What are the effects of an increase in altitude upon ESHP of a turboprop engine? This is one of the questions in the ATP material:


  What effect will an increase in altitude have upon the available
  equivalent shaft horsepower (ESPH) of a turboprop engine?
  
  A) Lower air density and engine flow will cause a decrease in power
  B) Higher propeller efficiency will cause an increase in usable
  power (ESHP) and thrust
  C) Power will remain the same but propeller
  efficiency will decrease


I chose (C) but the correct answer was (A). Could anyone explain why? Isn't the whole purpose of a turbocharged engine to sustain engine power at high altitude (below critical altitude)? Why does lower density at high altitude cause a decrease in power?
You have confused a turboprop engine with a turbocharged engine. They are very different and have very little in common. 

A turboprop engine is powered by a turbine engine geared to a propeller. Turbine engines have less power available at higher altitudes so answer "A" is correct. 

A turbocharged engine is a piston engine which is fitted with a small compressor driven by exhaust gas pressure. The compressor (turbocharger) can increase manifold pressure to restore sea level power at altitude. (Turbo-normalized) A turbo charger can also supply greater than sea level manifold pressure for more power. (turbo-supercharged)

Both types of turbocharged piston engines can maintain full rated power as they climb, but they will eventually reach a limit where full power can no longer be maintained.