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

  • $\begingroup$ 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 '20 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 '20 at 11:05
  • $\begingroup$ Hint: Does x=m/p1 or x=0 lead to higher utility? $\endgroup$ – Herr K. Apr 1 '20 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 '20 at 11:33
  • $\begingroup$ yep, done that, rather so I have two equations in terms of U $\endgroup$ – L W Apr 1 '20 at 11:41

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?

  • $\begingroup$ 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 '20 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 '20 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 '20 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 '20 at 12:04
  • 1
    $\begingroup$ Yes, that's correct. $\endgroup$ – Adam Bailey Apr 1 '20 at 12:31

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