RDP 2007-12: Dynamic Pricing and Imperfect Common Knowledge Appendix A: The Phillips Curve with Imperfect Common Knowledge

Appendix A: The Phillips Curve with Imperfect Common Knowledge

The price level follows

The optimal price of firm j is given by

or, equivalently,

The average reset price is then given by

Repeated substitution of Equations (A1) and (A4) into Equation (A3) and averaging across firms and substituting into (A1) yields

Use and to get

which can be rewritten as

Common knowledge of rationality implies that any order l of average expectations of current inflation can be written as

which allows us to substitute out all orders of current inflation expectations in Equation (A8) to get Equation (6) in the main text: