A consumer allocating an income of \text{₦}1,800 between Good X and Good Y faces market pr… — JAMB UTME

A consumer allocating an income of $\text{₦}1,800$ between Good $X$ and Good $Y$ faces market prices of $P_x = \text{₦}40$ and $P_y = \text{₦}30$ per unit, respectively. The consumer's Marginal Rate of Substitution of Good $X$ for Good $Y$ is given by $MRS_{xy} = \frac{2Y}{X}$ . Assuming the consumer maximizes satisfaction subject to their budget constraint, how many units of Good $X$ will be consumed at equilibrium?

Cevap: 30 units

Cevap

At consumer equilibrium under ordinal utility analysis, the optimal quantity of Good

X

consumed is 30 units.

At consumer equilibrium under ordinal utility, the tangency condition requires

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

. Substituting

MRS_{xy} = \frac{2Y}{X}

and prices

P_x = 40

P_y = 30

gives

\frac{2Y}{X} = \frac{40}{30}

, which simplifies to

Y = \frac{2}{3}X

. Substituting

Y = \frac{2}{3}X

into the consumer's budget constraint

40X + 30Y = 1800

yields

40X + 30\left(\frac{2}{3}X\right) = 1800 \implies 60X = 1800 \implies X = 30

units.

Adım Adım Çözüm

Equate the Marginal Rate of Substitution (

MRS_{xy}

) to the price ratio (

P_x / P_y

) to apply the tangency condition for ordinal utility equilibrium.

\frac{2Y}{X} = \frac{40}{30} \implies \frac{2Y}{X} = \frac{4}{3} \implies 6Y = 4X \implies Y = \frac{2}{3}X

Consumer equilibrium under ordinal utility requires that the slope of the indifference curve (

MRS_{xy}

) equals the slope of the budget line (

P_x / P_y

Substitute the expression for

Y

into the budget constraint equation

P_x X + P_y Y = I

40X + 30\left(\frac{2}{3}X\right) = 1800

To achieve maximum utility within income limits, the entire income of

\text{₦}1,800

must be spent on goods

X

and

Y

Simplify the equation and solve for the value of

X

40X + 20X = 1800 \implies 60X = 1800 \implies X = 30

Combining like terms gives a linear equation in

X

, yielding 30 units at equilibrium.

Anahtar Kavram

Consumer equilibrium under ordinal utility occurs where the highest attainable indifference curve is tangent to the budget line, satisfying

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

alongside the budget constraint

P_x X + P_y Y = I

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