Fiscal Policy within a Common Currency Area - Growth Implications in the Light of Neoclassical Theory

We examine the long-run impact of fiscal policy on economic growth under the conditions of an economic and monetary union (EMU). The analysis is based on the neoclassical growth model of a small (in economic terms) open economy in an EMU. The core assumptions are perfect capital mobility, which results in identical interest rates across the EMU, and perfect mobility of goods, which leads to the convergence of price levels. The model is based on standard neoclassical assumptions, i.e., the output is determined by the Cobb-Douglas production function with a Harrodneutral technical progress and constant returns to scale, capital and labor receive their marginal products, etc. We show that a unique long-run equilibrium exists and is characterized by the socalled natural rate of growth. The necessary and sufficient conditions of global asymptotic stability form a system of three non-trivial inequalities. We argue that in modern economies, these conditions are satisfied, except perhaps for very short periods of time. Furthermore, we show that the golden rules of fiscal policy have the form of an alternative optimal policy that crucially depends on the relation between the real interest rate and the natural rate of growth and on the relations between five other autonomous parameters.


Introduction
We are investigating the influence of fiscal policy on growth performance in the long run. The analysis is based on the neoclassical growth model of a small (in economic terms) country that participates in an economic and monetary union. Naturally, the model is inspired by the European Economic and Monetary Union. Nevertheless, the model rests on relatively general assumptions; hence, (perhaps after minor ad-justment) it may well be applied to almost any small economic entity with independent fiscal authorities in any monetary union. Therefore, our conclusions have the value of universality: they can be applied not only to existing monetary unions but also to those that have ceased to exist or will be born in the future.
The influence of fiscal policy on the long-run growth rate is the subject of many research papers, with the early contribution of Tobin (1965) to exogenous growth theory; there was another important paper by Barro (1990), who initiated a similar analysis in endogenous growth theory. Important contri-

A small economy in an EMU -basic assumptions
An economic union requires free movement of goods and services as well as all factors of production. In a monetary union, all the member countries use a common currency, and furthermore, the monetary policy is conducted by a common central bank. We assume that the home country is small relative to the entire EMU. In particular, any changes in its economy are trivial from the EMU's point of view, i.e., they have negligible influence on the union-wide level of prices, wage rates, interest rates, and etc. Furthermore, we treat an EMU as a closed economy. Henceforth, the terms "abroad" and "foreign" refer to "all other EMU member countries". (2012); Wolszczak-Derlacz (2010) have found that differences in price levels in many sectors within the European Union are slowly decreasing. Therefore, we assume that in the long run, the equilibrium price levels are equal. The financial capital is assumed to be perfectly mobile. Consequently, the nominal interest rates are equal everywhere, and due to identical inflation rates, the real interest rates are also uniform across an EMU. For a small economy, the real interest rate is exogenous: * r r = . Due to perfect capital mobility, the supply of capital in a small economy instantaneously adjusts to the demand. Let K be the domestic capital (the stock of capital used for production within the given country). Obviously, a certain part of K is owned by foreign citizens. Let KN be the national capital, i.e., all the capital that is owned by citizens of the country.
Then, a certain part of KN is allocated domestically, and the remainder is employed abroad. In addition, let E be the net foreign assets, so that Lastly, we assume that the labor L is immobile because every country uses only its own stock of L. This assumption has strong empirical support. Researchers argue that in the case of the European Union, the mobility of people is low. Indeed, apart from obvious language and cultural differences, there are significant institutional barriers to migration within Europe (Kahanec, 2012;Meardi, 2012;Zimmermann, 2009).

The public sector (government)
By assumption, public revenues are proportional to the domestic output (which may be measured by the GDP): Let B stand for the total outstanding public debt.
Then, the budget deficit, which takes into account the interest payments on the outstanding debt, is equal to where G represents all the government expenditures. By assumption, all public expenditures are classified as consumption. We assume that the government makes a decision on the specific value of the ratio of the public deficit to the domestic output (GDP), i.e., it decides on the value of the (non-negative) parameter ξ, which is defined as follows: A similar "fixed deficit" rule is used in some recent papers in the case of endogenous growth models of a closed economy, for example, Greiner, Semmler (2000), Groneck (2010), and Minea, Villieu (2009). It follows that Therefore, public expenditures are set according to the rule described by equation (3) with a crucial decision parameter ξ. The evolution of public debt is described by A certain part of the new emission of bonds is purchased by foreign investors (θ), and the rest is bought by domestic agents: Of course, at every moment where D B is the domestic debt and F B is the foreign debt of the government.

The national income account
The demand for domestic output consists of consumption and investment of the private sector, public expenditures and net exports, i.e., The real national income consists of four elements: • the compensation of labor, i.e., w L, where w is the average wage rate, which is equal to the marginal product of labor In the long-run state of equilibrium, the total revenues of firms are equal to the compensation of labor and the domestic capital K, i.e., firms make zero economic profits; for details see Konopczyński (2004), section 3. Hence, the real disposable national income (which is equivalent to the GNP) can be expressed as follows: Therefore, the real disposable income is equal to the volume of the domestic output plus the revenues from the net foreign assets and the interest from domestic and foreign bonds. After taxation, this income is devoted to consumption C and savings S. If γ stands for the average propensity to save, then Equations (8) -(11) imply that where Q is the current account balance, i.e., Equation (12) is a very well-known macroeconomic identity: the excess of private savings over investment is used to finance both the public deficit and the current account balance (Dornbusch, 1980, p. 23).
The dynamics of domestic capital (K) and national capital (KN) are described by standard rules: To keep things simple, we assume equal rates of depreciation for all types of productive capital. Otherwise, we would have to explicitly describe the dynamics of each of the four types of capital (see above), which would seriously complicate the model. The net foreign assets are equal to the difference between the national capital and the domestic capital: After taking into account (14) and (15), the evolution of E is described as follows: By assumption, the national investment (which is financed by the citizens of the country and augments the national capital) is proportional to the disposable income net of taxes, i.e., The savings of the private sector add up to the financial and real assets held by the citizens. There are three types of assets in the model that citizens can invest in: national capital (K N ), domestic bonds and foreign bonds. Hence, the usage of savings is described as follows: Using (12) and (16), after rearranging we obtain This equation gives important insight into the model: it implies that if the government borrows from abroad a certain amount of money ( F B ∆ ), then (if all the other conditions remain unchanged) the private sector will automatically lend the very same amount to these foreign governments (by purchasing O ∆ foreign bonds).
Accordingly, the domestic stock of money remains unchanged. Therefore, any government decisions regarding the proportion θ cannot influence the domestic absorption. The very same mechanism applies to the current account balance: if the current account balance increases (exogenously) by a certain amount of money, then (ceteris paribus) an identical amount of money will be invested in foreign bonds. Thus, neither the domestic supply of money nor the aggregate demand for domestic output will change.

The model
By summarizing all the above assumptions and using standard neoclassical technology (see Konopczynski (2004), section 4), we obtain the following system of equations: Fiscal policy within a common currency area -growth implications in the light of neoclassical theory where 0 > n with the following initial (endowment) conditions: As a result, the domestic output pel is fixed as well.
plies that the domestic investment pel is always proportional to k: Consequently, the taxes pel are also constant over time.
Hence, the model can be rewritten in the following very convenient recursive form (see Konopczyński (2004), section 4):

The dynamics of the model and the (global) asymptotic stability of the steady state
In the basic version of the model (without government), the dynamics of the economy were described by a single linear differential equation (see eq. 30 in Konopczyński (2004) where the first two functions are given in [b] and [b D ]: Because (27) is quite complicated, it does not allow for straightforward conclusions unless one moves to empirical research and calibrates the model with data. However, one interesting feature of the model is that the stability (or instability) is independent of all the fiscal parameters.
Substituting various values of the parameters into (27) leads to the somewhat pessimistic conclusion that for some realistic calibrations the equilibrium is unstable, whereas for other realistic calibrations it is stable. Nothing more conclusive can be derived from (27). However, from the economic point of view it is pointless to analyze unstable steady states. Whenever the economy is unstable (i.e., moving further away from the equilibrium), the decision parameters of the private sector (ψ, γ) have to be adjusted. Otherwise, the economy breaks down. In this sense, it is pointless to analyze theoretical unstable steady states. Henceforth, we assume that the stability conditions (27) are satisfied, and in the next part of the paper, we only consider stable equilibria.
However, in the following section we will closely examine the stability conditions. In particular, we will try to assess whether these conditions are satisfied in the real world.

The equilibrium
Obviously, in the dynamic equilibrium (hereafter: the DE) all the variables pel are constants. Accordingly, all the original variables (Y, K, C, I, and etc.) are growing exponentially at a constant rate that is equal to the natural rate of growth: σ. This rate also determines the speed of the growth of real wages (see Konopczyński (2004), section 6).
Before we examine the details, notice that four of the pel variables (domestic capital, output, domestic investment, and taxes) depend exclusively on the exogenous parameters and on the real (union-wide) interest rate r. This result follows directly from equations . . .
Vizja Press&IT www.ce.vizja.pl As a result, in the DE the public debt pel is proportional to the level of government deficit (expressed as the ratio of the public deficit to the GDP ξ), and it is also inversely proportional to the natural rate of growth σ + n . Interestingly, b is independent of τ, i.e., b does not depend on the size of the public sector. The index of indebtedness in the DE is equal to Formally, under our assumptions there is no guarantee that the steady-state consumption will be positive. Hence, (38) needs to be examined more closely. Notice that the first of the three conditions for stability (27) is equivalent to the following: . Therefore, the denominator of the ratio in (38)  The empirical estimates of β are usually approximately 2/3 (Balistreri et al., 2003;Konishi & Nishiyama, 2002;Willman, 2002). The level of taxation (at least in the OECD countries) only rarely approaches such a high level (though in some countries it is permanently above 50%). Hence, it seems fair (at least on empirical grounds) to assume that

The optimal fiscal policy
In this section, we will search for the golden rules for fiscal policy, i.e., we will solve the following problem: what values of the fiscal parameters guarantee maximum consumption (per capita) in the DE. It is important to remember that the overall consumption consists of two elements: privately financed consumption C and public expenditures G. Therefore, the optimization criterion is the total consumption per capita in the DE, i.e., L G C / ) ( + . Notice that this ratio grows at a constant rate σ. Hence, the first (rather intuitive) conclusion is as follows: in the DE, the overall consumption grows more rapidly with a higher rate of technological progress.
Inasmuch as the technological level A is exogenous, the maximization of consumption per capita L G C / ) ( + is equivalent to the maximization of consumption pel ( g c + ). Recall that in the DE, both public and private consumption pel are fixed according to the formula (38) and the following: If the DE is stable, the denominator of the above ratio is positive. Under this assumption, the sign of the partial derivative in (43)  In practice, all four of these situations may occur in various countries in different periods because of the natural variability of the parameters (i.e., the real interest rate r, the savings rate γ, the rate of investment ψ, and the natural rate of growth σ + n ).
To make things simpler and more tangible, notice that in real economies the rate of savings γ is usually close to the rate of investment ψ. In that case, . Hence, in the real world cases (b) and (d) will occur much more often than the remaining ones. The first of these two cases can be interpreted as follows: if the real interest rate r exceeds the natural rate of growth, then the level of taxation (as well as the public deficit, which was analyzed above) should be as low as possible to maximize the total consumption per capita in equilibrium. The opposite statement is true in case (d).

Summary
Introducing explicite the government sector allows for some new conclusions, and some of them significantly differ from the conclusions that were obtained in the simplified model (without government). For example, the necessary and sufficient conditions for the global asymptotic stability of the dynamic equilibrium are completely different than in the model without government. These conditions consist of three relatively complex inequalities that bind together virtually all the exogenous parameters of the economy; only the fiscal parameters are not connected thereby. This fact seems to be quite interesting, as it implies that the stability of an economy depends neither on the taxation rate nor on the deficit-to-GDP ratio.
It is relatively easy to come up with a set of parameters that violates the stability conditions. Nonetheless, we argue that any such set of parameters would be unrealistic because at least some of these parameters would diverge significantly from their real-world counterparts. For this reason, we argue that real-world economies easily satisfy the stability conditions. Next, we have been seeking the "golden rules" for the fiscal policy parameters. Strictly speaking, we sought parameter values that maximize the total (private plus public) consumption per capita in a dynamic equilibrium. The "golden rules" of fiscal policy have the form of an alternative optimal policy that crucially The model presented in our paper has some strengths (as well as weaknesses). First, it is based on very simple and general assumptions, and hence, it is very easy to analytically describe the state of equilibrium and investigate its properties. However, it is worth stressing that even in our very simple setting, the stability conditions are quite complex. Second, the dynamic equilibrium is globally asymptotically stable (for a very wide range of realistic values of the parameters), which enables straightforward applications of numerical methods for dynamic simulations of the transitory processes. For these two reasons, the model is perfectly suited to educational purposes. It can also serve as a starting point for more complex, applicable models.
Fiscal policy within a common currency area -growth implications in the light of neoclassical theory Appendix By making use of (23) -(26), equations of motion (22) , and e e − = e are the deviations from the steady state. Let D symbolize the square matrix in (A1). Because the dynamics are described by linear system of equations (22), a closer examination of the properties of the matrix D will provide us with the (necessary and sufficient) conditions of the global asymptotic stability of the steady state. Most likely the most convenient method is the Liénard-Chipart theorem (Gandolfo, 1980, p. 251 . Therefore, condition (iv) implies condition (v).
To summarize, the necessary and sufficient conditions of stability (A3) can be reduced to the following (equivalent) system of inequalities: