📖 边际分析与效用理论

Economics

I. Lesson Focus

This lesson examines how consumers make optimal choices under budget constraints using marginal analysis. Candidates must master the concepts of total utility, marginal utility, the law of diminishing marginal utility, consumer equilibrium conditions, indifference curves, budget lines, marginal rate of substitution, and consumer surplus. The focus is on applying these tools to predict changes in consumption when prices or income change.

II. The Problem

Li has a monthly income of CNY 8,000 to spend on food (X) and entertainment (Y). Food costs CNY 20 per unit and entertainment costs CNY 50 per unit. To maximize utility, how should Li allocate spending? If the price of food falls to CNY 15, how will the consumption bundle change? These questions lie at the heart of marginal analysis and utility theory: how a consumer, subject to a budget constraint, compares marginal utility per dollar across goods to reach the highest possible total utility. This framework is a classic topic in the CFA Level I Economics curriculum.

III. Utility and Marginal Utility

Utility is the satisfaction or happiness a consumer derives from consuming goods and services. Although utility is subjective and cannot be measured directly in cardinal terms, economists use ordinal utility to rank different consumption bundles according to preference.

Total Utility (TU) is the overall level of satisfaction obtained from consuming a given quantity of a good.

Marginal Utility (MU) is the additional satisfaction gained from consuming one more unit of a good: $$ MU_X = \frac{\Delta TU_X}{\Delta Q_X} $$ For discrete data we use differences; for continuous cases we use derivatives.

Law of Diminishing Marginal Utility: Holding other things constant, as the consumption of a good increases, the marginal utility derived from each additional unit eventually declines. This empirical regularity is the foundation of consumer theory.

IV. Consumer Equilibrium: The Core of Marginal Analysis

A consumer maximizes utility subject to a budget constraint when the marginal utility per dollar spent is equal across all goods: $$ \frac{MU_X}{P_X} = \frac{MU_Y}{P_Y} = \lambda $$ where $\lambda$ is the marginal utility of money. Equivalently: $$ \frac{MU_X}{MU_Y} = \frac{P_X}{P_Y} $$

This equality is called consumer equilibrium. At this point, the last dollar spent on any good yields the same additional utility, and total utility is maximized.

If $\frac{MU_X}{P_X} > \frac{MU_Y}{P_Y}$, the consumer should buy more of X and less of Y until the ratios are equalized.

V. Indifference Curves and Budget Lines

An indifference curve (IC) shows all combinations of two goods that yield the same total utility. Key properties: - Convex to the origin (reflecting diminishing marginal rate of substitution) - Downward-sloping - Higher curves (farther from the origin) represent higher utility levels

The marginal rate of substitution (MRS) measures how much of good Y a consumer is willing to give up for one more unit of good X: $$ MRS_{XY} = -\frac{\Delta Y}{\Delta X} = \frac{MU_X}{MU_Y} $$ At the optimal bundle, the indifference curve is tangent to the budget line, so: $$ MRS_{XY} = \frac{P_X}{P_Y} $$

The budget line represents the maximum combinations of two goods a consumer can afford given income and prices. Its slope is $-P_X/P_Y$.

An increase in income shifts the budget line outward in a parallel fashion. A decrease in the price of X rotates the budget line outward around the Y-intercept.

VI. Consumer Surplus

Consumer surplus is the difference between what a consumer is willing to pay and what is actually paid, summed across all units purchased. Graphically, it is the area below the demand curve and above the market price line.

When price falls, consumer surplus rises, explaining why consumers benefit from lower prices.

Worked Cases

Case 1: Diminishing Marginal Utility and Consumer Equilibrium (Discrete Case)

Li has CNY 100. Good X costs CNY 10 per unit, good Y costs CNY 20 per unit. Utility schedules are:

Question: How many units of X and Y should Li buy to maximize utility?

Solution: 1. Compute MU per dollar (MU/P) for each unit. 2. Purchase successive units starting with the highest MU/P until the budget is exhausted.

MU/P values: - X1: 40/10 = 4.0 - Y1: 60/20 = 3.0 - X2: 30/10 = 3.0 - Y2: 40/20 = 2.0 - X3: 20/10 = 2.0 - Y3: 30/20 = 1.5 - X4: 10/10 = 1.0 - Y4: 20/20 = 1.0

Optimal bundle: 4 units of X (CNY 40) and 3 units of Y (CNY 60). Total expenditure = CNY 100. The last dollar spent on each good yields MU/P = 1.0. Total utility = 100 + 130 = 230.

Case 2: Effect of a Price Change

In Case 1, the price of X falls to CNY 5 per unit. Recalculate the optimal bundle.

New MU/P for X: - X1: 40/5 = 8.0 - X2: 30/5 = 6.0 - X3: 20/5 = 4.0 - X4: 10/5 = 2.0 - X5: 5/5 = 1.0

Optimal purchase shifts to 5 units of X (CNY 25) and 3 units of Y (CNY 60), with remaining funds used to buy additional X until MU/P ratios equalize. Consumption of X rises from 4 to 5 units, illustrating both substitution and income effects.

Case 3: Calculating Consumer Surplus

A consumer’s demand for coffee is $Q_d = 20 - 2P$. Market price is CNY 4 per cup. - At P = 4, $Q_d = 12$ cups. - Maximum willingness to pay (when Q = 0) is CNY 10.

Consumer surplus = triangle area = $\frac{1}{2} \times (10 - 4) \times 12 = 36$ CNY.

If price falls to CNY 2, $Q_d = 16$ cups and new surplus = $\frac{1}{2} \times (10 - 2) \times 16 = 64$ CNY, an increase of 28 CNY.

Traps

Key Formulas

• Marginal utility: $MU = \frac{\Delta TU}{\Delta Q}$
• Consumer equilibrium: $\frac{MU_X}{P_X} = \frac{MU_Y}{P_Y}$
• Marginal rate of substitution: $MRS_{XY} = \frac{MU_X}{MU_Y}$
• Budget line slope: $-\frac{P_X}{P_Y}$
• Consumer surplus (linear demand): $\frac{1}{2} \times (P_{max} - P) \times Q$
• Utility maximization condition: $MRS = \frac{P_X}{P_Y}$

Practice Questions

Q1. According to the law of diminishing marginal utility, as consumption of a good increases, its marginal utility will: A. Increase
B. Remain constant
C. Decrease
D. First increase then decrease

Q2. The condition for a consumer to maximize utility is: A. MUx = MUy
B. MUx/Px = MUy/Py
C. TUx = TUy
D. Px = Py

Q3. If MUx/Px > MUy/Py, a rational consumer will: A. Increase consumption of X and decrease consumption of Y
B. Increase consumption of Y and decrease consumption of X
C. Increase consumption of both goods
D. Stop consuming

Q4. The absolute value of the slope of an indifference curve equals: A. The price ratio
B. The marginal rate of substitution
C. The level of income
D. The total utility ratio

Q5. A consumer has CNY 100 income. Good A costs CNY 10, good B costs CNY 20. Given MU_A = 40 and MU_B = 60, the consumer should: A. Buy more of A
B. Buy more of B
C. Maintain current consumption
D. Reduce purchases of both

Q6. Consumer surplus is most likely to increase when: A. The price of the good rises
B. The price of the good falls
C. Income decreases
D. Preferences change

Q7. When marginal utility of a good equals zero, total utility is: A. Maximized
B. Zero
C. Negative
D. Still increasing

Q8. If the MRS between two goods exceeds their price ratio (X in numerator), the consumer should: A. Consume more of X
B. Consume less of X
C. Leave the bundle unchanged
D. Increase income

Answers

Takeaways

• The law of diminishing marginal utility is the most important behavioral regularity in consumer theory.
• Utility maximization occurs when marginal utility per dollar is equal across all goods (MU/P equality).
• The optimal consumption bundle is where the indifference curve is tangent to the budget line, so MRS equals the price ratio.
• A price decrease affects quantity demanded through both substitution and income effects.
• Consumer surplus is the area below the demand curve and above the price; it increases when price falls.
• At the utility-maximizing quantity, marginal utility is typically zero or equal to the marginal utility of income.