The Nash bargaining solution in vertical relations with linear input prices

•We re-examine the Nash bargaining solution in vertical relations.•​We assume that up- and downstream firms bargain over a linear input price.•We show that the profit sharing rule is given by a simple and instructive formula.•We highlight the role of the elasticity of derived demand.

We re-examine the Nash bargaining solution when an upstream and N downstream firms bargain over a linear input price with unobservable contracts. 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.