Vertical Mergers and Bargaining Models : Simultaneous Versus Sequential Pricing

We consider an upstream firm U that supplies a key input to two symmetric downstream firms, A and B, that sell differentiated products. U negotiates bilaterally with A and B over a linear input price, and A and B set output prices. We assume Nash-in-Nash bargaining for input prices, and Bertrand competition for output prices. We compare two models. The simultaneous pricing model assumes that each price is determined holding all other prices constant (e.g., Crawford et al., 2018). The sequential pricing model assumes that input prices are determined first, and then output prices are determined given the input prices (e.g., Rey and Vergé, 2019). We compare the equilibria of the two models as well as their predictions for the effects of a vertical merger of U and A. For simplicity, we assume linear demand and no production costs. We show that input prices are lower, and hence the double marginalization problem is smaller, in the sequential pricing model than in the simultaneous pricing model. In both models, a merger of U and A leads to an input price increase to B (raising rival's cost) but the price increase is small in the simultaneous pricing model, while it can be very substantial in the sequential pricing model. We also show that, in the simultaneous pricing model, the merger leads to an output price reduction for both A and B, while in the sequential pricing model it leads to an increase in the output price of B (and also in the output price of A if the two products are relatively close substitutes). We then show that the bargaining model with simultaneous pricing actually produces merger effects that are similar to the effects obtained using a sequential pricing model without bargaining where U sets input prices by making take-it-or-leave-it offers to A and B