1
$\begingroup$

I have a max utility function, therefore; U(x,y)= max(2x,y) and I am trying to find the demand function x = x(𝑝x , 𝑝y , 𝑀), note this function cannot be differentiated. I am familiar that the utility function states that it is best to have x=0 and all of the good y, or vice versa. So I've been trying to solve with the budget line making one x=0 and then again y=0 but I am unsure what to do from this point?

Working so far: p1x+p2y=M when y=0 x=m/p1 U= 2m/p1

& when x=0 y=m/p2 U= m/p2

So now I have two equations in terms of U

$\endgroup$
  • $\begingroup$ Yeah, I actually do know what it is :) excuse my non-formal language, I'm asking what the demand function for x is for that given utility function... thanks $\endgroup$ – L W Apr 1 at 10:59
  • $\begingroup$ I get that, but you wrote "it is best to have x=0 and all of the good y, or vice versa." Surely it is very easy to check which one is better for any given M,p1,p2? $\endgroup$ – Giskard Apr 1 at 11:05
  • $\begingroup$ Hint: Does x=m/p1 or x=0 lead to higher utility? $\endgroup$ – Herr K. Apr 1 at 11:17
  • $\begingroup$ I can get the utility of when y=0 and x=0 I'm just getting stuck on showing which has greater utility $\endgroup$ – L W Apr 1 at 11:33
  • $\begingroup$ yep, done that, rather so I have two equations in terms of U $\endgroup$ – L W Apr 1 at 11:41
1
$\begingroup$

From your formula for $x$ when $y=0$, you should be able to find $U$ in terms of $M$ and $p_x$ when $y=0$. Similarly, $U$ in terms of $M$ and $p_y$ when $x=0$. The key then is to find the critical price ratio at which, to maximise $U$, the switch needs to occur from $y=0$ to $x=0$.

Can you take it from there?

| improve this answer | |
$\endgroup$
  • $\begingroup$ Sorry, I'm still struggling with this one, unfortunately, after finding the utility when y=0 and then x=0, U=2m/px and U=m/py respectively. I'm unfamiliar with the term critical price ratio. Thank you for your help so far :) $\endgroup$ – L W Apr 1 at 11:32
  • $\begingroup$ If you equate those two formulae for $U$ you obtain a condition under which utility is the same when $y=0$ or when $x=0$. You can then cancel $M$ and rearrange to obtain what I call the critical price ratio. One side of this ratio, utility is maximised with $y=0$; the other side, with $x=0$. $\endgroup$ – Adam Bailey Apr 1 at 11:46
  • $\begingroup$ Okay so I have 2py=px and then to establish which one provides high utility? how? Or am I supposed to use that in the demand function? sorry for being so lost $\endgroup$ – L W Apr 1 at 12:01
  • 1
    $\begingroup$ The demand function for $x$ will need to be in two parts: if $p_x/p_y\geq 2$ then ... and if $p_x/p_y\leq2$ then ... $\endgroup$ – Adam Bailey Apr 1 at 12:04
  • 1
    $\begingroup$ Yes, that's correct. $\endgroup$ – Adam Bailey Apr 1 at 12:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.