RDP 2009-01: Currency Misalignments and Optimal Monetary Policy: A Re-examination 4. Loss Functions and Optimal Policy

The loss function is derived for the cooperative monetary policy problem, which is the relevant criterion for evaluating world welfare. It is based on a second-order approximation to households' utility functions. Loss is measured relative to the efficient allocations.

The policy-maker wishes to minimise

This loss function is derived from household's utility, given in Equation (1). The discount factor, β, is the household's, and the per-period loss, X_t+j, represents the difference between the utility of the market-determined levels of consumption and leisure and the maximum utility achievable under efficient allocations.

The aim is to highlight the global inefficiency that arises from currency misalignments. For that reason, the difficult issues involved with deriving the loss function for a non-cooperative policy-maker and defining a non-cooperative policy game are set aside.

It is worth noting for future work that there are three technical problems that arise in the case of non-cooperative policy. One problem is discussed in detail in Devereux and Engel (2003). To consider policies in the non-cooperative framework requires an examination of the effects of one country changing its policies, holding the other countries' policies constant. In a complete markets world, to evaluate all possible alternative policies, it is necessary to calculate prices of state-contingent claims under alternative policies. In particular, Equation (22), which was derived assuming equal initial home and foreign wealth, cannot be assumed to hold under all alternative policies that the competitive policy-maker considers. Policies can change state-contingent prices and therefore change the wealth distribution. It is not uncommon for studies of optimal policy in open economies to treat Equation (22) as if it were independent of the policy choices, but it is not. There is a special case in which it holds in all states, which is the setup in CGG. When the law of one price holds, when home and foreign households have identical preferences and there are no preference shocks, and when preferences over home and foreign aggregates are Cobb-Douglas, Equation (22) holds in all states. This well-known outcome arises because in all states of the world, the terms of trade change in such a way as to leave home/foreign wealth unchanged.

Even if this technical challenge could be overcome,^[17] there are a couple of other technical challenges that appear in this framework that did not plague CGG. First, in the LCP model, it is not the case that , that is, the relative price of foreign to home goods is not the same in both countries. This relationship holds up to a first-order log-linear approximation in the LCP model (as long as we assume equal speeds of price adjustment for all goods,^[18] but only up to a first-order approximation. Complex first-order relative price terms (s_t and ) appear in the objective function of the non-cooperative policy-maker. However, those wash out in the objective function under cooperation. Second, CGG neatly dichotomise the choice variables in their model – the home policy-maker sets home PPI inflation and the home output gap taking the foreign policy choices as given, and vice versa for the foreign policy-maker. Such a neat dichotomy is not possible in the model with currency misalignments – we cannot just assign the exchange rate to one of the policy-makers.

In a sense, all of these technical problems are related to the real world reason why it is more reasonable to examine policy in the cooperative framework when currency misalignments are possible. The non-cooperative model assumes that central banks are willing and able to manipulate currencies to achieve better outcomes. However, in practice both WTO rules and implicit rules of neighbourliness prohibit this type of policy. Major central banks have typically been unwilling to announce explicit targets for exchange rates without full cooperation of their partners.

Even if the cooperative policy analysis is not a realistic description of actual policy decision-making, the welfare function is a measure of what could be achieved under cooperation.

Appendix B shows the steps for deriving the loss function when there are no currency misalignments, but with home bias in preferences. It is worth pointing out one aspect of the derivation. In closed economy models with no investment or government, consumption equals output. That is an exact relationship, and therefore the deviation of consumption from the efficient level equals the deviation of output from the efficient level to any order of approximation: . In the open economy, the relationship is not as simple. When preferences of home and foreign agents are identical, and markets are complete, then the consumption aggregates in home and foreign are always equal (up to a constant of proportionality equal to relative wealth). But that is not true when preferences are not the same. Equation (22) shows that does not hold under complete markets, even if the law of one price holds for both goods. Because of this, does not equal , except to a first-order approximation. Since a second-order approximation of the utility function is being used, the effect of different preferences (or the effects of the terms of trade) needs to be taken into account when translating consumption gaps into output gaps.

The period loss of the policy-maker under no currency misalignments is −X_t, where:

This depends on the squared output gap in each country, as well as the squared difference in the output gaps. The terms and represent the cross-sectional variance of prices of home goods and foreign goods, respectively. (Recall D ≡ σv(2 – v) + (v – 1)².)

Appendix B also shows the derivation of the loss function in the more general case in which currency misalignments are possible. Two aspects of the derivation merit attention. First, in examining the first-order dynamics of the model, the first-order approximation can be used. That is an exact equation when the law of one price holds, but it is not necessary that this relationship hold to a second-order approximation. The derivation of the loss function must take this into account. The second point to note is that, as is standard in this class of models, price dispersion leads to inefficient use of labour. But, to a second-order approximation, this loss depends only on the cross-section variances of , and , and not their co-movements (which would play a role in a third-order approximation.)

In the case of currency misalignments, X_t is given by:

It is important to recognise that this loss function and the loss function derived previously (Equation (33)) do not depend on how prices are set – indeed whether prices are sticky or not. The loss function of Equation (34) generalises Equation (33) to the case in which there are deviations from the law of one price, so that Δ_t ≠ 0. This can be seen by directly comparing the two equations. In Equation (34), is the cross-sectional variance of home goods prices in the home country, is the cross-sectional variance of home goods prices in the foreign country, etc. If there is no currency misalignment, then Δ_t = 0, so and for each firm f. In that case, and because the exchange rate does not affect the cross-sectional variance of prices. If we have Δ_t = 0, , and , then (34) reduces to (33).

Why does the currency misalignment appear in the loss function? That is, if both home and foreign output gaps are zero, and all inflation rates are zero, what problem does a misaligned currency cause?^[19] From Equation (31), if the currency is misaligned, then internal relative prices (s_t) must also differ from their efficient level if the output gap is zero. The home and foreign countries could achieve full employment, but the distribution of the output between home and foreign households is inefficient. For example, suppose Δ_t > 0, which from Equation (31) implies if both output gaps are eliminated. On the one hand, Δ_t > 0 tends to lead to overall consumption at home to be high relative to foreign consumption (Equations (29) and (30)). That occurs because financial markets make payments to home residents when their currency is weak. But home residents have a bias for home goods. That would lead to overproduction in the home country, were it not for relative price adjustments – which is why .

It is worth highlighting the fact that the loss functions are derived without specific assumptions about price setting not to give a false patina of generality to the result, but to emphasise that the loss in welfare arises not specifically from price stickiness but from prices that do not deliver the efficient allocations. Of course it is the specific assumptions of nominal price and wage setting that give rise to the internal and external price misalignments in this model, and indeed monetary policy would be ineffective if there were no nominal price or wage stickiness. But one could imagine a number of mechanisms that give rise to deviations from the law of one price, because the literature has produced a number of models based both on nominal stickiness and real factors. In the next section, the CGG model is modified in the simplest way – allowing LCP instead of PCP – to examine further the implications of currency misalignments.

Footnotes

The Appendix of Devereux and Engel (2003) demonstrates how this problem can be handled (available at <http://www.restud.com/uploads/suppmat/app0017.pdf>). [17]

See Benigno (2004) and Woodford (forthcoming) on this point. [18]

Optimal inflation targets are zero in this model because we have assumed zero inflation in steady state. [19]