C8

Comprehensive Study Notes on Price, Cross-Price, and Income Elasticity

Administrative Updates and Logistics

  • Schedule Changes:

    • Classes will meet twice this week (today and Friday).

    • Classes will also meet twice next week.

    • Swap Announcement: There is a conflict on October 31. The instructor and the assistant have swapped the class and the seminar.

    • Impact on Students: The time and the room remain identical. The only change is that the correction of Problem Set Number Two is postponed until the end of the month.

    • Notification: An official announcement will be posted on the Moodle page.

Fundamental Concepts of Elasticity

  • Definition: Elasticity is a measure of the sensitivity of one variable to a change in another variable.

  • General Application: It can be measured for any relationship between two variables, provided the relationship is logical. It evaluates the percentage change of one variable in response to the percentage change of the other.

  • Types of Elasticity Focused on in Class:

    • Own-Price Demand Elasticity: How the quantity demanded of a good changes relative to a percentage change in its own price.

    • Cross-Price Elasticity: The reaction of the demand for one good to a change in the price of another good.

    • Income (Revenue) Demand Elasticity: The sensitivity of demand to changes in consumer income.

    • Supply Elasticity: How the quantity supplied of a good changes relative to a percentage change in its price.

Own-Price Elasticity of Demand

  • Local Nature: Own-price elasticity is a local concept; it generally changes along the demand curve rather than remaining constant (though rare exceptions exist).

  • Elasticity vs. Slope:

    • In a linear demand curve, the slope is constant throughout.

    • However, the elasticity varies at different points on the same linear curve.

  • Inelastic Portion (Lower part of the curve):

    • The percentage change in quantity demanded is smaller in absolute terms than the percentage change in price.

    • Occurs where quantity is high and price is low. Changes are compared against a large base quantity, making the relative change small.

  • Elastic Portion (Upper part of the curve):

    • The percentage change in quantity demanded is larger in absolute terms than the percentage change in price.

    • Occurs where price is high and quantity is small. Changes are compared against a small base quantity, making the relative impact large.

  • Unit Elastic Point: Specifically at the midpoint of a linear demand curve, the percentage change in quantity demanded is exactly equal to the percentage change in price (∣elasticity∣=1|\text{elasticity}| = 1).

Elasticity and Total Revenue (TR)

  • Revenue vs. Expenditure: Total Revenue is the perspective of the seller (TR=P×QTR = P \times Q); Total Expenditure is the perspective of the buyer (TE=P×QTE = P \times Q).

  • Ambiguity of Price Increases: When price increases, there are two opposing effects on revenue:

    1. Positive Effect: Selling each unit at a higher price tends to increase TR.

    2. Negative Effect: The law of demand dictates that quantity sold will decrease, which tends to decrease TR.

  • Predicting the Outcome:

    • If Demand is Inelastic (|\text{elasticity}| < 1): The price effect dominates. The decline in quantity is small and doesn't offset the price increase. TR increases.

    • If Demand is Elastic (|\text{elasticity}| > 1): The quantity effect dominates. There is a huge reaction in quantity demanded that more than compensates for the price increase. TR decreases.

  • Maximizing Total Revenue: The TR is maximized at the point where the demand curve transitions from elastic to inelastic (the unit elastic point).

Mathematical Derivation of Revenue Change

  • Objective: To link the change in Total Revenue (TR2−TR1\text{TR}_2 - \text{TR}_1) to elasticity.

  • Step-by-step Transformation:

    1. Change in TR=P2Q2−P1Q1\text{Change in TR} = P_2 Q_2 - P_1 Q_1

    2. Manipulate by adding and subtracting P1Q2P_1 Q_2: (P2Q2−P1Q2)+(P1Q2−P1Q1)(P_2 Q_2 - P_1 Q_2) + (P_1 Q_2 - P_1 Q_1)

    3. Factor out terms: (P2−P1)Q2+P1(Q2−Q1)(P_2 - P_1)Q_2 + P_1(Q_2 - Q_1)

    4. Simplified for small changes: dTR=dP×Q+P×dQ\text{dTR} = \text{dP} \times Q + P \times \text{dQ}

    5. Divide by dP\text{dP}: dTRdP=Q+P×dQdP\frac{\text{dTR}}{\text{dP}} = Q + P \times \frac{\text{dQ}}{\text{dP}}

    6. Factor out QQ: dTRdP=Q×(1+PQ×dQdP)\frac{\text{dTR}}{\text{dP}} = Q \times (1 + \frac{P}{Q} \times \frac{\text{dQ}}{\text{dP}})

    7. Recognizing the second term as own-price elasticity (ep\text{ep}): dTRdP=Q×(1−∣ep∣)\frac{\text{dTR}}{\text{dP}} = Q \times (1 - |\text{ep}|)

  • Conclusion: If |\text{ep}| < 1, the bracket is positive, and revenue increases with price.

Real-World Application: Agricultural Markets

  • Scenario 1: Isolated/Closed Market:

    • A remote mountain village with poor transport produces its own agricultural goods. The demand curve is inelastic because there are no substitutes.

    • Shock: A bad weather event causes a negative supply shock (supply curve shifts left).

    • Result: Quantity decreases, but price increases sharply. Because demand is inelastic, the price increase more than compensates for the harvest loss, potentially increasing total revenue for local producers.

  • Scenario 2: Well-Connected Market:

    • A village with good roads has access to agricultural goods from elsewhere. The demand curve is flatter (more elastic) because substitutes are available.

    • Shock: Same negative weather shock (supply shifts left).

    • Result: Producers cannot raise prices significantly because consumers will switch to substitutes. The decline in quantity is large, leading to a decrease in revenue.

Cross-Price Elasticity of Demand

  • Definition: Measures the sensitivity of the quantity demanded of good XX to a percentage change in the price of good YY.

  • Formula:     Cross-Price Elasticity=Percentage change in Quantity of XPercentage change in Price of Y\text{Cross-Price Elasticity} = \frac{\text{Percentage change in Quantity of X}}{\text{Percentage change in Price of Y}}

  • Significance of the Sign: Unlike own-price elasticity where we often look at absolute values, the sign here is crucial:

    • Positive (+) Elasticity: The goods are substitutes (e.g., Meat and Fish). As the price of fish increases, the demand for meat increases.

    • Negative (-) Elasticity: The goods are complements (e.g., Meat and Potatoes, Entertainment and Food). As the price of entertainment increases, the demand for food decreases because they are consumed jointly.

Income Elasticity of Demand

  • Definition: Measures the sensitivity of demand to a percentage change in consumer income (II or Revenue).

  • Formula:     Income Elasticity=dQQdII\text{Income Elasticity} = \frac{\frac{\text{dQ}}{Q}}{\frac{\text{dI}}{I}}

  • Classification of Goods (Taxonomy):

    • Normal Goods: Income elasticity is positive (> 0). Demand increases as income increases.

      • Necessity Goods: Elasticity is between 00 and 11. The share of total expenditure on these goods declines as income rises.

      • Luxury Goods: Elasticity is greater than 11. The share of expenditure spent on these goods increases as income rises.

    • Inferior Goods: Income elasticity is negative (< 0). Demand decreases as income increases (e.g., Rice, Public Transportation, Potatoes).

Case Study: Manga Demand Calculation

  • Data:

    • Initial Income: 3,0003,000

    • New Income: 3,1503,150

    • Initial Quantity: 10,00010,000

    • New Quantity: 12,00012,000

  • Calculation:

    • \text{Percentage Change in Quantity} = \frac{12,000 - 10,000}{10,000} = 20\text{%}

    • \text{Percentage Change in Income} = \frac{3,150 - 3,000}{3,000} = 5\text{%}

    • \text{Income Elasticity} = \frac{20\text{%}}{5\text{%}} = +4

  • Result: Because the elasticity is +4+4 (positive and >1), Manga is a luxury good.

Engel’s Law and Empirical Observations

  • Ernst Engel's Observations (Late 19th Century):

    • Food: Identified as a necessity. Expenditure share decreases as income increases.

    • Clothing, Lighting, Heating, Housing: Expenditure share stayed relatively constant/independent of income (at that time).

    • Luxury items (Education, Health, Leisure): Expenditure share increases as income rises.

  • Historical Shift: In Engel's time, housing was a luxury; modern data classifies housing as a necessity (0 < \text{elasticity} < 1).

  • Graphical Representation (Logarithmic Scales):

    • Using a log-log scale (log(Quantity)\text{log(Quantity)} on one axis, log(Income)\text{log(Income)} on the other) makes the slope of the resulting line equal to the income elasticity itself.

    • This is based on the math rule: d(log x)=dxx\text{d(log x)} = \frac{\text{dx}}{x}.

Point vs. Arc Elasticity Calculation

  • The Reference Point Problem: Moving from point A to B can yield a different elasticity than moving from B to A because the base (denominator) changes.

    • Example: Moving A to B results in −1-1; moving B to A results in −1.22-1.22.

  • Solutions:

    1. Midpoint (Average) Method: Calculate using average prices and quantities as the base to ensure consistency regardless of direction.

    2. Point Elasticity (Derivative Method): Evaluate elasticity for an infinitesimal change in price. This is expressed as the derivative of the demand curve:         Elasticity=dQdP×PQ\text{Elasticity} = \frac{\text{dQ}}{\text{dP}} \times \frac{P}{Q}

    • For a linear demand curve (Q=a−bPQ = a - bP), the elasticity at any point is −b×PQ-b \times \frac{P}{Q}.

Elasticity of Supply

  • Definition: Measures the sensitivity of the quantity supplied by producers/sellers to a change in price.

  • Formula:     Supply Elasticity=Percentage change in Quantity SuppliedPercentage change in Price\text{Supply Elasticity} = \frac{\text{Percentage change in Quantity Supplied}}{\text{Percentage change in Price}}

  • Conceptual Consistency: The definition follows the same logic as demand elasticity, substituting the supply function for the demand function.