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For only max or min problems, I understand we should proceed they are complements but for that type of function, how do we really get demand functions? should we graph but can this be done without a computer?

$$ u\left( x,y\right) = \left( \max \left\{ x,y\right\} \right) ^{2}-\left( \min \left\{ x,y\right\} \right) ^{2} $$

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    $\begingroup$ 1. These are not really Leontief-preferences, the utility just contains the $\min$ operator. $\endgroup$
    – Giskard
    Nov 19 at 13:04
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    $\begingroup$ 2. Have you considered solving the utility maximization problem by looking at the cases $x \geq y$ and $x < y$ separately? $\endgroup$
    – Giskard
    Nov 19 at 13:05
  • $\begingroup$ to be honest I am not good at some parts of micro and I can't understand the nature of some assumptions. so basically I really don't know how to continue. If you don't mind, could you explain what do you mean by solving separately by imposing these restrictions? Can I even form a Lagrange here? $\endgroup$
    – Tatanik501
    Nov 19 at 16:58

1 Answer 1

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The utility function looks like this:

$(big)^2 - (small)^2$

Since $small$ is something that takes away utility, you want $small = 0$. Otherwise, you’re spending some money in getting unhappier, definitely not an optimal bundle.

This implies you wouldn’t consume anything at all from one good.

Therefore, you either spend everything on $x$ or everything on $y$.

  • If you spend everything on $x$:

$U(x,y) = (\max\{x,y\})^2 - (\min\{x,y\})^2 = x^2 - 0^2 = x^2 = (\frac{I}{p_x})^2$

  • If you spend everything on $y$:

$U(x,y) = (\max\{x,y\})^2 - (\min\{x,y\})^2 = y^2 - 0^2 = y^2 = (\frac{I}{p_y})^2$

Now you have to compare the prices of both goods.

  • $p_x < p_y \implies (\frac{I}{p_x})^2 > (\frac{I}{p_y})^2 \implies$ spend everything on $x \implies x^{m} = \frac{I}{p_x}, y^{m} = 0$

  • $p_x > p_y \implies (\frac{I}{p_y})^2 > (\frac{I}{p_x})^2 \implies$ spend everything on $y \implies x^{m} = 0, y^{m} = \frac{I}{p_y}$

  • $p:= p_x = p_y \implies (\frac{I}{p_x})^2 = (\frac{I}{p_y})^2 \implies$ indifferent between spending everything on x and spending everything on y $\implies (x^{m},y^{m}) = (0,\frac{I}{p})$ or $(\frac{I}{p},0)$.

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    $\begingroup$ That is a great explanation and thank you very much. And can I request any website or book to work on those? I appreciate any kind of source and have a nice day. $\endgroup$
    – Tatanik501
    Nov 20 at 12:51
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    $\begingroup$ @Tatanik501 well, the book that is in my Intermediate Microeconomics course’s syllabus is Nicholson Microeconomic Theory 11th edition but I haven’t seen the fancier utility functions (other than the ones that admit the Lagrangian method and the linear/standard Leontief). For these fancier ones, I think for a little bit and use some logic. $\endgroup$ Nov 20 at 14:57
  • $\begingroup$ Thanks for the recommendation. $\endgroup$
    – Tatanik501
    Nov 20 at 17:04

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