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I see the elasticity value(s) of demand being used frequently to determine the relation between demand and variables such as price and income. I have however been unable to find any uses for the slope of the demand curve. Are there any?

Also, is there any relationship between the slope and the elasticity?

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Price elasticity of demand is related (but not equal) to the inverse of the slope of the demand curve.

This Wikipedia article defines PED as:

Price elasticity of demand (PED or Ed) is a measure used in economics to show the responsiveness, or elasticity, of the quantity demanded of a good or service to a change in its price, ceteris paribus. More precisely, it gives the percentage change in quantity demanded in response to a one percent change in price. [emphasis added]

Look at the following (canonical) supply-demand graph.

enter image description here

Let's define $P^*$ the price of market equilibrium, $Q_D$ as the quantity demanded and $E_D$ as price elasticity of demand. Then:

$$ {slope} = \frac{rise}{run} = \frac{\Delta P}{\Delta Q_D} $$

And in mathematical terms, the verbal description of the PED reduces to:

$$ E_D = \frac{\frac{\Delta Q_D}{Q_D} \cdot 100}{\frac{\Delta P}{P^*} \cdot 100} $$

$$ E_D = \frac{P^*}{Q_D} \cdot \frac{\Delta Q_D} {\Delta P} $$

$$ E_D = \frac{P^*}{Q_D} \cdot \frac{1}{slope} $$

Edit

Constructive comments have been incorporated into this answer.

In particular, @Kontorus pointed out:

The PED is a percentage change, the slope of the demand curve is an absolute change.

And @denesp pointed out that:

... linear demand curves have constant slopes but changing elasticity. Cobb-Douglas derived demand (hyperbole) has changing slope but constant elasticity.

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    $\begingroup$ Not precisely. The PED is a percentage change, the slope of the demand curve is an absolute change. $\endgroup$ – Kontorus Jul 15 '16 at 17:20
  • $\begingroup$ The information in this answer is false. For example linear demand curves have constant slopes but changing elasticity. Cobb-Douglas derived demand (hyperbole) has changing slope but constant elasticity. $\endgroup$ – Giskard Jul 15 '16 at 17:31
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    $\begingroup$ I edited this answer to reflect the two above constructive comments. $\endgroup$ – Mowzer Jul 15 '16 at 18:33
  • $\begingroup$ If you give the PED formula so that relation is clear you'll get my + vote as well. $\endgroup$ – Giskard Jul 15 '16 at 18:35
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    $\begingroup$ @denesp: How'd I do? $\endgroup$ – Mowzer Jul 15 '16 at 19:09

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