<p>This is not supposed to be an answer, just a comment: Cobb-Douglas function is named after the paper <a href="https://www.jstor.org/stable/1811556" rel="nofollow noreferrer">https://www.jstor.org/stable/1811556</a> (1928).</p>
<p>Most cases of production functions have returns to scale equal to one, which is the Cobb-Douglas form. It is not special, it is common.</p>
<p>CES function was generalized by Dixit and Stiglitz in the paper <a href="https://www.jstor.org/stable/1831401" rel="nofollow noreferrer">https://www.jstor.org/stable/1831401</a> (1977).</p>

CC BY-SA 4.0
2023-11-12T23:37:46.670Z
2023-11-12T23:42:06.273Z
2023-11-12T23:42:06.273Z
André Luiz
<p>My question concerns CES utility functions, which have the form
<span class="math-container">$$u(x) = \left(\sum_{j=1}^n a_j x_j^{\rho} \right)^{1/\rho}$$</span>
for utility parameters <span class="math-container">$a_j$</span>, an elasticity parameter <span class="math-container">$\rho$</span>, and some allocation of goods given by <span class="math-container">$x$</span>.</p>
<p>When <span class="math-container">$\rho=1$</span> we have <strong>linear utility</strong>,  or <strong>perfect substitution</strong>, with <span class="math-container">$$u(x) = a^T x$$</span> The level sets of <span class="math-container">$u$</span> are straight lines, and it is always possible to achieve the same utility by exchanging one good for another in some fixed ratio. <strong>I can see this arising in real life:</strong> Two bags of flour are perfectly interchangeable, and the only thing that matters to the buyer is their size and cost.</p>
<p>When <span class="math-container">$\rho \to -\infty$</span> we have <strong>Leontief utility</strong>, or <strong>perfect complementarity</strong>, with <span class="math-container">$$u(x) = \min_j a_j x_j$$</span> The level sets of <span class="math-container">$u$</span> are L-shaped, and  optimal utility is always achieved by purchasing goods in a  target ratio given by <span class="math-container">$a$</span>. <strong>This also makes sense:</strong> If I am making a recipe that calls for two units of egg for every unit of sugar, having twice as much sugar makes no difference unless I also have twice as much egg.</p>
<p>Most situations are more complicated than this, however, and the nice thing about the CES form and its <span class="math-container">$\rho$</span> parameter is that it lets us capture situations that don't sit at one of these extremes. A baker can replace sugar with brown sugar, but not arbitrarily. Setting <span class="math-container">$\rho$</span> to some value between <span class="math-container">$1$</span> and <span class="math-container">$-\infty$</span> captures this imperfect substitutability. But the precise value of <span class="math-container">$\rho$</span> depends on the situation at hand: if the ingredients are pretty good substitutes, maybe <span class="math-container">$\rho = 0.8$</span>; if they are fairly poor substitutes, maybe <span class="math-container">$\rho = -10$</span>.</p>
<p>If we take the limit as <span class="math-container">$\rho \to 0$</span>, we get a utility function that has a nice name,
the Cobb–Douglas utility function. It also has a nice form:</p>
<p><span class="math-container">$$u(x) = \prod_j x_j^{a_j}$$</span></p>
<p>But is there anything that's truly <em>special</em> about this case?</p>
<p>From my standpoint, it looks like the Cobb–Douglas utility is just one arbitrary point along the continuum running from linear to Leontief. Perhaps it gets a name because it was discovered before the generalized CES family, but I can't see why there's anything especially meaningful about it beyond the simplicity of the form. Is this argument fair?</p>
<p>Is there a reason why we might assume <span class="math-container">$\rho$</span> is close to zero if we have ruled out the linear and Leontief cases?</p>
<p>Does Cobb–Douglas utility correspond to an &quot;idealized&quot; situation like the flour or egg/sugar examples above?</p>

CC BY-SA 4.0
2024-08-10T16:44:24.287Z
The demand functions only depend on own-price. Consumers always use a specific share of their income on each good. The share being $a_j$. For the CES the shares are functions of the other goods prices.
Jesper Hybel
<cobb-douglas><ces-function><measuring-utility><nonexpert>
What's special about Cobb–Douglas utility relative to the rest of the CES family?
Max
# What's special about Cobb–Douglas utility relative to the rest of the CES family?

(Asked by: username_0 on 2021-03-05)

My question concerns CES utility functions, which have the form
$$u(x) = \left(\sum\_{j=1}^n a\_j x\_j^{\rho} \right)^{1/\rho}$$
for utility parameters $a\_j$, an elasticity parameter $\rho$, and some allocation of goods given by $x$.

When $\rho=1$ we have **linear utility**, or **perfect substitution**, with $$u(x) = a^T x$$ The level sets of $u$ are straight lines, and it is always possible to achieve the same utility by exchanging one good for another in some fixed ratio. **I can see this arising in real life:** Two bags of flour are perfectly interchangeable, and the only thing that matters to the buyer is their size and cost.

When $\rho \to -\infty$ we have **Leontief utility**, or **perfect complementarity**, with $$u(x) = \min\_j a\_j x\_j$$ The level sets of $u$ are L-shaped, and optimal utility is always achieved by purchasing goods in a target ratio given by $a$. **This also makes sense:** If I am making a recipe that calls for two units of egg for every unit of sugar, having twice as much sugar makes no difference unless I also have twice as much egg.

Most situations are more complicated than this, however, and the nice thing about the CES form and its $\rho$ parameter is that it lets us capture situations that don't sit at one of these extremes. A baker can replace sugar with brown sugar, but not arbitrarily. Setting $\rho$ to some value between $1$ and $-\infty$ captures this imperfect substitutability. But the precise value of $\rho$ depends on the situation at hand: if the ingredients are pretty good substitutes, maybe $\rho = 0.8$; if they are fairly poor substitutes, maybe $\rho = -10$.

If we take the limit as $\rho \to 0$, we get a utility function that has a nice name,
the Cobb–Douglas utility function. It also has a nice form:

$$u(x) = \prod\_j x\_j^{a\_j}$$

But is there anything that's truly *special* about this case?

From my standpoint, it looks like the Cobb–Douglas utility is just one arbitrary point along the continuum running from linear to Leontief. Perhaps it gets a name because it was discovered before the generalized CES family, but I can't see why there's anything especially meaningful about it beyond the simplicity of the form. Is this argument fair?

Is there a reason why we might assume $\rho$ is close to zero if we have ruled out the linear and Leontief cases?

Does Cobb–Douglas utility correspond to an "idealized" situation like the flour or egg/sugar examples above?

**Tags:** cobb-douglas, ces-function, measuring-utility, nonexpert

> **Comments:**
> **username_1**: The demand functions only depend on own-price. Consumers always use a specific share of their income on each good. The share being $a_j$. For the CES the shares are functions of the other goods prices.  

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## 1 answers

### Answer by username_2 on 2023-11-12. Score: 0

This is not supposed to be an answer, just a comment: Cobb-Douglas function is named after the paper <https://www.jstor.org/stable/1811556> (1928).

Most cases of production functions have returns to scale equal to one, which is the Cobb-Douglas form. It is not special, it is common.

CES function was generalized by Dixit and Stiglitz in the paper <https://www.jstor.org/stable/1831401> (1977).


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