C13-S1

Microeconomics Seminar Notes: Elasticity and Market Surplus and Problem Set Solutions

Introduction to the Seminar Series on Microeconomics

• This document serves as a comprehensive record of the second installment of the seminar series for the Introduction to Microeconomics course.

• The primary focus of these seminars is the correction of assigned problem sets and a meticulous, step-by-step evaluation of every exercise to ensure a deep understanding of economic mechanisms.

Fundamental Concepts of Elasticity

Prior to addressing the specific exercises, the seminar established three foundational measures of elasticity that underpin all the subsequent problems:

• Own Price Elasticity of Demand: This is defined as the percentage variation in the quantity demanded of a specific commodity in response to a percentage change in the price of that same commodity.

  • Formula: $\epsilon_d = \frac{\% \Delta Q_d}{\% \Delta P}$

  • Assumption of Signs: In 99% of real-world scenarios, the own price elasticity of demand is negative. This follows the basic law of demand: if an object costs more, the quantity people are willing to purchase decreases.

• Cross-Price Elasticity of Demand: This measures the responsiveness of the quantity demanded for a specific good to a percentage change in the price of a different commodity. In this context, revenue or income is often denoted by $R$, but for cross-price contexts, we compare quantity of good 1 ($Q_1$) to the price of good 2 ($P_2$).

  • Formula: $\epsilon_p = \frac{\% \Delta Q_1}{\% Δ P_2}$

  • Significance of Sign:

  • Positive ( > 0 ): The goods are Substitutes. An increase in the price of one leads to an increase in demand for the other (e.g., if coffee prices rise, tea demand may increase).

  • Negative ( < 0 ): The goods are Complements. An increase in the price of one leads to a decrease in the demand for the other (e.g., phones and phone covers).

• Income Elasticity of Demand: This measures the change in quantity demanded relative to a change in the consumer's income (often denoted as $R$ for revenue or income in this seminar).

  • Formula: $\epsilon_r = \frac{\% \Delta Q}{\% \Delta R}$

  • Categorization of Goods:

  • Normal Goods: Income elasticity is positive (> 0). As income increases, consumption of the good increases (e.g., higher-quality electronics).

  • Inferior Goods: Income elasticity is negative (< 0). As income increases, consumption of the good decreases (e.g., public transportation, which consumers may trade for a private car as they become wealthier).

  • Luxury Goods: A sub-category of normal goods where the income elasticity is greater than one (> 1). Consumption increases more than proportionally to the increase in income.

  • Necessity Goods: A sub-category of normal goods where income elasticity is between zero and one (0 < \epsilon_r < 1). Consumption increases less than proportionally to income increases because basic needs are already met.

Relational Examples: Complements and Substitutes

• The Phone and Cover Metaphor: A smartphone and a phone cover are classic complementary goods. Without a phone, a cover has no utility; without a cover, a phone risks damage.

  • If the price of phones triples, consumers will buy fewer phones. Consequently, the demand for phone covers will drop.

  • This illustrates that the cross-price elasticity of covers with respect to the price of phones is negative.

• The Grain and Computer Example: In many cases, markets are unrelated. If the price of computers rises, it is unclear how this would affect the demand for grain. In such cases, the cross-price elasticity may be zero or negligible, as there is no direct relation in the consumer's budget allocation.

Exercise 1: Definition of Cross Price Elasticity

• The exercise requires selecting the correct definition of cross-price elasticity between Cinema Tickets and Online Streaming Services.

• Options and Analysis:

  • Definition A: Price elasticity of cinema tickets divided by the price elasticity of streaming services (Incorrect formula).

  • Definition B: Change in quantity of cinema tickets divided by change in quantity of streaming services (Incorrect formula).

  • Definition C (Correct): The percentage change in the quantity of cinema tickets demanded divided by the percentage change in the price of online streaming services.

• Conclusion: This directly follows the theoretical definition where elasticity is a ratio of percentage changes, not absolute levels ($\frac{\% \Delta Q_{\text{cinema}}}{\% \Delta P_{\text{streaming}}}$).

Exercise 2: Flight Tickets from Barcelona to Geneva

• This exercise involves calculating own price elasticity based on specific data points from Swiss Airlines (Swiss offers).

• Initial Conditions ($P_1, Q_1$): Price = $500 \text{ francs}$; Quantity = $10 \text{ passengers}$.

• New Conditions ($P_2, Q_2$): Swiss Airlines offers a Christmas special. Price = $300 \text{ francs}$; Quantity = $20 \text{ passengers}$ (an increase of 10 passengers).

• Calculation Method:

  • Percentage change in quantity: $\frac{20 - 10}{10} \times 100 = 100 \%$

  • Percentage change in price: $\frac{300 - 500}{500} \times 100 = -40 \%$

  • Elasticity: $\frac{100 \%}{-40 \%} = -2.5$

• Correction during Seminar: A student noted that the instructor initially swapped the denominators in the verbal explanation. The correct methodology requires using the second value minus the first, divided by the first: $\frac{Q_2 - Q_1}{Q_1} / \frac{P_2 - P_1}{P_1}$.

• Result: The elasticity of $-2.5$ indicates that the demand is elastic (|\epsilon| > 1).

Exercise 3: Soccer Balls and Basketballs

• Demand Function: $Q_d = 400 - 5P + 20P_b$

  • Where $P$ is the price of soccer balls ($20$) and $P_b$ is the price of basketballs ($15$).

• Current Quantity Calculation:

  • $Q_d = 400 - 5(20) + 20(15)$

  • $Q_d = 400 - 100 + 300 = 600$

• Method A: Infinitesimal Change (Partial Derivatives):

  • The cross-price elasticity is $\frac{\partial Q}{\partial P_b} \times \frac{P_b}{Q}$.

  • $\frac{\partial Q}{\partial P_b} = 20$

  • Elasticity = $20 \times \frac{15}{600} = 20 \times \frac{1}{40} = 0.5$.

• Method B: Discrete Unit Change:

  • Raise $P_b$ by 1 unit ($P_b = 16$).

  • New $Q_d = 400 - 5(20) + 20(16) = 620$.

  • $\Delta Q = 20; Q = 600; \Delta P_b = 1; P_b = 15$.

  • $\frac{20}{600} / \frac{1}{15} = \frac{1}{30} \times 15 = 0.5$.

• Conclusion: The correct answer is (c), and because the sign is positive, the goods are substitutes.

Exercise 4: Income Changes and Inferior/Normal Goods

• Scenario: Lilith loses her job; her income is cut in half ($R \downarrow$). Her consumption of comics decreases ($Q_{\text{comics}} \downarrow$) and her consumption of ramen increases ($Q_{\text{ramen}} \uparrow$).

• Comics Analysis: $\frac{\% \Delta Q (-) }{\% \Delta R (-) } = \text{Positive Sign}$. Therefore, comics are a Normal Good.

• Ramen Analysis: $\frac{\% \Delta Q (+) }{\% \Delta R (-) } = \text{Negative Sign}$. Therefore, ramen is an Inferior Good.

Exercise 5: Rental Housing in San Francisco (Linear Demand Properties)

• Demand Function: $Q = 2000 - P$ (derived from transcript data: $Q = 1000 + 1000 - P$).

• Point 1 ($P=1600$):

  • $Q = 2000 - 1600 = 400$.

  • Elasticity = $\frac{dQ}{dP} \times \frac{P}{Q} = -1 \times \frac{1600}{400} = -4$.

  • At this high price, demand is highly elastic (a 1% price increase leads to a 4% decrease in quantity).

• Impact on Total Expenditure: Since demand is elastic (|\epsilon| > 1), an increase in price will lead to a decrease in total revenue (expenditure).

  • Proof: At $P=1600$, $TE = 1600 \times 400 = 640000$. At $P=1700$, $Q=300$, $TE = 1700 \times 300 = 510000$.

• Point 2 ($P=1000$):

  • $Q = 2000 - 1000 = 1000$.

  • Elasticity = $-1 \times \frac{1000}{1000} = -1$.

  • This is the Unit Elastic point (midpoint of the linear demand curve).

• Point 3 ($P=800$):

  • $Q = 2000 - 800 = 1200$.

  • Elasticity = $-1 \times \frac{800}{1200} = -2/3$.

  • At this lower price, demand is Inelastic (|\epsilon| < 1).

Theoretical Distinction: Elasticity vs. Slope

• The Issue: Many students confuse the slope of a demand curve with its elasticity.

• Slope: The absolute change in quantity divided by the absolute change in price ($\frac{\Delta Q}{\Delta P}$). For a linear demand curve, the slope is constant.

• Elasticity: The percentage change comparison which is dimensionless and has no unit of measure.

• Why Elasticity is Superior:

  • It is independent of the goods being compared (standardizes comparison between oranges and houses).

  • It allows for a single number to categorize a good as elastic, inelastic, or unit-elastic regardless of units (CHF, kilograms, units).

  • For a linear demand curve, although the slope is constant, the elasticity changes at every single point. It is elastic at high prices, unit-elastic at the midpoint, and inelastic at low prices.

Market Surplus Theory

• Consumer Surplus (CS): The benefit consumers receive, measured as the difference between the maximum price they are willing to pay and the market price they actually pay. Graphically, it is the area below the demand curve and above the market price.

• Producer Surplus (PS): The benefit firms receive, measured as the difference between the market price and the minimum price at which they are willing to sell (marginal cost). Graphically, it is the area above the supply curve and below the market price.

• Total Surplus: The sum of CS and PS. It represents the total benefit society gains from market existence. Graphically, it is the area below the demand curve and above the marginal cost (supply curve).

Exercise 6: Pencil Market Equilibrium and Surplus

• Demand: $Q_d = 120 - 2P$

• Supply: $Q_s = 4P - 48$

• Equilibrium Calculation:

  • $120 - 2P = 4P - 48 \implies 168 = 6P \implies P = 28$.

  • $Q = 120 - 2(28) = 64$.

• Area Calculations using Geometry (Triangles):

  • Price intercept for demand ($Q=0$): $120 - 2P = 0 \implies P = 60$.

  • Price intercept for supply ($Q=0$): $4P - 48 = 0 \implies P = 12$.

  • Consumer Surplus: $\frac{1}{2} \times (60 - 28) \times 64 = 1024$.

  • Producer Surplus: $\frac{1}{2} \times (28 - 12) \times 64 = 512$.

• Result: The correct answer is (b).

Exercise 7: Technological Change and Supply Shifting

• Demand: $Q = 20 - P$.

• Initial Equilibrium ($S_1 = 3P$):

  • $3P = 20 - P \implies P=5, Q=15$.

  • Elasticity Demand: $-1 \times (5/15) = -1/3$.

  • Elasticity Supply: $3 \times (5/15) = 1$.

• New Equilibrium ($S_2 = 4P$):

  • $4P = 20 - P \implies P=4, Q=16$.

  • Elasticity Demand: $-1 \times (4/16) = -1/4$.

  • Elasticity Supply: $4 \times (4/16) = 1$.

• Observations:

  • The supply elasticity remains at 1 (unit elastic) because the supply curve is linear and passes through the origin.

  • The demand elasticity decreases in absolute value as the price decreases along the curve (from $-1/3$ to $-1/4$).

Exercises 8 to 11: Miscellaneous Topics

• Exercise 8 (Discrete Surplus): For 5 individual customers with unique willingness to pay with a market price of $75k$, only those with willingness to pay $\ge 75k$ buy.

  • Surplus = $(120 - 75) + (100 - 75) + (75 - 75) = 45k + 25k + 0 = 70k$.

• Exercise 9 (Qualitative Check): Given own-elasticities of $0.5$ and $0.55$ and a negative cross-price elasticity: The goods are Complements and the demand is Inelastic.

• Exercise 10 (Surplus Definition): Total surplus is the area below the demand curve and above the marginal cost (supply curve).

• Exercise 11 (Revenue Maximization): Total Revenue ($TR = P \times Q$) is maximized where elasticity is unit ($1$). Using the table provided, the combination of $P = 1.5$ and $Q = 95$ ($TR = 142.5$) yielded the highest value compared to other points on the curve.

Questions & Discussion

• Question on Formula Order: A student asked to confirm the formula for elasticity calculations during Exercise 2.

  • Response: The instructor confirmed that the standard formula uses the "new" value minus the "old" value divided by the "old" ($\frac{X_2 - X_1}{X_1}$).

• Question on Inelastic Definition: A student asked for the numerical definition of inelastic demand.

  • Response: Demand is inelastic when the absolute value of the price elasticity is between zero and one (0 < |\epsilon| < 1). This means a persistent price change of 1% results in a quantity change of less than 1%.

• Question on Constant Slope vs. Constant Elasticity: A student asked if the same slope implies the same elasticity.

  • Response: No. A linear demand curve with a constant slope has a varying elasticity at every point. To have constant elasticity throughout, the demand curve must be non-linear (curved), which simplifies to the form $Q = kP^\epsilon$.