<p><span class="math-container">$$\epsilon_D=\frac{\% \Delta Q_D}{\% \Delta P}$$</span>
A perfectly elastic demand curve is horizontal, meaning we quantity demanded can change by any amount without changing price (any quantity can be sold at the price corresponding the vertical intercept). Hence, for any change in quantity, <span class="math-container">$\% \Delta P=0$</span>. While we cannot strictly divide by zero, in the limit, the ratio is infinity. </p>

CC BY-SA 4.0
2019-03-20T22:31:49.163Z
2019-03-20T23:49:58.147Z
2019-03-20T23:49:58.147Z
dlnB
<p>Let demand elasticity be <span class="math-container">$\varepsilon$</span>. Then</p>

<p><span class="math-container">$$\varepsilon = (\frac{p}{D(p)})\frac{\mathrm{d}D(p)}{\mathrm{d}p} \ \ \text{(1)}$$</span></p>

<p>For a perfectly elastic demand curve, <span class="math-container">$\frac{\mathrm{d}D(p)}{\mathrm{d}p} = - \infty$</span>. Why? </p>

<p><span class="math-container">$$\frac{\mathrm{d}D(p)}{\mathrm{d}p} = \lim_{\Delta p \to 0} \frac{D(p + \Delta p) - D(p)}{\Delta p}$$</span></p>

<p>Now, <span class="math-container">$D(p + \Delta p) = 0 \ \forall \ \Delta p ≥ 0$</span>. Thus:</p>

<p><span class="math-container">$$\frac{\mathrm{d}D(p)}{\mathrm{d}p} = \lim_{\Delta p \to 0} \frac{-D(p)}{\Delta p} = D(p)\lim_{\Delta p \to 0}\frac{-1}{\Delta p} \implies \frac{\mathrm{d}D(p)}{\mathrm{d}p} = -\infty $$</span></p>

<p>Hence, substituting into <span class="math-container">$\text{(1)}$</span>, <span class="math-container">$\varepsilon = -\infty$</span>.</p>

CC BY-SA 4.0
2019-12-25T19:14:38.010Z
2019-12-25T19:14:38.010Z
Tejas Subramaniam
<p>When we have perfect elasticity, the demand curve is a horizontal line and the elasticity of demand coefficient is equal to infinity.</p>

<p>How do we arrive at a solution equal to infinity?</p>

<p>We know that </p>

<blockquote>
  <p>Elasticity of Demand Coefficient = Change in Quantity Demanded /
  Change in Price <span class="math-container">$x$</span></p>
</blockquote>

<p>If we change price by <span class="math-container">$x$</span>, how do we get what the change in quantity demanded is if there is no quantity demanded at the new price?</p>

CC BY-SA 4.0
2019-03-20T13:00:35.947Z
Your formula for elasticity of demand is incorrect. It should be the PERCENT change in quantity demanded divided by the PERCENT change in price.
David
<microeconomics><elasticity>
How to Calculate Price Elasticity of Demand When Perfectly Elastic?
econguy
# How to Calculate Price Elasticity of Demand When Perfectly Elastic?

(Asked by: username_0 on 2019-03-20)

When we have perfect elasticity, the demand curve is a horizontal line and the elasticity of demand coefficient is equal to infinity.

How do we arrive at a solution equal to infinity?

We know that

> Elasticity of Demand Coefficient = Change in Quantity Demanded /
> Change in Price $x$

If we change price by $x$, how do we get what the change in quantity demanded is if there is no quantity demanded at the new price?

**Tags:** microeconomics, elasticity

> **Comments:**
> **username_1**: Your formula for elasticity of demand is incorrect. It should be the PERCENT change in quantity demanded divided by the PERCENT change in price.  

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## 2 answers

### Answer by username_2 on 2019-03-20. Score: 1 | Accepted answer

$$\epsilon\_D=\frac{\% \Delta Q\_D}{\% \Delta P}$$
A perfectly elastic demand curve is horizontal, meaning we quantity demanded can change by any amount without changing price (any quantity can be sold at the price corresponding the vertical intercept). Hence, for any change in quantity, $\% \Delta P=0$. While we cannot strictly divide by zero, in the limit, the ratio is infinity.


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### Answer by username_3 on 2019-12-25. Score: 1

Let demand elasticity be $\varepsilon$. Then

$$\varepsilon = (\frac{p}{D(p)})\frac{\mathrm{d}D(p)}{\mathrm{d}p} \ \ \text{(1)}$$

For a perfectly elastic demand curve, $\frac{\mathrm{d}D(p)}{\mathrm{d}p} = - \infty$. Why?

$$\frac{\mathrm{d}D(p)}{\mathrm{d}p} = \lim\_{\Delta p \to 0} \frac{D(p + \Delta p) - D(p)}{\Delta p}$$

Now, $D(p + \Delta p) = 0 \ \forall \ \Delta p ≥ 0$. Thus:

$$\frac{\mathrm{d}D(p)}{\mathrm{d}p} = \lim\_{\Delta p \to 0} \frac{-D(p)}{\Delta p} = D(p)\lim\_{\Delta p \to 0}\frac{-1}{\Delta p} \implies \frac{\mathrm{d}D(p)}{\mathrm{d}p} = -\infty $$

Hence, substituting into $\text{(1)}$, $\varepsilon = -\infty$.


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