Download 135 Fiscal policy in a monetary union from the perspective of the

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
bF   y  (n   )bF
bD  (1   ) y  (n   )bD
y d  y  (r   )e  r (bK  o)
c  (1   )( y d  t )
s   ( yd  t)
yci g  xm
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
bF   y  (n   )bF
bD  (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)
bD  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  w13  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)
xm  y ci  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. Therefore, one should treat this model as
an idealistic picture of the economy, that provides only a very simplified, theoretic vision of
how an economy may evolve in the long run under a monetary union.
19
In our calibrations, they have to exceed 11% and 50%, respectively.
Under certain additional assumption – see section 6.
21
This weakness has been examined in detail in Kon (2004), p. 40.
22
See Kon (2004), footnote 23, p. 40.
20
19
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20