A rational consumer allocates income between Good X and Good Y, priced at \text{₦}400 and … — JAMB UTME

A rational consumer allocates income between Good $X$ and Good $Y$ , priced at $\text{₦}400$ and $\text{₦}100$ per unit respectively. If the consumer chooses a bundle on their budget line where the marginal rate of substitution of $X$ for $Y$ ( $MRS_{xy}$ ) equals $4$ , which condition must also hold to guarantee that this point of tangency achieves maximum satisfaction?

The indifference curve must be strictly convex to the origin at the tangency point.Answer
B
The marginal rate of substitution must remain constant along the entire indifference curve.
C
The total utility derived from Good $X$ must be equal to zero.
D
The budget line must shift parallel to the right.

Answer

The indifference curve must be strictly convex to the origin at the tangency point.

For a consumer to attain equilibrium under ordinal utility analysis, two conditions must be fulfilled simultaneously: first, the slope of the indifference curve (

MRS_{xy}

) must equal the slope of the budget line (

P_x / P_y

); second, the indifference curve must be strictly convex to the origin at the point of tangency (reflecting diminishing

MRS_{xy}

). Since

MRS_{xy} = 4

and

P_x / P_y = 400/100 = 4

, the first-order condition is met, and convexity guarantees maximum satisfaction.

Step-by-Step Solution

Calculate the price ratio of the two goods

\frac{P_x}{P_y} = \frac{400}{100} = 4

The slope of the budget line is determined by the ratio of market prices.

Verify the first-order condition for consumer equilibrium

MRS_{xy} = \frac{P_x}{P_y} = 4

The first-order necessary condition for ordinal equilibrium is that the marginal rate of substitution equals the price ratio.

Identify the second-order condition for a stable equilibrium

The indifference curve must be convex to the origin at the point of tangency (diminishing MRS_{xy}).

The first-order condition alone is insufficient unless the indifference curve is convex to the origin, which guarantees a unique point of maximum utility.

Key Concept

Conditions for Consumer Equilibrium under Ordinal Utility

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