Explain the consumer's equilibrium in cardinal approach. Price apple is Rs.20, price of banana is Rs. 10 and the consumer spends whole income Rs.110 on apple and banana. Find the consumer's…

Consumer’s Equilibrium in Cardinal Approach

Explanation of Consumer’s Equilibrium (Cardinal Utility Analysis)

Consumer’s equilibrium refers to the optimal allocation of a consumer’s income between different goods such that the utility derived from the last rupee spent on each good is equal. In the cardinal utility approach, utility is measured in utils (numerical units), and the law of diminishing marginal utility applies—each additional unit of a good consumed yields less additional satisfaction.

The condition for consumer’s equilibrium is given by: $$\frac{M{U}_x}{P_x}=\frac{M{U}_y}{P_y}=\frac{M{U}_z}{P_z}=\cdots =\frac{M{U}_n}{P_n}$$ where:

• $M{U}_x$ = Marginal utility of good $x$
• $P_x$ = Price of good $x$
• $M{U}_y$ = Marginal utility of good $y$
• $P_y$ = Price of good $y$

This means that the consumer maximizes utility when the marginal utility per rupee spent on each good is equal.

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Given Data

• Price of apple ($P_a$) = Rs. 20
• Price of banana ($P_b$) = Rs. 10
• Total income ($M$) = Rs. 110
• Marginal utilities (MU) of apples and bananas (from the table):

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Step 1: Find the Optimal Combination Where $\frac{M{U}_a}{P_a}=\frac{M{U}_b}{P_b}$

We need to find the quantities of apples ($Q_a$) and bananas ($Q_b$) such that: $$\frac{M{U}_a}{20}=\frac{M{U}_b}{10}$$ This simplifies to: $$M{U}_a=2\times M{U}_b$$

Now, we check the marginal utilities from the table to find where this condition holds:

From the table, the condition $M{U}_a=2\times M{U}_b$ holds at:

• $Q_a=1,Q_b=3$
• $Q_a=2,Q_b=4$
• $Q_a=3,Q_b=5$
• $Q_a=4,Q_b=6$

However, we must also ensure that the total expenditure equals the consumer’s income (Rs. 110).

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Step 2: Check Budget Constraint

The total expenditure on apples and bananas must satisfy: $$P_a\times Q_a+P_b\times Q_b=110$$ $$20\times Q_a+10\times Q_b=110$$

Let’s test the possible combinations:

1. For $Q_a=1,Q_b=3$: $$20(1)+10(3)=20+30=50\ne 110$$ Not feasible.

2. For $Q_a=2,Q_b=4$: $$20(2)+10(4)=40+40=80\ne 110$$ Not feasible.

3. For $Q_a=3,Q_b=5$: $$20(3)+10(5)=60+50=110$$ Feasible.

4. For $Q_a=4,Q_b=6$: $$20(4)+10(6)=80+60=140\ne 110$$ Not feasible.

Thus, the only feasible combination that satisfies both the equilibrium condition and the budget constraint is: $$Q_a=3\text{ apples},Q_b=5\text{ bananas}$$

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Verification of Consumer’s Equilibrium

At $Q_a=3$ and $Q_b=5$:

• $M{U}_a=600$
• $M{U}_b=300$

Check the equilibrium condition: $$\frac{M{U}_a}{P_a}=\frac{600}{20}=30$$ $$\frac{M{U}_b}{P_b}=\frac{300}{10}=30$$ Since $\frac{M{U}_a}{P_a}=\frac{M{U}_b}{P_b}$, the consumer is in equilibrium.

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Conclusion

The consumer’s equilibrium occurs when the consumer purchases:

• 3 apples
• 5 bananas

This allocation ensures that:

1. The marginal utility per rupee spent is equal for both goods.
2. The total expenditure equals the consumer’s income (Rs. 110).

Final Answer: The consumer’s equilibrium is achieved at 3 apples and 5 bananas.