Abstract:

With the improvement of production capacity of enterprises and the change of market structure， more and more high-tech enterprises use limited sales to provide consumers with new products. Limited sales mode is the main way for enterprises to use the scarcity effect for marketing. The limited sales approach is affected by a variety of factors， including price， fans， brand awareness， final volume， and marketing costs. However， there is a lack of research on this aspect. Therefore， a profit maximization model considering the network word-of-mouth effect is constructed， variational method is used to solve the limited optimal path， the influence of factors such as the size of fans， the ending number and price on the quantity path and the level of marketing effort is analyzed， and the design methods of the quantity path in two cases are given. And how to make decisions on the level of marketing effort under the environment of network word-of-mouth.

The model is hypothesized and described， then a differential equation is established to describe the dynamics of limited sales in the network environment， and a profit maximization model under limited sales is also established， which is solved and analyzed in two stages under the basic model of construction： Firstly， the optimal quantity path is obtained by the variational method， and then the parameter variables that affect the profit are analyzed and decided under this path. A profit maximization model considering the network word-of-mouth effect is further constructed， the influence of network word-of-mouth effect on profits， and how enterprises make decisions on marketing effort level are analyzed. Finally， a case is used to analyze and verify the practicability of the model. The conclusions point out that： 1） Under the condition that the quantity at time T is determined： first， the enterprise can set the final quantity N in advance， design the optimal quantity path according to the size of fans， and finally make a decision on the price to obtain the maximum profit； Second， by setting the price in advance， and then setting the final quantity N according to the price and the size of fans， the optimal quantity path is designed to obtain the maximum profit. By deducing the calculation， a final value N is determined which makes the enterprise profit maximum. 2） In the case of quantity uncertainty at time T： enterprises can determine the optimal quantity path through the combined effect of fan size and price. And it is given that under the situation of limited production capacity or limited edition， the optimal quantity supply decision of the enterprise is to implement the equivalent quantity in the cycle T. In both cases， the size of fans plays a key role in the design of the optimal quantity path， and efforts to expand the size of fans play an important role in the design of the acquisition of greater profits. 3） Enterprises should pay attention to the interactive impact of positive and negative word-of-mouth effect on profits. Negative word of mouth influence plays a key role in the decision-making of marketing effort level and the design of quantity path of enterprises， especially in enterprises with high brand awareness. 4） When the size of fans is fixed， in order to obtain the maximum profit， the higher the pricing， the larger the total number of sales. The results of paper will provide theoretical support and decision-making suggestions for enterprises to implement limited sales practice in scarcity marketing.

Key words: scarcity marketing, limited sales, the best quantity path, word-of-mouth effect, marketing effort