An individual consumes two commodities, X and Y, with market prices P_x = \text{₦}150 and … — JAMB UTME

An individual consumes two commodities, $X$ and $Y$ , with market prices $P_x = \text{₦}150$ and $P_y = \text{₦}50$ respectively. At a given point $K$ on their budget constraint, the marginal rate of substitution of $X$ for $Y$ ( $MRS_{xy}$ ) equals $3$ . Why is the condition $MRS_{xy} = \frac{P_x}{P_y}$ at point $K$ alone NOT sufficient to guarantee a stable consumer equilibrium under ordinal utility theory?

Tangency must be supported by the strict convexity of the indifference curve to the origin at point $K$ to satisfy the second-order condition for utility maximization.Answer
B
The marginal utility of money spent on Good $X$ must equal zero at point $K$ to confirm total satisfaction.
C
The slope of the indifference curve must remain constant along the entire length of the budget line.
D
The consumer must allocate equal monetary expenditure to Good $X$ and Good $Y$ at point $K$ .

Answer

The condition of tangency (

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

) must be fulfilled alongside the second-order condition that the indifference curve is strictly convex to the origin at the point of contact.

In ordinal utility theory, consumer equilibrium requires two conditions to be satisfied: (1) First-order necessary condition:

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

(tangency of the budget line and indifference curve), and (2) Second-order sufficient condition: the indifference curve must be strictly convex to the origin at the tangency point. Convexity ensures diminishing

MRS_{xy}

, guaranteeing that utility is maximized rather than minimized.

Step-by-Step Solution

Calculate the price ratio of Good

X

to Good

Y

\frac{P_x}{P_y} = \frac{150}{50} = 3

The slope of the budget line is given by the relative price ratio of the two commodities.

Compare the marginal rate of substitution (

MRS_{xy}

) to the price ratio

MRS_{xy} = 3 = \frac{P_x}{P_y}

This establishes that the necessary (first-order) condition for consumer equilibrium is met at point

K

Evaluate the second-order condition required for stable equilibrium

The indifference curve must be convex to the origin (diminishing

MRS_{xy}

If the indifference curve were concave or linear at the point of tangency, the consumer would minimize utility or achieve a corner solution rather than maximizing utility.

Key Concept

Consumer Equilibrium under Ordinal Utility (First-Order and Second-Order Conditions)

Estimated Time:2m 0s

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