📖 无差异曲线与预算线

Economics

I. Lesson Focus

This lesson explains how consumers make choices under constraints. Candidates must be able to draw and interpret indifference curves (representing preferences) and budget lines (representing constraints), identify the optimal consumption bundle where they are tangent, calculate marginal rates of substitution, and analyze the effects of income and price changes on consumer equilibrium.

II. The Problem

Wang has a monthly disposable income of CNY 8,000 and faces two goods: food (P = CNY 40 per unit) and entertainment (P = CNY 80 per unit). How should he allocate his limited income between the two goods to maximize his satisfaction? If the price of food rises by 20%, how will his optimal consumption bundle change? This is the central question of consumer theory—using indifference curves to represent preferences and budget lines to represent constraints in order to find the utility-maximizing equilibrium point.

III. Consumer Preferences and Indifference Curves

Consumer preferences reflect a consumer’s subjective ranking of different bundles of goods. CFA requires mastery of three fundamental assumptions: - Completeness: The consumer can compare any two bundles. - Transitivity: If bundle A is preferred to B and B to C, then A is preferred to C. - Non-satiation (more is better): Additional units of goods are preferred.

An indifference curve (IC) connects all combinations of two goods that provide the consumer with the same level of utility. Its key properties are: - Downward-sloping (negative slope): To keep utility constant, an increase in one good must be offset by a decrease in the other. - Convex to the origin: The marginal rate of substitution (MRS) is diminishing. - Higher curves (farther from the origin) represent higher utility levels. - Indifference curves never intersect each other.

Marginal Rate of Substitution (MRS)
MRS = –ΔY/ΔX = MUₓ / MUᵧ
MRS measures how much of good Y a consumer is willing to give up for one additional unit of good X while remaining equally satisfied. The absolute value of MRS declines as X increases, which produces the convex shape of the indifference curve.

IV. Budget Constraint and the Budget Line

The budget line (BL) shows all combinations of two goods that a consumer can afford given income (I) and prices. Its equation is:
PₓX + PᵧY = I

• Slope = –Pₓ/Pᵧ
• X-intercept = I/Pₓ (maximum units of X)
• Y-intercept = I/Pᵧ (maximum units of Y)

Points inside the line are feasible but do not exhaust income; points on the line exactly exhaust income; points outside are unattainable.

Income changes: An increase in income shifts the budget line outward in a parallel fashion; a decrease shifts it inward in parallel.
Price changes: A decrease in the price of one good rotates the budget line outward around the intercept of the other good; a price increase rotates it inward.

V. Consumer Equilibrium: Tangency of Indifference Curve and Budget Line

Utility is maximized where the budget line is tangent to the highest attainable indifference curve. At this point:
MRS = Pₓ / Pᵧ
or equivalently
MUₓ / Pₓ = MUᵧ / Pᵧ (equal marginal utility per dollar spent)

This tangency point is the optimal consumption bundle.

VI. Income-Consumption and Price-Consumption Curves

• Income-Consumption Curve (ICC): The locus of optimal bundles as income changes. For normal goods, the ICC slopes upward to the right.
• Price-Consumption Curve (PCC): The locus of optimal bundles as the price of one good changes.

Worked Cases

Case 1: Drawing the Budget Line and Finding Equilibrium

Li has income I = 1,200, Pₓ (food) = 30, Pᵧ (clothing) = 60.
(1) Write the budget-line equation and calculate intercepts.
(2) Given MRS = 2 – 0.05X, when X = 12, MRS = 1.4. Is the consumer at equilibrium?

Solution:
Budget equation: 30X + 60Y = 1,200 → X + 2Y = 40.
X-intercept = 40 units, Y-intercept = 20 units.
At X = 12, MRS = 2 – 0.05 × 12 = 1.4, while Pₓ/Pᵧ = 30/60 = 0.5.
Because MRS (1.4) > Pₓ/Pᵧ (0.5), the consumer values X more relative to its price and should increase X consumption until MRS equals the price ratio. Therefore, X = 12 is not the equilibrium.

Case 2: Effect of a Price Change on the Budget Line and Equilibrium

Income I = 1,000, initial Pₓ = 20, Pᵧ = 50. Optimal bundle is X = 30, Y = 8 (MRS = 0.4).
If Pₓ rises to 25, write the new budget equation, compute the new X-intercept, and qualitatively predict the change in X consumption.

Solution:
Initial: 20X + 50Y = 1,000 → 2X + 5Y = 100; X-intercept = 50, Y-intercept = 20.
New: 25X + 50Y = 1,000 → X + 2Y = 40; X-intercept = 40, Y-intercept = 20.
The rise in Pₓ rotates the budget line inward around the Y-intercept. Both substitution and income effects reduce X consumption (normal good), so new X must be less than 30 units.

Case 3: Marginal-Utility Equilibrium Calculation

Utility function U = 2X^{0.5}Y^{0.5}, I = 240, Pₓ = 8, Pᵧ = 12. Find optimal X and Y.

Solution:
MUₓ = Y^{0.5}/X^{0.5}, MUᵧ = X^{0.5}/Y^{0.5}.
Equilibrium: MUₓ/Pₓ = MUᵧ/Pᵧ
(Y^{0.5}/X^{0.5})/8 = (X^{0.5}/Y^{0.5})/12
12Y = 8X → X = 1.5Y.

Substitute into budget: 8(1.5Y) + 12Y = 240 → 24Y = 240 → Y = 10, X = 15.
Optimal bundle: X = 15, Y = 10. Utility ≈ 36.74.

Traps

Key Formulas

• Budget constraint: $P_X X + P_Y Y = I$
• Budget-line slope: $-P_X / P_Y$
• MRS = $\frac{MU_X}{MU_Y} = \frac{P_X}{P_Y}$ (at equilibrium)
• Equal marginal utility per dollar: $\frac{MU_X}{P_X} = \frac{MU_Y}{P_Y}$
• Cobb-Douglas example: For $U = X^a Y^b$, optimal $X = \frac{a}{a+b} \frac{I}{P_X}$

Practice Questions

Q1. The typical shape of an indifference curve is:
A. Upward-sloping
B. Convex to the origin
C. Concave to the origin
D. Horizontal straight line

Q2. If consumer income increases while prices remain constant, the budget line will:
A. Shift left in parallel
B. Shift right in parallel
C. Rotate around the X-intercept
D. Rotate around the Y-intercept

Q3. At the consumer’s equilibrium point, it must be true that:
A. MRS > Px/Py
B. MRS = Px/Py
C. MRS < Px/Py
D. MRS = 0

Q4. A consumer has MRS = 2.5 at X = 10, Y = 15, with Px = 5 and Py = 10. The consumer should:
A. Increase X and reduce Y
B. Increase Y and reduce X
C. Keep the bundle unchanged
D. Cannot be determined

Q5. For a normal good X, when its price falls, the substitution and income effects are:
A. Substitution increases X, income decreases X
B. Both substitution and income increase X
C. Substitution decreases X, income increases X
D. Both decrease X

Q6. The budget-line equation is 4X + 6Y = 48. Its slope is:
A. –4/6 = –2/3
B. –6/4 = –3/2
C. 4/6 = 2/3
D. 6/4 = 3/2

Q7. Which of the following is NOT a basic assumption of consumer preferences?
A. Completeness
B. Transitivity
C. Convexity
D. Non-satiation

Q8. A consumer has utility U = XY, I = 100, Px = 4, Py = 5. The optimal quantity of X is:
A. 10
B. 12.5
C. 15
D. 20

Answers

Takeaways

• Indifference curves represent equal utility, are convex to the origin, never intersect, and show diminishing MRS.
• Budget lines represent income and price constraints; slope = –Px/Py. Parallel shifts occur with income changes; rotations occur with price changes.
• Consumer equilibrium occurs at tangency of the budget line and highest reachable indifference curve: MRS = Px/Py or MUx/Px = MUy/Py.
• For normal goods, a price decrease raises consumption via both substitution and income effects working in the same direction.
• Giffen goods are an extreme inferior-good case where the income effect outweighs the substitution effect.
• Distinguishing the curved, convex shape of indifference curves from the straight budget line is essential for correct graphical analysis.