148
Coordinated Fiscal and Monetary Stabilization Policy
in the Manner of MMT
:
A Study by Means of
Dynamic Keynesian Model*
Toichiro Asada
Abstract
In this paper, we analyze the macroeconomic effects of the coordinated fiscal and monetary stabilization policy in the manner of MMT by using a dynamic Keynesian model that has been developed by ourselves and others. The MMT (Modern Monetary Theory or Modern Money Theory), which was proposed by L. R. Wray, W. Mitchell, M. Watts and others, has the following remarkable characteristics. (1) The role of the central bank is completely passive in the financing of government expenditure, which is actively determined by the government. (2) The amount of government debt is not subject to the constraint of fiscal policy, its only constraint being the rate of inflation. In this paper, we construct a mathematical model that incorporates these characteristics and study the dynamic stability/instability of the coordinated fiscal and monetary stabilization policy.
Keywords: Coordinated Fiscal and Monetary Stabilization Policy; Modern Monetary Theory (MMT); Dynamic Keynesian Model; Dynamic Stability
JEL Classification Number: E12; E31; E32; E52; E62
* [Acknowledgments] This research was financially supported by a Chuo University Personal Research Grant.
149
Ⅰ. Introduction
Recently, MMT (Modern Monetary Theory or Modern Money Theory) – a somewhat heterodox policy-oriented macroeconomic theory – has been developed by some economists belonging to a sub-group of the Post-Keynesian School.1 The proponents of
MMT, like L. R. Wray, W. Mitchell and M. Watts, acknowledge that their theory is heretical. In fact, their attitude toward the ‘mainstream’ economists, which include the “orthodox” Keynesians as well as “neoclassical” economists, is quite hostile. The proponents of MMT stress that the limit to the amount of government debt is not the constraint of fiscal policy, its only constraint being the rate of inflation, and that the purpose of the fiscal policy should be achieving full employment and price stability rather than “soundness” of the public finance. This way of thinking is greatly influenced by the idea in Lerner (1943) of “functional finance” as well as the theory of effective demand in Keynes (1936).2 Furthermore, the MMT literature presupposes that the role of the central bank is completely passive in the financing of government expenditure, which is actively determined by the government. In other words, MMT does not allow for a central bank’s monetary policy that is independent of the government’s fiscal policy.
1 One of the most active proponents of this line is L. R. Wray. See, for example, Wray (1998, 2005) and
Mitchell, Wray, and Watts (2019).
2 Exemplary are a couple of passages from Mitchell, Wray, and Watts (2019) and Lerner (1943).
Incidentally, we have no objection to the assertions made here.
“The most important conclusion reached by MMT is that the issuer of a currency faces no financial constraints. Put simply, a country that issues its own currency can never run out and can never become insolvent in its own currency. It can make all payments as they come due. For this reason, it makes no sense to compare a sovereign government’s finances with those of a household or a firm.” (Mitchell, Wray, and Watts 2019, p.13)
“The central idea is that government fiscal policy, its spending and taxing, its borrowing and repayment of loans, its issue of new money and its withdrawal of money, shall all be undertaken with an eye only to the results of these actions on the economy and not to any established traditional doctrine about what is sound or unsound. This principle of judging only by effects has been applied in many other fields of human activity, where it is known as the method of science as opposed scholasticism. The principle of judging fiscal measures by the way they work or function in the economy we may call Functional Finance.” (Lerner 1943, p.39, author’s italics)
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This is a typical example of “fiscal dominance”. In contrast to the “mainstream” economists, however, the proponents of MMT do not see “fiscal dominance” as an evil; they see it as the rule rather than the exception.
The MMT propositions are particularly challenging to the preconceptions on macroeconomic issues held by many “mainstream” economists as well as ordinary people. It seems, however, that so far there are rather few formal macrodynamic models that can verify the validity of the main MMT propositions. The purpose of this paper is to provide such a model, albeit a very tentative one. In this paper, we analyze the macroeconomic effects of the coordinated fiscal and monetary stabilization policy with an MMT approach, using a dynamic Keynesian model developed by ourselves and others. We construct a mathematical model that has the characteristics of MMT and study the dynamic stability/instability of the coordinated fiscal and monetary stabilization policy. The paper is organized as follows. In section Ⅱ, the full system of equations is presented in order. In section Ⅲ, the reduced form of the fundamental dynamic equations, which is a four-dimensional system of nonlinear differential equations, is presented. In section Ⅳ, the characteristics of the long-run equilibrium are investigated. In section Ⅴ, the local stability/instability of the long-run equilibrium point and the existence of the cyclical fluctuations are studied analytically. In Section Ⅵ, we offer an economic interpretation of the analytical results obtained in section Ⅴ. Section Ⅶ presents our concluding remarks. Detailed mathematical formulae and proofs are contained in the appendices.
Ⅱ. Formulation of the Model
The main symbols that are used throughout this paper are as follows, a dot over the symbol denoting the derivative with respect to time.
Y
real national income (real output).K
real capital stock. y Y K / output-capital ratio, which is a surrogate variable of the “rate of capital utilization”.3
3 In this paper we assume that the output-capital ratio at full capital utilization
( )
f
y
is fixed, but the actual output-capital ratio ( )y is a variable rather than constant, determined by the relationship;
f151
B
nominal stock of public debt (public bond).p
price level. wnominal wage rate./ p p
rate of price inflation. e expected rate of price inflation.
/ ( )
bB pK
public debt-capital ratio.
M
nominal money supply.
M M
/
growth rate of nominal money supply. m M / (pK)money-capital ratio.H
high-powered money./
v M H money multiplier, which is assumed to be a constant that is larger than 1.4
I
real private investment expenditure.5 i I K / rate of investment. G real government expenditure. g G K / government expenditure-capital ratio. K I real capital accumulation.C
w
workers’ real consumption expenditure.C
r
capitalists’ real consumption expenditure.C C
w
C
r total real consumption expenditure.
nominal rate of interest of public bonds).
e
expected real rate of interest of public bonds. W pre-tax real wage income.P
pre-tax real profit. r P K / pre-tax rate of profit.
T
w
real income tax on workers.T
r
real income tax on capitalists.t
w
T K
w/ .
t T K
r
r/ .
N labor employment. Ns labor supply./ s
e N N rate of employment
1
rate of unemployment.n
1
N N
s/
s
growth rate of labor supply
constant>0. a Y N / average labor productivity.n
2
a a
/
growth rate of average labor productivity (rate of technical progress)
constant≧0. 6
in this model through the effective demand condition.
4 Needless to say, in reality, v is a variable rather than constant. In particular,vis likely to be an
increasing function of the nominal rate of interest. In this paper, however, we do not consider this type of complication, because the dynamic behavior of the present model is not significantly influenced by this type of complication.
5 For the sake of simplicity, we disregard capital depreciation, so there is no distinction between gross
investment and net investment.
6
This type of technical progress is called “Harrodian neutral” exogenous technical progress. Here we do not treat the problem of “endogenous” technical progress, which is beyond the scope of this paper. Barro
152
1 2
n n n
natural rate of growth or “potential” rate of growth, which is assumed to be a positive constant.We shall now go on to present the building blocks of the model use in this paper in order.
1. Equilibrium condition of the goods market and the principle of effective demand The equilibrium condition of the goods market, i.e. the IS equation, becomes as follows.7
Y C I G C
w
C I G
r (1)As for the consumption expenditures of workers and capitalists, we adopt the following Kalecki (1971) postulate of the two-class economy, that is, the workers consume all of their disposable income, and the capitalists save a part of their disposable income.8
C
w
W T
wY P T
w (2) Cr (1 sr){P ( )B Tr} p (3)where
s
r is the capitalists’ propensity to save, which is assumed to be a constant such that 0<s
r<1.As for the firms’ investment expenditure, we adopt the following standard Keynesian investment function, which means that the rate of investment is the decreasing function of the expected real rate of interest (see, for example, Keynes (1936) and Asada and Ouchi (2015)).
and Sala-i Martin (2004) explains various versions of the neoclassical models of endogenous technical progress. On the other hand, Asada and Ouchi (2015) provides a version of the Keynesian model of endogenous technical progress that is based on the idea of the “technical progress function” in Kaldor (1957), implying that firms’ investment expenditure brings about technical progress.
7 In this paper foreign trade is disregarded for the sake of simplicity. 8
In equation (3), it is assumed that the capitalists own public bonds, and we disregard repayment of the public bond principal.
153
i I K i
/
(
e)
; 0 ( ) e e di i d (4)In this case, we can write the rate of capital accumulation
( / )
K K
as follows.
K K I K i
/
/
(
e)
(5)Dividing both sides of Equation (1) by
K
and substituting Equations (2), (3) and (4) into it, we obtain the following equation, which means that the rate of profit ( )r is determined by the principle of effective demand in this model.9
r
(1/ ){ (
s i
r
e)
g
(1
s
r)
b t
w(1
s t
r) }
r (6)Following the procedure of Chiarella and Flaschel (2000), Asada, Chiarella, Flaschel and Franke (2003, 2010), and Asada (2006), we adopt the simplifying assumption that
t
w andt
r are positive and constant. In reality, they will depend on the incomes of workers and capitalists respectively, but this type of complication will not alter the dynamic behavior of the model significantly.2. Equilibrium condition of the money market
Next, we will formulate the equilibrium condition of the money market, which is called the LM equation, as follows: the left hand side is the nominal money supply and the right hand side is the standard Keynesian nominal money demand function.
1
(
0)
2M h pY
h pK
;h
1
0,
h
2
0,
0
0
(7)In this formulation,
0 is the lower bound of the nominal rate of interest of the government bond.10 Dividing both sides of this equation by pK and solving with respect to
,
we obtain the following expression.
9 In fact, this is a variant of the formula for determination of the rate of profit in Kalecki (1971). 10
This type of money demand function was introduced by Chiarella and Flaschel (2000), Asada, Chiarella, Flaschel and Franke (2003, 2011), and Asada (2006).
154 0 1 2 h y m h
(8)Equation (8) is effective, however, only in the case of
h y m
1
0.
A more accurate form of this “LM equation” under the lower bound constraint of
becomes as follows.1 0 1 2 0 1 0 ( , ) 0 h y m if h y m h y m if h y m
(9)The case of
h y m
1
0
corresponds to the case of the ‘liquidity trap’, where the nominal rate of interest of the government bond is stuck to its lower bound. From Eq. (9), we have
y
/
y
0,
m
/
m
0.
Incidentally, differentiating the definitional equation of the money-capital ratio
/ ( )
m M pK with respect to time and substituting Eq. (5) into it, we have the following dynamic law of the motion of the money-capital ratio.
m M p K i( e)
m M p K
(10)
3. Dynamics of employment, wages and prices
In this subsection, we consider the dynamics of employment, wages and prices. By definition, we can write the labor employment as follows:
( / ) / Y K K yK N Y N a (11)
Dividing both sides of this equation by
N
s,
we have the following formula which determines the rate of employment( ).ee Ns yKs
N aN
; 0 e 1
(12)
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motion of the rate of employment.
( ) ( 1 2) ( ) s e e s e y K N a y y i n n i n e y K N a y
y
(13)As for the dynamic law of motion of money wages, we adopt the following standard type of the expectation-augmented wage Phillips curve, where is a positive parameter that represents the speed of wage adjustment and
e
is the “natural” rate of employment (1 – natural rate of unemployment), which is assumed to be exogenously given(0 e 1).11
w (e e) n2 e
w
(14)
As for the firms’ pricing behavior, we assume the following postulate of the mark-up pricing of the imperfect competitive economy as in Kalecki (1971), where z is the mark-up parameter that reflects the ‘degree of monopoly’ of the economy (see also Asada (2006), Asada (2014), Asada and Ouchi (2015), and Asada, Demetrian and Zimka (2019)).
p z(wN) zw
Y a
;
z
1
(15)The share of profit in national income ( ) is then determined by the following equation.
P Y W 1 ( ) 1 {W ( / )w p N} 1 ( )1
Y Y Y Y z
; 0 1 (16)The parameter is the increasing function of the mark up z. In this case, we have
r P Y y.
K K
(17)
This means that the rate of profit is proportional to the rate of capital utilization. Substituting Equation (9) and (17) into Equation (6), we have the following expression.
( 1 ){ ( ( , ) e) (1 ) ( , ) (1 ) } r w r r r y i y m g s y m b t s t s (18)
Solving this equation with respect to y, we obtain the following equation, which determines the rate of capital utilization by means of the principle of effective demand.
11
This type of formulation was also adopted by Asada (2006), Asada (2014), Asada, Demetrian and Zimka (2019), and Asada and Ouchi (2015).
156
y y g m b
( , , , )
e (19)Let us assume that the sensitivity of the investment expenditure with respect to changes in the expected real rate of interest is so great that the following inequality is satisfied.
(1 )
e r
i s b (20)
In this case, we have the following results concerning the partial derivatives of Equation (19) with respect to relevant variables.
( 0) ( ) / 1/ [ { e (1 ) } ] 0 g r r y or y y g s i s b (21) ( 0) ( 0) ( ) ( ) / [{ e (1 ) } ] / [ { e (1 ) } ] 0 m r m r r y or or y y m i s b s i s b (22) ( 0) ( ) / [(1 ) ] / [ { e (1 ) } ] 0 b r r r y or y y b s s i s b (23) ( 0) ( ) ( ) / [ ] / [ { (1 ) } ] 0 e e e e r r y or y y
i
s i s b
(24)In the special case of the ‘liquidity trap’ such that
0,
we haveyg 1/ (
sr) 1,y
m
0,
y
b
(1
s
r) / (
0s
r) 0,
( ) / ( ) 0. e e r y i s (25)We can derive the dynamic of the price level as follows. Differentiating Equation (15) with respect to time, we have
p w a w n2.
p w a w
(26)
Substituting Eq. (14) into Equation (26), we obtain the following expectation-augmented price Phillips curve.
(
e e
)
e ; 0 (27)4. Formulation of the coordinated fiscal and monetary stabilization policy
We are now in a position to apply a thoroughgoing MMT approach to formal treatment of the coordinated fiscal and monetary policy. The first important relationship for formal
157
treatment of the problem is the following equation.
H B pG
B pT pG
B p T ( wTr) (28)This equation implies that the government deficit must be financed through the issue of new high-powered money by the central bank or new government bonds, and usually it is called the “budget constraint” of the “consolidated government” including the central bank. The proponents of MMT also consider this equation, although they refuse to call it the “budget constraint” of the government since theoretically a central bank can issue high-powered money indefinitely, so there is no “constraint” in this equation.12 In this paper, we simply assume that the money supply is connected to the high-powered money through the relationship
M vH ; v1 (29)
where v is the money multiplier that is assumed to be a constant.13 Differentiating Equation (29) with respect time and substituting it into Equation (28), we have the following relationship.
(1/ )v M B G
B p T ( wTr) (30)Dividing both sides of this equation by pK, we obtain the following equation, where
/ . B B B
(1/ )
v m
Bb g
( , )
y m b t
(
w
t
r)
(31)Let us assume that the government controls the issue of public debt so as to keep the following condition.
b B b
pK
constant>0 (32)
This postulate means that
( ( , ) e) 0. B b B p K i y m b B p K (33)
Substituting equations (32) and (33) into Equation (31), we obtain the following expression.
12 See, for example, Wray (2015) and Mitchell, Wray, and Watts (2019). Their argument is greatly
influenced by Lerner (1943), with the idea of the “functional finance”.
158
m v g [ { ( , )
y m
i( ( , )
y m
e)}b (twtr)] (34)In this equation, the growth rate of money supply ( ) is an endogenous variable that
depends on the value of the government expenditure-capital ratio ( ),g which is determined by the government’s fiscal policy. This is a mathematical formalization of the MMT postulate on money financing of government expenditure. That is, the central bank’s monetary policy is completely passive in financing the independently determined government expenditure.14
Under the influence of the theory of “functional finance” in Lerner (1943), the proponents of MMT presuppose that the purpose of the government’s fiscal policy is attainment of “full employment and price stability” (see, for example, Wray (1998, 2015), and Mitchell, Wray, and Watts (2019)).15 We can formalize this type of fiscal policy as follows.
g
1(
e e
)
2(
)
;
1
0,
2
0,
0 (35)where is the target rate of inflation set by the government, which is assumed to be fixed at some positive value – for example, 2% per year. The parameters
1 and
2 are the measures of sensitivity of the government expenditure-capital ratio with respect to changes in the rate of employment and the rate of inflation. Unlike the standard theory of monetary policy that presupposes an independent monetary policy on the part of the central bank, the government rather than the central bank plays the leading role in inflation targeting in this MMT setting, although in this setting the central bank’s monetary policy and the government’s fiscal policy are inseparable.16
14 Hori (2017) studies another type of money finance fiscal stabilization policy in the context of a dynamic
Keynesian model. See also Asada (2019).
15 Lerner (1943) writes: “The first financial responsibility of the government (since nobody else can
undertake that responsibility) is to keep the total rate of spending in the country on goods and services neither greater nor less than that rate which at the current prices would buy all the goods and it is possible to produce. If total spending is allowed to go above this there will be inflation, and if it is allowed to go below this there will be unemployment.” (Lerner 1943, p. 39)
16 A representative of the standard type of monetary policy rule is the interest rate monetary policy rule
due to Taylor (1993), which is called the “Taylor rule”. As for the macroeconomic stabilization policy models that include Taylor rule monetary policy, see Asada (2014), Asada, Chiarella, Flaschel and Franke (2010), Asada, Demetrian and Zimka (2019), Asada and Ouchi (2015), Galí (2015) and Woodford (2003).
159
5. Dynamic law of price expectation formation
The final building block of this model is formulation of the dynamic law of the formation of expectations of the rate of price inflation. We adopt the following hypothesis of formation of price expectation, which is a mixture of “forward-looking” and “backward-looking” (adaptive) expectation formations, where is the parameter that represents the weight of the ‘forward-looking’ expectation formation.17
e
[ (
e) (1
)(
e)]
; 0, 0 1 (36)If the inflation target set by the consolidated government including the central bank is sufficiently “credible” for the public, the parameter value is sufficiently close to 1. For this reason, the parameter may be called the “credibility parameter”. The expectation formation equation closes the model in this paper.
Ⅲ. Reduced Form of a System of Fundamental Dynamic Equations
Equations (5), (6), (9), (10), (13), (14), (19), (27), (32), (34), (35), and (36) constitute a system of 12 independent equations that determines the dynamics of 12 endogenous variables ( ,K r, , m, e, w, y, , b, , g, and
e).
It will readily be seenthat this system of equations can be reduced to the following more compact four-dimensional system of nonlinear differential equations, which is a system of fundamental dynamic equations of the present model.
g(
1 2 )(e e )
2( e)F1( , ; , , )
e e
1 2 (37a)
e
[ ( e) (1
) (e e )]F2( , ; , , )
e e
(37b)
17 Equation (36) is reduced to the completely forward-looking expectation formation model in the case of 1,
and is reduced to the completely backward-looking expectation formation model in the case of
0.
This type of formulation was adopted by Asada (2014), Asada, Chiarella, Flaschel and Franke (2003, 2010), Asada, Demetrian, and Zimka (2019), and Asada and Ouchi (2015).
160 m v g [ { ( , , )
g m
e
(e e )
ei( ( , , )
g m
e
e)}b(twtr)] m[ (
e e )
ei( ( , , )
g m
e
e)]F g3( , , , ; )
e m e
(37c) [ ( , , ) 1( , ; ,1 2, ) 2( , ; , , ) ( , , ) ( , , ) e e g e e e e y y r m e e F e F e y g m y g m ( , , ) 3( , , , ; ) ( ( , , ) ) ] ( , , ) e e e e m e y g m F g m e i g m n y g m F g4( , , , ; , , , , )
e m e
1 2 (37d)Ⅳ. Characteristics of the Long-Run Equilibrium Solution
The long-run equilibrium of the dynamic system (37) satisfies the condition
g
e
m e
0.
(38)Substituting Equation (38) into Equation (37), we can characterize the long-run equilibrium
( *, *, *, *)
e
em g
of this system as follows:* e e (39a) * * e (39b)
i
( *)
n n
1 2n
,
( *, *, )g m * (39c) v g[ * ( * )
n b(twtr)]m*[
n] 0 (39d) Equation (39a) means that the “natural” rate of employment is attained at long-run equilibrium. Equation (39b) means that the target rate of inflation set by the government is attained at long-run equilibrium. The first part of Equation (39c) means that the equilibrium real rate of interest ( *) is determined by the condition that the equilibriumrate of investment ( *)i be equal to the ‘natural’ rate of growth
(
n n
1
2n
).
The second part of Equation (39c) tells us that the long-run equilibrium nominal rate of interest ( *) must be equal to *. Equations (39c) and (39d) jointly determine the161
long-run equilibrium values of g and .m It is important to note that the equilibrium values
( *, *, *, *)
e
eg m
are independent of the parameter values1
,
2,
and , .Incidentally, it can readily be demonstrated that the following two conditions are satisfied at long-run equilibrium.
( / )* ( / )*K K Y Y n n1 2 n (40)
( / )*
M M
* ( / )*
B B
n
(41)Equation (40) means that at the long-run equilibrium point of our model, both the rate of capital accumulation and the growth rate of real national income are equal to the exogenously given “natural rate of growth”. Equation (41) means that the equilibrium growth rate of nominal money supply and that of nominal public debt become increasing functions of the target rate of inflation and the natural rate of growth. In other words, the continual growth of money supply and public debt is necessary to support long-run economic growth.
It is worth noting that there will be no long-run equilibrium if the target rate of inflation set by the government is too low, for the following reason. The second part of Equation (39c) requires the condition
0
*
(42)because the equilibrium nominal rate of interest ( * *, )g m cannot be less than the lower bound of the nominal rate of interest
( ).
0 This means that there will be no long-run equilibrium if the target rate of inflation is so low that the following inequality is satisfied.
0
*
(43)In the remaining part of this paper, we assume that the economically meaningful long-run equilibrium point such that g* 0, m* 0,
( *, *, )
g m
*
0 exists uniquely, and we shall study the local stability/instability of this long-run equilibrium point.162
Ⅴ. Local Stability/Instability of the Long-Run Equilibrium Point and
the Existence of Cyclical Fluctuations
In this section, we study the local stability/instability of the long-run equilibrium point. We can write the Jacobian matrix of the four-dimensional dynamic system (37) at the equilibrium point as follows.
2 1 2 4 31 32 33 34 41 42 2 43 44 1 2 0 0 ( ) 0 0 (1 ) ( , , ) ( , , , ) J F F F F eG eG eG eG (44)
In Appendix A, the detailed expressions of the partial derivatives Fij and Gij in Equation (44) are presented.
We can write the characteristic equation of the Jacobian matrix (44) as follows.18
4 3 2
4( )
I J4
a1
a2
a3
a4 0, (45) where
a
1
traceJ
4,
(46)
a
2
sum of all principal second-order minors ofJ
4,
(47)a
3
(
sum of all principal third-order minors ofJ
4),
(48)
a
4
det .
J
4 (49)We adopt the concept of “stability” in the traditional sense. In other words, the equilibrium point of the dynamic system (37) is considered to be locally stable if and only if all of the roots of the characteristic equation (45) have negative real parts. If at least one root of Equation (44) has a positive real part, the equilibrium point is considered to be “unstable”.19
18
See, for example, the mathematical appendices in Asada, Chiarella, Flaschel and Franke (2003, 2010).
163
It is generally recognized that a set of necessary (but not sufficient) conditions for the local stability of the equilibrium point are given by the following set of inequalities (cf. Gandolfo (2009) Ch.16).
aj 0 (j1, 2,3, 4) (50)
We can now prove the following two important propositions that characterize the dynamic properties of the model.
[Proposition 1]
Suppose that the parameter values
1,
2,
and are sufficiently close to zero (including the case of
1
2
0),
and the parameter value is sufficiently large.The equilibrium point of the dynamic system (37) is then unstable. (Proof) See Appendix B.
[Proposition 2]
Suppose that the following properties (ⅰ) – (ⅱ) are satisfied.
(ⅰ) The parameter value is close to 1 (including the case of 1).
(ⅱ) At the equilibrium point, the nominal rate of interest ( ) is sufficiently insensitive
to the changes of y and m for * y
and * m
to be sufficiently close to zero, although * 0y
and
m* 0 are satisfied.20We then obtain the following properties (1) – (3).
literature, such as Galí (2015) and Woodford (2003), which assumes that the initial conditions of the endogenous variables can be freely “chosen” by the economic agents, so that the system can be considered to be “stable” even if some of the characteristic roots have positive real parts. This is called the “jump variables” approach. In our model without “jump variables”, the initial conditions of all endogenous variables are fixed, and they cannot be freely “chosen”. The stability concept in our model belongs to the “old Keynesian” tradition that was proposed by Tobin (1994).
20 This means that the parameter
2
164
(1) The equilibrium point of the dynamic system (37) is locally unstable for all parameter values
(
1
2) 0
that are sufficiently close to zero (including the case of1 2
(
) 0).
(2) The equilibrium point of the same system is locally stable for all sufficiently large parameter values
(
1
2) 0.
(3) At some intermediate range of the parameter value
(
1
2),
there exists a family of closed orbits. In other words, there exist cyclical fluctuations at some intermediate range of(
1
2).
(Proof) See Appendix C.
Ⅵ. An Economic Interpretation of the Analytical Results
In this section, we shall provide an economic interpretation of the analytical results presented in the previous section.
Proposition 1 means that the equilibrium point of the dynamic system in our model becomes unstable if the coordinated fiscal and monetary stabilization policy is appreciably inactive and the price expectation formation is highly backward-looking (adaptive). If there is no inflation targeting by the consolidated government including the central bank, the public’s price expectation formation will be highly backward-looking. We can interpret this instability proposition schematically by resorting to the following destabilizing positive feedback mechanism through the changes in the expected real rate of interest, which is called the “Mundell effect”.21
21 As for the “Mundell effect”, see, for example, Asada, Chiarella, Flaschel, and Franke (2003, 2010),
Asada (2006, 2014), Asada, Demetrian and Zimka (2019) and Asada and Ouchi (2015). The effect was originally proposed in Mundell (1971).
165
[
y
e
]
e(
e)
i
[
y
e
]
(ME) The effect of the causal chain e is strong if the parameter value issufficiently large.
Furthermore, this destabilizing effect is supplemented by another destabilizing positive feedback mechanism, which may be called “consumption demand effect”, as follows. [y e ] (( / ) ,W K r ( / ) )P K (cw,cr ) [y e ] (CDE)
On the other hand, if the inflation targeting by the consolidated government is sufficiently credible, the following stabilizing negative feedback mechanism, which may be called the “inflation targeting effect”, will dominate the destabilizing feedback mechanism (ME).
[
]
e(
e e)
(
e)
[
]
y
e
i
i
y
e
(ITE) If the inflation targeting by the consolidated government is sufficiently ‘credible’ for the public, the credibility parameter is close to 1. In this case, the causal chain(
e
e)
will be strong.Finally, we have the following stabilizing negative feedback mechanism that is due to the MMT type coordinated fiscal and monetary stabilization policy, which may be called “fiscal-monetary policy effect”.
[
y
( ,
e
)]
(
e e
,
)
(
g m
,
)
[
y
( ,
e
)]
(FMPE)If the fiscal-monetary policy effect is strong, the causal chain
(
e e
,
)
(
g m
,
)
will be strong.Proposition 2(1) means that the equilibrium point of the dynamic system in our model may be unstable even if the stabilizing price expectation feedback mechanism (ITE)
dominates the destabilizing price expectation feedback mechanism (ME),when the fiscal/monetary stabilization policy by the consolidated government is substantially inactive and the nominal interest rate is considerably insensitive to changes inyand .m
Proposition 2(2) means that the equilibrium point of our model becomes locally stable if both of the stabilizing feedback mechanisms (ITE) and (FMPE) are strong, even if the nominal interest rate is considerably insensitive to changes in the relevant variables.
166
Proposition 2(3) implies that the destabilizing and the stabilizing feedback mechanisms come into conflict with one another and, as a result, the cyclical fluctuations will occur at some intermediate range of the degree of strength of the fiscal/monetary stabilization policy by the consolidated government.
Ⅶ. Concluding Remarks
In this paper, we studied the effects of the MMT type coordinated fiscal and monetary stabilization policy of a consolidated government on macroeconomic stability. A remarkable postulate of the MMT that was proposed by R. Wray and others is that the government’s fiscal policy plays an active role and the role of central bank’s monetary policy is completely passive in financing the government’s fiscal policy. In this case, in a period of deflationary depression both the government’s fiscal policy and the central bank’s monetary policy are expansive, and in a period of the inflationary boom both the fiscal and monetary policies are contractive. That is to say, both fiscal and monetary policies are countercyclical. We showed that in our dynamic Keynesian model this policy mix can stabilize the intrinsically unstable economy.
On the other hand, the more orthodox Keynesian-oriented policy mix of fiscal and monetary stabilization policy, which was proposed by Blanchard, Krugman and others, presupposes an active role for the central bank’s monetary policy as well as an active role for the government’s fiscal policy. Asada (2014), Asada, Demetrian, and Zimka (2019) and Asada and Ouchi (2015) presented the following formulation as a typical example of such a policy mix rule.22
g
{ (
e e
) (1
)(
b b
)}
; 0, 0 1 (51) 1 2 1 2 ( ) ( ) 0 max[0, ( ) ( )] 0 e e if e e if ;
1
0,
2
0
(52)where is the weight of the employment consideration in contrast to the public debt consideration in the government’s fiscal policy. In this case, the public debt-capital ratio
22 The meanings of the symbols g, e, b, , and are the same as those in the previous sections in
this paper, and the mixed type price expectation formation such as Equation (36) in this paper is also adopted in these texts.
167
b becomes an endogenous variable rather than constant. Equation (52) is a standard interest rate monetary policy rule under the nonnegative constraint of the nominal interest rate, which is called the “Taylor rule” after Taylor (1993). Asada (2014), Asada, Demetrian, and Zimka (2019) and Asada and Ouchi (2015) showed that this policy mix can stabilize the intrinsically unstable macroeconomic system if is sufficiently close to 1, is sufficiently large,
1 and
2 are sufficiently large, and the central bank’s inflation targeting is sufficiently “credible” for the public. In this case, too, both the fiscal and the monetary policies are expansionary in the case of deflationary depression, and both policies are contractionary in the case of inflationary boom. In this respect, the orthodox (mainstream) Keynesian-oriented fiscal and monetary stabilization policy mix rule produces qualitatively almost the same result as that of the more “heretical” MMT stabilization policy mix rule. In spite of the fact that the writings by the proponents of MMT such as Mitchell, Wray, and Watts (2019) and Wray (1998, 2015) are usually hostile to “mainstream” economics, probably for philosophical and ideological reasons, the fiscal and monetary policy mix proposed by them and the policy mix proposed by “mainstream” Keynesian-oriented economists such as Blanchard (2016) and Krugman (2012) produce almost the same qualitative result.Finally, we shall consider two possible extensions of the present model. The first possible extension is to replace the simple Keynesian investment function (4) with the following more realistic but more complicated investment function, where re is the
expected rate of profit that is related to the Keynesian concept of “animal spirits”.
i I K i r
/
( ,
e
e)
; 0, e e r i i r e ( e) 0 i i (53)The relatively simple way to close the system is to introduce the following ‘adaptive expectation’ hypothesis following the procedure in Murakami (2018).
r
e
(
r r
e)
;0
(54)
The four-dimensional dynamic system (37) is then extended to a more complicated five-dimensional dynamic system. We may conjecture that this extension will have a destabilizing effect, because the large (small) value of r compared with re induces
further increase (decrease) of r through the increase (decrease) of i, which is due to the increase (decrease) of re. In this case, more active consolidated fiscal and monetary
168
The second possible extension is to reconsider the problem in the context of an open economy macrodynamic model. The proponents of MMT, such as Wray and others, insist that the MMT policy works well in an open economy under a flexible exchange rate system. To consider this problem theoretically, we must extend the model in this paper to the open economy model under flexible exchange rates. This is an aspect that remains to be studied in the future. It is worth noting that the analytical framework presented by Asada, Chiarella, Flaschel, and Franke (2003) will be of use in solving the problem. Appendix A: The Partial Derivatives
3 * * * 31 ( ) ( ) ( ) ( )* [1 (1 e) g ] * e 0 F F v i b m i g (A1) 3 * * * * 32 ( ) ( ) ( ) ( ) ( e)* [(1 e) e e] * e F F v i i b m i (A2) 3 * * * * 33 ( ) ( ) ( ) ( ) ( )* (1 e) m * e m ( ) 0 F F v i b m i n m (A3) 3 34 ( )* ( *) 0 F F v m e
(A4) 4 * * 41 31 ( ) ( ) ( ) ( ) ( ) 1 ( )( )* ( m/ *) e g* F G y y F i e g (A5) 4 * * * 42 2 2 32 ( ) ( ) ( ) ( ) (?) ( ) ( ) 1 ( , , ) ( )( e)* ( / *)g ( e/ *) ( m/ *) F G y y y y y y F e (A6) 4 * * * 43 33 ( ) ( ) ( ) ( ) ( ) 1 ( )( )* ( m/ *) e m 0 F G y y F i e m (A7) 4 * * 44 1 2 1 2 ( ) ( ) ( ) ( ) 1 ( , , , ) ( )( )* ( / *)(g ) ( e/ *) (1 ) F G y y y y e e * * * ( ) ( ) ( ) ( ) ( / *)(ym y v m*)
i e(
e 1) (A8)169
Appendix B: Proof of Proposition 1
We can express the coefficient
a
1 in the characteristic equation (45) as follows:a
1
traceJ
4
F
33
eG
44( , , , )
1 2 * 1 2 1 1 2 ( ) ( ) ( ) (ye/ *) (1y ) A a( , , , ) (B1)where A is independent of the parameters
1,
2,
and Thus, we have .* 1 ( ) ( ) (0,0, 0, ) ( e/ *) 0 a y y A (B2)
for all sufficiently large values of This means that the inequality .
a
1
0
is satisfiedfor all sufficiently large values of even if the parameter values of
1,
2,
and are positive, insofar as these parameter values are sufficiently close to zero, by continuity. In this case, one of the necessary conditions of local stability (50) is violated, so that the equilibrium point becomes unstable.
Appendix C: Proof of Proposition 2
Let us assume that 1.We can then rewrite the Jacobian matrix at the equilibrium point (44) as follows:23 2 1 2 4 31 32 33 34 41 42 2 43 44 1 2 0 0 ( ) 0 0 0 ( 1, ) ( , ,1, ) J F F F F eG eG eG eG (C1) 23 In fact, 44
( , ,1, )
1 2170
In this case, the characteristic equation (45) can be reduced to
3 2 4( )
I J4 (
) I J3 (
)( b1
b2
b3) 0 (C2) where 1 2 3 31 33 34 41 43 44 1 2 0 0 ( ) ( , ,1, ) J F F F eG eG eG (C3)and we can express the coefficients bj (j1, 2, 3) as follows:
* 1 3 33 44 1 2 33 1 2 ( ) (?) ( ) ( ) ( ) ( , ,1, ) {( g/ *)( ) } b traceJ F e G F e y y B (C4)
b
2
sum of all principal second-order minors ofJ
31 2 33 34 31 33 41 44 1 2 43 44 1 2 0 0 0 ( ) ( , ,1, ) ( , ,1, ) F F e e F F G G G G 1 2 41 33 44 1 2 34 43 (?) ( ) (?) ( ) ( ) [( ) ( , ,1, ) ] e G F G F F * * * * 31 33 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [{( / *)m e g ( / *)}(g ) e y y F i
F y y
* * * * * * 33 33 34 33 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( / *)(m *) e( e 1) {( / *)m e m}] F y y v m F i
F y y F i
(C5) 31 33 3 3 1 2 1 2 31 43 33 41 (?) ( ) (?) ( ) 41 43 det ( ) F F ( )( ) b J e e F G F G G G * * * 1 2 31 33 ( ) ( ) ( ) ( ) ( ) ( ) e( m g) 0 e
i F
F
(C6) 2 * * * * * 2 1 2 3 31 33 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ( / *){( / *)g m e g ( / *)}](g ) b b b e y y y y F i
F y y
D
(
1
2)
E
(C7)171
The characteristic Equation (C2) has a real root
4
0,
and the three other roots are determined by the following equation.3 2
3( )
I J3
b1
b2
b3 0 (C8) It is well known that all of the roots of the characteristic equation (C8) have negative real parts if and only if the following ‘Routh-Hurwitz stability conditions’ are satisfied (see, for example, Gandolfo (2009) Chap. 16 and Asada, Chiarella, Flaschel, and Franke (2003, 2010) mathematical appendices).
bj 0 (j1, 2, 3),
bb b
1 2
30
(C9)Eq. (C6) means that the condition
b
3
0
is always satisfied. Therefore, the localstability conditions of the equilibrium point in this system can be reduced to the following set of inequalities.
b
1
0,
b
2
0,
bb b
1 2
30
(C10)(1) Suppose that * y
and
m* are sufficiently close to zero for * * *, g y gy
* * * , e yy e
and * my also to be sufficiently close to zero (see Equation (22)). In this case,
b
2 becomes negative for all parameter values(
1
2) 0
that are sufficiently close to zero (including the case of(
1
2) 0).
The inequality2
0
b
violates one of the local stability conditions (C10).(2) It will readily be seen that the coefficient of
(
1
2)
in Equation (C5) and the coefficient of 21 2
(
) in Equation (C7) become positive if * y
and * m
are sufficiently close to zero. In this case, all of the local stability conditions (C10) are satisfied for all sufficiently large parameter values(
1
2) 0.
172
(3) The results (1) and (2) mean that there exists a “bifurcation point”
(
1
2)
0 at which the local stability of the equilibrium point is lost as the parameter value1 2
(
)
decreases, by continuity. At this “bifurcation point”, the characteristic equation (C8) has at least one root with zero real part. However, we cannot have a real root such that 0 because we have
3(0)
b
30
from Equations (C6) and (C8). Therefore, the characteristic Equation (C8) has a pair of pure imaginary roots at the bifurcation point, which is called the “Hopf bifurcation point”. The Hopf bifurcation theorem implies that there exists a family of closed orbits at some range of the parameter values(
1
2)
that are sufficiently close to(
1
2) .
0 24So far, our analysis has been confined to the case of 1. Even if 0 1,
however, the qualitative conclusion of the analysis is not changed as long as is sufficiently close to 1, by continuity.
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Toichiro Asada
Faculty of Economics, Chuo University, Tokyo, Japan
