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Cover designed by Hart McLeod Ltd

Cover illustration: Paolo Ucello’s `The Hunt’ is a masterpiece of perspective. Hunters follow their dogs in pursuit of quarry towards a vanishing point enveloped in darkness. Like the painting, Economics offers a perspective in this case, on human behavior, which like the painting’s vanishing point is enveloped in darkness. Photo courtesy of IanDagnall Computing / Alamy Stock Photo.

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Fundamentals of Microeconomics

RAKESH V. VOHRA

VO

HR

A

“A leader in the field provides a readable but rigorous introduction to microeconomics with clear, mathematical arguments that students will depend on to fill conceptual gaps in their understanding of economic markets.”

PAUL KLEMPEREREdgeworth Professor of Economics, University of Oxford

“This beautifully written textbook gives a masterfully innovative development of modern intermediate microeconomics, elegantly and concisely building core principles by moving from monopoly to imperfect competition and basic game theory, then to consumer theory and general equilibrium. Woven throughout are compelling and engaging examples drawn from classics, history, literature, and current events, making it as enjoyable to read as it is instructive, and ideally suited for learning modern economics.”

CHRIS SHANNONRichard and Lisa Steiny Professor of Economics and Professor of Mathematics, University of California–Berkeley

Rakesh V. Vohra offers a unique approach to studying

and understanding intermediate microeconomics

by reversing the conventional order of treatment,

starting with the topics that are mathematically

simpler and progressing to the more complex. The

book begins with monopoly, which requires single-

variable rather than multivariable calculus and allows

students to focus very clearly on the fundamental

trade-off at the heart of economics: margin vs.

volume. Imperfect competition and the contrast with

monopoly follows, introducing the notion of Nash

equilibrium. Perfect competition is addressed toward

the end of the book, where it is framed as a model

non-strategic behavior by firms and agents. The last

chapter is devoted to externalities, with an emphasis

on how one might design competitive markets to

price externalities and linking the difficulties to

the problem of efficient provision of public goods.

Real-life examples and anecdotes engage the reader

while encouraging them to think critically about the

interplay between model and reality.

RAKESH V. VOHRA is the George A. Weiss and

Lydia Bravo Weiss University Professor at the

University of Pennsylvania. He is the author of

Principles of Pricing: An Analytical Approach with

Lakshman Krishnamurthi (2012) and Mechanism

Design: A Linear Programming Approach (2011).

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 1

∆-substitutes and Indivisible Goods

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn)

May 11, 2020


What

Competitive equilibria (CE) with indivisible goods.

1. Extend single improvement property of Gul & Stachetti tonon-unit demand and non-quasi-linear preferences.

2. Extend unimodular theorem (Baldwin & Klemperer(2019)) to non-quasi-linear preferences.

3. Identify prices at which the excess demand for each goodis bounded by a preference parameter independent of thesize of the economy.


Why

CE outcomes are a benchmark for the design of markets forallocating goods and services.

When they exist they are pareto optimal and in the core.

Under certain conditions they satisfy fairness properties likeequal treatment of equals and envy-freeness.

When goods are indivisible, CE need not exist.


Prior Work

Restrict preferences to guarantee existence of a CE (eg grosssubstitutes/ M#-concavity).

Kelso & Crawford (1982), Gul & Stachetti (1999), Danilov,

Koshevoy & Murota (2001), Sun & Yang (2006)

Determine prices that ‘approximately’ clear the market;mismatch between supply and demand grows with size ofeconomy.

Broome (1971), Dierker (1970), Starr (1969)


Prior Work

Smooth away indivisibility by appealing to ‘large’ marketsassumption.

Azevedo & Weyl (2013)

Approximate CE outcomes based on cardinal notions ofwelfare; approximations scale slowly with size of economy.

Dobzinski et al (2014), Feldman et al (2014)


Notation

M = set of indivisible goods.

A bundle of goods is denoted by a vector x ∈ Zm+.

Utility for a bundle x and transfer t transfer is denotedU(x , t).

U(x , t) is continuous and non-increasing in t.

Quasi-linearity means U(x , t) = v(x) + t.


Notation

p ∈ Rm is a price vector.

Choice correspondence, denoted Ch(p):

Ch(p) = arg max{U(x , p · x) : x ∈ Zn+}.

(x − y)+ is vector whose i th component is max{xi − yi , 0}.

||x − y ||1 = ~1 · (x − y)+ +~1 · (y − x)+.


Single Improvement (quasi-linear)

Binary bundles only (no agent wants more than one unit ofany good). Suppose at price vector p:

Suppose U(x , p · x) < U(y , p · y).

Then, ∃ bundle z such that ||x − z ||1 ≤ 2 and

U(z , p · z) > U(x , p · x).


∆- Improvement (quasi-linear)

Suppose U(x , p · x) < U(y , p · y). Suppose at price vector p:

Then, ∃ bundle z such that ||x − z ||1 ≤ ∆ and

U(z , p · z) > U(x , p · x).

The case ∆ = 2 contains gross substitutes (Kelso & Crawford,M#-concave).


∆-Improvement (non-quasi linear)

Two price vectors p and p′,

x , y ∈ Ch(p), ||y − x ||1 > ∆ and (p′ − p) · y < (p′ − p) · x

1. ∃ a ≤ (x − y)+ and b ≤ (y − x)+

2. z := x − a + b ∈ Ch(p),

3. ||z − x ||1 ≤ ∆ and

4. (p′ − p) · z < (p′ − p) · x .

Preferences satisfy ∆-substitutes.


Approximate CE

Let N denote the set of agents each equipped with utilityfunction Uj(x , t).

si is the supply of good i ∈ M and s the supply vector.

TheoremIf all agent’s demand types are ∆-substitutes, there exists aprice vector p and demands x j ∈ Chj(p) for all j ∈ N suchthat ||

∑j x

j − s||∞ ≤ ∆.


Rounding Lemma

Polytope P binary if all of its extreme points are 0-1 vectorsand denote its set of extreme points by ext(P).

Binary polytope P is ∆-uniform if the `1 norm of each of itsedge directions is at most ∆.


Rounding Lemma

LemmaLet P1, . . . ,Pk be a collection of binary polytopes in Rn eachof which is ∆-uniform.

Let y ∈∑k

i=1 Pi be an integral vector.

Then, there exist vectors x i ∈ ext(Pi) for all i such that||∑k

1 xi − y ||∞ ≤ ∆.


Shapley-Folkman-Starr

Let P1, . . . ,Pk be a collection of binary polytopes in Rn withk > n.

Let y ∈∑k

i=1 Pi be integral.

Then, there exist vectors x i ∈ ext(Pi) for all i such that||∑k

i=1 xi − y ||∞ ≤ n.


Comparison

LemmaLet P1, . . . ,Pk be a collection of binary polytopes in Rn eachof which is ∆-uniform. Let y ∈

∑ki=1 Pi be an integral vector.

Then, there exist 0-1 vectors x i ∈ ext(Pi) for all i such that||∑k

1 xi − y ||∞ ≤ ∆.

TheoremLet P1, . . . ,Pk be a collection of binary polytopes in Rn withk > n. Let y ∈

∑i Pi be integral. Then, there exist vectors

x i ∈ ext(Pi) for all i such that ||∑k

i=1 xi − y ||∞ ≤ n.


Demand Type

Baldwin & Klemperer: characterize preferences over bundles ofindivisible goods in terms of how demand changes in responseto a small non-generic price change.

Danilov & Koshevoy (2004), tangent cone

Set of vectors that summarize the possible demand changes iscalled the demand type.

In quasi-linear setting, multiple equivalent definitions.

Discrete analog to the rows of a Slutsky matrix.


Demand Type (Baldwin & Klemperer)

Consider convex hull of Ch(p) denoted conv(Ch(p)).

The edges of conv(Ch(p)) are its 1-dimensional faces and arevectors of the form v − u for some pair v , u ∈ Ch(p).

If entries of v − u are scaled so that the greatest commondivisor of their entries is 1, we call it a primitive edgedirection.


Demand Type (Baldwin & Klemperer)

A set D ⊆ Zm is the demand type of an agent if it containsthe primitive edge directions of conv(Ch(p)) for all pricevectors p such that |Ch(p)| > 1.

∆-substitute preferences correspond to the vectors in thedemand type having `1 norm of at most ∆.


Unimodular Demand Type

Matrix is unimodular if determinant of every full ranksubmatrix has value 0,±1.

A demand type D is called unimodular if the matrix of itsvectors is unimodular.

Network matrix is a 0,±1 matrix with at most two non-zeroentries in each column and these being of opposite sign.

Gross substitutes/ M#-concave corresponds to demand typebeing a network matrix.


Unimodular Theorem

TheoremSuppose each agent is interested in consuming at most oneunit of each good. If all agent’s demand types are unimodular,there exists a price vector p and demands x j ∈ Chj(p) for allj ∈ N such that ||

∑j x

j − s||∞ = 0.


Documents

Advertising · Principles of Pricing: An Analytical Approach with Lakshman Krishnamurthi (2012) and Mechanism Design: A Linear Programming Approach (2011). Thanh Nguyen (Purdue) &