1. Preliminary Notations and Assumptions

• Consumer consumes many goods; analysis often simplifies to two goods (x and y) for graphical clarity.
• Basic assumptions: completeness (consumer can rank bundles), transitivity (consistent rankings), non-satiation (more is preferred to less), and continuity (no jumps in preferences).
• Notations: x and y are quantities of two goods; P_x and P_y are their prices; Y denotes income (budget).

2. Utility

2.1 Cardinal Utility Analysis

• Cardinal approach assumes utility (satisfaction) can be measured in cardinal numbers (utils).
• Two important measures:
• Total Utility (TU): total satisfaction from consuming n units of a good x.
• Marginal Utility (MU): additional satisfaction from consuming one extra unit of x: MU = ΔTU / Δx
• Law of Diminishing MU: as consumption of a good increases, MU eventually falls (other things equal).
• Consumer equilibrium (cardinal): consume where MU per rupee across goods equalizes (for two goods x and y):
MU_x/P_x = MU_y/P_y

2.2 Ordinal Utility Analysis

• Ordinal approach ranks bundles (more preferred, less preferred) without measuring utility numbers.
• Uses indifference curves (ICs) to represent combinations of goods delivering same satisfaction.
• Indifference maps: higher ICs represent higher utility levels.

3. Indifference Curves and Their Properties

• Indifference curve slopes downward from left to right — to keep utility constant, increase in one good requires decrease in the other.
• ICs are convex to the origin reflecting diminishing Marginal Rate of Substitution (MRS).
• Marginal Rate of Substitution (MRS) = rate at which consumer is willing to give up y for additional x while remaining equally satisfied: MRS_xy = Δy / Δx |_{utility constant}
• Diminishing MRS: as x increases, MRS (willingness to give up y for x) falls — consumer gives up less y for additional x.
• Higher IC ⇢ greater utility. Two ICs never intersect. ICs are continuous and downward sloping.

4. Representation of Law of Diminishing MRS

• At high y / low x, consumer willing to give up many units of y for small increments of x (high MRS).
• As x increases and y falls, consumer values additional x less relative to y (MRS falls).
• This produces convex ICs — mathematical implication of diminishing marginal utility across goods.

5. The Consumer’s Budget

5.1 Budget Set and Budget Line

• Budget constraint: all bundles the consumer can afford given prices and income.
• Budget line equation: P_xx + P_yy = Y
• Intercepts: x-intercept = Y / P_x (all income on x); y-intercept = Y / P_y (all income on y).
• Slope of budget line = − P_x / P_y (relative price or opportunity cost).

5.2 Changes in the Budget Set

• Income change (Y↑): parallel outward shift of budget line (same slope).
• Price change (P_x↓): pivot outward on x-axis (budget line rotates), changing slope.
• Price ratio change affects relative affordability and thus optimal choice.

6. Optimal Choice of the Consumer

• Consumer chooses bundle on budget line that lies on highest attainable indifference curve.
• Tangency condition (interior solution): slope of IC = slope of budget line:
MRS_xy = P_x / P_y
• Corner solutions occur when tangency not feasible (e.g., perfect substitutes or binding non-negativity).
• At optimum with interior solution: MRS equals relative price; marginal utility per rupee equalized across goods (ordinal analogue of cardinal rule).

7. Demand

7.1 Demand Curve and the Law of Demand

• Demand: relationship between price of a good and quantity demanded, ceteris paribus.
• Law of demand: price ↑ → quantity demanded ↓ (negative slope) due to substitution and income effects.

7.2 Deriving a Demand Curve from Indifference Curves and Budget Constraint

• For various prices of x (keeping income and P_y fixed), find consumer optimum (tangency or corner). Record corresponding x quantities → plot price-quantity pairs to obtain individual demand curve for x.
• Graphically: as P_x falls, budget line becomes flatter → higher optimum x (movement along demand curve).

7.3 Normal and Inferior Goods

• Normal good: demand ↑ as income ↑ (positive income elasticity).
• Inferior good: demand ↓ as income ↑ (negative income elasticity) — consumer substitutes better goods as income rises.
• Income changes shift demand curve (right for normal goods, left for inferior goods).

7.4 Substitutes and Complements

• Substitutes: goods where an increase in price of one raises demand for the other (positive cross-price elasticity).
• Complements: goods consumed together; price rise of one reduces demand for the other (negative cross-price elasticity).

7.5 Shifts in the Demand Curve

• Demand curve shifts due to changes in income, tastes, prices of related goods, expectations, population, or taxes/subsidies.
• Movement along the curve occurs only when own price changes (other factors constant).

8. Market Demand

• Market demand is horizontal summation of individual demands at each price.
• Market demand curve shifts when aggregate factors change (income distribution, population, preferences, policy).

9. Elasticity of Demand

9.1 Price Elasticity of Demand (PED)

• Price elasticity measures responsiveness of quantity demanded to price change:
PED = (% change in quantity demanded) / (% change in price)
• Point elasticity (differential): ε = (dQ/dP) × (P/Q)
• Interpreting ε: |ε| > 1 elastic; |ε| < 1 inelastic; |ε| = 1 unitary.

9.2 Income and Cross-Price Elasticities

• Income elasticity: responsiveness of demand to income changes: η = (%ΔQ)/(%ΔY)
• Cross-price elasticity: responsiveness of demand for good A to price of B: ε_AB = (%ΔQ_A)/(%ΔP_B)
• Signs indicate normal/inferior (income) and substitutes/complements (cross-price).

9.3 Elasticity along a Linear Demand Curve

• Along a straight-line demand curve, elasticity varies: elastic at high price/low quantity end, unitary at midpoint, inelastic at low price/high quantity end.
• Slope (ΔQ/ΔP) is constant but elasticity (ε) depends on P and Q values.

9.4 Factors Determining Price Elasticity

• Availability of close substitutes (more substitutes → more elastic).
• Proportion of income spent on the good (larger share → more elastic).
• Necessity vs luxury (necessities → inelastic; luxuries → elastic).
• Time horizon (long-run elasticity greater than short-run).
• Definition of market (narrowly defined goods → more elastic).

9.5 Elasticity and Expenditure

• Total expenditure (TE) = P × Q.
• If demand is elastic (|ε| > 1): price decrease increases TE; price increase reduces TE.
• If demand is inelastic (|ε| < 1): price increase raises TE; price decrease reduces TE.

10. Movements vs Shifts — Clarification

• Movement along demand curve: change in quantity demanded due to own-price change (ceteris paribus).
• Shift of demand curve: change in demand due to non-price factors (income, tastes, related goods, expectations, taxes).

11. Practical Examples & Short Applications

• Numerical MU example: TU sequence: 0, 10, 18, 24, 28 → MU: 10, 8, 6, 4 (diminishing MU).
• Deriving demand from ICs: at P_x=10, optimum x=2; at P_x=8, optimum x=3 → demand points (10,2) and (8,3).
• Elasticity and revenue: if P falls from 10 to 9 and Q rises from 5 to 7 → %ΔQ = 40%, %ΔP = −10% → PED = −4 (elastic) → TE increases.

12. Quick Summary Table