Economic Exchange
on Networks
Networked Life
CSE 112
Spring 2007
Prof. Michael Kearns
Exchange Economies
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Suppose there are a bunch of different goods orcommodities
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We may all have different initial amounts or endowments
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Of course, we may want to exchange some of our goods
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How should we engage in exchange?
What should be the rates of exchange?
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These are among the oldest questions in markets and economics
– wheat, milk, rice, paper, raccoon pelts, matches, grain alcohol,…
– no differences or distinctions within a good: rice is rice
– 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.
– 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 many sacks of rice per box of matches?
Cash and Prices
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Suppose we introduce an abstract resource called cash
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And now suppose we introduce prices in cash
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Then if we all believed in cash and the prices…
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But will there really be:
– no inherent value
– simply meant to facilitate trade, “encode” pairwise exchange rates
– 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
– we might try to sell our initial endowments for cash
– then use the cash to buy exactly what we most want
– others who want to buy all of our endowments? (demand)
– others who will be selling what we want? (supply)
Mathematical Microeconomics
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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:
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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
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, though
Market Equilibrium
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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 e*p = 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 (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:
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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
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A specific mechanism: bringing your endowments to the stage
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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 clears
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– what could go wrong? 1) stuff left on stage 2) not enough stuff on stage
Another Phone Call from Stockholm
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Arrow and Debreu, 1954:
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Intuition: suppose p is not an equilibrium
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The trickiness:
– There is always a set of equilibrium prices!
– Both won Nobel prizes in Economics
– 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
– changing prices can radically alter consumer preferences
• not necessarily a gradual process; see “bang for the buck” argument
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– everyone reacting/adjusting simultaneously
– utility functions may be extremely complex
May also have to specify “consumption plans”:
– who buys exactly what, and from whom 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 LC
Remarks
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A&D 1954 a mathematical tour-de-force
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Actual markets have been around for millennia
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Model abstracts away details of price adjustment/formation process
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Model can be augmented in various way:
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“Efficient markets” ~ in equilibrium (at least at any given moment)
– resolved and clarified a hundred of years of confusion
– proof related to Nash’s; (n+1)-player game with “price player”
– highly structured social systems
– it’s the mathematical formalism and understanding that’s new
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modern financial markets
pre-currency bartering and trade
auctions
etc. etc. etc.
– labor as a commodity
– firms producing goods from raw materials and labor
– etc. etc. etc.
Network Economics
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All of what we’ve said so far assumes:
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But there are many economic settings in which everyone is not
free to trade with everyone else
– that anyone can trade (buy or sell) with anyone else
– equivalently, exchange takes place on a complete network
– at equilibrium, global prices must emerge due to competition
– 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
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Still begin with the same framework:
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But now assume an undirected network dictating exchange
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Note: can “encode” network in goods and utilities
– k goods or commodities
– n consumers, each with their own endowments and utility functions
– 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
– 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
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Now prices are for each (g,i), not for just raw goods
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Each consumer must still behave rationally
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Market equilibrium still always exists!
– 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?
– 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
– set of prices (and consumptions plans) such that:
• all initial endowments sold (no excess supply)
• no consumer has money left over (no excess demand)
Network Structure and Outcome
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Q: How does the structure of a network influence the prices/wealths at equilibrium?
Need to separate asymmetries of endowments & utilities from those of NW structure
We will thus consider bipartite economies
Only two kinds of players/consumers:
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Equal numbers of Milks and Wheats
Network is bipartite --- only have edges between Milks and Wheats
When will such a network have variation in prices?
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“Milks”: start with 1 unit of milk, but have utility only for wheat
“Wheats” start with 1 unit of wheat, but have utility only for milk
exact form of utility functions irrelevant
An Example
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2
a
2/3
b
2/3
c
2/3
d
w
x
y
z
1/2
3/2
1/2
3/2
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Price = amount of the other good
received = wealth
Prices at opposite ends of an edge
always reciprocal: p and 1/p
Checking equilibrium conditions:
– only “cheapest” edges used
– supply and demand balance:
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a sends 1/2 each to w and y
b sends 1 to x
c sends 1/2 each to x and z
d sends 1 to z
w sends 1 to a
x sends 2/3 to b, 1/3 to c
y sends 1 to a
z sends 1/3 to c, 2/3 to b
Some edges unused at equilibrium
– exchange subgraph
1
a
1
b
1
c
1
d
w
x
y
z
1
1
1
1
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Suppose we add the single green edge
Now equilibrium has no wealth variation!
A More Complex Example
• 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. B2 vs. B11
– e.g. S5 vs. S14
Characterizing Price Variation
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Consider any bipartite “Milk-Wheat” network economy
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Necessary and sufficient condition for all equilibrium prices and wealths to be equal:
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What if there is no perfect matching subgraph? How large can the price variation be?
For any set of vertices S on one side (e.g. Milks), let N(S) be its set of neighbors on the other side
Find the S such that |S|/|N(S)| = p is maximized (here |S| is the number of vertices in S)
Then the largest price/wealth in the network will be p, and the smallest 1/p
Intuition: When S is very large but N(S) is small, consumers in S are “captives” of their neighbors N(S)
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Note: When network has a perfect matching, N(S) is always at least as large as S
Note: Finding the maximizing set S may involve some computation…
Now let’s examine price variation in a statistical network formation model…
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again, all endowments equal to 1.0, equal numbers of Milks and Wheats
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network has a perfect matching as a subgraph
a pairing of Milks and Wheats such that everyone has exactly one trading partner on the other side
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Can actually iterate: remove S and N(S) from the network, find S’ maximizing |S’|/|N(S’)|,…
A Bipartite Economy Network
Formation Model
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Consider economies with only two goods: milk and wheat…
…and only two kinds of players/consumers:
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Wheats and Milks added incrementally in pairs at each time step
Goal: bipartite network formation model interpolating between P.A. and E-R
Probabilistically generates a bipartite graph
All edges between buyers and sellers
Each new party will have n > 1 links back to extant graph
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Distribution of new buyer’s links:
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So (a,n) characterizes distribution of generative model
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Milks: start with 1 unit of milk, have utility only for wheat
Wheats: start with 1 unit of wheat, have utility only for milk
exact form of utility functions irrelevant
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note: n = 1 generates bipartite trees
larger n generates cyclical graphs
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with prob. 1 – a: extant seller chosen w.r.t. preferential attachment
with prob. a: extant seller chosen uniformly at random
a = 0 is pure pref. att.; a = 1 is “like” Erdos-Renyi model
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
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Wealth distribution at equilibrium:
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Price variation (max/min) at equilibrium:
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Random graphs result in “socialist” outcomes
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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
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Characterized by presence/absence of a perfect matching
Alternately: an expansion property
Theory of random walks
Economic vs. geographic isolation
An Amusing Case Study
U.N. Comtrade Data Network
Full Network
price
sorted equilibrium prices
vertex degree
USA: 4.42
Germany: 4.01
Italy: 3.67
France: 3.16
Japan: 2.27
European Union Network
Full Network
EU network
price
sorted equilibrium prices
vertex degree
USA: 4.42
Germany: 4.01
Italy: 3.67
France: 3.16
Japan: 2.27
EU: 7.18
USA: 4.50
Japan: 2.96