<p>The indifference curves are constructed by viewing the utility function as an equation (for a fixed utility index value per curve).  So from</p>

<p>$$U = U(x_1,x_2)$$ </p>

<p>where the left side is just a symbol, we move to</p>

<p>$$\bar U = U(x_1,x_2)$$
where now the left side is a specific number.</p>

<p>Take the total differential on both sides to obtain</p>

<p>$$0 = U_1dx_1 + U_2dx_2 \implies \frac {dx_2}{dx_1} = -\frac {U_1}{U_2}$$</p>

<p>Any straight line in the two-dimensional plane has a constant slope so also</p>

<p>$$\frac {dx_2}{dx_1} =c$$</p>

<p>Same for $y$-bundle. Etc.</p>

CC BY-SA 3.0
2015-09-02T00:50:51.363Z
Thanks for your answer Alecos! I'm still trying to understand it perfectly before accepting it. I'm not sure what would happen if $U$ wasn't differentiable. Thanks for helping! :D
Guilherme Salome
CC BY-SA 3.0
2015-09-02T02:02:38.513Z
@GuilhermeSalomé Good point. Think it over.
Alecos Papadopoulos
CC BY-SA 3.0
2015-09-01T11:36:59.100Z
2015-09-01T11:36:59.100Z
Alecos Papadopoulos
<p>Given a continuous preference relation $\succeq$ over $X=\mathbb{R}^2_{+}$ where all sets:
$$
I_x\equiv\{y\in X:y\sim x\}
$$
are lines on $X,\forall x\in X$, and are parallel to $I_y,\forall y\notin I_x$. </p>

<p><strong>How can I show that $\succeq$ has a linear representation?</strong></p>

<p>It seems intuitive that if every indifference curve is a line, then the utility itself must be a line, but I'm not sure how to go back to it. Would someone be willing to help me?</p>

<p>Thanks! Any helpful tips are appreciated! :D</p>

<utility><preferences>
If all indifference curves are parallel lines, then preference has linear representation
Guilherme Salome
# If all indifference curves are parallel lines, then preference has linear representation

(Asked by: username_0 on 2015-08-31)

Given a continuous preference relation $\succeq$ over $X=\mathbb{R}^2\_{+}$ where all sets:
$$
I\_x\equiv\{y\in X:y\sim x\}
$$
are lines on $X,\forall x\in X$, and are parallel to $I\_y,\forall y\notin I\_x$.

**How can I show that $\succeq$ has a linear representation?**

It seems intuitive that if every indifference curve is a line, then the utility itself must be a line, but I'm not sure how to go back to it. Would someone be willing to help me?

Thanks! Any helpful tips are appreciated! :D

**Tags:** utility, preferences

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## 1 answers

### Answer by username_1 on 2015-09-01. Score: 7 | Accepted answer

The indifference curves are constructed by viewing the utility function as an equation (for a fixed utility index value per curve). So from

$$U = U(x\_1,x\_2)$$

where the left side is just a symbol, we move to

$$\bar U = U(x\_1,x\_2)$$
where now the left side is a specific number.

Take the total differential on both sides to obtain

$$0 = U\_1dx\_1 + U\_2dx\_2 \implies \frac {dx\_2}{dx\_1} = -\frac {U\_1}{U\_2}$$

Any straight line in the two-dimensional plane has a constant slope so also

$$\frac {dx\_2}{dx\_1} =c$$

Same for $y$-bundle. Etc.

> **Comments:**
> **username_0**: Thanks for your answer Alecos! I'm still trying to understand it perfectly before accepting it. I'm not sure what would happen if $U$ wasn't differentiable. Thanks for helping! :D  
> **username_1**: @GuilhermeSalomé Good point. Think it over.  

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