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The tight money paradox—an alternative view
NISSAN LMATAN The
Hebrew
Linioersity
of Jerusalem
The Tight Money Paradox-An Alternative View The Tight Money Paradox refers to a result obtained by Sargent and Wallace that a reduction in the constant rate of monetary expansion may lead to inflation both in the short and long runs. This result was obtained for a which is dynamically unstable. In the present paper I show that the foregoing can be obtained within the framework of a stable system if we redefine “tight to mean a reduction in the share of money &mncing of the budget deficit,
stating higher system result money”
1. Introduction In a recent, much discussed, paper Sargent and Wallace [(SW), 19811 show that a tightening of monetary policy may lead to an increase in the level of inflation. This paradoxical result is based on the assumption that while the restrictive monetary policy is undertaken the government’s budget deficit remains constant in real terms. Moreover, the concept of the budget deficit is one which does not include (on the expenditure side) interest on government bonds. In a subsequent paper, Liviatan (1984), I showed how the SW result can be derived within the framework of a simple Sidrauski-Brock’ type of model. The way in which SW define restrictive monetary policy, or “tight money,” seems to be of a rather specific nature. They define tight money as a reduction in the constant rate of expansion of nominal money over a specified time interval (Stage I). A conceptual diEiculty which arises in the context of this approach is that it is based in Stage I on a dynamically unstable system which is stabilized eventually by a freeze of the stock of government bonds. At the point of the freeze, the system finds itself in a new steady state with a larger stock of bonds, a smaller stock of money and a permanently higher rate of inflation. In the present paper I will show that the dynamic instability is not essential for obtaining the “paradoxical” result if we use a different (but not uncommon) definition of tight money. Instead of ‘Sidrauski
Journal Copyright
(1967),
Brock
of Macroeconomics, 0
1986
by Wayne
(1974).
Winter State
1986, Vol. 8, No. University Press.
1, pp.
105-112
105
Nissan Liviatan defining tight money as a reduction in the rate of growth of nominal money I define it as a reduction of the share of money-financing of the government deficit over a specified time interval. We will show that with the latter definition we can reproduce the SW “paradoxical result” within the framework of a SidrauskiBrock model which is stable in the saddle point sense. In the modified model a temporary tightening of money will lead to an immediate, but temporary, increase in inflation while a permanent tightening will lead to an immediate and permanent increase in inflation.’
2. The Model The model is reproduced from Liviatan (1984). It is based on a representative individual with an infinite planning horizon, of the Sidrauski-Brock type. The utility function is logarithmic and additive in consumption and real balances. From the Euler conditions we obtain f2 -=p-$;p=r-n c
EC PI 0m
(1)
= i ; i = p + n + T = nominal interest rate ,
(2)
where r is real interest, n population growth, 7~ inflation rate, 6 time preference and pi are positive parameters. All variables are per capita. The demand functions (for constant p, real income y and transfers s) are given by c=i$w;m=-w;6,>0,
i m
61 + 82 = 1 .
e-P(“-f)(y + s) do = total wealth It a = m + I? = financial assets, where b denotes government bonds.
(3)
w(t) = a(t) +
“This refers to the case where the government’s ments (as in SW). It can be shown that the results includes interest payments.
106
indexed
deficit is net of interest are reversed when the
paydeficit
The Tight We define
the government
deficit
Money
(net of interest
Paradox payments)
as D=g+s=ni+(n+TT)m+1;-pb
(4)
where g denotes real government expenditures. We shall assume throughout that g is constant so that a constant D implies that s is constant. Note that by the national income identity (with y constant) y=c+g
(5)
so that a constant g implies that c is constant. By (4) this implies that d = 0 and hence p = 6. A steady state equilibrium is characterized by C = ni = & = 0. This equilibrium is represented by the system
(
y-g=&
m =
m+b+-
&i p+n+n
s+Y goods market
P 1
m+b+-
equilibrium
,
St-Y
P
money
market
equilibrium
, (7)
government
g + s = m(n + z) - pb
budget
equation
.
It can be shown that this system contains only two independent equations. The two independent equations can determine the values of IT and m if we treat b as given parameter or may determine n, nt and b if we add a policy equation relating b to m.
3. The Concept
of Tight
Money
The ratio of money finance to bond finance of the government deficit is given by the positive parameter y: 107
Nissan Liviatan
Y=
ni + (n + n)m
(9)
d + nb
By tightening monetary policy we mean a reduction in y over a specified time interval [0, T]. At time T, y returns to its original level. Denoting
the rate of monetary
expansion by p,
we have
ti + (n + n)m = pm. Hence
Note that in our formulation t.~ is endogenous.
4. The Dynamic
of tight
money
y is exogenous
while
System
Using (4) and (9) we obtain I;=--.-
D
(11)
l+Y
Treating D as fixed (11) provides a linear dynamic equation for b with fixed coefficients. It may be noted that this equation is independent of m. Since b is a state variable it can be seen that stability requires that the coefficient of b should be negative, i.e., y > p/n = E - 1. n Thus stability requires that the share of money finance should not be too small. It follows from (11) that this condition is necessarily satisfied if D and b are positive in the steady state equilibrium. Using (2), (4) and (9) we obtain ni=pm+-
” l+Y
b-&-+YD, Pl
where both D and c are fixed during the dynamic that c is fixed because y and g are fmed.) 108
(12)
l+Y process.
(Recall
The Tight Expressing the foregoing equations steady state equilibrium we have
Money
Paradox
in terms of deviation
from
d = U,l(b - b*) )
03)
ni = azl(b- b*) + a&n - m*) ,
(14)
vP - yn = and a22 = p > 0 where a star all = ~ 1 + y ’ ozl l+Y denotes steady state equilibrium values. The condition for the steady state to be a saddle point is that a11uZ2 < 0 and hence that a,, < 0 or y > p/n. We have shown already that this condition is satisfied under our assumptions. The phase diagram corresponding to the foregoing system is described in Figure 1, where the negative slope of the lit = 0 line utilizes the condition alI < 0. It is important to note that the slope of the saddle path QQ is equal to minus one. To see this note that since c is constant we must also have a constant w, by (3). However S+Y with s, y and p constant. Since (3) must be w=m+b+P satisfied along the saddle path it follows that (m + b) must be constant along this path, which proves that the slope of QQ is indeed minus one.
with
5. The Effect of Tight
Money
Let us consider first the effect of a permanent across steady states. Using (4) we have D -=,,I-;, nb
reduction
in y
(15)
which shows that a reduction in y must increase the steady state value of b, since D is a positive constant. Thus a reduction in y shifts the steady state equilibrium from EO to El in Figure 2 where m is reduced and m + b remains constant, since w remains constant. It then follows from (7) that a reduction in y must increase a. Thus, across steady states tighter money leads to higher intlation. If we consider a permanent reduction in y we know that the 109
Nissan Liviatan Figure
1.
m’
0
b
long-run equilibrium point will shift from E,, to El along the original saddle path QQ which h as a slope of minus one. The new saddle path is still along QQ in Figure 2 with E,, being an “initial condition” and El the new long-run equilibrium. The reduction of y will then lead to a dynamic adjustment along the segment EoEI. In view of (2) this implies that at the moment when y is reduced II remains momentary constant but then rises gradually towards its new steady state level at El. We may therefore state that the tight money “paradox” holds in the present model for T 4 to. Let us turn now to a temporary reduction of y over the interval [0, T]. It is clear that over this interval the economy must proceed along one of the trajectories which correspond to the system with a saddle point at El. However, we know that since wealth must remain constant (so as to keep c constant), we must also have m + b = constant. It also follows from the same considerations that m(t) cannot change with a jump at any point along the equilibrium 110
The Tight Money Figure
Paradox
2.
m
path. omy of T T-t
We may therefore infer that over the interval [0, T] the econmust move from E0 to ET along QQ. The larger is the value the closer will ET be to El. In fact El is the limit of ET as ~0. The foregoing analysis corresponds to phase I. Note that along this phase m is constantly declining which implies in view of (9, that IT is constantly increasing. This corresponds to the “paradoxical result” of SW. Phase II, where y returns to its original level, starts with the economy at ET as the initial condition and EO as the saddle point. The economy will then move along QQ from ET to E,,. Since m increases along this path we have a gradual reduction of IT to its original level at EO. Note however along the entire adjustment path a is higher than at the original steady state. We have thus shown that the paradoxical result of SW can be reproduced in a stable dynamic system with “tight money” defined appropriately. 111
Nissan Liuiatan Conclusion In this paper we define tight money as a reduction in the ratio (y) of money finance to debt finance of the budget deficit, where the latter is net of interest payments. It is shown that under the tight money phase inflation rises gradually while after the return to the original y inflation decreases gradually to its orginal value. In the present model the system is dynamically stable, which implies that the time span of the tight money regime can be of any length. Stability also implies that a temporary tight money policy will increase inflation only temporarily. In these aspects the conclusions of the present model differ from those of SW, where tight money is defined as a reduction in the constant rate of monetary growth. Received: January 1985 Final version: August 1985
References Brock, William A. “Money and Growth: The Case of Long Run Perfect Foresight. ” International Economic Review 15 (1974): 75077. Liviatan, Nissan. “Tight Money and Inflation.” Journal of Monetary Economics 13 (1984): 5-15. Sargent, Thomas J. and Neil Wallace. “Some Unpleasant Monetarist Arithmetic.” Federal Reserve Bank of Minneapolis Quarterly Reuiew, (1981): 1-17. Sidrauski, Miguel. “Rational Choice and Patterns of Growth in a Monetary Economy.” American Economic Review 57 (1967): 53444.
112
Hebrew
Linioersity
of Jerusalem
The Tight Money Paradox-An Alternative View The Tight Money Paradox refers to a result obtained by Sargent and Wallace that a reduction in the constant rate of monetary expansion may lead to inflation both in the short and long runs. This result was obtained for a which is dynamically unstable. In the present paper I show that the foregoing can be obtained within the framework of a stable system if we redefine “tight to mean a reduction in the share of money &mncing of the budget deficit,
stating higher system result money”
1. Introduction In a recent, much discussed, paper Sargent and Wallace [(SW), 19811 show that a tightening of monetary policy may lead to an increase in the level of inflation. This paradoxical result is based on the assumption that while the restrictive monetary policy is undertaken the government’s budget deficit remains constant in real terms. Moreover, the concept of the budget deficit is one which does not include (on the expenditure side) interest on government bonds. In a subsequent paper, Liviatan (1984), I showed how the SW result can be derived within the framework of a simple Sidrauski-Brock’ type of model. The way in which SW define restrictive monetary policy, or “tight money,” seems to be of a rather specific nature. They define tight money as a reduction in the constant rate of expansion of nominal money over a specified time interval (Stage I). A conceptual diEiculty which arises in the context of this approach is that it is based in Stage I on a dynamically unstable system which is stabilized eventually by a freeze of the stock of government bonds. At the point of the freeze, the system finds itself in a new steady state with a larger stock of bonds, a smaller stock of money and a permanently higher rate of inflation. In the present paper I will show that the dynamic instability is not essential for obtaining the “paradoxical” result if we use a different (but not uncommon) definition of tight money. Instead of ‘Sidrauski
Journal Copyright
(1967),
Brock
of Macroeconomics, 0
1986
by Wayne
(1974).
Winter State
1986, Vol. 8, No. University Press.
1, pp.
105-112
105
Nissan Liviatan defining tight money as a reduction in the rate of growth of nominal money I define it as a reduction of the share of money-financing of the government deficit over a specified time interval. We will show that with the latter definition we can reproduce the SW “paradoxical result” within the framework of a SidrauskiBrock model which is stable in the saddle point sense. In the modified model a temporary tightening of money will lead to an immediate, but temporary, increase in inflation while a permanent tightening will lead to an immediate and permanent increase in inflation.’
2. The Model The model is reproduced from Liviatan (1984). It is based on a representative individual with an infinite planning horizon, of the Sidrauski-Brock type. The utility function is logarithmic and additive in consumption and real balances. From the Euler conditions we obtain f2 -=p-$;p=r-n c
EC PI 0m
(1)
= i ; i = p + n + T = nominal interest rate ,
(2)
where r is real interest, n population growth, 7~ inflation rate, 6 time preference and pi are positive parameters. All variables are per capita. The demand functions (for constant p, real income y and transfers s) are given by c=i$w;m=-w;6,>0,
i m
61 + 82 = 1 .
e-P(“-f)(y + s) do = total wealth It a = m + I? = financial assets, where b denotes government bonds.
(3)
w(t) = a(t) +
“This refers to the case where the government’s ments (as in SW). It can be shown that the results includes interest payments.
106
indexed
deficit is net of interest are reversed when the
paydeficit
The Tight We define
the government
deficit
Money
(net of interest
Paradox payments)
as D=g+s=ni+(n+TT)m+1;-pb
(4)
where g denotes real government expenditures. We shall assume throughout that g is constant so that a constant D implies that s is constant. Note that by the national income identity (with y constant) y=c+g
(5)
so that a constant g implies that c is constant. By (4) this implies that d = 0 and hence p = 6. A steady state equilibrium is characterized by C = ni = & = 0. This equilibrium is represented by the system
(
y-g=&
m =
m+b+-
&i p+n+n
s+Y goods market
P 1
m+b+-
equilibrium
,
St-Y
P
money
market
equilibrium
, (7)
government
g + s = m(n + z) - pb
budget
equation
.
It can be shown that this system contains only two independent equations. The two independent equations can determine the values of IT and m if we treat b as given parameter or may determine n, nt and b if we add a policy equation relating b to m.
3. The Concept
of Tight
Money
The ratio of money finance to bond finance of the government deficit is given by the positive parameter y: 107
Nissan Liviatan
Y=
ni + (n + n)m
(9)
d + nb
By tightening monetary policy we mean a reduction in y over a specified time interval [0, T]. At time T, y returns to its original level. Denoting
the rate of monetary
expansion by p,
we have
ti + (n + n)m = pm. Hence
Note that in our formulation t.~ is endogenous.
4. The Dynamic
of tight
money
y is exogenous
while
System
Using (4) and (9) we obtain I;=--.-
D
(11)
l+Y
Treating D as fixed (11) provides a linear dynamic equation for b with fixed coefficients. It may be noted that this equation is independent of m. Since b is a state variable it can be seen that stability requires that the coefficient of b should be negative, i.e., y > p/n = E - 1. n Thus stability requires that the share of money finance should not be too small. It follows from (11) that this condition is necessarily satisfied if D and b are positive in the steady state equilibrium. Using (2), (4) and (9) we obtain ni=pm+-
” l+Y
b-&-+YD, Pl
where both D and c are fixed during the dynamic that c is fixed because y and g are fmed.) 108
(12)
l+Y process.
(Recall
The Tight Expressing the foregoing equations steady state equilibrium we have
Money
Paradox
in terms of deviation
from
d = U,l(b - b*) )
03)
ni = azl(b- b*) + a&n - m*) ,
(14)
vP - yn = and a22 = p > 0 where a star all = ~ 1 + y ’ ozl l+Y denotes steady state equilibrium values. The condition for the steady state to be a saddle point is that a11uZ2 < 0 and hence that a,, < 0 or y > p/n. We have shown already that this condition is satisfied under our assumptions. The phase diagram corresponding to the foregoing system is described in Figure 1, where the negative slope of the lit = 0 line utilizes the condition alI < 0. It is important to note that the slope of the saddle path QQ is equal to minus one. To see this note that since c is constant we must also have a constant w, by (3). However S+Y with s, y and p constant. Since (3) must be w=m+b+P satisfied along the saddle path it follows that (m + b) must be constant along this path, which proves that the slope of QQ is indeed minus one.
with
5. The Effect of Tight
Money
Let us consider first the effect of a permanent across steady states. Using (4) we have D -=,,I-;, nb
reduction
in y
(15)
which shows that a reduction in y must increase the steady state value of b, since D is a positive constant. Thus a reduction in y shifts the steady state equilibrium from EO to El in Figure 2 where m is reduced and m + b remains constant, since w remains constant. It then follows from (7) that a reduction in y must increase a. Thus, across steady states tighter money leads to higher intlation. If we consider a permanent reduction in y we know that the 109
Nissan Liviatan Figure
1.
m’
0
b
long-run equilibrium point will shift from E,, to El along the original saddle path QQ which h as a slope of minus one. The new saddle path is still along QQ in Figure 2 with E,, being an “initial condition” and El the new long-run equilibrium. The reduction of y will then lead to a dynamic adjustment along the segment EoEI. In view of (2) this implies that at the moment when y is reduced II remains momentary constant but then rises gradually towards its new steady state level at El. We may therefore state that the tight money “paradox” holds in the present model for T 4 to. Let us turn now to a temporary reduction of y over the interval [0, T]. It is clear that over this interval the economy must proceed along one of the trajectories which correspond to the system with a saddle point at El. However, we know that since wealth must remain constant (so as to keep c constant), we must also have m + b = constant. It also follows from the same considerations that m(t) cannot change with a jump at any point along the equilibrium 110
The Tight Money Figure
Paradox
2.
m
path. omy of T T-t
We may therefore infer that over the interval [0, T] the econmust move from E0 to ET along QQ. The larger is the value the closer will ET be to El. In fact El is the limit of ET as ~0. The foregoing analysis corresponds to phase I. Note that along this phase m is constantly declining which implies in view of (9, that IT is constantly increasing. This corresponds to the “paradoxical result” of SW. Phase II, where y returns to its original level, starts with the economy at ET as the initial condition and EO as the saddle point. The economy will then move along QQ from ET to E,,. Since m increases along this path we have a gradual reduction of IT to its original level at EO. Note however along the entire adjustment path a is higher than at the original steady state. We have thus shown that the paradoxical result of SW can be reproduced in a stable dynamic system with “tight money” defined appropriately. 111
Nissan Liuiatan Conclusion In this paper we define tight money as a reduction in the ratio (y) of money finance to debt finance of the budget deficit, where the latter is net of interest payments. It is shown that under the tight money phase inflation rises gradually while after the return to the original y inflation decreases gradually to its orginal value. In the present model the system is dynamically stable, which implies that the time span of the tight money regime can be of any length. Stability also implies that a temporary tight money policy will increase inflation only temporarily. In these aspects the conclusions of the present model differ from those of SW, where tight money is defined as a reduction in the constant rate of monetary growth. Received: January 1985 Final version: August 1985
References Brock, William A. “Money and Growth: The Case of Long Run Perfect Foresight. ” International Economic Review 15 (1974): 75077. Liviatan, Nissan. “Tight Money and Inflation.” Journal of Monetary Economics 13 (1984): 5-15. Sargent, Thomas J. and Neil Wallace. “Some Unpleasant Monetarist Arithmetic.” Federal Reserve Bank of Minneapolis Quarterly Reuiew, (1981): 1-17. Sidrauski, Miguel. “Rational Choice and Patterns of Growth in a Monetary Economy.” American Economic Review 57 (1967): 53444.
112