<p>Let $Q^d = D(p)$ be the market demand function, depending on price $p$. Let $p^*$ be equilibrium price (that depends also on supply of course). Then Consumer Surplus is usually defined as</p>

<p>$$\text{CS}=\int_{p^*}^\infty\!D(p)\,dp$$</p>

<p>i.e. the "area under the demand curve", starting from equilibrium price. So it appears, that if $D(p) =\bar q&gt;0$ (perfectly inelastic demand), then we would have</p>

<p>$$\text{CS}=\int_{p^*}^\infty\!\bar q\,dp =\bar q \cdot p\Big|^{\infty}_{p^*}\rightarrow \infty$$</p>

<p>Hmm, this is mathematically sound, but is it useful/meaningful from an economics perspective? </p>

<p>The critical point is the upper limit of the integral of course: by putting it equal to "infinity" we assume that for <em>any</em> price, however high, there will be <em>some</em> demand, however minuscule. In reality, after a price demand will drop to zero (simple linear downward sloping demand curves have this realistic property). Even for totally inelastic demand, after a price all consumers will be "budgeted out" of the market -that they "desperately need" to have the product does not mean that they will indeed get it if the price exceeds, say, their total wealth. In a graph, this means that the vertical line representing demand has an upper end, it does not extend to the sky. So if we realistically take into account this fact, and denote this supremum price by $P_s&lt;\infty$, we can redefine Consumer Surplus as</p>

<p>$$\text{CS}=\int_{p^*}^{P_s}\!D(p)\,dp$$</p>

<p>and for totally inelastic demand we obtain</p>

<p>$$\text{CS}=\int_{p^*}^{P_s}\!\bar q\,dp =\bar q \cdot (P_s-p^*)$$</p>

<p>Graphically,</p>

<p><a href="https://i.sstatic.net/Iu8hY.png" rel="noreferrer"><img src="https://i.sstatic.net/Iu8hY.png" alt="enter image description here"></a></p>

CC BY-SA 3.0
2015-08-29T23:43:00.553Z
2015-08-29T23:43:00.553Z
Alecos Papadopoulos
<p>From a purely theoretical perspective, if an <em>individual's</em> demand curve is perfectly inelastic, then her willingness to pay for the good is infinite. NB this also implies that she has an infinite budget. Thus, consumer surplus is well defined: it is the willingness to pay minus the price she pays, so as long as the price is finite her consumer surplus is finite.</p>

<p>In practice, no one has an infinite budget. So if the individual's demand curve is truly perfectly inelastic (i.e. the inverse demand is vertical), there exists a price such that beyond that price she can no longer afford to buy the good. This price is her willingness to pay, so consumer surplus is again well defined: the willingness to pay minus the price.</p>

CC BY-SA 3.0
2016-07-30T00:27:03.943Z
I don't quite understand your theoretical perspective. "consumer surplus" is "the willingness to pay minus the price she pays", "the price is finite", "her willingness to pay is infinite": how can these statements make a finite surplus?
xuhdev
CC BY-SA 3.0
2016-07-31T06:12:15.913Z
I have to second the query. Infinite willingness to pay minus finite price is infinity, which, sure, is well defined, but not finite.
Kitsune Cavalry
CC BY-SA 3.0
2015-08-29T01:31:21.997Z
2016-07-30T10:14:08.453Z
2016-07-30T10:14:08.453Z
Nick
<p>How do we define consumer surplus in the case of perfectly inelastic demand?</p>

<p>This question was inspired by the comments following this <a href="https://economics.stackexchange.com/a/5841/1601">answer</a>. For a graph of inelastic demand please also see the linked question.</p>

<p>A motivation for a definition would be that while consumer surplus would be strange in this special case, change in consumer surplus may still be measurable easily with quite reasonable definitions.</p>

CC BY-SA 3.0
2017-02-12T16:59:02.700Z
how can some one have an **infinite** budget.
lord_anonymous
CC BY-SA 3.0
2017-02-12T20:57:40.937Z
@lord_anonymous I don't think someone can. Nick also explains the problems with this in his answer. But why are you asking me? I did not write any such thing.
Giskard
<microeconomics><consumer-surplus>
Consumer surplus in case of perfectly inelastic demand
Giskard
# Consumer surplus in case of perfectly inelastic demand

(Asked by: username_0 on 2015-08-28)

How do we define consumer surplus in the case of perfectly inelastic demand?

This question was inspired by the comments following this [answer](https://economics.stackexchange.com/a/5841/1601). For a graph of inelastic demand please also see the linked question.

A motivation for a definition would be that while consumer surplus would be strange in this special case, change in consumer surplus may still be measurable easily with quite reasonable definitions.

**Tags:** microeconomics, consumer-surplus
> **Comments:**
> **username_1**: how can some one have an **infinite** budget.  
> **username_0**: @lord_anonymous I don't think someone can. Nick also explains the problems with this in his answer. But why are you asking me? I did not write any such thing.  

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## 2 answers

### Answer by username_3 on 2015-08-29. Score: 7 | Accepted answer

From a purely theoretical perspective, if an *individual's* demand curve is perfectly inelastic, then her willingness to pay for the good is infinite. NB this also implies that she has an infinite budget. Thus, consumer surplus is well defined: it is the willingness to pay minus the price she pays, so as long as the price is finite her consumer surplus is finite.

In practice, no one has an infinite budget. So if the individual's demand curve is truly perfectly inelastic (i.e. the inverse demand is vertical), there exists a price such that beyond that price she can no longer afford to buy the good. This price is her willingness to pay, so consumer surplus is again well defined: the willingness to pay minus the price.

> **Comments:**
> **username_4**: I don't quite understand your theoretical perspective. "consumer surplus" is "the willingness to pay minus the price she pays", "the price is finite", "her willingness to pay is infinite": how can these statements make a finite surplus?  
> **username_5**: I have to second the query. Infinite willingness to pay minus finite price is infinity, which, sure, is well defined, but not finite.  

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### Answer by username_2 on 2015-08-29. Score: 5

Let $Q^d = D(p)$ be the market demand function, depending on price $p$. Let $p^\*$ be equilibrium price (that depends also on supply of course). Then Consumer Surplus is usually defined as

$$\text{CS}=\int\_{p^\*}^\infty\!D(p)\,dp$$

i.e. the "area under the demand curve", starting from equilibrium price. So it appears, that if $D(p) =\bar q>0$ (perfectly inelastic demand), then we would have

$$\text{CS}=\int\_{p^\*}^\infty\!\bar q\,dp =\bar q \cdot p\Big|^{\infty}\_{p^\*}\rightarrow \infty$$

Hmm, this is mathematically sound, but is it useful/meaningful from an economics perspective?

The critical point is the upper limit of the integral of course: by putting it equal to "infinity" we assume that for *any* price, however high, there will be *some* demand, however minuscule. In reality, after a price demand will drop to zero (simple linear downward sloping demand curves have this realistic property). Even for totally inelastic demand, after a price all consumers will be "budgeted out" of the market -that they "desperately need" to have the product does not mean that they will indeed get it if the price exceeds, say, their total wealth. In a graph, this means that the vertical line representing demand has an upper end, it does not extend to the sky. So if we realistically take into account this fact, and denote this supremum price by $P\_s<\infty$, we can redefine Consumer Surplus as

$$\text{CS}=\int\_{p^\*}^{P\_s}\!D(p)\,dp$$

and for totally inelastic demand we obtain

$$\text{CS}=\int\_{p^\*}^{P\_s}\!\bar q\,dp =\bar q \cdot (P\_s-p^\*)$$

Graphically,

[![enter image description here](https://i.sstatic.net/Iu8hY.png)](https://i.sstatic.net/Iu8hY.png)


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