Network EconomicsExchange EconomiesCash and PricesMathematical MicroeconomicsMarket EquilibriumExamplesAnother Phone Call from StockholmRemarksSlide 9A Network Model of Market EconomiesNetwork EquilibriumNetwork Structure and OutcomeSlide 13Slide 14A More Complex ExampleCharacterizing Price VariationA Bipartite Economy Network Formation ModelSlide 18(Statistical) Structure and OutcomeSlide 20Slide 21Slide 22Slide 23Slide 24Behavioral Experiments in Networked TradeGame OverviewSlide 27Equilibrium Theory and Network StructureSlide 29Slide 30Slide 31Slide 32Slide 33Slide 34Network EconomicsNetworked LifeCIS 112Spring 2008Prof. Michael KearnsExchange Economies•Suppose there are a bunch of different goods orcommodities –wheat, milk, 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?–how many sacks of rice per box of matches?•These are among the oldest questions in markets and economicsCash and Prices•Suppose we introduce an abstract resource called cash–no inherent value–simply meant to facilitate trade, “encode” pairwise exchange rates•And now suppose we introduce prices in cash–i.e. rates of exchange between each “real” good and cash–e.g. a racoon pelt is worth $5.25, a box of matches $1.10•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 Microeconomics•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 subjective “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 (* = multiplication)•this is an example of a linear utility function•lots of other possibilities; e.g. diminishing utility as amount becomes large–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 moreMarket Equilibrium•Suppose we announce 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 C = e1*p1 + e2*p2 + … + ek*pk (* = times)–with this cash, they will then attempt to purchase x = (x1,x2,…,xk) that maximizes their utility U(x) subject to their budget C (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•A specific mechanism: –bring your endowments to the stage–I act as banker, distribute cash for endowments–return to stage, use cash to buy optimal bundle of goods•What could go wrong? –1) stuff left on stage 2) not enough stuff on stage•Say that the prices p are an equilibrium if there are exactly enough goods to accomplish all supply and demand constraints•That is, supply exactly balances demand --- market clearsExamples•Example 1: 3 consumers, 2 goods–Consumer A: utility 0.5*x1 + 0.5*x2 (indifferent)–Consumer B: utility 0.75*x1 + 0.25*x2 (prefers Good 1)–Consumer C: utility 0.25*x1 + 0.75*x2 (prefers Good 2)–All endowments = (1,1)•Claim: equilibrium prices = (1.0,1.0)–All three consumers receive 2.0 from sale of endowments–3 units of Good 1: •Consumer B buys as much as he can  2 units–3 units of Good 2:•Consumer C buys as much as he can  2 units–1 unit remains of each good•Consumer A is indifferent, buys both•Example 2:–Consumer A: utility 0.5*x1 + 0.5*x2 (indifferent)–Consumer B: 1.0*x1 (prefers Good 1)–Consumer C: 1.0*x1 (also prefers Good 1)•Claim: equilibrium prices = (2.0,1.0)–All three consumers receive 2+1 = 3.0 from sale of endowments–3 units of Good 1:•Consumer B buys as much as he can  1.5 units•Consumer C buys as much as he can  1.5 units•Supply of Good 1 is exhausted–3 units of Good 2•Consumer A can exactly purchase all 3•How did I figure this out? Guess that B and C must split Good 1  1.5*p1 = p1+p2•Note: even for centralized computation, finding equilibrium is challenging (but tractable)Another 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 famed “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, and from whom–in previous example, may have to specify how much of g2 and g3 to buy–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/formation process–modern financial markets–pre-currency bartering and