RDP 2009-01: Currency Misalignments and Optimal Monetary Policy: A Re-examination Appendix B: Welfare Functions and Other Derivations

B.1 Derivation of Welfare Function in Clarida-Galí-Gertler Model with Home Bias in Preferences

The object is to rewrite the welfare function, which is defined in terms of home and foreign consumption and labour effort, into terms of the squared output gap and squared inflation. The joint welfare function of home and foreign households is derived, since cooperative monetary policy is being examined.

Most of the derivation requires only first-order approximations of the equations of the model, but in a few places, second-order approximations are needed. If the approximation is first-order, the notation is used to indicate that there are second-order and higher terms left out, and if the approximation is second-order, is used. (a is notation for the log of the productivity shock.)

From Equation (1) in the text, the period utility of the planner is given by:

Take a second-order log approximation around the non-stochastic steady state. Allocations are assumed to be efficient in steady state, so . The fact that follows from the fact that in steady state C = N from market clearing and symmetry, and from the condition that the marginal rate of substitution between leisure and consumption equals one in an efficient non-stochastic steady state.

It follows that:

Since maximising an affine transformation of Equation (B2) is equivalent, it is convenient to simplify that equation to get:

Utility is maximised when consumption and employment take on their efficient values:

In general, this maximum may not be attainable because of distortions. Writing , where , it follows that:

or

The object is to write (B6) as a function of squared output gaps and squared inflation if possible. A second-order approximation of is needed. But for the rest of the terms, since they are squares and products, the first-order approximations that have already been derived will be sufficient.

Recalling that Δ_t = 0 in the PCP model, Equations (A7)–(A8) can be written as:

where .

It follows from (B7) and (B8) that:

Next, it is easy to show that:

These follow as in Equations (A2)–(A3) because and (and similarly in the foreign country).

Expressions for and are required. From Equations (A10)–(A11):

Using and , these can be written as

Turning attention back to the loss function in Equation (B6), focus first on the terms .

As noted above, these involve only squares and cross-products of , and . Equations (B9)–(B16) can be substituted into this expression. It is useful to provide a few lines of algebra since it is a bit messy:

Now return to the term in Equation (B6) and conduct a second-order approximation. Start with Equation (18), dropping the k⁻¹ term because it will not affect the approximation, and noting that in the PCP model, :

Then use Equation (22), but using the fact that and there are no deviations from the law of one price:

Substitute in to get:

Solve for C_t:

Take first and second derivatives, evaluated at the non-stochastic steady state:

Then this second-order approximation is obtained:

Symmetrically,

Since only is of interest, these can be added together to get:

Now take a first-order approximation for s_t to substitute out for . Equation (A9) implies:

Substituting into Equation (B28) leads to:

where .

Evaluating (B29) at flexible prices:

It follows from the fact that that

See Section B.3 below for the second-order approximations for and :

Substitute expressions (B31)–(B33) along with (B17) into the loss function (B6):

Some tedious algebra demonstrates that

So, finally it is possible to write:

This expression reduces to CGG's when there is no home bias (γ = 1). To see this from their expression at the top of p 903, multiply their utility by 2 (since they take average utility), and set their γ equal to ½ (so their country sizes are equal).

B.2 Derivation of Welfare Function under LCP with Home Bias in Preferences

The second-order approximation to welfare in terms of logs of consumption and employment of course does not change, so Equation (B6) still holds. As before, the derivation is broken down into two parts. First-order approximations to structural equations are used to derive an approximation to the quadratic term

Then second-order approximations to the structural equations are used to derive an expression for .

The quadratic term involves squares and cross-products of and . Expressions (B9)–(B10) still provide first-order approximations for and ; Equations (B13)–(B14) are first-order approximations for ñ_t and ; and Equations (B15)–(B16) are first-order approximations for and . But Equations (A7)–(A8) and (B11)–(B12) are required to derive:

With these equations, the derivation as in Equation (B17) can be followed. After tedious algebra, the same result is achieved, with the addition of the terms and . Note that the last term involves output levels, not output gaps. That is:

The derivation of is similar to the PCP model. However, one tedious aspect of the derivation is that the equality , that holds under PCP and flexible prices, cannot be used. The equilibrium conditions are expressed for home output, and its foreign equivalent, from Equations (18) and (19):

Directly taking second-order approximations of these equations around the efficient non-stochastic steady state implies:

Note that in a second-order approximation, cannot be imposed. However, can be imposed. Then adding Equations (B42) and (B43) together, leads to:

Next, Equations (A7), (A8) and (A9) can be used to get approximations for and . These equations are linear approximations for and s_t, but since the goal is to approximate the squares of these variables, that is sufficient. With some algebra, it can be shown that:

Then, substituting Equation (B45) into Equation (B44) and rearranging, it follows that:

Note that if Δ_t = 0, Equation (B46) leads to the second-order approximation for from the PCP model.

Then following the derivations as in the PCP model derivation of Equation (B31), it follows that:

As shown in Section B.3, the following second-order approximation can be made:

Equation (B47) and (B48), along with Equation (B39), can be substituted into the loss function (B6). Notice the cancellations that occur. The cross-product terms on in Equations (B39) and (B47) cancel. The other cross-product terms involving output gaps and efficient levels of output also cancel, just as in the PCP model, when Equation (B35) was used. Hence:

B.3 Derivations of Price Dispersion Terms in Loss Functions

In the PCP case, it is true that

where df. Taking logs:

It is the case that ,

where we define

Following Galí (2008), note that

By the definition of the price index . Hence, from (B53),

It is also the case that

It follows, using (B54):

Note the following relationship:

Using our notation for variances, ≡ var(p_Ht), and taking the log of (B56) leads to

Substituting this into Equation (B51), and recalling that , implies Equation (B32). The derivation of Equation (B33) for the foreign country proceeds identically.

For the LCP model, the following second-order approximation to the equation is used:

In the LCP model, it is possible to write:

where the definitions of V_Ht and are analogous to that of V_t in the PCP model. Taking a second-order log approximation to the Expression (B60):

The same steps as in the PCP model can be followed to conclude:

Substituting these expressions into Equation (B61) and cancelling higher-order terms, implies:

Then using Equation (B59), this implies that:

Keeping in mind that :

Following analogous steps for the foreign country,

Adding Equations (B66) and (B67) gives Equation (B48).

Finally, to derive the loss functions for policy-makers (Equation (65) for the PCP model and (69) for the LCP model), note that the loss function is the present expected discounted value of the period loss functions derived here (Equation (B36) for the PCP model and (B49) for the LCP model). That is, the policy-maker seeks to minimise .

Following Woodford (2003, Chapter 6), it can be seen that, in the PCP model, if prices are adjusted according to the Calvo price mechanism given by Equation (50) for P_Ht, then

Analogous relationships hold for in the PCP model, and for , and in the LCP model. This relationship can then be substituted into the present value loss function, , to derive the loss functions of the two models presented in the text.

B.4 Solutions for Endogenous Variables under Optimal Policy Rules

Assume that shocks follow the processes (where the W superscript on the wage mark-up shocks have been dropped):

s_t is determined by:

The solutions for the variables that appear in the loss function under the optimal policy described in the text are: