A Predator-Prey Model for Shares of Wealth Between Industry and Agriculture

Abstract

The problem is to explore the possible coordinates and stability conditions for equilibria between agricultural wealth and industrial wealth generation and distribution processes. Instability of the equilibria may establish the possibility of moving along trajectories to preferred positions. The behaviour of the trajectories approaching such positions, as oscillations periodic or otherwise, may be explored by studying the operation of bifurcation parameters to determine whether such positions are attractors or repellors. The problem extends to determining qualitative change in the dynamic properties of equilibrium shares of wealth under varying conditions of ecospheric sustainability. This paper is prompted by curiousity about the long future of agriculture as the ecosphere is pressured, and dependency on government supports and off-farm income is put in jeopardy by government deficits and economic recession. The problem definition and approach stem from the second generation of systems theory. A predator prey paradigm is used to model the relationship of agriculture to the rest of the economy and the ecosphere. In the event that the ecosphere continues to degrade and productivity differentials between agriculture and industry expand, could chaotic processes emerge to jeopardize the persistence of both agriculture and industry? The transfer of wealth between agriculture and industry is a central theme in development economics. The transfer takes places through shifts of value added attributable to labour, intellectual property, land and capital. Wealth is understood in Adam Smith's terms of the purchasing power for necessities, conveniences and enjoyments of human life. The term agriculture could be generalized to denote all natural resource sectors. These are usually located in rural economies. Industry refers to manufacturing and tertiary sectors, usually associated with urban economies. Debate is ongoing in development economics around sectoral investment priorities based on leading sector hypotheses, dualism and strategies for balanced and unbalanced growth. Comparative statics and market equilibrium modelling, premised on perfect competition, have driven many of the prescriptions for market structures, wages, productivity, extension, training and capital allocations between agriculture and industry. Debate over the prescriptions regularly spills over into trade friction when governments modify competitive advantages of one sector to the other through domestic support programs and border policies. Predator prey modelling appears to offer new insight into a number of persistent questions left unresolved by agricultural economists. Are there substantive economic reasons for government support of agriculture and industry? Government accompanies or directs agricultural development in many jurisdictions globally. Is their reluctance to disengage and allow competitive forces to determine development outcomes based on more than opportunistic lobbying of vested interests? What are the equilibrium and stability conditions underlying the relationships of agricultural to industrial wealth in the long run? What is the effect of ecospheric degradation on the properties of these relationships which could shed light on the sustainability question? Can human determinism influence attainment of these conditions? Are the conditions leading to and away from chaos predictable? The solution of the model presented in this paper suggests that three parametric relationships among economic variables determine the long run tendency either to impoverish or to enrich agriculture and industry. The first is that the terms of trade between agriculture and industry should at least favour agriculture to enable trajectories to move away from both axes in the long run. Second, a scaling parameter representing the productivity for agriculture relative to industry should be near to the value one. Third, the ratio of the growth rate to the rate of degradation of agricultural wealth should be greater than a threshold level of agricultural wealth determined as a composite parameter. This composite is calculated from a fixed industrial cost expressed in terms of a decline in industrial wealth in the absence of agriculture, the price index for industry, and the economic recovery rates for agriculture and industry. All possible trajectories are bounded by parametric asymptotes to the isoclines. The model is formulated mathematically so as to explore the predictability of effects of stability of equilibria on changes in agricultural wealth. This formulation provides for oscillations or \"equilibrium chases\" along trajectories. The outcomes suggest that for economies where industry buys and sells to agriculture, levels of wealth are indeed limited and that the fortunes of industry are favoured by a wealthy sustained agriculture. The alternative, subsistence agriculture, is associated with near zero levels of industrial wealth. Finally, the model demonstrates that agricultural and industrial wealth, while complementing each other in growth processes, may also rise and fall in sustained oscillation relative to each other on a periodic or near periodic basis.