Elasticity

Explain what the price elasticity of demand measures and write its formula using percentage changes: $E_p = \frac{\%\ \Delta Q_d}{\%\ \Delta P}$. What is the difference between elastic and inelastic demand?

The price of a music app rises by $10\ \%$ and subscriptions fall by $20\ \%$.

1. a) Calculate the price elasticity of demand in absolute value.
2. b) Classify the demand (elastic, unit elastic, or inelastic) and interpret the result.

The price of a streaming service rises by $5\ \%$ and subscriptions fall by $10\ \%$.

1. a) Calculate the price elasticity in absolute value.
2. b) Classify the demand and explain what that number means for the company.

When the price of sandwiches in the cafeteria rises from $\$4$ to $\$5$, daily sales fall from $100$ to $90$.

1. a) Calculate the percentage change in price and quantity.
2. b) Calculate the price elasticity in absolute value and interpret the result for the cafeteria.

The management of a private school that receives state funding (similar to a charter school) is running a deficit in the school cafeteria service. The daily lunch menu currently costs $\$5.00$ and has $400$ regular students. To balance the budget, management is considering raising the price to $\$5.50$ ($+10\ \%$), but the parent-teacher association warns that many parents will choose packed lunches or leaving the children with grandparents instead.

1. a) Design a data collection and analysis plan for the school to estimate the price elasticity of demand before implementing the measure. Explain which sources of information (internal or external) you would use and which contextual variables (family income level, type of school day schedule, prices of nearby cafeterias) you would definitely need to control for to avoid getting a biased estimate.
2. b) Assume the study yields a price elasticity of demand in that interval of $1.6$. Calculate the impact of the price increase to $\$5.50$ on the final number of users and on the total daily cafeteria revenue. In light of the result and considering the social and work-life balance function of the service, argue whether the price increase should be maintained and propose two alternative measures (cost management or segmented pricing) to resolve the deficit.

The figure shows two demand curves, D1 and D2, intersecting at point P: one very steep and the other very flat.

1. a) Which of the two curves corresponds to a more elastic demand? Explain your answer by looking at the slope.
2. b) Give an example of a good that could have each curve and justify it.

Explain in your own words what the price elasticity of demand measures. Without doing any calculations: if the price of bread rises by $10\ \%$ and the price of ice cream rises by $10\ \%$, in which one do you expect the quantity bought to fall more? Justify your answer.

Classify these demands as more elastic or more inelastic and justify each one considering substitutes and how necessary the good is: gas for a truck driver, movie tickets, a drug without a generic alternative, and a specific brand of cookies.

The same bottle of water costs $\$0.60$ in the supermarket and $\$2$ at the train station, and it sells in both places.

1. a) In which of the two contexts is demand more inelastic? Justify your answer by thinking about the substitutes available in each place.
2. b) Explain what this teaches us about why prices are so different for the same product.

What does it mean for the price elasticity of demand to equal $1$ (unit elastic)? Explain what happens to the quantity when the price rises by $8\ \%$ and give an example of how that affects the seller's revenue.

A movie theater lowers the ticket price from $\$9$ to $\$7$ and weekend revenue changes from $\$4,500$ to $\$5,600$.

1. a) Calculate how many moviegoers there were before and after the price cut.
2. b) With these data, what can you say about the elasticity of demand in that price range? Reason the relationship between elasticity and total revenue.

A gym raises the monthly fee from $\$30$ to $\$33$ and membership falls from $800$ to $760$.

1. a) Calculate the percentage change in price and quantity.
2. b) Calculate the price elasticity in absolute value and interpret it.
3. c) Calculate revenue before and after the increase and explain the relationship with elasticity.

A publisher lowers the price of an ebook from $\$12$ to $\$9$ and monthly sales rise from $1,000$ to $1,500$ copies.

1. a) Calculate revenue before and after the price cut.
2. b) Without further calculation, what does the result say about the elasticity of demand in that range? Was the cut worth it? Justify your answer.

A public metropolitan transit company operates at a loss. It currently charges $\$1.50$ per ticket and records $120,000$ daily trips. A technical report estimates that the price elasticity of demand in this range is $0.3$ in absolute value.

The City Councilmember for Finance proposes raising the ticket price by $20\ \%$ (up to $\$1.80$), stating: "Since the subway is an essential service, our daily revenue will increase by exactly $20\ \%$, which will allow us to restore the company to financial health while hardly losing any passengers." At the same time, a rider association maintains that any price increase will lead to a drop in total revenue because people will stop using public transit.

1. a) Prove mathematically that both parties are mistaken: calculate the new quantity of daily trips, the percentage change in demand, and the final total revenue. Determine by what exact percentage revenue changes, and explain why the increase in revenue does not match the $20\ \%$ fare hike.
2. b) Imagine that the company operated in a coastal tourist city during the summer, where the price elasticity of the subway was $1.4$ due to an abundance of alternatives (rental bicycles, walking routes, scooters). Predict what would happen to revenue if the same $20\ \%$ price increase were applied, and justify what pricing policy the company should adopt in this second scenario if its goal were to increase revenue.

A movie theater currently charges a flat rate of $\$8$ per ticket across all its showings, but it faces two extreme situations:

• Tuesday and Wednesday afternoon showings: the auditorium ($200$-seat capacity) registers an average attendance of $40$ viewers and the estimated price elasticity is $2.2$.

• Saturday night premiere showings: the auditorium is completely full ($200$ seats), tickets sell out at the box office, and the estimated price elasticity in that time slot is $0.4$.

1. a) Management proposes lowering the price to $\$6$ on weekdays and raising it to $\$10$ on Saturday night. Calculate, with reasoned arguments, how each of these decisions will affect revenue for each showing and predict whether Saturday's attendance will be at risk of falling significantly (taking into account the auditorium's maximum capacity).
2. b) An investor partner fears that Saturday customers will stop going on the weekend to take advantage of the Tuesday discount, or that a resale market for cheap tickets will emerge. Assess what economic conditions must be met to successfully implement this price discrimination, and design two concrete organizational mechanisms that the theater should implement to prevent arbitrage and the loss of audience in the most profitable showings.

To curb pollution, a state government introduced an environmental levy of $\$0.15$ per single-use plastic bag at supermarket checkouts (previously provided for free). During the first year, bag consumption plummeted by $85\ \%$ (falling from $2,000,000$ to $300,000$ units annually). Total revenue was $\$45,000$, while the administrative costs of managing the tax amounted to $\$50,000$.

In the state legislature, a representative criticizes the measure: "The levy is a budgetary fiasco: it generates a deficit of $\$5,000$, which proves its economic inefficiency. To correct this, I propose tripling the levy to $\$0.45$ per bag; that way we will triple revenue and guarantee a fiscal surplus".

1. a) Evaluate the representative's criticism: using the concept of price elasticity of demand and the logic of corrective (Pigouvian) taxes, justify why a low revenue intake is precisely the sign of success of the environmental policy and not its failure.
2. b) Predict what will happen to consumption and revenue if the proposal to raise the levy to $\$0.45$ is approved. Reason whether consumers who still buy bags at $\$0.15$ have more or less elastic demand than those who stopped using them initially, and argue whether revenue will increase in the proportion expected by the representative.

Income elasticity measures how demand reacts when consumers' income changes. If household income rises by $10\ \%$ and restaurant dinners rise by $15\ \%$, calculate the income elasticity and explain what type of good eating out is.

Cross elasticity relates the demand for one good to the price of another. If the price of coffee rises by $10\ \%$ and the demand for tea rises by $5\ \%$, calculate the cross elasticity and explain what relationship there is between the two goods.

A country's income rises by $8\ \%$ and sales change as follows: branded clothing rises by $12\ \%$, bread rises by $1\ \%$ and cheap fast-food value meals fall by $2\ \%$.

1. a) Calculate the income elasticity of each good.
2. b) Classify each good (luxury, basic normal, inferior) and justify the classification.

A city council raised the single urban bus fare from $\$1.20$ to $\$1.50$ ($+25\ \%$). At the end of the year, the annual number of passengers had fallen from $10,000,000$ to $7,500,000$ ($-25\ \%$). The municipal economic report concludes:

"The price elasticity of demand for the bus is exactly unitary ($|-25\ \% / 25\ \%| = 1$), which proves that future price increases will always keep revenue constant".

However, in that same year, three additional factors came into play: mandatory remote working was introduced for state government employees, private car gas prices fell by $15\ \%$, and a segregated network of free bike lanes was built along the main avenues.

1. a) Demonstrate the methodological flaw in the report based on the ceteris paribus assumption. Explain, with reasoned arguments, whether the true price elasticity of demand for the bus is greater or less than $1$ (that is, deduce whether the report is overestimating or underestimating passengers' price sensitivity due to uncontrolled variables).
2. b) Design a methodological proposal so that municipal technicians can empirically isolate the exclusive impact of price on demand. Set out what additional data they would need to collect and how they could use a similar neighboring city (as a control group) that has not changed its fares.

Demand for avocados in the United States suddenly soars.

1. a) Explain what happens to the price in the short run, when farmers cannot produce more from one year to the next (very inelastic supply).
2. b) What happens in the long run, when new avocado trees are planted? Relate the elasticity of supply to the volatility of farm prices.

A sudden tourism boom generates a $40\ \%$ increase in demand for two goods in a coastal region: locally produced strawberries (for restaurants) and residential rental housing. In the first month after the shock, prices of both goods double ($+100\ \%$).

Eighteen months later, the market shows two opposite realities:

• The price of strawberries has returned practically to its original level and marketed production has tripled.

• The price of residential rent remains $80\ \%$ above the initial level and the number of rental apartments has barely increased by $4\ \%$.

1. a) Interpret these data using the concept of price elasticity of supply. Explain how the elasticity of supply changes in the very short run (momentary run), the short run, and the long run in each of the two sectors, justifying the structural causes (production cycle, product perishability, zoning and building permits, and capital stock) that prevent housing from responding with the same agility as strawberries.
2. b) In the face of citizen protests, the city council plans to grant direct subsidies to construction companies, demanding that "within a maximum of six months, they double the supply of rental apartments". Predict the real effectiveness of this measure based on the short-run elasticity of supply, and propose two alternative supply-oriented measures that would be viable over a five-year horizon.

The Treasury Department is considering two tax measures to increase public revenue and address health problems:

• Measure 1: A $30\ \%$ tax on a specific brand of sugary soft drink ("Choco-Cola").

• Measure 2: A $30\ \%$ tax on all tobacco products in general.

A junior adviser writes in their report: "Since the percentage tax rate is identical in both cases ($30\ \%$), we can expect an equivalent revenue and a similar $15\ \%$ reduction in the consumption of both harmful goods".

1. a) Refute the adviser's report by analyzing how the breadth of the market definition (specific brand versus general category) and the availability of substitute goods influence the price elasticity of demand for each good. Predict, with reasoned arguments, in which of the two cases the drop in consumption will be drastic and in which it will be very small.
2. b) The public sector usually pursues two objectives with these taxes: a revenue-raising objective (generating income) and a corrective objective (reducing the consumption of goods with negative externalities). Explain the conflict or trade-off that arises between both objectives when taxing goods with elastic demand versus inelastic demand, and redesign the tax in Measure 1 so that it effectively reduces total consumption of sugary drinks rather than merely diverting buyers to competing brands.

A government approves a permanent increase of $\$0.30$ per liter in excise taxes on motor fuels. At a press conference, the Treasury Secretary states:

"The price elasticity of hydrocarbons in our country is $0.2$. Therefore, fuel consumption will barely decrease, ensuring us record and stable tax revenues over the next ten years to continuously finance public spending."

1. a) Refute the secretary's statement by explaining why price elasticity of demand is not a constant value over time and how it behaves in the short run versus the long run. Justify what technical adjustments, investments in durable goods, and lifestyle changes households and businesses can make over a decade that are impossible to carry out in the first few months.
2. b) Qualitatively predict how both the volume of liters consumed and the tax revenue from the levy will evolve in year 1 versus year 10 after the increase. Design an alternative fiscal policy proposal so that the government does not experience an unexpected drop in tax revenues as substitution toward zero-emission vehicles progresses.