<p>Isoelastic here means both the elasticity of demand to its own price and the elasticity of substitution are constant. The former is obvious given <span class="math-container">$x(v, t)=p^{x}(v, t)^{-1 / \beta} L$</span> and thus the markup is also a constant. The latter can be also shown directly by <span class="math-container">$\frac{x\left(v_{1}\right)}{x\left(v_{2}\right)}=\left(\frac{p\left(v_{1}\right)}{p\left(v_{2}\right)}\right)^{-1 / \beta} $</span>.</p>

CC BY-SA 4.0
2022-08-20T08:39:24.007Z
2022-08-20T14:56:59.957Z
2022-08-20T14:56:59.957Z
Alalalalaki
<p>I have a very simple question:</p>
<p>I'm studying Acemoglu's book the horizontal innovation paradigm of endogenous growth.</p>
<p>Here, the conditional demand function for the machine variety <span class="math-container">$\upsilon$</span>, i.e., <span class="math-container">$x(\upsilon,t)$</span> is given by <span class="math-container">$[13.6]$</span>, and the book says that it is isoelastic. Does this mean that the elasticity of substitution between any pair of machines is constant? Does this mean that after a change in the relative price of a machine <span class="math-container">$\upsilon$</span> with respect to <span class="math-container">$\upsilon'$</span>, for any <span class="math-container">$\upsilon \neq \upsilon'$</span> the ratio between <span class="math-container">$\upsilon$</span> and <span class="math-container">$\upsilon'$</span> changes at a constant rate regardless of the initial quantity of the pair of machines?
Finally, since machines are manufactured in monopolistic competition, does the isoelastic shape of <span class="math-container">$[13.6]$</span> ensure constant mark-up over the marginal cost?</p>
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<p>Further info:</p>
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<macroeconomics><microeconomics>
Isoelastic Demands
John M.
# Isoelastic Demands

(Asked by: username_0 on 2022-08-19)

I have a very simple question:

I'm studying Acemoglu's book the horizontal innovation paradigm of endogenous growth.

Here, the conditional demand function for the machine variety $\upsilon$, i.e., $x(\upsilon,t)$ is given by $[13.6]$, and the book says that it is isoelastic. Does this mean that the elasticity of substitution between any pair of machines is constant? Does this mean that after a change in the relative price of a machine $\upsilon$ with respect to $\upsilon'$, for any $\upsilon \neq \upsilon'$ the ratio between $\upsilon$ and $\upsilon'$ changes at a constant rate regardless of the initial quantity of the pair of machines?
Finally, since machines are manufactured in monopolistic competition, does the isoelastic shape of $[13.6]$ ensure constant mark-up over the marginal cost?

[![enter image description here](https://i.sstatic.net/xaJ4a.png)](https://i.sstatic.net/xaJ4a.png)

Further info:

[![enter image description here](https://i.sstatic.net/FqGp7.png)](https://i.sstatic.net/FqGp7.png)

[![enter image description here](https://i.sstatic.net/p0d60.png)](https://i.sstatic.net/p0d60.png)

**Tags:** macroeconomics, microeconomics


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## 1 answers

### Answer by username_1 on 2022-08-20. Score: 2 | Accepted answer

Isoelastic here means both the elasticity of demand to its own price and the elasticity of substitution are constant. The former is obvious given $x(v, t)=p^{x}(v, t)^{-1 / \beta} L$ and thus the markup is also a constant. The latter can be also shown directly by $\frac{x\left(v\_{1}\right)}{x\left(v\_{2}\right)}=\left(\frac{p\left(v\_{1}\right)}{p\left(v\_{2}\right)}\right)^{-1 / \beta} $.


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