<p>Basically, you're correct that elasticity is a summary statistic under the assumption that the curve "looks like" an ideal demand/supply curve, i.e one with constant elasticity. This assumption is usually pretty valid, e.g. here's <a href="https://bmcpublichealth.biomedcentral.com/articles/10.1186/s12889-017-4098-x" rel="nofollow noreferrer">empirical data</a> estimating the elasticity of the demand of soft drink across incomes groups (quintiles): </p>

<p><a href="https://i.sstatic.net/x4onI.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/x4onI.png" alt="enter image description here"></a></p>

<p>In general, (in the math-sense) the (point) elasticity of a function is <a href="https://en.wikipedia.org/wiki/Elasticity_of_a_function" rel="nofollow noreferrer">itself a function</a>, which may or may be constant, but it is so for the "usual" demand curves:</p>

<blockquote>
  <p>The elasticity of a function is a constant <span class="math-container">$\alpha$</span> if and only if the function has the form <span class="math-container">$f(x) = C x ^ \alpha$</span> for a constant <span class="math-container">$C&gt;0$</span>.</p>
  
  <p><a href="https://i.sstatic.net/oySbj.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/oySbj.png" alt="enter image description here"></a></p>
  
  <p>The elasticity at a point is the limit of the <a href="https://en.wikipedia.org/wiki/Arc_elasticity" rel="nofollow noreferrer">arc elasticity</a> between two points as the separation between those two points approaches zero.</p>
</blockquote>

<p>Finally the bit you said about "elastic and inelastic" regions is a bit of a confusion, as it's not about an elasticity curve itself in separation, but <a href="https://en.wikipedia.org/wiki/Price_elasticity_of_demand#Relation_to_marginal_revenue" rel="nofollow noreferrer">rather</a></p>

<blockquote>
  <p>On a graph with both a demand curve and a marginal revenue curve, demand will be elastic at all quantities where marginal revenue is positive. Demand is unit elastic at the quantity where marginal revenue is zero. Demand is inelastic at every quantity where marginal revenue is negative</p>
</blockquote>

<hr>

<p>Per [expert] protest below, an expert answer <a href="https://halshs.archives-ouvertes.fr/halshs-00973418/document" rel="nofollow noreferrer">should say</a> that estimating elasticities is a fairly difficult problem (for the obvious reason that we don't get to experience all the price points).</p>

<blockquote>
  <p>almost all price-elasticities which are presented in the literature are estimated on macro timeseries data by means of demand systems under Slutsky constraints. They are generally considered as being not robust to the specification of the demand system and they suffer from aggregation biases and lack of information on price variations. Moreover the stationarity conditions are generally rejected for long-term time-series. </p>
  
  <p>Second, estimation technics on cross-section individual data (such as surveys on consumer expenditures) are either based on separability assumptions over the utility function (Frisch, 1959, Deaton, 1974) which allows to calculate price elasticities in terms of income elasticities and the Frisch’s income flexibility (see Theil, 1986, Selvanathan, 1993), or on unit values
  (ratio of the value of expenditures over the corresponding quantities whenever both are recorded in the survey, generally for food expenditures, see Deaton, 1988). The first method is based on the assumption of strong separability which generally is not supported by the data. The second concerns only those rare datasets which contain both values and quantities of
  consumption, generally only for food commodities. </p>
  
  <p>Third, arc-elasticity can be computed between two periods (see Gardes, Merrigan, 2011
  for such an estimation for tobacco). In contrast to estimation on macro data, this method
  allows to compute price effects for different types of households, but it necessitates
  comparable repeated cross-sections or a panel over a period characterized by large price
  changes. Moreover, it concerns only the direct price effect for the commodity having
  experienced this price change.</p>
  
  <p>A fourth method consists in computing semi-aggregate price indexes for a set of products
  using the individual budget shares as a weight of the individual product prices: these
  aggregate prices are thereby individualized and can be used to estimate price elasticities on a
  cross-section survey. This method initiates in remarks by Hicks and Stone and was fully
  discussed by Lewbel (1989). It has been recently applied by Ruiz and Trannoy (2007) on the
  French Household Expenditures surveys and proved efficient, but perhaps not very robust
  (considering the volatility of estimates in Ruiz’s thesis, 2006). Those techniques imply the
  endogeneity of semi-aggregate prices (since they are defined by means of current budget
  shares) which can be corrected by instrumentation. More critical is the assumption that the set
  of detailed commodities (such as various durables for home cleaning) are sufficiently
  different to experience different price changes, but pertain to a homogenous group as
  concerns households’ preferences: whenever they differ sufficiently, the difference in the
  composition of the corresponding aggregate for two households implies that the semiaggregate does not play the same role in these households’ consumption, and thus the price
  change of the semi-aggregate is largely due to a change in its composition or quality. The
  hidden assumption of that method (commodities within an aggregate consumption play the
  same role in the household’s consumption) is thus intrinsically illogical: either the
  commodities are similar, and their prices must follow the same trend; or they differ, and the
  role of the aggregate in household’s consumption depends on the proportion of the detailed
  commodities in this aggregate. </p>
</blockquote>

<p>The kind of empirical graph I showed you as empirical "evidence" is based on the 2nd kind of method, if I'm not mistaken.</p>

CC BY-SA 4.0
2020-04-08T10:59:37.633Z
Your first figure is a bit misleading: it shows that price elasticity is nearly constant with respect to *income*, but not necessarily w.r.t. price.
Giskard
CC BY-SA 4.0
2020-04-08T11:55:51.863Z
I'm pretty sure the bit about "elastic and inelastic regions" refers to the linear demand curves of introductory micro textbooks.
VARulle
CC BY-SA 4.0
2020-04-08T10:54:38.417Z
2020-04-08T11:59:36.127Z
2020-04-08T11:59:36.127Z
fantastic peace and prosperity
<p>I know that the Demand curve has elastic and inelastic regions but when we say horizontal demand/supply curves are elastic or vertical curves are inelastic, in that case, does the elasticity still vary along the curve?</p>

<p>More specifically, one of my lectures gives Elasticities of various items such as 0.09 for Gasoline and 0.8 for soda drinks. Are these values just taken on a specific point? If yes, how can they be used to describe the complete curve of a specific item.</p>

<microeconomics><supply-and-demand>
Does elasticity vary along the Demand/Supply curve or is it a summary statistic of a curve?
Shahzeb
# Does elasticity vary along the Demand/Supply curve or is it a summary statistic of a curve?

(Asked by: username_0 on 2020-04-08)

I know that the Demand curve has elastic and inelastic regions but when we say horizontal demand/supply curves are elastic or vertical curves are inelastic, in that case, does the elasticity still vary along the curve?

More specifically, one of my lectures gives Elasticities of various items such as 0.09 for Gasoline and 0.8 for soda drinks. Are these values just taken on a specific point? If yes, how can they be used to describe the complete curve of a specific item.

**Tags:** microeconomics, supply-and-demand

---

## 1 answers

### Answer by username_1 on 2020-04-08. Score: 2

Basically, you're correct that elasticity is a summary statistic under the assumption that the curve "looks like" an ideal demand/supply curve, i.e one with constant elasticity. This assumption is usually pretty valid, e.g. here's [empirical data](https://bmcpublichealth.biomedcentral.com/articles/10.1186/s12889-017-4098-x) estimating the elasticity of the demand of soft drink across incomes groups (quintiles):

[![enter image description here](https://i.sstatic.net/x4onI.png)](https://i.sstatic.net/x4onI.png)

In general, (in the math-sense) the (point) elasticity of a function is [itself a function](https://en.wikipedia.org/wiki/Elasticity_of_a_function), which may or may be constant, but it is so for the "usual" demand curves:

> The elasticity of a function is a constant $\alpha$ if and only if the function has the form $f(x) = C x ^ \alpha$ for a constant $C>0$.
>
> [![enter image description here](https://i.sstatic.net/oySbj.png)](https://i.sstatic.net/oySbj.png)
>
> The elasticity at a point is the limit of the [arc elasticity](https://en.wikipedia.org/wiki/Arc_elasticity) between two points as the separation between those two points approaches zero.

Finally the bit you said about "elastic and inelastic" regions is a bit of a confusion, as it's not about an elasticity curve itself in separation, but [rather](https://en.wikipedia.org/wiki/Price_elasticity_of_demand#Relation_to_marginal_revenue)

> On a graph with both a demand curve and a marginal revenue curve, demand will be elastic at all quantities where marginal revenue is positive. Demand is unit elastic at the quantity where marginal revenue is zero. Demand is inelastic at every quantity where marginal revenue is negative

---

Per [expert] protest below, an expert answer [should say](https://halshs.archives-ouvertes.fr/halshs-00973418/document) that estimating elasticities is a fairly difficult problem (for the obvious reason that we don't get to experience all the price points).

> almost all price-elasticities which are presented in the literature are estimated on macro timeseries data by means of demand systems under Slutsky constraints. They are generally considered as being not robust to the specification of the demand system and they suffer from aggregation biases and lack of information on price variations. Moreover the stationarity conditions are generally rejected for long-term time-series.
>
> Second, estimation technics on cross-section individual data (such as surveys on consumer expenditures) are either based on separability assumptions over the utility function (Frisch, 1959, Deaton, 1974) which allows to calculate price elasticities in terms of income elasticities and the Frisch’s income flexibility (see Theil, 1986, Selvanathan, 1993), or on unit values
> (ratio of the value of expenditures over the corresponding quantities whenever both are recorded in the survey, generally for food expenditures, see Deaton, 1988). The first method is based on the assumption of strong separability which generally is not supported by the data. The second concerns only those rare datasets which contain both values and quantities of
> consumption, generally only for food commodities.
>
> Third, arc-elasticity can be computed between two periods (see Gardes, Merrigan, 2011
> for such an estimation for tobacco). In contrast to estimation on macro data, this method
> allows to compute price effects for different types of households, but it necessitates
> comparable repeated cross-sections or a panel over a period characterized by large price
> changes. Moreover, it concerns only the direct price effect for the commodity having
> experienced this price change.
>
> A fourth method consists in computing semi-aggregate price indexes for a set of products
> using the individual budget shares as a weight of the individual product prices: these
> aggregate prices are thereby individualized and can be used to estimate price elasticities on a
> cross-section survey. This method initiates in remarks by Hicks and Stone and was fully
> discussed by Lewbel (1989). It has been recently applied by Ruiz and Trannoy (2007) on the
> French Household Expenditures surveys and proved efficient, but perhaps not very robust
> (considering the volatility of estimates in Ruiz’s thesis, 2006). Those techniques imply the
> endogeneity of semi-aggregate prices (since they are defined by means of current budget
> shares) which can be corrected by instrumentation. More critical is the assumption that the set
> of detailed commodities (such as various durables for home cleaning) are sufficiently
> different to experience different price changes, but pertain to a homogenous group as
> concerns households’ preferences: whenever they differ sufficiently, the difference in the
> composition of the corresponding aggregate for two households implies that the semiaggregate does not play the same role in these households’ consumption, and thus the price
> change of the semi-aggregate is largely due to a change in its composition or quality. The
> hidden assumption of that method (commodities within an aggregate consumption play the
> same role in the household’s consumption) is thus intrinsically illogical: either the
> commodities are similar, and their prices must follow the same trend; or they differ, and the
> role of the aggregate in household’s consumption depends on the proportion of the detailed
> commodities in this aggregate.

The kind of empirical graph I showed you as empirical "evidence" is based on the 2nd kind of method, if I'm not mistaken.

> **Comments:**
> **username_2**: Your first figure is a bit misleading: it shows that price elasticity is nearly constant with respect to *income*, but not necessarily w.r.t. price.  
> **username_3**: I'm pretty sure the bit about "elastic and inelastic regions" refers to the linear demand curves of introductory micro textbooks.  

---

