I understand that quasi-linear functions have a general form

$U(x_1,x_2,...,x_n,y) = f(x_1,x_2,...,x_n) + y$

and that for a quasi-linear function, the income effect with respect to the other variables of the function ($x_1,x_2,...,x_n$) are all $0$, i.e., income has no effect on the consumption of those goods. I also read online that a if a preference is quasi-linear, then indifference curves are parallel.

However, I came across this function, $U(x,y)=e^xy$, where the income effect with respect to $y$ is $0$, when the price of $y$ changed. Is this also a quasi-linear function, since IE is $0$? Plotting the graph of this function, I see that the indifference curves are actually not parallel.

<span class=$U(x,y)$">


The indifference curves are not "parallel", as they are not straight lines. They are however shifted, that is they are supposed to maintain vertical distance regardless of the value of $x$.

The curves you map maintain horizontal distance regardless of $y$. That is because the non-linear variable here is $y$, not $x$. The curves are still shifted, but along the other axis.

Taking the logarithm of the utility function you get $$ x + \ln y $$ which is more clearly quasi linear.

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