### Utility Functions Close Up

A basic staple of microeconomics is the utility function, which is usually presented to students thus:

LetUrepresent consumer utility as a function of goodsxandy. MaximizeU(x,y) subject toI=p+_{1}xp._{2}yU(x,y) = ln(xy).

Sometimes the textbook writer tries something fancy, like

*U*(x,y

*) =*

_{}*α ln(x) + β ln(y), where*

^{}*α*and

*β*are exponents of any value.

The Lagrangian for this equation is

And the first order conditions are L(λ,x,y) = α ln(x) + β ln(y)+ λ(^{}I—p—_{1}xp_{2}y_{)}

=α/x— λp= 0_{1}

=β/y— λp= 0_{2}

=I—p—_{1}xp= 0_{2}y

These simplify tox* = andy* = , which means thatexpenditureon eitherxorywould always be exactly the same given a constant income; the actual amount consumed would vary inversely with price.

Map of Cobb-Douglas Utility function Click for larger image |

Linear Expenditure Utility function Click for larger image |

An alternative to the Cobb-Douglas Utility function is the linear expenditure function. This modifies the utility function to

*x*,

*y*)=

*α*ln (

*x - x*

_{0}) +

*β*ln (

*y - y*

_{0})

where

*x*

_{0}and y

_{0}are given constants, and

*α*+

*β*= 1.

In this case, the optimal values of

*x**,

*y** are

*p*=

_{1}x*αI*+

*βp*—

_{1}x_{0}*αp*

_{2}y_{0}*p*=

_{2}y*βI*—

*βp*+

_{1}x_{0}*αp*

_{2}y_{0}which can be contrived to alter the shape of indifference curves; say, if we wanted to discuss fuel versus everything else, and then changed the subject to another type of good that is a superior good.

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