Price Elasticity of Demand (PED)
A measure of the responsiveness of quantity demanded when there is a price change.
The Formula for PED
Price elasticity of demand is defined as the percentage change in quantity demanded divided by the percentage change in price.
$$ \mathrm{PED} = \frac{\%\Delta Q_d}{\%\Delta P}$$
Where:
- $\%\Delta Q_d$ = Percentage change in quantity demanded
- $\%\Delta P$ = Percentage change in price
To calculate percentage changes, use this formula:
$$\%\Delta X = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100$$
This gives you the percentage change from Old Value to New Value
The formula above works for linear (straight-line demand curves) and is a good approximation for others.
The exact percentage change and elasticity for non-linear curves concerns the derivative which is not part of this syllabus.
Therefore, using the above formulas, PED can be written as:
$$ \mathrm{PED} = \frac{\frac{Q_{new}-Q_{old}}{Q_{old}}\times 100}{\frac{P_{new}-P_{old}}{P_{old}}\times 100} = \frac{\frac{Q_{new}-Q_{old}}{Q_{old}}}{\frac{P_{new}-P_{old}}{P_{old}}}=\frac{\frac{\Delta Q}{Q_{old}}}{\frac{\Delta P}{P_{old}}} $$
If we know that
- At a price $P_1 = \$ 50$ people buy $Q_1=30$ bars of chocolate.
- At a price $P_2 = \$ 100$ people buy $Q_2 = 15$ bars of chocolate.
Then we can calculate the $PED$ by using the formula above.
- $\%\Delta P = \frac{100-50}{50}\times 100 = \% 100$, since the price doubled.
- $\%\Delta Q_d = \frac{15 - 30}{30}\times 100 = - \% 50$.
Hence the price elasticity of demand would be:
$$ PED = \frac{-\% 50}{\% 100} = -\frac{1}{2}$$
For the example above, we would get a different $PED$ if we used $P_2,Q_2$ as our initial values and $P_1,Q_1$ as the new one (you can try this yourself)
Hence, the above $PED$ would be written as the price elasticity of demand at $\mathbf{P_1}$ because at $P_2$ the value is different.
As we can see, the price elasticity of demand is negative.
- That is because of the law of demand, the percentage change of quantity demanded will be in the inverse of the percentage change of price.
- Therefore PED is mathematically always negative.
However, for easier interpretation, economists use the absolute value of PED such that the PED calculated will just be noted as $\frac{1}{2}$ instead of $-\frac{1}{2}$.
Degrees of PED: Theoretical Range of Values
PED values help classify demand into different categories (we take the absolute values):
Price Inelastic
- When $0 < PED < 1$, the demand is called price inelastic.
- Here, the percentage change in quantity demanded is less than the percentage change in price.
- Therefore $PED$ is smaller than 1 and the quantity demanded is highly unresponsive.
Price Elastic
- When $1 < PED < \infty$, the demand is called price elastic.
- Here, the percentage change in quantity demanded is more than the percentage change in price.
- Therefore $PED$ is greater than 1 and the quantity demanded is greatly responsive.
Special values of PED
Figure 1 below showcases the three special values the PED of a demand curve can take.
Unit Elastic
- When $PED = 1$, the demand is called unit price elastic.
- Here, the percentage change in quantity demanded is equal to the percentage change in price.
- Therefore $PED$ is equal to 1 and the quantity demanded is exactly as responsive to the price.
Why is the unit elastic demand curve in Figure 1 unit elastic if it appears steep and the x and y axis have the same scale?
Many students make the mistake of thinking the demand on the left diagram in Figure 1 is inelastic, since the axis have the same scale and it is clearly steep.
However, this is not the case. Remember:
PED is defined as the percentage change in quantity demanded divided by the percentage change in price.
The formula of PED is:
$$PED=\frac{\%\Delta Q_d}{\%\Delta P}
Where:
- $\%\Delta Q_d = \frac{Quantity Demanded_initial - Quantity Demanded_final}{Quantity Demanded_final} \times 100$
- $\%\Delta P = \frac{Price_intial - Price_final}{Price_final} \times 100$
If we calculate the PED for the demand curve shown:
- $\%\Delta Q_d = \frac{2-4} {2} \times 100 = \frac{-2}{2} \times 100 = -100\%$
- $\%\Delta P = \frac{6-3} {3} \times 100 = \frac{3}{3} \times 100 = 100\%$
Therefore, the PED is:
$$PED = \frac{-100\%} {100\%} = -1$$
As we explained, we always take the absolute value of the PED. Hence PED = 1.
This is because the PED is calculated as a percentage change, not as a change.
Perfectly Inelastic
- When $PED = 0$, the demand is called perfectly price inelastic.
- Here, the percentage change in quantity demanded is always at zero regardless of what happens to the price.
- Therefore $PED$ is 0 and the quantity demanded is perfectly unresponsive.
Perfectly Elastic
- When $PED = \infty$, the demand is called perfectly price elastic.
- A small price fall causes people to buy an infinite amount, and a small price increase causes people to buy nothing.
- Therefore $PED$ is infinite and the quantity demanded is perfectly responsive.
Why is the unit elastic demand curve curved?
In the given unit elastic demand curve, the price elasticity of demand (PED) is equal to 1.
This happens when the percentage change in quantity demanded is exactly equal to the percentage change in price at all points along the curve. Looking at the figure:
- When the price decreases from 4 to 2 (a 50% decrease), the quantity demanded increases from 1 to 2 (a 50% increase).
- Similarly, when the price falls from 2 to 1 (a 50% decrease), the quantity rises from 2 to 4 (a 50% increase).
Since the percentage change in quantity is equal to the percentage change in price in both cases, the PED is exactly 1 at all points.
Students often confuse PED with the slope of the demand curve. Remember, PED measures percentage changes, not the steepness of the curve.
Determinants of PED
Several factors influence how responsive consumers are to price changes:
- Number and closeness of substitutes.
- Degree of necessity.
- Proportion of income spent.
- Time.
Number and Closeness of Substitutes
The more substitutes available, the more elastic the demand.
- If price changes happen, people can easily switch over to the substitute goods. Hence, the quantity demanded is quite responsive (elastic).
- If fewer substitutes are available, there is not many alternatives to switch to and therefore the quantity demanded is less responsive (inelastic).
A specific type of toothbrush has many substitutes. If the prices of one noticeably increase, most people would just switch to a cheaper one.
Further, it is important to know how close these substitutes are to each other, since they are a determining factor on whether consumers want to switch.
- If two substitutes are closer, consumers wouldn't mind switching over. This makes the demand more elastic.
- If they aren't very good substitutes of each other, consumers still might not switch.
Sprite and 7UP are much closer substitutes of each other than Sprite and Almond Milk.
Another important factor is the broadness of the product.
- If we look at broad categories of products (Food, Water, Coffee etc.) then they will likely have fewer substitutes.
- If you focus into narrower products (specific type of food, brand of bottled water, a type of espresso) then they will likely have more substitutes.
- A specific type of coffee has more substitutes (other types of coffee)
- Coffee itself doesn't have many (maybe tea)
- A drink technically has none (you have to drink something!)
Degree of Necessity
Necessities
Goods or services crucial for ones' life and extremely difficult to live without.
Luxuries
Goods and services that are not essential for one's living and often bought for leisure purposes.
Necessities tend to have inelastic demand, as consumers cannot easily reduce consumption. Luxuries, on the other hand, have elastic demand.
Demand for food (a necessity) is less elastic than demand for high-end hand bags (a luxury).
If the price of food increases, people will still need to buy it to eat and survive. If it decreases, people don't just magically start eating more.
Addictive Goods
A unique case of necessity goods are those that people get addicted to. Even though it might not be a good they need physically, their addiction demands it.
- People can get addicted to goods and services like cigarettes, alcohol, gambling and many more.
- Even though these aren't crucial for living, it is hard for people to change their spending patterns even if the price rises due to their addiction.
Proportion of Income Spent
Goods that take up a large share of income (e.g., cars) tend to have elastic demand, while inexpensive items (e.g., chewing gum) are inelastic.
- If the price percentage increases by a lot, people might not be able to buy a car while chewing gum prices won't affect them much.
- If the price percentage falls, more people might demand cars or luxury cars while people won't just buy more chewing gum.
A price decrease might not mean people buy more cars, but instead more people buy cars.
Time
Time is an essential factor as consumers can adjust their behaviour and think more thoroughly about their decision, making demand more elastic.
If a person has a lot of time when given a choice to buy a car, they will consider all alternatives making them switch to the better option if the price rises.
However if they have to make a decision in a hurry, they might not know the other alternatives or choices and impulsively buy, making their demand inelastic.
PED and Total Revenue
Total Revenue
Amount of money received by firms when they sell a good (or service).
Total revenue is calculated as:
$$\text{TR} = \text{Price} \times \text{Quantity Sold}$$
- Different increases in $P$ lead to different decreases in $Q$, so the effect on total revenue will depend upon the elasticity of demand.
- As a result, understanding how to price their goods and services helps firms to increase their revenue.
Price Inelastic Demand (PED < 1)
When demand is inelastic, raising prices increases total revenue and decreasing prices causes total revenue to fall.
- Total revenue increases because the percentage drop in quantity demanded is smaller than the percentage increase in price.
- Total revenue decreases because the percentage increase in quantity demanded is smaller than the percentage decrease in price.
The figure above shows the points along the demand curve with: $P_1 = 10, P_2 = 15$ and $P_3 = 5$ alongside $Q_1 = 100, Q_2 = 75$ and $Q_3 = 125$.
At point $P_1$, whether we move up or down the curve, the demand is inelastic (verify yourself).
The total revenue at $(P_1,Q_1)$ is:
$$ TR = 10 \times 100 = \$ 1000 $$
- If the price increases to $P_2$ then total revenue will also increase to $P_2Q_2=15\times 75=\$ 1125$.
- If the price decreases to $P_3$ then the total revenue will also decrease to $P_3Q_3 = 5 \times 125 = \$ 625$.
Price Elastic Demand (PED > 1)
When demand is elastic, raising prices decreases total revenue and decreasing prices causes total revenue to increase.
- Total revenue decreases because the percentage drop in quantity demanded is larger than the percentage increase in price.
- Total revenue increases because the percentage increase in quantity demanded is larger than the percentage decrease in price.
The figure above shows the points along the demand curve with $P_1 = 10, P_2 = 12.5$ and $P_3 = 7.5$ alongside $Q_1 = 100, Q_2 = 50$ and $Q_3 = 150$.
At the point $P_1$, whether we go up or down the curve, the demand is elastic (verify yourself).
The total revenue at $(P_1,Q_1)$ is:
$$ TR = 10 \times 100 = \$ 1000 $$
- If the price increases to $P_2$ then total revenue will decrease to $P_2Q_2=12.5\times 50=\$ 625$.
- If the price decreases to $P_3$ then the total revenue will increase to $P_3Q_3 = 7.5 \times 150 = \$ 1150$.
Unit Elastic Demand (PED = 1)
When demand is unit elastic throughout the entire demand curve, changing prices doesn't affect total revenue.
- Total revenue doesn't change because the percentage change in quantity demanded is the same as the percentage change in price.
Changing PED Along a Linear Demand Curve (HL Only)
On a straight-line downward-sloping demand curve, PED varies along its length:
As observed in the figure above, for higher prices and lower quantities, the demand is elastic while for lower prices and higher quantities, it is inelastic.
- At high prices (low quantities), a change in price is worth a lower percentage of price, and a change in quantity is worth a higher percentage of quantity.
- This can be seen by high prices (lower quantities) leading to a smaller $\frac{\Delta P}{P}$ since the denominator is big.
- And a higher $\frac{\Delta Q}{Q}$ since the denominator is small.
- Therefore the responsiveness of quantity demanded to price is high and the demand is elastic.
The argument follows similarly for low prices (high quantities)
- At low prices (higher quantities), a change in price is worth a higher percentage of price, and a change in quantity is worth a lower percentage of quantity.
- This can be seen by low prices (higher quantities) leading to a larger $\frac{\Delta P}{P}$ since the denominator is small.
- And a lower $\frac{\Delta Q}{Q}$ since the denominator is big.
- Therefore the responsiveness of quantity demanded to price is low and the demand is inelastic.
Mathematical Derivation
We can rewrite the elasticity function as:
$$ PED = \frac{\frac{\Delta Q}{Q}}{\frac{\Delta P}{P}} = \frac{\Delta Q}{\Delta P}\times \frac{P}{Q} = \frac{1}{\frac{\Delta P}{\Delta Q}}\times \frac{P}{Q} = \frac{1}{slope} \times \frac{P}{Q}$$
Where $\frac{\Delta P}{\Delta Q}$ is the slope because $P$ is the $y$-axis and $Q$ is the $x$-axis allowing us to use $\frac{rise}{run}$ since we are focusing on a straight line.
The slope is constant for a straight line hence the only variable is $\frac{P}{Q}$.
- For higher prices, $\frac{P}{Q}$ is larger, and so more elastic.
- For lower prices, $\frac{P}{Q}$ is smaller, and so more inelastic.
Even though the slope of a linear demand curve is constant, PED changes because it depends on percentage changes, not absolute changes.
- Why do you think necessities like water have inelastic demand, while luxuries like vacations have elastic demand?
- How might cultural differences influence the elasticity of demand for certain goods?
- To what extent can firms manipulate PED through marketing or product differentiation?
Can you explain what might affect the PED of a good and showcase the different PEDs on a graph?
The following figure shows the demand curve in the market of cigarettes.
- Define price elasticity of demand. [2 marks]
- Calculate the price elasticity of demand at the price of $5$. [3 marks]
- Using your answer to part (2), explain why the demand might be facing this elasticity. [4 marks]
Solution
- A measure of the responsiveness of quantity demanded (1 mark) when there is a change in price. (1 mark)
- Calculate percentage change in $Q$
$$ \%\Delta Q = \frac{20-25}{25} \times 100 = \frac{-5}{25} \times 100 = -20\% $$
Calculate percentage change in $P$
$$ \%\Delta P = \frac{10-5}{5} \times 100 = 100\%$$
Calculate PED
$$ PED = \frac{-20}{100} = \frac{-1}{5} $$ - The elasticity found in part (2) shows that at $P=5$, hence the demand is inelastic (1 mark). Cigarettes are addictive goods (1 mark) and therefore even with substantial price increases, the quantity demanded will not fall significantly as consumers cannot change their consumption due to their addiction (1 mark). They may also be a small part of income for many consumers. (1 mark)
For the last (1 mark), student can give any other reason including:
- Fewer Substitutes
- Psychological Factors
- Any other valid factor
