From the Engel's Law we know that as income increases the share of income spent of foods decreases and the share of income spent on luxury goods increases. I wanted to represent this using Consumer Theory Models. Namely, I wanted to obtain an income consumption curve that was exponential, that is, as income increases the amount of the luxury good increases at a faster rate than income, and the amount of the necessity good increases at a slower rate than income. Like in the following Image.

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I want to reach this departing from the equation Marginal Utility of X/Marginal Utility of Y = Price of x/Price of y. However I haven't been able to find a mathematical example of an Utility Function that after taking the derivatives would allow me to reach an income consumption curve like for example y=x^2. Can you provide a practical example? Thank you.


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One example of the utility function that can give $y=x^2$ as income consumption curve is $u(x,y)=\min(x^2,y)$ enter image description here


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