Theory of Consumer Behaviour

The Law of Diminishing Marginal Utility states that as a consumer consumes more and more units of a commodity continuously, the marginal utility derived from each additional unit continuously declines.
Relationship between TU and MU:
1. When MU is positive and falling, Total Utility (TU) increases at a diminishing rate.
2. When MU becomes zero ($MU = 0$), Total Utility reaches its maximum peak (Point of Satiety).
3. When MU becomes negative, Total Utility declines.
4. Total Utility is the mathematical sum of all marginal utilities: $TU_n = \Sigma MU$.

Step 1: Check the Equi-Marginal Equilibrium condition ($\frac{MU_x}{P_x} = \frac{MU_y}{P_y}$):

$$\frac{MU_x}{P_x} = \frac{20}{4} = 5 \text{ utils per rupee}$$

$$\frac{MU_y}{P_y} = \frac{20}{5} = 4 \text{ utils per rupee}$$

$$\frac{MU_x}{P_x} > \frac{MU_y}{P_y} \quad (5 > 4)$$

Conclusion: The consumer is NOT in equilibrium. Every rupee spent on good $X$ yields greater satisfaction than on good $Y$.
Consumer Reaction: The consumer will reallocate spending by increasing consumption of Good $X$ and reducing consumption of Good $Y$. As consumption of $X$ increases, by the Law of DMU, $MU_x$ will fall. Simultaneously, as consumption of $Y$ decreases, $MU_y$ will rise until equality ($\frac{MU_x}{P_x} = \frac{MU_y}{P_y}$) is restored.

The 4 Geometric Properties:
1. Indifference Curves Slope Downward from Left to Right: To keep satisfaction constant, increasing good $X$ requires sacrificing good $Y$.
2. Indifference Curves are Strictly Convex to the Origin: Due to the Law of Diminishing Marginal Rate of Substitution ($MRS_{xy}$).
3. Higher Indifference Curves Represent Higher Satisfaction: Due to monotonic preferences (more of a good provides more utility).
4. Two Indifference Curves Can NEVER Intersect: Intersecting curves would violate the fundamental axiom of transitivity.
• Why Convex to Origin: As the consumer acquires more units of $X$, the marginal desire for $X$ diminishes while remaining units of $Y$ become increasingly valued. Therefore, the consumer is willing to surrender progressively fewer units of $Y$ for each extra unit of $X$ ($MRS_{xy} = -\Delta Y / \Delta X$ declines).

1. Condition 1 (Tangency Condition): The Marginal Rate of Substitution must equal the ratio of market prices:
   

$$MRS_{xy} = \frac{P_x}{P_y}$$

(The slope of the Indifference Curve must equal the slope of the Budget Line).
2. Condition 2 (Convexity Condition): The Indifference Curve must be strictly convex to the origin at the point of equilibrium (meaning $MRS_{xy}$ must be diminishing). If the IC were concave, the tangency would represent a point of minimum utility rather than maximum satisfaction.

The Budget Line represents all combinations of two goods ($X$ and $Y$) that a consumer can purchase by spending their entire money income ($M$): $P_x X + P_y Y = M$.
• (a) Consumer Income Increases: The budget line undergoes a parallel rightward shift outwards, because the consumer can now purchase proportionally more of both goods at unchanged relative prices.
• (b) Price of Good $X$ Falls: The budget line rotates outward along the horizontal X-axis (pivoting from the vertical Y-intercept), because lower $P_x$ allows more of Good $X$ to be purchased, while the maximum affordable quantity of Good $Y$ is unchanged.

Given:
$P_1 = 10, \quad P_2 = 12 \implies \Delta P = 12 - 10 = +2$
$Q_1 = 50, \quad Q_2 = 40 \implies \Delta Q = 40 - 50 = -10$
Formula:

$$E_d = (-) \frac{\Delta Q}{\Delta P} \times \frac{P}{Q}$$

Substitute values:

$$E_d = (-) \frac{-10}{2} \times \frac{10}{50} = 5 \times \frac{1}{5} = \mathbf{1}$$

Conclusion: Price elasticity of demand is unitary elastic ($E_d = 1$).

Suppose two indifference curves $IC_1$ and $IC_2$ intersect at point $A$.
Let point $B$ lie on $IC_1$ and point $C$ lie on $IC_2$, such that $B$ and $C$ contain the same quantity of good $X$ but $C$ has more of good $Y$.
• Since $A$ and $B$ lie on $IC_1$: $\text{Utility}(A) = \text{Utility}(B)$
• Since $A$ and $C$ lie on $IC_2$: $\text{Utility}(A) = \text{Utility}(C)$
By the axiom of transitivity, this implies $\text{Utility}(B) = \text{Utility}(C)$.
However, bundle $C$ contains more of Good $Y$ than bundle $B$, and by monotonic preferences, $\text{Utility}(C) > \text{Utility}(B)$.
This creates an impossible logical contradiction! Therefore, two indifference curves can never intersect.

The paradox observes that water is essential for human life yet has a negligible market price, while diamonds are useless for biological survival yet command colossal market prices.
Resolution via Marginal Utility: Market price is determined not by Total Utility ($TU$), but by Marginal Utility ($MU$).
Because water is abundant in supply, its marginal utility ($MU_{\text{water}}$) is close to zero, resulting in a low market price. Diamonds are exceptionally scarce, so their marginal utility ($MU_{\text{diamond}}$) is extraordinarily high, commanding an exorbitant price.