<p>The vertical bar notation is used to denote conditions/restrictions applied to the expression to its left. For example, <span class="math-container">$\frac{\mathrm dx_2}{\mathrm dx_1}\bigg\vert_{\text{$u$ constant}}$</span> reads:</p>
<blockquote>
<p>The derivative of <span class="math-container">$x_2$</span> with respect to <span class="math-container">$x_1$</span>, holding <span class="math-container">$u$</span> constant.</p>
</blockquote>
<p>This apparently refers to the slope of the indifference curve, holding utility constant at <span class="math-container">$I_1$</span>. Similarly, the RHS of [C.5] refers to the slope of the budget line. The inequality holds at <span class="math-container">$x^*$</span>, where the slope of the indifference curve is smaller (i.e. more negative) than the slope of the budget line.</p>

CC BY-SA 4.0
2022-07-04T15:47:36.800Z
Thanks - this makes sense. The one confusing thing is that the inequality has the same dx2/dx1 fraction on both sides. But, if I understand you correctly, it is implicit that the RHS fraction refers to the budget line because the bar notation refers to the budget line?
Tomas R
CC BY-SA 4.0
2022-07-04T16:25:42.950Z
@TomasR: Yes, that's my educated guess based on the snapshot you provide. The RHS has a restriction on income $M$ which usually occurs in the budget equation (e.g. $M=p_1x_1+p_2x_2$), while the LHS has a restriction on utility level, which naturally brings to mind indifference curves. Given utility function $u(x_1,x_2)$, the slope of the an indifference curve is given by $\frac{\mathrm dx_2}{\mathrm dx_1}=-\frac{\partial u(\cdot)/\partial x_1}{\partial u(\cdot)/\partial x_2}$ according to the implicit function theorem.
Herr K.
CC BY-SA 4.0
2022-07-04T14:26:00.083Z
2022-07-04T14:26:00.083Z
Herr K.
<p>I wonder if someone can help me interpret the vertical bar notation used in the picture. From the graph, it is apparent that the consumer will consume only good <span class="math-container">$x_1$</span>, since the indifference curve is steeper than the budget line at <span class="math-container">$x^*$</span>. I assume that is what eq. C.5 is expressing? I'm hoping that C.6 and C.7 will become apparent if I first understand C.5.</p>
<p>Appreciate any help!</p>
<p><a href="https://i.sstatic.net/PDzJK.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/PDzJK.png" alt="enter image description here" /></a></p>

<microeconomics><utility><consumer-theory><preferences>
Indifference curve - corner point - Q about notation
Tomas R
# Indifference curve - corner point - Q about notation

(Asked by: username_0 on 2022-07-04)

I wonder if someone can help me interpret the vertical bar notation used in the picture. From the graph, it is apparent that the consumer will consume only good $x\_1$, since the indifference curve is steeper than the budget line at $x^\*$. I assume that is what eq. C.5 is expressing? I'm hoping that C.6 and C.7 will become apparent if I first understand C.5.

Appreciate any help!

[![enter image description here](https://i.sstatic.net/PDzJK.png)](https://i.sstatic.net/PDzJK.png)

**Tags:** microeconomics, utility, consumer-theory, preferences

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## 1 answers

### Answer by username_1 on 2022-07-04. Score: 4 | Accepted answer

The vertical bar notation is used to denote conditions/restrictions applied to the expression to its left. For example, $\frac{\mathrm dx\_2}{\mathrm dx\_1}\bigg\vert\_{\text{$u$ constant}}$ reads:

> The derivative of $x\_2$ with respect to $x\_1$, holding $u$ constant.

This apparently refers to the slope of the indifference curve, holding utility constant at $I\_1$. Similarly, the RHS of [C.5] refers to the slope of the budget line. The inequality holds at $x^\*$, where the slope of the indifference curve is smaller (i.e. more negative) than the slope of the budget line.

> **Comments:**
> **username_0**: Thanks - this makes sense. The one confusing thing is that the inequality has the same dx2/dx1 fraction on both sides. But, if I understand you correctly, it is implicit that the RHS fraction refers to the budget line because the bar notation refers to the budget line?  
> **username_1**: @TomasR: Yes, that's my educated guess based on the snapshot you provide. The RHS has a restriction on income $M$ which usually occurs in the budget equation (e.g. $M=p_1x_1+p_2x_2$), while the LHS has a restriction on utility level, which naturally brings to mind indifference curves. Given utility function $u(x_1,x_2)$, the slope of the an indifference curve is given by $\frac{\mathrm dx_2}{\mathrm dx_1}=-\frac{\partial u(\cdot)/\partial x_1}{\partial u(\cdot)/\partial x_2}$ according to the implicit function theorem.  

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