Exchange Economies on NetworksExchange EconomiesCash and PricesMathematical EconomicsMarket EquilibriumAnother Phone Call from StockholmRemarksNetwork EconomicsA Network Model of Market EconomiesNetwork EquilibriumSlide 11Slide 12Our Experimental NetworkThe Theory Says…Slide 15Slide 16What Actually Happened?Some AnalysisPrizes: $10 EachA Preferential Attachment Model for Buyer-Seller NetworksA Sample Network and EquilibriumSlide 22(Statistical) Structure and OutcomeSlide 24Slide 25Slide 26Slide 27Exchange Economieson NetworksNetworked LifeCSE 112Spring 2006Prof. Michael KearnsExchange Economies•Suppose there are 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. etc. 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. etc. etc.•How should we engage in exchange?•What should be the rates of exchange?•These are among the oldest questions in economicsCash 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 g1, g2, … , gk•Have n consumers or players•Each player has an initial endowment e = (e1,e2,…,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(x1,x2,x3,x4) = 0.2*x1 + 0.7*x2 + 0.3*x3 + 0.5*x4–here g2 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, thoughMarket Equilibrium•Suppose we post prices p = (p1,p2,…,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 = e1*p1 + e2*p2 + … + ek*pk–with this cash, they will then attempt to purchase x = (x1,x2,…,xk) that maximizes their utility U(x) subject to their budget (demand)–example:•U(x1,x2,x3,x4) = 0.2*x1 + 0.7*x2 + 0.3*x3 + 0.5*x4•p = (1.0,0.35,0.15,2.0)•look at “bang for the buck” for each good i, wi/pi:–g1: 0.2/1.0 = 0.2; g2: 0.7/0.35 = 2.0; g3: 0.3/0.15 = 2.0; g4: 0.5/2.0 = 0.25–so we will purchase as much of g2 and/or g3 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 clearsAnother 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 LCRemarks•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. etc. etc.•Model can be augmented in various way:–labor as a commodity–firms producing goods from raw materials and labor–etc. etc. 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 Jenn is the same as wheat bought from Partha–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 prohibitedA 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 jNetwork Equilibrium•Now prices are for each (g,i), not for just raw goods–permits the possibility of variation in price for raw goods–prices