PhD Microeconomic Theory
Study Handbook — Part I: Individual Decision-Making
Consumer Theory, Producer Theory, and Choice Under Uncertainty
About this handbook. This is an original study guide written to consolidate the standard
first-semester PhD microeconomic theory curriculum on individual decision-making: rational
choice, consumer theory, producer theory, and decision-making under uncertainty. It is not
a summary or reproduction of any specific textbook — the definitions, theorems, and proof
techniques here are the common mathematical content of the field (the same core results
appear, in one arrangement or another, in every leading graduate microeconomics course).
It is intended as a compact reference for review and as a source of worked, qualifying-exam-
style practice problems. It is not a substitute for working through a full course text and
problem sets.
Contents
1 Notation and Conventions 2
2 Rational Preferences and Choice 2
3 Consumer Theory 4
3.1 The Utility Maximization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 The Expenditure Minimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.4 The Slutsky Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 Producer Theory 8
4.1 Profit Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 Cost Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5 Choice Under Uncertainty 10
5.1 Expected Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.2 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.3 Stochastic Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6 Practice Qualifying Questions with Full Solutions 13
,PhD Microeconomic Theory Handbook Part I: Individual Decision-Making
1 Notation and Conventions
n commodities; a bundle is x = (x1 , . . . , xn ) ∈ Rn+ .
p = (p1 , . . . , pn ) ∈ Rn++ is a price vector; w > 0 is wealth (income).
⪰ denotes a preference relation, ≻ strict preference, ∼ indifference.
u(·) denotes an ordinal utility function (Sections 2–3); u(·) denotes a Bernoulli (vNM) utility
index over money in Section 4 — context makes clear which is meant.
Superscript notation x(p, w) denotes Walrasian (Marshallian) demand; h(p, ū) denotes Hicksian
(compensated) demand.
“LNS” abbreviates local nonsatiation: for every x and ε > 0 there exists x′ with ∥x′ − x∥ < ε
and u(x′ ) > u(x).
2 Rational Preferences and Choice
Definition 2.1: Preference Relation
A preference relation ⪰ on a set of alternatives X is a binary relation. We interpret x ⪰ y as
“x is at least as good as y.” Strict preference is x ≻ y iff x ⪰ y and not y ⪰ x; indifference is
x ∼ y iff x ⪰ y and y ⪰ x.
Definition 2.2: Rationality
⪰ is rational if it is:
(i) Complete: for all x, y ∈ X, x ⪰ y or y ⪰ x (or both).
(ii) Transitive: for all x, y, z ∈ X, x ⪰ y and y ⪰ z imply x ⪰ z.
Definition 2.3: Continuity
⪰ is continuous if for any sequences xn → x, yn → y with xn ⪰ yn for all n, it holds that x ⪰ y.
Equivalently, the upper and lower contour sets {y : y ⪰ x} and {y : x ⪰ y} are closed for every
x.
Theorem 2.1: Utility Representation (Debreu)
If ⪰ is a rational, continuous preference relation on X = Rn+ , then there exists a continuous
function u : X → R representing ⪰, i.e. x ⪰ y ⇐⇒ u(x) ≥ u(y).
Remark
Representation is ordinal : any strictly increasing transform f (u(·)) represents the same pref-
erences. This is why, in Sections 2–3, utility levels themselves have no meaning — only the
ranking they induce. Contrast this with the cardinal vNM utility index of Section 4, which
is unique only up to a positive affine transformation because it must also respect probability
mixtures.
2
, PhD Microeconomic Theory Handbook Part I: Individual Decision-Making
Definition 2.4: Choice Structure and WARP
A choice structure (B, C(·)) consists of a family of budget sets B and a choice correspondence
C(B) ⊆ B for each B ∈ B. It satisfies the Weak Axiom of Revealed Preference (WARP)
if: whenever x, y ∈ B and x ∈ C(B), then for any B ′ with x, y ∈ B ′ and y ∈ C(B ′ ), it must
also hold that x ∈ C(B ′ ). Informally: if x is ever chosen when y was affordable, y can never
be chosen (to the exclusion of x) when x is also affordable.
Proposition 2.1: Rationality Implies WARP
If demand x(p, w) is generated by maximizing a rational preference relation, it satisfies WARP.
The converse — recovering a rationalizing preference relation from choice data satisfying WARP
— requires either a sufficiently rich family of budget sets (Richter’s theorem) or, for finite data
sets, is characterized by the Generalized Axiom of Revealed Preference (Afriat’s Theorem).
3
Study Handbook — Part I: Individual Decision-Making
Consumer Theory, Producer Theory, and Choice Under Uncertainty
About this handbook. This is an original study guide written to consolidate the standard
first-semester PhD microeconomic theory curriculum on individual decision-making: rational
choice, consumer theory, producer theory, and decision-making under uncertainty. It is not
a summary or reproduction of any specific textbook — the definitions, theorems, and proof
techniques here are the common mathematical content of the field (the same core results
appear, in one arrangement or another, in every leading graduate microeconomics course).
It is intended as a compact reference for review and as a source of worked, qualifying-exam-
style practice problems. It is not a substitute for working through a full course text and
problem sets.
Contents
1 Notation and Conventions 2
2 Rational Preferences and Choice 2
3 Consumer Theory 4
3.1 The Utility Maximization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 The Expenditure Minimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.4 The Slutsky Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 Producer Theory 8
4.1 Profit Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 Cost Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5 Choice Under Uncertainty 10
5.1 Expected Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.2 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.3 Stochastic Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6 Practice Qualifying Questions with Full Solutions 13
,PhD Microeconomic Theory Handbook Part I: Individual Decision-Making
1 Notation and Conventions
n commodities; a bundle is x = (x1 , . . . , xn ) ∈ Rn+ .
p = (p1 , . . . , pn ) ∈ Rn++ is a price vector; w > 0 is wealth (income).
⪰ denotes a preference relation, ≻ strict preference, ∼ indifference.
u(·) denotes an ordinal utility function (Sections 2–3); u(·) denotes a Bernoulli (vNM) utility
index over money in Section 4 — context makes clear which is meant.
Superscript notation x(p, w) denotes Walrasian (Marshallian) demand; h(p, ū) denotes Hicksian
(compensated) demand.
“LNS” abbreviates local nonsatiation: for every x and ε > 0 there exists x′ with ∥x′ − x∥ < ε
and u(x′ ) > u(x).
2 Rational Preferences and Choice
Definition 2.1: Preference Relation
A preference relation ⪰ on a set of alternatives X is a binary relation. We interpret x ⪰ y as
“x is at least as good as y.” Strict preference is x ≻ y iff x ⪰ y and not y ⪰ x; indifference is
x ∼ y iff x ⪰ y and y ⪰ x.
Definition 2.2: Rationality
⪰ is rational if it is:
(i) Complete: for all x, y ∈ X, x ⪰ y or y ⪰ x (or both).
(ii) Transitive: for all x, y, z ∈ X, x ⪰ y and y ⪰ z imply x ⪰ z.
Definition 2.3: Continuity
⪰ is continuous if for any sequences xn → x, yn → y with xn ⪰ yn for all n, it holds that x ⪰ y.
Equivalently, the upper and lower contour sets {y : y ⪰ x} and {y : x ⪰ y} are closed for every
x.
Theorem 2.1: Utility Representation (Debreu)
If ⪰ is a rational, continuous preference relation on X = Rn+ , then there exists a continuous
function u : X → R representing ⪰, i.e. x ⪰ y ⇐⇒ u(x) ≥ u(y).
Remark
Representation is ordinal : any strictly increasing transform f (u(·)) represents the same pref-
erences. This is why, in Sections 2–3, utility levels themselves have no meaning — only the
ranking they induce. Contrast this with the cardinal vNM utility index of Section 4, which
is unique only up to a positive affine transformation because it must also respect probability
mixtures.
2
, PhD Microeconomic Theory Handbook Part I: Individual Decision-Making
Definition 2.4: Choice Structure and WARP
A choice structure (B, C(·)) consists of a family of budget sets B and a choice correspondence
C(B) ⊆ B for each B ∈ B. It satisfies the Weak Axiom of Revealed Preference (WARP)
if: whenever x, y ∈ B and x ∈ C(B), then for any B ′ with x, y ∈ B ′ and y ∈ C(B ′ ), it must
also hold that x ∈ C(B ′ ). Informally: if x is ever chosen when y was affordable, y can never
be chosen (to the exclusion of x) when x is also affordable.
Proposition 2.1: Rationality Implies WARP
If demand x(p, w) is generated by maximizing a rational preference relation, it satisfies WARP.
The converse — recovering a rationalizing preference relation from choice data satisfying WARP
— requires either a sufficiently rich family of budget sets (Richter’s theorem) or, for finite data
sets, is characterized by the Generalized Axiom of Revealed Preference (Afriat’s Theorem).
3
