# Price Elasticity of Demand for Positive Price Increases

What does it mean when the price elasticity of demand %Qd/%P is greater than one? Typically I hear that it means the demand is elastic since if, say, the price decreases by 1% the demand for the good increases by more than 1%. But what happens if %P is +1% and %Qd still increases by more? Sure, this is elastic, but does it give us any more information? I can't see why %Qd/%P > 1 makes sense in this case.

• It is not clear what you are asking. The question in your third sentence, is answered in your second sentence. "Elastic demand" is a verbal way to say that the percentage increase in quantity demanded is higher (in absolute terms) than the percentage decrease in price. In what sense, it doesn't make sense? Please edit your question to clarify, so that we can answer it. – Alecos Papadopoulos Dec 21 '14 at 20:02
• @AlecosPapadopoulos Wait, so an increase in %P would be denoted -%P? Why are we talking about increase in quantity and decrease in price? To me that would imply the correct equation is %Qd/(-%P). If this is not the correct equation the curiosity to me is that 3%/1% for example does not denote that a 1% increase in price results in a 3% increase in quantity demanded. And if this is the case it is confusing to me that such a case would really occur very often. – user1501 Dec 21 '14 at 21:14
• Well it appears that you forget or are unaware of a standard convention here: algebraically, the price elasticity of demand is negative. Economists tend to talk about it in absolute terms, considering the algebraic sign as understood. I have prepared an answer as to where elastic demand emerges, so that the concept acquires some usefulness other than descriptive. – Alecos Papadopoulos Dec 21 '14 at 21:18
• @Alecos Thanks, that makes more sense then, so my revision of the equation is correct, it's just that we always consider the absolute value of Ed? – user1501 Dec 21 '14 at 22:19
• Verbally yes. Because otherwise we would have to say "the smaller the elasticity the more elastic demand is" which would obviously create more confusion. In other words, the concept of elasticity itself was conceived having absolute magnitudes in mind. – Alecos Papadopoulos Dec 21 '14 at 22:27

## 3 Answers

Except of the purely descriptive aspect, "elastic demand", or more accurately, regions of the demand schedule where demand elasticity with respect to price is higher than unity, in absolute terms, is linked to the basic monopoly theory, since the monopolist maximizes profits at a point of the demand schedule where "demand is elastic".

Define the demand point-elasticity with respect to price as

$$\eta = \frac {\partial Q }{ \partial P}\cdot \frac {P}{Q} \Rightarrow \frac {\partial Q }{ \partial P} = \eta \cdot \frac {Q}{P} \tag{1}$$

Note that algebraically, the elasticity is a negative number, with the sign indicating the direction of influence, since $\partial Q / \partial P <0$.

The profit function of a monopolist is

$$\pi = P\cdot Q(P) - C(Q(P)) \tag{2}$$

The first-order condition for a maximum with respect to price is

$$\frac {\partial \pi}{\partial P} = 0 \Rightarrow Q + P\frac {\partial Q }{ \partial P} - MC\cdot \frac {\partial Q }{ \partial P} = 0 \tag{3}$$

Inserting $(1)$ into $(3)$ we have

$$Q + P\cdot \eta \cdot \frac {Q}{P} - MC\cdot \eta \cdot \frac {Q}{P} = 0$$

$$\Rightarrow 1 + \eta - \eta \cdot \frac {MC}{P} =0$$

$$\Rightarrow - \eta \cdot \frac {MC}{P} = -\eta -1$$

$$\Rightarrow |\eta| \cdot \frac {MC}{P} = |\eta| -1$$

$$\Rightarrow P^* = \frac {|\eta|}{|\eta|-1} MC \tag{4}$$

$(4)$ is essentially an implicit relation since $\eta$ is a function of price also, but it provides a specific insight: Since we naturally expect that price will be positive, we see that we must have $|\eta| >1$: the price will necessarily be set at a level where "demand is elastic", i.e. at a point on the demand schedule where the point price elasticity of demand is higher than unity, in absolute terms.

Yes, if elasticity is greater than one we say that demand is elastic. This means that the percentage change in quantity demanded is greater than (in magnitude) the percentage change in price. More generally, if elasticity is e then the percentage change in quantity demanded is e times the percentage change in price. For example, if e=0.5, the percentage change in quantity demanded is half the percentage change in price.

If it's more than one, it means that the variation of the selled quantity is greater that the variation of the price. In other words a variation in price cause a bigger variation in production. So if price goes down, consumers will increase purchases in a bigger variation: the demand of that good is more than elastic to the price. If it's 1 it's elastic. If it's less than one, it is not very elastic or unelastic.