Question

A canonical uality function. Consider the utility function
co-1
u(c)=
1-0
where c denotes consumption of some arbitrary good and o (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>
(because "negative consumption” is an ill-defined concept). The parameter o is treated
as a constant.
a. Plot the utility function for o=0. Does this utility function display diminishing
marginal utility? Is marginal utility ever negative for this utility function?
b. Plot the utility function for o=1/2. Does this utility function display diminishing
marginal utility? Is marginal utility ever negative for this utility function?
Consider instead the natural-log utility function u(c)= ln(c). Does this utility
function display diminishing marginal utility? Is marginal utility ever negative
for this utility function?
d. Determine the value of o (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.)​

Answer 1

Explanation:

where c denotes consumption of some arbitrary good and o (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>

(because "negative consumption” is an ill-defined concept). The parameter o is treated

as a constant.

a. Plot the utility function for o=0. Does this utility function display diminishing

marginal utility? Is marginal utility ever negative for this utility function?

b. Plot the utility function for o=1/2. Does this utility function display diminishing

marginal utility? Is marginal utility ever negative for this utility function?

Consider instead the natural-log utility function u(c)= ln(c). Does this utility

function display diminishing marginal utility? Is marginal utility ever negative

for this utility function?

d. Determine the value of o (if any value exists at all) that makes the general utility

function presented above collapse to the natural-log utility function in part c.