Definition
Stone-Geary function, or linear expenditure system (LES):
Here 
is the necessary level (subsistence level, minimum level) of consumption for good i.
Each input can be regarded as two parts:
- γ: fixed amount of demand
- x - γ: behave as demand in C-D function.
Cost function of LES
Take a LES utility function as an example:
When we have 
, the (x - γ) part of demand behaves as in a C-D function with constant return to scale technology.
So for (x-γ) part, we have: cost function = u (quantity of utility ) × unit expenditure
From C-D function, we know the unit expenditure function is 
(Here θ is a constant term calculated from A and β. See C-D function for the detailed relationship between θ, A and β), then the cost function for (x-γ) part equals 
.
For γ part, when we have interior solution (x >= γ), we have expenditure as 
So the expenditure function equals
From Shephard's lemma, we can derived Hicksian demand as
If we have the expenditure equals to wealth w: 
, we can solve the indirect utility function u(p, w) as:
Then we can substitute u into the Hicksian demand to get the Marshallian demand as 
Alternatively, we can solve marshallian demand from indirect utility function with Roy's identity.
