Price Elasticity of Demand
What is price elasticity of demand?
Elasticity of demand is a measure used in economics to determine the sensitivity of demand of a product to price changes. In theory, this measurement can work on a wide range of products, from low priced items like pencils to more significant purchases like cars. Because of this diversity of products, elasticity of demand looks at percent changes in price rather than absolute changes; for example, a \$10 increase for a pack of pencils would be outrageous, while a \$10 increase for a new car would likely go unnoticed. Ultimately, the equation used to determine elasticity of demand can be simply thought of as: how do a price increase of X% affect the demand of product Y? A higher demand elasticity means that consumers are more responsive to changes in the price of the product.
How to determine elasticity of demand
Given that elasticity of demand calculates the relationship between change in price and change in demand, we can begin to derive the formula:
|$\frac{\text{Percent change in demand}}{\text{Percent change in price}}| = |\frac{∆q/q}{∆p/p}$|
This can be further simplified as:
|$\frac{∆q}{q} \cdot \frac{p}{∆p}| = |\frac{p}{q} \cdot \frac{∆q}{∆p}$|
In the above equation ∆p refers to change in price, while ∆q represents the corresponding change in the quantity of the product demanded. Absolute values are used when determining the coefficient of elasticity, because the correlation between price increase and quantity demand can be assumed to always be negative.
For small changes in price, $\frac{∆q}{∆p}$ can be approximated by the derivative $\frac{dq}{dp}$. This means that we can determine elasticity of demand, E, by substituting in the derivatives of ∆q and ∆p into the above formula.
Therefore, E = |$\frac{p}{q} \cdot \frac{dq}{dp}$|
Important values for elasticity of demand
The word “coefficient” is used to describe the values for price elasticity of demand (E). Different coefficient values have various implications for the price elasticity of demand of products:
- E = 0: demand is perfectly inelastic, meaning that demand does not change at all when the price changes.
- 0 < E < 1: in these cases, the % change in demand from is smaller than the percentage change in price, and the demand is inelastic.
- E = 1: here, the % change in demand is exactly the same as the % change in price, which means that the demand is unit elastic. For example, a price increase of %10 would lead to a 10% decrease in demand.
- E > 1: demand responds more than proportionately to a price increase, so the demand is elastic. For example if a 15% increase in the price of a product corresponds to a 45% drop in demand. In this specific case, E = 3.
The more the demand for a product decreases in relation to the change in price, the more elastic that good is considered.
Application
The demand curve for a product is given by $q = 2000−4p^2$, where p = price. What is the elasticity of the product when the price is \$10?
To solve this problem, first find $\frac{dq}{dp}$. Using the power rule, we know that $\frac{dq}{dp} = -8p$.
We plug this, as well as the price, into the equation, yielding:
$E = |\frac{10}{q} \cdot (-8)$|
To find q, we go back to our original equation.
$q = 2000−4p^2 = 2000−4(10)^2 = 1,600$
Now we have all of the components needed to calculate the price elasticity of demand at price = /$10.
$E = |\frac{10}{400} \cdot (-8)| = .2$
Considering the values of E described above, we know that the product is inelastic at p=10. Another way to think of this is that a 1% increase in price will correspond with an approximately .2% decrease in quantity demand.
Sources:
Applied Calculus 5th Edition (Hughes-Hallet, Gleason, Lock, Faith, et al.) – section 4.6
https://www.tutor2u.net/economics/reference/price-elasticity-of-demand
https://www.extension.iastate.edu/AGDM/wholefarm/pdf/c5-207.pdf
https://www.intelligenteconomist.com/price-elasticity-of-demand/
This post is well-thought out and engaging. You did an excellent job of explaining a complex concept. Defining price elasticity as “sensitivity of demand of a product to price changes,” you emphasize the interrelated nature between the two market variables, price and demand. I appreciated your example, explaining elasticity “a 1% increase in price will correspond with an approximately .2% decrease in quantity demand.” Price elasticity is an example of the market’s push-and-pull dynamic between buyers and sellers. Next, you clarify that price elasticity is relative, not absolute. It is important to note that the same absolute price change will not have the same effect on demand of different goods. I like how you illustrate this relative disparity in the pencil versus car example.
Also, you apply this topic to our course material. Specifically, you explain that we can find the demand’s elasticity (E) by utilizing the derivatives of changes in quantity and price. In class, we learned that the derivative is the rate of change of a function. By differentiating an expression, we are able to discern how fast that function’s path is increasing or decreasing. You apply the derivative concept beautifully here. First, you explain that price elasticity is similar to the derivative by stating its formula, where E = percent change in demand/ percent change in price and the derivative = dy/dx. Then you apply one of the differentiation methods, the power rule, to find the derivative of q=2,000-4p^2, multiplying the exponent (2) by the leading coefficient (4) and subtracting 1 from the exponent.
The values for each elasticity demand were also very helpful. In a market is it better to be unit elastic or inelastic? Which elasticity rate is considered too high, where the buyers are overreacting to a rise in price or causing a shortage because of attractive low prices? I feel if E=1 and goods are unit elastic, then the market remains balanced. Also, what other differentiation methods can you use to find price elasticity. For example, can a good’s quantity or price be expressed by a log or exponential function? Could you apply a sin or cosine function derivative to a cyclical sales cycle, perhaps holiday seasons. So, buyers would be more responsive to lower price changes on Black Friday, but less so in the summer months with higher prices?
I also suggest using the relative rate concept in here. We learned that the relative rate is rate of change in quantity/quantity itself. How can this concept be applied? It may be possible to track not price elasticity, but the change in overall prices in a good over time. Also, you can use economic terms like substitutes, normal goods, and inferior goods to further illustrate the effects in price changes.
Overall, I really enjoyed this post! I learned a lot about price elasticity and how it applies to calculus!
This is an interesting post! It made me think of the gradual increase in prices that I’ve seen with everyday products, and how there are probably people employed by large corporations to calculate the limits to how much the price of a product can increase. It also highlights the trends of product price increases, as I assume the PED is calculated based off of the most recent product? For example, the PED of the new iPhone is probably calculated based off the price of the most recent iPhone. It shows the mathematical process of finding a balance between price and demand in order to maximize profits
Also, I think it is important you emphasized that the PED is the absolute value of the coefficient, because the majority of PED’s are negative but this is ignored for simplicity purposes. The pencil vs. car example was a good introduction to the topic that represented the contrast between price increases between the two. A question I have is whether this equation could also be used for price decreases as well? I’m sure certain products probably sell better at a reduced price, therefore bringing in a higher profit.
I’d never heard of this concept before (never having taken an Econ class) and your post actually made me interested enough to look it up myself. Very well written!