SCHOOL OF FINANCE AND ECONOMICSUTS: BUSINESS

WORKING PAPER NO. 96 April 2003 Towards Applied Disequilibrium Growth Theory: IV. Numerical Investigations of the Core 18D Model Carl Chiarella Peter Flaschel Peiyuan Zhu ISSN: 1036-7373 http://www.business.uts.edu.au/finance/

Towards Applied Disequilibrium Growth Theory:ry Numerical investigations of the core lBD model.

Carl ChiarellaSchool of Finance and EconomicsUniversity of Technology, Sydney

Sydney, Australia

Peter FlaschelFaculty of EconomicsUniversity of Bielefeld

Bielefeld, Germany

Peiyuan ZhuSchool of Electricai and Information Engineering

University of SydneySydney, Australia

April 15, 2003

Abstract

In this paper we investigate, from the numerical perspective, the 18D core dynamicsof a theoretical 39D representation of an applied disequilibrium model of monetarygrowth of a small open economy. After considering the model from the viewpointof national accounting, we provide a compact description of the intensive form ofthe model, its laws of motion and accompanying aìgebraic expressions and its uniqueinterior steady state solution. We then give a survey of various types of subsystems thatcan be decomposed from the integrated 18D dynamics by means of suitable assumptionsand also survey the feedback channels that can typically be found in these decomposedor re-integrated model types. These subsystems and their partial or full integrationare investigated and compared in the remainder of the paper from the perspectiveof bifurcation diagrams that separate situations of asymptotic stability from stablecyclical behavior as well as pure explosiveness. In this \ryay we lay the foundationsfor future extensions of the paper, which will show, in contrast to what is generallybelieved to characterize structural macroeconometric models, that applied integratedmacrodynamical systems can have a variety of interesting attractors and transientsto them. Such attractors are obtained in particular when locally explosive situationsare turned into bounded dynamics by the addition of specifically tailored extrinsicnonlinearities.

Keywords: Structural macroeconometric models, decompositions, re-integration, feedback channels, stability basinsJEL Classification: EIz, F,32.

0

1 Introduction

Structural macroeconometric model building, viewed from today's perspective now looksback over a long gestation period with considerable ups and downs and a variety of alternativeprocedures, ranging from the early attempts after World War II to the huge models that werebuilt when this type of applied economic theory was ruling the roost up to microfoundedcontemporary approaches which stress optimizing and forward-looking behavior and therational expectations methodology to deal with the forward looking parts of the model. Thehistory of such model building is presented in Bodkin et al. (1991), while more recent viewson this subject are discussed in Whitley (1994). Recent approaches to structural modelbuilding often involve the market-clearing approaches to macrodynamics, as for exampleMcKibbin and Sachs (1991), but there are also approaches that allow for disequilibrium inthe goods market and within firms, see Powell and Murphy (1997), Fair (1994), Barnett,Gandolfo and Hillinger (1996) and Bergstrom et al. (199a) in this regard.

There is however also the well-established view, see Whitley (1994), that short-run restric-tions on the formulation of macroeconometric models are too arbitrary in nature in order tobe of real help and that at best only long-run restrictions as they are discussed in Garrattet al. (1993) and Deleau et al. (1990) can be justified by economic theory. If short-runbehavioral equations are used then this is only of the basis of equilibrium relationships, sincedisequilibrium is not at all properly understood by economic theory and often specified invery arbitrary terms.

This paper takes the following position in these matters. We believe that real markets(as opposed to financial markets) are generally in equilibrium and subject to sluggish dise-quilibrium adjustment processes for the specifications of which there is a long tradition ineconomic theorizing with a common core, but often with a fairly partial perspective. Thispaper indeed provides a long list of partial feedback channels which have been well knowna long time, but have never been analyzed from an integrated point of view. Had thatanalysis been done, as in the present paper and its theoretical counterparts, see Chiarella,Flaschel , Groh and Semmler (2000) in particular, the outcome that balanced growth pathsare likely to be surrounded by (moderate) centrifugal forces would not look so strange as itlooks from the perspective of for example the McKibbin and Sachs (1991) model that is ofshock-absorber type by its very construction (based on the rational expectations methodol-ogy). Unstable steady states are indeed observed when estimating structural macrodynamicmodels, explicitly in the Bergstrom model, see Barnett and He (1998, 1999a,b), or implicitlypresent in the Murphy model for the Australian, see Powell and Murphy (1997), as simula-tions of the model seem to imply. We therefore suggest that the findings on partial feedbackchains, when taken together, suggest that instability of balanced growth is more likely thanthe opposite and we suggest in this paper a variety of aspects that allow us to draw thisconclusion with more certainty. In sum this paper therefore attempts to demonstrate thatstructural macroeconometric model building should use small, but complete models at leastas theoretical reference point, should allow for disequilibrium in the real markets and withinfirms, should decompose and re-integrate their theoretical reference point in various waysto analyze the interaction of the important feedback structures that are summarized in thispaper and in the other works of Chiarella et al. that we cite, which in our view prove thatne\M progress can now be made in this area of research.

In this paper we will investigate the dynamical model of disequilibrium growth, with appliedorientation, introduced in Chiarella and Flaschel (1999a,b). This model is discussed in

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Chiarella, Flaschel and Zhu (2000) with respect to the various feedback loops it contains,from the analytical and the numerical point of view on various levels of generality, butalways as subdynamics of the simplified 18D core dynamics we have derived in Chiarellaand Flaschel (1999b) from the general 39D case. The first thing we do, in this introductorysection, is to repeat briefly the economic framework within which these dynamics have beenformulated. This will be done immediately on the intensive form level needed for steadystate analysis and for the final presentation of the laws of motion of the state variables tobe employed. We thereby also supply an introduction to the concepts (and their notation)we employ in this paper.

Section 2 then provides a short description of the interior steady state of the model, its lawsof motion and of various algebraic equations that supplement these dynamical laws. We dothis in a way which removes the cross-references still present between some of the 18 laws ofmotion we derived in Chiarella and Flaschel (1999b). We also reformulate the intensive formmodel in an order that is close to a representation for programming purposes. Section 3 willthen isolate the 9D real dynamics of these 18D dynamics by suppressing in an appropriateway the feedbacks from financial markets and from government policy rules.

It is then the task of sections 4 and 5, respectively, to add again, on the one hand, thedynamics obtained from the fiscal and monetary policy rules and, on the other hand, theinteraction with financial market dynamics employed in the general 18D dynamics. Thenumerical investigation of the full 18D dynamics, finally, is started in section 6. We therefind that these applied disequilibrium dynamics do not often support the view of relatedstructural macroeconometric modelling that the steady state of such models will be sur-rounded by centripetal forces, locally or even globally. Rather we find instead that locallycentrifugal forces are a typical outcome of such disequilibrium growth models and these canlead to persistent fluctuations or more complex dynamics around its steady state or even topurely expiosive movements. In this latter case the obtained dynamics must be regarded asincompletely specified and must be supplemented by forces that keep them bounded in aneconomically meaningful way. This additional task, up to one exception, will not be tackledin the present paper however, but is left for future reformulations and investigations of ourmodelling framework. Section 7 will summarize and put into perspective what has beenachieved in this paper with respect to the numerical properties of the 18D core dynamicsof the disequilibrium model of monetary growth of a small open economy as introduced inChiarella and Flaschel (1999a,b).

In summary, this paper continues the investigation of applied integrated disequilibrium mod-els of monetary growth begun in Chiarella, Flaschel, Groh and Semmler (2000). It deepensthe insights of that book, that such high order dynamical systems are already well repre-sented in their fundamental dynamical features by its prototype 6D KMG dynamics andthus basically add numerous interesting details to this working model of integrated disequi-Iibrium growbh. Adding descriptive detail to this model type therefore puts it into a broaderperspective without losing sight of the theoretical core that has been the starting point ofthis work, namely that of Chiarella and Flaschel (2000).

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1.1 The structure of the economy

In order to give an overview of the type of economic modelling made use of in the followingintensive form presentations and their numerical investigation we first of all consider theeconomy's structure by separating it in two parts: The real and the financial sector (whichof course interact in the following modelling of them). We begin with the real part ofthe economy. Note that all magnitudes considered in the following are already expressedin intensive form (denoted by lower case letters in the place of formerly capital ones), byrepresenting their analogs per unit of real or nominal capital (depending on whether weconsider real or nominal extensive expressions) and by using efficiency units in the case oflabor (due to the assumption of Harrod neutral technical progress in the fixed proportionstechnology employed in the sector of firms).

Labor Non traded Goods Exports Imports Dwellings

'Workers

Asset holders

Firms

Government

Ie : a¿ll

td,e tueol , tÍ

td,e -

td,wog -og

ci

sÊ

f ,Y,Sdk,t/K

I

ctn

""n, sdh

jd

Prices

Expectations

u"rw"ru)b"r1Du"

r:þlpa : (l * rr)pu p, : ep| p* : (L * r*)epi

*_,ae^-yu

PntPy

r:þlStocks

Growth

q

n

KIK:1,ru:N/K

sÍ-õ- ad)1"

l+n

i<:ñ:@

k¡:sf,-t¡

Table L: The real part of the economy

The columns of the table refer to the different goods in our model: labor, non traded good,exports, imports and dwellings. The first four rows refer to the considered sectors: privatehouseholds, firms, and the government (fiscal and monetary authority), with the privatesector split into asset holders and workers in addition. We distinguish between workers andasset holders to allow for a simple treatment of income distribution and its implications.Other important items of this table are the goods' prices and their expected rate of changeas well as the stocks of labor force, capital and houses and their growth rates. Note thatthe foreign countries do not appear explicitly in the table. But by allowing for exports andimports it is clear that imports for the home country implies that this goods are exports forthe foreign countries and vice versa. So we have to introduce prices for those goods thatmust be sold or bought abroad: p] denotes the price for the export good of the domesticeconomy, while pi denotes the price that firms pay for the imported good. Note that theseprices are considered as fixed in the following model economy.

Only the workers of the sector of private households supply labor. The amount of this supply

3

l" depends on the number of workers in working age Il and the given participation rate a¿.

Therefore the dimension of the supplied labor /" is a number of persons (representing thenormal working day and per unit of capital and measured in efficiency units). In contrastto this the dimension of lj', the labor demand, is hours actually worked. This distinctionis used for modelling over- and under-utilization of labor in the firms' sector. Intermediatebetween hours worked and labor supply is the workforce employed by firms l'i", that is thenumber of persons who work within firms. The column representing the labor market lacksan entry in the row of asset holders because asset holder do not supply labor nor do theydemand it. The government needs labor I!" for providing public goods. But in contrast tofirms we assume that there is no need for over- or under-utilization of this part of the laborforce which by assumption gives lj" - liThere is a set of price expressions for labor effort: t¿" is the nominal wage rate (before taxesand in efficiency units) that workers get for a time unit of labor. In contrast to this t¿b"

represents the amount that firms or the public sector have to pay for one unit of labor,because they have to pay payroll taxes in addition. The income of unemployed and workersbeyond working age is also considered as a kind of wage rate and thus represented in thelabor market column. They are denotedby w"" andw'" (where e stands again for efficiencyunit). Expectations about price and wage inflation are here simply based on expected priceinflation throughout. They will appear as medium run expectations zr¿ solely in the following.The growth rate of the stock of workers in working age (as well as the one of retired persons)is assumed to be a constant: n.

The non traded good serves for workers, but not for asset holders (due to our simplified18D dynamics), as consumption good in the amount c!. For the latter group it serves as

investment good for the supply of dwelling services. The firms' sector produces the quantityof the non-traded good gr restricted by a full capacity production of Ap. Secondly the firmsuse the domestic good for intended inventory investmentsllK as well as for business fixedcapital investments gf . The government uses the domestic good as public consumption good.

The prices for the non-traded good can be denoted inclusive or exclusive of a given valueadded tax, by pu and p, respectively, and expectations refer to the expected growth rate ofboth pr,po. Stocks of the domestic good are held only by the firms' sector. The businessfixed capital stock is K and the actual inventoriesper unit of capital are denotedby u.

The export good is the second output good of the firms. It cannot be sold in the domesticeconomy. We assume, that every amount ø of this good that is produced can be sold onthe world market at a price p, thal depends on the given price abroad pi, and the exchangerate e. The import good is only for use in the sector of firms. They need it as an inputfactor for production. Its price depends on the exchange rate e and the given foreign pricepl augmented by the rate of import taxation r-.The asset holders supply the dwelling services cf . For simplicity we assume that only workershave demand for dwelling services cf. The domestic good serves for gross investments intodwelling services gÍ.W" thus have to consider two prices in this sector of the econom/: p¿,

the rent for dwelling services, and po, the price per unit of investment into dwellings. Thereare no value added taxes on investment good purchases. The capital stock in the housingsector is fr¿ and its growth rate depends on gross investment in dwellings minus depreciation.

Next we have to consider the financial part of the economy. The rows of table 2 describe allfinancial assets of our model. They consist of short-term bonds, long-term bonds, equities

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Short-term Bonds Long-term Bonds Eauities Foreign Bonds

Workers A.l@"X): B-b-

a./@,x): Ê"b. a!l@,x)Asset holde¡s nl@"x) aLl@"x)

irl@"x)Firms

Government ø¡1p,X¡: þ6 Bt l@"K)

P¡ices

Expectations

1 ['] Pu: Ilr¿ Pe

r.: þ2

ePi : e'Llri

e: ê"nu: þ3

Stocks

Growth

b: Bl(p"K)

B

e : E l(p"K)

E

bL: BLI@"K)

BL

bt : B¿l(puK), öi : Bl/(p,K)

B',8t,

Table 2z The frnancial part of the economy

and foreign (long-term) bonds. Note that money is not considered as a store of value in thepresent model, see Chiarella and Flaschel (1999a) for the details and justifications. The firstfour rows show, how the sectors interact on all the asset markets. Note that only flows areconsidered in the first part of this table.

The first row has only one entry. We assume that the only way workers do participate inthe asset markets is by holding short-term bonds (saving deposits). In contrast to this thepure asset holders do spread their savings to all kinds of financial assets: bonds (domesticshort and long term bonds as well as foreign long term bonds), and equities. The latter areissued by the firms' sector and represent the only way of financing the deficits of firms inthe present model, i.e., bonds are issued only by the domestic and the foreign government.Short term bonds have a fixed price equal to unity and the flexible interest rate they offer isr. The long term bonds'price is Ifr and the interest consists of the annual payment of onedollar (secalled consols or perpetuities).

The above represents only a short description of the structure of the economy underlying itslaws of motion to be considered in the following section. The reader is referred to Chiarellaand Flaschel (1999a,b) for more details, also with respect to the following brief representationof the national accounts of the sectors allowed for in this approach to disequilibrium growththeory.

L.2 National Accounting (in intensive form)

The structure of the considered economy from the viewpoint of national accounting is thefollowing (everything being measured in nominal domestic currency units per gross value ofthe capital btock):

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L.2.I The sector of firms (Table 3)

The firms produce two kinds of output, the pure export good which is tradeable only on theworld market and the domestic good which can solely be sold in the domestic economy. Thedomestic good serves as the consumption good for the workforce and the government (inour simplified 18D dynamical version of the model). It can also be used for investments ininventories, in business fixed capital and in housing. Firms use three kinds of inputs for theirproduction: imports, capital, and labor. The capital stock in the firms' sector depreciatesby a given rate ô. Value added taxes (on consumption goods solely) appear on the leftside of the production account and have to be paid to the government. The balance of thisaccount is the profit of the firms' sector. Note again that all expressions are in intensive formas already discussed in the preceding subsection (they have all been measured in domesticcurrency units in Chiarella and Flaschel (1999a,b) and are divided here uniformly by p,K,the value of the capital stock (including value added taxation by assumption).1 W'e stressthat the profits are not subject to any direct tax. By assumption profits are only used tobe paid as dividends to asset holders (and then taxed) or to be used for planned inventoryinvestments. One can clearly see this in the income account. The accumulation accountdisplays again that investments in business fixed capital and in inventories are the onlystocks which can be accumulated by firms. There is no possibility to accumulate financialstocks, i.e., no holding of bonds by firms in the present context. The financial deficit offirms must be financed in our present model by selling new equities. This assumption is ofcourse not very realistic, and thus should be modified in future reconsideration of the modelto allow in particular for bond financing and loans of firms in addition.

1.2.2 Asset holders (Table 4a)

While firms produce and sell two types of goods, the sector of the private asset holderssells dwelling services. Hence there is a production account for this sector. The income ofthis sector consists of interest payments (long and short term bonds, the former also fromabroad), dividend payments from the sector of firms, and the profits from selling dwellingservices. This income is reduced through profit income taxation. The remaining amount isthe saving of this sector (since asset holders do not consume in the 18D core dynamics ofour general model to be considered in this paper). Savings plus depreciation is spiit intogross investment in housing and the financial surplus in the following account. The financialsurplus is distributed by asset owners to all kinds of financial assets that exist in our model.

L.2.3 Households ('Workers) (Table b)

This sector does not take part in private ownership production, but only provides the laborinput for firms. Therefore the production account remains empty. The income account in-cludes wages, unemployment benefits, and pensions. Worker's income is allocated to income

lNote that all investment and thus also the value of the capital stock and the measure of the rate ofprofit based on it are in prices pn net of value added tax, since these taxes are only applied to consumptionpurchases and not to investment purchases in the present model. Note also that the following uniformintensive form representation of the model does not immediately apply to the structural form of the modelin intensive form, since we do not need accounting homogeneity in this structural form as is necessary in thepresent subsection.

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taxes and consumption and savings. All savings is allocated to short-term bonds.

Production Account of Firms:

Uses Resources

Imports ephjd lp,

Depreciation 6polp"

Value Added Taxes ro(cf, * ùpa/p"

Taxes on imports r*ephjd /pu

Wages (excluding payroll taxes) w" f pulj"

Payroll Taxes row" f polt"

Profits (O'+I/X¡po¡p"

Consumption c!

Consumption g

Exports epirlp"

Gross Investment gflprf p,

Durables (Dwellings) Sfrpulp"

Inventory Investment puÑl@"K) : pa(U -vd)lp"

Income Account of Firms:

Uses Resources

Dividends p"pyfp,

Savings TlKpolp"

Profits (p'+îlK)pulp"

Accumulation Account of Firms:

Uses Resources

Gross Investment gftprf p,

Inventory Investment N /Kpr/p,

Depreciation õprlp,

Savings Si /(n"K)

Financial Deficit F D I @"K)

Financial Account of Firms:

Uses Resources

Equity Financing p"E l(p"K)Financial Deficit F D / (p"K)

Table 3: Accounts of Firms

7

Production Account of Households (Asset Owners/Housing Investment):

Uses Resources

Depreciation ô nlcnpu / p"

Earnings Tln/(p"K)

P"ent p¡co¡/n"

Income Account of Households (Asset Owners):

Uses Resources

Tbx payment r.rb

Tax payment r"b!

Tbxes rs(p¡cf /Fu - 6¡k¡pu/,p"¡

Tax payment Tcpepu/p!

Savings St /(p"K)

Interest payment rb

Interest payment ô¿,

Interest payment e(t - r)bt,

Dividend payment p'pu /p"

Earnings nn/(p"K)

Accumulation Account of Households (Asset Owners):

Uses Resources

Gross Investm ent glpu / p,

Financial Surplus FS / (p"K)

Depreciation ïnhnpu / p"

Savings St/(prK)

Financial Account of Households (Aeset Owners):

Uses Rnsources

Short-term bonds Bb

Long-term bonds p6.Ê{bl

Foreign nonas e A \bt, / riA

!.jqurtres pelr6

Financial Surplus F S / (p" K)

Table 4a; Accounts of Households (Asset Owners)2

2ExpressionssuchasBö(: È/@rX))areusedtoindicatethewaythelau'of motion,of here b: Bl(prK),

8

has to be derived.

Production Account of Households (Workers):

Uses Resources

Income Account of Households (Workers)r

Uses Resources

Tbxes r-þ"ld. + wu"7l" - tue) * r""til/p,

Consumption cfi * p¡rc"¡/O"

Savings Si,/(p,K)

wages weld" f nu - (w"tj" + u"tg.)/p!

Unemployment benefits u"(1" - l-")/p"

Pensions -"t13/p"

Accurnulation Account of Households (Workers):

Uses Resources

FinancialSurplus FS/(p"X) Savings Sß/(prK)

Financial Account of Households (\Morkers):

Uses Resources

Short-term bond accumulation .Ê-ô- Financial Surplus FS/ (p"K)

Table Abz Accounts of Households (Workers)

1.2.4 Fiscal and Monetary Authorities

The government sector's production account takes up the costless provision of public goodswhich is defined to be identical to self consumption of the government. To provide theeconomy with those provisions the government has to buy goods and pay wages to theworkers it employs.

The only sources of income for the government are the various taxes. They are used for in-terest payments, pensions, unemployment benefits and salaries. The balance of this accountare the savings of the government. Generally these savings are negative hence there is afinancial deficit in the accumulation account, rather than an financial surplus in general.

In financial accounting of the government one can see the sources from which the deficit isfinanced: issuing short'and long-term bonds.

9

Production Account of Fiscal and Monetary Authorities:

Uses Resources

Government expenditure for goods g

Salaries wb"t!"/n, = (w.I!" +hw"tg")/pu

Costless Provision of

public goods = self consumption

Income Account of Fiscal and Monetary Authorities:

Uses Resources

Interest payment rb

Interest payment öl + ôl-

Pensions ."t13/p"

Unemployment benefits wte (le - Pt) / po

self consumption g

Savings Si /(p"K)

'Wage income taxation rylwtld" + wte(le - ¿ue) t*"lll/P"

Profit and interest ta><ation rcp" pu / p, * r.rb * r"b\ *,.bf

Rent income taxation r.(p¡co¡/n" - õnknpu/p")

Payroll taxes (rrw"Ij" + bu"¿|.)/po

Value added tax r"(cfl * ùpu/p"

Import taxes r-epfljd f po

Accumulation Account of the Fiscal ,A,uthority:

Uses Resources

Savings Si /@"K)

Financial Defrcir F D / (ps K)

Financial Account of Fiscal and Monetary Authorities:

Uses Resources

Financial deñcit F D / (ps K) Short-term debt Bb

Long-term debt Êtb'/r¿

Table 5z Accounts of the Fiscal and Moneta,ry Authorities

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I.2.5 International relationships

The external account contains all transactions with the foreign countries. It exhibits theamounts of goods, capital, and interest payments that cross the borders.

External Account:

Uses Resources

Exports eplrlp"

Factor Income from Abroad e(L - r)bt2

Capital Imports ÊlUl lr¿

Imports epl^jd lp"

Factor Income to Abroad (I - r")b¿i

Capital Exports eAlrb¿, I r;

Table 6: International Relationships

This closes this section on the national accounts of the model to be investigated numericallyin the following sections.

2 Explicit representation and feedback structure of thecore 18D dynamical system

We will base our subsequent numerical investigation of the 18D core model of the generalmodel, see Chiarella and Flaschel (1999b), in this paper on the following condensed formof its 18 laws of motion (adjusted to and to be used for programming purposes in thefollowing) and the unique interior steady state (up to the level of nominal magnitudes) thatthis dynamical model exhibits. In order to simplify the notation to some degree we assumein the following, in addition to what is assumed in Chiarella and Flaschel (1999b), that theriskandliquiditypremium€:0andthuswillhaver:rI:yt*:p"forinterestandprofitin the steady state. For the same reason we also assume for the normal employment rateVi :1, and also C" :0,'i.e., there is no consumption goods demand of asset holders whothus save all of their income. All these assumptions have only slight influences on the steadystate position of the economy, and do not alter at all the dynamics around the steady state.

We consider the 18 steady state values of the model first. All these values have an index'o' (denoting their steady state character) when used for programming purposes. To notoverload the notation here we do not add this index to the following list of steady statevalues. Note again that all steady state values are expressed in per unit of capital form andif necessary in efficiency units.

a:,

uo,ue[ 1,o

r"

APU ITffi,' [a": a'Úl

þ"ouïIlZ: tsaeU [total employment: I!" : tî: + Iî:,tî: : dssa""]

(Ii: + assaï)lV

(1)

(2)

(3)

(4)

11

Pu (5)

(6)

(7)

(8)

(e)

(10)

(11)

(12)

( 13)

(14)

( 15)

( 16)

(17)

(18)

e"a

weo

*t/lo

pfl

k"h

bo

bi

P"b

*o/l bs

€3

ro

_oln¿

_ot 'u)

eo

I1-ru'ub,fPi

L*rr'0

: ei?i + õn)ltncz(u|G-s)-(.y+ô))

lp" arbitrarily given]

lr*:(L+ro)#:W,

c1(r[ * õùlG + r") + cz(t -t õn)

a'gbda:,

: ri1 - "!)d'u""

pzl

Phjs

1-Pi"ju

pEUnki

tlri0

0

ri [:Plrra -

c2(I + r,)pia?,t

so - lr.a?o, + h#Ii" + ffi(a8 - 0+ ô) - (z + ôn)kB)l

c9¡ : Unki

tZ : r"lrf I Q ]- ro) + robo * u'" + @ilpÐ":nl G ¡ ru) - 6hk"hlG+ r,)l

so:ga3+robo*b,._t".-&[o"(t""_ry)+a,L2(0)l¿1(0)¿å]

+ (r+rr)ffiaseu""-#uin : w""[ry" * a"(li - li"-) + a' L2(0)l L{o)l:,]l ((7 + r")pl)

r^pkjsU"l(Q + r,)pfl)

With respect to the last two of the above equations, for the taxation rate r- and for the rateof exchange e of. the model, we have to apply (besides the above definitions of Uo,lfr", andø!", see the above) the further defining expressions:

in order to have a determination of the steady state that is complete.

Note that the value of the exchange rate eo will be indeterminate when we have rm : 0 in thesteady state in which case the above formula for eo cannot be applied. Note furthermore thatthe parameters of the model have to be chosen such that le¡o,T-o(r^o), eo are all positive inthe steady state.s Note finally that the parameter o" must always be larger than 1 -lf B,lorr:pbte,p" in order to satisfy the restrictions established in Chiarellaand Flaschel (1999b).

SThere are furthe¡ simple restrictions on the parameters of the model due to the economic meaning ofthe variables employed. Note also that the steady state rate of wage taxation must be defined in a differentway when the housing sector is removed from the model.

L2

Equation 1 gives (the steady state solution of) expected sales per unit of capital K (and alsooutput per K) and eq. 2 provides on this basis the steady inventory-capital ratio N/K. Eq.3 provides the amount of workforce per K employed by the firms which in the steady stateis equal to the hours worked by this workforce (assuming that the normal working day orweek is represented by 1). It also shows total employment per K where account is taken ofthe employment in the government sector in addition. Eq. 4 is the full employment laborintensity (in the steady state). Eq. 5 provides the price level (net of value added tax) andeq. 6 gives the wage level (net of payroll taxes) on the basis of the steady state value forthe real wage øb". The steady state value of the inflation rate expected to hold over themedium run is zero, since the inflationary target of the central bank is zero in the presentformulation of the model.

Next we have the price level for housing rents (in eq. 8) and the stock of houses per unitof the capital stock K (in eq. 9). There follows the steady state value of. b: Bl(p"K) aswell as the one for long-term domestic bonds. The price of these bonds is given by the givenpricelfri offoreign long-term bonds in the steady state, see eq. 12. Since there is no steadystate inflation there is no change in the expected exchange rate and there is also (always) nochange in the price of long term bonds, i.e., both markets exhibit rational expectations inthe long-run. The steady state value of the short term rate of interest settles at its long-runequivalent as there is no risk or liquidity premium allowed for in the 18D version of thegeneral model. Import taxes r- just balance the trade balance in the steady state, see eq.16, while the wage tax rate r., must be calculated by means of gross steady wage income!.7 aîd the marginal propensity to spend this income for housing services, see eq. 17. Eq.18, finally, is the most complicated one and it provides the steady state value of the rate ofexchange which depends on nearly all of the parameters of the model, due to the definitionalterms shown that have still be inserted into the expression for e shown in eq. 18.

This closes the description of the interior steady state solution of our dynamical model. Nextwe present the 18 laws of motion which have been derived in Chiarella and Flaschel (1999b)and which of course also employ the state variables we have just discussed.

Making use of the formula:

Apu : þo - nt : n[ne(þ_,(v -v) + Ê.,(I\"lti" - t)) + þo@ly, -u)],

with rc :IlG- K.Kp), for the deviation of the actual inflation rate from the one expectedover the medium run, the laws of motion around the above steady state solutions of thedynamics read as follows:a

ù" : þo"(uo - a1+ (r - þl - Ð)a",ù, : a-ad-þl-6)r,

i';" : þ,U1" - ry1+ [r - (sfl - Ð]ry",aNote here that we ¿rssume zr : 0 for the target rate of inflation of the central bank which implies that

there is no inflation in the steady state. We therefore can use price levels (for goods and housing services)as state variables of the model. Furthermore, since money supply is driven by money demand in the caseof a Taylor interest rate policy rule we (implicitly) get that money supply will grow with the same rate asthe real economy in the steady state. Note also that the Tobin's q is a further state variable of the model(representing the dynamics of share prices in particular) which however does not feed back into the 18Dcore dynamics since neither investment nor consumption depends here on the evolution of share prices byassumption.

(1e)

(20)

(2r)

13

l"

û)"

Pa

ilt

Pn

îtn

b

bt

,ID

: t-þl-6),: rt + nlB.,(y" lt" - V) + p_,(tj" lti" - 1) + n-þp(alyp - U)),

: rt + nlnr(p.,(t" lt' - V) + p,,(t\" lti'- r)) + þo@ly, - u)1,

: Bnt(o'ntL,þ, * (1 - e,r.)(O - o')),n?: B^Ê, - Uh) + KhLþa + iTt ,

: gÍ- 6n- (gÍ- d),

: onalga" *rb*bI -t" -t" +g"l- (Lpr+r¿ + gl- 6)b,

: (t- "!)lpolgy"

trb+bt -to -t" + g"l- (Lpo+rt + gt- Ðbt,,d , b+p6bl: a"-r(ã- L), d - ;; ,

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(2e)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

These laws of motion make use of the following supplementary definitions and abbreviations,which provide the algebraic equations of the model:

: A" * þ"(\dA" - u) + 1Bnoy",: IuU,

Aul :

lT" : asgae,

tj" + tdf,,

I'i' + li",

,'Uo" * au (t" - t'") + t' L#rlilG t r,)pal,

.t(1 - r.)A.r,(L + r")pocr(I - r.)aø I pn,

u" - õ+ (epi,lpòraa - (Q+rr)w"lpòIl" - ((1 +r^)epilpr)joy,

"f ((1 - r")p'- ((1 - r")rt - n'¿)) + a\(r¿ - r),

"t@lao -Ú)+ j*6, rt:rlpu,

"f((1 - r")((pnlpò""n/ kn- ôr,) - ((L - r")r¿ - "ID + a|(r¿ - r),

"!(t - Uù t 1 -t ön, rt : rlpu,

""n+ gt+ gl,ko* gu",

os'trbs+(1 -a")þu,

u

t!"

tg.

P"

I-'

ci

ct¡

p"

st+

sÍ:

a'lTb

+

T4

go:

+4,tJ-

++Ct-

w"lau(l" - t ") + .,L#t" + (1 + rr)t!")l (r t r,)po,

r*w'lld" t au(t" - l-") * *'ffirll(G i r,)py) + row"ld" f((r + r,)pa)

ftfr' - gÍ - gÍt'n) -t r*epijra l(Q * r,)pa),

,"[p" I G t r,) + rb + bt + (pnlpòc?,/ Q + r,) - 6hkhl 0 + r")].

Inserting these equations into the above 18 laws of motion gives an explicit system of eighteenautonomous nonlinear differential equations in the 18 state variabies (19) - (36) shown above.Note that we have to supply as initial conditions the relative magnitude iudSì i" order to geta complete characterization of the dynamics and that the evolution of pricè'levels is subjectto hysteresis, since it depends on historical conditions due to our assumptions on costlesstransaction balances for the behavior of the four agents of the model.

In table 7 we break down the state vector X of the 18D dynamics into subsectors corre-sponding to the subsectors and their subdynamics that we investigate in sections 3,4 and5 below. These subsectors are: X,: (U",|'T",f ,w",pa), for the real core subsector (withseparate equations for wage and price inflation); X-und,: (trt), for the subsector engender-ing the Mundell effect; Xn: (pn,kn), for the housing subsector; XÍn: (b,bt,r-), for thefiscal policy subsectoîi X^o: (r), for the monetary policy subsector; Xd: (pb,rb), for thedomestic assets subsector; Xl : (T*,e,e"), for the foreign assets subsector (including im-port taxation). All of the statically endogenous variables are gathered in the vector Z. WiIhthese definitions the full 18D dynamics that contains all the complex feedbacks between thevarious sectors identified above is succinctiy represented by X : Frc(X,Z).

The methodology vue use to analyze such a high dimensional dynamical system is to switchoff most of these feedback mechanisms so as to focus on the core real part of the model. Afteranalyzing these subdynamics we gradually switch back on the other feedback mechanisms.Table 8 lays out what we call the on/off switches. These are the amendments that need tobe made to the 18D system in equations (19)-(36) to suppress the feedbacks from the varioussubsectors (by way of assumptions shown below).

We investigate the dynamics via numerical simulations that attempt to give the reader globalinformation. In particular we display (i) bifurcation diagrams of output with respect to keyparameters such as speed of adjustment of wages, prices, expectations on inflation and salesand inventories, (ii) eigenvalue diagrams, (iii) stability basins with respect to the same keyparameters, and (iv) some typical time series patterns of the key economic variables. Wedisplay in table 9 the common parameter set used in the simulations.

The stability basins indicate parameter combinations for which the system dynamics:-

1. are converging to the interior steady state,

2. exhibit sustained oscillations around the steady state, or

3. are totally explosive.

the initial values for all basin calculations were obtained by perturbing the steady state valueof sales expectations by five percent. It should be borne in mind that a different shock (andhence different initial conditions) could produce different looking basins.

15

We stress that the above dynamical system is intrinsically nonlinear due to:

o the growth rate formulations employed in the model, and

o due to various unavoidable products and fractions of the state variables of the model.

The state vector X:

T -lI

I

I

corel

J

Xn===Pn

!o= =b

6t

X

a"u

ry"F'tt)"

2a-

"t-

J*=r==pb,rb"

===Tm

e€s

real

housinqsector

X,

X^und

x¡n

X¡

Lt-t-t-I

!I

I

I

Lrl_t-I

Lt-

I

I

t_

-]

I

-lI

_ _ 13"9'Lp"liE X^o-l

domestio tlassets | '/rd

J----l

foreigrlassets

¡

The vector Z of. statically endogenous variables:

z : (a, I!", I!', \f,", Id", I'", u r, con, con, p", gÍ, gl, ud, rr b, go, t", t")

The dynamical system:

* : Fn(X, Z(X))

Table 7: The Structure of the 78D Dynamics

16

þn,:0, r¿: rl Mundell effect off

c2:0, gÍ:0, þn: Q,k¿(O) : ¡

p*ry - pkiu

Phjaþ": þr" :0; e : eo

þpo: Bno":0; Po: P8 + domestic assets off

fiscal policy off (up to irrelevantmovements of ó, b¿)

0rr: þrr: þrr:0; r : ro monetary policy off

Ktu: I Rose real wage effect off

Table 8:

In order to put the above into perspective and to show the relationship of the above 18Ddynamics to the general structure that can be associated with integrated models of dise-quilibrium growth we close this section with a general survey and a brief discussion of thepartial feedback chains that can be part of models of disequilibrium growth. Table 8a showsin this respect the feedback mechanisms that may be part of the dynamics of the real part ofthe economy (concerning goods and labor markets dynamics). This table shows the Keynesand the Mundell effects and the two types of Rose effects (all present in our 18D dynamics)and furthermore the Pigou and the Fisher debt effect (not present in the 18D dynamics dueto the neglect of wealth effects in consumption and the neglect of debt in consumption andinvestment behavior). W. also consider in table 8a certain real accelerator mechanisms ofwhich only the Metzlerian inventory accelerator is present in our model (as an improvementof Kaldor's dynamic multiplier trade cycle component). Harrod's investment acceleratingmechanism is however partly present in the 18D dynamics, since the rate of capacity utiliza-tion of firms influences their investment behavior in a proportional, but not yet in a derivativeway. 'We thus see that our 18D dynamics already contains a variety of mechanisms (but notall) that are typical for the Keynesian analysis of disequilibrium growth.

Let us consider next the partial feedback mechanisms shown in table 8b which basicallyconcern the financial sector of our economy.

=>housing off (up to irrelevantmovements of p¿ via r")

foreign assets off

+

+

The On/Otr Swiúcåes for the Analysis of the Subdynamics

ar-, :0, r* - 7flb: bo, bt : bL

77

Feedback Mechanisms in Models of AS-AD Growthin the Real Part of the Economy

Keynes Effect *1 +pî +r1 +ttcJ.+ YJ=+ t!+.!

known to bestabilizing

Pigou Effect *1+pî+Mlp!+cJ+YJ=+ t!+wI

known to bestabilizing

Normal RoseEffects

w/pI+tUcl+vltIwJ!pI+w/ pIor rJCllandY,L1 wT pllw/ pJ

can be stabilizing,depending on C,Iand adjustmentspeeds

Adverse RoseEffects

wtpl+¡llcl+vItJ+wJpll+wtpT

or.ItClandY,L1w1l plwI pI

2 unstable cases.remedy: sluggishwage and priceadjustments

Mundell Effectw1 + nî + n'1 r-f J,

+tTcl+Y,LÎ +nTwT

real interest raterule, kinked Phillipscurve

Fisher DebtEffect

w!+ ptr+ Dl pî+ tIcI+Y,LI+w,ptr

downward rigidwages and prices +..'?

Harrod TypeInvestmentAccelerat<¡rs

vÎ+t11+v'1v"1v1 fiscal policies ofPID controller type

Kaldor TypeDynamicMultiplierInstability

Yî+Yd11+r"1v1 nonlinearinvestment function

Metzler TypeInventoryAccelerator

Expected.Sales.Y' 1

P lanned. Inventories,s 1

Y =Y" +51c1 tlYd 1

Actual. Inventorie s I Y", 31

cautious inventoryadjustment fa¡ offthe steady state

Table 8az Partial Feed,baclc Mechan'isms in the Real Part of the Economg: Summary.

1B

Feedback Mechanisms in Models of AS-AD Growthin the Financial Part of the Econom

Capital GainAccelerator:Long-termBonds

pil Expected.Returnl

Bll pol p"rî

cautious adjustmentfor largediscrepancies inreturns

Capital GainAccelelerator:Equities

pi 1 Expected.Returnl

Eo 1 p"Î pil

cautious adjustmentfor largediscrepancies inreturns

Capital GainAccelerator:ForeignExchange

e" 1 Expected.Returnl

B'01e|e"1

Cautious adjustmentfor largediscrepancies inreturns

E.g.: Anti-CyclicalBehavi<lr ofInterest onLoans

Y1 Screening-cosrsJrJt,c1vo,v" 1v1

Taylor type interestrate policy rule?

E.g.: WealthEffects inMoney Demand

wtplMo/p|r1c,tIYo,Y',yLptwtpT

Pure moneyfinancing ofgovernment debt?

DisposøbleIncomeMeasarements

Changes inDisposableIncome,AggregateDemand andEconomicActivity

plf1YD =Y-T-r"W / pIcJvd,Y"IvJpI

is stabilizing, sinceinflation decreasesdisposable incomeand thus economicactivity

Table 8b: Partial Feedback Mechan'isms 'in the F'inanc'ial Part of the Economy: Summary.

19

The financial accelerator mechanisms of this table are all present in our model underlyingthe 18D dynamics, though the one concerning equity markets does not feed back into thesedynamics. They all state that expected returns exercise a positive feedback on actual returnsand are thus destabilizing to a certain degree. The real financial accelerator mechanism ishowever not part of the model underlying the 18D dynamics, since it concerns loans tofirms which may become cheaper in the boom and more expensive in the depression whichstrengthens booms and deepens depressions. Also not included in the dynamics are wealtheffects in asset and in particular money demand, due to our neglect of money transactionson the one hand and the neglect of portfolio considerations on the other hand. Finally, theconcept of disposable income we employ is still of the simple Keynesian type that does notyet consider the influence of inflation on the wealth of economic agents and thus on theirconcept of disposable income. This brief characterization of the financial elements containedor not contained in the 18D dynamics shows that its formulation of the dynamics of thefinancial part is still of a fairly preliminary nature.

Tables 8a,b therefore also indicate what remains to be done in order to arrive at a fully devel-oped descriptively oriented macrodynamics that incorporates all important feedback chainsof a modern market economy. Our development of theoretical representations of structuralmacroeconometric model buildings will continue to approach structures a^s surveyed in tables8a,b, see for example Chiarella, Flaschel, Groh, Köper and Semmler (1999a,b) for interme-diate steps in this direction. In the next section we now begin with the numerical analysis ofthe considered structural model. The reader interested in theoretical results on the stabilityand the loss of stability in models of this type is referred to Chiarella, Flaschel and Franke(2003) and Asada, Chiarella, Flaschel and Franke (2003), in particular with respect to atypical methodology that allows to establish asymptotic stability theorems in high orderdynamical systems.

3 Numerical simulations of the real part of 18D dy-namlcs

In this section we consider the dynamics of the real part of the economy on various levels ofgenerality,by switching off the feedbacks from the financial markets as well as from fiscal andmonetary policy. These aspects of the full 18D dynamics will be added back successively insubsequent sections. Table 10 lays out the way we develop the various real subdynamics byuse of the on/off switches.

Due to the fact that the laws of motion contain the housing capital stock in the denominatorin some places we have set adjustment speeds in this section only to very small magnitudes,but not to zero in order to avoid division by zero during the simulations. Note finally thatthe external rate of growth 7 has been chosen very high. In the current low dimensionalreal dynamics there exist stability problems when both the rate g, determining governmentexpenditures, and 'y are chosen reasonably low. It appears as if the dynamics is more rigidand explosive in such low dimensions than it is in a full 18D setup (as we shall see later on).

We start with the full 9D version of these real dynamics (which includes the nominal dy-namics of wages and prices and expectations about their rate of change).

20

3.1 The 9D real part of the economy

We separate the real part of the dynamics, i.e., labor and goods markets, from the rest ofmodel by switching off foreign assets, domestic assets, fiscal policy and monetary policy.

The condition for switching off foreign assets must be guaranteed via an appropriate choiceof the four parameters that govern the equation underlying it. This condition freezes thenominal exchange rate at its steady state value. The condition for switching off fiscal policysays that government does not care about the evolution of its debt position and keeps therate of wage taxation (and import taxation) fixed at its steady state value. The conditionfor switching off domestic assets freezes domestic asset prices at their steady state position.Finally, the condition for switching off monetary policy does the same for the short-termnominal rate of interest.

þ-tt)Ppit

þ.þn¿

ß,tDP13

asb

Q,U

ahgd.-

okg

1pUn

Iy

Pi"ô.Y

Tp

jyap

0.400.500.100.100.100.100.500.500.100.500.100.500.902.001.000.100.060.300.101.00

þ-zþoo

þ,þnôP12

agQt¡l

ah2O¿-

o¿kz

¿r(o)ll'uVotp

p;6n

riTa

trh

Pa

0.500.100.100.800.500.200.100.500.500.5020.000.500.900.001.000.100.080.150.501.00

þpnPna"

þnAt

þv'a¿

ahte,rotktQ.s

Lz(o)UdshockCl

ITc

C2

ru

0.700.100.200.501.000.500.100.500.100.505.000.900.601.050.500.330.500.330.20

Table 9z The Patameter Set

Using again as abbreviation:

Lpy : þr_ot: n[np(þ.,(p"lF _V)+ p.,(tl"lry" _ r)) + þo@la, _t)),

with rc:|lG- K*Kp), the 18 laws of motion of the economic dynamics around the steadystate solution are then reduced to the 9D real dynamics:

a" : þr.(yo - a1 + 0 - Øl- õ))a",

ù : a-ad-Øi-õ)r,i'1" : þ,(tj' - ry")+ [r - Øl - 6)]tî",

î" : "y-(gÍ-6),û)e : rt+rc[p,,(t-.lr -V)+ p-,(t\"1ry. -1) + n,\p(alap -(])1,Pn = rt+L,þ,,

(37)

(38)

(3e)

(40)

(41)

(42)

2L

ùt : Bn,(an,Lp, i (1 - a"r)(0 - n\),19

pn : p^Ê,- Uh) + KhLþa + llt,

î'n: gt-6n-(gÍ-d),

(43)

(44)

(45)

(46)

with the following supplementary definitions, abbreviations and statically endogenous vari-ables:

ad:a:

tdetl :tde[g:td,eL:

, u)eL:

"i+ st+ gtkn* sa",

a" * þ"(þda" - u) + 1Bnay",lsU,

lT" : ongu",td,e , tdeL¡ tLgt

li" + Ii",

"r(1 - rf)aø,(I + ru)pocr(1 - rf)an I pn,

,"lld" i au (1" - t-") + r' L#rlilG * ru)psl,

"f((1 - r")p' - ((1 - r")ri -n'')) + aE@lao - u) + t i õ,

y"-6-((t+r)w"fp)tj",

ci

c9¡

UuI :

sl:p":

c9¡

lcn - ûn) i- t -t 6n.

Inserting these equations into the above laws of motion gives a system of nine autonomousdifferential equations in the 9 state variables shown above. Note that we have to supply againas initial conditions the relative magnitud"r Í¡d3l in order to get a complete characterizationof these 9D dynamics.

As shown in Chiarella and Flaschel (1999b) the law of motion for real wages (in reducedform) reads:

û": nl(t- Ko)(þ-,(t*"lf -Ð+ p*,(tj"lti" - 1)) - (L- o*)þo@lu, -U)l (47)

Inspecting the above statically endogenous relationship then shows that - ignoring the hous-

ing sectors - it is only the expected inflation rate that brings about an influence of thenominal magnitudes on the real magnitudes of this real part of the economy. Therefore, ifinflationary expectations are stationary, we can decouple the real dynamics of the real partof the economy from the nominal dynamics in this subsystem as will be shown in more detailbelow.

swhich however can also be reformulated in terms of real magnitudes

sdh : o?((1 - r"¡(pnl!ì"cn - õn - ((7 - r")ri- n'¿)) + o!(

22

18D

foreign assets offdomestic assets off

fiscal policy offmonetary policy off

X : (Xr, X^una, Xn)

9D* : Fs(X, Z(X))

Sections 3.1 and 3.5

Mundell offa" : w" /Ps, Ôh : PnlPa

X, + *, : (U" , u,lî" ,1" , u.) - *n : (ôn, kn)y : (X,,X¡)

X : Fz(X, Z(X))Section 3.3

7D

housingotr;X-X,

X : Fs(X, Z(X))Section 3.2

5D

Table 10: Súructure of the Rea,l Pa,rt of the lBD Dynamicf

The solution for th" i"tu.io. tt"udy sta.le or point of rest of these dynamics is obtained inthefollowi'gway@ptythatgÍ:l+6,gÍ:l+ô¿muStholdinthesteady state. The remaining adjustment equations for quantities then imply: A!: U|,Ao:A! + equation (43) equat to zero implies furthermo re: A,þfl : tff

ntwhic 2), set equal to zero, implies that rt must be zero. Equations (41),(42), impty two equations in the unknowns I!" lti-V ,AolAo -ü, wLichare linearly independent of each other and which therefore imply li" lt? : V ,ao : apú. Thisprovides us with the steady state value of gro and therefore also with the ones for If|, ti: , ty" ,

since we have according to the above Uo : U3 * ^yuo,Uo : US * þ"(p^oUS - uo) i lþ*oAE and,thus gj : ffi,uo: þndU,!. The equation lî"lt|:7 ttt.n provides us with the steadystate value of /j.

6We add here that turning housing off before Mundell is turned of gives ¡ise to another 7D subdynamicswhere however the price level does not feed back into the remaining 6D system, see also our discussion ofthis type of subdynamics below.

c

c

)l

t

l()(

o

23

Due to what has been shownfor yo we get from the equation for gfl the equality pZ: ri and

thus as real wage u3: wïlpfl since all other expressions that define the rate of profit pihavealready been determined. Inserting this real wage into the definition of y.1o then provides

us with the steady state value of this part of workers' income, since again all other steady

state expressions that form this expression have already been determined. From this income

value we immediately ger ci and thus from goods market equilibrium

a!:a?: c9s+"y+õ + (r+ 6n)kn.* guï, (48)

the steady state value of. le¡o. Equation (44), set equal to zero, next implies that co : khoUh

must hold true, which finally implies via the investment function in the housing sector:

(e'nl ei)c:rl kno - õ¡ - ri : g,

and provides us with the steady state value of p"nlpi.

This is however all that can be deduced for the steady state positions of this economy, since

the above system of differential equations and its static definitional equations all depend onlyon the relative prices u" : w' /pa, ón : pnlp, and thus do not imply anything for the absolute

levels of the prices shown in these expressions. The laws of motion for u" : w" / Pa , Ón : pn f pa

are given byt)' : ût" - þa, ôn: pn - py.

By inserting the above nominal laws of motion into these dynamical equations would indeed

reduce the above dynamical system to a system with dimension 8, with the law of motionfor þo as an appended dynamics that does not feed back into the now truly real part of theeconomy.

The interior steady state of the dynamics of this section is therefore only uniquely determinedup to the level of pi which can be preset to any positive value. Flom the above we also

conclude that the determinant of the Jacobian J of the dynamics at the steady state mustbe zero (the matrix J has rank 8), which in addition implies that the system is subject tohysteresis in that all of its nominal price magnitudes depend on historical conditions and

the shocks to which the system is subjected. The actual steady state values are the ones

determined in the preceding section if one neglects those of the state variables not involvedin the 9D dynamics here under consideration. Finall¡ we conjecture, on the basis of theknowiedge on the dynamics of related, but smaller dynamical models considered in Chiarellaand Flaschel (2000) and Chiarella, Flaschel and Flanke (2003) that the steady state of thedynamics will be asymptotically stable for low adjustment speeds of prices, low adjustmentspeeds of inventories and a fast sales expectations mechanism, but that such stability willget lost (via Hopf-bifurcations, implying the birth or death of periodic orbits at the Hopfbifurcation point) as the speed of adjustment of the slow variables is increased. However,

these are all issues which shall be investigated in the simulations reported in the rest of thepaper.

3.2 The Keynes-Metzler-Goodwin core 5D dynamics

The 9D dynamicsT can be reduced to a 7D dynamical system by switching off the Mundelleffect (i.e. by setting þn, :0 and r¿ set to its steady state value) and formulating the model

TThe Keynes-Metzler-Goodwin core dynamics to which we refer in this section is a special case of theKeynes-Metzler model of Chiarella and Flaschel (2000) and the Keynes-Metzler-Goodwin model of Chiarella,

24

in real terms by introducing the real wage u"(: u" lp,) and real rental prices ôn(: pnlpo)The resulting 7D dynamical system is

a":

teL-

ìuteL¡ :J

ù":

u

n[(1 - Kò(þ.,(t "lr -7)+p.,(t\"lti" -Ð)-(1 - n*)þp(aluo-t)1,

-["f(t -r")(u' - ô- (r+r)u"to¡'-rÐ+o!(ala, -t)],þ,(t1' - ti")- ["f(r - r")(a" - ô - (r + rr)u"Ij" - ri)+ot@ lao - ú)1ry",

þ,"(ao -a)- ["f(t - r")(a" -ô- (L+ro)u"Ij" -ri)+"t@lao - t)\u",u - ud- ["f(t - r")(a" - 6 - (r + r)a"t\" - ri)+"!@lao -(J)+11u,

P^(*- uh) i (on - r)õpo,

sÍ-6n-þÍ-t),

ci+ol+sÊko*su",a" i þ"(þ*oa' - u) + lBnay',lsU,

U' : asgu",

I!" + I!",: li" + li',

Lz(o)¿'(o)awL : r'ud' I a"(1" - P") + a' rll0 i ru),

: "r(1 - ri)U.,,

(4e)

(51)

(52)

(50)

(53)

(54)

(55)

where

ón

îth

Ud

a

4"tt"

P"

lt"

c"n

ci

sl

gl

: (t + rr)29,4Qn

: "f((1 -r")(a" - d- (L+r)a't!" -rÐ)+o!(alao -u)+-yt õ,

: o?((1 - ")T - 6n - (r - r")rf) * "!(#,- uh) * t * 6n

with steady state solution as in the case of the 9D system (given by the subsystem of steadystate values of the preceding section that corresponds to the state variables here considered).Note that cf,, co¡ do not represent steady state values in this set of algebraic equations, butdenote concepts of desired consumption of goods and housing services which are no longersubject to an error correction process.

This 7D system is reduced to the Keynes-Metzler-Goodwin (or KMG) core 5D dynamicsby switching off the housing sector by setting c2 : 0, gl : 0, þn : 0, k,,(0) : 0. These

Flaschel, Groh and Semmle¡ (2000) in that inflation is frozen at its steady state value. Also real balancesare treated differently here because of use of the Taylor interest rate rule

25

imply that the ratio k¡ stays at zero, that equations where divisions through k¡ occur are

suppressed and that the then still given, but purely formal evolution of the price level p¡does not matter for the rest of the dynamics' Due to c2 :0 there is then of course also nodemand for housing services. It is likely for the present formulation of the dynamics that thehousing sector is not of central importance for the overall dynamical features of the full 18D

or real 9D dynamics as far as interesting feedback mechanisms are concerned. This however

is in part due to the approach chosen to model it in the present series of papers and may be

different if other formulations of housing investment and housing services are attempted.

This 5D system with which we are dealing becomes

u)

îeL:

iuleLt :J

ù':

u:

ad:

a:tdeLr :tdeLg:

tdeL:tu)e

ci:Uutl :

K[(1 - Kò(þ*,(l'" ll" - V)

¡8.,(tj" lli" - 1)) - (1 - n.)1p@ lu, - t)1,

-["f(t - r")(a' - ô - (t + rr)u"11" - rÐ + o!(a lao - u)]

0,(t1" - ti") - [af (r - r"¡@" - 6- (1 + rr)a"tj" - ri) + a8@lao - Ù)1ry'

þo"(ao - a) - [af (t - r")

@" - õ- (1 + rr)a'tj' - ri) + d!(alap - Ù)la"

a-ad -["f(t -r")(a' -ô- (r+r)u"tj"-ri)+ot@ lao - û) + ju

ci + af(t - r")(y" - õ - (t + r)u"tl' - ri)+ot@la, -u)+6+ttsa",a" t 0"(0,oa" - u) + 1Bnoy",

lva,

IT" : asgue,

Ij" + t!",li" + l[",

"r(1 - rio)U-,,

,"Ud" r au(t" - Y") * r'ffiFl Q + r,)

(56)

(57)

(58)

(5e)

(60)

with the following supplementary definitions for the statistically endogenous variables

Note that the foregoing expression for c!, which is not restricted to the state value of thismagnitude, makes again use of the steady state value of the rate of wage taxation whichhowever can no longer be given by eq. (17) in the preceding section, since we now have notonly ,k¿ : 0, but also c2 : g. Instead, we now take from eq. (48) in the steady state theexpression ci: Q - g)AE - 0 + d) and determine the steady state value of r- by:

rfo: I _ cilGtui,)

with gfl, the steady state value of wage income (as determined in section 2).

26

s

St¡ble I Cyclic¡l

þ.2= 0'25

lfut

È¿= 1.0

I Exploding

Þ'¿= 0'50

I þ¡r

Þr¡= l '5

Ê,

I

0

0

$

0

0

2

0

ifu,20

pe

I

tfu,2

Figure I: Stabili,ty regions for the KMG core 5D dgnam'ics; þp as. 0.

In the special case K., : 1 this core model consists of a Goodwin (1967) type accumulationand income distribution mechanism, coupled with a Keynesian goods market demand blockthat is here based on sluggish quantity adjustment as in Metzler (1941). This version of theKMG model therefore represents a very basic way of marrying the Goodwin growth cycleidea (also with inside labor) with the Keynesian problem of deficient aggregate demand onthe market for goods and a sluggish quantity adjustment of Metzlerian type. This specialcase we label the KMG core 5D dynamics of our general 18D dynamics.

In the more general case K. 1L the KMG core 5D dynamics are augmented by the Rose real\t/age effect as formulated in Chiarella and Flaschel (2000) which integrates goods marketdynamics into the subdynamics of income distribution and grorvr¡th (but not yet the Mundelleffect of inflationary expectations which would add their law of motion to the 5D dynamicsand also the dynamics of the price level pr).

The steady state values of the state variables of the dynamical system (56) - (60) are givenby'

Iaaeu lli" : liå + as9U1l,

(Ii: + dssyS)lV,

a3-õ-riI'if Q + r)'

Next we analyze the KMG 5D dynamics augmented with Rose goods market effects.

,uteLlo :,eLo:

ui:

27

2

f__-l St¡ble

þP =o'l

I Cycllcel

pir

I Explodlng

Þp= l'0

p"r

0

Ênl

01 , Þr.,

Pp = l's

2 W.3

45

a5

0'l z Pr.l

Þp = 2'0

t5

PJ

0 r 2p!.3 . s

Figure 2: The stability regions for the KMG core 5D d,ynamics; þn as. þa.

Figure 1 shows the (Bo,Br) stability basin at various values of þ-, for the 5D core with theRose effect turned on (nu < 1). We see a stable region at low þ-r, in the stable region at agiven level of wage flexibilit¡ increasing price flexibility leads to greater stability. The effect

of increasing þ., is to reduce (and slightly distort) the stable region.

Figure 2 shows the (8o., B") stability basin at various values of þe. A relatively high value of

B, is required before a stable region emerges. In the stable region, at a fixed þn an increase

in þa. is destabilizing, indicating that a strong Metzlerian quantity adjustment process is

destabilizing for such values of. Bn.

It appears that the nonlinearities of the 5D dynamics, which âre all intrinsic in nature, are

still too weak to bound the dynamics globally once the steady state has become a repeller.

We have also computed figures 1 and 2 for the case when K- : l, the corresponding Good-

winian type of dynamics. However the stability regions are totally explosive in this case andso we have not bothered to reproduce them here.

3.3 The KMG core dynamics u/ith a housing sector

Next we augment the 5D dynamics by switching on the housing sector and consider the 7D

dynamics that are generated thereby. The relevant differential equations are equations (49)

- (55).

Figure 3 displays the (Bo,dr, ) stabitity regions for various values of þn, the speed of response

of housing prices to excess capacity. Compared to the corresponding (at þr":0.5) 5D case

in figure 1 we see that an increase in B¿ has very little effect on stability.

Figure 4 displays the stability trade-off between þn and d,Ìo (the relative strength of excess

28

l-__-] Stable I Cycllcal I Exploding

= 1.02

pp

1

9e

I

0

0 lfur2 19rr2

Figure 3: Stability regi,on (þ., us. þò fo, the KMG (core and housi,ng) 7D dynamics

capacity on housing investment) at various values of þ*, and Pp, with the stable regionseeming almost invariant to these latter parameters. We see from these figures that at agiven B¿, increasing o¡, fon¡lc fn lro rloctohilizino

f__-l Sl¡ble ICtcllcol I Explodlng

Êp = 1.0 Þrr= 0.25 Ê¡ = l.OÊ'r = o.z

pr

I

pr

1

Ê¡,|

0

0 I oa, 2 Ioh

for= 0.2 9p = 1.5þ,r0.2 S =0.8

p¡

2

fotr2o lc¡¡2

Figure 4: Stability region (þn ot. a!) for the KMG (core and housing) 7D dgnamics

3.4 The KMG 5D dynamics and the Mundell effect

If we now add to the KMG 5D dynamics (with the housing sector switched off in the same

way as in section 3.2 and the same steady state formula for the wage taxation rate) thedynamic equation for inflationary expectations (i.e. the Mundell effect is switched on) then

29

St¡ble

Þ'¡= 0'5

I Cyclic¡l f Erploding

p,,=0.05p,r=o.s h=1.0

p"

f

hI

0 lLt2 0 lßy.2

Figure 5: Stabi,li,ty regi,on (þ-, us. Bo) and (þn ,t. þà for the KMG (real) 9D dynamics

we are considering the 7D dynamical system (37) - (43).t

At the present stage of the investigation \/e might expect that the addition of the Mundelleffect (þn, > 0) is generally destabilizing. This is so since from a local point of view - whichonly involves intrinsic nonlinearities - the Mundell inflationary positive feedback mechanismseems to imply not only additional cyclical explosiveness to the plots so far shown, but alsoleads to saddlepath effects in the sense of a superimposed positive or negative trend aroundwhich the cycles occur (and this also in real magnitudes which therefore fluctuate arounda path that is diverging from the steady state). Adding the Mundell effect of inflationaryexpectations as a sixth law of motion (and price inflation as an appended seventh law) tothe real 5D dynamics in fact means that one adds a positive nominal feedback mechanismwithout any other nominal feedback mechanism that can keep this mechanism bounded,since nominal interest rates are still fixed at their steady state values.

We have computed the stability regions corresponding to figure 1 and 2 with the Mundelleffect switched on. There is very little change to the stabitity regions displayed in figures 2and 3, since Brr is still chosen relatively small, so we have not bothered to reproduce themhere. 'We

also note that a sufficiently large increase in this parameter value will make thedynamics purely explosive.

3.5 The integrated dynamics of the real part of the economy

We turn now to the full 9D dynamics of the real sector of the economy expressed in real andnominal terms in equations (37) - (45). This essentially considers the interaction of all thefeedback mechanisms of the real sector; the 5D core (Rose effect), the Mundell effect andthe housing sector.

8We stress here again that the evolution of p, does not influence any of the other lau¡s of motion if nominalwage dynamics are reformulated as real wage dynamics as in Chiarella and Fìaschel (2000):

ù' : n[(r - Kp)(þ.,(y' lt' - 7) + p-"(4" lti" - 1) - (r - n-)þp@la, - ü)].

The 5D real part ofthe economy (and the evolution ofinflationary expectations) then depend on the evolutionof this real wage, but nowhere on the evolution of the price level itself, which in particular means that thedynamical system based on the state variables y"ru,ll',l",e",ps,zr¿ has a vanishing sixth column in itsJacobian at the steady state.

30

Figure 5 displays the Br,É.,r stability region f.or 8.": 0.5. We see that the stabitity region isquite small. A very similar picture is obtained for a wide range of 0.* Figure 5 also displaysthe Bn, Br" stability region for B.r: 0.05, þ., :0.5 and þp : t.0. Overall these stabilityregions indicate that the interaction of all the mechanisms of the real sector is destabilizing.

4 Adding policy issues to the real dynamics

In this section we consider the impact of fiscal and monetary policy on the stability basinsof the 9D real dynamics studied in section 3.5. Tables 11,12 and 13 summarize the varioussubmodels we consider in this regard and how they are obtained from the full 18D model.Thus in table 11 we see that by turning off foreign assets, domestic assets and fiscal policy,the 18D model is reduced to a 10D system which consists of the gD real dynamics togetherwith the Taylor interest rate rule (equation 31). Table 12 shows that when foreign assets,domestic assets and monetary policy are switched off, the 18D model reduces to a 12D systemconsisting of the 9D real dynamics plus the 3D fiscal policy dynamics (equations 28,29 and30). Finally table 13 shows how the 9D real dynamics with both the Taylor interest ratepolicy rule and fiscal policy dynamics (resulting in a 15D system consisting of equations(19)-(33)) is obtained from the 18D dynamics by switching off foreign assets and domesticassets. In the following subsections we investigate in turn each of the foregoing subdynamics.

Table 11: Reducrng the 78D model to the 9D rcaI dynamics with the Taylor interest raterule

18D

,åX3.Tn::i,"f-fiscal policy off

X: (Xr,Xmund,Xn,X^o): (a, Ij", tl", li", Id", I-", u *, c8, c9n, p", gÍ, gf, , ud, T b, g", to, t")

X : Frc(X, Z(X))

real dynamics (9D) * Taylor interest rate rule (10)Section 4.1

31

18D

foreign assets off

'*:iå'ffi"if;:ii%X: (Xr,X^und,,X¡,X¡)z : (a, tl', tg", \f,", ld", I'", u., cf, , c9¡, p', gl, gf, , ud, ir b, go, t", t")

* : Fn(X, Z(X))

real dynamics (9D) * fiscal policy dynamicsSection 4.2

Table L2z Reducing the 18D model to the 9D real dynamics with frscal policy dynamics

Table 1-3: Reducing the 18D model to the 9D real dynamics with both the Taylor interestrate rule and frscal. policy dynamics

18D

foreign assets offdomestic assets off

X : (X. Xmrnd, X¡, X^o, X ¡t)Z : (u, lf , lg", If", ld', I-", u., con, con, p", gÊ, gf, , ud,,r b, go, t", t")

X : Frc(X, Z(X))

real dynamics (9D) * Taylor interest rate rule * fiscal policySection 4.3

32

4.L Interest rate policy rules

The subdynamics of this subsection consist of the 9D real dynamics of section 3 plus theinterest policy rule of the central bank, viz.

i : -þ,,(r - ri) + þ,,(Lþo +,,rt) + 0,,(a lao - U) (61)

This brings back the negative feedback effects of the short-term rate of interest on fixedbusiness and housing investment, at present only compared with a given rate of interest onlong-term bonds ri through the a2 terms in the two investment functions. We now considerhere a situation where the Mundell effect is at work (i.e., at least a 7D dynamical system) andwhere the system would experience breakdown if the interest rate policy would be switchedoff (even for very sluggish adjustments of inflationary expectations). By having this policyrule present, we would expect that a positive and increasing rate of inflation is counteracted,since the rule will work against economic expansion and further increases in the rate ofinflation and expectations about it in such cases. This policy - as we know already fromChiarella, Flaschel and Zhu (2000) - should reduce, and indeed does significantly reduce, theextent of nominal instability inherent in the real part of private sector of the economy, sinceit works against the Mundell-effect of a positive feedback structure between the expectedand the actual rate of inflation, which we found to be very destabilizing and problematic inthe observations made in the last subsection.

Figure 6 displays lhe Bo vs. þq and Bn vs. þa" stability regions. Both stability regionsindicate that, compared to the 9D real dynamics (see figure 5) without the interest ratepolicy rule, the Taylor interest rate policy rule is stabilizing.

[-l St¡ble I Cycllc¡l I Unst¡ble

9D + Monelery Pollcy

2

p, Àr2

0

0 I 2fu,3 1 5 0t z Pt.c a5

Figure 6: The 9D real dgnamics with the Taylor rule switched on

We stress, but do not prove this here that a Taylor rule of the type:

r : ,re + þ,r(r" - fr) + þ,r(l'" ll" - V), þ,,, þ,, ) 0.

would be even more successful in fighting the explosiveness caused by the Mundell effect.This rule states that the central bank sets the expected real rate of interest according to thediscrepancy that exists between the expected rate of inflation n" and the target rate zr of thecentral bank and the deviation of the actual rate of employment from the NAIRDratee and

eThe Non-Accelerating-Infl ation Rate of Employment.

33

this in such a way that interest rates counteract what is observed at high or low economic

activity and inflation.1O This rule is not based on a dynamic law, but concerns levels and

thus reduces the dimension of the system of differential equations considered by one. Inaddition it directly attempts to steer the expected real rate of interest and thus appears tobe more powerful as it immediately attacks the source of the Mundell effect, and is not onlycounteracting it via the Keynes-effect.

4.2 Fiscal policy rules

We have so far ignored the role of the government budget constraint, since it did not exerciseany influence on the real dynamics of the model as considered in the preceding section 3.

This is however problematic, since the accumulation of government debt may follow an

explosive path in the background of the dynamics that has been explicitly considered so far.F\rrthermore it may be of a kind which would not be tolerated by the present or a subsequentgovernment. We therefore have to consider the evolution of government debt explicitly andwill do this of course subject to the hopefully stabilizing influence that may come fromthe assumed adjustment in the wage taxation rate in the pursuit of a given target ratio ofgovernment debt per unit of an appropriate index for the social product, of the type shownin equation (30). The dynamics now consist of equations (19)-(30). Thus bond dynamicshave thereby been integrated again into the dynamics of the real part of the economy as

shown in section 2.

This is a decisive extension of the dynarnics of the model, since it brings back into theconsidered dynamics the complicated evolution of short and long term bonds per unit ofcapital, ö, ö¿, together with the law of motion of the taxation rale r-. Figure 7 shows the

þp vs. 0q and B¡ vs. Éy" stability regions. Compared to the 9D stability regions with nofiscal policy dynamics rvve see that if anything instability has increased. The previous stableregions in figure 5 have disappeared. The intuition that the bond dynamics a,re highlydestabilizing seems to be borne out by these stability regions.

l-l Stable ICycllc¡l I Unstablc

9D + Flscrl Pollcy6

I

pP Ar2

0 f 2h,3 4 5 0r 2 s.3 45

Figure 7: The 9D real dynamics with the fi,scal policy dynamics switched on:

Employing the wage income taxation rule in the place of the interest rate policy rule is thusnot stabilizing in the 9D real dynamics in contrast to what might be expected from such a

rule according to the comments made in Powell and Murphy (1997). This seems to be due to

l0See Flaschel and Groh (1998) for a further discussion of the properties of this monetary policy rule.

34

the cumulative effect that the evolution of government debt has on the change in the wagetaxation rate (which makes things worse instead of better).

Quite the contrary to what we expect on the basis of Chiarella, Flaschel and Zhu (2000)and its treatment of the GBR even small positive parameters ar- contribute significantly tothe instability of the steady state and are therefore problematic. This may also be due tothe complicated government bond feedback mechanism which so far did not influence thedynamics shown and which may not have the properties found to hold (Chiarella, Flascheland Zhu (2000)) where it worked in isolation. The evolution of the government debt basedon our complicated formulation of the GBR is however always there and must be integratedinto the full dynamics at some stage of the investigation. The question can then only bewhether its evolution is less or more problematic in its consequences for the whole systemwhen the taxation rule is switched on with the aim of stabilizing government debt at acertain target ratio.

4.3 Fiscal and monetary policy rules in interaction

The next and final figures of this section show the joint working of the tax policy rule andthe interest rate policy rule. The dynamical system now consists of equations (19)-(33).Figure 8 displays the stability regions for this case. We see that they are very similar to thecorresponding regions for the 9D plus Taylor interest policy rule in figure 6. So the monetarypolicy is also able to stabilize the explosiveness of the fiscal policy dynamics. There are ofcourse many further possibilities for feedback policy rules that have not yet been includedinto the general model of this pa,per, but which merit further research.

f---l St¡ble I Cyclic¡l I Unst¡ble

9D + Mooct¡ry ond Fiscol Policics

Êil

5

1

3

p,2

0

0 I 2h,3 a s 0l t q.' 45

Figure 8: The 9D real dynamics with monetary and, fiscal pol'icy rules

5 Adding asset price dynamics to the real dynamics

In this section we consider the interaction of the 5D real case and the asset sets, bothdomestic and foreign.

This extension of the real dynamics adds first of all and most irnportantly long-term interestrate movements (expected and actual long term bond price dynamics) through their influenceon the investment in fixed capital and housing and thus on aggregate demand and the output

35

of firms. We therefore now integrate into the real dynamics the two dynamic equations (32)

and (33) namely:11

I - þpo(I - a,): þ^,"(Pu - rur)

and their two (opposing) effects on the two types of investment just considered, via profitabil-ity differentials, here shown for fixed business investment (l-r")p"-((I-r")r¿-Tt), rt : llpa,and via the interest rate spread rt - r.This extension would generally be expected to addinstability to the real dynamics, since it represents a positive feedback loop between theexpected and the actual increase in the growth rate of long-term bond prices, if the adaptivecomponent in the expectations mechanism works with sufficient strength. We stress thatthese asset market dynamics are independent of the movements in the real part of the econ-

omy as long as the central bank keeps the short-term rate of interest fixed to its steady statevalue, in which case there is only a one way route leading from the market for long-termbonds to the real part of the economy.

A similar observation does not so obviously hold, if we allow the exchange rate e to influencethe evolution of the real part of the dynamics, by removing the assumption that the rateof import taxation is always set such that the trade account of firms is balanced (whenmeasured in domestic prices). In this latter case, the expected rate of profit of firms doesnot depend on their exports and imports levels and thus on exchange rate changes. As longas there are no wealth effects in the model and as long as the individual allocation of bondson the various sectors does not matter, there is indeed only this one channel through whichthe nominal exchange rate can influence the real economy (besides of course through theGBR which includes the tax income of the government deriving from import taxation, butwhich does not play a role for the real part of the model unless wage taxation is responsiveto the evolution of government debt as v¡e have seen in the preceding section). To have thisinfluence of the exchange rate we thus have to extend the 9D real dynamics by the followingthree laws of motion (34)-(36) namelyl2

Po

ira,

0ourc - r");i d.,rb" - (I - r")rl

p"l(I - r")ri f 0,6" - ((1 - ")*+

rr)1,e 7-p"(L-a,): 0r"(ê - e")'

The exchange rate dynamics is more difficult to analyze, since their two laws of motion need

the influence of the bond dynamics in order to be meaningful. Otherwise these two lawsof motion would imply monotonic implosion or explosion of exchange rate expectations andthe actual exchange rate depending on whether the adjustment speed of the exchange rateis smaller or larger than one (for a" : 1). The financial dynamics is therefore in this respect

llNote that o" has been assumed to be larger than (1 - Il þrr) in the presentation of the structural formof the model in Chiarella and Flaschel (1999a,b) which makes the parametric expression in front of the firstlaw of motion positive, Note also that the parameter rc caa be neglected in the numerical simulations thatfollow.

l2Where the first one is independent of the changes of the exchange rate.

es

36

immediately of dimension 5 and it also needs input from the real dynamics to get the effectsfrom the exchange rate e on bond prices p6 and thus an interdependent dynamics and notone of the appended monotonic form just discussed. Yet, the effect of changes in e via therate of profit p" of firms and the investment decisions that are based on it, needs to extenda long way in order to reach the market for long term bonds. Changes in investment leadto changes in aggregate goods demand and thus to changes in sales expectations and actualoutput. This leads to changes in capacity utilization of firms and domestic price inflationwhich - if and only if monetary policy responds to them - are transferred to changes in theshort-term rate of interest and thus to changes in the long-term rate of interest. In this waythere is a feedback of a change in the exchange rate on its rate of change which has to beanalyzed if the full dynamics are investigated

Taken together the above two extensions which integrate the financial dynamics with thereal dynamics will lead us to a 14D dynamics of the real financial interaction, but withno feedbacks from government policy and the GBR yet. This system will be investigatednumerically on various levels of generality, i.e., by means of appropriate subcases, in thissection.

Clearly the bond dynamics is the more important one from among these two possibilities ofmaking the real dynamics dependent on what happens in the financial part of the economy.lsWe will therefore investigate next how independent monotonic or cyclical movements in long-term bond prices act by themselves (with no coupling with the exchange rate dynamics) onthe real part of the dynamics and how they can be bounded in an economically sensibleway in the case where their steady state solution is surrounded by centrifugal forces. Weshall assume here, as discussed in Chiarella, Flaschel and Zhu (2000), that locally explosiveasset market dynamics can give rise to limit or even limit limit cycle behavior (relaxationoscillations) in the bond market and thus to more or less fast, persistent fluctuations inthe long-term rate of interest and expectations about its rate of change. This result is ofinterest in its own right, but of course also important when studying its consequences for theeconomy as a whole, without (or with) feedback from the real side to the financial markets.

Arriving at such a situation thus provides an interesting intermediate step in the analysis ofthe full 18D dynamics, since we can study here the role of fluctuations in long-term interestrates (and the exchange rate) on the real dynamics in isolation before coming to a real-financial interaction of these two fundamental modules of our model. The obtained resultcan be usefully contrasted with the one way investigation of the real-financial interaction ofFranke and Semmler (2000), who study the behavior of a fully specified set of asset marketsin its dependence on a given wave form of the business cycle in the real sector, whereas thissection considers how the opposite situation can be investigated as a natural subcase of ourgeneral model of the real-financial interaction, where asset market fluctuations only work onthe functioning of the goods and the labor markets of the economy.

13The third asset, equities, does not have any impact on the dynamics of the model of this paper, sinceneither consumption nor investment depends on sha¡e prices here, see Chiarella and Flaschel (1999a,b) fordetails.

37

18D

foreign assets offfiscal policy off

monetary policy offX: (XrrX^und,,Xn,Xa)

Z : (y, tj", tÍ", l[", ld", l, u., con, con, g", gÍ, gÊ, ud,'r b, go, t", t")

X : Fn(X, Z(X))

X : (X,,Xd)

Mundell offhousing offX, -* *,

X : Fz(X, Z(X))

Core real dynamics (5D)

* domestic asset market dynamics (2D)

Table t4: The 5D Real Core plus Domestic Asset Ma,rket

Table Líz The 5D rcaI core plus domestic and foteign asseús

18D

fiscal policy offmonetary policy off

Mundell offhousing offXr-*,

X: (*,,Xa,X¡)

X : Frc(X, Z(X))

Core real dynamics (5D)

* domestic asset market dynamics (2D)

* foreign asset market dynamics (3D)

38

'We first apply these observations to the numerical investigation of the 5D real dynamics (thecore dynamics of this paper) augmented by the 2D dynamics in long-term bond prices andinterest rates and their impact on the real part of the economy. Table 14 shows how the7D system consisting of the 5D real core plus the domestic asset dynamics is obtained fromthe full 18D dynamics. This is done by switching off foreign assets, fiscal policy, monetarypolic¡ the Mundell effect and the housing sector.

Figure 9 shows the Bo vs. þq and þn vs. þs" stability regions for these situations at þ.r: 0.5and 1.0. We observe that there is very little change compared to the corresponding 5D realcore situation of figures 1 and 2. In the 0p vs. Ét,1 region a cyclical region appears before theonset of instability. In the B¡ vs. þs. region there is some contraction of the stable regionf.or B.r: 1.5.

We stress with respect to the simulations shown in figure 9 that they are based on the 5Ddynamics with which we began the numerical investigations of the 18D dynamics in thispaper. There are thus no housing activities involved, no Rose or Mundell effects at work andno policy rules implemented in the dynamics shown. This closes our considerations of thebasic case of a c rension 7D.

Stable I Cyctlcal I Unstable

fuz: l'OOÊ-z: O'5

FP pe

2

oo I Þ-,

[-_-_-l stabte

9o : l'S

2or9-t

I Cyclical I Unstable

þP :2'O

2

2.0

p,

t.3

t.0

0.5

o.o

2.O

Þ.t.3

t.0

o.3

0.o

o r 2 3 . þr.s o ! t Þr.3

Figure 9: The 5D core real d,gnamics with domestic assets.

39

pe

2

0

Stable

Þ.2= O'5

I Þ*,

Stable

9p : l'5

I Cycllcal f Unstable

Arz = f 'OO

pe

o2OlÞrr

tr Cyclical I Unstable

þ" :2'o

2

2o

2.O

p,

1.5

1.0

0.5

0.0

2.O

p.

t.5

1.0

0,s

0-o

2 t a Þr.5 o t t 3 4 Þr.5

Figure I0: The 5D core real dAnl,rnics with domestic and foreign o,sset rna,rlcets.

We consider next the integrated financial market interaction (between domestic and foreignbonds and their expected rates of return) which are of the following final form:

po : þool! - r")! Ira"- (I - r")ri - 0,þ- eo)],Pa

'ira" : þno"(pr-7rös),

þ - P"lG - r")ri +€, - ((1 - ");+ 7rö,)1,

è" : þ,"(ê-er),

Trn t : fryAt jd : jaA,

Table 15 shows the derivation of the 10D dynamics consisting of the 5D core real dynamicstogether with domestic and foreign asset market dynamics from the 18D dynamics by switch-ing off both policy rules, the Mundell effect and the housing sector. The system consists ofequations (56)-(60) and equations (32)-(36). Figure 10 displays the stability regions. Weobserve that these are very little changed from figure 9 which involved only the domesticasset market.

We conjecture that this system, with appropriate nonlinearities added, will give rise to twocoupled relaxation oscillations of the type we have considered in Chiarella, Flaschel and Zhu(2000). It is therefore to be expected that the fluctuations in financial markets and theirimpact on the real part of the economy will become significantly more complicated in such

situations of coupled (relaxation) oscillations and their effect on the real part of the economywithout or with feedback on the financial sector via the interest rate policy rule of the centralbank. In this regard we refer the reader to Asada, Chiarella, Flaschel and Flanke (2003).

40

6 Numerical investigations of the full 18D dynamics

'We have so far discussed in this paper various possibilities for a systematic approach towardsan investigation of the numerical properties of the full 18D dynamics by mean of appropriatesubdynamics. Before \Me now start the numerical investigation of the full 18D system wesummarize the discussion so far by means of a flow diagram that shows the various feedbackstructures and feedback policy rules involved in dynamic interaction. The following thusprovides a graphical representation of what we have discussed so far and it also gives aguide as to how \rye can go back and forth between appropriate subsystems and the full 18Ddynamics in order to understand the outcome of the feedback chains this'system contains.We refer the reader to section 3 and Chiarella, Flaschel and Zhu (2000) for a detailed analysisof the partial feedback mechanisms these disequilibrium growth structures in fact integrates.

Note also with respect to the following graphical representation that there are some feedbackmechanisms included (for reasons of completeness) that are not yet contained in the presentlyconsidered dynamics (namely the Fisher debt effect, based on investment behavior or alsodifferent consumption propensities of creditors and debtors) and the Pigou real balances orwealth effect (which would introduce wealth as an argument into the consumption functionsof the model). Note also with respect to our basic 5D dynamics of KMG type (discussedin section 3) that it brings together the Keynesian goods market view augmented by theMetzlerian inventory adjustment mechanism and the Goodwin real wage - capital stockgrowbh dynamics augmented by Rose (1967) goods-market effects of the real wage on priceinflation. The full downward causal nexus of Keynes (1936, ch.19) from asset via goods toIabor markets extends these real dynamics in the way we have analyzed in the precedingsection and it also allows for the influence of monetary policy rules besides fiscal policy rulesas shown in the graph. The question of course again is (see Chiarella, Flaschel, Groh andSemmler (2000) for detailed discussions) whether the shown feedback mechanisms increase ordecrease the stability features of the full dynamics close to the steady state (leading towardsor away from NAIRU'full'employment positions) and whether the downward causal nexusshown or the supply side real wage dynamics dominate the dynamics in the medium andIonger run should the economy depart from their steady state due to centrifugal forces aroundit.

Let us begin our numerical investigations of the full 18D dynamics by showing a situationwhere all equations of the 18D system interact with each other, but where adjustmentspeeds in the asset markets, concerning asset revaluations (long-term bonds, exchange rate)and expectations on their rate of change, are still low so that there is not much movementpresent in this part of the model. Larger fluctuations, which are of a simple limit cycle type,therefore basically concern the interaction of prices and quantities on the real markets, as

figure 12 shows.

The simulation of the full 18D dynamics in figure 12 (the parameters of this simulationrun are shown in table 14) provides a first impression of a type of persistent economicfluctuations (here in fact a fairly simple limit cycle) as it may be generated by the intrinsicnonlinearities characterizing the dynamics. Of course, there can exist supply bottlenecksin the case of larger fluctuations, as discussed in Chiarella, Flaschel, Groh and Semmler(2000, ch.5), which must be taken into account in the formulation of the dynamics if certainthresholds are passed, but which are ignored in the present section.la

laSee Chiarella, Flaschel, Groh and Semmler (2000, ch.6) for a treatment of such supply side restrictions.

41

Traditional Keynesian Theory: Summary

Market HierarchiesSupply Side Features

Moneysupplyrule

Taylor interest rate rule

Keynæeffed

realwagedynamim

Feedback MechanismsFeedback Policy Rules

Dombusdr rate

Euilyandbond

dynamim

lnvestment

MeElerian

spdd

Mundelleffect

dynamæ

short

plofit

sa espriæ

nflatlon

I{,age

oriæ

bpiral

Roæefie6 Prduolion

func'tion efiæt

lrage

nflalion Fiscalpolicy rules

How dominantis the downward influence? How Strong are the Repercuss¡ons?

How Dominant is the Supply-Side Dynamics?Can Policy Shape the Attractors / the Transients of the Full Dynamics?

Figure lI: An oueruiew of the integrated dynamics

42

py

we

t

ó

<i

d

¡

d

0fo?03040ææ70øorc1Oo

01020JO40æ607080m1OO

10 20 Ð ao 50 æ 70 90 10c

0 10 20 J0 ¡o 50 æ 70 ao 90 10t

Figure 12:

Conuergence to a l,imit cycle for the full 18D dynam,ics

Table L4z The parameters corresponding to figure 12

Table 14 shows that parameters that were critical with respect to the dynamic behaviorof certain subdynamics, like the speed of adjustment for the wage taxation rate r.,, needno longer be restricted to small values in order to obtain a meaningful dynamic evolution.However, the table also shows that asset prices still adjust very sluggishly with respect tothe relevant interest rate differentials, which leaves for future research the task of investi-gating in more detail what thresholds must be applied to these dynamics in order to getbounded or viable dynamics also for larger adjustment speeds of asset prices and capital

I t

k¡

v

tftb. llb

þ-, :0'4 þ-r:l þp: o'7 B¡ :0.6 þpa : o't þnr, :0.t þ. :0.7 þ, :0.1

þn:0.2 Bna :0.I Þy' :1 þn :0.8 þ¿ :0.5 þr, : o-7 þr, : o'5 1rs : g'1

orr :0.1 0s : 0.5 0rrr :0'1 ahz :0'5 ah" :0'I 0Ì"1 :0,1 0r", : 0.5 0r"" : 0.1

0r- :0.5 0¡- :0.5 as :0'2 al: o's 0, :0.5 0r : 0'5 .ú1(0) : 2gggg .02(0) : 5ggg

Kp :0'5 Æu :0'5 Kn:0'5 U":0'9 Un :0.9 V :0.9 d:0.6 9: 0'33

Ph: I Pi: I rÍ :0'08 d:0.1 ô¿ : 0.1 Îc :0'5 Tu :0.L5 Tp:0'3

7: 0.06 cr : 0.5 cz:0.33 la :2 fia :0'2 iu :0'l AP:I Pa 1

43

gains expectations around the steady state of the dynamics. Note also that rates of growthand of interest are no\M chosen in a more plausible range than was the case in some of thesubdynamics considered in the preceding sections. The simulations of figure 12 and furtherones (not shown) suggest that the full dynamics behaves more smoothly with respect toparameter changes than the various subdynamics we have investigated beforehand.

Increasing the parameter 8., to 1.14, the adjustment speed of nominal wages due to theemployment rate of inside workers, stabilizes the dynamics further in the sense of makingthe limit cycle shown in figure 13 a smaller one. In fact, further increases of this parameterwill remove the limit cycle totally and will create the situation of an asymptotically stable

steady state or point attractor, as shown in figure 14. This indicates that a supercriticalHopf bifurcation is occurring from stable limit cycles back to convergence to the steady stateas the parameter /wz goes beyond 1.14. This situation will be confirmed by a subsequenteigenvalue diagram calculation.

I

I

d

o

t0 20 JO ¡o æ 70 s 100

2010.0æ607080s100

oro20na050æ70809010c

0ro20Jo¡oá0æ70æ90loc

Figure 13:

Shrinking limit cgcles when the parameter 8., is increased.

We note with respect to figure 13 that there is a long transient behavior shown in thisfigure with irreguiar fluctuations and varying cycle lengths of the time series of the 18 statevariables that are shown. Note however that this is partly caused by the enormous shockthat is here applied (a thirty percent increases in sales expectations).

In the situation shown in figure 14 we may increases the adjustment parameters on theasset markels, /pu, þn0", þ", þ, rp to 0.6 and will find that fluctuations will now occur in thecorresponding state variables (still of a minor degree), but quite astonishingly accompanied

by a further increase in stability, i.e., by a more rapid convergence to the steady state. Assetprices and capital gain expectations thus do not always destabilize the dynamics when theircorresponding adjustment speeds are increased. This may be due to the Taylor rule, thesteering of the short-term rate of interest by the central bank, which may move the termstructure of returns on assets in a way that increases the stability of the steady state.

Tm

w"

t

e

b

t

eI

k¡

nArV\¡-

v

Tw

trb, ne

44

Tm

py

o

0r0?0J0.oil607080æ100

I

I

ci

I

o1020С050æ70êo9010c

0 10 20 fo ¡0 50 æ 70 E0 90 10t

Figure 14:

Establishment of a point attractor as the parameter 0., is further increased (to the ualue 3).

However, if the four parameters just considered are all in fact increased to 0.6 and if wechange the portion orr of people who form adaptive price inflation expectations from 0.1 to0.5 the fluctuations of the economy, and also the transient behavior, are significantly changedas figure 15 shows. These fluctuations still converge to a limit cycle which however is onlyrevealed when the economy is simulated over a much longer time horizon than is here shown(100 years).

Next we come to the calculation of eigenvalue diagrams for speeds of adjustment and im-portant other parameters characterizing fiscal or monetary policy and the behavior of theprivate sector of the economy. These eigenvalue diagrams show the maximum real part ofthe eighteen eigenvalues of the 18D core dynamics and they are based on the parametervalues given in table 14, with þ., : 1.14 however. Note that due to the indeterminacy ofthe level of nominal magnitudes one eigenvalue must always be zero in these 18D dynamics,in distinction to the dynamics we have considered in Chiarella, Flaschel, Groh and Semmler(2000, ch.s 7/8). Therefore, Iocal asymptotic stability of the remaining variables is givenwhen rrye see a horizontal portion (at zero) in the eigenvalue diagrams shown below. The de-gree of asymptotic stability therefore cannot be seen from the depicted eigenvalue diagrams,but only the points where stability gets lost, presumably by way of a Hopf-bifurcation.

The eigenvalue diagrams shown in figure 16 are remarkable in that they confirm, in a verystraightforward way, what intuition from the partial lD or 2D perspectives would suggest,despite the fact that the partial stability analysis is often quite easy to understand sincedestabilizing feedback mechanism very often sit in the trace of the Jacobian of the dynamicsat the steady state while they are distributed in the full 18D Jacobian in a very uninformativeway at first glance. We thus see that the system very often behaves in a very simple way

e

r

l/

tV

t

e v

Tw

fibt t

45

even though it integrates Rose type price adjustment, Metzler type quantity adjustment,Goodwin type growth cycles, a housing sector related to the Goodwin - Rose approach tothe employment cycle, the dynamics of the government budget constraint, asset marketdynamics of Dornbusch type, and monetary and fiscal policy rules.

0r020)o409æ7080æro0

100

I

I

I

o 10 20 30 ¡0 50 æ 70 eo 90 !0c

o lo 20 30 a0 50 æ 70 Eo 90 loc2010 lo 40 æ æ 70

Figure 15:

A more dominant role for price'inflation and adaptiue etpectations

Inspection of the parameter set underlying these eigenvalue diagrams, which is given by table14, with 0ru2 eeuralto 1.14, first of all shows that wage flexibility (on the outside labor market)should be destabilizing and price flexibility on the market for goods should be stabilizing,since broadly speaking aggregate demand gd depends positively on the real wage, due to verylow marginal propensities to invest as far as profitability component of investment behavioris concerned. These two diagrams therefore concern what has been called Rose effects in thispaper. Indeed, this is what is shown in the first two diagrams in figure 16 over the range (0, 1)

in the case of the parameter B*, and the range (0, 2) in the case of þe.The Hopf-bifurcationvalue for these two parameter values, where stability gets lost, is slightly below (respectively

above) the parameter values þ-, :1.14 and þp:0.7 since the parameter values of figure 13

already provide a stable limit cycle around an unstable steady state.

e

\I'v,\Ï-qva%v.- \v\vü/

t

e v

n" t

46

Figure 16:

Eigenualue calculations for adjustment speed parameters

2

1.5

47

In the second row of figure 16 we see again what has already been demonstrated in relationto figures 12 and 13, namely that larger flexibility of the money wage with respect to theemployment rate within firms is stabilizing. We also see in this row that increasing flexi-bility of adaptively formed inflationary expectations is stabilizing, which stands in strikingcontrast with what we know about the role of Mundell effects from the smaller KMG modelsconsidered in this paper. It is however easy to understand why this adverse situation arises

here. The parameter characterizing the portion of adaptively behaving agents is, as table14, shows in the present situation equal lo a¡: 0.1 which means that the other, regressive,component of inflationâry expectations is the dominant one which is stabilizing. Increasingthe parameter dr.r to its extreme value of 1 indeed reverses this situation and gives for the Bnteigenvalue diagram the same form as for the 8., diagram and thus implies that the Mundelleffect is working, as usual, in a destabilizing way when the adaptive expectations of priceinflation become faster.

The third row in figure 16 shows very low bond price adjustment speeds turn the stablelimit cycle situation given by the base parameter set into convergence to the steady state,while an increase has only moderate effect on instability for a while, until a point is reached(approximately Bru : 1) where instability increases significantly with the parameter Boo.

Modifying the speed of adjustmen| Bno" of the adaptive part of expectations formation inthe market for long-term bonds, on the other hand, provides no way of obtaining stabilityin the present situation, i.e., the limit cycle will not shrink to zero in this case for eitherhigh or low values of this expectational parameter. Similar conclusions hold in the case ofexchange rate dynamics, where however a small middle range of adjustment speeds for theexchange rate provides local asymptotic stabilit¡ while the system becomes unstable againfor very iow adjustment speeds of exchange rate dynamics. Asset markets thus behave byand large as expected for isolated changes towards higher adjustment speeds of prices andexpectations. Note here however that we have found in connection with figure 14 that a

simultaneous increase in the speeds of adjustment here involved could improve the rate ofconvergence of the dynamics.

Turning to the fourth row of figure 16 we see that there is a small range for inventoryadjustment speeds B,, where local asymptotic stability holds, while there is instability belowand above this range. Not only do faster inventory adjustments destroy stability, as expectedfrom the 2D presentation of the Metzler dynamics in Chiarella, Flaschel, Groh and Semmler(2000, ch.2), but now also for very slow adjustments of inventories. The finding for sales

expectations, Bo", is as expected from the 2D situation, i.e., the stable limit cycle situationunderlying the parameters of table 14 is turned into local asymptotic stability when theparameter þo. i, increased, since the marginal propensity to spend is broadly speaking smallerthan one in the considered situation and the dynamic multiplier process, here in expectedsales, is therefore stabilizing.

Finall¡ the interest rate policy rule works as it is expected to work. Increasing inflation oractivity levels here lead to increasing short-term nominal interest rates and this counteractsthe increases in inflation and economic activity. Increasing the adjustment speeds withwhich the central bank reacts to inflation or economic activity changes thus leads to localasymptotic stability and makes the stable limit cycle around the then unstable steady stateagain disappear. We furthermore note, but do not demonstrate this here, that increasingadjustment speed B¿ of the price level for housing services (from a certain point onwards) willdestabilize the econom¡ as will increasing adjustment speeds in the employment policy offirms, B¿. However, in both cases, this will also occur if these adjustment speeds are decreased

48

to a sufficient degree which again means that there is only a certain corridor for which it canbe expected in the present situation that convergence to the steady state is assured.

Our next set of eigenvalue diagrams in figure 17 concerns important policy parameters ofthe 18D core model.

Figure 17:

Eigenualue calculations for policg parameters.

In the first row of figure 17 we see that an increase of the adjustment speed of the wagetaxation rate (in order to approach a target level of 60 percent for the debt to (sold) outputratio) is destabilizing further when started from the reference case of the limit cycle situationin figure 13, while a decrease of this speed will produce convergence to the steady state. Bycontrast, increasing the targeted debt to (sold) output ratio d removes the limit cycle andleads to asymptotic stability. The presently considered case therefore leads to the remarkableconclusion that the Maastricht criterion for the ratio d should be relaxed and I or the speedof adjustment towards this ratio be reduced if asymptotic stability of the steady state is adesired objective

The second row of diagrams in figure 17 shows to the left that (further) increases in thepercentage of unemployment benefits, and also pension payments (not shown), as compared

98

49

to the limit cycle reference situation tend to be destabilizing, while reductions in both of these

ratios bring asymptotic stability and thus convergence to the steady state of the dynamics.To the right this row provides the eigenvalue diagram for the percentage of governmentexpenditures per unit of (expected) sales, which shows that there is a small corridor forthis ratio below the reference situation where local asymptotic stability of the steady stateis given. Variations in this expenditure ratio therefore generally do not add much to thestability features of the reference situation.

Finally, in the last row of eigenvalue diagrams in figure 17, we consider to the left the shift indebt financing of government expenditures a\May from short-term bonds towards long-termbonds and find that this is stabilizing in the current situation. By contrast, in the diagrambottom right, we see again that there is a range of parameter values for the payroll-taxparameter ro, and similarly increase in capital income taxes r" and value added taxes rr,tothe right of the reference situation where convergence to the steady state is obtained, i.e.,increasing payroll taxes in the reference situation will produce asymptotic stability, whiledecreases from there will be destabilizing. Payroll tax increases are therefore only in alimited way comparable to increases in the adjustment speed of nominal \Mages with respectto the external labor market and thus must be considered as an independent event fromthe proposal that the (downward) adjustment speed of nominal wages should be increasedsomewhat.

Note that we here only consider stability issues, and not how steady state values themselvesmay be changed through those of the here considered parameters that do not concern adjust-ment speeds, which do not affect steady state positions. Such steady state comparisons haveto use the set of steady state values presented at the beginning of this section. Note also

that the stability assertions made are generally not confined to very limited basins aroundthe steady state, but can in most cases be tested by means of considerable shocks out of thesteady state.

We note that the parameter values e.^ and a¿, the speed of adjustment of import taxationand the participation rate of the labor forces, do not influence the eigenvalues of the Jacobianof the dynamics at the steady state, and that variations in the ratio of heterogeneity incapital gains expectations on the asset markets do not produce asymptotic stability in thepresently considered situation. Not unexpectedly there is a band of intermediate rangesfor the marginal propensities of workers to consume goods and housing services (below thereference ratio) where convergence is established, but low as well as high values of these ratiosbetween zero and one do not produce such results. Note here that both ratios may exceed 1

in sum and thus give rise to unstable multiplier dynamics and also to the possibility of debtdeflation since workers then become debtors of asset holders in and around the steady state.Finally, and also not demonstrated by an explicit presentation of such a numerical result,we have that a portion of adaptively formed expectations, d7rr, that lies between 0.12 and0.84 provides convergence instead of the limit cycle situation shown for the value o'r : 0.1.

In the last set of eigenvalue diagrams (figure 18) we consider further important parameters ofthe 18D core dynamics, characterizing business fixed investment, labor productivity, externalgrowbh and the external labor market.

50

Figure 18:

E'igenualue calculations for inuestment, growth, the NAIRU and labor productiu,itg

The first row in the diagrams in figure 18 shows that increased sensitivity with respect toboth the profit / required interest differential and the sensitivity towards the term structureof interest rates increase the stability'of the steady state as far as convergence towards it isconcerned. The same however does not hold true for the impact of capacity utilization rateson the rate of investment which when varied does not create situations of local asymptoticstability (see second row to the left). On the right hand side of the second row we considerthe ratio lr, the labor coefficient which is the inverse of labor productivity. Increasing thisratio adds convergence to the dynamics, a thing one would have expected for the reciprocalratio, the labor productivity of the economy. At the bottom left of figure 18 we consider thegrowth rate of the world economy which when lowered, starting from the reference situationof table 14, adds asymptotic stability to the dynamical system, unless it comes too close tozero. Finally, a higher NAIRU level for the employment rate,V, equal to 0.9 in the referencesituation, produces convergence, that is a smaller corridor for nominal wage increases onthe external labor market adds to the stability of the economy, see the diagram bottomright. The same holds true for the NAIRU rate for capacity utilization of firms as well asfor housing services (not shown).

l6---

o4

51

T--_l Stoble

Ê'r =05

I Cycllc¡l I Explodlng

Ê¡r= l'0

3

pP

2

3

pP

2

0 I 2p,¡ aa 0 t 2A,3 | 5

Figure 19: The full 18D dynamics: Global consi,derations

All of these stability investigations are of great importance since in particula,r in macroe-

conometric work convergence back to the steady state, if not enforced by the sccalled jumpvâriâble technique, is a basic requirement in these types of approaches, not however in thepresent modelling framework. Nevertheless, adjustment speeds are difficult to estimate withrespect to their most plausible range, and are therefore to be studied intensively in their roleof creating or destroying convergence. As the figures of this section show the outcome forour 18D core dynamics, though basically only a single example in this direction, looks quitereasonable compared to the discussion of the basic feedback mechanism of such a model typethat we have conducted on various levels of generality in parts I and II in Chiarella, Flaschel,

Groh and Semmler (2000).

We conclude this section with an example of the global simulation studies we have used

extensively in the preceding sections for studying the subsystems of the full 18D dynamics.The parameter set underlying the figures 19 and 20 is the one provided in table 9.

We see again, in figure 19, that price flexibility is stabilizing in the present situation, whilewage flexibility, concerning the outside labor market, is not. However increasing the reactionspeed of wages with respect to the inside employment rate improves the stability region forwage and (outside) wage flexibility. Figure 20, finall¡ shows that increased price flexibil-ity does not significantly alter the domain where the quantity adjustment process exhibitsconvergence to the steady state. AII these stability results heavily depend on the fact thatthe consumption propensity c1 is situated in a certain economically meaningful range (ofapproximately 0.4 to 0.6).

7 Conclusions

We have considered in this paper the 18D core dynamics of the approach of Chiarella and

Flaschel (1999a,b) from a variety of perspectives, in particular with respect to the various

economically meaningful subdynamics it contains. Our general finding was that the impli-cations of the 6D working KMG model, derived and investigated in Chiarella and Flaschel(2000) and Chiarella, Flaschel, Groh and Semmler (2000) from various perspectives, are

confirmed if more structural details such as a housing sector, more complete asset market

dynamics, exchange rate dynamics and fiscal and monetary policy rules are added to the pic-

ture. Though the descriptive relevance of the considered dynamics is considerably improved

52

T__--l Stâble

ÊP =os

I Cyclic¡l

0nl

I Exploding

ßr=2'o2

Ênl

01'9r,'

1 6 0 1 2h, 316

Figure 20: The full 18D dynamics: Global considerations

thereby, we still often simply find a set of three possible outcomes, namely convergence tothe steady state, limit cycle behavior, or pure explosiveness as long as the dynamics are onlyintrinsically nonlinear and not augmented by extrinsic mechanisms that capture the factthat such economies will change their behavior far off the steady state. F\rrthermore, therange of persistent fluctuations found was often very small, so that increasing adjustmentspeed soon led us from convergence to explosive behavior around the steady state.

The paper has in addition discussed a variety of feedback chains that characteúze the con-sidered dynamics as well as others that are not yet present in it. It has provided a discussionof how the partial feedback mechanisms and their known (de-)stabilizing potential can be

investigated from a partial as ìMell as a more or less integrated perspective, giving rise tothe general impression that the considered dynamics will more often be locally repellingthan convergent. The study of extrinsic nonlinearities that bound the dynamics is thereforean important next step in the investigation of the disequilibrium growth model - with anapplied orientation - introduced in Chiarella and Flaschel (1999a,b), Chiarella, Flaschel andZhu (2000) and extended further in a variety of ways in Chiarella, Flaschel, Groh, Köperand Semmler (1999a,b) and Chiarella, Flaschel and Zhu (2003).

The general outcome of our investigation in the present paper is that such models of dise-quilibrium groÌv'th, due to the fact that most of their important feedback chains are morelikely to be destabilizing rather than stabilizing their uniquely determined interior steadystate solution, that macroeconometric applications of the considered disequilibrium dynam-ics have to be prepared to find local instability of the steady state that is turned into globallybounded business fluctuations by important behavioral nonlinearities known to exist far offthe steady state. The most prominent example here maybe is an asymmetric (strictly con-vex) money-wage Phillips curve that is nearly horizontal for low rates of wage inflation, as

it was recently again confirmed to exist in the paper by Hoogenveen and Kuipers (2000).

The new challenging task thus is, on the one hand, that m,;,;rodynamics now have to have

- from the mathematical point of view - a high order orieutation in order to understandintegrated feedback systems with respect to local as well as global stability, with the under-lined topic a still much neglected area, since knowledge about behavioral nonlinearities - tobe associated with certain destabilizing feedback channels - is at best developed in a veryrudimentary fashion. Dynamic macroeconometrics, on the other hand, has to consider, likein the work of Hoogenveen and Kuipers (2000), how such nonlinearities can be confirmedby the data, and - if so - that the business cycle is an endogenous phenomenon driven by

53

local instabilities, global bounds and stochastic shocks, implying that the Frisch paradigm isnot a good guide in this area of research, on this point see see also Chen (1996, 1999, 2001).

Structural macroeconomic model building must place its main emphasis on the importantfeedback channels that drive the macroeconomy (away from the steady state),15 and it musthandle their decomposition and re-integration (as demonstrated in this paper from the for-mal as well as the numerical point of view).16 Finally, structural macroeconomic buildingmust approached with the view that if business cycles are endogenous components in theworking of modern macroeconomies, then tools must be correspondingly chosen rather thantools determining what is to be investigated and what not.

15See Flaschel, Gong and Semmler (2001, 2002) for actual examples.l6see Chiarella, Flaschel and Semmler (2001), Asada et al. (2003), Chia,rella, Flaschel and Fïanke (2003)

with respect to the analytical possibilities that here meanwhile exist.

54

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56

Appendix: Notation

The following list of symbols contains only domestic variables and parameters. Foreign magnitudesare defined analogously and are indicated by an asterisk (x). To ease verbal descriptions we shallconsider in this paper the 'Australian Dollar' as the domestic currency (A$) and the 'US Dollar'($) as a representation of the foreign currency (currencies).

A. Statically or dynamically endogenous variables:

vadyp

ydP

yn

a"ail,y?tilet'2

t6

ld.td.eúft'de

- tdeog -og,ue,Í

l-evTatv:ld.lpc- (c?')c"(cZ)c:c-*ccci^dvh

sÍsf,

Output (per K) of the domestic goodAggregate demand (per K) for the domestic goodPotential output (per K) of the domestic goodNormal sales (per K) of the domestic goodNormal output (per 1{) of the domestic goodExpected sales (per K) for the domestic goodReal disposable income (per K) of workers and asset-holdersPopulation aged 16 - 65 in efficiency units (EU: x exp(rz¿ú), per K)Population aged 66 - ... in EU (per K)Population aged 0 - 14 in EU (per K)Total employment of the employed in EU (per K)Total employment of the work force of firms in EU (per K)Total government employment in EU (per K)Work force of firms in EU (per K)Total active work forceEmployment rate of those employed in the private sectorParticipation ¡ate of the potential work forceRate of employment (7 the employment-complement of the NAIRU)Real (equilibrium) goods consumption of workers (per K)Real (equilibrium) goods consumption of asset owners (per K)Total goods consumption (per K)Supply of dwelling services (per K)Demand for dwelling services (per K)Gross business fixed investment (per /()Gross fixed housing investment (per /()Gross (net) actual total investment (per K)Planned inventory investment (per If)Actual inventories (per K)Desired inventories (per K)Nominal short-term rate of interest (price of bonds pu : L)

Nominal long-term rate of interest (price of bonds pb : Llrt)expected appreciation in the price of long-term domestic bondsRequired rate of interestPrice of equitiesexpected appreciation in the price of equities

ü lp"x + Si ln"K + S! ln"K Total nominal savings (per p"K)Si,/p"K + S! lp,K Nominal savings of households (per p,K)Nominal savings of firms (: puY¡ lp"K, the income of firms) per poKGovernment nominal savings (pet p"K)Nominal (real) taxes puK,KReal government expenditure (per K)Expected short-run rate of proût of firmsActual short-run rate of profit of firmsNormal operation rate of profit of firmsExpected long-run rate of profit of firmsActual rate of return for housing services

Expected long-run rate of return for housing services

Capital stockCapital stock in the housing sector (per K)

rT¿

îtb : þ3p'Pe

I" lK (1"" lK)TIKN/Kud

7t. : þ3S"fP,K:

lpoK :lp"Klp'xlp"K (TlK)

siS';siTnIp'po

pnp¿

Pn

P¿ì,

Kle¡

õt

P,a-ePi.,id

ncgTu

d

B. Parameters of the model

þ3

êe

u)b'u)e

u)ue

u)fe

Pu

Py

P,Pm

Pnl¡l:

e

e:le

b

b.b.6t

bL

€

nntTrn

Tjd

nr

õ

6n

aliþ,"Yttnlú1Û,, Kp

K

apraIy

i.ud¿

¿Se

Tc

Ta

Tp

Cl

C2

Pt

Nominal wages including payroll tax (in EU)Nominal wages before taxes (in EU)Unemployment benefit per unemployed (in EU)Pension rate (in EU)Price level of domestic goods including value added taxPrice level of domestic goods net of value added taxPrice level of export goods in domestic currencyPrice level of import goods in domestic currency including taxationRent per unit of dwellingExpected rate of inflation (over the long run)Exchange rate (units of domestic currency per unit offoreign currency: A$/$)Expected rate of change of the exchange rateLabor supply (per K)Stock of domestic short-term bonds (index d: stock demand) (per p"K)Short-term debt held by workers (: Blp"K)Short-term debt held by asset owners (per : B"lprK)Stock of domestic long-term bonds, of which b¿1 are held (: Bllp"K)by domestic asset-holders (index d: demand)and å¡r- by foreigners (index d: demand)Foreign bonds held by domestic asset-holders (index d: demand) (: BLlp"K)Equities (index d: demand) (: Elp"K)Natural growth rate of the labor force (adjustment towards ñ)Rate of Harrod neutral technical change (adjustment towards ñ.¿

Tax rates on imported commoditiesExports (per K)Imports (per K)Net exports in terms of the domestic currency (per p"K)Net factor export payments (per p"K)Net capital exports (per p"K)tax rate on wages, pensions and unemployment benefitsActual public debt / output ratio

Depreciation rate of the capital stock of fi¡msDepreciation rate in the housing sectorAll o-expressions (behavioral or other parameters)All B-expressions (adjustment speeds)Steady growth rate in the rest of the worldNormal rate of capacity utilization of firmsNormal rate of capacity utilization in housingWeights of short- and long-run inflation (n-n, I L): (1 - n-Kp)-rOutput-capital ratioExport-output ratioLabor-output ratio (labor in efficiency units)Import-output ratioDesired public debt / output ratioRisk and liquidity premium of long-term over short-term debtRisk premium of long-term foreign debt over long-term domestic debtTax rates on profit, rent and interestValue added tax ratePayroll taxPropensity to consume goods (out of wages)

Propensity to consume housing services (out of wages)

nlæ

58