Question

MTH133 Lab 5: Pre-Lab Learning Goals In class, typical examples of divergent sequences and series are those whose values fly off to infinity. In real life, however, divergence is more frequently associated with meandering behavior that doesn't fly off to infinity but also doesn't settle down toward one number. The simplest example of this is the eyclical behavior of the sequence 2 1,-1,1, 1, 1,-1,... which remains always bounded but nevertheless does not converge. Real-life examples of bounded divergent sequences don't usually exhibit the same level of regu larity, and this is what makes it hard to predict the future. Today you will learn about convergence and divergence, especially the cyclical booms and busts in a model of economics. You will see examples of the butterfly effect that comes up in chaos theory and see why it is difficult to profit from short-term investments in the stock market. A model of the market We describe here a mathematical model that captures some essential features of a capitalist free market. The main unknown is X), the number of producers in a given industry Here counts the number of financial quarters since a fixed date (say first quarter of 1980). so is always a whole number Since financial reports are released quarterly in the United States, www on the reported earn ings of the previons porter une companies may decide to enter the market, and some may decide to leave Our model aims to describe the following process i n weitional mumbers will enter . When the number of proces i below market the marot When the number of producers in a marko market l ibrium, some fraction will exit the Fundamental to the model in the response curve R(X) that relates the member of industry players in the next quarter to the m er active in the current quarter. In other words, if at the

Answer 1

Please give me thumbs up. Thank you.

Solution I = We expect R(x) to be a decreasing function Because we want one X(t) to converse and AS X(++) = R(X(t)) x (t) if RCW) is increasing , then X (t) will never conuerge. solution 2 = R(x) is always nen negative because number of industry players can not be negative. As X (tt) = R(X(t)) X(t) if R(x) is negative then X(t) is negative. Solution 3 & for market equillibrium we want that number of industry player doesn't change ) x (t+1) = x which is true if R(x)=1. Therefore, solution to R(x)=1 is the market equilibrium.