A CHAOTIC ECONOMIC GROWTH MODEL AND THE AGRICULTURAL SHARE OF AN OUTPUT

The agricultural share of a total output generally declines in the process of economic growth. The major reason for this is that consumer demand for food increases only slightly with rising incomes. However, a small, open economy can overcome this constraint to the growth of agricultural production by expanding its net exports. The basic aim of this paper is to set up a chaotic growth model of the gross domestic product that is capable of generating stable equilibria, cycles, or chaos depending on parameter values.

accompanied by a marked increase in total production and associated with the spread of techniques based upon empirical science.All three elements are important in the process; but the spread of modern technology, physical and social , was the base; growth in production its most important result; and the diversion away from agriculture a corollary consequence, even though superficially most conspicuous (K u z n e t s, 1971).
The economic growth can thus be defined on the quantitive side as a sustained increase in total output; and on the structural side, as the extension of scientific technology resulting in diversification away from agriculture, the emergence of a host of new industries of urban character, rapid growth of productivity in all industries including agriculture (K u z n e t s, 1971).
Chaos theory is used to prove that erratic and chaotic fluctuations can indeed arise in completely deterministic models.Chaos theory reveals structure in aperiodic, dynamic systems.The number of nonlinear business cycle models use chaos theory to explain complex motion of the economy.Chaotic systems exhibit a sensitive dependence on initial conditions: seemingly insignificant changes in the initial conditions produce large differences in outcomes.This is very different from stable dynamic systems in which a small change in one variable produces a small and easily quantifiable systematic change.
Chaos theory started with L o r e n z ' s (1963) discovery of complex dynamics arising from three nonlinear differential equations leading to turbulence in the weather system.L i and Y o r k e (1975) discovered that the simple logistic curve can exhibit very complex behaviour.F u r t h e r , M a y (1976)

The model
Irregular movement of the gross domestic product (GDP) , Y , can be analyzed in the formal framework of the chaotic growth model .
The autonomous rate of the gross domestic product (GDP) , Y , growth is α , or ) 1 ( Further , it is supposed that the increase in the GDP , (Y t+1 -Y t )/Y t , depends on productivity (Y/L) and the share of agriculture , P, in the gross domestic product ( GDP) ,Y, (P/Y) ,i.e.
Further, it is assumed that the current value of the gross domestic product (GDP), Y, is restricted by its maximal value in its time series, Y m .This premise requires a modification of the growth law.Now, the GDP growth rate depends on the current size of the GDP , Y, relative to its maximal size in its time series Y m .We introduce y as y = Y/Y m .Thus y range between 0 and 1. Again we index y by t, i.e. write y t to refer to the size at time steps t=0,1,2,3,... Now growth rate of the gross domestic product is measured This model given by equation ( 7) is called the logistic model.For most choices of α , β , γ, λ and δ, there is no explicit solution for (7).Namely, knowing α , β , γ, λ and δ, and measuring y 0 would not suffice to predict y t for any point in time, as was previously possible.This is at the heart of the presence of chaos in deterministic feedback processes.L o r e n z (1963) discovered this effect -the lack of predictability in deterministic systems.Sensitive dependence on initial conditions is one of the central ingredients of what is called deterministic chaos.
This kind of difference equation ( 7) can lead to very interesting dynamic behavior , such as cycles that repeat themselves every two or more periods, and even chaos, in which there is no apparent regularity in the behavior of y t .This difference equation (7) will possess a chaotic region.Two properties of the chaotic solution are important:firstly, given a starting point y 0 the solution is highly sensitive to variations of the parameters α , β , γ, λ and δ ; secondly, given the parameters α , β , γ, λ and δ the solution is highly sensitive to variations of the initial point y 0 .In both cases the two solutions are for the first few periods rather close to each other, but later on they behave in a chaotic manner.
It is possible to show that iteration process for the logistic equation is equivalent to the iteration of growth model ( 6) when we use the identification Using ( 9) and ( 7) we obtain On the other hand, using ( 8) and ( 9) we obtain Thus we have that iterating Consider the first order, nonlinear difference equation The steady-state points, z e , are obtained by solving Because the slope is negative but less than 1 in absolute value, z t converges to the equilibrium value z e from either direction within a neighborhood but the approach path will oscillate locally.As z t approaches the neighborhood of z e , the slope becomes negative, causing z t to begin oscillating as it converges to the  (z t ) is negative in this case so that paths oscillate in the region of the stationary point.Besides, f '(z t ) being smaller than -1 is outside the stable range.In this sense, paths do not converge to the steady stae point, z e .They also do not diverge to zero or infinity.The reason is that f ( z ) is nonmonotonic, causing the phase curve to be hill -shaped.As a path diverges from z e , z 1 is further away from z e than was z 0 ; z 2 is even further away.Further, the path next runs into the positively sloped region of the phase curve, causing it to bounce back towards z e .This returns the path immediately to the negatively sloped region of the phase curve, which leads once again to diverging oscillations.
If 1 < π < 3 than z t converges to the steady-state point and if π ≥ 3 than z t will not converge to the steady-state point but neither will it diverge endlessly.If π is slightly larger than 3, z t converges to a stable limit cycle, two periods long.Namely, regardless of the starting value, z t converges to a path that cycles back and forth in successive periods between two values.The system bifurcates at π = 3 , which means that it changes from having one steady-state value, z e , to having an equilibrium in which there are two values between which the path of z oscillates.If π is as large as about 3.5 , the two-period limit cycle itself becomes unstable and the system bifurcates again.Now, a stable limit cycle being four periods long.z t+1 z t F i g. 3. -When π = 3.2 .there is a stable limit cycle of two periods Figure 3. shows the phase diagram for π = 3.2.At this value , we get a limit cycle of two periods.
Higher values of π produce repeated bifurcations and hence, repeated doublings of the period of the limit cycles.At some values of π , the time paths do not have cycles of any length, although they continue to be bounded.This is said to be chaos.This means that the path of z t fluctuates in an apparently random fashion over time, not settling down into any regular pattern whatsoever.
More precisely, as π increases through the range 3 < π < 3.57 , stable cycles of lengths 1,2,4.8,16,... are generated.As π increases further , 3.57 < π < 4, an infinite number of bifurcations arise, leading to chaos ( see It is obtained that : (i) For parameter values 0 < π < 1 all solutions will converge to z = 0; (ii) For 1 < π < 3.57 there exist fixed points the number of which depends on π; (iii) For 1<π< 2 all solutions monotnically increase to z = (π -1 ) / π; (iv) For 2 < π < 3 fluctuations will converge to z = (π -1 ) / π; (v) For 3 < π < 4 all solutions will continuously fluctuate; (vi) For 3.57 < π < 4 the solution becomes "chaotic" which means that there exist totally aperiodic solution or periodic solutions with a very large, complicated period.This means that the path of z t fluctuates in an apparently random fashion over time, not settling down into any regular pattern whatsoever.

C o n c l u s i o n
The agricultural share of a country's total output generally declines in the process of economic growth.There are two main reasons which explain this tendency: firstly, the Engel's Law (Ernest Engel, 19th century), i.e. demand for food and agricultural products tends to be income-inelastic ; secondly, economic growth generates improvements in agricultural productivity.Further, labor is released from agriculture.Also, as incomes rise, industry's share in the gross domestic product (GDP) is increased.
This paper suggests conclusion for the use of the economic growth model in predicting the fluctuations of the gross domestic product (GDP ) .The model (7) has to rely on specified parameters α , β , γ, λ and δ and initial value of the gross domestic product (GDP), y 0 .But even slight deviations from the values of parameters α , β , γ, λ and δ and initial value of the gross domestic product (GDP), y 0 , show the difficulty of predicting a long-term behavior of the gross domestic product (GDP).
A key hypothesis of this work is based on the idea that the coefficient α π + = 1 -γ δ plays a crucial role in explaining economic stability , where αthe autonomous rate of the GDP growth , δ -the proportion of the gross domestic product assigned to agriculture, γ-the influence of the agricultural share of a total output (P/Y) on the GDP growth rate.
described chaos in population biology.Chaos theory has been applied in Economics by B e n h a b i b and D a y (1981,1982), D a y (1982, 1983), G r a n d m o n t (1985), G o o d w i n (1990), M e d i o (1993), L o r e n z (1993), among many others.

)
It is important because the dynamic properties of the logistic equation (8) have been widely analyzed (L i and Y o r k e (1975), M a y (1976)).

z e -π z e ( 1
steady-state equilibrium exists only if π > 1.If π ≤ 1, then the steady states are 0 and negative , respectively.It is important to evaluate the derivative of equation (8) at the points z e We obtain π at z e = 0 We can draw the phase diagram for this difference equation.The graph cuts through the 45 o line at the points 0 and π π 1 − .The graph peaks at z =1/2 where the slope is 0. If the second derivative of the function is negative ( = -2 π ) then the function is concave.Whether the peak of the graph occurs to the left or right π is larger or smaller than 2. If 1 < π < 2 then the graph intersects the 45 o line to the left of the peak and the slope of the graph is positive at the stable steady -state point.If π > 2 then the slope of the graph is negative at the steady-state point.Figure 1.shows the phase diagram for the case in which 2 < π < 3.This fulfils the condition for local stability.On the other hand, the slope of the graph is negative at the stable steady-state point. .1. -The path of z t converges monotonically at first, but eventually oscillates as it converges to the steady state z e

5 Figure 2 .
Figure 2.  shows the phase diagram for the case of π = 3.5 .It is known that f' (z t ) is negative in this case so that paths oscillate in the region of the stationary point.Besides, f '(z t ) being smaller than -1 is outside the stable range.In this sense, paths do not converge to the steady stae point, z e .They also do not diverge to zero or infinity.The reason is that f ( z ) is nonmonotonic, causing the phase curve to be hill -shaped.As a path diverges from z e , z 1 is further away from z e than was z 0 ; z 2 is even further away.Further, the path next runs into the positively sloped region of the phase curve, causing it to bounce back towards z e .This returns the path immediately to the negatively sloped region of the phase curve, which leads once again to diverging oscillations.If 1 < π < 3 than z t converges to the steady-state point and if π ≥ 3 than z t will not converge to the steady-state point but neither will it diverge endlessly.If π is slightly larger than 3, z t converges to a stable limit cycle, two periods long.Namely, regardless of the starting value, z t converges to a path that cycles back and forth in successive periods between two values.The system bifurcates at π = 3 , which means that it changes from having one steady-state value, z e , to having an equilibrium in which there are two values between which the path of z

Figure 4
Figure 3. shows the phase diagram for π = 3.2.At this value , we get a limit cycle of two periods.Higher values of π produce repeated bifurcations and hence, repeated doublings of the period of the limit cycles.At some values of π , the time paths do not have cycles of any length, although they continue to be bounded.This is said to be chaos.This means that the path of z t fluctuates in an apparently random fashion over time, not settling down into any regular pattern whatsoever.More precisely, as π increases through the range 3 < π < 3.57 , stable cycles of lengths 1,2,4.8,16,... are generated.As π increases further , 3.57 < π < 4, an infinite number of bifurcations arise, leading to chaos ( seeFigure 4.)