Price Dynamics of Maize in Ghana: An Application of Continuous –Time Delay Differential Equations

This paper seeks to study the price dynamics of maize in Ghana, in the context of mathematical modelling using continuous-time cobweb models derived from linear and nonlinear delay differential equations. The stability conditions of two cobweb models: continuous-time linear and nonlinear models are discussed. The data is obtained from the Ministry of Food and Agriculture, Statistical Directorate Kumasi-Ghana, from 1994 to 2013.
The models performed on the assumptions that, maize has no equal substitutes and there are no exogenous shocks needed to generate price fluctuations so that market price would be determined by only the available supply in a single market.
From the results of the analysis performed on the real economic price and production data using numerical approach, the nonlinear delay differential model (formulated from linear demand and quadratic supply functions) showed oscillations between and around two equilibrium points and would neither converge. This result seems realistic as it appears to reflect real market conditions since an equilibrium price would not be compatible with the prevailing situation of high inflation, food insufficiency and/or producers’ sensitivity towards price.
The linear model (formulated from linear demand and linear supply functions) on the other hand showed little oscillations and then converged towards an equilibrium point (zero equilibrium price). However, this seems quite unrealistic.
Effects of delay parameter on oscillations are also discussed. It is observed that oscillations (price fluctuations) are suppressed for ≤0.5, using the nonlinear delay equation. This is an indication that price fluctuations are reduced, if and only if, factors affected by time lag, such as the time necessary for increasing supply, buying new inputs, hiring workers or transporting commodities to market centres, building warehouse and reducing effects of natural constraints on crop yields are improved. On the contrary, the linear delay differential equation, for >0.5 or ≤0.5, would still be in stable equilibrium.
We draw inferences from this study that researchers should rather use nonlinear models instead of linear models in solving most real-life economic problems to avoid misleading conclusions.