Appendix for: Price Setting in Forward-Looking Customer Markets

It is straightforward to show that the steady state price with full commitment is p(z) = θ θ − 1 S, where variables without subscripts denote steady state values. Notice furthermore that equation (1) in the paper implies that C = (1 − γ)c(z) and equation (2) in the paper implies that (1 − γβ)P = p(z). A second order Taylor series approximation of the value of the firm around the steady state of the solution to the firm's problem with commitment is given by E 0 ∞ t=0 β t c(z)(p t (z) − p(z)) + 1 θ p(z)(c t (z) − c(z)) + (p t (z) − p(z))(c t (z) − c(z)) − θ − 1 θ p(z) S (S t − S)(c t (z) − c(z)) + c(z) β t (p t (z) − p(z))(M 0,t − β t) + 1 θ p(z) β t (c t (z) − c(z))(M 0,t − β t) + ex. terms + O(||ξ|| 3), (A.1) where " ex. terms " stands for terms that are exogenous to the firm's decision problem, ξ stands for a vector of the exogenous variables and O(||ξ|| 3) denotes higher order terms. The exposition of our results is simplified if we make a change of variables. Letˆc t (z) = log(c t (z)/c(z)) and define hatted versions of all other variables in the same way. Making use of the fact that c t (z) = c(z) 1 + ˆ c t (z) + 1 2 ˆ c t (z) + O(||ξ|| 3),