Strategic Interactions Among Central Bank and National Fiscal Authorities in a Monetary Union Subject to Asymmetric Country Shocks

Abstract

In this paper we extend Nordhaus’ (Brookings Pap Econ Act (2):139–199, 1994) results to an environment which may represent the current European situation, characterised by a single monetary authority and several fiscal bodies. We show that, even assuming that the monetary and the fiscal authorities share the same ideal targets, in the presence of asymmetric shocks the “symbiosis” result found by Dixit and Lambertini (J Int Econ 60:235–247, 2003) no longer obtains. Thus, fiscal rules as those envisaged in the Maastricht Treaty and in the Stability and Growth Pact may work as monetary/fiscal coordination devices that improve welfare. The imposition of common targets, however, may work as a substitute for policy coordination only if these are made state contingent, an aspect that the recent version of the Stability and Growth Pact takes into account in a more appropriate way than its original version.

Appendix: derivation of the reaction functions

The reaction function of the monetary authority is obtained by substituting into Eq. 9 from Eqs. 1–5, by differentiating with respect to r and by imposing the optimality condition $\frac{\partial L_{M}}{\partial r}=0$, which yields:

$$S_{1}=-\frac{\varepsilon \mu _{2}+\delta \nu _{2}}{\delta \mu _{1}+\varepsilon \nu _{1}}S_{2}-\frac{\mu _{r}}{\delta \mu _{1}+\varepsilon \nu _{1}}r-\frac{\delta \zeta }{\delta \mu _{1}+\varepsilon \nu _{1}} -\,\frac{ \epsilon z}{\delta \mu _{1}+\varepsilon \nu _{1}}+\frac{u^{\ast \ast }+\alpha \beta ^{\ast \ast }(k-p^{\ast \ast })}{\left( \delta \mu _{1}+\varepsilon \nu _{1}\right) \left( 1+\alpha ^{2}\beta ^{\ast \ast }\right) }.$$

(40)

As for the fiscal authority of country 1, by substituting into Eq. 7 from Eqs. 1–5 and by setting $\frac{ \partial L_{F_{1}}}{\partial S_{1}}=0$, we obtain:

$$S_{1} =-\,\mu _{r}\frac{\mu _{1}+\alpha ^{2}\beta _{1}^{\ast }\left( \delta \mu _{1}+\varepsilon \nu _{1}\right) (\delta +\epsilon )}{\mu _{1}^{2}+\alpha ^{2}\beta _{1}^{\ast }\left( \delta \mu _{1}+\varepsilon \nu _{1}\right) ^{2}+\gamma _{1}^{\ast }}r -\,\frac{\mu _{1}\nu _{2}+\alpha ^{2}\beta _{1}^{\ast }\left( \delta \mu _{1}+\varepsilon \nu _{1}\right) \left( \varepsilon \mu _{2}+\delta \nu _{2}\right) }{\mu _{1}^{2}+\alpha ^{2}\beta _{1}^{\ast }\left( \delta \mu _{1}+\varepsilon \nu _{1}\right) ^{2}+\gamma _{1}^{\ast }}S_{2} -\,\frac{\mu _{1}+\alpha ^{2}\beta _{1}^{\ast }\delta \left( \delta \mu _{1}+\varepsilon \nu _{1}\right) }{\mu _{1}^{2}+\alpha ^{2}\beta _{1}^{\ast }\left( \delta \mu _{1}+\varepsilon \nu _{1}\right) ^{2}+\gamma _{1}^{\ast }} \zeta -\,\frac{\alpha \beta _{1}^{\ast }\varepsilon \left( \delta \mu _{1}+\varepsilon \nu _{1}\right) }{\mu _{1}^{2}+\alpha ^{2}\beta _{1}^{\ast }\left( \delta \mu _{1}+\varepsilon \nu _{1}\right) ^{2}+\gamma _{1}^{\ast }} z +\,\frac{\mu _{1}u_{1}^{\ast }+\alpha \beta _{1}^{\ast }\left( \delta \mu _{1}+\varepsilon \nu _{1}\right) (k-p_{1}^{\ast })+\gamma _{1}^{\ast }S_{1}^{\ast }}{\mu _{1}^{2}+\alpha ^{2}\beta _{1}^{\ast }\left( \delta \mu _{1}+\varepsilon \nu _{1}\right) ^{2}+\gamma _{1}^{\ast }}.$$

(41)

By substituting into Eq. 8 from Eqs. 1–5 and by setting $\frac{\partial L_{F2}}{\partial S_{2}}=0$, we obtain the reaction function of the fiscal authority of country 2:

$$S_{2} =-\,\mu _{r}\frac{\mu _{2}+\alpha ^{2}\beta _{2}^{\ast }\left( \delta \nu _{2}+\varepsilon \mu _{2}\right) (\delta +\epsilon )}{\mu _{2}^{2}+\alpha ^{2}\beta _{2}^{\ast }\left( \delta \nu _{2}+\varepsilon \mu _{2}\right) ^{2}+\gamma _{2}^{\ast }}r -\, \frac{\mu _{2}\nu _{1}+\alpha ^{2}\beta _{2}^{\ast }\left( \delta \mu _{1}+\varepsilon \nu _{1}\right) \left( \delta \mu _{2}+\varepsilon \nu _{2}\right) }{\mu _{2}^{2}+\alpha ^{2}\beta _{2}^{\ast }\left( \delta \nu _{2}+\varepsilon \mu _{2}\right) ^{2}+\gamma _{2}^{\ast }}S_{1} -\, \frac{\mu _{2}\nu _{1}+\alpha ^{2}\beta _{2}^{\ast }\left( \delta \mu _{1}+\varepsilon \nu _{1}\right) \left( \delta \mu _{2}+\varepsilon \nu _{2}\right) }{\mu _{2}^{2}+\alpha ^{2}\beta _{2}^{\ast }\left( \delta \nu _{2}+\varepsilon \mu _{2}\right) ^{2}+\gamma _{2}^{\ast }}S_{1} -\,\frac{\mu _{2}+\alpha ^{2}\beta _{2}^{\ast }\epsilon \left( \delta \nu _{2}+\varepsilon \mu _{2}\right) }{\mu _{2}^{2}+\alpha ^{2}\beta _{2}^{\ast }\left( \delta \nu _{2}+\varepsilon \mu _{2}\right) ^{2}+\gamma _{2}^{\ast }} z -\,\frac{\alpha \beta _{2}^{\ast }\delta \left( \varepsilon \mu _{2}+\delta \nu _{2}\right) }{\mu _{2}^{2}+\alpha ^{2}\beta _{2}^{\ast }\left( \delta \nu _{2}+\varepsilon \mu _{2}\right) ^{2}+\gamma _{2}^{\ast }}\zeta +\,\frac{\mu _{2}u_{2}^{\ast }+\alpha \beta _{2}^{\ast }\left( \varepsilon \mu _{2}+\delta \nu _{2}\right) (k-p_{2}^{\ast })+\gamma _{2}^{\ast }S_{2.}^{\ast }}{\mu _{2}^{2}+\alpha ^{2}\beta _{2}^{\ast }\left( \delta \nu _{2}+\varepsilon \mu _{2}\right) ^{2}+\gamma _{2}^{\ast }}. $$

(42)

If the two countries are mirror images, which implies that

$$ \mu _{1}=\mu _{2}=\mu _{s};\text{ }\nu _{1}=\nu _{2}=\nu _{s};\text{ }\delta =\varepsilon =\frac{1}{2};\text{ }\zeta =z=Z;\text{ }S_{1}=S_{2}=S $$

and that they have the same targets and parameters in their loss functions, by setting, without loss of generality, $\left( \mu _{s}+\nu _{s}\right) =1$ , $u=\frac{u_{1}+u_{2}}{2}$ and p ^∗ ∗ = − αu ^∗ ∗ + k,²³ Eq. 40 becomes Eq. 10 in the text.

As for the fiscal authority, by applying the restrictions above to Eq. 41, and by setting p ^∗ = − αu ^∗ + k, we obtain:

$$S_{1} =-\,\mu _{r}\frac{2\mu _{s}+\alpha ^{2}\beta ^{\ast }}{2\mu _{s}^{2}+ \frac{\alpha ^{2}\beta ^{\ast }}{2}+2\gamma ^{\ast }}r-\frac{2\mu _{s}\nu _{s}+\frac{\alpha ^{2}\beta ^{\ast }}{2}}{2\mu _{s}^{2}+\frac{\alpha ^{2}\beta ^{\ast }}{2}+2\gamma ^{\ast }}S_{2} +\,\frac{\left( 2\mu _{s}+\alpha ^{2}\beta ^{\ast }\right) u^{\ast }+2\gamma ^{\ast }S^{\ast }}{2\mu _{s}^{2}+\frac{\alpha ^{2}\beta ^{\ast }}{2}+2\gamma ^{\ast }}-\frac{2\mu _{s}+\alpha ^{2}\beta ^{\ast }}{2\mu _{s}^{2}+\frac{ \alpha ^{2}\beta ^{\ast }}{2}+2\gamma ^{\ast }}Z,$$

(43)

which is the reaction function for the fiscal authority of country 1; symmetrically, for country 2 Eq. 42 becomes:

$$S_{2} =-\,\mu _{r}\frac{2\mu _{s}+\alpha ^{2}\beta ^{\ast }}{2\mu _{s}^{2}+ \frac{\alpha ^{2}\beta ^{\ast }}{2}+2\gamma ^{\ast }}r-\frac{2\mu _{s}\nu _{s}+\frac{\alpha ^{2}\beta ^{\ast }}{2}}{2\mu _{s}^{2}+\frac{\alpha ^{2}\beta ^{\ast }}{2}+2\gamma ^{\ast }}S_{1} +\,\frac{\left( 2\mu _{s}+\alpha ^{2}\beta ^{\ast }\right) u^{\ast }+2\gamma ^{\ast }S^{\ast }}{2\mu _{s}^{2}+\frac{\alpha ^{2}\beta ^{\ast }}{2}+2\gamma ^{\ast }}-\frac{2\mu _{s}+\alpha ^{2}\beta ^{\ast }}{2\mu _{s}^{2}+\frac{ \alpha ^{2}\beta ^{\ast }}{2}+2\gamma ^{\ast }}Z.$$

(44)

Recalling the assumption that the two countries are mirror images so that S ₁ = S ₂ = S, we can derive a unique reaction function for the fiscal authority, which, by applying the restrictions above and by setting p ^∗ = − αu ^∗ + k, becomes Eq. 11 in the text.

Equations 32, 33 and 34 are obtained by using the small country restrictions (δ→0, ε→1 and ν ₁ = 0, which implies μ ₁ = 1) to Eqs. 40, 41 and 42, respectively.