A canonical utility function. Consider the utility function u c c () , = 1 1 1 where c denotes consumption

A canonical utility function. Consider the utility function u c c () , = −

−

1− 1 1

σ

σ

where c denotes consumption of some arbitrary good and σ (Greek lowercase letter

“ sigma ” ) is known as the “ curvature parameter ” because its value governs how curved the utility function is. In the following, restrict your attention to the region c > 0

(because “ negative consumption ” is an ill-defined concept). The parameter σ is treated as a constant.

a. Plot the utility function for σ = 0 . Does this utility function display diminishing marginal utility? Is marginal utility ever negative for this utility function?

b. Plot the utility function for σ = 1 2/ . Does this utility function display diminishing marginal utility? Is marginal utility ever negative for this utility function?

c. Consider instead the natural-log utility function uc c ( ) ln( ) = . Does this utility function display diminishing marginal utility? Is marginal utility ever negative for this utility function?

d. Determine the value of σ (if any value exists at all) that makes the general utility function presented above collapse to the natural-log utility function in part c.

(Hint: Examine the derivatives of the two functions.)