5.2 The Price Elasticity of Demand

Concept of Elasticity:

• Measures the responsiveness of one variable (in percentage terms) to a percentage change in another variable.

• Calculated as the ratio of those percentage changes: e_{y,x} = \frac{\text{% change in y}}{\text{% change in x}}

• A larger percentage change in 'y' relative to 'x' indicates more elasticity.

Price Elasticity of Demand (e_D):

• A specific type of elasticity that measures how much the quantity demanded (QD) of a good or service changes (in percentage terms) for each one-percent change in its price (P), holding all other things constant.

• Calculation Formula: e_D = \frac{\text{% change in quantity demanded}}{\text{% change in price}}

• Example (Lawnmowers): If a 10% price decrease leads to a 12% increase in lawnmower sales: e_D = \frac{+12\%}{-10\%} = -1.2

Key Characteristics of Price Elasticity of Demand:

• Generally Negative: Due to the inverse relationship between price and quantity demanded (Law of Demand). Economists sometimes ignore the negative sign, but it's important to keep track of it.

• Unit-less: It is a ratio of two percentage changes, so it has no units.

• Interpretation: An elasticity of -1.2 means a 1% decrease in price will cause a 1.2% increase in quantity demanded.

Categorization Based on Absolute Value of e_D:

• Price Elastic Demand (|e_D| > 1):

• Small percentage changes in price lead to larger percentage changes in quantity demanded.

• Example (Lawnmowers): e_D = -1.2, meaning demand is elastic because |-1.2| > 1.

• Unit Price Elastic Demand (|e_D| = 1):

• Changes in price lead to equal and opposite percentage changes in quantity demanded.

• Example (College Tuition): If a 4% price increase leads to a 4% decrease in students: e_D = \frac{-4\%}{+4\%} = -1.0.

• Price Inelastic Demand (|e_D| < 1):

• Large percentage changes in price lead to smaller percentage changes in quantity demanded.

• Example (Netflix): If a 5% price increase leads to a 3% decrease in subscribers: e_D = \frac{-3\%}{+5\%} = -0.6, meaning demand is inelastic because |-0.6| < 1.

The Challenge of Simple Percentage Changes:

• Directly calculating percentage changes between two points (e.g., A to B vs. B to A) can yield different elasticity values, leading to inconsistency.

• Example (Transit Tickets):

• From A ($0.80, 40,000$) to B ($0.70, 60,000$): e_D = \frac{0.5}{-0.125} = -4

• From B ($0.70, 60,000$) to A ($0.80, 40,000$): e_D = \frac{-0.333}{0.1429} = -2.33

Arc Elasticity (Midpoint Method):

• A measure of elasticity that uses average values for price and quantity as the baseline for calculating percentage changes, ensuring the same result regardless of the direction of movement between two points.

• Formula: e{D{\text{Arc}}} = \frac{\Delta Q / \bar{Q}}{\Delta P / \bar{P}} where \bar{Q} is the average quantity and \bar{P} is the average price.

• Calculation Steps:

• Calculate average quantity and change in quantity: \Delta Q = Q{\text{New}} - Q{\text{Initial}}, \bar{Q} = \frac{Q{\text{New}} + Q{\text{Initial}}}{2}.

• Calculate average price and change in price: \Delta P = P{\text{New}} - P{\text{Initial}}, \bar{P} = \frac{P{\text{New}} + P{\text{Initial}}}{2}.

• Compute percentage change in quantity demanded: \frac{\Delta Q}{\bar{Q}}.

• Compute percentage change in price: \frac{\Delta P}{\bar{P}}.

• Divide the percentage change in quantity by the percentage change in price.

• Example (Transit Tickets using Arc Elasticity): From A ($0.80, 40,000$) to B ($0.70, 60,000$):

• Arc % Change in QD: \frac{60,000 - 40,000}{(60,000 + 40,000)/2} = \frac{20,000}{50,000} = 0.4

• Arc % Change in P: \frac{0.70 - 0.80}{(0.70 + 0.80)/2} = \frac{-0.10}{0.75} \approx -0.133

• e{D{\text{Arc}}} = \frac{0.4}{-0.133} \approx -3

Elasticity vs. Slope:

• Elasticity is the ratio of percentage changes, while slope is the ratio of absolute changes (\Delta y / \Delta x). Slope is constant for a linear demand curve, but elasticity varies.

Varying Elasticity:

• For a linear demand curve, the price elasticity of demand changes along its length.

• Top End (High Prices, Low Quantities):

• Demand is highly elastic (|e_D| > 1). At this region, a given dollar change in price represents a smaller percentage change, while the corresponding quantity change represents a larger percentage change relative to the initial low quantity.

• Example: Between points A and B in Figure 5.2, demand is elastic (e.g., -3.00).

• Midpoint:

• Demand is unit elastic (|e_D| = 1). The percentage change in quantity equals the percentage change in price.

• Example: Around points C and D in Figure 5.2, demand is unit elastic (e.g., -1.0).

• Bottom End (Low Prices, High Quantities):

• Demand is highly inelastic (|e_D| < 1). Here, a small dollar change in price represents a larger percentage change, while the corresponding quantity change represents a smaller percentage change compared to the initial high quantity.

• Example: Between points E and F in Figure 5.2, demand is inelastic (e.g., -0.33).

Summary for Linear Demand Curves:

• Highly elastic at the top.

• Unit elastic at the midpoint.

• Highly inelastic at the bottom.

Total Revenue (TR):

• The total amount of money a seller receives from selling goods or services.

• Formula: TR = P \times Q

Relationship Between e_D and TR Changes (Using the "Arrow" Method):

• Elastic Demand (|e_D| > 1):

• When price increases (\uparrow P), quantity demanded decreases significantly (\downarrow\downarrow\downarrow Q), leading to a decrease in total revenue (\downarrow\downarrow TR).

• When price decreases (\downarrow P), quantity demanded increases significantly (\uparrow\uparrow\uparrow Q), leading to an increase in total revenue ($\uparrow\uparrow TR).

• Fact: If demand for rides is elastic, reducing fares will increase revenues.

• Inelastic Demand (|e_D| < 1):

• When price increases ($\uparrow\uparrow\uparrow P), quantity demanded decreases slightly (\downarrow Q), leading to an increase in total revenue (\uparrow\uparrow TR).

• When price decreases ($\downarrow\downarrow\downarrow P), quantity demanded increases slightly (\uparrow Q), leading to a decrease in total revenue ($\downarrow\downarrow TR).

• Fact: If demand for rides is inelastic, increasing fares will increase revenues.

• Unit Elastic Demand (|e_D| = 1):

• When price increases (\uparrow P), quantity demanded decreases proportionally ($\downarrow Q), leading to no change in total revenue (\bar{TR}).

• When price decreases ($\downarrow P), quantity demanded increases proportionally (\uparrow Q), leading to no change in total revenue ($\bar{TR}).

• Fact: When demand is unit elastic, total revenue is maximized and cannot be increased by changing prices.

Visual Summary (Figure 5.3):

• At very high or very low prices, total revenue is zero.

• As price decreases from high levels (elastic region), revenue rises.

• As price increases from low levels (inelastic region), revenue rises.

• Total revenue is maximized at the midpoint of the demand curve where demand is unit elastic.

Perfectly Inelastic Demand (e_D = 0):

• Definition: Price changes have no effect on quantity demanded.

• Graphical Representation: A vertical demand curve.

• Example: Insulin for diabetics (theoretical extreme; slight responsiveness likely in reality).

Perfectly Elastic Demand (e_D = -\infty):

• Definition: Even the smallest price changes have enormous (infinite) effects on quantity demanded.

• Graphical Representation: A horizontal demand curve.

• Example: A single wheat farmer in a perfectly competitive market; cannot sell above the market price, can sell all output at market price.

Non-linear Demand Curves with Constant Elasticity: Some non-linear demand curves (e.g., a hyperbola) can have a constant price elasticity of demand throughout their range, such as -1.00 or -0.50.

Availability of Substitutes:

• The more close substitutes available for a good or service, the greater the absolute value of its price elasticity of demand.

• Fact: If consumers can easily switch to alternatives, they are more sensitive to price changes.

• Examples:

• Potato chips: Highly elastic due to many substitutes (corn chips, tortilla chips, etc.).

• Gasoline (short-term): Inelastic due to few immediate substitutes (-0.2).

• Broader vs. Narrower Categories:

• Demand for broad categories (e.g., "food") is less elastic (e.g., -0.5).

• Demand for narrower categories (e.g., "breakfast cereal") is more elastic (e.g., -0.9).

• Demand for specific brands (e.g., "Cap'N Crunch") is highly elastic (e.g., -2.28) due to even more substitutes.

Importance in Household Budgets (Budget Share):

• The greater the proportion of a consumer's income spent on a product, the more elastic their demand for that product is likely to be.

• Fact: If an item represents a large budget share, consumers are more sensitive to price changes.

• Example: An increase in clothing prices (large budget share) will cause a greater cutback than an increase in salt prices (small budget share).

Time Horizon:

• The longer the time consumers have to adjust to a price change, the more elastic demand is expected to be.

• Fact: Consumers need time to find substitutes, change habits, or make long-term adjustments.

• Example (Crude Oil):

• Short-run price elasticities (e.g., -0.05 to -0.20) are typically lower (more inelastic).

• Long-run price elasticities (e.g., -0.11 to -0.36) are higher (more elastic) as people can change vehicles, use public transport, or insulate homes.

• OPEC Strategy: OPEC exploits the inelastic nature of demand for oil in the short run to restrict output and raise prices, increasing revenues and decreasing costs.

Case in Point: Price Elasticity of Home Water Demand in Phoenix, Arizona:

• Findings: Water demand in Phoenix was inelastic in the short-run (-0.66 from 2000-2002) but became more elastic in the long-run (-1.155 from 2000-2008), as consumers adjusted by investing in water-efficient appliances and redesigning yards.

• Income Effect: Low-income/low-water users show greater price responsiveness due to a larger budget share of water bills. Policy implications suggest tiered pricing to influence high-income/high-water users more effectively.