<p><span class="math-container">$$u(x, y) = 2$$</span>
is a simple example of a continuous utility function with thick indifference curve.</p>

<p><span class="math-container">\begin{eqnarray*} u(x, y)= \begin{cases} x &amp; x &lt; 1 \\ 1  &amp; 1 \leq x &lt; 2 \\ x - 1 &amp; x \geq 2 \end{cases} \end{eqnarray*}</span></p>

<p>is another continuous utility function with thick indifference curve for <span class="math-container">$u=1$</span>.</p>

CC BY-SA 4.0
2020-01-13T01:13:55.500Z
2020-01-13T01:13:55.500Z
Amit
<p>How about a utility function that only enjoys discrete quantities. This takes discrete values but allows for continuous inputs.</p>

<p><span class="math-container">$$U(x,y) = (floor(x))^{\alpha} \cdot (floor(y))^{1-\alpha}$$</span></p>

<p><a href="https://i.sstatic.net/feC8j.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/feC8j.png" alt="enter image description here"></a></p>



<pre><code>
    import numpy as np
    import seaborn as sns
    import pandas as pd
    import matplotlib.pyplot as plt

    x = np.arange(0, 10, 0.01)
    y = np.arange(0, 10, 0.01)
    alpha = 0.7

    xv, yv = np.meshgrid(x, y)

    def U(x1,y1, alpha1):
        return(np.floor(x1)**alpha1 * np.floor(y1)**(1-alpha1))

    uv = U(xv,yv, alpha)

    fig, ax = plt.subplots()
    CS = ax.contourf(xv, yv,uv, levels=np.unique(uv))
    ax.set_xlabel('x')
    ax.set_ylabel('y')
    plt.colorbar(CS)
    plt.show()
</code></pre>

<p>If it must be strongly monotonic and differentiable, then no, I believe this is impossible. An agent's preferences are said to be strongly monotonic if, given a consumption bundle <span class="math-container">$x$</span>, the agent prefers all consumption bundles <span class="math-container">$y$</span> that have more of at least one good, and not less in any other good. That is <span class="math-container">$y\geq x$</span> and <span class="math-container">$y\neq x$</span>
imply <span class="math-container">$y\succ x$</span>. If this is a continuous and differentiable utility function then this means:
<span class="math-container">$$ \frac{\partial U}{\partial X_{i}} &gt; 0 \: \forall i$$</span>
This implies that the indifference curves cannot be fat. Consider being on a thin indifference curve and, holding fixed x, increasing y by an infinitesimal amount. This will increase utility by  <span class="math-container">$\frac{\partial U}{\partial y}$</span>, which we just stipulated is positive. So you must be moving onto a higher indifference curve and therefore it is not "fat" in the y direction. Similarly, taking an infinitesimal step in the x direction holding fixed y increase utility by  <span class="math-container">$\frac{\partial U}{\partial x}$</span>, which we also just stipulated is positive. So this utility function cannot have fat indifference curves. </p>

<p>I am unsure if this also is true for non-differentiable utility functions.  </p>

CC BY-SA 4.0
2020-02-14T18:05:35.280Z
2020-02-15T14:07:39.700Z
2020-02-15T14:07:39.700Z
BKay
<p>The question in the title sounds like a trick question, due to the monotonicity property that indifference curves have, such that for two goods x and y, strong monotonicity implies y > x. </p>

<p>Possible that you could answer the question by stipulating the area under a curve? </p>

<microeconomics><mathematical-economics><utility><preferences>
For the case of two goods, give an example of preferences that are represnted by a continuous utility function that allows for fat indifference curves
CorporateNationalism
# For the case of two goods, give an example of preferences that are represnted by a continuous utility function that allows for fat indifference curves

(Asked by: username_0 on 2020-01-12)

The question in the title sounds like a trick question, due to the monotonicity property that indifference curves have, such that for two goods x and y, strong monotonicity implies y > x.

Possible that you could answer the question by stipulating the area under a curve?

**Tags:** microeconomics, mathematical-economics, utility, preferences

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## 2 answers

### Answer by username_1 on 2020-01-13. Score: 2

$$u(x, y) = 2$$
is a simple example of a continuous utility function with thick indifference curve.

\begin{eqnarray\*} u(x, y)= \begin{cases} x & x < 1 \\ 1 & 1 \leq x < 2 \\ x - 1 & x \geq 2 \end{cases} \end{eqnarray\*}

is another continuous utility function with thick indifference curve for $u=1$.


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### Answer by username_2 on 2020-02-14. Score: 2

How about a utility function that only enjoys discrete quantities. This takes discrete values but allows for continuous inputs.
$$U(x,y) = (floor(x))^{\alpha} \cdot (floor(y))^{1-\alpha}$$


    import numpy as np
    import seaborn as sns
    import pandas as pd
    import matplotlib.pyplot as plt

    x = np.arange(0, 10, 0.01)
    y = np.arange(0, 10, 0.01)
    alpha = 0.7

    xv, yv = np.meshgrid(x, y)

    def U(x1,y1, alpha1):
        return(np.floor(x1)**alpha1 * np.floor(y1)**(1-alpha1))

    uv = U(xv,yv, alpha)

    fig, ax = plt.subplots()
    CS = ax.contourf(xv, yv,uv, levels=np.unique(uv))
    ax.set_xlabel('x')
    ax.set_ylabel('y')
    plt.colorbar(CS)
    plt.show()

If it must be strongly monotonic and differentiable, then no, I believe this is impossible. An agent's preferences are said to be strongly monotonic if, given a consumption bundle $x$, the agent prefers all consumption bundles $y$ that have more of at least one good, and not less in any other good. That is $y\geq x$ and $y\neq x$
imply $y\succ x$. If this is a continuous and differentiable utility function then this means:
$$ \frac{\partial U}{\partial X_{i}} > 0 \: \forall i$$
This implies that the indifference curves cannot be fat. Consider being on a thin indifference curve and, holding fixed x, increasing y by an infinitesimal amount. This will increase utility by  $\frac{\partial U}{\partial y}$, which we just stipulated is positive. So you must be moving onto a higher indifference curve and therefore it is not "fat" in the y direction. Similarly, taking an infinitesimal step in the x direction holding fixed y increase utility by  $\frac{\partial U}{\partial x}$, which we also just stipulated is positive. So this utility function cannot have fat indifference curves. 
I am unsure if this also is true for non-differentiable utility functions.


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