We now investigate the optimal choice of monetary policy. Home and foreign governments choose monetary policy rules to maximize welfare for their domestic residents. Due to the one-period ahead pricing policies of firms, and the presence of complete markets, the equilibrium is stationary. Optimal monetary policies are obtained by maximizing expected utility for any period. Read about history of money at Speedy Payday Loans - http://speedy-payday-loans.com/speedy-payday-loans-the-history-of-dollar.html. From a welfare perspective, the utility of real balances is likely to be of minor significance. Hence, following OR (1995, 1998) and Corsetti and Pesenti (1998), we ignore this term. Expected utility may be represented as The choice of monetary policy rules may be represented as a game between home and foreign governments. The home government chooses the degree of foreign exchange rate intervention g and its response to home and foreign productivity shocks, taking the monetary rule of the foreign country government as given. The foreign government chooses its response to home and foreign productivity shocks, taking the home country monetary rule as given. The pair of monetary rules that satisfy these conditions form a Nash equilibrium of the monetary policy game. 3.a Producer Currency Pricing Under producer currency pricing, consumption and expected employment are identical in the home and the foreign country, so that both governments will face the same objective function. Using (2.5), (2.8) and (3.1), we may write common expected utility as Using the expressions for the mean and variance of consumption from the Appendix, the objective function under PCP can be written as Equation (3.3) indicates that the objective function of the monetary authority under PCP is represented in a very simple form; minimize a linear function of consumption variance, exchange rate variance, and the covariance of consumption and the exchange rate with home and foreign productivity shocks. Given the simple assumptions of our model, the objective function for monetary policy becomes a quadratic, as in much of the earlier literature on optimal monetary rules based on more ad-hoc models and objective functions. The appeal of the present model however is that it delivers the appropriate welfare-based set of variables to be included in the objective function, and their relative weighting. Note also that monetary policy should explicitly take account of its effect on average consumption (and output). A Taylor series approximation to (3.2) around a constant average consumption level would lead us to conclude that minimizing consumption alone was the best objective for monetary policy; exchange rate variance and the covariance’s with productivity shocks would be ignored. Given the dependence of consumption and the exchange rate on the monetary rules, we may implicitly write expected utility as: EU = EU (g, a1, a 2, bj, b2). A Nash equilibrium in monetary policies is defined as the set {gN, a^, aN, bjN, bN } which satisfies the problem P1, Both countries face the same objective function. So in fact it is clear that the solution to P1 is identical to the equilibrium of a cooperative monetary policy game, where the governments of both countries jointly choose the monetary rules to maximize joint welfare. Therefore, in the PCP environment, there are no gains to monetary policy coordination. We characterize the solution to P1 in two stages. First, we describe the solution for the monetary rules when there are only productivity shocks, holding the home country intervention rule g fixed. Then, we solve for the optimal intervention rule in the presence of monetary shocks. Productivity Shocks The values of al, a2, bj, b2 that satisfy P1 are given by For the foreign country government, the optimal monetary rule is to respond positively to foreign productivity shocks, but respond negatively (positively) to home country productivity shocks as e > 1 (e < 1). For the home country government, the optimal response to productivity shocks depends on the degree of foreign exchange market intervention. With zero intervention, the rule is the mirror image of the foreign rule: respond positively to home country shocks and either negatively or positively to foreign shocks, depending on e. But when there is positive exchange rate intervention, the response to productivity shocks for the home country may be in the opposite direction. Monetary authorities attempt to use the rules (3.4) and (3.5) to sustain the flexible price allocation. Essentially, the economy has three departures from full efficiency. Prices are sticky, so consumption and employment do not respond to money and productivity shocks as they would in a flexible price environment. There are monopolistic competitive markups. Finally, there is a departure from the Friedman rule of zero nominal interest rates. The second loss cannot be affected by the monetary rules (2.1) and (2.2), since these rules cannot influence the markups obtained in Table 1. The third loss is neglected from the analysis because we ignore the utility of real balances. Thus, the best that the monetary policy rules can do is to attempt to replicate the flexible price equilibrium. To attain the flexible price allocation, the authorities must ensure that consumption equals (2.4). This requires that in the aggregate, monetary policy should react in a pro-cyclical manner to productivity shocks. Take the case where there is only a home country productivity shock. Then in face of a u shock, if the world money supply responds by n +i, consumption variance is the same as in the flexible price equilibrium. But expected consumption would not necessarily be at its desired level. For instance, if both home and foreign monetary authorities reacted in the identical manner, so as to achieve an average monetary policy response given by the exchange rate would have zero variance. From (2.8) however, expected consumption would then be too low. In order to achieve the desired level of expected consumption, it is necessary that the exchange rate responds positively to u shocks. This is facilitated by the monetary rules (3.4) and (3.5), which ensure that the home country money supply response is positive and exceeds that of the foreign country. An alternative intuition on the role of exchange rate changes may be described in terms of the movements in employment. In order to achieve the optimal employment response in the home and foreign economy, it is necessary to alter world relative prices, through movements in the nominal exchange rate. In response to a home country productivity shock, the nominal exchange rate must depreciate, to ensure world demand is substituted towards the home country’s products. For this, it is necessary that the home country monetary reaction to a home productivity shock is positive and exceeds that of the foreign country. Optimal monetary policy therefore relies on the expenditure switching mechanism of the exchange rate under the PCP pricing regime. The exchange rate movement in fact replicates the required terms of trade movement that would take place under fully flexible prices. It is easy to establish that with the monetary rules given in (3.4) and (3.5), the equilibrium under PCP replicates the flexible price allocation under productivity shocks alone. This can be shown by substituting (3.4) and (3.5) into (2.7) and deriving the expression for expected utility (3.2) with the optimal monetary rules, when u = u* = 0 . An immediate implication of this is that exchange rate flexibility is a requirement for an optimal monetary policy rule, under PCP, in the presence of productivity shocks. The optimal monetary policy is one that must not only ensure that distribution of consumption is identical to that under flexible nominal prices, but also that the distribution of the exchange rate is identical to that of the terms of trade under flexible prices. Exchange rate adjustment is necessary to achieve the desired terms of trade adjustment. This is the essence of Friedman’s argument for floating exchange rates quoted in the introduction. A critical aspect of the optimal monetary rules given in (3.4) and (3.5) is that they require participation by both monetary authorities. It is interesting to contrast problem P1 with an alternative case, where only the home monetary authority chooses optimal feedback rules, assuming that the foreign monetary authority follows a purely passive monetary policy, where b=0, i =1,2. Problem P2 is thus defined as It can easily be established that a1 is consistent with the solution for P1 only when e = 1, and a2 is in general never consistent with the solution for P1. A policy maker optimizing alone cannot support the flexible price allocation. Take the case where e > 1. Under problem P1, the home monetary authority will expand money supply in response to a home country productivity shock, while the foreign authority will contract money supply. This achieves the right combination of increasing world demand and exchange rate depreciation. But with problem P2, the rule given by (3.4) would entail too high a response in consumption, since the foreign government does not respond to the disturbance. The optimal rule under P2 is to respond to domestic productivity shocks less than is indicated in rule (3.4). Consumption still rises by too much, while the exchange rate rises by too little, relative to P1. Also, in response to a foreign technology shock, the home government cannot adequately respond. The reason is that the desire to increase domestic consumption calls for a monetary expansion (a2 > 0 ), but the desire to generate a terms of trade improvement calls for a monetary contraction (a2 < 0). The optimal a2 may therefore be either positive or negative. Monetary Shocks Given optimal choices ai, the home country government chooses the degree of exchange rate intervention according to P1. From (2.6), we see that the optimal intervention rule is that which minimizes the following function of consumption variability and exchange rate variability: The rule depends solely on monetary variability. It is independent of the distribution of real shocks. The optimal intervention depends on the strength of home country monetary variability relative to foreign monetary variability. When foreign monetary variability tends to zero, g tends to negative infinity; i.e. the exchange rate is prevented from moving at all in response to monetary disturbances. This supports the traditional result that a fixed exchange rate regime is optimal when the only shocks arise from the domestic monetary sector. One might think then that a floating exchange rate ( g = 0 ) would be preferable when home country monetary variability becomes arbitrarily small. But in fact we see that as home country monetary variance gets arbitrarily small, the intervention coefficient approaches a limit of The optimal intervention in face of foreign monetary shocks will lie between a fixed exchange rate and a free float, when pf2 > 1. The reason that a free float is not in general desirable is that the authorities are not trying to minimize consumption variance only, but are also concerned with exchange rate variance. Floating exchange rates would tend to minimize the extent to which foreign monetary disturbances contribute to consumption variance, but would leave exchange rate variance too high. Excessive exchange rate variance has a welfare cost through the reduction in expected consumption. An optimal intervention rule will trade off the benefits of exchange rate adjustment in reducing consumption variance against the welfare costs of exchange rate variability itself. 3.b Local Currency Pricing Under LCP, home and foreign consumption are not equated, because the real exchange rate is no longer constant. This implies that the objective functions for home and foreign governments will differ. For the home government, the objective function depends on expected consumption, consumption variance, and expected employment. Substituting (2.13) into expected utility given in (3.1), gives These functions may again be decomposed into means and variances of consumption. Then substituting for these values from the Appendix, we can establish: The optimal monetary rule with LCP requires policymakers to target a combination of two separate functions of consumption variance and the covariance between consumption and productivity shocks. But because under LCP, foreign consumption is independent of the parameters of the home country monetary rule, the home country problem amounts to simply minimizing the expression: An essential difference from the PCP case is that exchange rate volatility is no longer an independent argument in the objective function. The determination of monetary rules in the LCP case can be described as the Nash equilibrium {gN, a’N, aN, bjN, bN } Again, we characterize the solution to P3 first in the presence of productivity shocks, and then we allow for monetary shocks and optimal intervention. Productivity Shocks The solution to P3, holding g constant, is given by Again, as in the case of PCP, these rules ensure that consumption is identical across countries, and has the same variance as in the flexible price equilibrium. But unlike the case of PCP, these rules do not entail exchange rate adjustment. Each government responds in an identical way to a home or foreign productivity shock. The optimal monetary rules are consistent with a fixed exchange rate. The reason that optimal monetary rules do not require exchange rate adjustment in the LCP case is that the exchange rate plays no role in determining expected consumption. Exchange rate changes do not pass-through to price levels and therefore do not influence wages. Thus, they do not influence either marginal costs or expected prices and expected consumption. An equivalent perspective on the monetary rules (3.10) and (3.11) is that in the LCP case, exchange rate adjustment does not play a role in altering patterns of expenditure between home and foreign goods. Therefore, it cannot influence the determination of employment in either country. Optimal monetary policy no longer tries to move the exchange rate so as to achieve an optimal terms of trade adjustment, since the terms of trade do not influence demand or employment. The absence of exchange rate movement in the optimal rules in the LCP environment is at variance with the Friedman/Mundell argument for exchange rate flexibility, and more generally with the conventional statement of the costs of a single currency area. In the conventional argument, countries that are subject to idiosyncratic real shocks will benefit from exchange rate adjustment in face of these shocks. This argument is borne out in the PCP case. But with LCP, even country-specific shocks do not require exchange rate movement as part of an optimal policy response. Thus, even in the presence of distinct country specific real disturbances, there is no sacrifice of macroeconomic adjustment by eliminating the exchange rate as a tool of macroeconomic management (when the predominant disturbances are country specific productivity shocks), if we take the institutional structure of goods pricing as given. Although optimal monetary policy under LCP does not require exchange rate adjustment, a natural guess would be that the absence of exchange rate pass-through comes at a cost. This is correct. The optimal monetary policy under LCP does not support the full flexible price optimum. While the monetary rules (3.10) and (3.11) keep the response of home and foreign consumption equal to that at the flexible price optimum, they cannot ensure that employment in each country is identical to the flexible price optimum. Equivalently, the monetary rules lead consumption variance to match the flexible price equilibrium, but the mean level of consumption falls short of the flexible price equilibrium. Thus, the presence of LCP imposes welfare costs on the world as a whole, by preventing terms of trade adjustment from having macroeconomic effects. Problem P3 describes an environment where both domestic and foreign governments choose an optimal response to productivity shocks. What happens when the home country alone chooses an optimal monetary policy, assuming that the foreign government follows a purely passive monetary growth rule? Defining a choice for the home country government analogous to P2 gives optimal monetary rules equal to: Examining (3.12) allows us to establish that the home country consumption variance is identical to problem P3. But the foreign country consumption variance is not the same as under P3. Thus, as regards home country consumption, the home country government, acting alone, can do as well as it can when the foreign government also chooses optimal monetary policy. Unlike the PCP case, there is no alteration in the policy rules when governments act alone - the optimal rules support the same consumption process for the home government. Of course, because expected utility does depend upon both home and foreign consumption, overall welfare must be lower when the foreign country government does not follow an optimal rule. Monetary Shocks Under LCP, the government’s choice of exchange rate intervention is designed simply to minimize the variance of consumption. The optimal exchange rate intervention rule is given by Again, the rule trades off the benefit of a fixed exchange rate, insulating the economy from domestic monetary shocks, against the benefits of exchange rate adjustment, cushioning the economy from foreign monetary disturbances. Unlike the rule under PCP, the intervention rule under LCP will vary between fully fixed exchange rates to fully floating exchange rates, as the importance of home and foreign monetary disturbances change in relative terms. In particular, if there are no home monetary shocks at all, then the monetary authority should follow a clean float. This is because, unlike the PCP case, there are no independent costs of exchange rate volatility under LCP. The optimal intervention rule does not have to take account of the way in which intervention affects exchange rate volatility. When there are only foreign monetary disturbances, a floating exchange rate achieves the maximum insulation of home country consumption from the effects of the shocks. When productivity shocks dominate, we have shown that welfare under LCP must be lower than that under PCP. But the same result does not apply in the case of money shocks. A simple example may be used to illustrate this. Say that the variance of productivity shocks is zero in both countries, and that money shocks have identical variance equal to sv in both countries. Finally, imagine that the home country follows a zero exchange rate intervention rule (this is easily relaxed - see below). Then in the PCP case, expected utility is a positive function of Under the same conditions, expected utility under LCP is a positive function of From these two expressions, we see that the welfare comparison between PCP and LCP contains two elements. On the one hand, consumption variance is lower under PCP, because part of a monetary shock is dissipated through exchange rate adjustment. On the other hand, the exchange rate volatility driven by money shocks imparts a welfare cost under PCP, but does not under LCP. Expected utility is higher under LCP (PCP) whenever f2p > 1 (<1). In general, either inequality may be satisfied. Therefore, the welfare comparison across the two pricing specifications is ambiguous; the adjustment benefits of the exchange rate in response to monetary shocks are tempered by the welfare costs of exchange rate volatility.
