C7

Comprehensive Study Notes on Price Elasticity of DemandAdministrative Announcements and Course Context

Friday Schedule: There will be no class and no seminar held this Friday.
University Events: Students are encouraged to participate in university-organized events taking place on Friday. These events often feature high-quality guest speakers and serve as significant community-building moments.
Information Access: Details regarding these events can be found on the university webpage (the instructor noted they missed copying the announcement to the Open Board).
Previous Topics: The course is continuing the discussion on elasticity, a critical analytical device used by policy authorities to determine the impact of market interventions.

The Ambiguity of Total Revenue and Price Changes

The Producer's Dilemma: When a producer increases the price of a good, the effect on total revenue is ambiguous.
Total Revenue ( $TR$ ): Defined as the product of the unit price ( $P$ ) and the quantity sold ( $Q$ ).
- $TR = P \times Q$
Opposing Effects of a Price Increase:
- The Price Effect: Increasing the unit price tends to increase revenue because each unit is sold for more.
- The Quantity Effect: According to the Law of Demand, moving along the demand curve in response to a price increase results in fewer units being sold. This higher price is applied to a smaller number of units.
The Role of Elasticity: Own-price elasticity of demand solves this ambiguity by determining which of these two effects (price or quantity) is stronger.
Terminology by Perspective:
- Seller/Producer Perspective: Reffered to as Total Revenue.
- Buyer/Demand Perspective: Referred to as Total Expenditure or Expenses.
- Both are calculated identically: $P \times Q$ .

Real-World Applications of Elasticity

Policy Interventions in Illegal Markets:
- Supply-Side Measures: In the United States, significant portions of the budget are spent fighting the supply of drugs (e.g., law enforcement and interdiction).
- Demand-Side Measures: In countries like Switzerland and various European nations, interventions focus on prevention and reducing consumption (the demand side).
- Effectiveness: The success of these alternative measures depends heavily on the elasticity of both supply and demand for drugs.
GDP and Food Expenditure:
- Observations show an inverse relationship between log per capita GDP and the log of expenditure on food.
- The slope of this relationship (the derivative of the log-log line) provides the Income Elasticity of Demand.
- Inflation/Deflation Impact: Knowing this elasticity allows central banks to anticipate which consumer groups are most affected by price increases. For example, it helps determine if the rich or the poor are more severely impacted by rising food prices due to inflation.
Types of Elasticity on the Demand Side:
- Own-Price Elasticity: Sensitivity of a good's demand to its own price change.
- Cross-Price Elasticity: Sensitivity of a good's demand to the price change of a related good (e.g., coffee vs. tea).
- Income (Revenue) Elasticity: Sensitivity of demand to changes in consumer income.

Elasticity vs. Slope: Why Elasticity is Preferred

The Limitation of Slope: While the slope of demand and supply curves measures sensitivity, it is considered a poor metric for two primary reasons:
1. Units of Measure: The slope depends on the specific units used (e.g., tons, kilos, liters). This makes it impossible to compare different goods directly.
2. Magnitude Clarity: Simply knowing that a $1 change in coffee price results in a 10-ton demand change doesn't indicate the "importance" or relative weight of that change.
The Advantages of Elasticity:
- Pure Number: Elasticity is a dimensionless value (a "pure number") because it uses percentage changes, allowing the units of measure to cancel out.
- Comparability: It allows economists to compare the sensitivity of vastly different markets, such as meat, vegetables, oil, gas, or drugs.

Mathematical Definition of Own-Price Elasticity of Demand

General Definition: Elasticity is the ratio of the percentage change in one variable to the percentage change in another.
Notation: $\epsilon_{d,p}$ (Epsilon with $(d)$ for demand and $(p)$ for price).
Formula: $\epsilon_{d,p} = \frac{\%\Delta Q}{\%\Delta P} = \frac{\Delta Q / Q}{\Delta P / P}$

Methods of Calculation

There are several ways to calculate the percentage change, depending on the reference point:

Initial Point Method (Standard): $\frac{(Q_1 - Q_0) / Q_0}{(P_1 - P_0) / P_0}$ Note: This is the most common definition used in textbooks and exams unless specified otherwise.
Final Point Method: $\frac{(Q_1 - Q_0) / Q_1}{(P_1 - P_0) / P_1}$
Midpoint (Arc) Elasticity Method: $\frac{(Q_1 - Q_0) / ((Q_1 + Q_0) / 2)}{(P_1 - P_0) / ((P_1 + P_0) / 2)}$ Benefit: This provides a unique number regardless of whether the price is increasing or decreasing by using the average as a reference point.

Numerical Example: Pizza Market

Data:
- Initial Price ( $P_0$ ) = $16$
- New Price ( $P_1$ ) = $18$
- Initial Quantity ( $Q_0$ ) = $300$
- New Quantity ( $Q_1$ ) = $240$
Percentage Change in Quantity ( $\%\Delta Q$ ): $\frac{240 - 300}{300} = -0.2$
Percentage Change in Price ( $\%\Delta P$ ): $\frac{18 - 16}{16} = 0.125$
Resulting Elasticity: $\frac{-0.2}{0.125} = -1.6$
Interpretation of the Sign:
- The negative sign reflects the First Law of Demand (the inverse relationship between price and quantity).
- Because the slope of the demand curve is negative, the own-price elasticity of demand is always negative.
- In practice, economists often use the absolute value ( $|\epsilon|$ ) to compare magnitudes more easily (e.g., comparing $1.6$ to $2.0$ is clearer than comparing $-1.6$ to $-2.0$ ).

Terminology and Categories of Elasticity

Elastic Demand (|\epsilon| > 1): The percentage change in quantity is greater than the percentage change in price. The quantity demanded is very sensitive to price changes.
Inelastic Demand (|\epsilon| < 1): The percentage change in quantity is smaller than the percentage change in price. Quantity is not very sensitive.
Unit Elastic Demand ( $|\epsilon| = 1$ ): The percentage change in quantity exactly equals the percentage change in price. Total expenditure remains constant.
Perfectly Inelastic Demand ( $\epsilon = 0$ ):
- Graph: A vertical line.
- Meaning: Consumers buy the same quantity regardless of price.
- Examples: Life-saving medicine (insulin), water, or goods that create strong addiction (cigarettes).
Perfectly Elastic Demand ( $\epsilon = \infty$ ):
- Graph: A horizontal line.
- Meaning: Any price increase causes demand to drop to zero; any price decrease leads to infinite demand for that specific producer.
- Example: A single producer in a perfectly competitive market selling a homogeneous product.

Factors Influencing Own-Price Elasticity

Availability of Substitutes: The higher the number of good alternatives, the more elastic the demand (e.g., oil vs. nutmeg).
Luxury vs. Necessity: Necessities (food) are inelastic; luxuries (flowers) are more elastic.
Definition of the Market: Narrowly defined markets are more elastic (e.g., chicken legs vs. meat in general).
Time Horizon: Demand is more elastic over the long run as consumers have more time to adjust their behavior.
Share of Income: Goods that represent a large portion of a consumer's budget (housing) are more elastic than those representing a small portion (matches).

Elasticity Along a Linear Demand Curve

Local vs. Global Concepts: In most cases, elasticity is a "local concept," meaning it changes depending on the specific point on the demand curve.
The Linear Curve Paradox: On a linear demand curve, the slope is constant, but elasticity is not.
Variable Elasticity Regions:
- Top Portion (High Price, Low Quantity): Elastic region (|\epsilon| > 1). A change in quantity is large relative to the small starting quantity.
- Bottom Portion (Low Price, High Quantity): Inelastic region (|\epsilon| < 1). A change in quantity is small relative to the large initial quantity.
Special Cases (Global Elasticity): Only perfectly vertical, perfectly horizontal, and equilateral hyperbola (constant unit elastic) curves have the same elasticity at every point.

Questions & Discussion

Question on the negative sign: A student asked why we get a negative number.
Response: It captures the First Law of Demand. When price goes up ( $+\Delta P$ ), quantity goes down ( $-\Delta Q$ ), and vice versa. Therefore, the ratio is always negative.
Question on using Absolute Values: When comparing different demand curves, we use absolute values so that a higher number always indicates higher sensitivity.
Predictive Power Example: If we know $\%\Delta P = 20\%$ and estimated $|\epsilon| = 1.5$ , we can calculate $\Delta Q$ as $30\%$ ( $1.5 \times 0.20$ ). In this predictive context, we re-apply the sign to show that an increase in price causes a decline in demand.