<p>We can show if a utility function exhibits complementarity or substitutability just from its function, without having any information about prices. The utility function gives information about the preferences.</p>
<p>Take a utility function and calculate the <span class="math-container">$MRS = - U’x_1 / δU’x_2 $</span>:</p>
<ul>
<li><p>Perfect Substitutes:  <span class="math-container">$U(x1,x2) = a x_1 + b x_2$</span> with</p>
<p><span class="math-container">$ MRS = -a/b$</span>. This means that there is a substitutive relation of <span class="math-container">$a:b$</span> between the bundles <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span>.</p>
</li>
<li><p>Perfect Complements: <span class="math-container">$U(x1,x2) = min(a x_1, b x_2)$</span> with</p>
<p><span class="math-container">$MRS = 0/b= 0$</span> for the horizontal part <span class="math-container">$(a x_1 &gt; b x_2)$</span></p>
<p><span class="math-container">$MRS = a/0 = - ∞$</span> for the vertical part <span class="math-container">$(a x_1 &lt; b x_2)$</span> and</p>
<p><span class="math-container">$MRS= undef.$</span> at the kink.</p>
</li>
<li><p>Cobb-Douglas: <span class="math-container">$U(x_1,x_2) = x_1^{0.5}  x_2^{0.5}$</span> with <span class="math-container">$MRS = - x_2/ x_1$</span></p>
<ul>
<li>The MRS is not constant, hence there is substitutability between the two bundles which can vary depending on the values. Thus, the C-D preferences <strong>show imperfect substitutes</strong>.</li>
<li>The <span class="math-container">$MRS=0$</span> (or <span class="math-container">$U(x_1,x_2)=0$</span>) if <span class="math-container">$x_i=0$</span>. Meaning that both bundles must be consumed for a positive utility, but the ratio can vary. Thus, the C-D preferences <strong>show imperfect complements</strong>.</li>
</ul>
</li>
</ul>

CC BY-SA 4.0
2022-04-08T13:01:47.623Z
Thanks for your contribution which highlights interesting aspects of the differenct utility functions and actually anwers most parts of my first question. With regard to my second question, I am particularly interested in a concrete utility function (neither perfect substitutes nor perfect complements), in which the goods are either gross complements or gross substitutes, after having solved for the demand functions. I have edited my question and described in more detail what I look for. Thanks again.
Sceptical_Economist
CC BY-SA 4.0
2022-04-05T10:00:45.257Z
2022-04-05T10:00:45.257Z
enthusiastic economist
<p>So the most prominent preferences are perfect substitutes, perfect complements and cobb-douglas preferences. Perfect complements and perfect substitutes are extreme cases and I was asked whether there are preferences (modelled by a corresponding utility function) which are e.g. complements but not perfect complements or substitutes but not perfect substitutes. I immediately thought of Cobb-Douglas preferences, since a special property of Cobb-Douglas preferences is that convexity is strictly fulfilled here: A bundle of goods formed as a weighted average of two goods on the same indifference curve is always better than any of the extreme bundles of goods. So at first sight, this appeared to be a quite reasonable candidate for &quot;a bit of complements but not perfect complements.&quot;</p>
<p>However, the common definition for good 1 to be a <strong>gross</strong> complement of good 2 is <span class="math-container">$\frac{\Delta x_{1}}{\Delta p_{2}}&lt;0$</span> while for substitutes it must be <span class="math-container">$\frac{\Delta x_{1}}{\Delta p_{2}}&gt;0$</span> (see e.g. slide 2 and 3 on this ressource: <a href="https://web.stanford.edu/%7Ejay/micro_class/lecture7.pdf" rel="nofollow noreferrer">https://web.stanford.edu/~jay/micro_class/lecture7.pdf</a>). When deriving the demand function of good 1 (using Lagrange with the corresponding budget constraint), demand of good 1 only depends on price of good 1 (plus income) and not on the price of good 2. This is a well documented fact and property of the Cobb-Douglas function. The underlying reason is that the substitution and income effect of an increase in price 2 cancel out here (see e.g. page 2 on this ressource: <a href="https://cpb-us-w2.wpmucdn.com/campuspress.yale.edu/dist/3/352/files/2012/01/Week-3-Lecture2.pdf" rel="nofollow noreferrer">https://cpb-us-w2.wpmucdn.com/campuspress.yale.edu/dist/3/352/files/2012/01/Week-3-Lecture2.pdf</a>)</p>
<p>Consequently the goods are neither <strong>gross</strong> complements nor <strong>gross</strong> substitutes when applying the definitions of &quot;gross substitutes&quot; or &quot;gross complements&quot; on the resulting demand functions derived from Cobb-Douglas preferences. My questions:</p>
<ol>
<li><p>Do you agree that this result is somehow contraintuitive (especially from a learner perspective) since Cobb-Douglas preferences imply strict convexity and one would rather think of a <strong>gross</strong> complementary relation of these goods when solving for the demand functions? How can one make sense of this result?</p>
</li>
<li><p>I would be interested to know whether you have some utility function proposals (except for perfect complements and perfect substitutes) which lead to <strong>gross</strong> complements or <strong>gross</strong> subsitutes according to the definitions above when solving for the demand functions? For the CES production function shown on page 5 on these slides <a href="https://economics.mit.edu/files/9016" rel="nofollow noreferrer">https://economics.mit.edu/files/9016</a> it is pointed out that the factors are gross substitutes for sigmas greater than one and gross complements for sigma smaller than one. I am sure that this should also apply for the CES utility function and that one can derive a concrete utility function (based on sigma values smaller or greater than one), solve for demand of good one and two and hence illustrate that the goods are indeed gross substitues or gross complements. I am especially curious to see how the indifference curves will look like. I am not so familiar with the CES function which is the reason why I cannot do it myself at the moment. Do you have a good reading advice for an understandable explanation of the CES function with its properties? Thanks!</p>
</li>
</ol>

<demand><ces-function><perfect-complements><perfect-substitutes>
A utility function (neither perfect substitues nor perfect complements) which stems from a CES f. and leads to gross complements or gross substitutes
Sceptical_Economist
# A utility function (neither perfect substitues nor perfect complements) which stems from a CES f. and leads to gross complements or gross substitutes

(Asked by: username_0 on 2022-03-30)

So the most prominent preferences are perfect substitutes, perfect complements and cobb-douglas preferences. Perfect complements and perfect substitutes are extreme cases and I was asked whether there are preferences (modelled by a corresponding utility function) which are e.g. complements but not perfect complements or substitutes but not perfect substitutes. I immediately thought of Cobb-Douglas preferences, since a special property of Cobb-Douglas preferences is that convexity is strictly fulfilled here: A bundle of goods formed as a weighted average of two goods on the same indifference curve is always better than any of the extreme bundles of goods. So at first sight, this appeared to be a quite reasonable candidate for "a bit of complements but not perfect complements."

However, the common definition for good 1 to be a **gross** complement of good 2 is $\frac{\Delta x\_{1}}{\Delta p\_{2}}<0$ while for substitutes it must be $\frac{\Delta x\_{1}}{\Delta p\_{2}}>0$ (see e.g. slide 2 and 3 on this ressource: [https://web.stanford.edu/~jay/micro\_class/lecture7.pdf](https://web.stanford.edu/%7Ejay/micro_class/lecture7.pdf)). When deriving the demand function of good 1 (using Lagrange with the corresponding budget constraint), demand of good 1 only depends on price of good 1 (plus income) and not on the price of good 2. This is a well documented fact and property of the Cobb-Douglas function. The underlying reason is that the substitution and income effect of an increase in price 2 cancel out here (see e.g. page 2 on this ressource: <https://cpb-us-w2.wpmucdn.com/campuspress.yale.edu/dist/3/352/files/2012/01/Week-3-Lecture2.pdf>)

Consequently the goods are neither **gross** complements nor **gross** substitutes when applying the definitions of "gross substitutes" or "gross complements" on the resulting demand functions derived from Cobb-Douglas preferences. My questions:

1. Do you agree that this result is somehow contraintuitive (especially from a learner perspective) since Cobb-Douglas preferences imply strict convexity and one would rather think of a **gross** complementary relation of these goods when solving for the demand functions? How can one make sense of this result?
2. I would be interested to know whether you have some utility function proposals (except for perfect complements and perfect substitutes) which lead to **gross** complements or **gross** subsitutes according to the definitions above when solving for the demand functions? For the CES production function shown on page 5 on these slides <https://economics.mit.edu/files/9016> it is pointed out that the factors are gross substitutes for sigmas greater than one and gross complements for sigma smaller than one. I am sure that this should also apply for the CES utility function and that one can derive a concrete utility function (based on sigma values smaller or greater than one), solve for demand of good one and two and hence illustrate that the goods are indeed gross substitues or gross complements. I am especially curious to see how the indifference curves will look like. I am not so familiar with the CES function which is the reason why I cannot do it myself at the moment. Do you have a good reading advice for an understandable explanation of the CES function with its properties? Thanks!

**Tags:** demand, ces-function, perfect-complements, perfect-substitutes

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## 1 answers

### Answer by username_1 on 2022-04-05. Score: 1

We can show if a utility function exhibits complementarity or substitutability just from its function, without having any information about prices. The utility function gives information about the preferences.

Take a utility function and calculate the $MRS = - U’x\_1 / δU’x\_2 $:

* Perfect Substitutes: $U(x1,x2) = a x\_1 + b x\_2$ with

  $ MRS = -a/b$. This means that there is a substitutive relation of $a:b$ between the bundles $x\_1$ and $x\_2$.
* Perfect Complements: $U(x1,x2) = min(a x\_1, b x\_2)$ with

  $MRS = 0/b= 0$ for the horizontal part $(a x\_1 > b x\_2)$

  $MRS = a/0 = - ∞$ for the vertical part $(a x\_1 < b x\_2)$ and

  $MRS= undef.$ at the kink.
* Cobb-Douglas: $U(x\_1,x\_2) = x\_1^{0.5} x\_2^{0.5}$ with $MRS = - x\_2/ x\_1$

  + The MRS is not constant, hence there is substitutability between the two bundles which can vary depending on the values. Thus, the C-D preferences **show imperfect substitutes**.
  + The $MRS=0$ (or $U(x\_1,x\_2)=0$) if $x\_i=0$. Meaning that both bundles must be consumed for a positive utility, but the ratio can vary. Thus, the C-D preferences **show imperfect complements**.

> **Comments:**
> **username_0**: Thanks for your contribution which highlights interesting aspects of the differenct utility functions and actually anwers most parts of my first question. With regard to my second question, I am particularly interested in a concrete utility function (neither perfect substitutes nor perfect complements), in which the goods are either gross complements or gross substitutes, after having solved for the demand functions. I have edited my question and described in more detail what I look for. Thanks again.  

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