Comprehensive Study Guide: Price, Income, Cross Elasticity of Demand and Supply

Fundamentals of Elasticity

• Definition of Elasticity: A quantitative measure of the responsiveness or sensitivity of a dependent variable to changes in a relevant independent variable.
• Elasticity Coefficient: The ratio representing the percentage change in a dependent variable when the independent variable changes by one percent ($1\%$).
• General Elasticity Formula:   $\text{Elasticity} = \frac{\%\,\Delta\,\text{in dependent variable}}{\%\,\Delta\,\text{in independent variable}}$
• Examples of Economic Variable Relationships:
  • Dependent Variable ($DV$) = Size of maize crop; Independent Variable ($IV$) = Rainfall
  • Dependent Variable ($DV$) = Investment; Independent Variable ($IV$) = Interest Rate
  • Dependent Variable ($DV$) = Quantity demanded ($Q_d$) of rice; Independent Variable ($IV$) = Price of rice
  • Dependent Variable ($DV$) = Quantity of labour supplied ($Q_s$); Independent Variable ($IV$) = Wage rate
• Four Primary Types of Economic Elasticity:
  1. Price elasticity of demand
  2. Income elasticity of demand
  3. Cross elasticity of demand
  4. Price elasticity of supply

Price Elasticity of Demand

• Definition: The percentage change in the quantity demanded of a product divided by the percentage change in the price of the product, $ceteris\,paribus$. It measures consumer responsiveness to price changes.
• Basic Formula for Price Elasticity of Demand ($e_p$):   $e_p = \frac{\%\,\Delta\,\text{in quantity demanded of a product}}{\%\,\Delta\,\text{in price of the product}}$
• Step-by-Step Derivation of the Point Elasticity Formula:
  • Express percentage changes explicitly using initial quantity ($Q$) and initial price ($P$):     $e_p = \frac{\frac{\Delta Q}{Q} \times 100}{\frac{\Delta P}{P} \times 100}$
  • Cancel out the factor of $100$ from numerator and denominator:     $e_p = \frac{\frac{\Delta Q}{Q}}{\frac{\Delta P}{P}}$
  • Rewrite division as multiplication by the reciprocal:     $e_p = \frac{\Delta Q}{Q} \times \frac{P}{\Delta P}$
  • Regroup the variables to separate quantity changes over price changes:     $e_p = \frac{\Delta Q}{\Delta P} \times \frac{P}{Q}$
• Structural Components of Point Elasticity:
  • The slope of a linear demand curve is defined as $\frac{\Delta P}{\Delta Q}$. The first term in the elasticity equation, $\frac{\Delta Q}{\Delta P}$, is the exact inverse of the slope of the linear demand curve.
  • The second term, $\frac{P}{Q}$, represents the ratio of price to quantity at a specific coordinate point on the demand curve.
  • Because the inverse slope $\frac{\Delta Q}{\Delta P}$ is constant along a straight-line demand curve, but the price-to-quantity ratio $\frac{P}{Q}$ changes continuously at every point, the price elasticity coefficient $e_p$ varies continuously along the length of a linear demand curve.

Units, Relative Changes, and Comparative Analysis

• Unit-Free Characteristic of Elasticity:
  • Elasticity evaluates relative (percentage) changes rather than absolute units.
  • Absolute measures express price in monetary terms (Rands, Pounds, Euros) and quantity in physical units (kilograms, boxes, bags).
  • Standardizing metrics into percentage changes removes measurement units entirely, allowing direct cross-product elasticity comparisons.
• Comparative Example — Cars vs. Bread:
  • Consider an identical absolute price increase of $\text{R}1.00$ applied to both a car (priced at $\text{R}100\,000.00$) and a loaf of bread (priced at $\text{R}10.00$).
  • Evaluating absolute changes ($\text{R}1.00$) implies equal impact, which is misleading.
  • Percentage calculation for the car:     $\%\,\Delta P_{\text{car}} = \frac{\text{R}1.00}{\text{R}100\,000.00} \times 100 = 0.001\%$
  • Percentage calculation for the loaf of bread:     $\%\,\Delta P_{\text{bread}} = \frac{\text{R}1.00}{\text{R}10.00} \times 100 = 10\%$
  • By focusing on percentage adjustments, elasticities reveal that a $\text{R}1.00$ increase causes a negligible percentage change for cars ($<1\%$) but a major percentage change for bread ($10\%$), enabling accurate comparative analysis of consumer sensitivity across different goods.

Point Elasticity vs. Arc (Midpoint) Elasticity

• Point Elasticity Application and Limits:
  • Used for calculating elasticity at a single specific point for infinitesimally small price changes.
  • Point Elasticity Formula:     $e_p = \frac{\Delta Q}{\Delta P} \times \frac{P}{Q}$
  • Variations Along a Linear Demand Curve (where $\frac{\Delta Q}{\Delta P} = -2$):
  • At vertical axis intercept Point A ($P = 10, Q = 0$):       $e_p = -2 \times \frac{10}{0} = -\infty$
  • At midpoint Point C ($P = 5, Q = 10$):       $e_p = -2 \times \frac{5}{10} = -2 \times \frac{1}{2} = -1$
  • At horizontal axis intercept Point E ($P = 0, Q = 20$):       $e_p = -2 \times \frac{0}{20} = 0$
  • Practical Limitation: Extremes where price drops to zero or quantity drops to zero are rarely encountered in practice; only realistic intermediate segments are observed.
• The Direction-Dependence Problem in Point Calculations:
  • Consider a section on a beef demand curve between Point A ($P_1 = 48, Q_1 = 2$) and Point B ($P_2 = 36, Q_2 = 4$), where price is in Rand per kilogram and quantity is in thousands of kilograms.
  • Inverse slope ratio calculation:     $\frac{\Delta Q}{\Delta P} = \frac{4 - 2}{36 - 48} = \frac{2}{-12} = -\frac{1}{6}$
  • Calculating elasticity moving from Point A to Point B (using Point A as base):     $e_p = -\frac{1}{6} \times \frac{48}{2} = -\frac{1}{6} \times 24 = -4$
  • Calculating elasticity moving from Point B to Point A (using Point B as base):     $e_p = -\frac{1}{6} \times \frac{36}{4} = -\frac{1}{6} \times 9 = -1.5$
  • This creates two different elasticity values ($-4$ vs $-1.5$) for the exact same segment of the demand curve depending on the direction of price change.   
• Arc (Midpoint) Elasticity Solution:
  • To eliminate direction dependency and ensure calculations are independent of the starting point, average price and average quantity values are used.
  • Arc Elasticity Formula:     $e_p = \frac{\Delta Q}{\Delta P} \times \frac{\text{Average } P}{\text{Average } Q} = \frac{Q_2 - Q_1}{\frac{Q_1 + Q_2}{2}} \div \frac{P_2 - P_1}{\frac{P_1 + P_2}{2}}$
  • Simplified Arc Expression:     $e_p = \frac{Q_2 - Q_1}{Q_1 + Q_2} \times \frac{P_1 + P_2}{P_2 - P_1}$
  • Step-by-Step Arc Example ($P_1 = 10, Q_1 = 17$ and $P_2 = 8, Q_2 = 19$):
  • Quantity change: $Q_2 - Q_1 = 19 - 17 = 2$
  • Average quantity: $\frac{17 + 19}{2} = 18$
  • Price change: $P_2 - P_1 = 8 - 10 = -2$
  • Average price: $\frac{10 + 8}{2} = 9$
  • Substitution and evaluation:       $e_p = \frac{2}{18} \div \frac{-2}{9} = \frac{2}{18} \times \frac{9}{-2} = \frac{1}{9} \times \left(-\frac{9}{2}\right) = -\frac{1}{2} = -0.5$

Categories of Price Elasticity of Demand

• Absolute Value Rule: By convention, the negative sign resulting from the inverse relationship of the law of demand is ignored; only the absolute value of the coefficient ($|e_p|$) is evaluated when assigning categories.
• Categories Overview:
  • Perfectly Inelastic Demand ($e_p = 0$):
  • Meaning: Quantity demanded ($Q$) does not change at all when price ($P$) changes. Demand is completely unresponsive to price.
  • Revenue Effect: Total revenue ($TR = P \times Q$) moves in the exact same direction as price ($P$). Suppliers have a strong incentive to increase prices.
  • Graphical Shape: A vertical straight line parallel to the price axis.
  • Product Examples: Insulin for diabetics, essential life-saving medications, highly addictive substances.     
  • Inelastic Demand ($0 < e_p < 1$):
  • Meaning: Percentage change in quantity demanded is smaller than the percentage change in price (\%\,\Delta Q_d < \%\,\Delta P).
  • Numerical Example: A $10\%$ increase in price triggers only a $2\%$ decrease in quantity demanded ($10\% > 2\%$).
  • Revenue Effect: Total revenue ($TR$) moves in the same direction as price ($P$). Raising prices increases total revenue; suppliers have an incentive to raise prices.
  • Graphical Shape: A steep downward-sloping curve.
  • Product Examples: Basic necessities such as staple foods, housing, electricity, health care, education, petrol.     
  • Unitarily Elastic Demand ($e_p = 1$):
  • Meaning: Percentage change in quantity demanded equals the percentage change in price (\%\,\Delta Q_d = \%\,\Delta P).
  • Numerical Example: A $10\%$ decrease in price yields an exact $10\%$ increase in quantity demanded.
  • Revenue Effect: Total revenue ($TR$) remains completely unchanged following price adjustments.
  • Graphical Shape: A rectangular hyperbola.     
  • Elastic Demand ($1 < e_p < \infty$):
  • Meaning: Percentage change in quantity demanded is greater than the percentage change in price (\%\,\Delta Q_d > \%\,\Delta P). Highly responsive demand.
  • Numerical Example: A $10\%$ increase in price triggers a $15\%$ decrease in quantity demanded ($15\% > 10\%$).
  • Revenue Effect: Total revenue ($TR$) moves in the opposite direction to price ($P$). Lowering prices increases total revenue; suppliers have an incentive to lower prices.
  • Graphical Shape: A relatively flat downward-sloping curve.
  • Product Examples: Luxuries and non-essentials such as sports cars, CDs, magazines, fast food.     
  • Perfectly Elastic Demand ($e_p = \infty$):
  • Meaning: Consumers purchase an indeterminate quantity at a given price, but if price rises fractionally above that level, quantity demanded drops to zero.
  • Revenue Effect: Increasing price above market level causes quantity demanded to fall to zero, reducing total revenue to zero.
  • Graphical Shape: A horizontal straight line parallel to the quantity axis.
  • Example: Two identical vending machines located side by side selling identical products at identical prices. If one machine increases its price slightly, its quantity demanded drops to zero as consumers purchase exclusively from the cheaper machine.     

Price Elasticity of Demand and Total Revenue

• Consumer Expenditure vs. Supplier Revenue: Total expenditure by consumers on a product ($P \times Q$) is numerically identical to total revenue ($TR$) received by firms supplying that product.
• Strategic Significance: Knowing the price elasticity category enables firm managers and economic policymakers to predict how total revenue will change following a price change.
• Total Revenue Behavior Along a Linear Demand Curve:
  • Elastic Region ($e_p > 1$): Price decreases lead to increases in total revenue ($TR$).
  • Unitary Point ($e_p = 1$): Total revenue ($TR$) reaches its maximum point.
  • Inelastic Region ($e_p < 1$): Price decreases lead to decreases in total revenue ($TR$).
• Numerical Cappuccino Schedule Example:
  • At $P = \text{R}20$, $Q = 0\text{k}$ $\rightarrow$ $TR = \text{R}0$
  • At $P = \text{R}18$, $Q = 2\text{k}$ $\rightarrow$ $TR = \text{R}36\,000$ ($e_p > 1$)
  • At $P = \text{R}14$, $Q = 6\text{k}$ $\rightarrow$ $TR = \text{R}84\,000$ ($e_p > 1$)
  • At $P = \text{R}10$, $Q = 10\text{k}$ $\rightarrow$ $TR = \text{R}100\,000$ (Maximum TR where $e_p = 1$)
  • At $P = \text{R}6$, $Q = 14\text{k}$ $\rightarrow$ $TR = \text{R}84\,000$ ($e_p < 1$)
  • At $P = \text{R}2$, $Q = 18\text{k}$ $\rightarrow$ $TR = \text{R}36\,000$ ($e_p < 1$)
  • At $P = \text{R}0$, $Q = 20\text{k}$ $\rightarrow$ $TR = \text{R}0$

Market Equilibrium and Demand Elasticities

• Impact of Supply Shifts Across Elasticities:
  • When supply increases (supply curve shifts rightward from $S$ to $S'$):
  • Relatively Elastic Demand Curve: A rightward supply shift causes a small drop in equilibrium price ($P_1$ to $P_2$) and a large increase in equilibrium quantity ($Q_1$ to $Q_2$).     
  • Relatively Inelastic Demand Curve: A rightward supply shift causes a large drop in equilibrium price ($P_1$ to $P_3$) and a small increase in equilibrium quantity ($Q_1$ to $Q_3$).     

Determinants of Price Elasticity of Demand

• 1. Substitution Possibilities:
  • Availability of close substitutes increases elasticity. Price increases trigger a strong substitution effect, encouraging consumers to switch to cheaper alternative goods (e.g., switching from train tickets to taxi services).
  • Broad vs. Narrow Definitions: Broadly defined goods have few substitutes and low elasticity (e.g., total meat or toothpaste), while narrowly defined brand products have many substitutes and high elasticity (e.g., beef vs pork, Colgate vs Aquafresh).
• 2. Degree of Complementarity:
  • Goods with high complementarity exhibit low price elasticity (inelastic demand), such as cars and tires, petrol and cars, or CD players and CDs.
  • Low elasticity often stems from the overall system lacking viable substitutes rather than joint consumption itself.
• 3. Type of Want Satisfied:
  • Necessities (basic food, medical care, petrol, electricity) satisfy basic needs and have inelastic demand ($e_p < 1$).
  • Luxuries (swimming pools, entertainment) satisfy non-essential wants and have elastic demand ($1 < e_p < \infty$).
• 4. Time Period Under Consideration:
  • Demand is more price elastic in the long run than in the short run due to the time required to adjust consumption habits and substitute technologies.
  • Historical Example: Crude oil price increases in the 1970s forced minimal short-run consumption changes, but led to long-run developments in fuel-efficient vehicles that significantly increased petrol demand elasticity.
  • Durability: Durable goods can be reused, making demand more elastic because consumers can postpone new purchases when prices rise.
• 5. Proportion of Income Spent on the Product:
  • Products consuming a tiny fraction of income (salt, matches) have inelastic demand because price increases have minimal budget impact.
  • Products consuming a large fraction of income (housing rent, vehicle payments, insurance) exhibit higher price elasticity.
• 6. Number of Uses for the Product:
  • Products with multiple uses (electricity for heating, lighting, cooking) have elastic demand because consumers can selectively eliminate secondary uses when prices rise.
• 7. Addiction:
  • Highly addictive products (tobacco, alcohol, drugs) lower price elasticity. For severely addicted consumers, demand can become virtually perfectly price inelastic ($e_p = 0$).
• Net Determinant Effect: The observed price elasticity of demand for any good represents the combined interaction of all these factors.

Income Elasticity of Demand

• Definition: Measures the responsiveness of quantity demanded to changes in consumer income.
• Formula:   $e_y = \frac{\%\,\Delta\,\text{in quantity demanded}}{\%\,\Delta\,\text{in consumer income}}$
• Categories Based on $e_y$ Values:
  • Normal Essential Good ($0 < e_y < 1$): Quantity demanded increases as income increases, but at a slower percentage rate than income growth.
  • Normal Luxury Good ($1 < e_y$): Quantity demanded increases at a faster percentage rate than income growth.
  • Inferior Good ($e_y < 0$): Quantity demanded decreases as consumer income increases (negative income elasticity), as consumers shift toward higher-quality alternatives.

Cross Elasticity of Demand

• Definition: Measures the responsiveness of quantity demanded of a specific good ($Good\,A$) to a price change in another good ($Good\,B$).
• Formula:   $e_c = \frac{\%\,\Delta\,\text{in quantity demanded of Good A}}{\%\,\Delta\,\text{in price of Good B}}$
• Categories Based on $e_c$ Values:
  • Substitute Goods ($e_c > 0$): Positive cross elasticity. A price increase in Good B increases quantity demanded of Good A (e.g., butter and margarine).
  • Complementary Goods ($e_c < 0$): Negative cross elasticity. A price increase in Good B decreases quantity demanded of Good A (e.g., coffee and sugar, cars and petrol).
  • Unrelated Goods ($e_c = 0$): Zero cross elasticity. Price changes in Good B have no effect on quantity demanded of Good A (e.g., shoes and apples).

Price Elasticity of Supply

• Definition: The ratio between the percentage change in quantity supplied of a product and the percentage change in its price. It measures producer responsiveness to price adjustments.
• Formula:   $e_s = \frac{\%\,\Delta\,\text{in quantity supplied}}{\%\,\Delta\,\text{in price}}$
• Categories of Price Elasticity of Supply:
  • Perfectly Inelastic Supply ($e_s = 0$): Quantity supplied is completely fixed and unaffected by price. Graphical representation: Vertical line.   
  • Inelastic Supply ($0 < e_s < 1$): Percentage change in quantity supplied is smaller than percentage change in price. Supply curve intersects horizontal quantity axis.   
  • Unitary Elastic Supply ($e_s = 1$): Percentage change in quantity supplied equals percentage change in price. Any straight-line supply curve passing directly through the origin has an elasticity of 1.   
  • Elastic Supply ($1 < e_s < \infty$): Percentage change in quantity supplied is greater than percentage change in price. Supply curve intersects vertical price axis.   
  • Perfectly Elastic Supply ($e_s = \infty$): Producers supply an infinite quantity at a specific price, but supply falls to zero at any price below that level. Graphical representation: Horizontal line.   

Determinants of Price Elasticity of Supply

• 1. Time Elapsed Since Price Change:
  • Short Run: Supply is inelastic because firms face production bottlenecks, fixed plant capacities, and limited resource adaptability.
  • Long Run: Supply is elastic because firms have enough time to expand production facilities, hire labor, acquire capital, or enter/exit industries.
• 2. Price Expectations:
  • If producers expect a price increase to be long-term or permanent, they expand production capacity, increasing supply elasticity. Temporary price changes result in minimal output adjustments.
• 3. Stockpiling and Storage Capacity:
  • Storable non-perishable goods have higher supply elasticity because existing inventories can be released rapidly when prices rise. Highly perishable items cannot be stockpiled, resulting in inelastic supply.
• 4. Excess Production Capacity:
  • Firms operating with spare production capacity can rapidly scale up output when prices rise (elastic supply). Firms operating at full capacity cannot increase output without expanding plant size (inelastic supply).
• 5. Availability and Mobility of Inputs:
  • If raw materials, machinery, and skilled labor are readily available and mobile, firms can expand output quickly (elastic supply). If essential inputs are scarce or restricted, supply remains inelastic.