<p>The indifference curves are not "parallel", as they are not straight lines. They are however shifted, that is they are supposed to maintain vertical distance regardless of the value of <span class="math-container">$x$</span>. </p>

<p>The curves you map maintain horizontal distance regardless of <span class="math-container">$y$</span>. That is because the non-linear variable here is <span class="math-container">$y$</span>, not <span class="math-container">$x$</span>. The curves are still shifted, but along the other axis. </p>

<p>Taking the logarithm of the utility function you get
<span class="math-container">$$
x + \ln y
$$</span>
which is more clearly quasi linear. </p>

CC BY-SA 4.0
2019-03-01T11:44:02.757Z
2019-03-01T18:55:46.917Z
2019-03-01T18:55:46.917Z
Giskard
<p>I understand that quasi-linear functions have a general form</p>

<p><span class="math-container">$U(x_1,x_2,...,x_n,y) = f(x_1,x_2,...,x_n) + y$</span></p>

<p>and that for a quasi-linear function, the <strong>income effect</strong> with respect to the other variables of the function (<span class="math-container">$x_1,x_2,...,x_n$</span>) are all <span class="math-container">$0$</span>, i.e., <strong>income has no effect</strong> on the consumption of those goods.
I also read online that a if a <strong>preference is quasi-linear, then indifference curves are parallel</strong>. </p>

<hr>

<p>However, I came across this function, <span class="math-container">$U(x,y)=e^xy$</span>, where the income effect with respect to <span class="math-container">$y$</span> is <span class="math-container">$0$</span>, when the price of <span class="math-container">$y$</span> changed. Is this also a quasi-linear function, since IE is <span class="math-container">$0$</span>? Plotting the graph of this function, I see that the indifference curves are actually not parallel. </p>

<p><a href="https://i.sstatic.net/DiIZJ.png" rel="noreferrer"><img src="https://i.sstatic.net/DiIZJ.png" alt="<span class="math-container">$U(x,y)$</span>"></a></p>

<microeconomics><utility>
Quasi-Linear Functions
gpa
# Quasi-Linear Functions

(Asked by: username_0 on 2019-03-01)

I understand that quasi-linear functions have a general form

$U(x\_1,x\_2,...,x\_n,y) = f(x\_1,x\_2,...,x\_n) + y$

and that for a quasi-linear function, the **income effect** with respect to the other variables of the function ($x\_1,x\_2,...,x\_n$) are all $0$, i.e., **income has no effect** on the consumption of those goods.
I also read online that a if a **preference is quasi-linear, then indifference curves are parallel**.

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However, I came across this function, $U(x,y)=e^xy$, where the income effect with respect to $y$ is $0$, when the price of $y$ changed. Is this also a quasi-linear function, since IE is $0$? Plotting the graph of this function, I see that the indifference curves are actually not parallel.

[![<span class=](https://i.sstatic.net/DiIZJ.png)$U(x,y)$">](https://i.sstatic.net/DiIZJ.png)

**Tags:** microeconomics, utility

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## 1 answers

### Answer by username_1 on 2019-03-01. Score: 6 | Accepted answer

The indifference curves are not "parallel", as they are not straight lines. They are however shifted, that is they are supposed to maintain vertical distance regardless of the value of $x$.

The curves you map maintain horizontal distance regardless of $y$. That is because the non-linear variable here is $y$, not $x$. The curves are still shifted, but along the other axis.

Taking the logarithm of the utility function you get
$$
x + \ln y
$$
which is more clearly quasi linear.


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