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A consumer has the utility function $u(x_1; x_2) = \min(x_1; x_2) + 5 \max(x_1; x_2)$. Find its Walrasian demand $x^*(p; w)$.

I've tried searching it up when we have two Leontief functions summed together but to no luck. If this was a simple case we'd just have $x_1=x_2$ and substitute it to the budget constraint to find the demand for both. Should we take each function separately or as a whole together? Kinda lost here.

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  • $\begingroup$ You could start by drawing indifference curves for the utility function given by $v(x_1,x_2)=\max\{x_1,x_2\}$. This might give you the crucial insight to solve your problem. $\endgroup$ – Michael Greinecker Nov 29 '20 at 20:47
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    $\begingroup$ Hint: $\min \{x,y\} + \max \{x,y\} = x + y$. $\endgroup$ – Walrasian Auctioneer Nov 29 '20 at 21:02
  • $\begingroup$ I understand that the indifference curves together would sort of form rectangle shapes but honestly even with the hints I'm kinda lost. It's my first time seeing something like this and don't know where to start. If you could give one more hint I feel like I'd be able to start working on it. $\endgroup$ – Ana Ellis Nov 30 '20 at 11:41
  • $\begingroup$ Can you draw a single indifference curve? $\endgroup$ – Michael Greinecker Nov 30 '20 at 12:24
  • $\begingroup$ Is the solution supposed to be something like this? drive.google.com/file/d/1jQD67U8orP0oFmvqRL511vbSNv9TTRPn/… $\endgroup$ – Ana Ellis Nov 30 '20 at 14:01
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Why don't you just plug in some values for $x_1$ and $x_2$? Start with something like

$x_1=1, x_2=1$ and find utility $u(1,1)=1+5*1$.

Then increase $x_1$ or $x_2$ and let

$x_1=2, x_2=1$ and find utility $u(2,1)=1+5*2=11$, and

$x_1=1, x_2=2$ and find utility $u(1,2)=1+5*2=11$.

The utilities of both bundles are identical, and the goods seem to be substitutable. In fact for any $x_1<x_2$ you will find $u(x_1,x_2) = x_1 + 5 x_2$, and, similarly, for any $x_1>x_2$ you will find $u(x_1,x_2) = x_2 + 5 x_1$.

Now, just for now, assume both goods have the same price $p_1=p_2=1$ and you have a budget of $w=3$ such that you can buy both bundles giving utility 11 up there. You can see that it would be better to put all your money in one of the goods and get $u(3,0)=u(0,3) = 0 + 5*3=15$. Now think about different prices and a general income $w$. Does it again make sense to buy only one of the goods? Is it the cheaper one?

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