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).

A benchmark demand point with both prices equal and demand for y equal to twice the demand for x. Find values for which are consistent with optimal choice at the benchmark. Select these parameters so that the income elasticity of demand for x at the benchmark point equals 1.1. 3. Consider the utility function: U(x,L) = (αLρ +(1−α)xρ)1/ρ

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