<p>Consider a population of individuals <span class="math-container">$\{1,\ldots, n\}$</span> with individual demands <span class="math-container">$x_i(p)$</span> (depending on the price of the good). Market demand is then defined by:
<span class="math-container">$$
X(p) = x_1(p) + \ldots + x_n(p) = \sum_{i = 1}^n x_i(p).
$$</span>
Taking the derivative on both sides gives:
<span class="math-container">$$
\frac{\partial X}{\partial p} = \sum_{i = 1}^n \frac{\partial x_i}{\partial p}.
$$</span>
Now multiply both sides by <span class="math-container">$p$</span> and divide by <span class="math-container">$X$</span> to obtain:
<span class="math-container">$$
\frac{p}{X} \frac{\partial X}{\partial p} = \sum_{i = 1}^n \frac{x_i}{X} \frac{p}{x_i} \frac{\partial x_i}{p}.
$$</span>
So:
<span class="math-container">$$
E^X_p = \sum_{i = 1}^n w_i e^{x_i}_p.
$$</span>
Where <span class="math-container">$E^X_p$</span> is the price elasticity of the market demand. <span class="math-container">$e^{x_i}_p$</span> is the price elasticity of individual <span class="math-container">$i$</span>'s demand and <span class="math-container">$w_i = \frac{x_i}{X}$</span> is the share of <span class="math-container">$i$</span>'s demand in overall market demand.</p>

CC BY-SA 4.0
2023-07-30T07:52:52.413Z
2023-07-30T07:52:52.413Z
tdm
<p>I have heard that the market elasticity of demand for a good is the weighted sum of the individual price elasticities of demand for that good but cannot find any information about it online. Could someone please supply a proof that shows this?</p>

<demand>
Proof for calculating market elasticity of demand
Aryan Arora
# Proof for calculating market elasticity of demand

(Asked by: username_0 on 2023-07-29)

I have heard that the market elasticity of demand for a good is the weighted sum of the individual price elasticities of demand for that good but cannot find any information about it online. Could someone please supply a proof that shows this?

**Tags:** demand


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## 1 answers

### Answer by username_1 on 2023-07-30. Score: 2

Consider a population of individuals $\{1,\ldots, n\}$ with individual demands $x\_i(p)$ (depending on the price of the good). Market demand is then defined by:
$$
X(p) = x\_1(p) + \ldots + x\_n(p) = \sum\_{i = 1}^n x\_i(p).
$$
Taking the derivative on both sides gives:
$$
\frac{\partial X}{\partial p} = \sum\_{i = 1}^n \frac{\partial x\_i}{\partial p}.
$$
Now multiply both sides by $p$ and divide by $X$ to obtain:
$$
\frac{p}{X} \frac{\partial X}{\partial p} = \sum\_{i = 1}^n \frac{x\_i}{X} \frac{p}{x\_i} \frac{\partial x\_i}{p}.
$$
So:
$$
E^X\_p = \sum\_{i = 1}^n w\_i e^{x\_i}\_p.
$$
Where $E^X\_p$ is the price elasticity of the market demand. $e^{x\_i}\_p$ is the price elasticity of individual $i$'s demand and $w\_i = \frac{x\_i}{X}$ is the share of $i$'s demand in overall market demand.


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