<p>Quasilinear preference is defined to be:</p>

<p><span class="math-container">$x \sim y \Rightarrow x+ae_1 \sim y+ae_1$</span> and <span class="math-container">$x + ae_1 \succ x$</span> with <span class="math-container">$e_1 = (1,0,0,...)$</span>,</p>

<p>Given a quasilinear preference, if f <span class="math-container">$x  \succeq y - ae_1$</span>, does it mean <span class="math-container">$x + ae_1 \succeq y$</span>? </p>

<p>I know that <span class="math-container">$x + ae_1 \succ x$</span> but how do I compare <span class="math-container">$x+ae_1$</span> with <span class="math-container">$y$</span>? Drawing a diagram makes me see it but I can't prove it rigorously. </p>

<p>EDIT: Included the definition for quasilinear preference.</p>

CC BY-SA 4.0
2019-09-26T05:26:26.890Z
Could you please define what you mean by "quasilinear preference". Doing so may help you answer your own question.
Giskard
CC BY-SA 4.0
2019-09-26T21:16:35.807Z
Edited the question.
Rainroad
CC BY-SA 4.0
2019-09-26T21:35:56.510Z
Now that you have included your definition, are you sure you do not know how to proceed? There is no trick at all, you just apply the definition.
Giskard
CC BY-SA 4.0
2019-09-26T23:15:26.830Z
Could you elaborate on it?
Rainroad
CC BY-SA 4.0
2019-09-27T07:15:30.487Z
If the OP had used the utility function definition of quasilinear preferences, the question would be easy. However, the OP is using a different definition which makes the problem more difficult and more interesting.
Angela Pretorius
CC BY-SA 4.0
2019-09-27T13:03:07.603Z
If the preference isn't continuous we aren't sure we can use utility function either... so we're stuck with this definition.
Rainroad
CC BY-SA 4.0
2019-09-27T19:21:07.937Z
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.
CC BY-SA 4.0
2019-09-28T12:30:53.807Z
The definition applies to  $\sim$, does it apply to $ \succ$?
Rainroad
CC BY-SA 4.0
2019-09-29T04:39:02.007Z
Sure it does. If not, you'll have $x\succ y$ implying $y+a\mathrm e_1\succsim x+a\mathrm e_1$, which is a contradiction when $a=0$.
Herr K.
<microeconomics><mathematical-economics>
Quasilinear utility: if $x \succeq y - ae_1$, does it mean $x + ae_1 \succeq y$?
Rainroad
# Quasilinear utility: if $x \succeq y - ae_1$, does it mean $x + ae_1 \succeq y$?

(Asked by: username_0 on 2019-09-26)

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.

**Tags:** microeconomics, mathematical-economics

> **Comments:**
> **username_1**: Could you please define what you mean by "quasilinear preference". Doing so may help you answer your own question.  
> **username_0**: Edited the question.  
> **username_1**: Now that you have included your definition, are you sure you do not know how to proceed? There is no trick at all, you just apply the definition.  
> **username_0**: Could you elaborate on it?  
> **username_2**: If the OP had used the utility function definition of quasilinear preferences, the question would be easy. However, the OP is using a different definition which makes the problem more difficult and more interesting.  
> **username_0**: If the preference isn't continuous we aren't sure we can use utility function either... so we're stuck with this definition.  
> **username_3**: 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.  
> **username_0**: The definition applies to  $\sim$, does it apply to $ \succ$?  
> **username_3**: Sure it does. If not, you'll have $x\succ y$ implying $y+a\mathrm e_1\succsim x+a\mathrm e_1$, which is a contradiction when $a=0$.  

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## 0 answers

