Lecture Notes on Constant Elasticity Functions

1 CES Utility In many economic textbooks the constant-elasticity-of-substitution (CES) utility function is defined as: U (x, y) = (αx ρ + (1 − α)y ρ) It is a tedious but straightforward application of Lagrangian calculus to demonstrate that the associated demand functions are: x(p x , p y , M) = α p x σ M α σ p 1−σ x + (1 − α) σ p 1−σ y and y(p x , p y , M) = 1 − α p y σ M α σ p 1−σ x + (1 − α) σ p 1−σ y. The corresponding indirect utility function has is: V (p x , p y , M) = M α σ p 1−σ x + (1 − α) σ p 1−σ y 1 σ−1 Note that U (x, y) is linearly homogeneous: U (λx, λy) = λU (x, y) This is a convenient cardinalization of utility, because percentage changes in U are equivalent to percentage Hicksian equivalent variations in income. Because U is linearly homogeneous, V is homogeneous of degree one in M : and V is homogeneous of degree-1 in p: Furthermore, linear homogeneity permits us to form an exact price index corresponding to the cost of a unit of utility: e(p x , p y) = α σ p 1−σ x + (1 − α) σ p 1−σ y 1 1−σ The indirect utility function can then be written: