Comprehensive Notes on Price Elasticity of Demand, Supply, Income, and Cross-Price Elasticities

Fundamentals of Price Elasticity of Demand

• Definition of Price Elasticity of Demand:

  • Price elasticity of demand measures the sensitivity or responsiveness of buyers (consumers) to changes in price.

  • Formally defined as the percentage change in quantity demanded resulting from a $1\%$ change in price.

• Standard Percentage Change Calculation & Asymmetry Problem:

  • Standard formula for percentage change:     $\text{Percentage Change} = \frac{\text{End Value} - \text{Initial Value}}{\text{Initial Value}}$

  • Standard percentage change calculations depend entirely on the initial starting value, producing asymmetrical results depending on the direction of movement along the demand curve.

• Demonstration of Asymmetry (Cell Phone Demand Example):

  • Market data for cell phones (iPhones):

  • At $P = \$250$, quantity demanded $Q = 8$ units.

  • At $P = \$200$, quantity demanded $Q = 12$ units.

  • Case 1: Movement from Point A ($P_1 = \$250, Q_1 = 8$) to Point B ($P_2 = \$200, Q_2 = 12$):

  • Percentage change in quantity demanded:       $\%\Delta Q_d = \frac{12 - 8}{8} = \frac{4}{8} = 0.5 = 50\%$

  • Percentage change in price:       $\%\Delta P = \frac{200 - 250}{250} = \frac{-50}{250} = -0.2 = -20\%$

  • Elasticity of demand calculation:       $E_d = \frac{50\%}{-20\%} = -2.5$

  • Taking the magnitude yields $2.5$.

  • Case 2: Movement from Point B ($P_1 = \$200, Q_1 = 12$) to Point A ($P_2 = \$250, Q_2 = 8$):

  • Percentage change in quantity demanded:       $\%\Delta Q_d = \frac{8 - 12}{12} = \frac{-4}{12} = -0.3333 = -33.33\%$

  • Percentage change in price:       $\%\Delta P = \frac{250 - 200}{200} = \frac{50}{200} = 0.25 = 25\%$

  • Elasticity of demand calculation:       $E_d = \frac{-33.33\%}{25\%} = -1.33$

  • Taking the magnitude yields $1.33$

  • Negative Correlation: The negative sign in price elasticity reflects the Law of Demand, indicating that price and quantity demanded move in opposite directions.

The Midpoint Method for Calculating Elasticity

• Purpose of the Midpoint Method:

  • To avoid obtaining two different elasticity values for the same segment of a demand curve, economists use the Midpoint Method.

  • The Midpoint Method calculates percentage changes relative to the average (midpoint) of the initial and final values, ensuring a unique number regardless of the direction of movement.

• Midpoint Formula for Price Elasticity of Demand:   $E_d = \frac{Q_2 - Q_1}{P_2 - P_1} \times \frac{P_2 + P_1}{Q_2 + Q_1}$

  • $Q_1, Q_2$: Initial and final quantities demanded.

  • $P_1, P_2$: Initial and final prices.

• Step-by-Step Calculation Example (Hotel Room Demand):

  • Market data:

  • At $P_1 = \$70$, $Q_1 = 5,000\,\text{rooms}$.

  • At $P_2 = \$90$, $Q_2 = 3,000\,\text{rooms}$.

  • Plugging values into the Midpoint Formula:     $E_d = \frac{3,000 - 5,000}{90 - 70} \times \frac{90 + 70}{3,000 + 5,000}$     $E_d = \frac{-2,000}{20} \times \frac{160}{8,000}$     $E_d = -100 \times 0.02 = -2$

  • Economic Interpretation: An $E_d = -2$ means that if the price of hotel rooms increases by $1\%$, the quantity demanded for hotel rooms decreases by $2\%$.

Determinants of Price Elasticity of Demand

• Availability of Close Substitutes (Lesson 1):

  • Demand is more elastic when a good has many close substitutes available because consumers can easily switch to alternatives if the price rises.

  • Breakfast Cereal vs. Sunscreen Example: Breakfast cereal has many close substitutes (including non-cereal options or leftover party pizza), making consumers highly price sensitive. Sunscreen has very few substitutes, making buyers less price sensitive.

  • Water Example: Water has virtually zero close substitutes; therefore, its elasticity is extremely low.

  • Rule: Elasticity is higher when more close substitutes are available.

• Necessities vs. Luxuries (Lesson 2):

  • Necessities have inelastic demand because consumers require them regardless of price increases. Luxuries have elastic demand because consumers can easily forgo them when prices rise.

  • Insulin vs. Caribbean Cruises Example: Insulin is a vital necessity for diabetics, so price changes cause minimal drop in quantity demanded. A Caribbean cruise is a luxury, so a price increase causes a large drop in quantity demanded.

  • Electrical Power Example: Electrical power is a necessity with few practical energy substitutes for daily appliances, resulting in a very small elasticity value.

  • Rule: Elasticity is higher for luxury goods than for necessities.

• Time Horizon (Lesson 3):

  • Demand is more elastic over longer time horizons because consumers have more time to adjust their behavior and adopt alternatives.

  • Gasoline Short Run vs. Long Run Example: If gasoline prices rise by $20\%$ overnight (e.g., from $\$2.00$ to $\$2.40$ per gallon), short-run behavior changes minimally because driving is necessary. Over a long-run period (e.g., 1 year), consumers adjust by carpooling, relocating closer to campus, taking public transit, or buying bicycles.

  • Rule: Elasticity is higher in the long run than in the short run.

• Definition of the Market (Broad vs. Narrow Categorization):

  • Narrowly defined markets have more substitutes and higher elasticity than broadly defined markets.

  • Nike Running Shoes Example: Nike running shoes have a higher price elasticity of demand than running shoes as a broad category because consumers can switch between specific brands (e.g., Nike vs. Adidas).

Elasticity, Slope, and Classification of Demand Functions

• Relationship Between Elasticity and Curve Slope:

  • Price elasticity of demand is related to the inverse of the slope of the demand curve.

  • Rule of Thumb: The flatter the demand curve, the higher the elasticity; the steeper the demand curve, the lower the elasticity.

• The Five Categories of Demand Elasticity:

  • 1. Perfectly Inelastic Demand ($E_d = 0$):

  • Visual Representation: Vertical line.

  • Consumer Response: A $10\%$ drop in price results in a $0\%$ change in quantity demanded.

  • Price Sensitivity: Buyers have zero price sensitivity.

  • Example: Water in extreme survival conditions (e.g., approaching an oasis in a desert).

  • 2. Inelastic Demand ($0 < |E_d| < 1$):

  • Visual Representation: Relatively steep curve.

  • Consumer Response: A $10\%$ price change leads to a quantity change of less than $10\%$.

  • Price Sensitivity: Buyers have relatively low price sensitivity.

  • Examples: Necessities, short-run demand, sunscreen, eggs ($0.1$), healthcare ($< 1$), rice ($0.5$), housing ($< 1$).

  • 3. Unit Elastic Demand ($|E_d| = 1$):

  • Visual Representation: Intermediate slope.

  • Consumer Response: Percentage change in quantity demanded equals the percentage change in price.

  • Price Sensitivity: Intermediate price sensitivity.

  • Example: Spirits / alcoholic liquors.

  • 4. Elastic Demand ($|E_d| > 1$):

  • Visual Representation: Relatively flat curve.

  • Consumer Response: A $10\%$ price change leads to a quantity change of greater than $10\%$.

  • Price Sensitivity: Buyers have relatively high price sensitivity.

  • Examples: Luxuries, long-run demand, beef ($1.6$), restaurant meals ($2.3$), Mountain Dew ($4.4$).

  • Mountain Dew Application: An elasticity of $4.4$ means a $10\%$ increase in price reduces quantity demanded by $44\%$ ($\%\Delta Q_d = 4.4 \times 10\% = 44\%$).

  • 5. Perfectly Elastic Demand ($|E_d| = \infty$):

  • Visual Representation: Horizontal line.

  • Consumer Response: Slope is zero, making the elasticity approach infinity. Any price increase drops quantity demanded to zero.

  • Price Sensitivity: Buyers have extreme/infinite price sensitivity.

Elasticity Along a Linear Demand Curve

• Mathematical Decomposition of Elasticity:

  • Elasticity can be rewritten as:     $E_d = \left(\frac{\Delta Q}{\Delta P}\right) \times \left(\frac{P}{Q}\right) = \left(\frac{1}{\text{slope}}\right) \times \left(\frac{P}{Q}\right)$

• Variation Along a Straight-Line Demand Curve:

  • On a linear demand curve, the slope ($\frac{\Delta P}{\Delta Q}$) remains constant throughout.

  • However, the price-to-quantity ratio ($\frac{P}{Q}$) changes continually as you move down the curve.

  • Top Portion of Demand Curve: Price $P$ is high and quantity $Q$ is low, making $\frac{P}{Q}$ large. Demand is highly elastic ($E_d > 1$; e.g., $E_d = 5$ moving from Point A to Point B).

  • Middle Portion of Demand Curve: Demand is unit elastic ($E_d = 1$; moving from Point B to Point C).

  • Bottom Portion of Demand Curve: Price $P$ is low and quantity $Q$ is high, making $\frac{P}{Q}$ small. Demand is inelastic ($E_d < 1$; e.g., $E_d = 0.2$ moving from Point C to Point E).

• Real-World Commodity Examples:

  • Table Salt Example (Inelastic Bottom Region):

  • Salt costs approximately $\$0.50$ to $\$1.00$ per pound at Walmart, amounting to less than $1\,$ cent per pinch.

  • Because price is already extremely low and quantity consumed is high, salt sits at the bottom of the demand curve with a tiny $\frac{P}{Q}$ ratio.

  • A $100\%$ price increase (doubling from $\$1.00$ to $\$2.00$ per pound) causes virtually no change in consumption habits.

  • Microeconomics Textbook Example (Elastic Top Region):

  • Textbooks cost approximately $\$180.00$.

  • Positioned at the upper end of the demand curve, a $20\%$ price increase raises the price to $\$216.00$.

  • Because of the high $\frac{P}{Q}$ ratio, students are very price sensitive and choose alternative options such as renting.

Price Elasticity of Demand and Total Revenue

• Total Revenue Definition:

  • Total Revenue ($TR$) earned by firms equals price per unit multiplied by quantity sold:     $TR = P \times Q$

  • Total revenue earned by sellers is mathematically identical to total expenditure ($TE$) spent by consumers.

• The Two Opposing Forces of a Price Increase:

  • Price Effect: Selling each unit at a higher price increases revenue.

  • Quantity Effect: Selling fewer units due to the Law of Demand decreases revenue.

  • Elasticity determines which effect dominates.

• Total Revenue Rules Based on Elasticity:

  • When Demand is Elastic ($|E_d| > 1$):

  • $\%\Delta Q > \%\Delta P$

  • Price and Total Revenue move in opposite directions.

  • If price increases ($P \uparrow$), Total Revenue falls ($TR \downarrow$).

  • If price decreases ($P \downarrow$), Total Revenue rises ($TR \uparrow$).

  • Example Calculation: At $P = \$200$, $Q = 12$, $TR = \$2,400$. Raising price to $P = \$250$ causes $Q$ to drop to $8$ ($E_d = 1.8$). New $TR = \$2,000$. Revenue loss from lower quantity outweighs revenue gain from higher price.

  • Bar Happy Hours Application: Lowering prices during competitive happy hours increases overall revenue because demand is elastic.

  • When Demand is Inelastic ($|E_d| < 1$):

  • $\%\Delta Q < \%\Delta P$

  • Price and Total Revenue move in the same direction.

  • If price increases ($P \uparrow$), Total Revenue rises ($TR \uparrow$).

  • If price decreases ($P \downarrow$), Total Revenue falls ($TR \downarrow$).

  • Example Calculation: At $P = \$200$, $Q = 12$, $TR = \$2,400$. Raising price to $P = \$250$ causes $Q$ to drop to $10$ ($E_d = 0.82$). New $TR = \$2,500$. Revenue gain from higher price outweighs revenue loss from lower quantity.

• Applied Policy & Market Scenarios:

  • Pharmacies Raising Insulin Prices: Insulin has inelastic demand. A $10\%$ price increase causes total expenditure on insulin to rise.

  • Airline Luxury Cruise Fare Wars: Luxury cruises have elastic demand. A $20\%$ drop in cruise fares causes cruise line total revenue to rise.

Price Elasticity of Supply

• Definition of Price Elasticity of Supply:

  • Price elasticity of supply ($E_s$) measures sellers' sensitivity to price changes.

  • Defined as the percentage change in quantity supplied resulting from a $1\%$ change in price.

• Calculation Formula:

  • Uses the same Midpoint Method formula as demand:     $E_s = \frac{Q_2 - Q_1}{P_2 - P_1} \times \frac{P_2 + P_1}{Q_2 + Q_1}$

  • Sign Convention: The sign of $E_s$ is always positive due to the Law of Supply (price and quantity supplied move in the same direction).

Determinants and Classification of Supply Elasticity

• Determinants of Supply Elasticity:

  • Flexibility / Ease of Adjusting Production: The easier it is for sellers to alter production or exit/enter a market, the more elastic the supply. Beachfront property is fixed and difficult to vary (inelastic supply); car manufacturing is flexible and easier to expand or contract (elastic supply).

  • Time Horizon: Supply is more elastic in the long run than in the short run because firms can build new factories, hire workers, or adopt new technologies.

• Five Categories of Supply Curves:

  • 1. Perfectly Inelastic Supply ($E_s = 0$):

  • Vertical line. Quantity supplied is fixed regardless of price.

  • Example: Original Picasso paintings (fixed stock because Picasso is deceased).

  • 2. Inelastic Supply ($0 < E_s < 1$):

  • Steep curve. Percentage change in quantity supplied is less than percentage change in price.

  • Example: Housing market, beachfront property.

  • 3. Unit Elastic Supply ($E_s = 1$):

  • Percentage change in quantity supplied equals percentage change in price.

  • 4. Elastic Supply ($E_s > 1$):

  • Flat curve. Percentage change in quantity supplied exceeds percentage change in price.

  • Examples: Manufactured goods, automotive industry, processed food industry.

  • 5. Perfectly Elastic Supply ($E_s = \infty$):

  • Horizontal line. Long-run supply curve in perfectly competitive markets under specific standard assumptions.

Market Equilibrium Dynamics and Case Studies

• Market Dynamic Comparison: Beachfront Property vs. New Cars (Demand Doubling):

  • Population growth causes demand for both beachfront property and new cars to double (demand curve shifts right from $D$ to $D'$).

  • Beachfront Property Market (Inelastic Supply): Steep supply curve leads to a small increase in equilibrium quantity ($Q_1 \rightarrow Q_2$) and a massive surge in equilibrium price ($P_1 \rightarrow P_2$).

  • New Car Market (Elastic Supply): Flat supply curve leads to a massive increase in equilibrium quantity ($Q_1 \rightarrow Q_2$) and a small increase in equilibrium price ($P_1 \rightarrow P_2$).

  • Fargo, North Dakota Fracking Boom Real-World Case: A natural gas fracking boom caused a population surge in Fargo. Because short-run housing supply was steep and inelastic, home prices skyrocketed while housing quantity expanded slowly. Workers earning over $\$100,000$ per year were sleeping in Walmart parking lots awaiting new home construction.

• Case Study 1: Good News or Bad News for Farmers? (Corn Market):

  • Initial Equilibrium: Corn demand is inelastic (food staple). Initial price $P = \$3$ per bushel, quantity $Q = 100\,\text{million bushels}$. Initial farm revenue = $\$300\,\text{million}$.

  • Technological Progress: Monsanto introduces a genetically modified (GMO) crop resistant to plant pathogens with higher yields, lowering production costs (reducing fertilizer and pesticide needs).

  • Market Impact: Positive technology shock shifts supply right from $S$ to $S'$.

  • New Equilibrium: Price falls to $P = \$2$ per bushel, quantity increases slightly to $Q = 110\,\text{million bushels}$.

  • New Farm Revenue: \2 \times 110\,\text{million} = \$220\,\text{million}$.

  • Conclusion: Technological advancement is good news for production costs, but bad news for farmer revenue. Inelastic demand causes price to drop severely, outweighing the small quantity gain and dropping farm sector revenue from $\$300\,\text{million}$ to $\$220\,\text{million}$.

• Case Study 2: Drug Interdiction vs. Drug Education Policies:

  • Demand for illegal drugs (e.g., heroin, opioids, methamphetamine) is extremely inelastic due to addiction acting as a necessity.

  • Policy 1: Drug Interdiction (Supply-Side Enforcement):

  • Police arrest drug dealers and confiscate cartel supplies, shifting supply left ($S \rightarrow S'$).

  • Because demand is inelastic, drug quantity drops very little, while drug price surges drastically.

  • Result: Uncaught drug dealers make significantly higher revenue, turning the drug trade into a more lucrative business and attracting new drug dealers. Drug interdiction is an ineffective policy.

  • Policy 2: Drug Education (Demand-Side Intervention):

  • Educational programs (e.g., the Montana Meth Project in the 1990s) inform youth about drug dangers, shifting demand left ($D \rightarrow D'$).

  • Result: Both equilibrium drug quantity and equilibrium drug price fall.

  • Outcome: Total revenue for drug dealers declines, reducing market profitability and discouraging new suppliers from entering. Drug education is an effective policy.

Income Elasticity of Demand

• Definition of Income Elasticity of Demand:

  • Income elasticity of demand ($E_I$) measures the percentage change in quantity demanded resulting from a $1\%$ change in consumer income.

• Calculation Formula (Midpoint Method):   $E_I = \frac{Q_2 - Q_1}{I_2 - I_1} \times \frac{I_2 + I_1}{Q_2 + Q_1}$

  • $I_1, I_2$: Initial and final income levels.

  • $Q_1, Q_2$: Initial and final quantities demanded.

• Classification of Goods Based on Income Elasticity:

  • Inferior Goods ($E_I < 0$):

  • Income and quantity demanded move in opposite directions. As income rises, consumption drops.

  • Example: Hamburgers relative to steak (filet mignon or ribeye).

  • Normal Goods ($E_I > 0$):

  • Income and quantity demanded move in the same direction. As income rises, consumption increases.

  • Superior / Luxury Goods ($E_I > 1$):

  • Demand increases more than proportionally relative to income growth.

Cross-Price Elasticity of Demand

• Definition of Cross-Price Elasticity of Demand:

  • Cross-price elasticity of demand ($E_{xy}$) measures the percentage change in the quantity demanded of Good 1 ($Q_1$) resulting from a $1\%$ change in the price of Good 2 ($P_2$).

• Calculation Formula (Midpoint Method):   $E_{xy} = \frac{Q_{1,2} - Q_{1,1}}{P_{2,2} - P_{2,1}} \times \frac{P_{2,2} + P_{2,1}}{Q_{1,2} + Q_{1,1}}$

  • $Q_{1,1}, Q_{1,2}$: Initial and final quantity demanded of Good 1.

  • $P_{2,1}, P_{2,2}$: Initial and final price of Good 2.

• Classification of Good Relationships Based on Cross-Price Elasticity:

  • Complements ($E_{xy} < 0$):

  • An increase in the price of Good 2 reduces the quantity demanded of Good 1.

  • Example: Bacon and eggs.

  • Substitutes ($E_{xy} > 0$):

  • An increase in the price of Good 2 increases the quantity demanded of Good 1.

  • Example: Coca-Cola and Pepsi.

  • Unrelated Goods ($E_{xy} = 0$):

  • Price changes in Good 2 have zero impact on the quantity demanded of Good 1.