Lecture Notes Supplement on CES Economics of the Environment and Natural Resources/Economics of Sustainability K Foster, CCNY, Spring 2011

 

 

Note on CES utility functions

Some papers use a Constant Elasticity of Substitution (CES) utility function in the economic theory.  These are common but used not so much for their realism but to simplify some of the formulas.

 

A CES utility has a general form of

,

which looks a bit awful but is "simple" insofar as it makes later formulas so.

 

Since people make decisions on the margin, we look at marginal utility, which is the derivative,

. 

 

At optimum, C is chosen to make the price, P, equal to Marginal Utility (MU), so

,

which is the demand curve.  Plot it and note it has a general shape as:

which looks "like a demand curve should."

 

The elasticity of demand at any point is , and we can find  by taking the derivative of the demand curve, , .  Substitute into the elasticity equation for P and the derivative, to get

.

A constant elasticity, as promised.