The Differential and Delay Differential Approach in the Analysis of Stable State Equilibrium Prices Using Characteristic Equation Techniques
The study compares the stability states of price adjustment differential models with and without delay parameter using roots of characteristic equations. The states of stability of both models were simulated using their particular solutions with inputs from same source. It was found that irrespective of initial prices set for the commodity, the current price for the differential models will always have the propensity to move monotonically to the equilibrium price defined for the system. On the other hand, the current price for the delay-differential models tends to oscillate and move away from the initial price, then with time, decreases and turns towards equilibrium, different from the system defined equilibrium price, due to the delay associated with the supply. It was deduced that whilst the equilibrium prices in the delay-differential models are not predictable due the time delay parameters associated with them, its counterpart without the delay are predictable, since differential models converge to the equilibrium price points defined for the system. Since most economic and natural phenomena are associated with delays, it is recommended that such systems are modeled using delay-differential equations to reflect realities of the models.
Keywords - Stability analysis, delay differential equations, differential equations and equilibrium prices.