<p>In general, any utility function that produces non-linear and downward sloping indifference curves will feature imperfect substitutability between goods. (By considering only <em>goods</em>, we're implicitly assuming &quot;more is preferred to less&quot;, or positive marginal utility, for each good.) In other words, look at the marginal rate of substitution and see if it depends on the quantity of either good; if it does, then there is imperfect substitutability.</p>
<h1>Examples</h1>
<ul>
<li>Quasi-linear utility: <span class="math-container">$u(x,y)=v(x)+y$</span>, where <span class="math-container">$v(x)$</span> is non-linear in <span class="math-container">$x$</span> (e.g. <span class="math-container">$v(x)=\sqrt x$</span>)</li>
<li>Additively separable utility: <span class="math-container">$u(x,y)=g(x)+h(y)$</span>, where <span class="math-container">$g,h$</span> are non-linear functions</li>
<li>Cobb-Douglas: <span class="math-container">$u(x,y)=x^\alpha y^\beta$</span>, where <span class="math-container">$\alpha,\beta$</span> are constants</li>
<li><s>Leontief (perfect complement): <span class="math-container">$\style{text-decoration:line-through}{u(x,y)=\min\{\alpha x,\beta y\}}$</span></s> [Leontief utility function features <em>non</em>substitutability rather than imperfect substitutability.]</li>
<li>CES utility: <span class="math-container">$u(x,y)=\left(\alpha x^\rho+\beta y^\rho\right)^{1/\rho}$</span>, where <span class="math-container">$\rho\ne1$</span>. Note that the class of CES utility functions includes Cobb-Douglas (as <span class="math-container">$\rho\to0$</span>), Leontief (as <span class="math-container">$\rho\to-\infty$</span>), and linear utility (as <span class="math-container">$\rho=1$</span>) as special cases.</li>
</ul>
<p>This <a href="http://www.econ.ucla.edu/sboard/teaching/econ11_09/econ11_09_lecture2.pdf#page=15" rel="nofollow noreferrer">lecture note</a> by Simon Board has examples and indifference curves corresponding to the cases above.</p>

CC BY-SA 4.0
2019-02-20T06:20:58.510Z
Leontief utilities represent imperfect substitution? Not non-substitution?
Giskard
CC BY-SA 4.0
2019-02-20T06:21:48.610Z
Similary, I don't think that if $U(x,y) = x -y $ then goods $x$ and $y$ are substitutes. Having a negative slope for the indifference curve seems like a logical think to want.
Giskard
CC BY-SA 4.0
2019-02-20T16:57:59.110Z
@denesp: Thanks. Your point on Leontief utility is well taken. I've also added an explanation making clear that we're considering only *goods* here, which would rule out the case of negative marginal utility.
Herr K.
CC BY-SA 4.0
2019-02-19T22:23:23.793Z
2019-02-20T16:54:21.590Z
2020-06-17T09:40:54.643Z
Herr K.
<p>Cobb-Douglas utility does NOT reflect substitute goods. They are unrelated. The demand equations are <span class="math-container">$x_i=\frac{a_i \cdot m}{p_i}$</span> for each good <span class="math-container">$i$</span>. Note that only <span class="math-container">$p_i$</span> affects the demand for <span class="math-container">$x_i$</span>; other goods' prices don't enter into demand. Thus, all other goods are unrelated: neither substitutes nor complements.</p>

CC BY-SA 4.0
2023-10-04T18:48:01.313Z
2023-10-04T18:48:01.313Z
marcoecon
<p>The utility function for perfect substitutes is defined as U(X,Y) = aX + bY. If the two goods X&amp;Y are imperfect substitutes what would be their utility function?</p>

<microeconomics><utility><consumer-theory>
Imperfect Substitutes and Utility Funcitions
Kautilya
# Imperfect Substitutes and Utility Funcitions

(Asked by: username_0 on 2019-02-19)

The utility function for perfect substitutes is defined as U(X,Y) = aX + bY. If the two goods X&Y are imperfect substitutes what would be their utility function?

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

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## 2 answers

### Answer by username_1 on 2019-02-19. Score: 3 | Accepted answer

In general, any utility function that produces non-linear and downward sloping indifference curves will feature imperfect substitutability between goods. (By considering only *goods*, we're implicitly assuming "more is preferred to less", or positive marginal utility, for each good.) In other words, look at the marginal rate of substitution and see if it depends on the quantity of either good; if it does, then there is imperfect substitutability.

# Examples

* Quasi-linear utility: $u(x,y)=v(x)+y$, where $v(x)$ is non-linear in $x$ (e.g. $v(x)=\sqrt x$)
* Additively separable utility: $u(x,y)=g(x)+h(y)$, where $g,h$ are non-linear functions
* Cobb-Douglas: $u(x,y)=x^\alpha y^\beta$, where $\alpha,\beta$ are constants
* ~~Leontief (perfect complement): $\style{text-decoration:line-through}{u(x,y)=\min\{\alpha x,\beta y\}}$~~ [Leontief utility function features *non*substitutability rather than imperfect substitutability.]
* CES utility: $u(x,y)=\left(\alpha x^\rho+\beta y^\rho\right)^{1/\rho}$, where $\rho\ne1$. Note that the class of CES utility functions includes Cobb-Douglas (as $\rho\to0$), Leontief (as $\rho\to-\infty$), and linear utility (as $\rho=1$) as special cases.

This [lecture note](http://www.econ.ucla.edu/sboard/teaching/econ11_09/econ11_09_lecture2.pdf#page=15) by Simon Board has examples and indifference curves corresponding to the cases above.

> **Comments:**
> **username_2**: Leontief utilities represent imperfect substitution? Not non-substitution?  
> **username_2**: Similary, I don't think that if $U(x,y) = x -y $ then goods $x$ and $y$ are substitutes. Having a negative slope for the indifference curve seems like a logical think to want.  
> **username_1**: @denesp: Thanks. Your point on Leontief utility is well taken. I've also added an explanation making clear that we're considering only *goods* here, which would rule out the case of negative marginal utility.  

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### Answer by username_3 on 2023-10-04. Score: 0

Cobb-Douglas utility does NOT reflect substitute goods. They are unrelated. The demand equations are $x\_i=\frac{a\_i \cdot m}{p\_i}$ for each good $i$. Note that only $p\_i$ affects the demand for $x\_i$; other goods' prices don't enter into demand. Thus, all other goods are unrelated: neither substitutes nor complements.


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