question about optimization

[mhamini]

Optimization
The level of satisfaction an employe has on its job is determined by many factors. Obviously, the salary plays an important role, but also the fact that a work is "clean"(relaxed) or "dirty"(stressful) are also important. Suppose that for a specific employe the level of satisfaction can be expressed by the following utility function: u = √w − 2x, where w
is the wage per hour and the parameter x ∈ [0, 1] denotes the level of "pollutionöf the job: x = 0 corresponds to a "clean"job and x = 1 a "dirty"job. "Dirty"jobs are usually paid better than "clean"jobs. The relation between w and x is given by w = 2+6x.
a) Is this employe more happier in a complete "clean"job or a complete "dirty"job?
b) Determine the level of dirtiness x this employe likes most and determine the corresponding
wage.

 

[steve10]

The utility function becomes

u=√(2+6x)-2x

Now, take the first derivative

((du)/(dx))=(3/(√(2+6x)))-2.

Set ((du)/(dx))=0, we obtain

(3/(√(2+6x)))-2=0,

which produces

x=(1/(24)).

Therefore, the answer to (a) is negative.
He like x=1/24 the most and wage is

w=2+6/24=9/4.

 

[me2please]

complete clean: x=0
wage w= 2 + 6(0) = 2
satisfaction u = √2 -2(0) = √2 ≈1.4142

complete dirty: x=1
wage w=2 + 6(1) = 8
satisfaction u = √8 -2(1) ≈2.8284 -2 ≈0.8284

a) Is this employe more happier in a complete "clean"job or a complete "dirty"job?
Since utility of clean job 1.4142 > utility of dirty job 0.8284, this empolyee is more happier in a complete clean job.