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Let $u(x,y)=f(x)+y$ be a quasilinear utility. Now we rotate it by 45 degrees, (such that the $x-$axis becomes the direction of $(1,1)$)

$v(x,y)=f(x-y)+x+y$. Is $v$ also a quasilinear utility? What is its name? It share many similarity with quasilinear utility such as parallel indifference curves. Being a rotation of quasilinear utility seems like the iff condition of parallel indifference curve.

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    $\begingroup$ It's not clear what you mean by rotation by 45 degrees. $u$ has two arguments, so in which direction are you rotating it. In any case, a (differentiable) utility function $g(x_1,\ldots,x_n)$ is quasi-linear in $x_i$ if and only if $\frac{\partial g}{\partial x_i}$ is a constant. We have $\frac{\partial v}{\partial x}=f'(x-y)+1$, which is constant only if $f$ is linear. A similar point holds for $y$. $\endgroup$ – Ubiquitous Jun 2 at 17:09
  • $\begingroup$ @Ubiquitous In my case, $\nabla_v u$ is constant for some directional vector $v$. If $v$ is a basis vector, then it reduces to your definition of quasilinear utility: $\nabla_i g$ for some $i$. $\endgroup$ – High GPA Jun 2 at 19:39

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