Dynamic monopoly with demand delay

Abstract

Implicit in the text-book monopoly is an assumption of complete and instantaneous information or knowledge available to economic agents at free of charge. Under such circumstances, knowing the certain price and cost functions, the monopolist can make an optimal decision of price and output to maximize its profit and realize it. As a result, the text-book monopoly model becomes static in nature. There are, however, many empirical works to indicate that such an assumption of rational economic agents goes too far. In reality the monopolist is boundedly rational and adjusts its price and output as a function of its limited knowledge and past experiences. To fill this gap, we propose, in this study, to relax this assumption and develop a dynamic monopoly model. In particular, we assume first that the monopolist has only partial information about the market condition and second that the monopolist obtains it with time delay. In natural consequence of these alternations, the monopolist cannot jump to the optimal point but searches for it with using the actual data obtained through the market. The modified model becomes dynamic in nature. This is the issue far outside the scope of the text book monopoly and it is what we will consider in this study. In the recent literature, various learning processes of the boundedly rational monopolist have been extensively studied. Puu [1995, CSF] constructs a discrete-time monopoly model in which price function is cubic and cost function is linear. It is shown that the gradient learning or search process based on locally obtained information might behave in an erratic way under the condition that the price function has an inflection point. Assuming that the monopolist uses a rule of thumb to determine quantity to produce, Naimzada and Ricchiute [2008, AMC] reconsider Puu's model with a linear cost function and a cubic price function without the inflection point. Their model is then generalized by Askar [2013, EM] who replaces the cubic function with higher-order polynomials. Matsumoto and Szidarovszky [2013] further generalize Asker's model by introducing the more general type of the cost function. Since those models are described by one dimensional difference equation, chaotic dynamics can arise via period-doubling bifurcation. In this study we reconsider a dynamic monopoly model from two different points of view. First, to detect the effect caused by non-instantaneous information, the dynamic process is constructed in continuous-time scales and a fixed time delay is introduced. Second, we discretize the continuous process to obtain a 'delay' discrete process and analyse the delay effect on discrete dynamics. In both models, local stability of a stationary state is analytically considered and global dynamics is numerically examined.