Definition
Constant Elasticity of substitution (CES) function represents CES relationship.
Formula:
Where:
F: utility (consumption) or quantity (production)
α: factor productivity / scaling parameter
x: commodity (consumption) or inputs (production)
β: share parameter. 
(See the note of β for CET function)
ρ: exponent
: CES function with different sigma and rho
-1< ρ < 0 
σ > 1 elastic
ρ > 0 σ < 1 inelastic
When ρ = -1, we have 
and the function becomes linear function with perfect substitution:
When ρ approaches 0, the function become Cobb-Douglas function.
When 
, σ = 0, we get the Leontief function or perfect complements function, that commodities / inputs cannot substitute each other.
Relationship between σ and ρ, shown in a figure:
CES function is:
- linear when ρ = -1
- concave when ρ > -1
Modification of CES function
To make a CES function with decreasing return to scale, we can add a positive constant:
or having an additional input as fixed
or conduct a exponential transformation with γ, 0 < γ < 1
Property
CES is increasing in each of its arguments and concave (strictly only in the case of decreasing return to scale)
See also: