# Thread: optimization

1. ## optimization

hello,

i have a little question to ask, so here it is,

"A consumer has £ 60 to spend on purchases of two goods with prices £ 5
and £ 10. If x is the quantity of the first good bought then the quantity of
the second good that can be bought is y = (60 􀀀 5x)=10 = 6 􀀀 0:5x. The
consumer0s preferences are represented by the utility function
u(x; y) = ln(x) + 2 ln(y) = ln(x) + 2 ln(6 􀀀 0:5x)
Find that x which is the global maximum of this function. Justify your
answer."

when i do this question i keep getting a negative answer which seems wrong, people can't buy negative amounts of stuff.

(hello bye the way, its my first post so hello all)

2. Originally Posted by happyhudson
hello,

i have a little question to ask, so here it is,

"A consumer has £ 60 to spend on purchases of two goods with prices £ 5
and £ 10. If x is the quantity of the first good bought then the quantity of
the second good that can be bought is y = (60 �� 5x)=10 = 6 �� 0:5x. The
consumer0s preferences are represented by the utility function
u(x; y) = ln(x) + 2 ln(y) = ln(x) + 2 ln(6 �� 0:5x)
Find that x which is the global maximum of this function. Justify your
answer."

when i do this question i keep getting a negative answer which seems wrong, people can't buy negative amounts of stuff.

(hello bye the way, its my first post so hello all)
Could you please edit this to use only standard ASCII characters, so we
don't have to guess what you are asking.

Use + for addition, - for subtraction, * for multiplication and / for divisions,
^ to indicate raising to a power, and sufficient brackets to make the meaning clear.

RonL

3. sorry about that, i didn't check to see if it came out alright.

"A consumer has £60 to spend on purchases of two goods with prices £5
and £10.
If x is the quantity of the first good bought then the quantity of
the second good that can be bought is y = (60-5x)/10 which can be re written as y = 6- 0.5x.

The consumers preferences are represented by the utility function
u(x,y) = ln(x) + 2 ln(y) which can be rewritten as u(x)= ln(x) + 2 ln(6- 0.5x)

Find the value of x which is the global maximum of this function"

(and i only just noticed the edit button...never mind)

4. Originally Posted by happyhudson
sorry about that, i didn't check to see if it came out alright.

"A consumer has £60 to spend on purchases of two goods with prices £5
and £10.
If x is the quantity of the first good bought then the quantity of
the second good that can be bought is y = (60-5x)/10 which can be re written as y = 6- 0.5x.

The consumers preferences are represented by the utility function
u(x,y) = ln(x) + 2 ln(y) which can be rewritten as u(x)= ln(x) + 2 ln(6- 0.5x)

Find the value of x which is the global maximum of this function"

(and i only just noticed the edit button...never mind)
A global maximum of:

$
u(x)= \ln(x) + 2 \ln(6- 0.5x)=\ln(2x(6-0.5x))
$

Is also a global maximum of:

$
U(x)= 2x(6-0.5x)
$

This is either a calculus type maximum where $U'(x)=0$ at an interior point of the feasible region $[0,12]$ or occurs at an end point of this interval.

Put:

$
U'(x) = 2(6-0.5x)-x = 0
$

this has an extremum at x=6 and the second derivative test tells us this is a local maximum. At the end points of $[0,6],\ U(x)=0$ so $x=6$ is corresponds to the global maximum.
RonL

5. thank you for the help, its always nice to have someone else go over a question, i never thought of just putting the 2 lns together i just differentiated both straight off and went from there. so thank you.