Two scenarios:

Elasticity measures this price sensitivity precisely. Essential for pricing, taxation, and policy!

Key properties:

• \(\varepsilon_d\) is almost always negative (law of demand: \(\uparrow P \Rightarrow \downarrow Q\))
• We often report the absolute value: \(|\varepsilon_d|\)
• It’s unit-free: same whether measuring in euros or dollars, trips or thousands of trips

Example: If \(\varepsilon_d = -2\), then a 1% increase in price causes a 2% decrease in quantity demanded.

💡 Elasticity tells us the proportional response, not the absolute change!

We classify demand based on \(|\varepsilon_d|\):

Key insight: If \(|\varepsilon_d| > 1\) (elastic), consumers are very responsive. If \(|\varepsilon_d| < 1\) (inelastic), they are not very responsive.

Method 1: Point Elasticity (at a specific point on the demand curve)

\[\varepsilon_d = \frac{dQ}{dP} \cdot \frac{P}{Q}\]

• Use when you have a demand function and want elasticity at a particular \((P, Q)\)
• For linear demand \(Q = a - bP\) \(\Rightarrow\) \(\frac{dQ}{dP} = -b\)

Method 2: Arc Elasticity (between two points)

\[\varepsilon_d = \frac{\Delta Q / Q_{avg}}{\Delta P / P_{avg}} = \frac{\frac{Q_2 - Q_1}{(Q_1 + Q_2)/2}}{ \frac{P_2 - P_1}{(P_1 + P_2)/2}}\]

• Use when you observe two discrete points (e.g., before/after price change)
• Uses midpoints to avoid asymmetry issues (increasing 10% \(\neq\) decreasing 10%)

Demand for hotel rooms in Porto: \(Q = 500 - 2P\) (rooms per night, \(P\) in €)

Question: What is the price elasticity of demand when \(P = €100\)?

A museum in Sintra raises ticket prices from €10 to €12. Visitors fall from 1,000/day to 800/day.

Question: What is the arc elasticity of demand?

Five key determinants:

1. Availability of substitutes 🔄
   • More/closer substitutes → more elastic
   • Example: Brand-name hotels (many alternatives) vs. only hotel in remote village
2. Share of budget 💰
   • Larger expense → more elastic
   • Example: International vacation vs. coffee
3. Necessity vs. luxury 💎
   • Luxuries → more elastic; Necessities → less elastic
   • Example: Business travel (necessity) vs. leisure tourism (luxury)

4. Time horizon 🕐
   • Long run → more elastic (more time to adjust)
   • Example: After fuel price increase, tourists eventually switch to closer destinations
5. Definition of the market 🗺️
   • Narrowly defined → more elastic
   • Example: “TAP flights to Paris” (elastic) vs. “all flights to Paris” (less elastic) vs. “all air travel” (even less elastic)

Summary: Demand is more elastic when consumers have options, time, and the good is less essential.

💡 Tourism managers must understand their market’s elasticity to price optimally!

Why?

• Elastic: \(\%\Delta Q > \%\Delta P\) → quantity effect dominates
• Inelastic: \(\%\Delta Q < \%\Delta P\) → price effect dominates

Gulbenkian Museum charges €8/ticket, sells 10,000 tickets/month. Managers estimate \(\varepsilon_d = -1.5\).

Should they raise the price to €10?

Interpretation:

• \(\varepsilon_{ij} > 0\): Goods \(i\) and \(j\) are substitutes (e.g., flights vs. trains)
• \(\varepsilon_{ij} < 0\): Goods \(i\) and \(j\) are complements (e.g., flights and hotels)
• \(\varepsilon_{ij} = 0\): Goods are independent (e.g., milk and concert tickets)

Example: If \(\varepsilon_{\text{hotels, flights}} = -0.4\), a 10% increase in flight prices causes 4% decrease in hotel demand.

Interpretation:

• \(\varepsilon_M > 1\): Luxury good (tourism, fine dining)
• \(0 < \varepsilon_M < 1\): Normal good (most goods)
• \(\varepsilon_M < 0\): Inferior good (budget accommodations, bus travel)

Tourism application: International tourism has high income elasticity (\(\varepsilon_M \approx 1.5 - 2.5\)). During recessions, tourism demand falls sharply!

How airlines and hotels use elasticity:

Revenue maximization: Charge different prices to segments with different elasticities!

This is called price discrimination.

Who really pays a tourism tax?

Today’s Key Takeaways:

1. Price elasticity of demand (\(\varepsilon_d\)): percentage change in quantity per 1% price change
2. Elastic (\(|\varepsilon_d| > 1\)): quantity very responsive; Inelastic (\(|\varepsilon_d| < 1\)): not very responsive
3. Calculation: Point formula \(\varepsilon_d = \frac{dQ}{dP} \cdot \frac{P}{Q}\) or arc formula (midpoint method)
4. Determinants: substitutes, budget share, necessity vs. luxury, time, market definition
5. Revenue: If elastic, price ↑ → revenue ↓; If inelastic, price ↑ → revenue ↑; Maximized at \(|\varepsilon_d| = 1\)
6. Other elasticities: cross-price (substitutes/complements), income (normal/inferior/luxury)
7. Tourism applications: dynamic pricing, tax incidence, demand forecasting

Connection: This builds on demand curves (L8) and leads into supply and market equilibrium (L11+).

Next (after Test 1): Producer theory🏭: costs, profits, and supply!

⚠️ Test 1 is TOMORROW, March 13! Covers Lectures 1–8 (Fundamentals + Consumer). Good luck! 🍀

Question: The demand for luxury cruises from Lisbon is estimated to have a price elasticity of \(\varepsilon_d = -2.5\). If cruise operators raise prices by 8%, what happens to total revenue?

A. Total revenue increases
B. Total revenue decreases
C. Total revenue stays constant
D. Cannot determine without knowing the initial price

Question: Airbnb and traditional hotels are substitutes. If the cross-price elasticity between Airbnb and hotels is \(\varepsilon_{AH} = 0.8\), and Airbnb prices increase by 10%, what happens to hotel demand?

A. Hotel demand increases by 8%
B. Hotel demand decreases by 8%
C. Hotel demand increases by 10%
D. Hotel demand increases by 1.25%

The Algarve Tourism Authority is studying demand for beach resorts. They have the following demand function:

\[Q = 10{,}000 - 20P\]

where \(Q\) is the number of tourists per month and \(P\) is the average price per night (in €).

a) Calculate the price elasticity of demand when \(P = €100\).

b) Is demand elastic or inelastic at this price? What does this mean for revenue if resorts raise prices?

c) At what price is demand unit elastic (\(|\varepsilon_d| = 1\))? What is the quantity demanded and total revenue at this price?

d) The income elasticity for Algarve tourism is estimated at \(\varepsilon_M = 1.8\). If European incomes rise by 5% next year, by what percentage will demand for Algarve tourism increase?

a) Price elasticity at P = €100:

Demand: \(Q = 10{,}000 - 20P\)

At \(P = 100\): \(Q = 10{,}000 - 20(100) = 8{,}000\) tourists

Calculate elasticity: \(\varepsilon_d = \frac{dQ}{dP} \cdot \frac{P}{Q}\)

\[\frac{dQ}{dP} = -20\]

\[\varepsilon_d = (-20) \cdot \frac{100}{8000} = -\frac{2000}{8000} = -0.25\]

b) Elastic or inelastic?

\(|\varepsilon_d| = 0.25 < 1\) → Demand is inelastic at \(P = €100\).

Revenue implication: Since demand is inelastic, price and revenue move in the same direction. If resorts raise prices, total revenue will increase. Quantity falls, but by a smaller percentage than price rises, so \(TR = P \times Q\) increases.

c) Unit elastic demand (\(|\varepsilon_d| = 1\)):

For linear demand \(Q = 10{,}000 - 20P\) (or inverse: \(P = 500 - 0.05Q\)), unit elasticity occurs at the midpoint.

Method 1 (midpoint of inverse demand):

Choke price (intercept): \(P = 500\) when \(Q = 0\)

Maximum quantity: \(Q = 10{,}000\) when \(P = 0\)

Midpoint: \(P^* = \frac{500}{2} = €250\), \(Q^* = \frac{10{,}000}{2} = 5{,}000\) tourists

Verify: \(\varepsilon_d = (-20) \cdot \frac{250}{5000} = -\frac{5000}{5000} = -1\) ✓

Total Revenue at unit elasticity:

\[TR = P^* \times Q^* = 250 \times 5{,}000 = €1{,}250{,}000\]

This is the maximum possible revenue on this demand curve!

d) Income elasticity:

\(\varepsilon_M = 1.8\) means a 1% increase in income causes a 1.8% increase in demand.

If incomes rise by 5%:

\[\% \Delta Q = \varepsilon_M \times \% \Delta M = 1.8 \times 5\% = 9\%\]

Demand for Algarve tourism will increase by 9%.

💡 Since \(\varepsilon_M > 1\), tourism is a luxury good, highly responsive to income changes!