RDP 2009-01: Currency Misalignments and Optimal Monetary Policy: A Re-examination 9. The Interest Rate Reaction Functions

An interest rate rule that will support the optimal policies given by Equations (79) and (80) can be derived.

Substituting Equation (80) into Equations (57)–(58) and (59)–(60), and using the definitions of CPI inflation given in Equations (63) and (64) implies:

Assuming that u_t and are each AR(1) processes, independently distributed, with a serial correlation coefficient given by ρ. Under these assumptions, Equation (84) can be solved as^[20]:

Similarly, under the optimal monetary policy, the solution for ‘world’ inflation is:

Under the assumption that the mark-up shocks follow AR(1) processes, it follows that:

Substituting these equations into the Euler equations for the home and foreign country, given by Equations (61) and (62), making use of the consumption Equations (29) and (30), it can be shown that:

Here, represents the real interest rate in the efficient economy. Equations (88) and (89) can be used to write:

Surprisingly, these interest rate reaction functions are identical to the ones derived in CGG for the PCP model, except that the inflation term that appears on the right-hand side of each equation is CPI inflation here, while it is PPI inflation in CGG.

This finding starkly highlights the difference between monetary rules expressed as ‘target criteria’ (or targeting rules) and monetary rules expressed as interest rate reaction functions (or instrument rules). The optimal interest rate reaction functions presented in Equations (90) and (91) appear to give no role for using monetary policy to respond to deviations from the law of one price. However, the ‘target criteria’ show the optimal trade-off does give weight to the law of one price deviation. The key to understanding this apparent conflict is that the reaction functions, such as (90) and (91) are not only setting inflation rates. By setting home relative to foreign interest rates, they are also prescribing a relationship between home and foreign output gaps, the law of one price deviation, and home relative to foreign inflation rates.

If central bankers really did mechanically follow an interest rate rule, their optimal policy rules would have the nominal interest rate responding only to CPI inflation. In practice, however, central bankers set the interest rates to achieve their targets. We have shown that the optimal target criteria involves trade-offs among the goals of achieving zero inflation, driving the output gaps to zero, and eliminating the law of one price gap.

The optimal policy indeed is not successful in eliminating the currency misalignment. Nor does policy drive inflation to zero or eliminate the output gap. Monetary policy-makers do not have sufficient control over the economy to achieve the efficient outcome. Appendix B.4 displays the solutions for inflation, the output gap, and the currency misalignment under optimal policy.

Footnote

See Appendix B for the complete solutions of the model under optimal policy. [20]