 hz electricity, such as in Britain, then it will work even safer, since the current (.213 amps) will be half of that at 110/120 vts.  In that case, use a .5 amp, 220vt fuse.
 
NOTE:  While this WILL work, if you are putting it inside of a wall, changing the fuse may be problematic unless you make some allowance for accessing the wire.  For that reason, If you cannot make such an allowance, then I recommend you use 14ga wire, since if there is a short the wire is heavy enough to trip the breaker (if it is 30 amp or less) before burning the wire in two.
 
Sorry, freedom, the breaker only determines a maximum current that can flow, however the resistance of the load determines what the actual current turns out to be. The load will be the same, whether the breaker is a 10 amp, or 200 amp.
 
 

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