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Comprehensive Study Guide: Consumer
Theory
Based on Intermediate Microeconomics by Hal R. Varian
- 1 Introduction Contents
- 2 The Budget Constraint (Chapter 2)
- 2.1 The Budget Line
- 2.2 Key Concepts
- 2.3 Comparative Statics
- 2.4 The Numeraire
- 3 Preferences (Chapter 3)
- 3.1 Axioms of Rational Preferences
- 3.2 Indifference Curves
- 3.3 Well-Behaved Preferences
- 3.4 Marginal Rate of Substitution (MRS)
- 4 Utility (Chapter 4)
- 4.1 Fundamental Concepts
- 4.2 Relationship between MRS and Marginal Utility
- 4.3 Classic Utility Functions Examples
- 4.3.1 1. Cobb-Douglas (Standard Preferences)
- 4.3.2 2. Perfect Substitutes
- 4.3.3 3. Perfect Complements
- 4.3.4 4. Quasilinear Preferences
- 5 Optimal Choice (Chapter 5)
- 5.1 Interior Solution (Tangency Condition)
- 5.2 Corner Solutions (Boundary Optima)
- 5.3 Lagrange Method (For Advanced Calculus)
- 6 Demand (Chapter 6)
- 6.1 Classification of Goods
- 6.1.1 Relative to Income (Engel Curve)
- 6.1.2 Relative to Own Price (Demand Curve)
- 7 The Slutsky Equation (Chapter 8)
- 7.1 Substitution Effect (∆xs)
- 7.2 Income Effect (∆xn)
- 8 Summary of Cobb-Douglas Demand Formulas
- 9 Conclusion
2.4 The Numeraire
We can set one of the prices (or income) to 1 to serve as a unit of account. If we set p 2 = 1 (good 2 is the numeraire), the constraint becomes:
p 1 x 1 + x 2 = m
Here, p 1 represents the relative price of good 1. This simplification is useful because microeconomics focuses on relative prices, not nominal ones.
3 Preferences (Chapter 3)
Preferences describe how a consumer ranks different bundles of goods. We use the symbol ⪰ to denote ”weakly preferred to.”
3.1 Axioms of Rational Preferences
For consumer behavior to be consistent and modelable, we assume three axioms:
- Completeness: For any two bundles A and B, the consumer can determine if they prefer A to B, B to A, or if they are indifferent. They are never ”paralyzed” by indecision.
- Reflexivity: Any bundle is at least as good as itself (A ⪰ A).
- Transitivity: If A ⪰ B and B ⪰ C, then A ⪰ C. This ensures there are no cycles in preferences.
3.2 Indifference Curves
An indifference curve connects all bundles that provide the same level of satisfaction to the consumer.
- Indifference curves cannot cross (this would violate transitivity).
- Bundles on higher curves (further to the northeast) are strictly preferred.
3.3 Well-Behaved Preferences
To facilitate mathematical analysis (calculus), we usually assume two additional prop- erties: 1. Monotonicity: ”More is better.” This implies that indifference curves have a negative slope. 2. Convexity: ”Averages are preferred to extremes.” The consumer prefers a balanced mix of goods over consuming just one type. This implies that the Marginal Rate of Substitution is diminishing.
3.4 Marginal Rate of Substitution (MRS)
The MRS is the slope of the indifference curve at a specific point.
Definition of MRS
M RS =
dx 2 dx 1
∆x 2 ∆x 1
Economic Interpretation: The MRS measures the psychic willingness to trade. How much of x 2 is the consumer willing to give up to gain an extra unit of x 1 while maintaining the same satisfaction level?
4 Utility (Chapter 4)
Utility is a numerical way to represent preferences. If A ≻ B, then U (A) > U (B).
4.1 Fundamental Concepts
- Ordinality: The numerical value of utility does not matter, only the rank order matters. A utility of 20 is not ”twice as good” as 10; it is simply ”better.”
- Monotonic Transformation: Any transformation that preserves the rank or- der (e.g., adding a constant, multiplying by a positive number, cubing if u > 0) represents the same preferences. Example: U (x 1 , x 2 ) = x 1 x 2 represents the same preferences as V (x 1 , x 2 ) = ln(x 1 ) + ln(x 2 ).
- Marginal Utility (M U ): The change in total utility resulting from an additional unit of a good. M U 1 =
∂U
∂x 1
4.2 Relationship between MRS and Marginal Utility
The slope of the indifference curve can be derived from marginal utilities:
MRS via Marginal Utility
M RS = −
M U 1
M U 2
Intuitive Proof: Along an indifference curve, total utility is constant (dU = 0).
dU = M U 1 dx 1 + M U 2 dx 2 = 0 =⇒
dx 2 dx 1
M U 1
M U 2
4.3 Classic Utility Functions Examples
4.3.1 1. Cobb-Douglas (Standard Preferences)
U (x 1 , x 2 ) = xc 1 xd 2
5.1 Interior Solution (Tangency Condition)
For ”well-behaved” preferences (like Cobb-Douglas), the optimum occurs where the in- difference curve is tangent to the budget line.
The Optimization Condition
|M RS| =
p 1 p 2 M U 1 M U 2
p 1 p 2
or
M U 1
p 1
M U 2
p 2
Economic Meaning: At the optimal point, the rate at which the consumer is willing to trade goods (internal valuation) equals the rate at which the market allows the trade (opportunity cost).
- If |M RS| > p 1 /p 2 : The consumer values good 1 more than the market does. They should buy more good 1.
- If |M RS| < p 1 /p 2 : The consumer values good 1 less than the market cost. They should buy less good 1.
To solve a problem, solve the system of two equations: 1. M RS = −p 1 /p 2 2. p 1 x 1 + p 2 x 2 = m (Budget Line)
5.2 Corner Solutions (Boundary Optima)
These occur when the consumer spends their entire income on just one good. This is common with Perfect Substitutes or concave preferences.
- If p 1 < p 2 (and utility is x 1 + x 2 ): The consumer buys only good 1.
- The tangency condition does not necessarily hold here.
5.3 Lagrange Method (For Advanced Calculus)
To maximize U (x 1 , x 2 ) subject to p 1 x 1 + p 2 x 2 = m, we set up the Lagrangian:
L = U (x 1 , x 2 ) − λ(p 1 x 1 + p 2 x 2 − m)
First Order Conditions (FOC):
- (^) ∂x∂L 1 = M U 1 − λp 1 = 0
- (^) ∂x∂L 2 = M U 2 − λp 2 = 0
- ∂ ∂λL = m − p 1 x 1 − p 2 x 2 = 0
Dividing (1) by (2), we recover the tangency condition: M U M U^12 = p p^12.
6 Demand (Chapter 6)
The consumer’s demand functions, x 1 (p 1 , p 2 , m) and x 2 (p 1 , p 2 , m), show the optimal quan- tities for every set of prices and income.
6.1 Classification of Goods
6.1.1 Relative to Income (Engel Curve)
- Normal Good: Demand increases when income increases ( (^) ∂m∂x > 0).
- Inferior Good: Demand decreases when income increases ( (^) ∂m∂x < 0). Example: Instant noodles (when rich, you switch to steak).
6.1.2 Relative to Own Price (Demand Curve)
- Ordinary Good: Demand falls when price rises (∂x∂p < 0). This is the Law of Demand.
- Giffen Good: Demand rises when price rises (∂x∂p > 0). This is a rare theoretical anomaly. It happens when a good is so inferior that the income effect (loss of purchasing power) overwhelms the substitution effect.
7 The Slutsky Equation (Chapter 8)
When the price of a good changes, two effects occur simultaneously. Slutsky decomposes the total change in demand (∆x) into these two components.
The Slutsky Equation
∆x 1 = ∆xs 1 + ∆xn 1 Where:
- ∆xs 1 : Substitution Effect.
- ∆xn 1 : Income Effect.
7.1 Substitution Effect (∆xs)
This is the change in demand due to the change in relative prices, holding purchasing power constant.
- In the Slutsky decomposition, we pivot the budget line around the original bundle.
- The Substitution Effect is always negative (opposite to price change). If P 1 rises, xs 1 falls, because the good is now relatively more expensive.
