Price Elasticity of Demand - elastic demand and impact of price reductions on total revenue

Question

A question in relation to Price Elasticity of Demand.

My understanding is that when demand is elastic, a price reduction will increase total revenue.

If the price of a product changes from \$100 to \$80 (20% decrease) and quantity demanded increases from 1000 to 1240 (24% increase), PED = 24/-20 = -1.2 (elastic).

Total revenue at the original price (\$100) and quantity demanded (1000) = \$100,000 Total Revenue after the price change is \$80 x 1240 = \$99,200 so total revenue has actually fallen

Can anyone reconcile this with the point that when demand is elastic a price reduction should increase total revenue?

Answer 1

Like anything derived from calculus, the statement is guaranteed to hold true only for infinitesimal change. Unfortunately, since I don't know the entire demand function, I can't say much more than this.

A clear example of why this wouldn't work: starting at the same point \$100 and Q = 1000, we can decrease price to zero and total revenue would go down.

Answer 2

My understanding is that when demand is elastic, a price reduction will increase total revenue.

Some readers may want to know that an elastic demand can either be unitary elastic ($E_d = −1$) or relatively elastic ($−∞ < E_d < −1$). When unitary elastic, the above statement is not true, strictly speaking.

---

Can anyone reconcile this with the point that when demand is elastic a price reduction should increase total revenue?

Actually, to be logically compatible with your "point" (in which the observed change in total revenue is the objective reference of accuracy), your elasticity should be smaller than $1$ (in absolute value). Which you do not get since you are using an approximation formulae, not good enough in this case. Indeed, as outlined in the other answer, this approximation formulae is only good for infinitesimal changes.

That being explained, note that the mathematical construct behind elasticities is based on continuous sets. A better approximation is then

$E_d = \frac{\ln(1240/1000)}{\ln(80/100)} = \frac{.215}{-.223} = -.964$

so $−1 < E_d < 0$, which caracterizes a relatively inelastic demand. In this case, as summarized on wikipedia, when the price goes down, the total revenu decreases... which is what you see. Put differently, the "reconciliation" you are searching lies in the fact that you are actually not dealing with an elastic demand.

---

NB: The better approximation here lies in computing variations as $\left[\ln\frac{a}{b}\right]$ instead of $\left[-1 + \frac{a}{b}\right]$. See this answer of mine for explanations.

Answer 3

Like @Art said, I think the key thing to realise is that the elasticity is only well defined for small intervals. When you have a big change in price and quantity, the statement of "revenue increases when price decreases if demand is elastic" are no longer accurate / well defined.

We can see the problem mathematically. Let R, P, Q be the old revenue, price, and quantity respectively. Let R' be the new revenue, and $\Delta P$, $\Delta Q$ to be the change in price and quantity, respectively. Let $\epsilon_D$ be the price elasticity of demand.

\begin{align} R & \equiv P Q \\ R' & \equiv (P + \Delta P)(Q + \Delta Q) \\ \frac{R'}{R} & = 1 + \frac{\Delta Q}{Q} + \frac{\Delta P}{P} + \frac{\Delta P \Delta Q}{PQ} \\ \end{align}

Meanwhile, \begin{align} \epsilon_D & \equiv \frac{\Delta Q / Q}{\Delta P / P} \leqslant 0 \\ \frac{\Delta Q}{Q} & = \frac{\Delta P}{P} \epsilon_D \end{align}

Substituting:

\begin{align} \frac{R'}{R} & = 1 + \frac{\Delta P}{P}(\epsilon_D+1)+(\frac{\Delta P}{P})^2\epsilon_D \end{align}

Suppose the price decreased and the revenue also decreased. The textbooks say this means demand is inelastic. What do maths say about $\epsilon_D$?

\begin{align} -1 < \frac{\Delta P}{P} & < 0 \\ \frac{R'}{R} & < 1 \\ \frac{\Delta P}{P}(\epsilon_D+1)+(\frac{\Delta P}{P})^2\epsilon_D & < 0 \\ \epsilon_D+1+\frac{\Delta P}{P}\epsilon_D & > 0 \\ \epsilon_D(1+\frac{\Delta P}{P}) & > -1 \\ \epsilon_D & > -\frac{1}{1+\frac{\Delta P}{P}} \\ \epsilon_D & > -\frac{P}{P+\Delta P} \neq -1 \end{align}

Suppose the price changed from 11 to 10. This means the elasticity could have any value that's greater than -1.1. If the elasticity is -1.05, which means demand is technically elastic, the revenue would still decrease in this scenario.

Notice that such discrepancy is reconciled when the changes are small:

\begin{align} \lim_{\Delta P \rightarrow 0} (-\frac{P}{P+\Delta P}) = -1 \end{align}

What does this say about real world?

• We will never have a smooth, true-to-reality mathematical expression of the demand curve irl.
• We will never change the price by an infinitesimal amount irl.
• We likely don't care if the total revenue stays exactly the same. As long as we have a good idea of which direction it's going.
• If the demand is very elastic, the total revenue increases when you decrease the price. Vice versa.
• When the demand elasticity is roughly -1, the total revenue will stay roughly the same when you change the price.