Stabilizing an Unstable Economy:

Fiscal and Monetary Policy, Stocks, and

the Term Structure of Interest Rates

Matthieu Charpe

International Labor OrganizationGeneva, Switzerland

Peter Flaschel

Department of Business Administration and EconomicsBielefeld University, Bielefeld, Germany

Florian Hartmann∗

Department of Economics

University of Osnabruck, Osnabruck, Germany

Christian ProanoMacroeconomic Policy Institute (IMK)

Dusseldorf, Germany

February 10, 2010

Abstract

Currently, many monetary and fiscal policy measures are aimed at preventing thefinancial market meltdown that started in the US subprime sector and has spread worldwide as a great recession. Although some slow recovery appears to be on the horizon, itis worthwhile exploring the fragility and potentially destabilizing feedbacks of advancedmacroeconomies in the context of Keynesian macro models. In this paper we in particularfocus on the financial markets. We use a Tobin-like macroeconomic portfolio approach,and the interaction of heterogeneous agents on the financial market to characterize thepotential for financial market instability. We show that specific fiscal and monetarypolicies have to be used to stabilize such unstable macroeconomies.

Keywords: Monetary Business Cycles, Portfolio Choice, (In-)Stability, StabilizingPolicy Measures

JEL classifications: E12, E24, E31, E52.

∗Corresponding Author: fhartman@uos.de

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1 Introduction

The financial crisis starting in the US subprime sector, has spread world wide as a greatrecession. A hyperactive monetary and fiscal policy since the end of 2007 has aimed atpreventing a further financial meltdown in the advanced countries. Some observers meanwhilemaintain that a slow recovery appears to be on the horizon. It is still worthwhile however toexplore the fragility and potentially destabilizing feedbacks of advanced macroeconomies, inparticular in the context of a Keynesian disequilibrium model where such issues have rarelybeen treated so far.

As the history of macroeconomic dynamics and business cycles – which recently have beendeveloped as boom - bust cycles – has taught us, fragilities and destabilizing feedbacks areknown to be potential features of all markets – the product markets, the labor market, andthe financial markets. In this paper we will focus on the financial market. We use a Tobin-likemacroeconomic portfolio approach, coupled with the interaction of heterogeneous agents onthe financial market, to characterize the potential for financial market instability.

Though the study of the latter has been undertaken in many partial models, we focus hereon the interconnectedness of the goods and the financial markets. Furthermore, we studywhat potential fiscal and monetary policies can have in stabilizing unstable macroeconomies.It was in particular Minsky (1982) who has put forward many ideas to stabilize an unstableeconomy. Besides specific Tobin type taxes we in particular propose a countercyclical monetarypolicy that focuses on the market for stocks as the measure of the state of confidence of aneconomy. Conventional types of Taylor rules are by contrast not too effective in stabilizingthe economy, unless they are reformulated and take account of the risky assets traded in thefinancial markets. Dynamic tools and stability analysis are brought to bear to demonstratethe stabilizing effects of those suggested policies.

In the paper we extend the portfolio model of Flaschel, Malikane and Proano (2009), whichcoupled a dynamic multiplier process with the dynamic of the stock market, towards theadditional treatment of long-term bonds as a further risky asset and the term structure ofinterest rates dynamics this gives rise to. We then study the role that monetary (interest rate)policy can play in such a framework, which was completely ineffective in the portfolio approachof the earlier paper. To a certain degree this result also holds in the present framework, thoughthis ineffectiveness can be overcome when the central bank is willing to act on the long endof the bond market in place of its short end or – in the present context even better – on theequity market as the market that here impacts the investment behavior of firms in particularand economic activity in general.

We show in this respect that the private sector of the economy is likely to be an unstable one,since its steady state is generally surrounded by centrifugal forces. These forces thus have tobe tamed by appropriate fiscal and monetary policy measures. We show in this respect thata Tobin type tax of capital gains together with a long-term oriented monetary policy and anopen market policy that trades in long-term bonds are capable of stabilizing the steady stateof the economy.

In the next section we comment on and investigate the asset market structure considered in ourmodel type by contrasting it with a traditional Keynesian approach presented in Turnovsky(1995). The model that is based on this structure of the financial markets is presented insection 3. We then investigate in section 4 the stability of the real-financial market interaction

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of the model. In section 5 we use this analysis to investigate what policy measures can improvethe working of the private sector of the economy. Section 6 concludes.

2 Financial markets and the effectiveness of monetary

policy

We start here from (and extend) the modeling framework of Flaschel, Malikane and Proano(2009) which is based on the widely employed simplifying practice, whether implicitly as inthe presentations of the UIP condition or explicitly as in Sargent (1987), to use as array offinancial assets the set: money M (issued by the central bank), fix-price bonds B (issued by thegovernment, with pb = 1) and equities E (issued by firms at price pe). This has the advantagethat money and bonds can be aggregated without any price interference. Moreover there isthen only one risky asset (equities) for which capital gains and capital gain expectations haveto be taken into account. Sargent (1987, p.12) notes with respect to the variable B that it

‘is a variable-coupon bond that is issued by the government. The bond is essentiallya savings deposit, changes in the interest rate altering the coupon, but leaving thedollar value of bonds outstanding unchanged.’

This characterization implies that the supplies of money and bonds M+B can be characterizedas providing the money supply M2 from the various definitions of such money supplies.

But should we really use as baseline financial structure for a model of the real-financial marketinteraction a scenario where government can issue money (perfectly liquid – and thus riskless– fixed price interest bearing bonds)? There of course exist some financial assets, issued by thegovernment, which exhibit a high degree of liquidity (due to their short-term maturity horizon),but the bulk of government expenditures is not financed in this way, and in particular not at aconstant price for the new issue, since this not only interferes with the objectives of the centralbank, but would also be an ideal objective for rising liquidity preference (hoarding), due tothe positive yield this asset provides (especially when it is considered as perfect substitutefor equities as in Sargent (1987)). Moreover, we have shown in a Tobinian portfolio choiceframework under mild assumptions on the employed portfolio demand functions for such arange of financial assets that monetary policy is then completely ineffective, since it onlyinfluences in this case the cash management process between money and bonds, but does notat all reach the equity market and thus the financing structure of firms, see Flaschel, Malikaneand Proano (2009) for details.

As representation and investigation of the implication of traditional macroeconomicsTurnovsky (1995, part I) also makes use of a dynamic portfolio balance macroeconomic modelwhere he in particular considers the following representation of the financial part of his port-

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folio model.1

Md = fm(Y, r, ρ)(M + B + pK) (1)

Bd = fb(Y, r, ρ)(M + B + pK) (2)

pK = fk(Y, r, ρ)(M + B + pK) (3)

with

fm2(Y, r, ρ) + fb2(Y, r, ρ) + fk2(Y, r, ρ) ≡ 0, fm3(Y, r, ρ) + fb3(Y, r, ρ) + fk3(Y, r, ρ) ≡ 0,

in particular (and Md +Bd + pK = M +B + pK) and with the gross substitution assumptionbeing made. In the representation of the asset structure of the economy, even the only riskyasset we have considered above is removed from sight and replaced by the current value ofthe stock of real capital pK, that is only the denominator of Tobin’s average q = peE

pK. The

financial markets are therefore represented in a way completely devoid of risk and capital gainexpectations which in particular in view of Keynes’ (1936) General theory is a very strangeoutcome for a modeling approach labeled ‘traditional Keynesianism’.

We concentrate in this section on the financial markets and therefore treat output Y as aparameter and consider the three asset markets in equilibrium with the corresponding supplies.Due to Walras’ Law of Stocks we need to consider only two of these equilibrium conditionsand use them to determine the two endogenous variables r, the rate of interest and ρ, the rateof profit.

This is a very tranquil representation of financial markets where expectations play no roleat all, since there are no capital gains, and where the rate of profit is determined through aresale market for physical capital, an assumption usually not made in Keynesian approachesto macrodynamics (see Sargent (1987, ch.III) for details of this ‘classical’ type of a Keynesianmodel). In this model monetary policy can influence the rate of profit ρ of the economy directlythrough the financial market structure of the economy, which is moreover here independent ofthe conflict about income distribution between capital and labor.

We view the above structure as not being adequate for building a Keynesian portfolio modelof the real-financial market interaction, in particular when the current situation on the actualfinancial markets is taken into account. As Keynes wrote already in 1936:

Speculators may do no harm as bubbles on a steady stream of enterprise. But theposition is serious when enterprise becomes the bubble on a whirl-pool of specu-lation. When the capital development of a country becomes a by-product of theactivities of a casino, the job is likely to be ill-done. The measure of success at-tained by Wall Street, regarded as an institution of which the proper social purposeis to direct new investment into the most profitable channels in terms of futureyield, cannot be claimed as one of the outstanding triumphs of laissez-faire capital-ism – which is not surprising, if I am right in thinking that the best brains of WallStreet have been in fact directed towards a different object. Keynes (1936, p. 159).

1In order to make it comparable to our subsequent modeling of such an approach we ignore here howeverinflation (assume a single price level p instead) and also transfer the Turnovsky model to the notation used inthis paper.

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It is in our view impossible to make sense out of this quotation on the basis of the aboveportfolio approach. As in Asada, Flaschel, Mouakil and Proano (2009), we therefore employnow – as baseline modification of the asset demand structure of the Turnovsky approach –equities E and share prices pe in place of the value of the physical capital stock. We assumefor analytical simplicity that all profits are paid out as dividends to equity owners. This paperwill moreover extend its asset market structure towards a treatment of the term structure ofinterest rates and will show in this regard that, on the one hand, the insights gained in Asada,Flaschel, Mouakil and Proano (2009), chapters 3 and 4, concerning safe and risky assets,can be preserved and meaningfully extended while, on the other hand, the role of monetarypolicy can be reformulated now in significant ways such that the effectiveness of an interestrate policy rule will now be given. The enhancement of the financial markets of the economyby one further risky asset does not restructure their working in a fundamental way, yet theworking of monetary policy can here be redesigned in a way that appears to be capable ofovercoming certain problematic features of its past improper working in the given world ofunleashed financialization.

Since macroeconomics has to economize on the use of financial instruments in order to remaintractable we will now however use the theoretically seen polar case to fix-price bonds, namelyperpetuities in place of the short-term bonds for the financing the government deficit. Thisis the other typical bond configuration that is used in macroeconomics.2 However, solelyreplacing B by long-term bonds Bl with a variable price pb and with rate of return the rate ofinterest augmented by capital gains: 1/pb + pb, raises the question how the short-term interestrate is now provided, since there is no longer a market for it.

We therefore go one important step further and assume as perspective in the following thatthe central bank is issuing short-term bonds B in place of money M and is therefore payinginterest on its money supply B. This may at first sight look unorthodox and even exceptional,but will indeed prove adequate for the model types we are considering in this paper.

As novel structure of the financial markets we consequently assume now: money B, with rateof interest r, issued by the central bank, perpetuities Bl with an expected rate of return 1

pb+πe

b ,issued by the government, and equities E, issued by firms, with an expected rate of returnpY −wLd

peE+ πe

e, with profits pY − wLd as the dividend payments of firms.

We use as in Flaschel, Malikane and Proano (2009) an explicit stock demand (portfolio)approach to financial markets, but no longer with three equilibrium conditions, for short-termand long-term bonds and equities, but integrate now in addition the hypothesis that there areno full stock adjustments at each moment in time. Instead we assume that only a portionof actual stock excess demand is impacting the asset market at each instant, so that there isnow imperfect stock adjustment occurring, due to adjustment costs and other considerationsof asset holders in their articulation of flow asset demands at the asset market.

Since the central bank has full control over the short-term rate of interest we assume in thefollowing either an interest rate peg or an interest rate policy rule as the monetary policyof the central bank. Note that the central bank can set the short-term rate of interest herewithout any open market operation (through simple announcement), since it pays this intereston the money it has issued. The money supply is therefore under the full control of the centralbank even in the case of an interest rate peg.

2See Turnovsky (1995, p.22) for a brief discussion of bonds with a finite maturity date T in a continuoustime framework.

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This is by and large the modeling framework we will use in this paper and it will be augmentedagain, as in Flaschel, Malikane and Proano (2009), by a simple dynamic multiplier processas far as the real markets are concerned. The paper then investigates the stability of theformulated real-financial market interaction. On this basis it shows (if the stability of theprivate sector is not given) what monetary (and fiscal) policy can do in order to stabilize suchan unstable economy.

3 Keynesian asset price dynamics and the multiplier

In this section we again simplify the real part of the Turnovsky (1995) model by ignoringinflation and growth altogether and by representing the quantity adjustment process by asimple dynamic multiplier approach. This simplifies the Metzlerian inventory acceleratormechanism of the KMG model of Chiarella and Flaschel (2000) (and thus suppresses it asa source of instability besides the wage price spiral). It makes the real part of the economyalways a stable one (from this partial perspective) if the propensity to spend is less thanone. We take only the stock market effect on investment (and consumption) behavior hereinto account, by the impact of Tobin’s q on these goods demand functions, since we conceivethe share prices here as measuring the state of confidence in the economy and thus use itas argument in the investment (and consumption) function in place of the commonly usedshort-term rate of interest.

This gives as representation of the real side the law of motion3

Y = βy[(Yd − Y )/Y ] = βy[(ayY + aq(q − qo) + A − Y )/Y ] (4)

where ay ∈ (0, 1), aq > 0 and where A summarizes autonomous expenditures (fiscal policyand more). This is a standard textbook dynamic multiplier process, with a change howeverin the employed aggregate demand function where Tobin’s average q, in its deviation from itssteady state value qo, is now used in place of the short-term interest rates. Since prices, thecapital stock and the stock of equities are considered as given in this section we assume E

pKto

be equal to one for simplicity so that Tobin’s average q = peEpK

is equal to the share price pe inthe following.

We save complexity on the real side of the economy, but add complexity on its financial sideby assuming for the financial assets B, Bl, E that they are imperfect substitutes and thatcapital gain expectations are also imperfect and working in a portfolio structure as in Tobin(1982). We assume that the stocks of the financial assets B, Bl, E are exogenously given inthe following Tobinian representation of the asset markets, despite a given interest rate peg(see the argument in the preceding section):

B = Bd = fb(r, ree, re

b)W np , fb(r, r

ee , re

b) + fbl(r, ree, re

b) + fe(r, ree, re

b ) ≡ 1, r given (5)

peE = peEd = fe(r, ree, re

b)W np , re

e =pY − wLd

peE+ πe

e = ρ/q + πee (6)

pbBl = pbB

ld = fbl(r, ree, re

b)W np , re

b = 1/pb + πeb , W n

p = B + pbBl + peE (7)

3In contrast to Flaschel, Malikane and Proano (2009) we now consider the case where no equity pricesmoothing occurs, i.e., we assume that the parameter βe used in Flaschel, Malikane and Proano (2009) equalsinfinity.

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We have for the rate of profit the expression:

ρ =pY − wLd

pK=

Y

K(1 − ω/z), ω = w/p < z = Y/Ld = const

where we assume a constant value for the labor productivity coefficient z > w/p that has toexceed the real wage ω = w/p of course.

Only two of these equations are independent from each other and can be used to determine thevariables pb, pe. Note that the expression re

e defines the rate of return on equities (the sum ofthe dividend rate of return ρ/q and expected capital gains on equities πe

e), since all profits areassumed to be paid out as dividends. We moreover assume that the Central Bank is fixing therate of interest r on its outstanding bonds at each moment in time and changes the number ofinterest bearing bonds B (if desired) through open market operations of the type dB = pbdBl.We assume finally that the demands for financial assets exhibit the gross substitute propertyand therefore depend positively on their own rate on return and negatively on the rates ofreturn of the other assets, i.e. in particular, money demand Bd depends positively on the rateof interest and negatively on the rates of return on equities and long -term bonds. Note that– in contrast to Flaschel, Malikane and Proano (2009) – there is no longer a hierarchy presentin this setup of financial assets and the portfolio choice of asset holders.

In contrast to Flaschel, Malikane and Proano (2009), its nested expectation formation andan equilibrium representation of asset markets, we are now considering adjustment processesfor bond prices as well as equity prices based on the following stock disequilibria, similar towhat we have already modeled for an open economy in Asada, Flaschel, Mouakil and Proano(2009):

pe = βe[αe(fe(r, ree, r

eb)W

np − peE)], αe ∈ (0, 1) (8)

pb = βb[αb(fbl(r, ree, r

eb)W

np − pbB

l)], αb ∈ (0, 1) (9)

We are therefore assuming that only a fraction αb of current stock disequilibria,fbl(r, re

e, reb)W

np − pbB

l in the case of bonds, is entering the asset markets as demand or supply(based on adjustment costs arguments), i.e., 1/αe for example represents the delay with whichfor example the stock imbalance fe(r, r

ee, r

eb)W

np − peE is intended to be balanced. Such an

approach was seen to be necessary in an open economy, where flows but not stock imbalancesenter the capital account of the balance of payments. It is definitely also a plausible modelingprocedure in a continuous time setup in the case of closed economies. The flow processes onthe asset markets are then translated into asset price changes by using speed parameters β inthe front of these expressions.

We now show the existence of an overall equilibrium for the above equations and demonstratethereafter that this equilibrium is stable under the above adjustment of asset prices in aneighborhood of the steady state position. In the steady state we have (since expected capitalgains are zero then):

B = fb(r, ρ/q, 1/pb)(B + pbBl + peE), r given

peE = fe(r, ρ/q, 1/pb)(B + pbBl + peE)

pbBl = fbl(r, ρ/q, 1/pb)(B + pbB

l + peE)

We use the first and the third equation in order to determine the steady state. The firstequation generally defines a negative relationship between the prices pe, pb, due to the implicit

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function theorem, the slope of which is given by:

∂pb

∂q= −

fb2(·)(−ρ/q2)W np + fbE

fb3(·)(−1/p2b)W

np + fbBl

< 0,

For the slope of the function that is implicitly defined by the third equilibrium condition weget in a similar way:

∂pb

∂q= −

fbl2(·)(−ρ/q2)W np + fblE

fbl3(·)(−1/p2b)W

np + (fbl − 1)Bl

> 0.

This time the slope of the curve defined by the implicit function theorem is positive, sincethere holds fbl2(·) < 0, fbl3(·) > 0 and fbl ∈ (0, 1). We note that these two features also holdfor all other capital gain expectations as long as these expectations are given (are static).

We thus have that the stationary state q, pb on the financial markets is uniquely determinedand assume that the parameters behind the functions fb, fbl are such that it exists everywherein the neighborhood of the steady state qo, po

b of the full dynamical system (where Y o = A1−ay

holds in addition).

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4 Stable real-financial market interactions?

We get from the above the following proposition for the stability of the asset markets whencapital gain expectations are static.

Proposition: Stable Financial Markets Interaction

Assume that capital gain expectations are static. Then: The dynamics

q = βe[αe(fe(r, ree, r

eb)W

np − qE)], αe ∈ (0, 1) (10)

pb = βb[αb(fbl(r, ree, r

eb)W

np − pbB

l)], αb ∈ (0, 1) (11)

converges to the current stationary state of the asset markets for all adjustmentspeeds of asset prices pe, pb.

Proof: The matrix of partial derivatives of the considered two laws of motion is given by:

J =

(

βeαe[fe2(·)(−ρ/q2)W np + (fe − 1)E] βeαe[fe3(·)(−1/p2

b)Wnp + feB

l]βbαb[fbl2(·)(−ρ/q2)W n

p + fblE] βbαb[fbl3(·)(−1/p2b)W

np + (fbl − 1)Bl]

)

The trace of this matrix is obviously negative, while for the determinant we obtain the ex-pression:

J = βeαeβbαb

∣

∣

∣

∣

−fe2(·)ρ/q2W np + (fe − 1)E −fe3(·)/p

2bW

np + feB

l

−fbl2(·)ρ/q2W np + fblE −fbl3(·)/p

2bW

np + (fbl − 1)Bl

∣

∣

∣

∣

We have

−(fe2 + fbl2)(·)ρ/q2W np + (fe + fbl − 1)E = fb2(·)ρ/q2W n

p − fbE < 0

and−(fe3 + fbl3)(·)/p

2bW

np + (fe + fbl − 1)Bl = fb3(·)/p

2bW

np − fbB

l < 0

and thus get that the negative entries in the diagonal dominate the positive entries in theoff-diagonal. This implies that the determinant of J must be positive and thus proves thevalidity of the Routh-Hurwitz stability conditions for such a planar dynamical system.

We therefore in sum have that the real and the financial markets, when considered in isolation(and with sufficiently tranquil capital gain expectations) are both stable. The next steptherefore is to investigate what happens when they are interacting as a full 3D dynamicalsystem.

Proposition: Stable 3D Real-Financial Markets Interaction

Assume that the parameter βy is sufficiently large and the parameter βe sufficientlysmall. Assume moreover that the parameter ay is sufficiently close to one (butsmaller than one). Then: The dynamics

Y = βy[(ay − 1)Y + aq(q − qo) + A] (12)

q = βe[αe(fe(r, ree, r

eb)W

np − qE)] (13)

pb = βb[αb(fbl(r, ree, r

eb)W

np − pbB

l)] (14)

is locally asymptotically stable around its steady state position.

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Proof: The Jacobian of the full 3D system at the steady state is given by:

J =

βy(ay − 1) βyaq 0βeαefe2ρ

′/qW np βeαe[−fe2ρ/q2W n

p + (fe − 1)E] βeαe[−fe3/p2bW

np + feB

l]βbαbfbl2ρ

′/qW np βbαb[−fbl2ρ/q2W n

p + fblE] βbαb[−fbl3/p2bW

np + (fbl − 1)Bl]

=

− + 0+ − +− + −

The trace of J is obviously negative and the principal minor of order 2

J1 =

∣

∣

∣

∣

βeαe[−fe2ρ/q2W np + (fe − 1)E] βeαe[−fe3/p

2bW

np + feB

l]βbαb[−fbl2ρ/q2W n

p + fblE] βbαb[−fbl3/p2bW

np + (fbl − 1)Bl]

∣

∣

∣

∣

is positive according to the proposition of financial markets interaction as is the principalminor J2. And for the remaining principal minor of order 2 we get

J3 =

∣

∣

∣

∣

βy(ay − 1) βyaq

βeαefe2ρ′/qW n

p βeαe[−fe2ρ/q2W np + (fe − 1)E]

∣

∣

∣

∣

= βyβeαefe2/qWnp

∣

∣

∣

∣

ay − 1 aq

ρ′ −ρ/q

∣

∣

∣

∣

+ βy(ay − 1)(fe − 1)E

which is positive if the speed parameter βe is chosen sufficiently small. Note however that theRouth-Hurwitz conditions only demand that the sum of principal minors of order 2 is to bepositive which provides a much weaker condition then the one just stated.

For the determinant of the Jacobian J one gets from the above:

|J | = βyβeαeβbαb

∣

∣

∣

∣

∣

∣

ay − 1 aq 0fe2ρ

′/qW np −fe2ρ/q2W n

p + (fe − 1)E −fe3/p2bW

np + feB

l

fbl2ρ′/qW n

p −fbl2ρ/q2W np + fblE −fbl3/p

2bW

np + (fbl − 1)Bl

∣

∣

∣

∣

∣

∣

= J3J33 + βyβeαeβbαb(fe3/p2bW

np − feB

l)

∣

∣

∣

∣

ay − 1 aq

fbl2ρ′/qW n

p −fbl2ρ/q2W np + fblE

∣

∣

∣

∣

In order to get a negative determinant we therefore have to show that the determinant

∣

∣

∣

∣

ay − 1 aq

fbl2ρ′/qW n

p −fbl2ρ/q2W np + fblE

∣

∣

∣

∣

is positive – in addition to the already assumed positivity of the minor J3. The last expressionhere shows that this for example holds if the marginal propensity to purchase goods ay ∈ (0, 1)is sufficiently close to 1.

The condition (−trJ)(J1 + J2 + J3) − |J | > 0 can be fulfilled by choosing the adjustmentspeed of the dynamic multiplier process sufficiently large, since it enters the product termwith power 2 and the determinant only in a linear form.

Summing up we thus get in the case of static chartist capital gain expectations (βπec= βπbc

= 0)the above stability proposition. This proposition and its proof show however that the coupling

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of two stable, but partial processes need not provide a stable interaction of the two partialprocesses.

We now extend the model by endogenizing capital gain expectations. We distinguish betweenfundamentalists f and chartists c and assume for the former that they expect capital gains toconverge back with speeds βπef

, βπbfto their steady state position which is zero. Chartists by

contrast (for analytical simplicity) make use of a simple adaptive mechanism to forecast theevolution of capital gains in the equity market and the market for long-term bonds. Marketexpectations are then an average of fundamentalist and chartist expectations with weight γ.De Grauwe and Grimaldi (2005) employ this kind of scheme to characterize the behaviorof agents on the foreign exchange market. We stress here that these simple expectationsformation mechanisms are chosen to make the dynamics analytically tractable. They can ofcourse be replaced by much more refined forward and backward looking expectation ruleswhen the model is treated numerically. We do not expect that this changes the results in asignificant way if these learning mechanisms are built in the spirit of the ones we introduce andemploy below. For empirical evidence on the chartist-fundamentalist framework in explainingexpectational heterogeneity see Menkhoff et al.(2009).

πeef = βπef

(0 − πeef ) (15)

πeec = βπec

(q − πeec) (16)

πebf = βπbf

(0 − πebf ) (17)

πebc = βπbc

(pb − πebc) (18)

πee = γeπ

eef + (1 − γe)π

eec (19)

πeb = γcπ

ebf + (1 − γb)π

ebc (20)

On the basis of the 3D stability proposition one then easily gets:

Corollary: Stable 5D Real-Financial Markets Interaction

Assume that the conditions of the preceding proposition are met and that the speedparameters for chartists are sufficiently small (or the parameter γe, γb sufficientlyclose to 1. Then: The dynamics

Y = βy[(ay − 1)Y + aq(q − qo) + A]

q = βe[αe(fe(r, ρ/q + πee , 1/pb + πe

b)Wnp − qE)]

pb = βb[αb(fbl(r, ρ/q + πee , 1/pb + πe

b)Wnp − pbB

l)],

augmented by the above adjustment rules for the two processes describing capitalgain expectations, is locally asymptotically stable around its steady state position.

The proof of this corollary is straightforward, since the absence of chartists implies a 5DJacobian matrix given by the investigated 3D one and the negative entries J44 = βπef

γe, J55 =βπbf

γb. Fundamentalists – if sufficiently dominant – may therefore calm down the situationthat chartists may create. By contrast if there are only chartists present, we get positiveentries in the entries J44, J55 in the trace of J, if the parameters βe, βb are sufficiently large.These entries will outperform the other entries in the trace of J if the parameters βπec

, βπbc

12

are chosen sufficiently large. Depending on the choice of other parameters the behavior ofchartists may however be neutralized to a larger degree if all the other stability conditionsapply.

If the determinant of the considered three dimensional dynamics is made negative (by anappropriate choice of the parameter ay), we get by appropriate row manipulations in the full5D Jacobian that its determinant must be negative as well. Increasing adjustment speeds canthen imply loss of stability only by way of so-called Hopf-bifurcations, i.e. in general, by wayof the death of a stable corridor around the steady state position of the economy or the birthof a stable persistent fluctuations around it. The dynamical system of this paper thereforecan provide a theory of business fluctuations caused through the interaction of the real withthe financial markets.

Note also that the positive feedback loop (for example in the case of the stock market) between:

q = βe[αe(fe(·, ... + πee , ·)W

np − qE)]

πeec = βπec

(q − πeec)

is of no importance (if present in the minors of order higher than 1), since the implied rowsin the Jacobian matrix J can be simplified as indicated by the following reduced form laws ofmotion:

q = βe[αe(fe(·, ... + 0, ·)W np − qE)]

πeec = βπec

(0 − πeec)

Depending on its parameter sizes the economy may therefore at one time be fairly robust tochartists’ centrifugal behavior, while it may be fairly vulnerable in other constellations.

The above row reduction methods again show that the upper 4D minor must be positive andthe determinant of the full 5D Jacobian negative if the latter also holds for the upper 3Dminor. As already indicated, stability through excessive chartist behavior can therefore onlyget lost by way of Hopf-bifurcations, i.e., in a cyclical fashion, by the birth of a stable persistentoscillation or by the death of a stability basin around the steady state (or in exceptional casesby passing through a center type dynamics).

We briefly contrast the above analysis with the case where neoclassical perfectness is given.This perfectness assumes that there holds βe, βb = ∞, αe, αb = 1, assumes perfect asset sub-stitution, and assumes myopic perfect foresight with respect to capital gains (βπbcβπec = ∞).These assumptions taken together imply that there always holds:

r = ρ/q + q = 1/pb + pb.

The dynamical system is in its core dynamics then reduced to

Y = βy[(ay − 1)Y + aq(q − qo) + A] (21)

q = r − ρ(Y )/q (22)

which is of the usual saddlepoint type if the 2D determinant in the former expression J3

is positive and thus there stabilizing for all considered adjustment speeds. Yet the perfectlimit case is then unstable, but it exhibits a stable submanifold that – when the economyis on it – leads it back to the stationary state (see Blanchard (1981)). Since Sargent and

13

Wallace (1973), the neoclassical procedure was to assume that the economy always jumps tothis stable submanifold and is thus made stable by assumption (in the case of anticipatedshocks via a bubble in the old dynamics until the shock actually occurs). The jump variabletechnique of the rational expectations school therefore simply excludes the possibility of anunstable economy by assumption. The reader is referred to Chiarella, Flaschel, Franke andSemmler (2009) for detailed evaluations of this methodology, its attractiveness and its variousanomalies.4

In the following section we therefore return to the case of imperfect asset substitution andimperfect capital gain expectations and will investigate for this case what fiscal and monetarypolicy can do in order to (further) stabilize the economy, in particular in cases where the privatesector is generating a repelling steady state. We conclude this section with the observationthat the model, we are going to study further below, provides one example for a properKeynesian representation of the real- financial market interaction, in contrast to both theexpectations-free Turnovsky (1995) representation of the financial markets at the one extremeand the superperformance of rational expectations agents at the other one.

4I shall argue that the postulates of the classical theory are applicable to a special case only and not tothe general case, the situation which it assumes being a limiting point of the possible positions of equilibrium.Moreover, the characteristics of the special case assumed by the classical theory happen not to be those of theeconomic society which we actually live, with the result that its teaching is misleading and disastrous if weattempt to apply it to the facts of experience. Keynes, J.M. (1936, p.3)

14

5 Stabilizing an unstable economy: Tobin type taxes,

interest rate rules and open market policies

As we approach the last decade of the twentieth century, our economic world is inapparent disarray. After two secure decades of tranquil progress following WorldWar II, in the late 1960s the order of the day became turbulence - both domesticand international.(...)What is needed now is a new approach, a policy synthesisfundamentally different from the mix that results when today’s accepted theory isapplied to today’s economic system.Minsky (1982, p.3)

The present section is only a simple illustration of how this can be done in our model economy.It first of all suggests that the feedback structure of the economy must be taken into accountin order to find the proper measures or rules for monetary policy. Moreover, significantinstitutional change may be needed for the implementation of the simple theoretical rulesproposed below. Though simple in its theoretical structure, this section nevertheless breaksnew grounds for monetary policy making, concerning the role of money and the interest ratepolicy of the central bank.

In view of the considered 5D extension of our model where expectations were no longer static,but are impacting the asset market in significant ways, the most obvious thing to do as policymaker is to raise taxes on capital gains in order to reduce the casino-component in the marketfor equities and for long-term bonds. This is necessary since these markets have or should havethe primary function to channel savings into investment in an efficient way, and not be shapedthrough the gambling activities of asset holders. This was already clearly stressed in Keynes(1936, p.159) and of course an important subject in the works of Hyman Minsky (1982, 1986).

We therefore now apply the Tobin type tax rates τe, τb ∈ (0, 1) to the capital gains achievedby chartists and get on this basis as reformulation of their expectations formation process thenet capital gains expectation rules:5

πee = γeπ

eef + (1 − γe)π

eec = γeβπef

(0 − πeef) + (1 − γe)βπec

(τeq − πeec) (23)

πeb = γbπ

ebf + (1 − γb)π

ebc = γbβπbf

(0 − πebf ) + (1 − γb)βπbc

(τbpb − πebc) (24)

In view of the 5D stability analysis of the preceding section we immediately get the propositionthat Tobin tax rates that are chosen sufficiently high will eliminate all instability caused throughthe expectations formation of the model. It therefore then suffices to concentrate on the 3Dcore dynamics of the model and to investigate what policy options are conceivable in thesesubdynamics that allow to increase the range of parameter choices where the private sector ofthe economy is subject to converging dynamics.

We consider first a conventional type of Taylor rule, based on the output gap solely, sincethere is no inflation gap in this model. This gives the simple interest rate policy rule. Westress again that this rule can be applied without any need for open market operations by the

5The Tobin tax was originally formulated as a transactions tax which should have similar stabilizationeffects, but which acts on the α−parameters of the capital market adjustment rules in a way that wouldintroduce a kink at α = 0 and thus a mathematical complexity we try to avoid in this paper.

15

central bank in our model.

r = ro + βr(Y − Y o), βr > 0.

The considered 3D Jacobian of the private sector of the model

J =

− + 0+ − +− + −

is then augmented by a matrix of the qualitative form

J+ =

0 + 0− − +− + −

with

∣

∣

∣

∣

∣

∣

0 + 0− − +− + −

∣

∣

∣

∣

∣

∣

< 0

The original determinant is thereby made a smaller one and thus more likely to be negative ascompared to the situation without a Taylor rule. Choosing βr large enough will in particularmake any initially given determinant negative.

The only further change in the minors of the original J happens in the principal minor J3

which is augmented by

J+3 =

(

0 +− −

)

with

∣

∣

∣

∣

0 +− −

∣

∣

∣

∣

> 0

We thus get the further proposition that the stability condition we have imposed on the matrixJ3 is thereby relaxed and this to a significant degree if the parameter βr is sufficiently increased.All other minors of the original matrix J are the same as before. The change in the RouthHurwitz condition

(−trJ)(J1 + J1 + J3 + J+3 ) + (|J | + |J+|)

is however ambiguous and must eventually be improved by hte choice of a higher adjustmentspeed in the employed dynamic multiplier process. This condition should however not be of areally significant nature.

At least from the theoretical perspective one may also consider and investigate the followingTobin’s q-based extension of the Taylor rule:

r = ro + βry(Y − Y o) + βrq(q − qo), βry, βrq > 0.

This rule now raises the interest rate further if the stock market is strong and vice versa andthus takes the stock market index now as some further expression for the state of confidencein the economy. Note that this rule is here superior to a rule that concentrates on long-termbond prices in place of share prices, since aggregate demand does not depend on the long-termrate of interest here.

The 3D Jacobian J + J+ is thereby augmented by a matrix of the qualitative form

J++ =

− 0 0+ − +− − −

with

∣

∣

∣

∣

∣

∣

− 0 0− − +− − −

∣

∣

∣

∣

∣

∣

< 0

16

The original determinant is thereby made again a smaller one and thus more likely to benegative as compared to the situation without a Taylor rule. In this situation all minors oforder 2 are also improved in their stability implications and also the trace of J. Moreover theaddition to the 3D determinant is now of the qualitative type J11J1 and thus does not questionthe positivity of the final Routh-Hurwitz condition:

(−trJ)(J1 + J1 + J3) + |J |.

We thus get the proposition that an interest rate policy which takes the state of the equitymarket into account and tries to make it less volatile is unambiguously successful in its impli-cations for the working of the economy. This result is not so obvious when the central bankuses the gap pb−po

b in place of the stock market gap which moreover is not considered with somuch public interest and emphasis as is the market for shares as far as the state of confidencefor the considered macroeconomy is concerned.6

Concerning long-term bonds the central bank might however use open market operations andthus for example the following static open market policy rule which changes the supply ofshort-term against long-term bonds according to the rule:

B(pb) = −pbBl(pb), dB(pb)/dpb < 0.

This rule is assumed to impact supply conditions solely, while demand is still articulated onthe basis of the original wealth position of the asset holders. In this case, the 3D Jacobian Jis augmented by a matrix of the qualitative form

J+++ =

− + 0+ − 0− + −

with similar properties as the case of equities we have just considered (if the upper principalminor of order 2 is positive). We thus finally get the proposition that trading in long-termbonds in the above way therefore provides additional stability to the considered dynamics ofthe private sector. We therefore in sum get that monetary policy can do a lot in this stillsimple framework in order to improve the working of the economy through its impact on thefinancial sector of the model.

6 Conclusions and Outlook

We have reached in this paper a stage where the structure of financial markets (with the excep-tion of credit relationships) can be considered from the macroeconomic perspective as beingmodeled in a basically complete – and nevertheless basic – way, where therefore all fundamen-tal financial instruments (except credit) are present regarding the financing decisions of firms(if the assumed long-term bonds are also considered as corporate bonds besides governmentbonds).

We have considered in such a framework the stability properties of the private sector of theeconomy and – in the case of its instability – the policy measures that are capable of overcoming

6This may change however if the prices of long-term bonds act on output dynamics in the same way wehave assummed for share prices q.

17

the centrifugal forces around the steady state of the economy. These policy measures weregiven by a Tobin type tax on capital gains for long term bonds as well as equities, certaininterest rate policy rules concerning equities (and long-term bonds) and also open marketoperations of the central bank in the market for long-term bonds for example.

In future investigations of the macroeconomic interaction of the real with the financial marketsone must however reintroduce again the advanced treatment of the real markets considered andanalyzed in Asada, Flaschel, Mouakil and Proano (2009). The present paper only indicatesthe direction into which the chapters 3 and 4 of Asada, Flaschel, Mouakil and Proano (2009)have to be developed further, including their treatment of Metzler-Harrod type acceleratormechanisms in the market for goods, the wage-price spiral, the interaction of growth withincome distribution and the stock-flow consistency that is provided by an explicit treatmentof the budget equations of the private households, firms and the government as well as thestock-flow interaction this implies.

These central topics remain however for future research to be reconsidered from the perspectiveof the financial market structure we have designed in this paper.

So here’s what I think economists have to do. First, they have to face up to theinconvenient reality that financial markets fall far short of perfection, that theyare subject to extraordinary delusions and the madness of crowds. Second, theyhave to admit - and this will be very hard for the people who giggled and whisperedover Keynes - that Keynesian economics remains the best framework we have formaking sense of recessions and depressions. Third, they’ll have to do their best toincorporate the realities of finance into macroeconomics.Many economists will find these changes deeply disturbing.(Paul Krugman, New York Times, September 6, 2009).

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7 References

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Asada, T., Flaschel, P., Mouakil, T., Proano, C.R., 2009. Macroeconomic Stock-Flow Inter-action and the Business Cycle. Chuo University, Tokyo. Unpublished book manuscript.

Blanchard, O. 1981. Output, the Stock Market, and Interest Rates. American EconomicReview, 71(1), 132-143.

Chiarella, C., Flaschel, P., 2000. The Dynamics of Keynesian Monetary Growth: Macrofoun-dations. Cambridge University Press, Cambridge.

Chiarella, C., Flaschel, P., Franke, R., Semmler, W., 2009. Financial Markets and the Macroe-conomy. A Keynesian Perspective. Routledge, New York.

De Grauwe, P., Grimaldi, M., 2005. Heterogeneity of agents and the exchange rate: A non-linear approach. In: De Grauwe, P. (Ed.), Exchange Rate Economics: Where Do We Stand?.CESifo Seminar Series, MIT Press, Cambridge, pp. 125-168.

Flaschel, P., Malikane, C., Proano, C., 2009. Output, Stock Markets and Macro-Policy Mea-sures in a Keynesian Portfolio Model. CEMM, Bielefeld University. Working paper.

Keynes, J.M. 1936. The General Theory of Employment, Interest and Money. Macmillan,London.

Menkhoff, L., Rebitzky, R., Schroder, M., 2009. Heterogeneity in Exchange Rate Expecta-tions: Evidence on the Chartist-Fundamentalist Approach. Journal of Economic Behaviour& Organization, 70(1-2), 241-252.

Minsky H., 1982. Can ”It” Happen Again? M.E. Sharpe, New York.

Minsky H., 1986. Stabilizing an Unstable Economy. Yale University Press, New Haven, CT.

Sargent, T., 1987. Macroeconomic Theory. Academic Press, New York, 2nd edn.

Sargent, T., Wallace, N., 1973. The stability of models of money and growth with perfectforesight. Econometrica, 41, 1043-48.

Tobin, J., 1982. Money and the Macroeconomic Process. Journal of Money, Credit andBanking, 14, 171-204.

Turnovsky, S., 1995. Methods of Macroeconomic Dynamics. MIT Press, Cambridge, MA.