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PRICES AND QUANTITIES

PRICES AND QUANTITIES

Fundamentals of Microeconomics RAKESH V. VOHRA

VOHRA

“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 KLEMPERER Edgeworth 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 SHANNON Richard 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

• n 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

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 2

What

Competitive equilibria (CE) with indivisible goods.

• 1. Extend single improvement property of Gul & Stachetti to

non-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 good

is bounded by a preference parameter independent of the size of the economy.

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 3

Why

CE outcomes are a benchmark for the design of markets for allocating goods and services. When they exist they are pareto optimal and in the core. Under certain conditions they satisfy fairness properties like equal treatment of equals and envy-freeness. When goods are indivisible, CE need not exist.

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 4

Prior Work

Restrict preferences to guarantee existence of a CE (eg gross substitutes/ 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 of economy.

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

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 5

Prior Work

Smooth away indivisibility by appealing to ‘large’ markets assumption.

Azevedo & Weyl (2013)

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

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

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 6

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 denoted U(x, t). U(x, t) is continuous and non-increasing in t. Quasi-linearity means U(x, t) = v(x) + t.

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 7

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 ith component is max{xi − yi, 0}. ||x − y||1 = 1 · (x − y)+ + 1 · (y − x)+.

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 8

Single Improvement (quasi-linear)

Binary bundles only (no agent wants more than one unit of any 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).

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 9

∆- 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).

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 10

∆-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.

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 11

Approximate CE

Let N denote the set of agents each equipped with utility function Uj(x, t). si is the supply of good i ∈ M and s the supply vector. Theorem If all agent’s demand types are ∆-substitutes, there exists a price vector p and demands xj ∈ Chj(p) for all j ∈ N such that ||

j xj − s||∞ ≤ ∆.

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 12

Rounding Lemma

Polytope P binary if all of its extreme points are 0-1 vectors and denote its set of extreme points by ext(P). Binary polytope P is ∆-uniform if the ℓ1 norm of each of its edge directions is at most ∆.

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 13

Rounding Lemma

Lemma Let P1, . . . , Pk be a collection of binary polytopes in Rn each

• f which is ∆-uniform.

Let y ∈ k

i=1 Pi be an integral vector.

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

1 xi − y||∞ ≤ ∆.

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 14

Shapley-Folkman-Starr

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

i=1 Pi be integral.

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

i=1 xi − y||∞ ≤ n.

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 15

Comparison

Lemma Let P1, . . . , Pk be a collection of binary polytopes in Rn each

• f which is ∆-uniform. Let y ∈ k

i=1 Pi be an integral vector.

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

1 xi − y||∞ ≤ ∆.

Theorem Let P1, . . . , Pk be a collection of binary polytopes in Rn with k > n. Let y ∈

i Pi be integral. Then, there exist vectors

xi ∈ ext(Pi) for all i such that || k

i=1 xi − y||∞ ≤ n.

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 16

Demand Type

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

Danilov & Koshevoy (2004), tangent cone

Set of vectors that summarize the possible demand changes is called the demand type. In quasi-linear setting, multiple equivalent definitions. Discrete analog to the rows of a Slutsky matrix.

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 17

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 are vectors of the form v − u for some pair v, u ∈ Ch(p). If entries of v − u are scaled so that the greatest common divisor of their entries is 1, we call it a primitive edge direction.

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 18

Demand Type (Baldwin & Klemperer)

A set D ⊆ Zm is the demand type of an agent if it contains the primitive edge directions of conv(Ch(p)) for all price vectors p such that |Ch(p)| > 1. ∆-substitute preferences correspond to the vectors in the demand type having ℓ1 norm of at most ∆.

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 19

Unimodular Demand Type

Matrix is unimodular if determinant of every full rank submatrix has value 0, ±1. A demand type D is called unimodular if the matrix of its vectors is unimodular. Network matrix is a 0, ±1 matrix with at most two non-zero entries in each column and these being of opposite sign. Gross substitutes/ M#-concave corresponds to demand type being a network matrix.

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 20

Unimodular Theorem

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

j xj − s||∞ = 0.

Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 21