Elasticity (economics) Study Guide

📖 Core Concepts Elasticity – a unit‑less ratio that measures how %‑wise a dependent variable reacts to a %‑wise change in an independent variable (ceteris paribus). Point (infinitesimal) elasticity – uses differentials: \(E=\dfrac{dQ/Q}{dP/P}= \dfrac{dQ}{dP}\cdot\frac{P}{Q}\). Arc elasticity – uses finite changes: \(E{arc}= \dfrac{\Delta Q/Q{avg}}{\Delta P/P{avg}}\). Price Elasticity of Demand (PED) – \(\displaystyle Ed=\frac{\%\Delta Qd}{\%\Delta P}\). Price Elasticity of Supply (PES) – \(\displaystyle Es=\frac{\%\Delta Qs}{\%\Delta P}\). Income Elasticity – \(\displaystyle Ey=\frac{\%\Delta Qd}{\%\Delta Y}\) (positive → normal good, negative → inferior). Cross‑Price Elasticity – \(\displaystyle E{AB}=\frac{\%\Delta QA}{\%\Delta PB}\) ( > 0 → substitutes, < 0 → complements). Elasticity of Scale – \(\displaystyle E{scale}=\frac{\%\Delta Q}{\%\Delta L}\) ( = 1 → CRS, > 1 → IRS, < 1 → DRS). Elasticity of Substitution – measures how easily one factor can replace another while output stays constant. --- 📌 Must Remember |E| > 1 → elastic (more than proportional response). |E| = 1 → unit elasticity (exactly proportional). |E| < 1 → inelastic (less than proportional). Total‑revenue rule: If \(|Ed|>1\), a price cut raises revenue. If \(|Ed|<1\), a price raise raises revenue. Revenue is maximized when \(|Ed|=1\). Tax incidence: the side (demand or supply) that is more inelastic bears the larger share of a per‑unit tax. Determinants of PED: (1) availability of close substitutes, (2) necessity vs. luxury, (3) time horizon, (4) share of income, (5) brand vs. category substitutability. Determinants of PES: (1) scarcity of inputs, (2) number of competitors, (3) production capacity/flexibility. --- 🔄 Key Processes Compute point PED for a linear demand \(Q=a-bP\): \[ Ed = -b\frac{P}{a-bP} \] Arc elasticity (discrete change): Find average quantity \(Q{avg}=\frac{Q1+Q2}{2}\). Find average price \(P{avg}=\frac{P1+P2}{2}\). Plug into \(E{arc}= \dfrac{\Delta Q/Q{avg}}{\Delta P/P{avg}}\). Revenue decision: Calculate \(|Ed|\). Apply the total‑revenue rule (see Must Remember). Tax‑burden analysis: Compute \(|Ed|\) and \(|Es|\). The side with the smaller absolute elasticity pays the larger tax share. --- 🔍 Key Comparisons PED vs. PES – Both use %‑change formula, but PED usually negative (downward‑sloping demand) while PES is positive (upward‑sloping supply). Elastic vs. Inelastic demand – Elastic: many substitutes, luxury, large income share → \(|E|>1\). Inelastic: few substitutes, necessity, tiny income share → \(|E|<1\). Perfectly elastic vs. perfectly inelastic – Perfectly elastic: \(|E|=\infty\) (horizontal line); perfectly inelastic: \(|E|=0\) (vertical line). Income elasticity (normal vs. inferior) – Positive → normal good; Negative → inferior good. --- ⚠️ Common Misunderstandings Elasticity ≠ slope. A steep curve can be inelastic because elasticity also depends on the \(P/Q\) ratio. Sign matters for interpretation (PED is negative by convention; we often report the absolute value). Using % change without a base → results in asymmetric values; arc elasticity corrects this. Assuming “elastic” always means “good for firms.” It only tells you how quantity reacts, not whether the firm’s profit rises. --- 🧠 Mental Models / Intuition Stretchiness analogy: Imagine a rubber band linking price (pull) to quantity (stretch). A loose band (high elasticity) stretches a lot for a small pull; a tight band (low elasticity) barely moves. Revenue seesaw: Price and quantity sit on opposite ends. When the band is elastic, moving the price a little causes a big swing in quantity, tipping the seesaw toward higher revenue when price falls. Tax burden tug‑of‑war: The side that resists being pulled (more inelastic) ends up carrying most of the tax weight. --- 🚩 Exceptions & Edge Cases Perfectly elastic demand – horizontal demand curve; any price above the market price yields zero quantity. Perfectly inelastic demand – vertical curve; quantity never changes regardless of price. Zero elasticity (E = 0) – total insensitivity; occurs for perfectly inelastic supply or demand. Long‑run vs. short‑run elasticity – demand (and supply) are usually more elastic in the long run because consumers/producers have time to adjust. --- 📍 When to Use Which Point elasticity – when you have a specific price‑quantity pair and the change is infinitesimal (e.g., marginal analysis). Arc elasticity – when dealing with discrete price‑quantity changes (e.g., before/after a tax or price change). Income elasticity – to classify a good as normal or inferior and to forecast demand shifts with income growth. Cross‑price elasticity – to assess substitution/complementarity between two products (useful in merger analysis). Elasticity of scale – in production theory to determine returns to scale for a firm or industry. --- 👀 Patterns to Recognize Many close substitutes → high PED (look for “availability of substitutes” in the stem). Luxury goods, large share of income → elastic demand. Long‑run scenarios → more elastic than short‑run. Tax‑incidence questions often give relative elasticities; the side with the lower absolute value bears the burden. Revenue‑maximization problems will have the phrase “unit elasticity” or ask where \(|E|=1\). --- 🗂️ Exam Traps Sign omission: Reporting PED as “2” instead of “‑2” (or forgetting to take absolute value when the question asks for magnitude). Using simple % change instead of arc formula: Leads to asymmetric elasticity when price rises vs. falls. Confusing slope with elasticity: A steep slope does not imply high elasticity; remember the \(P/Q\) scaling factor. Assuming “elastic” ⇒ “higher revenue” – only true when the firm can lower price; a price increase on elastic demand reduces revenue. Mixing up income vs. cross‑price elasticity signs – income elasticity can be negative (inferior), but cross‑price elasticity is positive for substitutes and negative for complements. ---