1. David Ellerman. University of California. Riverside www.ellerman.org ... good with one of the parties adding a cash ... $1 fence for 1/5 foot more length.
Towards an Arbitrage Approach to Optimization Theory
David Ellerman University of California Riverside www.ellerman.org
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Research Program Given any optimization problem, find the “market” in the mathematics where First-order conditions in opt. problem = equilibrium conditions in “market” and Lagrange multipliers = competitive equilibrium prices in the “market.” 2
Starting Insight: Kirchhoff = Cournot Kirchhoff’s Voltage Law (KVL): Voltages sum to zero around any circle if and only if there are potentials at nodes so that voltages are potential differences. (1847) P=5
V=3
V=2
P=2 V = –1
P=3
Cournot’s Arbitrage-free Law: exchange rates multiply to one around any circle if and only there are prices at the nodes so that exchange rates are price ratios. (1838) P=5
R = 5/2
P=2
R = 2/3
R = 5/3
P=3
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Calculus Version of KVL KVL is the discrete version of the calculus theorem that the following conditions are equivalent: 1. A vector field is the gradient of a potential function, 2. the line integral of the vector field around any closed path is zero, and 3. a line integral of the vector field between two 4 points is path-independent.
Economic Version of Additive KVL Suppose a market where only one unit of a good can be exchanged for one unit of another good with one of the parties adding a cash “boot” (e.g., trading apartments in a city). Then there exists a system of prices for the goods such that the boots are the price differences if and only if the sum of boots around any circle is zero. 5
Example: Assemblies of Gears or Wheels A gear assembly can move if and only if the product of gear ratios around any circular gear train is one. Circular gear train all in same plane can move if and only if even number of gears.
?
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The Economist seems to think Global Factory won’t work 7
Example: Clique Theory Nodes are people and arrows between people has +1 = like and -1 = dislike. Nodes can be partitioned into 2 cliques with all likes inside a clique and all dislikes between cliques iff product around any circle is +1. 8
Example: Utility Maximization Max U(x1,…, xn) Subject to: p1x1+…+ pnxn= B. “Market” has 2 “goods”, dollars B and utility U. There are n ways to transform an extra $ into U, namely buy more of xi for i = 1,…,n. An extra dollar buys 1/pi units of xi each of which yields MUi more utility so that marginal rate of transformation is MUi/ pi . Arbitragefree condition on “market” is: MU1/ p1 = …= MUn/ pn and that common rate is “the” transformation rate of money into utility, i.e., marginal utility of income.
MU1/p1
$B
MU2/p2 z z z
U
MUn/pn
"Income-Utililty Market” Arbitrage Diagram
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Example: Area Maximization Find maximum area of fenced rectangular field where fence on one side is $4 a foot and only $1 a foot on other 3 sides, given a fixed budget for fence. Two ways to transformdollars to area in the “market”: add to width or to length. $1 more on width gives 1/2 foot more width so L/2 more area. $1 more length uses $0.80 on $4 fence and $0.20 on $1 fence for 1/5 foot more length and W/5 more area.
$4 per foot
Width = W ft.
$1 per foot on other three sides
Length = L ft.
L/2 Square Feet of Area
Cost Dollars W/5
Arbitrage Diagram
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Optimization Results: I Max: w = f(x,y,z) subject to g1(x,y,z) = b1 and g2(x,y,z) = b2. What are marginal rates of transformation of bi→ w for I = 1,2? Form the matrix: fx 1 A = − g x 2 − g x
fy − g 1y − g 2y
fz 1 − gz 2 − g z
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Optimization Results: II What are the transformation rates resulting from changes in (say) y and z? They are the rates resulting from the “prices” which are the cofactors of the first column, i.e., the cofactors P0, P1, P2 in: Thus r1 = P1/ P0, r2 = P2/ P0. ? f y fz 1 1 ? − g y − g z 2 2 ? g g − − y z 12
Optimization Results: III Changing other variables, e.g., x and y or x and z, give other “cofactor prices” and other transformation rates.
b1 P2/P1
r1 = P1/P0
r2 = P2/P0
w
b2 Arbitrage Diagram for "market" between b1, b2, and w.
Arbitrage-free conditions with these rates = necessary 1st-order conditions with r1 and r2 as the Lagrange multipliers.
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Conclusions In an abstract (uninterpreted) optimization problem, we can construct a “market” so that the arbitrage-free conditions on the market are the 1st order conditions for the problem and the normalized prices on the market are the Lagrange multipliers. 14
Unexpected Conclusions “Price” interpretation of cofactors. From economics, we know prices encode transformation rates s.t. quantities transforming according to those rates have zero net value by those prices. Given any n x n matrix, cofactors of a column encode all the transformations made possible by the other columns. Thus value of determinant = cofactor expansion of that column says if that column has new transformations, i.e., det = 0 says “No” and det ≠ 0 says “Yes” 15
