The quantitative implications for welfare of the above results are illustrated in Figure 1. This figure plots the difference between the level of world welfare in the two Nash equilibria (with positive values indicating higher welfare with money supply rules).19 The figure plots the welfare difference against values of η and for three values of ε. Other parameter values are β = 0.95 and σ2²K = σ2²K∗ = 0.1 and ξK = ξK∗ = 0. Figure 1 shows that the welfare difference between the two types of policy rules is never very large even for quite extreme values of ε. For instance the welfare gap at its widest is only -0.01 percent of steady state consumption. However, it is important to note that (as has been emphasised in much of the recent literature) the welfare difference between coordinated and non-coordinated policy is very small in a model of the form used here. In itself this is likely to suggest that the welfare difference between different forms of non-coordinated policy are likely to be even smaller. Some recent contributions have analysed cases where the welfare gains from coordination can be relatively large. For instance, Benigno and
19The welfare difference between equilibria is expressed in terms of the change in consumption (as a percentage of consumption in a non-stochastic steady state) that would produce the equivalent change in aggregate utility.
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Benigno (2001) and Sutherland (2002b) analyse the implications of allowing the elasticity of substitution between home and foreign goods to differ from unity. In this case the welfare difference between policy instruments is also likely to be much larger. In Section 5 we briefly discuss the implications of allowing the elasticity of substitution between home and foreign goods to differ from unity.20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
−2.5
−2
−1.5
−1
−0.5 0 0.5x 10−3
η
Welfare
ε = 10 ε = 2 ε = 1 ε = 0.1
Figure 1: Welfare under the money supply rule minus welfare under the interest rate rule.
4 Endogenous Choice of Policy Instruments
In the previous section the choice of policy instrument is assumed to be exogenously given.
This section analyses the endogenous choice of policy instrument.
The setting of monetary policy now has two elements. The first is the choice of policy instrument - either the money supply or the nominal interest rate. The second is the choice of parameters for feedback rules. It is convenient to think of these decisions being made as part of a two-stage game where, in the first stage, the policymaker in each country
20The welfare gains from coordination (and thus the welfare difference between policy instruments) may also be larger when there is greater cross-country asymmetry in economic structures. Furthermore, as emphasised by Canzoneri, Cumby and Diba (2001), asymmetric shocks between traded and non-traded sectors may increase the welfare gains from coordination.
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chooses a policy instrument and, in the second stage, feedback parameters are chosen. In the first-stage game each policymaker faces a binary choice over instruments (either the money supply or the nominal interest rate). There are thus four possible outcomes from the first stage game. The four possible outcomes are summarised in Table 5.
Home country
M-rule R-rule
Foreign M-rule Ω˜t= ˜Ω∗t = W1 Ω˜t= W3, ˜Ω∗t = W2 country R-rule Ω˜t= W2, ˜Ω∗t = W3 Ω˜t= ˜Ω∗t = W4
W1 = −(1−η)2[4−4(1−∆)η+[5−(1+∆)(3−∆)]η2]
4[2−(3−∆)η+(2−∆)η2]2
W2 = − 2(1−η)2[2−(1−∆)η]
2(2−2η+η2)
[8−4(4−∆)η+2(7−3∆)η2−(4−3∆)η3]2
W3= −(2−η)2(1−η)2[4−4(1−∆)η+[5−(1+∆)(3−∆)]η2]
[8−4(4−∆)η+2(7−3∆)η2−(4−3∆)η3]2
W4 = −(1−η)2(2−2η+η2)
2(2−3η+2η2)2
Table 5: Welfare payoffs for game over policy instrument choice
The payoffs for each outcome from the first-stage game are the equilibrium welfare levels yielded by the relevant equilibrium in the second stage game. Thus, for instance, when both countries choose to follow money-supply rules, the payoff to each policymaker is the welfare level yielded by a Nash equilibrium in the choice of feedback parameters for money supply rules. Alternatively, when both countries choose to follow interest rate rules, the payoff to each policymaker is the welfare level yielded by a Nash equilibrium in the choice of feedback parameters for interest rate rules. Each cell in Table 5 shows the payoffs to the home and foreign policymakers for each of the four possible outcomes from the first stage game (including the payoffs which result when one country chooses to follow a money supply rule and the other country chooses to follow an interest rate rule).
It is simple to check that the payoffs to all four outcomes are identical in any of the four special cases discussed in the previous section. Thus if η = 0 or η = 1 or β = 1 or ε = 1 the two policymakers are indifferent about their choice of policy instrument and therefore any one of the four outcomes to the first stage game can be regarded as a Nash equilibrium. The outcome of the general case, where 0 < η < 1, 0 < β < 1 and ε 6= 1 is summarised in the following proposition.
Proposition 1 When 0 < η < 1, 0 < β < 1 and ε < 1 there is a single Nash equilibrium to the first-stage game where both policymakers choose the money supply to be the policy instrument. When 0 < η < 1, 0 < β < 1 and ε > 1 there is a single Nash equilibrium to the first-stage game where both policymakers choose the nominal interest rate to be the policy instrument.
Proof. It is sufficient to show that each player has a dominant strategy in the two cases. The choice of the money supply as the policy instrument is the dominant strategy for
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both policymakers when W1 > W3and W2> W4. Using the definitions for W1, W2, W3and W4 it is simple to show that this is true when ε < 1. The choice of the interest rate as the policy instrument is the dominant strategy for both policymakers when W1 < W3 and W2< W4. It is simple to show that this is true when ε > 1.
Proposition 1 shows that, provided ε 6= 1, there is a unique Nash equilibrium to the first-stage game where both policymakers choose the same policy instrument.21 Notice that the case where the choice of money supply rules is a Nash equilibrium to the first stage game (i.e. ε < 1) corresponds to the case where money supply rules yield higher world welfare in the second-stage game and the case where the choice of interest rate rules is a Nash equilibrium to the first stage game (i.e. ε > 1) corresponds to the case where interest rate rules yield higher world welfare in the second-stage game.
5 Robustness
In the previous sections we have discussed the instrument-choice problem under a set of simplifying assumptions. In particular, we have assumed log preferences in consumption and linear preferences in labour. Furthermore, we have assumed a unit elasticity of substi-tution between home and foreign goods. In this section we briefly discuss the implications for our results of relaxing these assumptions.
CRRA preferences
At this point, it should be clear that the instrument-choice problem arises only when there is strategic interdependence in monetary policy making and the policy makers do not coordinate their decisions. Imposing a degree of relative risk aversion in consumption different from unity (i.e. adopting a general CRRA functional form) does not eliminate strategic interdependence, nor does the assumption of non-linear preferences in labour supply.22 The special cases discussed in Section 3.3 also continue to apply with this more general structure of preferences. The results derived in Section 3 are therefore robust to generalisations of this type.
Nevertheless, one might wonder whether the first-stage game (the choice of instru-ments) still yields the same unique Nash equilibrium. Direct calculation of the payoff functions under general CRRA preferences shows that the ordering of the payoffs changes only in a very special way (Cf Table 5). For certain combinations of values of the con-sumption and labour risk-aversion coefficients only the pairs (W1, W2) and (W3, W4) cross more than once (at ε = 1 and at ε > 1). This implies that Proposition 1 still holds under general CRRA preferences in consumption and labour.
21The symmetry of the Nash equilibrium in the choice of policy instrument is likely to be a consequence of the symmetric structure of the model. In an asymmetric world Nash equilibria may exist where the two countries choose different policy instruments.
22With a degree of relative risk aversion different from one the consumption risk-sharing condition does not hold automatically. In this case we follow Devereux and Engel (2000) by assuming the existence of a full set of contingent assets that household can purchase in both countries so to insure themselves against consumption risk.
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Full exchange-rate pass-through
Turnovsky and d’Orey (1989) discuss the instrument-choice problem in a setting where the “law of one price” (and purchasing power parity - PPP) holds. In contrast, we have seen in our model that under PPP (i.e. the case where there is full exchange rate pass-through, η = 1), there is no strategic interdependence in monetary policy and, hence, the instrument-choice problem ceases to exists. Strategic interdependence does arise, however, in the PPP case in a more general version of our model where the elasticity of substitution between domestic and imported goods is greater than unity.23 For reasons of space we do not develop this model here. We simply report the implications of such a model.24 The more general model shows that, even in the full pass-through case, the equilibrium allocations differ between money supply rules and interest rate rules when the elasticity of substitution between home and foreign goods is greater than unity (and also ε and β are not equal to unity). Furthermore, the payoff functions for the first-stage game cross (more than once) only in pairs (i.e. (W1, W2) and (W3, W4)) leaving the Nash equilibrium in the choice of policy instrument unchanged. Therefore, we can claim that Proposition 1 can be generalized to the full-pass-through case when the elasticity of substitution between home and foreign goods is greater than unity.
An important feature of this more general model is that (as shown in Sutherland (2002b)) the welfare gains from policy coordination are potentially much higher when the elasticity of substitution between home and foreign goods is greater than unity. The welfare differences between the equilibria for the money supply rules and interest rate rules are thus also potentially much larger in this case.
6 Conclusion
This paper has compared state-contingent money supply rules with state-contingent inter-est rate rules in a two-country sticky-price general equilibrium model. It has been shown that, in general, the equilibrium outcome of non-coordinated policymaking can differ de-pending on whether monetary policy is specified in terms of money supply rules or interest rate rules. A number of special cases are identified where the two types of rule yield the same equilibrium. The endogenous choice of policy instrument was analysed as part of a two-stage game and it was shown that a unique Nash equilibrium exists in the choice of monetary instrument. This paper has concentrated on deriving explicit results for a fairly restrictive model (which is nevertheless representative of the recent literature). The results are, however, robust to a number of important generalisations of the model.
23Turnovsky and d’Orey (1989) use a Lucas-supply function to describe output. They assume that the elasticity of output with respect to the relative price can differ from unity.
24Sutherland (2002b) uses a model of this form to analyse the welfare gains from policy coordination.
Sutherland does not, however, analyse the instrument-choice problem.
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References
[1] Bacchetta, Philippe and Eric van Wincoop (2000) “Does Exchange Rate Stability Increase Trade and Welfare?” American Economic Review, 90, 1093-1109.
[2] Benigno, Gianluca and Pierpaolo Benigno (2001) “Price Stability as a Nash Equilib-rium in Monetary Open Economy Models” CEPR Discussion Paper No 2757.
[3] Canzoneri, Matthew B and Dale W Henderson (1991) Monetary Policy in Interde-pendent Economies: A Game Theoretic Approach MIT Press, Cambridge MA.
[4] Canzoneri, Matthew B, Robert E Cumby and Behzad T Diba (2001) “The Need for International Policy Coordination: What’s Old, What’s New, What’s Yet to Come?”
unpublished manuscript, Georgetown University.
[5] Clarida, Richard H, Jordi Gali and Mark Gertler (2001) “A Simple Framework for International Monetary Policy Analysis” Journal of Monetary Economics, 49, 879-904.
[6] Corsetti, Giancarlo and Paolo Pesenti (2001a) “Welfare and Macroeconomic Interde-pendence” Quarterly Journal of Economics, 116, 421-446.
[7] Corsetti, Giancarlo and Paolo Pesenti (2001b) “International Dimensions of Optimal Monetary Policy” NBER Working Paper No 8230.
[8] Devereux, Michael B and Charles Engel (2000) “Monetary Policy in an Open Econ-omy Revisited: Price Setting and Exchange Rate Flexibility” NBER Working Paper No 7665.
[9] Hamada, Koichi (1976) “A Strategic Analysis of Monetary Independence” Journal of Political Economy, 84, 677-700.
[10] Lane, Philip (2001) “The New Open Economy Macroeconomics: A Survey” Journal of International Economics, 54-235-266.
[11] Obstfeld, Maurice and Kenneth Rogoff (1995) “Exchange Rate Dynamics Redux”
Journal of Political Economy, 103, 624-660.
[12] Obstfeld, Maurice and Kenneth Rogoff (1998) “Risk and Exchange Rates” NBER Working paper 6694.
[13] Obstfeld, Maurice and Kenneth Rogoff (2002) “Global Implications of Self-Oriented National Monetary Rules” Quarterly Journal of Economics, 117, 503-536.
[14] Oudiz, Gilles and Jeffrey Sachs (1984) “Macroeconomic Policy Coordination among the Industrial Countries” Brookings Papers on Economic Activity, 1, 1-64.
[15] Poole, William (1970) “Optimal Choice of Monetary Policy Instruments in a Simple Stochastic Macro Model” Quarterly Journal of Economics, 84, 197-216.
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[16] Sutherland, Alan (2002a) “Incomplete Pass-Through and the Welfare Effects of Ex-change Rate Variability” CEPR Discussion Paper No 3431.
[17] Sutherland, Alan (2002b) “International Monetary Policy Coordination and Financial Market Integration” European Central Bank Working Paper No 174.
[18] Turnovsky, Stephen J and Vasco d’Orey (1989) “The Choice of Monetary Policy In-strument in Two Interdependent Economies under Uncertainty” Journal of Monetary Economics, 23, 121-133.
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Appendix
The full set of equations needed for the solution of the model is repeated here.
Pt= PH,t1/2PF,t1/2, Pt∗ = PH,t∗1/2PF,t∗1/2 (A1) The model is closed with the addition of the appropriate set of policy rules. All the equations of the model are linear in logs with the exception of the money-market equations (A6) and the aggregate output equations (A9). These equations can be replaced by second order approximations and the model can be solved on the assumption that all the variables of the model are approximately log-normal.
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21
The following Discussion Papers have been published since 2003:
Series 1: Studies of the Economic Research Centre
January 2003 Testing mean-variance efficiency in CAPM Marie-Claude Beaul with possibly non-gaussian errors: an Jean-Marie Dufour exact simulation-based approach Lynda Khalaf
January 2003 Finite-sample distributions of
self-normalized sums Jeong-Ryeol Kim
January 2003 The stock return-inflation puzzle and the asymmetric causality in stock returns,
inflation and real activity Jeong-Ryeol Kim
February 2003 Multiple equilibrium overnight rates
in a dynamic interbank market game Jens Tapking
February 2003 A comparison of dynamic panel data estimators: Monte Carlo evidence and
an application to the investment function Andreas Behr
March 2003 A Vectorautoregressive Investment
Model (VIM) And Monetary Policy Joerg Breitung Transmission: Panel Evidence From Robert S. Chirinko
German Firms Ulf von Kalckreuth
March 2003 The international integration of money markets in the central and east European accession countries: deviations from covered
interest parity, capital controls and inefficien- Sabine Herrmann Cies in the financial sector Axel Jochem
March 2003 The international integration of
foreign exchange markets in the central and east European accession countries:
speculative efficiency, transaction costs Sabine Herrmann
and exchange rate premiums Axel Jochem
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March 2003 Determinants of German FDI: Claudia Buch
New Evidence from Jörn Kleinert
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April 2003 Determinants of German Foreign
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Asian Emerging Markets in the 1990s Torsten Wezel
June 2003 Active monetary policy, passive fiscal policy and the value of public debt:
some further monetarist arithmetic Leopold von Thadden
June 2003 Bidder Behavior in Repo Auctions Tobias Linzert without Minimum Bid Rate:Dieter Nautz
Evidence from the Bundesbank Jörg Breitung
June 2003 Did the Bundesbank React to Martin T. Bohl Stock Price Movements? Pierre L. Siklos
Thomas Werner
15 2003 Money in a New-Keynesian model Jana Kremer
estimated with German data Giovanni Lombardo Thomas Werner
16 2003 Exact tests and confidence sets for the Jean-Marie Dufour tail coefficient of α-stable distributions Jeong-Ryeol Kurz-Kim
17 2003 The Forecasting Performance of B R Craig, E Glatzer, German Stock Option Densities J Keller, M Scheicher 18 2003 How wacky is the DAX? The changing Jelena Stapf
structure of German stock market volatility Thomas Werner
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1 2004 Foreign Bank Entry into Emerging Economies:
An Empirical Assessment of the Determinants
and Risks Predicated on German FDI Data Torsten Wezel
2 2004 Does Co-Financing by Multilateral Development Banks Increase “Risky” Direct Investment in Emerging Markets? –
Evidence for German Banking FDI Torsten Wezel
3 2004 Policy Instrument Choice and Non-Coordinated Giovanni Lombardo Monetary Policy in Interdependent Economies Alan Sutherland
4 2004 Inflation Targeting Rules and Welfare
in an Asymmetric Currency Area Giovanni Lombardo
Series 2: Banking and Financial Supervision
1 2003 Measuring the Discriminative Power B. Engelmann,
of Rating Systems E. Hayden, D. Tasche
2 2003 Credit Risk Factor Modeling and A. Hamerle,
the Basel II IRB Approach T. Liebig, D. Rösch
Visiting researcher at the Deutsche Bundesbank
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