Policy uncertainty, symbiosis, and the optimal fiscal and monetary conservativeness

Abstract

This paper extends a well-known macroeconomic stabilization game between monetary and fiscal authorities developed by Dixit and Lambertini (American Economic Review 93: 1522–1542) to multiplicative (policy) uncertainty. We find that even if fiscal and monetary authorities share a common output and inflation target (i.e., the symbiosis assumption), the achievement of the common targets is no longer guaranteed; under multiplicative uncertainty, in fact, a time consistency problem arises unless policymakers’ output target is equal to the natural level.

Appendices

Appendix A

This appendix contains some equations used in the discussion; all of them can be easily derived after tedious algebra.

By considering different orders of moves, the Stackelberg (fiscal leadership) solution is:

$$ E\left( y \right) = \frac{{B_{1} y^{ * } + \sigma_{\mu }^{2} B_{2} \bar{y}}}{{B_{1} + \sigma_{\mu }^{2} B_{2} }}\;{\text{and}}\;E\left( \pi \right) = \pi^{ * } + b\theta_{B} \left( {\frac{{\sigma_{\mu }^{2} B_{2} }}{{B_{1} + \sigma_{\mu }^{2} B_{2} }}} \right)\left( {y^{ * } - \bar{y}} \right) $$

(A.1)

where $B_{1} = a^{2} (\theta_{G} + \theta_{B}^{2} b^{2} )$ and $B_{2} = A_{2} (1 + \theta_{B} b^{2} )$.

The Stackelberg (central bank’s leadership) solution is:

$$ E\left( y \right) = \frac{{C_{1} y^{ * } + \sigma_{\mu }^{2} C_{2} \bar{y}}}{{C_{1} + \sigma_{\mu }^{2} C_{2} }}\;{\text{and}}\;E\left( \pi \right) = \pi^{ * } + b\theta_{B} \left( {\frac{{\sigma_{\mu }^{2} B_{2} }}{{C_{1} + \sigma_{\mu }^{2} C_{2} }}} \right)\left( {y^{ * } - \bar{y}} \right) $$

(A.2)

where: $\begin{gathered} C_{1} = A_{1} A_{2} \sigma_{\mu }^{2} + a^{2} \left[ {\mathop {\left( {a + cb} \right)}\nolimits^{2} \theta_{G}^{2} + c^{2} \theta_{B} } \right]\rm {,} \hfill \\ C_{2} = A_{2}^{2} \sigma_{\mu }^{2} + c^{2} \mathop {\left( {ab\theta_{G}^{2} + c} \right)}\nolimits^{2} + c^{2} D\;{\text{and}}\;D = a\left( {a + b^{3} c\left( {\theta_{G} + \theta_{B} } \right)} \right)\theta_{G} + B_{2} - c^{2} > 0 \hfill \\ \hfill \\ \end{gathered}$.

In the Nash equilibrium described in the main text, the derivatives of the equilibrium outcomes are:

$$ \frac{\partial E\left( y \right)}{{\partial \theta_{G} }} = \frac{{ac^{2} \sigma_{\mu }^{2} (a + b^{3} c\theta_{B} + bc)\left( {y^{ * } - \bar{y}} \right)}}{{\mathop {\left( {\theta_{G} a^{2} + c^{2} \sigma_{\mu }^{2} + abc\theta_{G} + b^{2} c^{2} \theta_{G} \sigma_{\mu }^{2} - abc\theta_{B} } \right)}\nolimits^{2} }} $$

(A.3)

$$ \frac{\partial E\left( \pi \right)}{{\partial \theta_{G} }} = - \frac{{\theta_{B} bc^{2} \sigma_{\mu }^{2} (a + b^{3} c\theta_{B} + bc)\left( {y^{ * } - \bar{y}} \right)}}{{\mathop {\left( {\theta_{G} a^{2} + c^{2} \sigma_{\mu }^{2} + abc\theta_{G} + b^{2} c^{2} \theta_{G} \sigma_{\mu }^{2} - abc\theta_{B} } \right)}\nolimits^{2} }} $$

(A.4)

$$ \frac{\partial E\left( y \right)}{{\partial \theta_{B} }} = - \frac{{abc^{3} \sigma_{\mu }^{2} (1 + b^{2} \theta_{B} )\left( {y^{ * } - \bar{y}} \right)}}{{\mathop {\left( {\theta_{G} a^{2} + c^{2} \sigma_{\mu }^{2} + abc\theta_{G} + b^{2} c^{2} \theta_{G} \sigma_{\mu }^{2} - abc\theta_{B} } \right)}\nolimits^{2} }} $$

(A.5)

$$ \frac{\partial E\left( \pi \right)}{{\partial \theta_{B} }} = \frac{{bc^{2} \sigma_{\mu }^{2} (1 + b^{2} \theta_{B} )\left( {\theta_{G} a^{2} + abc\theta_{G} + c^{2} \sigma_{\mu }^{2} + \theta_{G} b^{2} c^{2} \sigma_{\mu }^{2} } \right)\left( {y^{ * } - \bar{y}} \right)}}{{\mathop {\left( {\theta_{G} a^{2} + c^{2} \sigma_{\mu }^{2} + abc\theta_{G} + b^{2} c^{2} \theta_{G} \sigma_{\mu }^{2} - abc\theta_{B} } \right)}\nolimits^{2} }} $$

(A.6)

$$ \frac{\partial E\left( y \right)}{{\partial \sigma_{\mu }^{2} }} = - \frac{{ac^{2} (1 + b^{2} \theta_{B} )A_{1} \left( {y^{ * } - \bar{y}} \right)}}{{\mathop {\left( {\theta_{G} a^{2} + c^{2} \sigma_{\mu }^{2} + abc\theta_{G} + b^{2} c^{2} \theta_{G} \sigma_{\mu }^{2} - abc\theta_{B} } \right)}\nolimits^{2} }} $$

(A.7)

$$ \frac{\partial E\left( \pi \right)}{{\partial \sigma_{\mu }^{2} }} = \frac{{abc^{2} \theta_{B} (1 + b^{2} \theta_{B} )A_{1} \left( {y^{ * } - \bar{y}} \right)}}{{\mathop {\left( {\theta_{G} a^{2} + c^{2} \sigma_{\mu }^{2} + abc\theta_{G} + b^{2} c^{2} \theta_{G} \sigma_{\mu }^{2} - abc\theta_{B} } \right)}\nolimits^{2} }} $$

(A.8)

Appendix B

Consider a general welfare function (see Dixit and Lambertini (2003a: Appendix A-B):

$$ L_{W} = E_{0} \left[ {\frac{1}{2}\mathop {\left( {\pi - \pi^{ * } } \right)}\nolimits^{2} + \frac{{\theta_{W} }}{2}\mathop {\left( {y - y^{ * } } \right)}\nolimits^{2} + \vartheta_{W} x} \right] $$

(B.1)

where π ^*, y ^*, θ _W $,\;\vartheta_{W}$ are directly derived from the fundamentals of the economy.

The optimal degrees of conservativeness that can be derived after simple algebra are:

$$ \theta_{G}^{ * * } = \frac{{c^{2} \sigma_{\mu }^{2} \left[ {a\theta_{W} \left( {y^{ * } - \bar{y}} \right) - \vartheta_{W} } \right]}}{{c^{2} \sigma_{\mu }^{2} \left[ {a + bc\left( {1 + b^{2} \theta_{W} } \right)} \right]\left( {y^{ * } - \bar{y}} \right) + \left[ {a\left( {a + bc} \right) + b^{2} c^{2} \sigma_{\mu }^{2} } \right]\vartheta_{W} }} < \theta_{G}^{ * } $$

(B.2)

$$ \theta_{B}^{ * * } = 0 $$

(B.3)

The above result confirms the conclusion reached in the main text: the central bank should take account of inflation stabilization only, whereas the government should target both real output and inflation deviations and adopt a degree of conservativeness higher than the social one. By introducing a tax-distortion cost in the welfare function, the optimal degree of government’s conservativeness should be even higher. The economic intuition is trivial.