That equation in the Appendix is clearly wrong. The correct equation is equation (11) in www2.bc.edu/matteo-iacoviello/research_files/AER_TA_2005.pdf. You can see on page 7 of that document that there are various timing errors when translating equations (6) and (7) into (xx). Therefore the equation in the Appendix of the AER paper on page 760 is wrong, while the linearized FOC is correct again.