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Exchange Economies and Networks Networked Life CSE 112 Spring 2005 Prof. Michael Kearns

Market Economies • Suppose there a bunch of different goods – wheat, rice, paper, raccoon pelts, matches, grain alcohol, … – no differences or distinctions within a good: rice is rice • We may all have different initial amounts or endowments – I might have 10 sacks of rice and two raccoon pelts – you might have 6 bushels of wheat, 2 boxes of matches – etc. • Of course, we may want to exchange some of our goods – I can’t eat 10 sacks of rice, and I need matches to light a fire – it’s getting cold and you need raccoon mittens – etc. • How should we engage in exchange? • What should be the rates of exchange? • These are among the oldest questions in economics

Cash and Prices • Suppose we introduce an abstract good called cash – no inherent value – simply meant to facilitate trade, encode exchange rates • And now suppose we introduce prices in cash – i. e. rates of exchange between each “real” good and cash • Then if we all believed in cash and the prices… – we might try to sell our initial endowments for cash – then use the cash to buy exactly what we most want • But will there really be: – others who want to buy all of our endowments? (demand) – others who will be selling what we want? (supply)

Mathematical Economics • • Have k abstract goods or commodities g 1, g 2, … , gk Have n consumers or players Each player has an initial endowment e = (e 1, e 2, …, ek) > 0 Each consumer has their own utility function: – – assigns a personal valuation or utility to any amounts of the k goods e. g. if k = 4, U(x 1, x 2, x 3, x 4) = 0. 2*x 1 + 0. 7*x 2 + 0. 3*x 3 + 0. 5*x 4 here g 2 is my “favorite” good --- but it might be expensive generally assume utility functions are insatiable • always some bundle of goods you’d prefer more – utility functions not necessarily linear, though

Market Equilibrium • Suppose we post prices p = (p 1, p 2, …, pk) for the k goods • Assume consumers are rational: – they will attempt to sell their endowment e at the prices p (supply) – if successful, they will get cash e*p = e 1*p 1 + e 2*p 2 + … + ek*pk – with this cash, they will then attempt to purchase x = (x 1, x 2, …, xk) that maximizes their utility U(x) subject to their budget (demand) – example: • U(x 1, x 2, x 3, x 4) = 0. 2*x 1 + 0. 7*x 2 + 0. 3*x 3 + 0. 5*x 4 • p = (1. 0, 0. 35, 0. 15, 2. 0) • look at “bang for the buck” for each good i, wi/pi: – g 1: 0. 2/1. 0 = 0. 2; g 2: 0. 7/0. 35 = 2. 0; g 3: 0. 3/0. 15 = 2. 0; g 4: 0. 5/2. 0 = 0. 25 – so we will purchase as much of g 2 and/or g 3 as we can subject to budget • Say that the prices p are an equilibrium if there are exactly enough goods to accomplish all supply and demand steps • That is, supply exactly balances demand --- market clears

The Phone Call from Stockholm • Arrow and Debreu, 1954: – There is always a set of equilibrium prices! – Both won Nobel prizes in Economics • Intuition: suppose p is not an equilibrium – if there is excess demand for some good at p, raise its price – if there is excess supply for some good at p, lower its price – the “invisible hand” of the market • The trickiness: – changing prices can radically alter consumer preferences • not necessarily a gradual process; see “bang for the buck” argument – everyone reacting/adjusting simultaneously – utility functions may be extremely complex • May also have to specify “consumption plans”: – who buys exactly what from whom – example: • A has Fruit Loops and Lucky Charms, but wants granola • B and C have only granola, both want either FL or LC (indifferent) • need to “coordinate” B and C to buy A’s FL and LC

Remarks • A&D 1954 a mathematical tour-de-force – resolved and clarified a hundred of years of confusion – proof related to Nash’s; (n+1)-player game with “price player” • Actual markets have been around for millennia – highly structured social systems – it’s the mathematical formalism and understanding that’s new • Model abstracts away details of price adjustment process – – modern financial markets pre-currency bartering and trade auctions etc. • Model can be augmented in various way: – labor as a commodity – firms producing goods from raw materials and labor – etc. • “Efficient markets” ~ in equilibrium (at least at any given moment)

Network Economics • All of what we’ve said so far assumes: – – that anyone can trade (buy or sell) with anyone else wheat bought from Nikhil is the same as wheat bought from Elliot equivalently, exchange takes place on a complete network global prices must emerge due to competition • But there are many economic settings in which everyone is not free to trade with everyone else – geography: • perishability: you buy groceries from local markets so it won’t spoil • labor: you purchases services from local residents – legality: • if one were to purchase drugs, it is likely to be from an acquaintance (no centralized market possible) • peer-to-peer music exchange – politics: • there may be trade embargoes between nations – regulations: • on Wall Street, certain transactions (within a firm) may be prohibited

A Network Model of Market Economies • Still begin with the same framework: – k goods or commodities – n consumers, each with their own endowments and utility functions • But now assume an undirected network dictating exchange – each vertex is a consumer – edge between i and j means they are free to engage in trade – no edge between i and j: direct exchange is forbidden • Note: can “encode” network in goods and utilities – for each raw good g and consumer i, introduce virtual good (g, i) – think of (g, i) as “good g when sold by consumer i” – consumer j will have • zero utility for (g, i) if no edge between i and j • j’s original utility for g if there is an edge between i and j

Network Equilibrium • Now prices are for each (g, i), not for just raw goods – permits the possibility of variation in price for raw goods – prices of (g, i) and (g, j) may differ – what would cause such variation at equilibrium? • Each consumer must still behave rationally – attempt to sell all of initial endowment, but only to NW neighbors – attempt to purchase goods maximizing utility within budget – will only purchase g from those neighbors with minimum price for g • Market equilibrium still always exists! – set of prices (and consumptions plans) such that: • all initial endowments sold (no excess supply) • no consumer has money left over (no excess demand)

A Sample Network and Equilibrium • Solid edges: – exchange at equilibrium • Dashed edges: – competitive but unused • Dotted edges: – non-competitive prices • Note price variation – 0. 33 to 2. 00 • Degree alone does not determine price! – e. g. B 2 vs. B 11 – e. g. S 5 vs. S 14

Price Variation vs. a and n n=1 n = 250, scatter plot n=2 Exponential decrease with a; rapid decrease with n

(Statistical) Structure and Outcome • Wealth distribution at equilibrium: • Price variation (max/min) at equilibrium: • Random graphs result in “socialist” outcomes • Price variation in arbitrary networks: – Power law (heavy-tailed) in networks generated by preferential attachment – Sharply peaked (Poisson) in random graphs – Grows as a root of n in preferential attachment – None in random graphs – Despite lack of centralized formation process – – Characterized by an expansion property Connections to eigenvalues of adjacency matrix Theory of random walks Economic vs. geographic isolation