Abstract
Based on a nonlinear demand function and a market-clearing price, a cobweb model is introduced in this paper. A gradient mechanism that depends on the marginal profit is adopted to form the 1D discrete dynamic cobweb map. Analytical studies show that the map possesses four fixed points and only one attains the profit maximization. The stability/instability conditions for this fixed point are calculated and numerically studied. The numerical studies provide some insights about the cobweb map and confirm that this fixed point can be destabilized due to period-doubling bifurcation. The second part of the paper discusses the memory factor on the stabilization of the map’s equilibrium point. A gradient mechanism that depends on the marginal profit in the past two time steps is adopted to incorporate memory in the model. Hence, a 2D discrete dynamic map is constructed. Through theoretical and numerical investigations, we show that the equilibrium point of the 2D map becomes unstable due to two types of bifurcations that are Neimark–Sacker and flip bifurcations. Furthermore, the influence of the speed of adjustment parameter on the map’s equilibrium is analyzed via numerical experiments.
Highlights
Different disciplines such as biology, engineering, and economy are characterized by real-life models which possess complex dynamic behaviors and multistability criteria
Our contribution in this paper focuses on further investigating on complex dynamic characteristics of the cobweb model constructed based on the nonlinear demand function and gradient mechanism
We have investigated a cobweb model whose producers do not possess complete knowledge about the market and update their outputs according to the mechanism of bounded rationality. e producers have estimated their marginal profit by observing profit variations that might have occurred at the beginning of production
Summary
Different disciplines such as biology, engineering, and economy are characterized by real-life models which possess complex dynamic behaviors and multistability criteria. Our contribution in this paper focuses on further investigating on complex dynamic characteristics of the cobweb model constructed based on the nonlinear demand function and gradient mechanism. In order to brief the outcomes of this paper, we divide the contribution into two important parts: the first part deals with the studies and investigations on a one-dimensional nonlinear discrete-time map describing the change in price. It has four nonzero real fixed points and only one of them.
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