RDP 2009-01: Currency Misalignments and Optimal Monetary Policy: A Re-examination 6. Log-linearised Phillips Curves

Under PCP, a New Keynesian Phillips curve for an open economy can be derived:

or:

where .

Similarly for foreign producer price inflation:

In the LCP model, the law of one price deviation is not zero, so that:

There are also price adjustment equations for the local prices of imported goods:

Equations (57)–(58) and (59)–(60) imply that . Assuming a symmetric initial condition, so that , leads to the conclusion that as noted above. That is, the relative price of foreign to home goods is the same in both countries. It is worth emphasising that this is true in general for a first-order approximation.

The efficient allocations cannot be obtained with monetary policy alone because of the sticky-price externality, and because the policy-maker is assumed to not have access to fiscal instruments aside from setting a constant subsidy rate to firms.

The policy-maker has home and foreign nominal interest rates as instruments. As is standard in the literature, the policy-maker can be modelled as directly choosing output gaps, inflation levels, and (in the LCP case) deviations from the law of one price, subject to constraints. From the first-order conditions, the optimal choice of nominal interest rates can be backed out using a log-linearised version of Equation (12) and its foreign counterpart, given by:

In these equations, π_t and refer to home and foreign consumer price inflation, respectively: