5.2 Polar Cases of Elasticity and Constant Elasticity

Polar Cases of Elasticity

Elasticity measures how sensitive quantity is to a change in price. The values range from zero (completely unresponsive) to infinity (infinitely responsive). Understanding these two extremes, the "polar cases," makes it much easier to interpret all the real-world scenarios that fall between them.

Constant elasticity means the elasticity value stays the same at every point along a curve. Most demand curves have elasticity that changes as you move along them, but the polar cases and unitary elastic curves are special because their elasticity is the same everywhere.

Polar Cases of Elasticity

Infinite and zero elasticity concepts

Perfectly elastic (infinite elasticity) describes a situation where even the tiniest price change causes quantity to change by an unlimited amount.

• The curve is a perfectly horizontal line on a standard price-quantity graph.
• For demand: consumers will buy as much as they want at the market price, but if the price rises even slightly, quantity demanded drops to zero. This fits markets with perfect substitutes, like a single farmer's wheat in a competitive grain market.
• For supply: producers will supply any quantity at a given price but nothing below it. Digital goods like e-book copies come close, since the marginal cost of one more copy is nearly zero.
• The elasticity coefficient equals infinity: $|E| = \infty$

Perfectly inelastic (zero elasticity) describes a situation where quantity does not change at all, no matter what happens to price.

• The curve is a perfectly vertical line.
• For demand: consumers buy the same quantity regardless of price. Life-saving medication like insulin is the classic example, since patients need a specific dose whether the price doubles or drops.
• For supply: the quantity available is fixed and cannot increase. Think of land in a specific location or an original painting. No price increase can create more of it.
• The elasticity coefficient equals zero: $|E| = 0$

Identifying elasticity in graphs

Constant unitary elasticity ($|E| = 1$) is the third constant-elasticity case. On a graph, the demand curve forms a rectangular hyperbola, not a straight line.

• A 1% increase in price leads to exactly a 1% decrease in quantity demanded (and vice versa).
• Because the percentage changes perfectly offset each other, total revenue stays constant at every point along the curve. That's the defining feature of unitary elasticity.
• The slope of this curve is not constant; it's the elasticity that's constant. Don't confuse slope with elasticity.

Quick visual summary for identifying these on a graph:

| Elasticity Type | Curve Shape | $|E|$ Value | Total Revenue Effect | |---|---|---|---| | Perfectly elastic | Horizontal line | $\infty$ | Revenue drops to zero if price rises | | Perfectly inelastic | Vertical line | $0$ | Revenue changes proportionally with price | | Unitary elastic | Rectangular hyperbola | $1$ | Revenue stays constant |

Constant Elasticity

Elasticity in supply vs demand

Price elasticity of demand measures how responsive quantity demanded is to a price change:

$E_d = \frac{\%\Delta Q_d}{\%\Delta P}$

This value is technically negative (price up, quantity down), but we typically use the absolute value.

• Elastic demand ($|E_d| > 1$): Quantity changes by a larger percentage than price. Luxury cars are a good example, since buyers can easily delay or skip a purchase.
• Inelastic demand ($|E_d| < 1$): Quantity changes by a smaller percentage than price. Tobacco fits here because addictive products see little drop in consumption when prices rise.
• Unit elastic demand ($|E_d| = 1$): Quantity changes by the same percentage as price.

To avoid getting different elasticity values depending on whether you calculate from a price increase vs. a decrease, use the midpoint formula:

$E_d = \frac{(Q_2 - Q_1) / [(Q_2 + Q_1)/2]}{(P_2 - P_1) / [(P_2 + P_1)/2]}$

Price elasticity of supply works the same way but for producers:

$E_s = \frac{\%\Delta Q_s}{\%\Delta P}$

• Elastic supply ($E_s > 1$): Producers can ramp up output easily. Mass-produced goods like t-shirts fit here.
• Inelastic supply ($E_s < 1$): Output is hard to change quickly. Oil production is constrained by drilling capacity and geology.
• Unit elastic supply ($E_s = 1$): Quantity supplied changes by the same percentage as price.

Factors that affect demand elasticity:

• Availability of substitutes: More substitutes means more elastic demand. Pepsi and Coke are close substitutes, so a price hike on one pushes buyers to the other.
• Proportion of income: Goods that take up a large share of your budget (housing) tend to have more elastic demand than cheap items (gum).
• Necessity vs. luxury: Necessities like food are inelastic; luxuries like jewelry are elastic.
• Time horizon: Demand for gasoline is inelastic in the short run (you still need to drive), but more elastic in the long run (you can buy a fuel-efficient car or move closer to work).

Factors that affect supply elasticity:

• Time horizon: Farmers can't instantly grow more crops, making short-run supply inelastic. Over several seasons, they can adjust.
• Spare production capacity: A factory running below capacity can increase output easily (elastic). One running at full capacity cannot (inelastic).
• Inventory levels: Retailers with large stockpiles can respond to price changes quickly.
• Ease of switching inputs: Service industries can often adjust more flexibly than capital-heavy manufacturing.

Other types of elasticity

Cross-price elasticity of demand measures how quantity demanded of one good responds to a price change in a different good:

$E_{cross} = \frac{\%\Delta Q_A}{\%\Delta P_B}$

• A positive value means the goods are substitutes (price of Coke rises, demand for Pepsi rises).
• A negative value means the goods are complements (price of printers rises, demand for ink falls).

Income elasticity of demand measures how quantity demanded responds to a change in consumer income:

$E_{income} = \frac{\%\Delta Q_d}{\%\Delta Income}$

• Positive value = normal good (income rises, you buy more). Above 1 means it's a luxury; between 0 and 1 means it's a necessity.
• Negative value = inferior good (income rises, you buy less, like switching from instant noodles to restaurant meals).

Price discrimination is a pricing strategy where firms charge different prices to different groups based on their willingness to pay. This works because different consumer groups have different price elasticities. For example, airlines charge business travelers (inelastic demand) more than vacation travelers (elastic demand) for the same flight.