6 Elasticity

6.2 Elasticity = Escape

Price elasticity of demand measures how sensitive the quantity demanded of a good is to changes in its price. For example, the price elasticity of demand for apples is -1.159, while for potatoes, it is -0.13. Both values are negative because an increase in price typically leads to a decrease in the quantity demanded. However, the absolute value of the elasticity for apples is higher than for potatoes, meaning the demand for apples is more elastic, or more sensitive to price changes, compared to potatoes. This makes sense because apples have many close substitutes (snack on an orange instead, or make a pie out of blueberries), making it easier for consumers to switch when prices rise.

In contrast, potatoes have a much lower absolute price elasticity of demand at -0.13, indicating that demand is not very sensitive to price changes. Potatoes are a staple in many diets, whether they’re boiled, mashed, roasted, baked, or fried, and are essential in many processed foods and animal feed. Therefore, consumers are less likely to reduce their consumption significantly, even if prices increase. This is inelastic demand.

Several factors influence the price elasticity of demand for a good:

Broader categories tend to be more inelastic compared to more specific categories. For instance, “alcoholic beverages” have a more inelastic demand compared to “red wine” because if the price of red wine rises, consumers can easily switch to other alcoholic options like white wine, liquor, or beer. However, if the price of all alcoholic beverages increases, consumers find it harder to avoid the higher prices and are more likely to continue purchasing them.
Elasticity of demand tends to be higher over longer periods. Over time, consumers can adjust their behavior more easily in response to price changes. Gasoline is a common example: in the short term, people have few alternatives and must continue buying gas despite price increases, making the demand inelastic. However, in the long run, people might switch to more fuel-efficient vehicles, move closer to work, or use alternative modes of transportation, making the demand for gasoline more elastic over time.

6.3 Elasticity = $\frac{\% \Delta Q}{\% \Delta P}$

I’ll use the notation $\% \Delta$ to refer to “percentage change”: if the price of a movie ticket goes from $5 to $6, how much has it changed in percentage terms? The formula for percentage change is “new minus old, over old”:

\[\frac{\text{new} - \text{old}}{\text{old}} = \frac{6 - 5}{5} = \frac{1}{5} = 20\%\]

So we’d say the price of the movie ticket increased by 20%.

Price elasticity of demand is defined by the percentage change in quantity demanded divided by the percentage change in price. So if, when the price of hamburgers increased by 10%, the quantity demanded of hamburgers fell by 20%, we’d say that hamburgers has a price elasticity of demand of $\frac{-20\%}{10\%} = -2$. Note that the units $\%$ cancel: elasticity is a unitless measure, and no matter whether price is measured in dollars or euros, or whether quantity is measured in lbs or kilograms, the elasticity will be the same.

If something is perfectly inelastic, that means that the quantity demanded does not change at all as the price of the good changes. Because elasticity is the percent change in quantity demanded divided by the percent change in price, in the case of a perfectly inelastic good, we’d have:

\[\varepsilon = \frac{\% \Delta Q \text{ is very small}}{\% \Delta P \text{ is large}} = 0\]

Examples of goods that might have perfectly inelastic demand are goods that are very hard for people to escape: no matter the price, people are willing to pay: life-saving medications, utilities like water and electricity, and addictive substances.

The opposite is a good that is perfectly elastic: even the smallest changes in price leads to extremely large changes in quantity demanded:

\[\varepsilon = \frac{\% \Delta Q \text{ is large}}{\% \Delta P \text{ is very small}} = -\infty\]

Examples of goods that might have perfectly elastic demands are perfect substitutes: if two brands of bottled water are identical and one raises its price by even a cent, consumers would switch entirely to the other brand. If a digital product, like a particular e-book or software, is available from multiple sellers at the same price, the demand could be perfectly elastic. If one seller raises the price, consumers will instantly switch to another seller offering the same product at the original price.

6.4 Elasticity is Related to the Slope of the Demand Curve

We’ll explore this concept in Classwork 6.

6.5 Classwork 6

Type this up in a quarto document. Make sure to show your work in LaTeX.

1) Over a linear demand curve $p = -mq + b$, the price elasticity of demand ranges from $0$ to $-\infty$.

Show this is true by calculating the elasticity at different points on the demand curve:

a) Let $q$ increase and find $\varepsilon$ for $q$ going from 0 to a very small number $\alpha$. That’s the elasticity at the y-intercept.

b) Then let $q$ go from a large number near the x-intercept to the x-intercept itself. What’s the elasticity at the x-intercept?

c) For what $q$ value does a linear demand curve have $\varepsilon = -1$, and what is the significance of that $q$ value?

2) Question 1 highlights a problem with estimating a linear demand curve for a good: depending on where you are on the demand curve, you’ll get very different estimates for the elasticity of the good. A solution is to instead estimate a constant elasticity demand curve: instead of $p = -m q + b$, fit the model $p = a q^b$. The elasticity is guaranteed to be constant over the entire curve.

a) Take logs of both sides of the constant elasticity curve $p = a q^b$. Show that it simplifies to this, so this is also a constant elasticity demand curve: $\log(p) = a + b \log(q)$.

b) Consider these data points for the quantity demanded at any given price. Use lm() to fit the model $\log(p) = a + b \log(q)$. What are the coefficient estimates? Fill in the blanks to draw the constant elasticity demand curve that best fits the data.

library(tidyverse)

demand <- tibble(
  price = 14:1,
  quantity_demanded = 1:14
)

demand %>%
  lm(___)

ggplot() +
  stat_function(fun = function(q) ___ * q ^ ___, color = "red") +
  xlim(0, 5) +
  ylim(0, 5)

c) For the demand equation $p = a q^b$, what is the constant elasticity of demand? To answer this question, we’ll use a slightly different formula for elasticity: instead of $\frac{\% \Delta Q}{\% \Delta P}$, we’ll use $\frac{\frac{dq}{q}}{\frac{dp}{p}}$. Fill in the blanks to finish the proof.

\[\begin{align} \varepsilon &= \frac{\frac{dq}{q}}{\frac{dp}{p}}\\ &= \frac{dq}{q} \times \frac{p}{dp}\\ &= \frac{p}{q} \times \frac{dq}{dp}\\ &= \frac{p}{q} \times \left(\frac{dp}{dq}\right)^{-1}\\ &= \frac{aq^b}{q} \times \left(\frac{d}{dq}(\underline{\phantom{aq^b}})\right)^{-1}\\ &= \frac{aq^b}{q} \times (\underline{\phantom{abq^{b-1}}})^{-1}\\ &= \underline{\phantom{aq^b}} \times \frac{1}{abq^{b-1}}\\ &= \frac{q^b}{\underline{\phantom{bq^b}}}\\ &= \frac{1}{\underline{\phantom{b}}} \end{align}\]

d) Given your answer to the previous question, what did you estimate the elasticity to be in the dataset demand?

6.1 Introduction

6.2 Elasticity = Escape

6.3 Elasticity = \(\frac{\% \Delta Q}{\% \Delta P}\)

6.5 Classwork 6