# Exponential Income Consumption Curve

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. 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.

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