Definition
Expenditure minimization problem (EMP):
Problem of the decision maker:
Here 
is an exogenous set utility level.
It can be solved with KT condition.
We suppose
- u is continuous
- u is locally non-saturated
- constraint set is non-empty
Proposition
From Duality:
If 
is optimal in the UMP when w > 0, then is optimal in the EMP when the required utility is 
. And the minimized expenditure level in the EMP is w.
If is optimal in the EMP when the required utility is 
then is optimal in the UMP when wealth is 
. And the maximized utility level is in the UMP is 
.
Implication: UMP and EMP give the same solution, given the same inputs.