# Quasilinear utility: if $x \succeq y - ae_1$, does it mean $x + ae_1 \succeq y$?

Quasilinear preference is defined to be:

$$x \sim y \Rightarrow x+ae_1 \sim y+ae_1$$ and $$x + ae_1 \succ x$$ with $$e_1 = (1,0,0,...)$$,

Given a quasilinear preference, if f $$x \succeq y - ae_1$$, does it mean $$x + ae_1 \succeq y$$?

I know that $$x + ae_1 \succ x$$ but how do I compare $$x+ae_1$$ with $$y$$? Drawing a diagram makes me see it but I can't prove it rigorously.

EDIT: Included the definition for quasilinear preference.

• Hint: Let $z=y-a\mathbf e_1$, so that $x\succsim y-a\mathbf e_1$ becomes $x\succsim z$. Apply the definition of quasilinear preference. – Herr K. Sep 27 '19 at 19:21