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I utilize a differentiable dynamical system á la Lotka-Voletrra and explain monetary and fiscal interaction in a supranational monetary union. The paper demonstrates an applied mathematical approach that provides useful insights about the interaction mechanisms in theoretical economics in general and a monetary union in particular. I find that a common central bank is necessary but not sufficient to tackle the new interaction problems in a supranational monetary union, such as the free-riding behaviour of fiscal policies. Moreover, I show that supranational institutions, rules or laws are essential to mitigate violations of decentralized fiscal policies.

This paper studies the theoretical implications of monetary and fiscal interaction in a monetary union. This is an urgent and interesting topic, especially since the European sovereign debt crisis in 2010. I utilize an approach from applied mathematics in order to model the economic interactions in a supranational monetary union. A dif- ferentiable dynamical system, similar to a Lotka-Volterra model, turns out to be well suited for studying this pro- blem. Overall, my mathematical model is literally interdisciplinary and links two, up to now, hardly unconnect- ed areas: the theory of differential equations and monetary economics.

It is not surprising that there are relatively few economic models that capture the sophisticated subspace of monetary-fiscal interaction. This has to do with the complexity and dynamics in this field of economics. So far, it is common practice in the economics literature to apply a game theoretic approach to study this question [

The remainder of the paper is structured as follows: Section 2 explains the model and discusses some propo- sitions. I show the existence, stability and solution of the model as well as the economic implications in general. Finally, Section 3 concludes the paper.

The model’s framework consists of three interacting institutions. The first institution is the European Central Bank (ECB) that is centralized in a monetary union. The primary objective of monetary policy is to maintain price-stability according to Article 105 in the Treaty on the Functioning of the European Union (TFEU). The ECB, however, interacts with the decentralized fiscal policies, the second institution in the model. Both insti- tutions, in particular the central bank, determine the common interest rate. At the moment, there are 18 member countries and thus fiscal policies in the euro area. The main difference between monetary and fiscal policy is that the fiscal authorities retain full sovereignty at the national level. The third institution is supranational law or governance, such as the Stability and Growth Pact (SGP), the European Stability Mechanism (ESM) and other legal constraints [

a) Monetary policy interacts with fiscal policy. The decision about the level of public deficits and debts have an impact on the common central bank.

b) Fiscal policy in one member country interacts with the other fiscal policies in the monetary union. There is competition about the public good “price-stability” provided by ECB. One fiscal policy can undermine the supranational objective and transfer the cost to all countries; i.e. through free-riding.

c) Supranational law defines the level playing field for all institutions. These rules interact with both fiscal policies and the central bank. The main objective is the mitigation of fiscal heterogeneity as well as free-riding and moral hazard.

The paper analyzes these interaction channels in a monetary union in general. I utilize a mathematical model that consists of differential equations. Until now, economic literature has studied these interactions mainly in game theoretic models [

First of all, I model the interaction of fiscal policies and the supranational law respectively. Suppose, ^{1}. The dynamic yields

The

The solution of the model is^{2}. A more comprehensive modeling of the sanction scheme is

The sanction payment

where

Finally, I solve the equation for

This solution has the following boundaries for

If supranational law is fully effective, i.e. the detection probability

Next, I study monetary policy. The main instrument of a common central bank is the interest rate level

where ^{3}. Hence, in this case the parameter

where

where

Analyzing the complete dynamics of the fiscal-monetary-law interaction reveals new insights about the ne- cessary and sufficient conditions for a long-run stable and sustainable monetary union. Using Equations (3) and (7) together with conditions (8) and (9), yields the following system:

Interestingly, this system of two differential equations is similar to a so-called “Lotka-Volterra” model, de- veloped by Alfred James Lotka (1880-1949) and Vito Volterra (1860-1949), and is an useful concept in appli- ed mathematics [

The asymptotic stability or instability of the model can be studied. I define the function

The first derivative for the two solutions yields

Consequently, the eigenvalues of

This implies

instability can also be seen from

solve the following problem

The second solution of the model

or vice versa. But if

This implies no real solution. Finally, I describe the solution behaviour of the model near a point

Next, I integrate and obtain,

where

The integration constant

solution in the environment

where

The second-order Taylor approximation of the solution

This shows that that the specific solution solves the model for

Finally, I study the full model of Equation (10) with

The general model has four solutions:

and

Applying Cramer’s rule, I obtain

For later computation purposes, I define

The point

if

eigenvalues of Equation (20) for

where

The first constellation becomes a reality if free-rider incentives are small

Proposition 1 The number of fiscal policies

The proof of this proposition follows from Equation (21). First, the constellation

nomically not realistic because the common interest rate converges for

Proposition 2 For

a) the detection probability,

b) the marginal sanction fee,

c) central bank commitment, d, is high (i.e. c is low).

Proof: The proof follows by direct differentiation of

Part (ii), follows by differentiation in respect to

Part (iii) is shown by

This paper explains the unique fiscal-monetary-law interaction in a supranational monetary union. I conclude the paper by discussing some generalizations and by touching on some issues that the model did not address. First, the argument is much more general than initially considered. The results reveal new insights about the interac- tion of the key institutions in a monetary union. The model demonstrates that without effective laws and fiscal and economic governance, a monetary union is doomed to fail. Consequently, the fiscal and economic govern- ance scheme, together with the common cental bank, plays an important role in a monetary union. Second, the model is well designed to analyze the institutional drawbacks and interaction relationships in the EMU. The re- sult suggests a tough sanction scheme for unsound fiscal policies. Only this can mitigate the potential benefits of free-riding. The major omission of the model is an endogenous economic-political element that considers for in- stance strategic policy decisions or veto power. Furthermore, an empirical investigation of the proposition is also an important study object in future research. Moreover, I do not consider the fact that small and weak agents ty- pically pay more attention to supranational law than powerful agents do.

I would like to thank for comments Mr. Gassmann and the two anonymous referees. I gratefully acknowledge financial support from the RRI-Reutlingen Research Institute.