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Fiscal policy in a monetary union from the perspective of the neoclassical growth model Michał Konopczyński Abstract We examine the long-run impact of fiscal policy on the economy under 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 EMU. The core assumptions are perfect capital mobility resulting in identical interest rates across EMU, and perfect mobility of goods which leads to the convergence of price levels. The model is based on standard neoclassical assumptions, i.e. output is determined by the Cobb-Douglas production function with Harrod-neutral technical progress and constant returns to scale, capital and labor receive their marginal products, etc. We show that the unique long-run equilibrium exists and is characterized by the so-called natural rate of growth. The necessary and sufficient conditions of the global asymptotic stability form the system of three non-trivial inequalities. We argue that in modern economies these conditions are satisfied, except perhaps for very short periods of time. This proposition is supported by numerical simulations. We also show that the “golden rules” of fiscal policy have the form of an alternative – optimal policy crucially depends on the relation between the real interest rate and the natural rate of growth and on the relation between five other independent parameters. Finally, we analyze transitory dynamics of the model. JEL Classification Numbers: E13, E62, F43, H60 Keywords: monetary union, golden rule, neoclassical growth theory, fiscal policy Poznań University of Economics, 61-875 Poznań, al. Niepodległości 10, Poland. Contact: [email protected] 1 Introduction In this paper we are investigating the influence of fiscal policy on the growth performance in the long run. The analysis is based on the neoclassical growth model of a small (in economic terms) country, which participates in an economic and monetary union. The model is, of course, inspired by the European Economic and Monetary Union. Nevertheless, it rests on relatively general assumptions, hence (perhaps after minor adjustment) it may well be applied to almost any small economic entity with independent fiscal authorities in any monetary union. Therefore, our conclusions have the valor of universality – they can be applied not only to existing monetary unions, but also to those that seized to exist, or will be born in the future. One of the first growth models with explicit public sector was developed by Tobin (1965). It was, however, confined to the closed economy. Carlberg (1997) describes a generalized version for an open economy with perfect mobility of capital. We present certain modification of his model. By imposing small economy assumptions, and introducing both domestic and international bonds, we obtained some new conclusions. We are building on our previous paper Konopczyński (2004)1, where we presented the neoclassical growth model without explicit public sector. Now, we will distinguish two separate sectors: private and public. The first one, generally speaking, produces goods and services, which are sold in the market, and the income (after taxation) is partially saved and invested. The public sector collects taxes and finances public expenditures. The budgetary deficit is financed exclusively by bonds, sold indiscriminately to domestic and foreign investors. We also introduce foreign bonds, issued by foreign governments. Most other assumptions are similar to those in Kon (2004), for example the neoclassical aggregate production function, and classic assumptions regarding perfectly clearing labor and capital markets. We prove that the dynamic equilibrium is unique and derive the necessary and sufficient conditions for its global asymptotic stability. We also analyze the so-called golden rules regarding the fiscal parameters. Last but not least, we investigate the transition processes between two steady states (the transitory dynamics of the economy), which are associated with two different sets of fiscal characteristics. These processes are illustrated with computer simulations, based upon realistic (“calibrated”) values of parameters. 1. 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 member countries use a common currency, and 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 point of view – they have negligible influence on the union-wide level of prices, wage rates, interest rates etc. We also treat an EMU as a closed economy, i.e. the outside world is 1 Hereafter, we refer to this paper as Kon (2004). 2 neglected. Henceforth, the terms “abroad” and “foreign” refer to “all other EMU member countries”. By assumption, all goods move freely within EMU. Therefore, price levels and inflation rates in all member countries are equal. We admit that at least in case of the European Monetary Union this assumption is disputable2, but we expect that as the process of economic integration gains speed, price levels converge faster. The financial capital is assumed to be perfectly mobile. Consequently, nominal interest rates are equal everywhere, and – due to identical inflation rates – real interest rates are also uniform across 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 country). Obviously, a certain part of K is owned by foreign citizens. Let KN be the national capital, i.e. all of capital owned by citizens of the country. Certain part of KN is allocated domestically, and the remainder is employed abroad (see table). Let E be the net foreign assets: E KN A KFD . Obviously, KN K E . The structure of capital in an EMU located domestically (K) abroad citizens of the country (KN) KND KNA foreigners (KF) KFD KFA owned by: KN – national capital, KF – foreign capital, KN D – national capital located domestically, KN A – national capital located abroad, KFD – foreign capital located in the country, KFA – foreign capital located abroad (outside of the country) Last but not least, we assume that labor L is immobile – every country uses only its own stock of L. This assumption is certainly disputable, but it is a useful starting point. Besides it has some empirical support3. 2. The public sector (government) By assumption, public revenues are proportional to the domestic output (which may be measured by the GDP): T Y 2 (1) In Europe prices are converging more slowly than expected. See for example Angeloni and Ehrmann (2004), Lutz (2002). Also, Cecchetti et.al. (2000) document significant and persistent differences in prices of tradable goods in the U.S. 3 Some researchers argue that in case of the European Union mobility of people is low – especially in comparison to the U.S. Probably language and cultural barriers constitute serious obstacles to migration within Europe. See Bertola (1999), Krueger (2000). 3 where 0 1 . If public expenditures exceed revenues, the difference is financed by bonds sold (indiscriminately) to domestic and international investors. For simplicity, lets assume that the specific risk associated with lending money to all governments in EMU is uniform. Hence, the real interest rate on public bonds in all EMU countries is equal to a certain number r. Let B stand for the total outstanding public debt. The budget deficit, which takes into account the interest payments on the outstanding debt, is equal to G rB T , where G is 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 public deficit to the domestic output (GDP), i.e. it decides on the value of the (nonnegative) parameter defined as: (G rB T ) / Y (2) G T rB Y (3) It follows that Public expenditures are therefore set according to the rule described by the equation (3), with a crucial decision parameter . The evolution of the public debt is described by B G rB T Y (4) A certain part of the new emission of bonds is purchased by foreign investors (θ), and the rest is bought by domestic agents: B F Y (5) B D (1 )Y (6) where 0 1 . Of course, in every moment B BD BF (7) where BD is the domestic debt, and BF is the foreign debt of the government. 3. The national income account The demand for domestic output consists of consumption and investment of the private sector, public expenditures and net exports. For simplicity we assume that the domestic market is permanently in equilibrium, i.e. the actual output is equal to domestic absorption plus net exports: Y C I G X M (8) The real national income (i.e. income of the citizens of the country) consists of compensation of labor and rent from the capital owned by citizens (the national capital). It also includes interest payments on domestic and foreign bonds. In the long-run equilibrium the total revenues of firms are equal to the compensation of labor and capital (firms make only usual economic profits, for details see Kon (2004), section 3). Summing up, the real disposable national income (equivalent of the GNP) can be expressed as follows: 4 Yd wL wK KN r ( BD O) wL wK ( K E ) r ( BD O) Y wK E r ( B D O ) Y ( r ) E r ( B D O ) (9) The real disposable income is therefore equal to the volume of domestic output plus revenues from the net foreign assets and interest from domestic and foreign bonds. This income, after taxation, is devoted to consumption C and savings S. If stands for the average propensity to save, then S (Yd T ) (10) C (1 ) (Yd T ) , where 0 1 (11) The equations (8) – (11) imply that C S T C I G X M (r ) E r ( BD O) Combining the above with (4) and (7), and rearranging yields the well-known identity: S I B Q , (12) where Q is the current account balance, i.e. Q X M (r ) E rO rBF (13) The equation (12) is the very well known macroeconomic identity: the excess of private savings over investment is used to finance the public deficit and the current account balance4. The dynamics of domestic capital (K) and national capital (KK) is described by standard rules: K I K (14) KN IN KN (15) Recall that the net foreign assets are equal to the difference between the national capital and the domestic capital: E KN K (see Kon (2004), p. 9). Taking into account (14) and (15), the evolution of E is described as follows: E IN I E (16) By assumption, the national investment (financed by citizens of the country, and augmenting the national capital) is proportional to the disposable income net of taxes, i.e. IN (Yd T ) , where 0 1 (17) The savings of the private sector add up to the financial and real assets held by citizens. There are three kinds of assets in the model that citizens can invest in: national capital ( KN ), domestic bonds and foreign bonds. Hence, the usage of savings is described as follows: S IN B D O (18) Using (12) and (16), after rearranging we obtain: O B F Q ( E E ) 4 See Dornbusch (1980), p. 23, or Sachs, Larrain (1993), p. 194. 5 (19) This equation brings important insight into the model. It implies that if the government borrows from abroad certain amount of money ( B F ), then (if all other conditions remain unchanged) the private sector will automatically lend the very same amount to the 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. If the government borrows additional amount of money from abroad (by issuing and selling more bonds to foreign investors) and injects the money into the economy, the very same amount will be automatically withdrawn by private agents – they will invest the very same amount of money into foreign bonds. The very same mechanism applies to the current account balance. If, for some reasons, 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. Neither the domestic supply of money, nor the aggregate demand for domestic output will change. 4. The model Summarizing all of the above assumptions, and using standard neoclassical technology (see Kon (2004), section 4), we get the following system of equations: 1 [i] [ii] K r AL Y K (AL) where 0 1 where 1 , , 0 [iii] K w A AL [iv] T Y where 0 1 [v] G T rB Y where 0 [vi] B Y , B F Y , [vii] Yd Y (r ) E r ( BD O) [viii] C (1 )(Yd T ) [ix] S (Yd T ) [x] Y C I G X M [xi] Q X M (r ) E rO rBF [xii] IN (Yd T ) [xiii] K I K [xiv] O S IN B D [xv] E IN I E [xvi] A A B D (1 )Y where 0 1 where 0 1 where 0 1 where 0 6 [xvii] L nL where n 0 with the initial (endowment) conditions: K (0) K 0 0 , E (0) E 0 , B(0) B0 0 , BF (0) BF 0 0 , BD (0) B0 BF 0 0 , O(0) O0 , A(0) A0 0 , L(0) L0 0 It’s convenient to divide all variables through the quantity of effective labor (AL), and transform the system of equations into the following (equivalent), where small letters stand for the original variables per effective labor (hereafter pel, for short): [i’] r k 1 [ii’] y k w A y [iii’] [iv’] [v’] [vi’] [vii’] [viii’] [ix’] [x’] [xi’] [xii’] [xiii’] [xiv’] [xv’] [xvi’] t y g t rb y b y (n )b bF y (n )bF bD (1 ) y (n )bD y d y (r )e r (bK o) c (1 )( y d t ) s ( yd t) yci g xm q x m (r )e ro rbF in ( y d t ) k i k o s in (1 ) y (n )o e in i e A A The initial conditions are: k (0) k 0 0 , e(0) e0 , b(0) b0 0 , bF (0) bF 0 0 , bD (0) b0 bF 0 0 , o(0) o0 , A(0) A0 0 . For simplicity, assume that the real interest rate r and the depreciation rate are constants. Then the stock of domestic capital pel is fixed, which can be seen from equation [i’]: 1 k const. r (20) As a result, the domestic output pel is fixed as well, whereas the real wage rate grows exponentially at the constant rate equal to the rate of technical progress, which can be seen from equation [iii’]. Since k 0 in every moment, equation [xiii’] implies, that the domestic investment pel is always proportional to k i k , where n 7 (21) Consequently, taxes pel are also constant over time. Hence, the model can be rewritten in the very convenient recursive form (see Kon (2004), section 4): 1 [k] [y] [w] [i] [t] [g] [b] [ bF ] [ bD ] [yd] [c] [s] [ x m] [q] [ik] [o] [e] [A] k r y k w A y i k where n t y g t rb y b y (n )b where 0 bF y (n )bF bD (1 ) y (n )bD y d y (r )e r (bD o) where 0 1 c (1 )( y d t ) s ( yd t) where 0 1 x m y c i g q x m (r )e ro rbF where 0 1 in ( y d t ) o s in (1 ) y (n )o e in i e A A e(0) e0 , A(0) A0 0 . where: where 0 1 b(0) b0 0 , bF (0) bF 0 0 , bD (0) b0 bF 0 0 , o(0) o0 , 5. The dynamics of the model and (global) asymptotic stability of the steady state In the basic version of the model (without government) the dynamics of the economy was described by a single, linear differential equation (see eq. 30 in Kon (2004), page 22). Introducing explicite the government sector and foreign assets results in a significant complication of the dynamics. It is now described by a system of four (though still linear) differential equations: b f 1 (b, bD , o, e) bD f 2 (b, bD , o, e) o f 3 (b, bD , o, e) (22) e f 4 (b, bD , o, e) Where the first two functions are given in [b] and [ bD ]: f 1 (b, bD , o, e) (n )b y 8 (23) f 2 (b, bD , o, e) (n )bD (1 ) y (24) and y const. 0 . The function f 3 can be obtained by substituting [s], [ik] and [yd] into equation [o]. After some algebraic manipulation we get: f 3 (b, bD , o, e) ( )rbD ( )r (n ) o ( )( r ) e ( )( y t ) (1 ) y (25) Obtaining the explicit form of the function f 4 requires similar substitutions and manipulations, utilizing the following equations: [e], [in], [yd], [t]. At the end, we get: f 4 (b, bD , o, e) rbD ro (r ) e (1 ) y i (26) The dynamic equilibrium (or, in other words, the steady state) is defined as the situation in which all the variables pel (except for A and w) are constant in time. Technically, in the steady state the right-hand sides of the equations (22) are equal to zero. By making use of (23) – (26), the equations of motion (22) can be written in the following convenient form: b 0 0 0 (n ) 0 (n ) 0 0 b D o 0 ( )r ( )r (n ) ( )( r ) 0 r r (r ) e b b D, o e (27) where b b b , b D bD bD , o o o , e e e . Let D symbolize the square matrix in (27). Since the dynamics is described by the linear system of equations (22), a closer look at the properties of the matrix D will provide us with the (necessary and sufficient) conditions of the global asymptotic stability of the steady state5. Probably the most convenient method is the Liénard-Chipart theorem6. Let the characteristic equation of matrix D be det( D I ) w0 4 w13 w2 2 w3 w4 0 (28) Define the following determinants: R1 w1 , R2 w1 w3 w1 w0 , R3 w3 w2 0 w0 w2 w4 w1 0 w w1 , R4 3 0 w3 0 w0 w2 w4 0 0 w1 w3 0 0 w0 w2 w4 According to the Liénard-Chipart theorem, the steady state is globally asymptotically stable if and only if the following conditions are satisfied7: (i) w0 0 (ii) w4 0 (iii) w2 0 (29) 5 The steady state is globally asymptotically stable if the economy is asymptotically attracted towards the equilibrium, regardless of how far away is the starting point. See Berck, Sydsæter (1993), p. 51, or Panek (2000), p. 193, 848-849. 6 See Gandolfo (1980), p. 251. 7 There are also several other alternative sets of conditions, but this one is the most convenient in our case. 9 (iv) R3 0 (v) R1 0 Obviously (i) is satisfied, since w0 1 . It can easily be demonstrated that w4 det D (n ) 2 (n r ) (r n ) Therefore condition (ii) is satisfied iif r ( ) (n )( ) (30) Inequality (iii) can be transformed to the following form: r ( ) 3 (n ) 3(n ) (1 ) 2(n ) (31) On the other hand, R3 , after some manipulation can be written as8 R3 2(n ) r 2(n ) (1 ) ... 2 Hence condition (iv) can be written in the following equivalent form: r 2(n ) (1 ) (32) Also R1 trD r 4(n ) (1 ) Notice that if (32) is satisfied then R1 0 . Therefore, condition (iv) implies condition (v). To summarize, the necessary and sufficient conditions of stability (29) can be reduced to the following (equivalent) system of inequalities: r ( ) (n )( ) r ( ) 3 (n ) 3(n ) (1 ) 2(n ) r 2(n ) (1 ) (33) These are the necessary and sufficient conditions of the global asymptotic stability of the steady state. Since (33) is quite complicated, it does not allow for clear-cut conclusions. Substituting various values of parameters into (33) leads to the somewhat pessimistic conclusion that for some realistic (!) calibrations the equilibrium is unstable, whereas for some others it is stable. Nothing more conclusive can be derived from (33). From the economic point of view, however, it is pointless to analyze unstable steady states. Henceforth, we assume that the above conditions (33) are satisfied, and in the next part of the paper we take care of stable equilibria only9. 6. The equilibrium Obviously, in the dynamic equilibrium (hereafter: DE) all variables pel are constants. Accordingly, all of the original variables (Y, K, C, I, etc.) are growing exponentially at a 8 Dots have replaced a very long and complex expression. In the next section we will also take a closer look the stability conditions. In particular, we will try to assess, whether these conditions are satisfied in the real world. 9 10 constant rate equal to the natural rate of growth: n .10 In other words, all variables expressed in units per capita ( Y / L , C / L etc.) are growing at the speed equal to the rate of technical progress σ. This rate also determines the speed of growth of real wages (see Kon (2004), section 6). Before we take to details, notice that four variables (domestic capital, output, domestic investment and taxes – all expressed pel) depend exclusively on exogenous parameters and on the real (union-wide) interest rate r. It follows directly from equations [k], [y], [i], [t]. Hence, not only in equilibrium, but also elsewhere: 1 k const. r (34) y k const . (35) i k const . (36) t y const. (37) The equations [b], [ bF ], and [ bD ] lead directly to the steady state values of the public debt b , decomposed to the domestic ( bD ) and the foreign debt ( bF ): b y , n bD (1 )b , bF b (38) In the DE the public debt pel is therefore proportional to the level of government deficit (expressed as the ratio of public deficit to GDP; ), and at the same time inversely proportional to the natural rate of growth n . Interestingly, b is independent of , i.e. it does not depend on the size of the public sector. The index of indebtedness in the DE is equal to B b Y y n (39) The only decision parameter that influences the level of indebtedness in the DE is the government deficit parameter ξ. For illustration, suppose that the government plans to keep the ratio of the public debt to GDP at a constant level of, say, 50%11. According to (39), the deficit in relation to GDP would have to be maintained at a stable level of one-half of the natural rate of growth. For example, if the long run natural rate of growth n 3% , than the ratio of public deficit to GDP has to be maintained at about 1,5%. The obvious conclusion is: countries which are characterized by high rates of growth (of population and technical progress) may safely allow themselves for relatively high public sector deficits. Put differently, the lower the rate of growth of population, the lower the safe level of government deficit. However, if the speed of technical progress is very high, then even relatively high levels of public deficit do not endanger the country’s public finances. It follows from equations [g] and (38) that 10 11 Except for L and A, which grow at exogenous rates n and σ, respectively. The Stability and Growth Pact sets the maximum level of public debt at 60% of GDP. 11 T G rB (n ) B (40) which means, that in the DE primary budget surplus is just enough to cover the interest on public debt adjusted for the factor (n ) B . In other words, the ratio of the primary surplus to the public debt in the DE is equal to the difference between the real interest rate and the natural rate of growth. In order to find the stationary values of other important variables, we need to solve the system of equations that will show up, if we set to zero the right-hand sides of equations (25) and (26). Solving the resulting system of equations brings the following values: o ( ) y t (r ) e bD n r ( ) (41) where e (1 )( n ) r n r ( ) (n r ) (r n ) y (42) Substituting these values into other equations, we obtain the DE value of the disposable income (GNP) pel: y d y (r )e r (bD o ) (43) It follows from equation [c] that c (1 )( y d t ) . Taking into account formulas (38), (41) – (43), and rearranging results in the strikingly simple formula for the DE consumption pel: c (1 )(n )( ) y (n r ) (r n ) (44) where domestic output y const. 0 . Formally, under our assumptions, there is r no guarantee that the steady-state consumption will be positive. Hence (44) needs to be examined more closely. Notice that the first of the three conditions for stability (33) is equivalent with the following: (n r ) (r n ) 0 . Therefore the denominator of the ratio (44) is positive. Meanwhile, the numerator in (44) is positive if and only if . Empirical estimates of β are usually around 2/312. The level of taxation (at least in the OECD countries) only rarely approaches such high level (though in some countries it is permanently above 50%). Hence it seems fair (at least on empirical grounds) to assume that (45) This assumption, along with all others previously taken, ensures positive consumption in the DE. The above can also be interpreted in the following (a little perverse) way. If the government in the small economy under monetary union lifted the taxation too much (above 12 See for example Balisteri et. al. (2002), Konishi, Nishiyama (2002), and Willman (2002). 12 β), then private consumption would shrink to zero. The only remaining consumption would come from public expenditures g . This would, of course, be socialism. The equations [x–m], and (34) – (38) as well as (41), (42) and (44), allow to calculate the DE levels of the trade balance, the balance on current account, and the national capital: r (1 )(n )( ) x m y 1 n r (n r ) (r n ) (46) q x m (r )e ro rbF (47) kn k e (48) 7. The optimal fiscal policy Now we will investigate how the steady state depends on the decisions taken by the government, regarding its fiscal policy. In particular, we will search for the golden rules for fiscal policy, i.e. we will solve the following problem: what values of fiscal parameters guarantee maximum consumption (per capita) in the DE. This section focuses on consumption, whereas the next one will cover other important characteristics of the economy. It’s 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. (C G ) / L . Notice, that this ratio grows at a constant rate σ. Hence the first – rather intuitive – conclusion is: in the DE the overall consumption grows the faster, the higher the rate of technological progress. Since the technological level A is exogenous, the maximization of consumption per capita (C G ) / L is equivalent to the maximization of consumption pel ( c g ). Recall that in the DE both public and private consumption pel are fixed, according to the formula (44) and the following: r g y , where n y r (49) Notice, that neither public, nor private consumption depends on θ. Therefore, the composition of the public debt is of no importance, whatsoever. It doesn’t matter what share of the public debt is actually held by foreigners, and what share remains in hands of domestic investors13. Importantly, the steady-state consumption is a function of the level of public deficit (ξ) c 0, and the rate of taxation ( ). Lets examine these functions in detail. First, notice, that so private consumption in the DE is independent of the level of the deficit. On the other hand, g r 1 y n 13 This is a straightforward implication of the balance identity (19), described in section 2. 13 (50) Clearly, three cases can be distinguished: (a) If r n , then the level of deficit ξ should be as low as possible, (b) If r n , then the level of deficit ξ should be as high as possible, (c) If r n , then the steady-state consumption per capita is independent of ξ. Intuitively, in case (a), the cost of the public debt (the interest rate) is relatively high, hence it is not worthwhile to borrow, in order to finance public consumption. The exactly opposite situation occurs in case (b). It’s important to remember, that in reality all three parameters r, n , and σ, may change (and r may actually be quite volatile), hence all three cases may turn out in various countries, in different periods, and under varying circumstances. Lets turn out to the level of taxation . It follows from (44) that (1 )(n ) c y (n r ) (r n ) (51) For any positive values of parameters, that satisfy the stability conditions (33), this partial derivative is negative. Hence the function c ( ) is downward sloping. The smaller the redistribution of national income through the public sector, the bigger the private g y 0 , hence the higher the level of taxation, consumption in the DE. On the other hand, the bigger the public consumption in the DE. Both of these observations are quite obvious. The interesting problem is: which of these effects is stronger? In other words, what is the relation of the total consumption in the DE to the level of taxation? The above derivatives can be combined into: (c g ) (r n ) ( ) y (n r ) (r n ) (52) If the DE is stable, the denominator of the above ratio is positive. Under this assumption, the sign of the partial derivative in (52) is determined by the signs of two expressions in the numerator. Four cases can be distinguished14: (a) If r n and 0 , then the derivative in (52) is positive, (b) If r n and 0 , then the derivative in (52) is negative, (c) If r n and 0 , then the derivative in (52) is negative, (d) If r n and 0 , then the derivative in (52) is positive. Each of these cases is realistic from the empirical point of view, and none contradicts the stability conditions (33).15 Therefore the level of taxation should be either as high, or as low as possible – depending on the actual characteristics of the economy (the values of the parameters) In practice, all four situations may turn out in various countries, in different periods etc., because of natural variability of parameters (the real interest rate r, the savings rate γ, the rate of investment ψ, and last but not least the natural rate of growth n ). Therefore, somewhat pessimistic conclusion: there is no simple, universal recipe regarding the level of taxation. 14 15 We neglect the very special cases, when either one, or both expressions in the numerator are equal to zero. This can be verified by numerical experiments. 14 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. If so, ( ) (n ) 0 . Hence in the real world, cases (b) and (d) will occur much more often than the remaining ones. The first of these two 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 public deficit, which was analyzed above) should be as low as possible, in order to maximize the total consumption per capita in equilibrium. The opposite is true in case (d). 8. The transitory dynamics The dynamics of the model is described by the system of no less than four differential equations. For that reason it’s not possible to illustrate the transition between two equilibria with a simple diagram, like in the model without government (see Kon (2004), section 9). We will only present some simulations illustrating the process of transition from one steady state to another. For that purpose it’s convenient to use the discrete version of the model [k] – [A]16: k r y k w A y [k’] [y] [w] [i] [t] [g] [b] [ bF ] [ bD ] [yd] [c] [s] [ x m] [q] [ik] 1 1 k 2 r 2 2 where n i k 2 (1 )k t y g t rb y b2 (1 n )b y bF 2 (1 n )bF y bD2 (1 n )bD (1 ) y y d y (r )e r (bD o) c (1 )( y d t ) s ( yd t) xm y ci g q x m (r )e ro rbF in ( y d t ) o2 o s in (1 ) y (n )o [e] e2 (1 )e in i [A] A2 (1 ) A Lets set the following parameter values (we will refer to this as to the base scenario). [o] 1 3 (i.e. 2 3 ), n 0,1% , 2% , 3% , r 5% , 35% , 1% 0,5 , 28% , 25% along with the following starting values (for t 1) : 16 For greater transparency, we neglect the time index t wherever it is equal to 1. 15 (53) e(1) 3,5 , b(1) 1,2 , bF (1) 0,5 , bD (1) 0,7 , o(1) 0,5 , A(1) 1 First of all, we need to establish the stability region. It is easy to verify that the base set of parameters satisfies the necessary and sufficient conditions (33). By modifying the parameters one-by-one we can find the upper and lower bounds of stability. For example, it turns out that the DE becomes unstable, when the real interest rate exceeds about 13,5% (keeping all other parameters at the base level). It is not entirely unimaginable, that the cost of capital goes up so high, but it’s very hard to find supportive empirical evidence. On the other hand, the stability conditions are violated, if the savings rate exceeds about 50,5%, which – from the point of view of empirical data – is a purely theoretical possibility. The equilibrium is unstable if the natural rate of growth ( n ) remains smaller than about 0,37%. Once again, the chance of this happening in reality is extremely low, since it would require both the very slow technological progress (or regress), and the minute growth of population (or shrinking population). Interestingly, all remaining parameters have neither lower nor upper stability bounds, i.e. they can take any reasonable values without putting stability at risk. It’s worth stressing that all fiscal parameters (τ, ξ, θ) have no influence on stability whatsoever – simply because none of them shows up in the necessary and sufficient conditions (33). Therefore, no matter how high is the level of taxation, or how large is the budget deficit, under a monetary union there is no danger of instability17. The graphs below illustrate the transition from one steady state to another, which occurs when the government increases it’s deficit ξ from 1% in the period starting at t 0 until t 100 , to 2% since t 101 . Fig. 1. The public debt (b), foreign bonds held by the private sector (o), private consumption (c), and public consumption (g) – trajectories of these variables in the transition caused by an increase in public deficit (ξ). 2 0,9 b o 1,5 c g 0,8 1 0,7 0,5 t 0 1 51 101 151 201 251 t 0,6 1 51 101 151 201 251 In the first 100 periods the economy is asymptotically converging towards the steady state corresponding to the base set of parameters. The public debt pel is slowly decreasing, whereas public expenditures pel are growing. Beginning at t 101 the government loosens up its fiscal policy – from that moment on public deficit jumps from 1% to 2% of GDP. The dynamic equilibrium shifts to a completely different position, and the economy begins the 17 Of course this proposition hinges on a simplifying assumption that the real interest rate is not responding to the fiscal stance of the government. 16 slow process of convergence towards this new steady state. Since government deficit becomes permanently twice as big as before, the stationary value of b increases twofold, which is apparent in (39). The public debt grows fast, while the ratio of foreign to domestic debt remains constant. Since the citizens have to lend more and more to government, they invest less and less into international bonds (o). However, this has no impact at all on the real income of the private sector, because the foregone interest earnings on foreign bonds are exactly compensated by an extra income from domestic bonds. Hence, importantly, the real disposable income of the private sector, and consequently its consumption, remains intact. (This is the illustration of the fact that it is not dependent on the level of public deficit – not only in DE, but also elsewhere.) The same cannot be said about public expenditures g. A loosening of fiscal policy brings a temporary increase in public consumption, but it vanishes with time. The reason is the fast growth of public debt, which devours more and more money to pay interest, and leaves less for public expenditures. As public debt swells, government expenditures (except for interest payments) shrink – soon they fall below the previous steady-state level, and slowly converge towards the new equilibrium level which is about 4,1% lower than previously. It’s worth stressing that the fiscal loosening has no impact whatsoever (not even a transitory one) on many important variables including: domestic output, domestic capital, national capital, domestic investment, public revenues, disposable income, consumption and savings of the private sector. To summarize, in our illustrative example, the results of a permanent increase in public deficit are dire. There is no boom in the economy (not even a temporary one), and the total consumption in equilibrium is lower than before, because of higher public debt. The above simulation is of course compliant with the “optimal fiscal policy” analysis presented in section 7. Our simulation is the illustration of case (a), with r n . The next several charts illustrate the transition process caused by a permanent increase in taxation τ (fig. 2 and 3). In the first 100 periods the economy is asymptotically attracted towards the initial equilibrium corresponding to the base set of parameters. The budget deficit is being steadily reduced, and both private and public consumption goes up (pel). In time t 101 the taxation rate τ is lifted from 35% to 45%. The economy starts to converge towards the new steady state. Higher taxation leaves unchanged several important variables: domestic investment and capital, and domestic output – see (34) – (36). Since private sector pays higher taxes, a smaller proportion of disposable income can be devoted to private consumption and savings, which are in turn used to finance national investment and purchase international bonds. Consequently, private holdings of these assets are shrinking, resulting in lower earnings. To sum up, higher taxation leads to a substantial reduction in private savings as well as consumption. In the new equilibrium private consumption is lower than in the old one by as much as 32%. A substantial increase in public expenditures (by about 30%) is not enough to compensate for this loss. Altogether, the total consumption is lower by about 4,5%.18 Last but not least, the change in τ has no impact (not even a transitory one) on the public debt, which is 18 This is the illustration of the ‘golden rule’ derived in section 6. In our example we have case (b), because r n and 0 . 17 a direct result of our assumptions, and can be clearly seen from (34), (35), and the laws of motion (22). Fig. 2. The public debt (b), public revenues (t), private consumption (c), and public consumption (g) – trajectories of these variables in the transition process caused by an increase in the tax rate ( τ) 1,4 b c 1 t 1,2 g 0,9 1 0,8 0,8 0,7 0,6 0,6 t 0,4 t 0,5 1 51 101 151 201 251 1 51 101 151 201 251 Fig. 3. The disposable income (yd), private savings (s), domestic capital (kn), and foreign bonds held by the private sector (o) – trajectories of these variables in the transition process caused by an increase in the tax rate ( τ) 2,5 2 yd 7 kk s 6 o 5 1,5 4 1 3 2 0,5 t 1 t 0 0 1 51 101 151 201 251 1 51 101 151 201 251 Summary Introducing explicite the government sector allows for some new conclusions, some of them significantly differing from those 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. They consist of three relatively complex inequalities, binding together virtually all exogenous parameters of the economy, except for fiscal parameters. This seems to be quite interesting, since it implies that the stability of an economy depends neither on the taxation rate, nor the deficit-to-GDP ratio. It’s relatively easy to come up with the set of parameters that violates the stability conditions. Nonetheless, we argued that any such set of parameters would be unrealistic – at least some parameters would diverge significantly from their real-world counterparts. For example, the stability conditions are violated if the real interest rate or the savings rate are 18 (persistently) very high – at levels almost never seen19. For this reason we argue that the realworld economies easily satisfy the stability conditions. Next, we have been looking for the “golden rules” for fiscal policy parameters. Strictly speaking, we’ve been searching for such parameter values that maximize the total (private plus public) consumption per capita in the dynamic equilibrium. The “golden rules” of fiscal policy have the form of the alternative – optimal policy crucially depends on the relation between the real interest rate and the natural growth rate and on the relation between five other independent parameters. For example, if r n , then public deficit and taxation should be as low as possible20. However, if r n , then the opposite is true. Therefore, the “golden rules” for fiscal policy are hardly applicable in reality, since they recommend “switching” between completely different strategies. Last but not least, we have demonstrated that in the small open economy in an economic and monetary union it is completely indifferent from the point of view of the steady-state wealth, whether the public debt is financed by domestic or by foreign investors. The model presented in our paper has some strengths (as well as severe weaknesses). First, it is based on very simple and general assumptions, hence it is very easy to analytically describe the state of equilibrium and investigate its properties. It’s worth stressing, though, that even in our very simple setting, the stability conditions are non-trivial and quite complex. Second, the dynamic equilibrium is globally asymptotically stable (for a very wide range of realistic values of parameters), which enables straightforward application of numerical methods for dynamic simulations of transitory processes. For these two reasons, the model is perfectly suited for educational purposes. It can also serve as a starting point for more complex, applicable models. Now, lets turn to the weaknesses. First, some assumptions are certainly disputable, for example the assumption about perfect mobility of physical capital within a monetary union, which enables perfect (in theory infinitely fast) adjustment of the stock of physical capital to the level desired from the point of view of profit maximization.21 Moreover, equations [k’] and [i] imply that in the discrete version of the model firms know the real interest rate (i.e. both nominal interest rate, and inflation) one period ahead. This is certainly doubtful.22 Apart from these specific features, the model (and the entire group of the neoclassical growth models) can obviously be criticized for extremely strong assumptions (axioms) regarding permanent clearing of all markets. Specifically, it is very unrealistic to assume perfect (longrun) equilibrium in the markets for labor and capital. 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