A time-consistent open-loop Stackelberg equilibrium of shelf-space allocation

We propose a differential game to study retailer's allocation strategy of shelf-space shares between the manufacturers of two competing brands. Each manufacturer can influence the allocation decision by her advertising spending to improve her brand's goodwill which in turn affects the demand for her product. The game is played à la Stackelberg with the manufacturers as leaders and the retailer as follower. Stackelberg open-loop equilibrium is characterized and shown to be time-consistent.