The Nash bargaining solution in vertical relations with linear input prices

We re-examine the Nash bargaining solution when an upstream and a downstream firm bargain over a linear input price. We show that the profit sharing rule is given by a simple and instructive formula which depends on the parties' disagreement payoffs, the profit weights in the Nash-product and the elasticity of derived demand. A downstream firm's profit share increases in the equilibrium derived demand elasticity which in turn depends on the final goods' demand elasticity. Our simple formula generalizes to bargaining with N downstream firms when bilateral contracts are unobservable.