Demand Topic 4 Notes

4.3 Learning Outcomes

• Use the consumer theory to derive the demand curve.

• Describe and analyse the effects of an increase in income and price.

• Describe how the substitution and income effects impact on consumer preferences to maximise utility.

• Recognise how changes in price levels lead to inflation.

• Recognise the indices used in the UK to measure inflation.

• Recognise how consumer revealed preferences lead to strong and weak axiom.

4.4 The Unit Roadmap

• Deriving Demand Curve:

  • System of Demand Functions

  • Graphical Interpretation

  • Application

• Effects of an Increase in Income:

  • How Income Changes Shift Demand Curve

  • Consumer Theory and Income Elasticity

  • Application

4.5 The Unit Roadmap (continued)

• Effects of a Price Increase:

  • Income and Substitution Effects with Normal Goods

  • Income and Substitution Effects with An Inferior Good

  • Compensated Demand Curve

  • Slutsky Equation

4.6 The Unit Roadmap (continued)

• Cost of Living Adjustment:

  • Inflation Indices

  • Effects of Inflation Adjustments

• Revealed Preference:

  • Recovering Preferences

  • Substitution Effect

4.7 Deriving the Demand Curve

• Demand can be defined as the effective demand.

• Factors that determine demand include:

  • Price of the product

  • Prices of other goods

  • Consumer income

  • Demographic trends in the population

  • Consumer tastes, fashion and preferences

  • Weather conditions

4.8 Functional Relationship

• Quantity Demanded of Good n: Q_n

• Price of Good n: P_n

• Price of Substitute: P_sub

• Consumer Income: I

• All other factors: X

• Price of Complements: P_comp

• Functional form: Qn = f(Pn, Psub, I, X, Pcomp)

4.9 Law of Demand

• The Law of Demand states that at a higher price, a smaller quantity is demanded.

• Due to the negative relationship between quantity demanded and price, the demand curve is downward sloping.

4.10 Demand Curve

• The demand curve slopes downwards due to the inverse relationship between price and quantity demanded.

• Example coordinates (illustrative): Price £ vs Quantity D (illustrative points shown on slides)

4.11 Income Elasticity of Demand

• The level of demand for a commodity is highly responsive to a given change in income.

• When consumer income increases, demand for the commodity rises. These goods are normal goods.

• If consumer income increases, they would buy less of the product as it becomes an inferior good to them.

4.12 Income Elasticity

• Income elasticity is a measure of the responsiveness of the quantity demanded of a product to a given change in consumer income, when all factors other than the changing factor are held constant.

• Formula:
  E_I = rac{ ext{%
  Change in Quantity Demanded}}{ ext{%
  Change in Income}} = rac{ rac{ riangle Q}{Q}}{ rac{ riangle I}{I}}

4.13 Consumer Theory and Income Elasticity

• Consumer theory is understanding how people select what goods and services to buy.

• Income elasticity measures how responsive those choices are to changes in income.

• This information benefits businesses and governments when understanding how to market their products or allocate resources.

4.14 Application

• Original Quantity Demanded, Original Income, New Quantity Demanded, New Original Income (dataset):

  • a: Q0 = 100, I0 = 10, Q1 = 120, I1 = 14

  • b: Q0 = 15, I0 = 6, Q1 = 20, I1 = 7

  • c: Q0 = 50, I0 = 25, Q1 = 40, I1 = 35

  • d: Q0 = 12, I0 = 100, Q1 = 15, I1 = 125

  • e: Q0 = 200, I0 = 10, Q1 = 250, I1 = 11

  • f: Q0 = 25, I0 = 20, Q1 = 30, I1 = 18

• Income elasticity can be calculated as:
  $E<em>I = rac{ rac{Q</em>1 - Q<em>0}{Q</em>0}}{ rac{I<em>1 - I</em>0}{I_0}}$

• Example calculation (row a):

  • 
    $E_I^{(a)} = rac{(120 - 100)/100}{(14 - 10)/10} = rac{0.20}{0.40} = 0.50$

• Use this method for each row to obtain elasticities; interpret as normal vs. inferior based on sign and magnitude.

4.15 Price Decrease of Normal Goods with Income and Substitution Effect

• Demonstrates how a fall in price of a normal good X affects the quantity demanded of Y (and X) via income and substitution effects.

• Points A and B on the indifference curves show movement along budget constraints; budget line shifts reflect price change.

• Total Effect from moving from A to B equals the change in quantity demanded (e.g., 16 - 2 = 14 units in the slide example).

4.16 Price Decrease of Normal Goods with Income and Substitution Effect (Substitution vs. Income)

• To identify the effects, draw a new budget line R3 parallel to R2, tangent to IC1 at point C.

• Substitution effect: 8 - 2 = 6 units

• Income effect: 16 - 8 = 8 units

• Total effect: 6 + 8 = 14 units

4.17 Price Increase of Normal Goods with Income and Substitution Effect

• With a price increase of X, budget line shifts inward and new tangent point B on R2 yields total effect (e.g., 16 - 2 = 14 units).

4.18 Price Increase of Normal Goods with Income and Substitution Effect

• Identical framework: use budget line L3 to isolate effects; C is the tangent point for the compensated (substitution) path.

4.19 Price Increase of Normal Goods with Income and Substitution Effect

• At Point C: substitution effect = 8 - 16 = -8 units; income effect = 2 - 8 = -6 units; total effect = -8 + (-6) = -14 units.

4.20 Price Increase of an Inferior Good with Income and Substitution Effect

• Inferior goods respond to income effects opposing price effects.

• Example coordinates: A, B, C with lines L1, L2, L3; total effect from A to B is 6 - 4 = 2 units.

4.21 Price Increase of Inferior Goods with Income and Substitution Effect

• Use a parallel budget line L3 to isolate effects; tangent to IC1 at C.

4.22 Price Increase of Inferior Goods with Income and Substitution Effect

• At Point C: substitution effect = 10 - 4 = 6 units; income effect is negative = 6 - 10 = -4 units; total effect = 6 + (-4) = 2 units.

4.23 Price Decrease of an Inferior Good with Income and Substitution Effect

• Price decrease of X leads to reduced quantity demanded for inferior good Y? (Described through A, B, C, budget lines R1, R2, R3.)

• Total effect moving from B to A: 7 - 10 = -3 units.

4.24 Price Decrease of Inferior Goods with Income and Substitution Effect

• Isolate using R3; C tangent to IC1.

4.25 Price Decrease of Inferior Goods with Income and Substitution Effect

• At Point C: substitution effect = 6 - 10 = -4 units; income effect = 7 - 6 = 1 unit; total effect = -3 units.

4.26 Compensated Demand Curve

• Concept: relates price and quantity purchased when substitution effects are considered and income effects are ignored.

• Based on Hicksian demand curve: Q^H = h(P, U) with other prices and utility held constant.

4.27 Marshallian Demand Curve

• Marshallian (uncompensated) demand: shows the effect of price on quantity demanded, accounting for both income and substitution effects.

• The Marshallian demand curve is elastic.

• It reflects total effects (income + substitution).

4.28 Difference Between the Substitution and the Income Effect

• Substitution effect: change in consumption due to changes in prices of other goods, holding utility constant.

• Income effect: change in consumption due to a change in purchasing power from price changes.

4.29 The Hicksian and Marshallian Demand Curve (graph reference)

• At point A, initial optimization on budget line;

• When X price falls, budget line shifts outward; move to a higher indifference curve (UA) at point B on RB.

• A second indifference curve, UB, tangent to budget line RB.

• The Hicksian curve connects A to C (compensated path); the Marshallian curve connects A to B (uncompensated path).

4.30 The Hicksian and Marshallian Demand Curve

• Lower graph: quantity of X vs price of X.

• Marshallian demand: connected by A-B.

• Hicksian demand: connected by A-C.

• Welfare rises when the price of X falls.

4.31 The Hicksian and Marshallian Demand Curve (summary)

• Differences: Hicksian uses compensated income to keep utility constant; Marshallian uses actual income.

• The two curves diverge when income effects are non-negligible.

4.32 The Hicksian and Marshallian Demand Curve (diagram interpretation)

• Visual interpretation with price Q_X on vertical axis and X on horizontal;

• Points A, B, C correspond to different decompositions of price changes.

4.33 Slutsky Equation

• Slutsky identifies changes in demand arising due to a price change; the change is the combination of substitution and income effect.

• Substitution effect: movement along the indifference curve.

• Income effect: change in demand from the effective increase in income.

• Notation: xi(p, y) is Marshallian; xi^H(p, u) is Hicksian.

4.34 Slutsky Equation (diagrammatic)

• Notation and points: IC1, IC2, IC3; A original optimum, R after price change; AB parallel to PL2; C tangent to AB; S, SE, IE indicated.

4.35 Slutsky Equation (steps)

• Original budget line PL1; original choice Q on IC1 (M, M1);

• Price of X decreases; budget line shifts to PL2; new equilibrium at R with IC2 tangent to PL2 (N, N1);

• The new budget line AB is drawn parallel to PL2 and passes through Q;

• At point S, IC3 is tangent to AB.

4.36 Inflation Indices

• Inflation in the UK is measured by:

  • Consumer Price Index (CPI)

  • Retail Price Index (RPI)

4.37 Effects of Inflation Adjustments

• Effects include:

  • Reduction in the purchasing power of the consumer

  • Lower-income consumers spend a larger share on consumption

  • Rapid inflation can move away from deflation; risk of recession

  • Higher interest rates; negative impact on Bonds and stock market

  • Can boost real estate, energy and value stocks

4.38 Assumptions of Revealed Preference Theory

• Rationality

• Unchanged Taste

• Consistency

• Transitivity

• Revealed Preference Axiom

• Price Effect

4.39 Transitivity

• Transitivity: preferences are transitive if a consumer prefers basket X to basket Y, and Y to Z, then prefers X to Z; and does not prefer Z to X.

4.40 Transitivity Axiom (diagrammatic)

• Budget lines and bundles X1, X2, Y1, Y2, Z1, Z2 illustrate the axiom; the indifference curve should lie above X1 and X2.

4.41 Revealed Preference

• Assumptions:

  • Preferences are revealed by purchasing habits.

  • Preferences are stable over the observation period.

  • If a consumer purchases bundle A with given income/prices over another bundle, A is the revealed preference.

4.42 Revealed Preference (diagram interpretation)

• Superior Set: bundles affordable on the budget line PL; consumer prefers A over B, C, D, E, F, G within the region.

4.43 Recovering Preference

• Assumes rationality and utility maximisation under budget; combinations A is superior; other bundles become inferior; points within triangle OPL are inferior to A; above A, superior bundles exist.

4.44 Strong Axiom Revealed Preference (SARP)

• Strong ordering requires consistency; points where the consumer is indifferent are removed; a rational, affording consumer will not prefer B over A under any circumstance; SARP is satisfied through optimizing actual choice and preference.

4.45 Transitivity (recap)

• Revisit: X1, X2 revealed preferences for bundles in area AOB; A-B-D-C arrangement demonstrates transitivity.

4.46 Strong Axiom Revealed Preference

• SARP: underlying preferences and revealed preferences must be transitive.

• SARP is necessary for welfare maximisation; a rational consumer chooses the best affordable bundle.

4.47 Weak Axiom Revealed Preference (WARP)

• Two goods X and Y; budget BC1 and inside-budget bundles show that if A is preferred to B and B preferred to C, then B never prefers A (consistent with WARP).

4.48 Substitution Effect

• Samuelson describes quasi-substitution: when the price of X declines, rational consumers may still purchase the same amount of goods; this is counterintuitive under older formulations.

• Referencing Samuelson and Perloff for foundational definitions.

4.49 Substitution Effect (continued)

• The new budget line passes through X and is parallel to AC; A1C1 is the new line.

• Revealed preference approach resolves inconsistencies under inflation.

• Inflation raises cost of living, motivating index construction for cost of living.

4.50 Substitution Effect (continued)

• Consumers choose a combination of X and Y from the budget set; the chosen bundle reveals preferences.

• Changes in income reduce real income, affecting choices.

4.51 Topic Summary

• When product prices change, consumers face utility maximisation under nominal income.

• Demand patterns change with income; products may be normal or inferior depending on income changes.

4.52 Topic Summary (continued)

• A price change leads to a change in quantity demanded.

• Price effects consist of substitution and income effects.

• Revealed preference is a behaviourist approach to analysing consumer behaviour.

• Welfare is maximised by choosing a bundle that satisfies strong axiom revealed preference.

4.53 References

• Salvatore, D. (2019). International Economics. 13th edition. John Wiley.

• Wong, S. (2006). Foundations of Paul Samuelson's Revealed Preference Theory: A study by the method of rational reconstruction. 2nd ed. Routledge. https://doi.org/10.4324/9780203462430

End of Topic 4 Notes

4.3 Learning Outcomes

• Use the consumer theory to derive the demand curve.

• Describe and analyse the effects of an increase in income and price.

• Describe how the substitution and income effects impact on consumer preferences to maximise utility.

• Recognise how changes in price levels lead to inflation.

• Recognise the indices used in the UK to measure inflation.

• Recognise how consumer revealed preferences lead to strong and weak axiom.

4.4 The Unit Roadmap

• Deriving Demand Curve: System of demand functions, graphical interpretation, and application.

• Effects of Income Changes: How income shifts demand, consumer theory, and income elasticity.

• Effects of Price Changes: Income and substitution effects for normal and inferior goods, compensated demand, and the Slutsky Equation.

• Cost of Living Adjustment: Inflation indices and their effects.

• Revealed Preference: Recovering preferences and the substitution effect.

4.5 Deriving the Demand Curve

• Demand is effective demand, determined by factors such as: product price, prices of other goods, consumer income, demographic trends, tastes, and weather.

4.6 Functional Relationship

• Quantity Demanded of Good n: $Q_n$

• Price of Good n: $P_n$

• Price of Substitute: $P_{sub}$

• Consumer Income: $I$

• All other factors: $X$

• Price of Complements: $P_{comp}$

• Functional form: $Q<em>n = f(P</em>n, P<em>{sub}, I, X, P</em>{comp})$

4.7 Law of Demand & Demand Curve

• The Law of Demand states a negative (inverse) relationship between price and quantity demanded; thus, the demand curve is downward sloping.

4.8 Income Elasticity of Demand

• Income elasticity measures the responsiveness of quantity demanded to changes in consumer income.

• Normal goods: Demand rises with increased income.

• Inferior goods: Demand falls with increased income.

• Formula:
  $E_I = \frac{\%\text{ Change in Quantity Demanded}}{\%\text{ Change in Income}} = \frac{\frac{\Delta Q}{Q}}{\frac{\Delta I}{I}}$

• Consumer theory uses income elasticity to understand purchasing choices and allocate resources, benefiting businesses and governments.

4.9 Application of Income Elasticity

• Income elasticity is calculated using the formula:
  $E<em>I = \frac{\frac{Q</em>1 - Q<em>0}{Q</em>0}}{\frac{I<em>1 - I</em>0}{I_0}}$

• Example: For changes from $Q<em>0 = 100, I</em>0 = 10$ to $Q<em>1 = 120, I</em>1 = 14$, the elasticity is $E_I = \frac{(120 - 100)/100}{(14 - 10)/10} = \frac{0.20}{0.40} = 0.50$.

• Interpret elasticities (sign and magnitude) to classify goods as normal or inferior.

4.10 Income and Substitution Effects of Price Changes

• Price changes affect quantity demanded through two effects:

  • Substitution Effect: Change in consumption due to a change in relative prices, keeping utility constant (movement along an indifference curve). Always negative for a price increase (buy less of the now relatively more expensive good) and positive for a price decrease.

  • Income Effect: Change in consumption due to a change in purchasing power (real income) resulting from the price change (shift to a new indifference curve).

• Normal Goods:

  • When price falls: Substitution effect increases quantity demanded, and income effect also increases quantity demanded (as real income rises). Both effects reinforce each other.

  • When price rises: Substitution effect decreases quantity demanded, and income effect also decreases quantity demanded (as real income falls). Both effects reinforce each other.

• Inferior Goods:

  • When price falls: Substitution effect increases quantity demanded, but income effect decreases quantity demanded (as real income rises, consumers buy less of the inferior good). The effects oppose each other.

  • When price rises: Substitution effect decreases quantity demanded, but income effect increases quantity demanded (as real income falls, consumers buy more of the inferior good). The effects oppose each other.

• The total effect of a price change on quantity demanded is the sum of the substitution and income effects. Effects are isolated by constructing a hypothetical budget line parallel to the new relative price line, tangent to the original indifference curve.

4.11 Compensated and Marshallian Demand Curves

• Hicksian (Compensated) Demand Curve: Shows the relationship between price and quantity while keeping utility constant, isolating only the substitution effect. Functional form: $Q^H = h(P, U)$.

• Marshallian (Uncompensated) Demand Curve: Shows the relationship between price and quantity, incorporating both substitution and income effects (total effect). It typically reflects changes in actual income.

4.12 Relationship between Hicksian and Marshallian Demand Curves

• When price of a good falls, both curves depict the increase in quantity demanded, but differently:

  • The Hicksian curve ($A \to C$) represents the compensated path, keeping utility constant to show only the substitution effect.

  • The Marshallian curve ($A \to B$) represents the uncompensated path, showing the total effect (substitution + income effect) as actual income changes.

• They diverge when income effects are significant, with Hicksian using compensated income (constant utility) and Marshallian using actual income.

4.13 Slutsky Equation

• The Slutsky Equation mathematically decomposes the total change in quantity demanded due to a price change into the substitution and income effects.

• It formalizes how Marshallian demand (actual choice $x<em>i(p, y)$) relates to Hicksian demand (compensated choice $x</em>i^H(p, u)$):
  $\frac{\partial x<em>i(p, y)}{\partial p</em>j} = \frac{\partial x<em>i^H(p, u)}{\partial p</em>j} - x<em>j(p,y) \frac{\partial x</em>i(p, y)}{\partial y}$

• The substitution effect is represented by the Hicksian derivative, and the income effect is captured by the term involving the income derivative and the quantity of the good.

4.14 Inflation and its Effects

• UK Inflation Indices: Measured by the Consumer Price Index (CPI) and Retail Price Index (RPI).

• Effects of Inflation: Reduces consumer purchasing power, disproportionately affects lower-income consumers, can lead to higher interest rates negatively impacting bonds and stocks, but may boost real estate, energy, and value stocks.

4.15 Revealed Preference Theory

• A behaviorist approach to consumer choice, assuming rationality, unchanged taste, consistency, and transitivity.

• Assumptions: Preferences are revealed by actual purchasing habits and are stable. If bundle A is chosen over bundle B when both are affordable, then A is revealed preferred to B.

• Transitivity: If a consumer prefers X to Y, and Y to Z, then they must prefer X to Z.

• Weak Axiom of Revealed Preference (WARP): If bundle A is directly revealed preferred to bundle B, then B cannot be directly revealed preferred to A. That is, if A is chosen when B is available, B cannot be chosen when A is available.

• Strong Axiom of Revealed Preference (SARP): Requires underlying and revealed preferences to be transitive. If A is revealed preferred to B, and B to C, then A must be revealed preferred to C. SARP is necessary for welfare maximization and implies a rational consumer chooses the best affordable bundle.

• Substitution Effect in Revealed Preference: Describes how consumer choices change due to relative price changes, even if the absolute quantity might seem counterintuitive to older theories, and is relevant for understanding cost of living adjustments and inflation.

4.16 Topic Summary

• Consumer behavior involves utility maximization under changing prices and nominal income.

• Demand for products varies with income (normal or inferior goods).

• Price changes lead to changes in quantity demanded, decomposed into substitution and income effects.

• Revealed preference theory provides a behaviorist framework for analyzing consumer choices, where welfare is maximized by choosing bundles consistent with the Strong Axiom of Revealed Preference (SARP).

4.17 References

• Salvatore, D. (2019). International Economics. 13th edition. John Wiley.

• Wong, S. (2006). Foundations of Paul Samuelson's Revealed Preference Theory: A study by the method of rational reconstruction. 2nd ed. Routledge. https://doi.org/10.4324/9780203462430