<p>I will address the situation in which a) price falls by a given quantity, and b) the fall in price is more than merely marginal. This I think is the most natural interpretation of the question. Thus, in contrast to the answer by user47965, I shall not be concerned with the (correct) point that the effect on price of a shift in the supply curve will depend on the elasticity of demand.  Similarly, in contrast to the answer by smcc, I shall not be concerned with the (correct) point that the change in consumer surplus resulting from an infinitesimally small fall in price will be the product of that infinitesimal quantity and the quantity demanded at the initial price, and therefore will not depend on the elasticity of demand.</p>
<p>Since the consumer surplus equals the area below the demand curve and above the price, a fall in price will always increase, not reduce, consumer surplus. If <span class="math-container">$D$</span> is the demand function and <span class="math-container">$p_0$</span> and <span class="math-container">$p_1$</span> are respectively the initial and lowered prices, then the increase in consumer surplus will be given by:</p>
<p><span class="math-container">$$\Delta CS = \int_{p_0}^{p_1} D(x)dx$$</span></p>
<p>If demand is perfectly inelastic over the range of the price change, then we will have:</p>
<p><span class="math-container">$$D(p_1)=D(p_0)$$</span></p>
<p>and therefore:</p>
<p><span class="math-container">$$\Delta CS = (p_0-p_1)D(p_0)$$</span></p>
<p>If however demand is fairly elastic, we will have:</p>
<p><span class="math-container">$$D(p_1) &gt; D(p_0)$$</span></p>
<p>and therefore:</p>
<p><span class="math-container">$$\Delta CS &gt; (p_0-p_1)D(p_0)$$</span></p>
<p>So the increase in consumer surplus is greater when demand, over the relevant range, is elastic.</p>

CC BY-SA 4.0
2024-09-08T20:25:23.650Z
2024-09-08T20:25:23.650Z
Adam Bailey
<p>Let <span class="math-container">$D$</span> denote the demand function. The consumer surplus when the price is <span class="math-container">$p$</span> is:</p>
<p><span class="math-container">$$CS(p)=\int_p^\infty D(x)dx$$</span></p>
<p>The derivative (using the <a href="https://en.wikipedia.org/wiki/Leibniz_integral_rule" rel="nofollow noreferrer">Leibniz integral rule</a>) is</p>
<p><span class="math-container">$$CS'(p)=-D(p)$$</span></p>
<p>So the (absolute) change in consumer surplus (with a marginal change in price) is equal to the quantity demanded at the initial price. It follow that the absolute change in consumer surplus cannot be related to the price elasticity of demand (without further assumptions).</p>
<p>The proportional change in consumer surplus (the elasticity of consumer surplus with respect to price) is:</p>
<p><span class="math-container">$$\frac{pCS'(p)}{CS(p)}=-\frac{pD(p)}{CS(p)} \tag{1}$$</span></p>
<p>So the proportional change in consumer surplus is equal to the negative of the ratio of expenditure to consumer surplus (and cannot be related directly to the price elasticity of demand).</p>
<hr />
<p>This first paragraph of your question is incorrect (at least without further assumptions about demand and about the supply side):</p>
<blockquote>
<p>I know that the total consumer surplus is higher for more inelastic P.E.D values, which makes intuitive sense: People would be willing to pay more than they actually do for inelastic goods e.g. a lifesaving drug as compared to toothpicks.</p>
</blockquote>
<p>The magnitude of consumer surplus depends on more than just the elasticity. In particular, it depends on the price (and the equilibrium price depends on the cost function(s) and market structure).</p>
<p>One case where we <em>can</em> say that consumer surplus decreases with an increase in price elasticity of demand (at the initial equilibrium price) is when:</p>
<ul>
<li>There is perfect competition</li>
<li>Demand is linear</li>
<li>Demand becomes more elastic by rotating it anticlockwise about the initial equilibrium price (so that the demand becomes flatter and so more elastic at the initial equilibrium price)</li>
</ul>
<p>(That consumer surplus rises in this case can easily be verified diagrammatically.)</p>
<p>In particular, when demand is perfectly elastic consumer surplus will be zero and as demand becomes steeper it grows without bound (consumer surplus is undefined when demand is perfectly inelastic).</p>

CC BY-SA 4.0
2024-05-07T19:11:15.733Z
2024-05-07T19:27:51.260Z
2024-05-07T19:27:51.260Z
smcc
<p>I know that the total consumer surplus is higher for more inelastic P.E.D values, which makes intuitive sense: People would be willing to pay more than they actually do for inelastic goods e.g. a lifesaving drug as compared to toothpicks.</p>
<p>But what about the <em>change</em> in consumer surplus. Some sources i've seen say the change is larger for elastic P.E.D with other sources saying the opposite, all with minimal explanation of the reason.</p>
<p>Would any of you be able to give a definitive answer with a reason behind it because nothing I seem to</p>

<microeconomics><elasticity><consumer-surplus><surplus>
Assuming a fall in price, does consumer surplus fall more with elastic or inelastic price elasticity of demand
Hallojvjknvve
# Assuming a fall in price, does consumer surplus fall more with elastic or inelastic price elasticity of demand

(Asked by: username_0 on 2024-05-07)

I know that the total consumer surplus is higher for more inelastic P.E.D values, which makes intuitive sense: People would be willing to pay more than they actually do for inelastic goods e.g. a lifesaving drug as compared to toothpicks.

But what about the *change* in consumer surplus. Some sources i've seen say the change is larger for elastic P.E.D with other sources saying the opposite, all with minimal explanation of the reason.

Would any of you be able to give a definitive answer with a reason behind it because nothing I seem to

**Tags:** microeconomics, elasticity, consumer-surplus, surplus

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## 2 answers

### Answer by username_1 on 2024-09-08. Score: 2

I will address the situation in which a) price falls by a given quantity, and b) the fall in price is more than merely marginal. This I think is the most natural interpretation of the question. Thus, in contrast to the answer by user47965, I shall not be concerned with the (correct) point that the effect on price of a shift in the supply curve will depend on the elasticity of demand. Similarly, in contrast to the answer by smcc, I shall not be concerned with the (correct) point that the change in consumer surplus resulting from an infinitesimally small fall in price will be the product of that infinitesimal quantity and the quantity demanded at the initial price, and therefore will not depend on the elasticity of demand.

Since the consumer surplus equals the area below the demand curve and above the price, a fall in price will always increase, not reduce, consumer surplus. If $D$ is the demand function and $p\_0$ and $p\_1$ are respectively the initial and lowered prices, then the increase in consumer surplus will be given by:

$$\Delta CS = \int\_{p\_0}^{p\_1} D(x)dx$$

If demand is perfectly inelastic over the range of the price change, then we will have:

$$D(p\_1)=D(p\_0)$$

and therefore:

$$\Delta CS = (p\_0-p\_1)D(p\_0)$$

If however demand is fairly elastic, we will have:

$$D(p\_1) > D(p\_0)$$

and therefore:

$$\Delta CS > (p\_0-p\_1)D(p\_0)$$

So the increase in consumer surplus is greater when demand, over the relevant range, is elastic.


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### Answer by username_2 on 2024-05-07. Score: 2

Let $D$ denote the demand function. The consumer surplus when the price is $p$ is:

$$CS(p)=\int\_p^\infty D(x)dx$$

The derivative (using the [Leibniz integral rule](https://en.wikipedia.org/wiki/Leibniz_integral_rule)) is

$$CS'(p)=-D(p)$$

So the (absolute) change in consumer surplus (with a marginal change in price) is equal to the quantity demanded at the initial price. It follow that the absolute change in consumer surplus cannot be related to the price elasticity of demand (without further assumptions).

The proportional change in consumer surplus (the elasticity of consumer surplus with respect to price) is:

$$\frac{pCS'(p)}{CS(p)}=-\frac{pD(p)}{CS(p)} \tag{1}$$

So the proportional change in consumer surplus is equal to the negative of the ratio of expenditure to consumer surplus (and cannot be related directly to the price elasticity of demand).

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This first paragraph of your question is incorrect (at least without further assumptions about demand and about the supply side):

> I know that the total consumer surplus is higher for more inelastic P.E.D values, which makes intuitive sense: People would be willing to pay more than they actually do for inelastic goods e.g. a lifesaving drug as compared to toothpicks.

The magnitude of consumer surplus depends on more than just the elasticity. In particular, it depends on the price (and the equilibrium price depends on the cost function(s) and market structure).

One case where we *can* say that consumer surplus decreases with an increase in price elasticity of demand (at the initial equilibrium price) is when:

* There is perfect competition
* Demand is linear
* Demand becomes more elastic by rotating it anticlockwise about the initial equilibrium price (so that the demand becomes flatter and so more elastic at the initial equilibrium price)

(That consumer surplus rises in this case can easily be verified diagrammatically.)

In particular, when demand is perfectly elastic consumer surplus will be zero and as demand becomes steeper it grows without bound (consumer surplus is undefined when demand is perfectly inelastic).


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