Price Elasticity of Demand - Worked Example

Intuitively it makes sens that if the price of something goes up, people buy less of it. There are, of course, some exceptions, but it's usually assumed this is true.

Revenue an Elasticity

This behavior can be quantified with a the price elasticity of demand (PED), which is defined as

$$E = \frac{(dQ/Q)}{(dP/P)}$$

Where E is the elasticity, Q and dQ are the demand and a change in the demand respectively. P and dP are the price and change in price respectively.

This can be read as: If price increases by, say, 10%, then demand decreases by -E*10%.

If we look at revenue R = Q*P, we see that total revenue can go up if prices go down. The relative change in revenue becomes:

$$dR/R = (dQP + dpQ)/PQ$$

In response to a price decrease (dP < 0) the change in revenue is positive if

$$\frac{dQP + dpQ}{PQ} > 0$$

This becomes

$$-\frac{dQ/Q}{dp/P} > 1 \rightarrow -E>1$$

That is, the elasticity must be less than -1.

This makes sense, if price is reduced by, say 5%, the quantity needs to increase by the same percentage to stay revenue neutral.

Profit Margin and Elasticity

Let's assume that companies in some hypothetical industry operate with a fixed profit margin. This is a reasonable assumption: companies compete on price and will lower their prices to the point that they achieve the minimal profit margin needed to raise capital to enter this industry.

Let's say, for example that a product sells at EUR 100, where EUR 90 are various costs like production and marketing and EUR 10 are profit. This equates to a 10% profit margin.

If a company can reduce costs from EUR 90 to EUR 81 (a 10% reduction), the new price will be EUR 90. The profit margin is still 10%. That is, profit margin is forced to stay constant because competition drives prices down to the point that the margin equals the minimum for a given industry.

The total profit is given by R*m = P*Q*m.

If the quantity stays the same and m is constant, the profit also reduces by 10%.

But we saw, above that quantity will change according the PED. In this case, total profit will increase if PED is less than -1.

For example, if PED = -2, and costs are reduced by a fraction r the new profit will be

$$\begin{align} \begin{split} P' &= P(1-r) \rightarrow dP/P = -r\\ Q' &= Q+dQ \\ &= Q + QEdP/P \\ &= Q(1-Er) \end{split} \end{align}$$ $$\begin{align} \begin{split} R' &= P'Q' \\ &= PQ(1-r)(1-Er) \\ &= R(1-r)(1-Er) \end{split} \end{align}$$

Assuming that r is small this approximates to

$$R' = R(1-r-Er)\\ R'/R = 1 - (1+E)r$$

So if E is, for example, -2, and r is 0.1, then R increases by 10%. Since profit margin is constant, profit also increases by 10%

Important Pitfall

It would be tempting to think that a at a 10% profit margin, a 10% cost reduction would double profits. That is all cost reduction becomes profit. This would be true if one company in the industry managed to reduce cost in a way that other companies cannot replicate.

However, if all companies reduce costs by 10%, prices drop and changes in profits are fully governed by the price elasticity of demand. Profits can go up or down depending on if PED is above or below -1.