how to derive marshallian demand functions from leontief preferences?

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?

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

• 1. These are not really Leontief-preferences, the utility just contains the $\min$ operator. Nov 19, 2022 at 13:04
• 2. Have you considered solving the utility maximization problem by looking at the cases $x \geq y$ and $x < y$ separately? Nov 19, 2022 at 13:05
• 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? Nov 19, 2022 at 16:58

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 $$(\frac{I}{p},0)$$.

• 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. Nov 20, 2022 at 12:51
• @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. Nov 20, 2022 at 14:57
• Thanks for the recommendation. Nov 20, 2022 at 17:04