Before revision:  We study the optimization problem with a utility function of a CARA type, i.e. with negative and positive exponential utility functions (EXPn and EXPp) .
After revision:  We study the optimization problem with a utility function of a CARA type, i.e. with negative and positive exponential utility function. We prove that both cases are connected by a one-to-one analytical substitution and are identical from the economic, analytical, or Lie algebraic points of view .