Explicit Solutions of Optimization Models and Differential Games with Nonsmooth (Asymmetric) Reference-Price Effects

Abstract

Models in marketing with asymmetric reference effects lead to nonsmooth optimization problems and differential games which cannot be solved using standard methods. In this study, we introduce a new method for calculating explicitly optimal strategies, open-loop equilibria, and closed-loop equilibria of such nonsmooth problems. Application of this method to the case of asymmetric reference-price effects with loss-aversive consumers leads to the following conclusions: (1) When the planning horizon is infinite, after an introductory stage the optimal price stabilizes at a steady-state price, which is slightly below the optimal price in the absence of reference-price effects. (2) The optimal strategy is the same as in the symmetric case, but with the loss parameter determined by the initial reference-price. (3) Competition does not change the qualitative behavior of the optimal strategy. (4) Adopting an appropriate constant-price strategy results in a minute decline in profits.

Optimal pricing policy with inventory related costs and reference effects

“…The exponential smoothing model is widely used for such purpose [12]. Therefore, the reference price formation is given by the following ordinary differential equation [14]:…”

“…The current price and the reference price at time are denoted by ( ) and ( ), respectively. The model in Equation (9) was employed in [14], [17], [18], [32], [33].…”

“…In fact, this three-fold pricing strategy will be shown to be optimal in all cases: After an initial penetration or a skimming strategy depending on the initial reference price, the price increases until the end of the season and there is a discount at the end to sell of the inventory. This policy is similar to the zero holding cost case in [14], except the price increase after the initial stage. [14] showed that after a penetration or a skimming strategy, the price resides at a constant price, which was shown to be the steady state price of the infinite horizon problem.…”

“…This policy is similar to the zero holding cost case in [14], except the price increase after the initial stage. [14] showed that after a penetration or a skimming strategy, the price resides at a constant price, which was shown to be the steady state price of the infinite horizon problem. The steady state price value is shown to be independent of the initial reference price.…”

“…For the linear demand models, we provided explicit closed form solutions for the optimal pricing policy. [14] provided explicit closed form solutions as well, however our model is an extension of their model by introducing inventory as the state and incorporating the inventory holding cost. Finally, we resort to numerical studies for sensitivity analysis of the price.…”