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Economics

I. Lesson Focus

This lesson derives the market demand curve from individual consumer utility maximization. It covers the conditions for consumer equilibrium, the decomposition of price effects into substitution and income effects using the Slutsky equation, the distinction among normal, inferior, and Giffen goods, the construction of market demand by horizontal summation, and the calculation of consumer surplus. Candidates must be able to numerically separate substitution and income effects and identify exceptions to the law of demand.

II. The Problem

A consumer with monthly income of 5,000 yuan faces two goods: X (food) and Y (other goods). When the price of X falls from 10 yuan per unit to 6 yuan per unit, the consumer’s purchases of X rise from 200 to 280 units. Exam questions frequently ask: How much of the increase is due to the substitution effect and how much to the income effect? Would the demand curve for X slope upward if X were a Giffen good? This lesson derives the demand curve rigorously from utility maximization and decomposes price changes into the two component effects.

III. Consumer Equilibrium and the Micro Foundation of the Demand Curve

Consumers maximize utility subject to the budget constraint:
$P_X \cdot Q_X + P_Y \cdot Q_Y = I$

Utility is represented by $U(Q_X, Q_Y)$. At the optimum, the marginal rate of substitution equals the price ratio:
$\text{MRS}_{XY} = \frac{MU_X}{MU_Y} = \frac{P_X}{P_Y}$

where $MU_X = \partial U / \partial Q_X$ is the marginal utility of X.

Because marginal utility diminishes as consumption of X rises, a lower $P_X$ requires a higher $Q_X$ to restore equality of marginal-utility-per-dollar across goods. This produces a downward-sloping individual demand curve. Summing all individual demand curves horizontally at each price yields the market demand curve, which is also downward-sloping but flatter when many consumers are present.

IV. Income and Substitution Effects: Hicksian and Slutsky Decompositions

A price change produces two distinct effects:

• Substitution effect: When $P_X$ falls, holding utility constant (Hicksian) or real purchasing power constant (Slutsky), the consumer substitutes toward the now-cheaper X.
• Income effect: The fall in $P_X$ increases real income; the consumer buys more or less X depending on whether X is normal or inferior.

The Slutsky equation is:
$\frac{\partial Q_X}{\partial P_X} = \frac{\partial Q_X^c}{\partial P_X} - Q_X \cdot \frac{\partial Q_X}{\partial I}$

The first term on the right is the compensated (pure substitution) effect; the second term is the income effect.

• Normal good: income effect > 0, total effect < 0 → downward-sloping demand.
• Inferior good: income effect < 0, but if substitution effect dominates, demand still slopes down.
• Giffen good: income effect < 0 and larger in absolute value than the substitution effect → total effect > 0 and demand slopes upward (rare).

V. Consumer Surplus and Market-Demand Applications

Consumer surplus (CS) is the integral of the difference between willingness to pay and actual price:
$\text{CS} = \int_{0}^{Q^} (P(Q) - P^) \, dQ$

For a linear demand $Q = a - bP$, CS equals the triangular area:
$\text{CS} = \frac12 \times (P_{\max} - P^) \times Q^$

Market demand is also used to analyze tax incidence, price ceilings, and deadweight loss.

Worked Cases

Case 1: Numerical Decomposition of Substitution and Income Effects (Slutsky Method)

Utility function $U = Q_X^{0.6} Q_Y^{0.4}$, income $I = 1,000$. Initial prices $P_X = 10$, $P_Y = 5$.

Initial equilibrium:
$\frac{MU_X}{MU_Y} = \frac{0.6/0.4 \times Q_Y}{Q_X} = \frac{10}{5} \Rightarrow Q_Y = \frac43 Q_X$
Budget: $10Q_X + 5 \times \frac43 Q_X = 1,000 \Rightarrow Q_X = 60$, $Q_Y = 80$.

New equilibrium after $P_X$ falls to 5: $Q_X' = 90$, $Q_Y' = 80$. Total change = +30.

Slutsky compensated income to keep original bundle affordable: $I' = 5\times60 + 5\times80 = 700$.
At new prices and $I' = 700$, compensated quantity $Q_X^c = 72$.

Substitution effect = 72 − 60 = +12
Income effect = 90 − 72 = +18
Total effect = +30 (X is normal).

Case 2: Extreme Giffen-Good Scenario

A low-income consumer spends mainly on potatoes (X) and a little meat (Y). $P_X$ falls from 2 yuan/kg to 1 yuan/kg. Initial purchases: X = 400 kg, Y = 20 kg. After the price drop the consumer reduces potato purchases to 350 kg to buy more meat. Income effect = −50, substitution effect = +30, total effect = −20. Demand for X slopes upward; X is a Giffen good.

Case 3: Market Demand Aggregation and Consumer Surplus

1,000 identical consumers, each with $Q_d = 50 - 2P$. Market demand: $Q_M = 50,000 - 2,000P$.

At $P = 10$, $Q_M = 30,000$, $P_{\max} = 25$.
Consumer surplus = $\frac12 \times (25 - 10) \times 30,000 = 225,000$.

A per-unit tax of 2 raises price to 12, $Q_M = 26,000$, new CS = $\frac12 \times (25 - 12) \times 26,000 = 169,000$. Consumer loss = 56,000.

Traps

Key Formulas

• Consumer equilibrium: $\frac{MU_X}{MU_Y} = \frac{P_X}{P_Y}$ or $\frac{MU_X}{P_X} = \frac{MU_Y}{P_Y}$
• Slutsky equation: $\frac{\partial Q_X}{\partial P_X} = SE - Q_X \cdot \frac{\partial Q_X}{\partial I}$
• Linear demand: $Q = a - bP$
• Consumer surplus (linear): $\text{CS} = \frac12 (P_{\max} - P^) Q^$
• Market demand: $Q_M = \sum Q_i(P)$ (horizontal summation)
• Income elasticity: $E_I = \frac{\%\Delta Q}{\%\Delta I}$ (> 0 normal, < 0 inferior)
• Cross-price elasticity: $E_{XY} = \frac{\%\Delta Q_X}{\%\Delta P_Y}$ (> 0 substitutes, < 0 complements)

Practice Questions

Q1. At the consumer’s equilibrium, which condition must hold?
A. $MU_X = MU_Y$
B. $MU_X / P_X = MU_Y / P_Y$
C. $MRS = 1$
D. Income effect equals substitution effect

Q2. When the price of good X falls, its substitution effect is:
A. Always positive
B. Always negative
C. Positive for normal goods and negative for inferior goods
D. Dependent on income level

Q3. Which good is most likely to exhibit Giffen behavior?
A. Luxury watch
B. Imported wine
C. Staple food of low-income households
D. Smartphone

Q4. For the linear market demand $Q = 120,000 - 4,000P$, consumer surplus at $P = 15$ is closest to:
A. 225,000
B. 450,000
C. 675,000
D. 900,000

Q5. If X is an inferior good and its price falls, then:
A. Income effect positive, substitution effect positive
B. Income effect negative, substitution effect positive, total effect ambiguous
C. Income effect positive, substitution effect negative
D. Both effects negative

Q6. Market demand is obtained by:
A. Vertical summation of individual demands
B. Horizontal summation of individual demands
C. Using only the highest-income consumer’s demand
D. Averaging individual demands

Q7. The purpose of Slutsky compensation is to:
A. Keep utility constant
B. Keep the purchasing power of the original bundle constant
C. Make the income effect zero
D. Eliminate cross-price effects

Q8. Price falls from 10 to 8 and quantity demanded of X rises from 50 to 65 units. Given a substitution effect of +10 units, the income effect is:
A. −5 units
B. +5 units
C. +15 units
D. +10 units

Answers

Takeaways

• Consumer equilibrium rests on $\frac{MU_X}{P_X} = \frac{MU_Y}{P_Y}$; this is the fundamental reason individual demand curves slope downward.
• Substitution effect is always positive when own price falls; income effect depends on whether the good is normal (> 0) or inferior (< 0).
• Only when the negative income effect exceeds the positive substitution effect does a Giffen good appear and demand slope upward.
• Market demand is the horizontal sum of individual demands at each price.
• For linear demand, consumer surplus is the triangle above the price line.
• The Slutsky decomposition (constant purchasing power) is the version typically required at Level I.