Economics: Class 12 (Part 2) — Chapter 2: Theory of Consumer Behaviour

For simplicity, we assume a two-good world consisting of bananas ($x_1$) and mangoes ($x_2$).

• A consumption bundle is represented as $(x_1, x_2)$, where $x_1 ≥ 0$ and $x_2 ≥ 0$.
• Example: Bundle $(5, 10)$ consists of 5 bananas and 10 mangoes.

Cardinal utility analysis assumes that utility can be measured and expressed in numbers (utils).

• Total Utility ($TU$): The total satisfaction derived from consuming a given quantity of a commodity $x$. $$TU_n = MU_1 + MU_2 + ⋯ + MU_n$$
• Marginal Utility ($MU$): The change in total utility due to the consumption of one additional unit of a commodity. $$MU_n = TU_n - TU_{n-1}$$

The Law of Diminishing Marginal Utility states that as a consumer consumes more units of a commodity, the marginal utility derived from each successive unit declines, while keeping consumption of other commodities constant.

Ordinal utility analysis assumes that utility cannot be measured in numbers, but consumers can rank different consumption bundles based on preference.

• Indifference Curve (IC): A curve joining all points representing bundles among which the consumer is indifferent, as they provide the same level of satisfaction.
• Marginal Rate of Substitution (MRS): The rate at which a consumer is willing to substitute one good for another while keeping total utility constant. $$MRS = \frac{|Δ y|}{|Δ x|}$$
• Law of Diminishing MRS: As the consumer gets more of good 1 (bananas), the quantity of good 2 (mangoes) she is willing to sacrifice for an additional unit of good 1 declines. This makes the indifference curve convex to the origin.

• Monotonic Preferences: A consumer always prefers a bundle that has more of at least one good and no less of the other good.
• Indifference Map: A family of indifference curves. Higher indifference curves represent higher levels of utility.
• Features of Indifference Curves:
  1. Indifference curves slope downwards from left to right.
  2. Higher indifference curves give greater levels of utility.
  3. Two indifference curves never intersect each other.

The budget set is the collection of all bundles that the consumer can buy with her income at prevailing market prices: $$p_1 x_1 + p_2 x_2 ≤ M$$

The budget line represents all bundles that cost exactly equal to $M$: $$p_1 x_1 + p_2 x_2 = M$$ This can be rewritten as: $$x_2 = \frac{M}{p_2} - \frac{p_1}{p_2} x_1$$

• Horizontal Intercept: $\frac{M}{p_1}$ (entire income spent on bananas)
• Vertical Intercept: $\frac{M}{p_2}$ (entire income spent on mangoes)
• Slope of the Budget Line: $-\frac{p_1}{p_2}$ (the price ratio)

• Income Changes: An increase in income ($M' > M$) causes a parallel outward shift of the budget line. A decrease in income causes a parallel inward shift.
• Price Changes: An increase in the price of bananas ($p_1$) pivots the budget line inwards around the vertical intercept, making it steeper. A decrease in $p_1$ pivots it outwards, making it flatter.

The relation between the consumer's optimal choice of quantity and its price is the demand function: $$X = f(P)$$

• Law of Demand: Other things being equal, there is a negative relationship between the demand for a commodity and its price. When price increases, demand falls; when price decreases, demand rises.
• Linear Demand Curve: $$d(p) = a - bp \text{ (where } 0 ≤ p ≤ \frac{a}{b}\text{)}$$

• Normal Goods: Demand increases as consumer income increases (e.g., high-quality food).
• Inferior Goods: Demand decreases as consumer income increases (e.g., coarse cereals).
• Giffen Goods: A special category of inferior goods for which the demand is positively related to its price because the income effect dominates the substitution effect.

• Substitutes: Goods used in place of each other (e.g., tea and coffee). An increase in the price of a substitute increases the demand for the good.
• Complements: Goods consumed together (e.g., tea and sugar). An increase in the price of a complementary good decreases the demand for the good.

• Movement along the Demand Curve: Caused by a change in the price of the good itself (expansion or contraction of demand).
• Shift in the Demand Curve: Caused by changes in other factors like income, prices of related goods, or tastes and preferences.

• Inelastic Demand ($|e_D| < 1$): Percentage change in demand is less than percentage change in price.
• Elastic Demand ($|e_D| > 1$): Percentage change in demand is greater than percentage change in price.
• Unitary Elastic Demand ($|e_D| = 1$): Percentage change in demand equals percentage change in price.
• Perfected Inelastic ($e_D = 0$): Demand does not change at all with price (vertical demand curve).
• Perfected Elastic ($e_D = ∞$): Demand drops to zero with any price increase (horizontal demand curve).

For a linear demand curve $q = a - bp$, elasticity varies at different points: $$e_D = -\frac{bp}{a - bp}$$

• At $p = 0$, $e_D = 0$.
• At $q = 0$, $e_D = ∞$.
• At the midpoint $p = \frac{a}{2b}$, $e_D = -1$.

The relationship between price changes, elasticity, and total expenditure ($E = P \times Q$) is summarized below: