1 Cubic Equation 2 Newton-Raphson Method

November 27, 2002The method was generalized to the n variable case by Leonid Kantrovich (Nobel Prize in1975):x (i+1) = x (i) − (J F (x (j) )) −1 · F (x (i) ),where J F (x (j) ) is the Jacobian of F with respect to x evaluated at x (i) .3 Gale-Nikaido’s Proof of Existence of General EquilibriumWe prove the following system of equations has a solution, using the Brouwer Fixed PointTheorem.⎧e 1 (p 1 , p 2 , . . . , p n ) ≤ 0,⎪⎨ e 2 (p 1 , p 2 , . . . , p n ) ≤ 0,(GE)· · ·⎪⎩e n (p 1 , p 2 , . . . , p n ) ≤ 0,where e i (p) ≡ d i (p) − s i (p) is the excess demand function for commodity i, with p ≡(p 1 , p 2 , . . . , p n ). We know e i ’s are continuous on the non-negative orthant of R n , and homogeneousof degree zero. We also have the Walras Law:n∑p i · e i (p) = 0.i=1Therefore, a solution to the system (GE), p, satisfies the following:(1) if p i > 0, then e i (p) = 0, and (2) if e j (p) < 0, then p j = 0.We define the (n − 1)-simplex S to beS ≡ {x | x ∈ R n +,We construct the following map T from S into itself:n∑x i = 1}.i=1p i + max(0, e i (p))T : p ∈ S → (1 + ∑ ni=1 max(0, e i (p)) ).This map T is continuous, and the set S is compact convex, and so there is at least one fixedpoint p ∗ . This should be an equilibrium price vector. First we show at the fixed point p ∗ , thedenominator is 1. Suppose the denominator is k, and k > 1. (It is evident k ≥ 1.) Then weget(k − 1) · p i = max(0, e i (p)) for all i.This means that if p i > 0, then e i (p) > 0, which is a contradiction to the Walras Law.✷Uzawa, H. (1962): “Walras’ Existence Theorem and Brouwer’s Fixed Point Theory”, EconomicStudies Quarterly, vol. 31.Exercise:Q01. Try to devise out your own map whose fixed point can be an equilibrium.