Consumer Theory: Concept & Formula
Guide
Comprehensive Cheat Sheet based on Varian’s Microeconomics
1. The Budget Constraint ∆x2 M U1
M RS = =−
∆x1 M U2
This represents the limits of your choices. It defines the
set of affordable bundles given prices and income. Intuition: The MRS is your psychic trade-off. It asks:
”How much of Good 2 are you willing to give up to get
one more unit of Good 1 and stay equally happy?”
The Equation
Watch Out!
p1 x1 + p2 x2 ≤ m Diminishing MRS: As you have more of Good 1,
you value it less on the margin, so you are willing
Where p represents prices, x quantities, and m income. to give up fewer units of Good 2 for it. This is
equivalent to Convex Preferences.
Key Concepts
• Opportunity Cost (Slope): The slope is − pp12 . It 4. Utility Functions
tells you the market rate of exchange. To consume 1
more unit of good 1, you must give up p1 /p2 units of Utility is an ordinal concept (ranking matters, magnitude
good 2. does not). Monotonic Transformation: Applying a
function like ln(u) or u2 (for u > 0) preserves the order of
• The Numeraire: A price set to 1. If we set p2 = 1,
preference. It does not change the MRS or the optimal
the slope becomes −p1 . This simplifies analysis to
choice.
relative prices.
• Quantity Tax: A tax on the amount bought (p1 +t). Common Functions Table
Makes the budget line steeper.
• Lump-Sum Tax: A tax on income (m − t). Shifts Name Function u(x1 , x2 ) MRS (−M U1 /M U2 )
the budget line inward parallelly. Cobb-Douglas xc1 xd2 − dc xx12
Perf. Subst. ax1 + bx2 −a/b (Constant)
2. Preferences & Axioms Perf. Compl. min{ax1 , bx2 } Not Defined (Vertex)
Quasilinear v(x1 ) + x2 −v ′ (x1 )
Economists assume consumers are rational. This is defined
by three axioms:
1. Completeness: You can rank any two bundles (A ≻
B, B ≻ A, or A ∼ B). You are never ”confused.” 2.
Reflexivity: A bundle is at least as good as itself. 3.
Transitivity: If A ≻ B and B ≻ C, then A ≻ C. This 5. Optimal Choice
prevents circular logic.
The goal: Find the highest Indifference Curve (IC) that
touches the budget line.
Well-Behaved Preferences
To use calculus, we usually assume two more traits: A. Interior Solutions (Standard)
• Monotonicity (More is Better): Implies indiffer- Occurs when the consumer buys some of both goods. The
ence curves have a negative slope. mathematical condition is Tangency.
p1
• Convexity (Averages ≻ Extremes): Implies con- M RS = −
sumers prefer a balanced mix of goods. Mathemati- p2
cally, the indifference curve bows toward the origin. p 1 x1 + p 2 x2 = m
The Intuition:
3. Marginal Rate of Substitution (MRS)
• M RS: How much you want to trade (internal value).
The single most important concept in consumer theory. It • p1 /p2 : How much the market makes you trade (mar-
is the slope of the indifference curve. ket cost).
1
Guide
Comprehensive Cheat Sheet based on Varian’s Microeconomics
1. The Budget Constraint ∆x2 M U1
M RS = =−
∆x1 M U2
This represents the limits of your choices. It defines the
set of affordable bundles given prices and income. Intuition: The MRS is your psychic trade-off. It asks:
”How much of Good 2 are you willing to give up to get
one more unit of Good 1 and stay equally happy?”
The Equation
Watch Out!
p1 x1 + p2 x2 ≤ m Diminishing MRS: As you have more of Good 1,
you value it less on the margin, so you are willing
Where p represents prices, x quantities, and m income. to give up fewer units of Good 2 for it. This is
equivalent to Convex Preferences.
Key Concepts
• Opportunity Cost (Slope): The slope is − pp12 . It 4. Utility Functions
tells you the market rate of exchange. To consume 1
more unit of good 1, you must give up p1 /p2 units of Utility is an ordinal concept (ranking matters, magnitude
good 2. does not). Monotonic Transformation: Applying a
function like ln(u) or u2 (for u > 0) preserves the order of
• The Numeraire: A price set to 1. If we set p2 = 1,
preference. It does not change the MRS or the optimal
the slope becomes −p1 . This simplifies analysis to
choice.
relative prices.
• Quantity Tax: A tax on the amount bought (p1 +t). Common Functions Table
Makes the budget line steeper.
• Lump-Sum Tax: A tax on income (m − t). Shifts Name Function u(x1 , x2 ) MRS (−M U1 /M U2 )
the budget line inward parallelly. Cobb-Douglas xc1 xd2 − dc xx12
Perf. Subst. ax1 + bx2 −a/b (Constant)
2. Preferences & Axioms Perf. Compl. min{ax1 , bx2 } Not Defined (Vertex)
Quasilinear v(x1 ) + x2 −v ′ (x1 )
Economists assume consumers are rational. This is defined
by three axioms:
1. Completeness: You can rank any two bundles (A ≻
B, B ≻ A, or A ∼ B). You are never ”confused.” 2.
Reflexivity: A bundle is at least as good as itself. 3.
Transitivity: If A ≻ B and B ≻ C, then A ≻ C. This 5. Optimal Choice
prevents circular logic.
The goal: Find the highest Indifference Curve (IC) that
touches the budget line.
Well-Behaved Preferences
To use calculus, we usually assume two more traits: A. Interior Solutions (Standard)
• Monotonicity (More is Better): Implies indiffer- Occurs when the consumer buys some of both goods. The
ence curves have a negative slope. mathematical condition is Tangency.
p1
• Convexity (Averages ≻ Extremes): Implies con- M RS = −
sumers prefer a balanced mix of goods. Mathemati- p2
cally, the indifference curve bows toward the origin. p 1 x1 + p 2 x2 = m
The Intuition:
3. Marginal Rate of Substitution (MRS)
• M RS: How much you want to trade (internal value).
The single most important concept in consumer theory. It • p1 /p2 : How much the market makes you trade (mar-
is the slope of the indifference curve. ket cost).
1
