RDP 2009-01: Currency Misalignments and Optimal Monetary Policy: A Re-examination Appendix A: Log-linearised Model

The log-linear approximations to the models presented above are presented in this Appendix.

Use is made of the first-order approximation , as explained in the text. That relationship is obvious in the flexible-price and PCP models, but will require some explanation in the LCP case. That explanation is postponed until later.

In this Appendix, all of the equations of the log-linearised model are presented, but those that are used in the derivation of the loss function (which do not involve price setting or wage setting) are separated from those that are not.

Equations used for derivation of loss functions

The log of the deviation from the law of one price is defined as:

In the flexible-price and PCP models, Δ_t = 0. In the LCP model, because , the law of one price deviation is the same for both goods: .

In all three models, to a first order, . That allows Equation (23) and its foreign counterpart to be approximated as:

The market-clearing conditions, (18) and (19) are approximated as:

The condition arising from complete markets that equates the marginal utility of nominal wealth for home and foreign households, Equation (22), is given by:

For use later, it is helpful to use Equations (A4)–(A6) to express , in terms of y_t and and the exogenous disturbances, , and :

where D ≡ σv(2 − v) + (v − 1)².

Under a globally efficient allocation, the marginal rate of substitution between leisure and aggregate consumption should equal the marginal product of labour times the price of output relative to consumption prices. To see the derivation more cleanly, insert the shadow real wages in the efficient allocation, and into Equations (A10) and (A11) below. So, the efficient allocation would be achieved in a model with flexible wages and optimal subsidies. These equations then can be understood intuitively by looking at the wage-setting equations below (Equations (A14)–(A15), and (A16)–(A17)) assuming the optimal subsidy is in place. But they do not depend on a particular model of wage setting, and are just the standard efficiency condition equating the marginal rate of substitution between leisure and aggregate consumption to the marginal rate of transformation.

Equations of wage and price setting

The real home and foreign product wages, from Equation (38), are given by:

w_t − p_Ht and can be expressed in terms of y_t and and the exogenous disturbances, , and :

A.1 Flexible Prices

The values of all the real variables under flexible prices can be solved by using Equations (A2), (A3), (A7), (A8), (A9), (A14) and (A15), as well as the price-setting conditions, from (43):

A.2 PCP

Log-linearisation of Equations (49) and (50) leads to the familiar New Keynesian Phillips curve for an open economy:

where δ = (1 − θ)(1 − βθ)/θ.

This equation can be rewritten as:

or, using (A14) and (A10):

where .

Similarly for foreign producer price inflation:

A.3 LCP

Equation (A18) holds in the LCP model as well. But in the LCP model, the law of one price deviation is not zero. It follows that:

In addition, Equations (18) and (51) imply:

This can be rewritten as

Similarly:

From Equations (A7)–(A8) and (A14)–(A15), it can be seen that . Assuming a symmetric initial condition leads to the conclusion that as noted above. That is, the relative price of foreign to home goods is the same in both countries. I emphasise that this is true in general for a first-order approximation.