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Advertising “A leader in the field provides a readable but Rakesh V. Vohra offers a unique approach to studying rigorous introduction to microeconomics with and understanding intermediate microeconomics clear, mathematical arguments that students by reversing the conventional order of treatment, VOHRA PRICES AND will depend on to fill conceptual gaps in their starting with the topics that are mathematically understanding of economic markets.” simpler and progressing to the more complex. The book begins with monopoly, which requires single- PAUL KLEMPERER variable rather than multivariable calculus and allows Edgeworth Professor of Economics, students to focus very clearly on the fundamental University of Oxford QUANTITIES trade-off at the heart of economics: margin vs. volume. Imperfect competition and the contrast with PRICES AND QUANTITIES “This beautifully written textbook gives a monopoly follows, introducing the notion of Nash masterfully innovative development of modern equilibrium. Perfect competition is addressed toward 9781108488938 VOHRA – PRICES AND QUANTITIES PPC C M Y K intermediate microeconomics, elegantly and the end of the book, where it is framed as a model concisely building core principles by moving from non-strategic behavior by firms and agents. The last Fundamentals of monopoly to imperfect competition and basic chapter is devoted to externalities, with an emphasis game theory, then to consumer theory and general on how one might design competitive markets to equilibrium. Woven throughout are compelling price externalities and linking the difficulties to Microeconomics and engaging examples drawn from classics, the problem of efficient provision of public goods. history, literature, and current events, making it Real-life examples and anecdotes engage the reader as enjoyable to read as it is instructive, and ideally while encouraging them to think critically about the suited for learning modern economics.” interplay between model and reality. CHRIS SHANNON Richard and Lisa Steiny Professor of Economics RAKESH V. VOHRA is the George A. Weiss and RAKESH V. VOHRA and Professor of Mathematics, University of Lydia Bravo Weiss University Professor at the California–Berkeley 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). 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. Cover designed by Hart McLeod Ltd 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 ∈ Z m + . 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 ∈ R m is a price vector. Choice correspondence, denoted Ch ( p ): Ch ( p ) = arg max { U ( x , p · x ) : x ∈ Z n + } . ( x − y ) + is vector whose i th component is max { x i − y i , 0 } . 1 · ( x − y ) + + � || x − y || 1 = � 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 U j ( x , t ). s i 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 x j ∈ Ch j ( p ) for all j ∈ N such j x j − s || ∞ ≤ ∆ . that || � 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 ℓ 1 norm of each of its edge directions is at most ∆. Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 13

Rounding Lemma Lemma Let P 1 , . . . , P k be a collection of binary polytopes in R n each of which is ∆ -uniform. Let y ∈ � k i =1 P i be an integral vector. Then, there exist vectors x i ∈ ext ( P i ) for all i such that 1 x i − y || ∞ ≤ ∆ . || � k Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 14

Shapley-Folkman-Starr Let P 1 , . . . , P k be a collection of binary polytopes in R n with k > n . Let y ∈ � k i =1 P i be integral. Then, there exist vectors x i ∈ ext ( P i ) for all i such that i =1 x i − y || ∞ ≤ n . || � k Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 15

Comparison Lemma Let P 1 , . . . , P k be a collection of binary polytopes in R n each of which is ∆ -uniform. Let y ∈ � k i =1 P i be an integral vector. Then, there exist 0-1 vectors x i ∈ ext ( P i ) for all i such that 1 x i − y || ∞ ≤ ∆ . || � k Theorem Let P 1 , . . . , P k be a collection of binary polytopes in R n with k > n. Let y ∈ � i P i be integral. Then, there exist vectors x i ∈ ext ( P i ) for all i such that || � k i =1 x i − 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 ⊆ Z m 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 ℓ 1 norm of at most ∆. Thanh Nguyen (Purdue) & Rakesh Vohra (Penn) 19