Tilman C. Dette Prof. Alex Barnett Math 53 Dec. 5, 2007 A Single Agent Might Make a Difference, … but so doe s a single variation in our model 1. Introduction The well-known economic “Cobweb Model”1 explains cyclical market fluctuations as a result of the time lag in the producers’ response to a change in prices. Producers have to plan their future production (xt+1) ahead of time, basing it only on current production (xt) and prices (pt). Thus, for some supply function f, ! xt +1= f xt, pt( ) Prices, on the other hand, are determined by current output. Hence for some demand function g, ! pt= g xt( ) and thus ! xt +1= f xt,g xt( )( ), defining a discrete mapping. In their essay, “Stability, chaos and multiple attractors: a single agent makes a difference,” Onozaki et al. specify two different simple supply functions and one demand function to prove mathematically that adding a single agent to a large market consisting only of producers of the other supply function can change the qualitative behavior of the market. To investigate into Onozaki et al.’s theory, I wrote a Matlab program that evaluates their model and various alterations numerically. We will first use this program to illustrate Onozaki et al.’s argument that generalizations about the supply function easily misrepresent market behavior. We then use the program to show that the simplifications of the model reduce the model’s accuracy at least as much. 2. Onozaki et al.’s M odel For a given price p, producers choose the output that maximizes their profits ! ", thus ! maxx" x( )= px # C x( )$ C x( )= px =$ C #1p( ) 1 Discussed as early as in 1938 in Mordecai Ezekiel’s The Cobweb Theorem2.1 Naïve Optimizers For simplicity, Onozaki et al. assume the same quadratic cost function C for all producers. They then define the “naïve optimizer” producer to determine his future output xt+1, basing it solely on current prices pt, and thus: so ! C x( )= ax22axt +1= ptxt +1=pta They combine this supply function with an isoelastic2 demand function ! p = g X( ); that is, for any total X produced a marginal change in X will lead to the same marginal change in p: ! g X( )=bX", for ! "# 0. And thus a market consisting entirely of n naïve optimizers has the following discrete mapping: ! Xt +1= xit +1= npta" # $ % & ' =i=1n(nabXt)" # $ % & ' let ! nba= k, and so ! Xt +1=kXt" Consequently, if σ < 1, X will converge to the stable equilibrium X = ! k, if σ = 1, for all X we will have stable period-2 orbits and for σ > 1 the output cycles will explode:3 Fig1: σ = 0.8 Fig2: σ = 1 Fig3: σ = 1.2 2 Constant elasticity, as elasticity ! = "dgdx#xg$ % & ' ( ) "1=*"1bx*+1#xx*b$ % & ' ( ) =*"1 3 Which you can test using the model graph of the program, changing inv. el. above the graph and selecting the cobweb plot of agent 1 or simply proof, as demand is isoelastic and supply is linear.2.2 Cautious Adapters For the second type of producers, Onozaki et al. introduce the “cautious adapter” that chooses to adopt this optimum based on current prices only partially, such that, for c ∈ [0,1), ! xt +1= xt+ c˜ x " xt( ), where ! ˜ x =pta (the quantity the n.o. choose). So for a market consisting only of n cautious optimizers with the same c, we have ! Xt +1= n xt+ c˜ x " xt( )( )= n xt+ cbaXt#" xt$ % & ' ( ) $ % & ' ( ) = Xt+ cnbaXt#" Xt$ % & ' ( ) = Xt+ ckXt#" Xt$ % & ' ( ) as nxt = Xt. Onozaki et al. showed in an earlier paper4 that this mapping converges to X =! k if ! "< 2 # c( )/c, at ! "= 2 # c( )/c, the fixed point undergoes period doubling, for greater σ the map has one attracting periodic or chaotic orbit. Choosing c = 0.5, we use the program to generate the bifurcation diagram of X versus σ: Fig4 (The first period doubling happens at σ = 3 = (2 - .5)/.5, as Onozaki et al. predict) We can use the program to investigate further, noticing that for smaller c, σ needs to be higher for period doubling and higher periodic orbits to occur, which follows our intuition, as the ‘more-cautious’ adapters need a greater variation in price to change their output significantly. 4 Onozaki et al. (2000)2.3 Different Adjustment Strategies in one Market Suppose we have a market with n naïve optimizers that each produce u and m cautious adapters with the same c ∈ [0,1) that each choose to produce v, then from before, we have ! ut +1vt +1" # $ % & ' =baXt(vt+ cbaXt() vt* + , - . / " # $ $ $ $ % & ' ' ' ' 0nut +1mvt +1" # $ % & ' =nbaXt(m vt+ cbaXt() vt* + , - . / * + , - . / " # $ $ $ $ % & ' ' ' ' where Xt = nut + mvt. We can scale this equation such that n+m = 1 with m ∈ (0,1), moreover, we let k = b/c and so we have ! 1" m( )ut +1mvt +1# $ % & ' ( =1" m( )k1" m( )ut+ mvt( ))m vt+ ck1" m( )ut+ mvt( ))" vt* + , , - . / / * + , , - . / / # $ % % % % % & ' ( ( ( ( ( Letting ! x = 1 " m( )u and ! y = mv, we can define the map F to iterate the above equation such that ! F x, y( )=1" m( )kx + y( )#, y + cmkx + y( )#" y$ % & & ' ( ) ) $ % & & ' ( ) ) Suppose m = 0.01, and c = 0.5 (one cautious adapter in a market of naïve optimizers). We use the program to find the bifurcation diagram for σ from 0 to 2.5: Fig5 (blue: naïve optimizers, green: cautious optimizers – both scaled by their inverse frequency) Even though as σ increases, the maximum output still grows very fast (after σ = 2.8, output reaches 70,000), the price no longer explodes, but is bound. The cautiousness of the 0.01 producers prevents output from approaching zero and thus puts a limit on the maximum the price can achieve.In fact, in their preposition 2, Onozaki et al. prove the above intuition, proving that for our map F, “for m ∈ ! R++2, every trajectory starting from ! R++2is trapped into a compact region in ! R++2.” Instead of exploding, trajectories get attracted to periodic or chaotic orbits. Suppose in the market of cautious adapters we discussed before, one naïve optimizer enters, such that m = 0.99: Fig6 (black: …