Lecture Notes on Constant Elasticity Functions

Lecture Notes on Constant Elasticity FunctionsThomas F. RutherfordUniversity of ColoradoNovember, 20021 CES UtilityIn many economic textbooks the Constant - Elasticity -of-substitution (CES) utility function isdefined as:U(x, y) = ( x + (1 )y )1/ It is a tedious but straight-forward application of Lagrangian calculus to demonstrate that theassociated demand Functions are:x(px, py, M) =( px) M p1 x+ (1 ) p1 yandy(px, py, M) =(1 py) M p1 x+ (1 ) p1 corresponding indirect utility function has is:V(px, py, M) =M( p1 x+ (1 ) p1 y)1 1 Note thatU(x, y) is linearly homogeneous:U( x, y) = U(x, y)This is a convenient cardinalization of utility, because percentage changes inUareequivalent to percentage Hicksian equivalent variations in income. BecauseUis linearlyhomogeneous,Vis homogeneous of degree one inM:V(px, py, M) = V(px, py, M)andVis homogeneous of degree -1 inp:V( px, py, M) =V(px, py, M) .Furthermore, linear homogeneity permits us to form an exact price index corresponding tothe cost of a unit of utility:e(px, py) =( p1 x+ (1 ) p1 y)11 The indirect utility function can then be written:V(px, py, M) =Me(px, py)1 Conceptually, this equation states that the utility which can be realized with incomeMandpricespxandpyis equal to the income level divided by the unit cost of utility.

Lecture Notes on Constant Elasticity Functions Thomas F. Rutherford University of Colorado November, 2002 1 CES Utility In many economic textbooks the constant-elasticity-of-substitution (CES) utility function is defined as: U(x,y) = (αxρ +(1−α)yρ)1/ρ It is a tedious but straight-forward application of Lagrangian calculus to demonstrate ...

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