# Rotation of Quasilinear Utility

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.

• 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 $y$. – Ubiquitous Jun 2 '19 at 17:09
• @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 $i$. – High GPA Jun 2 '19 at 19:39