The CES utility function applies to consumer theory the same functional form proposed by K.J. Arrow 1 to describe a system of consumer preferences characterized by an elasticity of substitution between differentiated goods (the function’s arguments) which is constant. In general, this kind of utility function is used in economic theory as a function of the aggregate consumption of a given economy. Analytically, the level of utility U associated to the consumption of the n-goods  expressed by a CES utility function is given by:

Deriving the utility function with respect to the quantity of good i consumed, we obtain the marginal utility with respect to :

The marginal rate of substitution (MRS) between good i and good j is equal to the ratio between the marginal utility associated respectively to good i and good j:

The elasticity of substitution between the two goods σ is equal to the inverse of the elasticity of the MRS with respect to the ratio between the level of consumption of good i and good j . It is also equal to the logarithm of the ratio between the two goods

Divided by the logarithm of the MRS:

As supposed, the elasticity of substitution σ

is equal to a constant term depending on the value of .


1.Arrow, K.J., H.B. Chenery, B.S. Minhas, and R.M. Solow. 1961. "Capital-Labor Substitution and Economic Efficiency," Review of Economics and Statistics 53, (August), pp. 225-251.

Editor: Bianca GIANNINI
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