SCHOOL OF FINANCE AND ECONOMICS
UTS:BUSINESS
WORKING PAPER NO. 94 December, 1999
Towards Applied DisequilibriumGrowth Theory: II Carl Chiarella Peter Flaschel ISSN: 1036-7373 http://www.business.uts.edu.au/finance/
Towards Applied Disequilibrium Growth rheory:II. Intensive form and steady state analysis of the model
Carl ChiarellaSchool of Finance and EconomicsUniversity of Technology, Sydney
Sydne¡ Australia
December 1, 1999
Peter FlaschelFaculty of EconomicsUniversity of Bielefeld
Bielefeld, Germany
_ Towards Applied Disequilibrium Growth Theory:II. Intensive form and steady state analysis of the model
Carl ChiarellaSchool of Finance and EconomicsUniversity of Technology, Sydney
Sydney, Australia
Peter FlaschelFaculty of EconomicsUniversity of Bielefeld
Bielefeld, Germany
Abstract
In this paper we investigate further the 34D applied structural model whose exten-sive form we introduced in Chiarella and Flaschel (1999c). Here we express the modelin terms of intensive form state variables, thereby abstracting from the underlyinggrowth trend. We explain the dynamic (and static) laws of the model directly in termsof the intensive form variables, and then determine the steady state and its charac-teristics' Finally we show how a small number of simplifying assumptions, concerningin particular consumption of asset holders and some secondary adjustment processes,reduce the 34D model to an 18D core model. It is this latter core model whose detailedstructure, steady state characteristics and dynamical behavior will become the objectof study in the remaining papers of this sequence,
1 Introduction
In this paper we derive and investigate the 34D intensive (state variable) form of the appliedstructural model of disequilibrium growth we have introduced and discussed on its origlnallyextensive form level in great detail in Chiarella and Flaschel (lgggc). We will ,"pr".urrt thãresulting 34D dimensional dynamical system from various perspectives, providing compactintensive form representations of the real and the financial sector of this economy in tableform and also in form of a system of national accounts. We will then discuss to some extentthe economic contents of the obtained laws of motion f¡om their intensive form perspective,thereby showing that the model can be understood from the outset on this intensive formIevel.
Presenting the system from these various perspectives serves the pupose of making thereader acquainted with the notation and the relationships that applt on th" intensive formlevel of the model. We hope that this increases the readability of
-the laws of motion for
quantities (including rates of growth), for prices (including wages, asset prices, and alsoexpectations), financial asset accumulation and feedback fiscal ánd monetary policy rulesto be presented and discussed in section 3 of the paper, Section 4 then calculates the (rrpto the determination of nominal variables) ruriquely determined steady state solution ofthis dynamical system and briefly considers its comparative dynamic properties which aregenerally very simple in nature. We then go on and show that the dimension of the dynamicscan be significantly decreased by only a few simplifying assumptions (leading us from 34D to18D dynamics) whereby we obtain what we will call the 18D core dynamics of our approachto disequilibrium growth.
We shall briefly compare these dynamics in section 6 of the paper with the fourteen equationssecond order system of Bergstrom, Nowman and Wymer (1994), a prominent example fromthe literature on continuous time macroeconometric model buitding and testing.t \Ale the.,use an approach as the one by Barnett and He (1998), who reconsider the 14 equationmodel just mentioned from the numerical perspective, in order to also study the numericalproperties of our 18D core dynamics in particular with respect to the role played by speedsof price and quantity adjustment. In the present paper, however, we shall fàr the tiåe teingonly use eigenvalue calculations based on eone-parameter changes in order to see whicùadjustment speeds (and their corresponding feedback chains) are stabilizing in the full lgDdynamics, and which are destabilizing when they are increased. In future work we will alsocalculate as in Barnett and He (1998) bifurcation boundaries in two-parameter spaces (whichbound the regions of local as¡rmptotic stability of the system) and will then show that theoccurring bifurcations are essentiatly of the Hopf type (which at present is only a conjecturebased on earlier work on such disequilibrium growth dynamics).
In a concluding section we will finally briefly describe the ways in which the model introducedin Chiarella and Flaschel (1999c) will be investigated, both analytically and numerically, ina series of further papers that a¡e in preparation, see our work cited in the list of ràfer-ences, thereby indicating what can be done at present in the analysis of reasonably detailedapplicable macroeconomic models of disequilibrium growth.
lsee also Bergstrom (1996) for a brief survey on this literature,
1
2 The Real and the Financial structure on the rnten-sive Form Level
The following two tables provide a survey of the structure of the economy to be investigatedin the following2 and they do this on the basis of what has b"en pr"rerrted and discussedin (1999c) with respect to the extensive structural form of a generaldis growth model b¡transferring this discussion to the intensive formlev state calculations.s This päper therefore continues the analysisbegun in Chiarella and Flaschel (1999c) by showing inåt tnir model type has a well-definedintensive-form state variable representation and also a basically (,rp to ttre level of nominalvariables) uniquely determined interior steady state or balanced gràwth path solution.
2.t rhe structure of the real sector of the economy
central bank follows an interest rate policy rule with a zeto target rate of inflation, whichrestricts the steady state solution of the dynamics to zero.2see the appendix to this paper for the employed notation.3In order to clarifu the notation used and the contents it represents the reader should therefore utilizethis original presenta tion of the model.
2
Table 1 describes the real sector of the considered economy. We have a labor market, threecommodity markets and the housing market. Domestic production g:ylK, per únit ofcapital, concerns one good that is only domestically used (for all privatl consumpti on c. j-c.,al] investment gf,,gl,TlK, also into housing, ana ÀI government consumption g.: C¡k aiawhich uses up all the imports id us intermediate goods) and one gooã thui is only usedfor exports ø. There is thus only a single commodity used in domestic absorption -" up tothe housing services cfl demanded by workers. We denote the demand for this domesticallyproduced and absorbed commodity by Ade ,dlK)
Our model exhibits th¡ee domestic sectors: Households, firms and the government, but withheterogeneous agents in the household sector, workers and (prue) *r.t holders, the formersupplying their labor l" (measured here in efficiency units) at the gross v/age level u;öu (whichincludes payroll taxes) and the latter the housing services cf, for the workeis. Firms producea non-traded domestic and an exported commodity and employ labor li" (with varying ratesof utilizati"! l1:) and imports jd (besides their capital stock K) foi these pruposes, andinvest into fixed business capital gÊ (pe, unit of capital) and invent ories T f K . Gãvernmentfinally provides public consumption goods g, pays rents ü/'e and unemployment benefits ,r¿,"and also employs part of the workforce I!". There is endogenous growtlr-n of the potentiallabor force .L1, of the capital stock lc, bi gÍ - ó and of the stock of housing Kn, bi gÊ - 6n,(supplied at price p¡ for rental services) and also actual change of inventories z : Nik tnitis different from l,heir desired rate of change I / K.
Labor Non traded Goods Exports Imports Dwellings
Workers
Asset holders
Fi¡ms
Government
I" : o¿ll,13
¿1",ri'
tÍ.
cu
cc
AP rAr gxrî /K
sd:(l-òa"
û
cl
ctn, gn
jd
Prices
Expectations
,tJ)" frub" , wr" , wu.)
7t pi
Pu:(llr,)Pu
lr:þi
Pa: ePI Pm:
(l + r*)epi
PntPs
n:þiStocks
Growth
lî,13,13
L,.: n
v : NIKIK :71
k:go-6N g- u
cI: kn
k¡: g¡ - 6¡
Table 1: The real part of the economy.(Foreign country data: 1,pi,pîn,r: : r")4
3
4û the growth rate of a variable ø.
2.2 The structure of the financial sector in intensive form
Let us next consider the financial part of the economy. Note that all stock variables B, ..., E(and their rates of change) appearing here are mea^sured relative to the gross value of thecapital stock puK based on prices pu thal include value added tax. They are then denotedby lowercase Latin letters (and by e in the case of equities .Ð).
Money Short-term Bonds Long-term Bonds Equities Foreign Bonds
'Workers
Asset holders
Firms
Government
a-u.Í: #lB"b"[: ,,xL)
aul: #J
aiu\t: #l
ntUtl:41L pþnJ
Ee[: ou-E-1
ørl:#)a5u5¡:
,u-4-1
Prices
Expectations
1 1 ["] k: Llrt
,f A: þ3
P.
t": þ3
ePi: e'7lri
e:ê
Stocks (lp"K)
G¡owth
b:b-lb.
B, Ê-, B"
ut:b\+bl
Bt, Bl
€
E
bt,
BL
Table 2: The financial part of the econonxy.(Foreign country data: rf , rI : r")
The first column in the above table shows that we do not consider money holdings in themodel of this paper, see Chiarella and Flaschel (1999c) for details. At present therÃre only(four) interest-bearing financial assets in our model that can be held by the (p're) asseiowners and by the workers of our economy (as shown in the above tabte). As in the lieynes-Metzier model of monetary growth of closed and open economies, see Chiarella and Flaschel(1999a,b), we here ¿tssume, in order to start with a simple representation of financial fl.ows,that bonds are only issued by the government, that firms use only equity financing and payout expected earnings as dividends, and that there exist also long-term bonds issued by the'foreign government'. Financial flows between the sectors of our economy are therefo.. .r"rynarrowly defined. Note that we allow for savings out of wages in the present model (in aKaldorian way)_and that workers save only in the form of short-term debt (interest-bearingsaving depositss held at the local branches of the central bank6). All other assets (plulthe remainder of short-term debt) are exclusively held by the (pure) asset holders of ourmodel. We stress that this formulation has served the purpose of simplifying the budget
5or fixprice bonds, which are perfectly liquid, while the other type of bonds: long-term bonds (here consolsor perpetuities, held by asset holders) cannot be redeemed at a given price from the viewpoint of the sectorof a,sset holders as a whole.
6for the purpose of financing government expenditures with no explicit reserve requirements
4
constraints of the agents in Chiarella and Flaschel (19g9c), but should be extended in futurereformulations of the modei.
This is the basic structure we assume for our economy which will be further explained inthe next section from the viewpoint of national accounting before we present and discuss theintensive form of the model of chiareila and Flaschel (1g9gc).
2.3 The Structure of the Economy from the Viewpoint of the Sys-tem of National Accounts
We shall consider in this subsection the production accounts, income accounts, accumulationaccounts and financial accounts of the fou internal agents in our economy:7 firms, assetholders, workers, and the government (including the monetary authority). These accounts -plus the balance of payments - provide basic information on what is assumed for these foursectors and which of their activities are still excluded from the present theoretical frameworkof a small open economy.
2.3.1 The sector of firms
We start with the accoults of the flrms - shown in table 3 - which organize production gr
(per unit of capitat), employment lt", in efficiency units, of their *orÈfor.u li" and gros"sbusiness fixed investment g¡ (per unit of capital) and which use (in the present formulationof the model) only equities e (per value unit of capital) as financing instrument (no debtin the form of bank loans or bonds issued by firms). There
"r" lruirr" added taxes ru onconsumption goods, import taxes r* and payroll taxes ro with respect to hours worked lj",but no further taxation in the sector of firms and there are also no subsidies paid to tÉissector.
Firms build dwellings - which are of the same type as all other domestic (non-traded) preduction - and sell them to the asset holders (as investors) and thus have no own investmentoccurring in the housing sector. They sell consumption goods to workers, asset holders andthe government, export goods to the world economy, organize fixed gross investments withrespect to their capital stock (as well as voluntary inventory changes TIK with respect tofinished goods) and experience involuntary inventory changes NIK: (y -Ad)/, (plr ¡nitof capital) due to the deviation of aggregate demand yd from output y (which is Ùased onexpected sales 3t" and planned inventoriæ llK) (everything per unit of capital).
Firms use up all imports as intermediate goods which thereby become part of the uniquehomogeneous good that is produced for domestic purposes. They have replacement costs withrespect to their capital stock, pay indirect taxes and wages including payroll taxes. Theiraccounting profit is therefore equal to expected profits (based on sales expectations and paidout as dividend to equity owners) and retained profits (equal to planned inventories). As
TThe fifth agent, the foreign economy, is represented by the balance of payments at the end of thissubsection and is confined to steady state behavior in the present form of the model. All demands of thisforeign sector are indexed by *, while its supply of long-term bonds ¡|2 to domestic residents is indexed by2.
5
is obvious from the narrow income account of firms, firms thus only save an amount equalto their intended inventory changes. The accumulation account is self-explanatory as is ìhefinancial account which repeats our earlie¡ statement that the financial deficit of firms isfinanced by the issuing of new equities solely.
Note that all investment is valued (and performed) net of value added tax and thus atpro-ducer nric^es py in the place of the consumer prices p, : (l + Tu)ps.Indirect taxes (valueadded taxes)8 thus only fall on consumption activities and not on'lross investment (of anykind, i.e., here also not on housing investments and the inventoiy investment of hr-.i.There are furthermore no direct (capital) taxes in the sector of firms, neither o., prop..iynor on profits, since our model wilt be formulated in such a way that all expected profitsare distributed to asset holders and since there are no taxes on windfall profits (*nexpectedretained earnings - or losses - of firms that help to finance investment). Note however thatthe wages wb' (inefficiency units) paid by firms include payroll tæies ,r*. çfo, unemploymentinsurance, health and other social insurance, and retirement pension.)
".rd that waje income
w" oL workers is taxed at the rate r- in addition. Note finally that the accumulation accountof firms is based on realized magnitudes and thus refers to their intended inventories Z/Konly indirectly (via the expected rate of profit p", to be distinguished from the actual rateof profit po, which^iTl:9q y]ndfall profits or losses, due to unfulfilled sales expectations y"of firms, NIK: ñNlK,NlK - u).
Production Account of Firms:
Uses Resources
Imports eph,jd lp,
Depreciation 6polpu
Indirect Tarces z,(c,, * c" + gd)polp" * r^epijd f p,
Wages (including payroll taxes) wb.Ij"
Profits p"pv/Itu + (IlK).(pnlp,): popv/pu -t ñvpolp,
Consumption c,,
Consumption c"
Consumption gd
Exports p,a/p,
Gross Investme\t gkf pu
Durables (Dwellings) 9nk*ps I p,
Inventory Investment Ñrpulp"
sThere is however a rax on the imports made by the firms.
6
fncome Account of Firms:
Uses Resources
Dividends p"pslpu
Savings Si l@,K) : (I I K)(pu lp,)
Profits p"prlp,
Accumulation Account of Firms:
Financial Account of Firms:
Uses Resources
Financial Deficit F D I (p"K) Equity Financing p".Ee
Table 3
Note with respect to table 3 that profit rates are calculated per net value of the capitalstock PyK, since there is no value added tax rrpn on all ptuchases intended for investmentpurposes' while the relative variables and the intensive form of the model is calculated bydefinition per gross value of the capital stock prK,po: (1 + rr)py,which implies that theralio pufp, has to be used in various places in order to correct fór such diflerences in thevaluation of the capital stock. Note also that we do not always supply short-cut expressionsfor the intensive form variables, in those cases where such u io*"r-"*e representation is notneed in the presentation of the laws of motion of the model in section 3. Note finally thatrù/e use e in the place of e for denoting E lK since e is used to denote the nominal exchangerate of our model.
2,3.2 The sector of households: Asset holders
Table 4 gives the accounts of the sector of a.sset-holders for whom we already know that theinvestment in housing as well a^s the supply of housing services ha^s been exclusively allocated
7
Uses Resources
Gross Investment g¡prf pu
Inventory Investment Ñ rp, / p,
Depreciation 6polpu
Savings Si l(n,K)
Financial Deficit F D I @"K)
to this sector' The production account in this regard shows the actual sale (not the potentialsale) of housing services (: demand for housing úrvices by assumption) which is divided intoreplacement costs and actual earnings or profits on the uses side of the production account.
Income of asset holders comes from various sources: interest pa¡rments on short_ and long_term domestic bonds and on long-term foreign bonds (net oflax payments which must bepaid abroad), dividend payments of firms (based on thàir expected jrofit) and profits fromhousing rents. AII domestic profit income is subject to tax payments at the rate r"and aftertax income by definition is divided into the consumption of domestic commoditieslincludinghouses, but not housing services) and the nominal savings of asset owners.
The accumulation account shows the sources for gross investment of asset-holders in thehousing sector, namely depreciation and savings, the excess of which (over housing invest-ment) is then invested into financial assets as shown in the financial account. Note here thatshort-term bonds are fixed price bonds ?a
: 1 (which are perfectly liquid), while long-termbonds have the variable pricee p6 -_ Lf rt (an ayments of one unitof money per period) which shows that they or perpetuities (thesame holds true for imported foreign bonds,
" ,ot"ty¡.
Production Account of Households (Asset Owners including Housing fnvestment):
Uses Resources
Depreciation ïnknpv I p,
Earnings ÍInlp,K
Rent p¡cflf p,
Income Account of Households (Asset Owners):
Uses Resources
Tax payment r"rb.
Tax payment r"ò¿1
Tar<es r"(p¿cllp, - ínknpulp,)
Tax payment r"p'pu/pu
Consumption c"
Savings S! lp"K
Interest payment rö"
Interest payment å!
Interest payment e(l - ri)bl2
Dividend payment p"py fpu
Earnings flnlprK
9and are thus not perfectly liquid, since there is no 'money back' guarantee here for the sector of assetowners as a whole,
8
Accumulation Account of Households (Asset Owners):
Uses Resources
Gross Investment g¡le ¡pof pu
Financial Surplus F S I p,K
Depreciation ïnlcnpa I pu
Savings S? lp"K
Financial Account of Households (Asset Owners):
Uses Resources
Short-term bonds B"ö"
Long-term bonds p¿a{al
Foreign Bonds ee;A[UI,
Equities p".Ee
Financial Surplus F S I p"K
Table I
2,3.3 The sector of households: Workers
The next set of accounts, the ones of worker households, are fairly simple and easy to explainand are given in table 5. First, there is no production account in this sector. Income of themembers of the workforce, which may be employed, unemployed or retired, thus derivesfrom wages, unemployment benefits or pension pa¡rments wherã I" : atll denotes the totalnumber of persons in the crurent workforce (l the part that is emplåyed) and a¿tfi thenumber of retired people which have access to pension funds (o¡:const. the participÁtionrate of the potential workforce lf). To this we have to add the interest income on savingdeposits (short-term bonds) which is taxed at the general rate used for fi¡ancial a.sset income.All wage type incomes are subject to taxation at the rate r. and are by definition dividedinto nominal consumption (consumption goods and housing services) and savings. Note herethat the employment ld'of the employed l*" can difier from their normal employment whichis measuredby l, the number of persons that are employed (reduced to efficiency unitsand per unit of capital). Note also that wages ru" (in efficiency units) a¡e calculated net ofpayroll taxes.
We assume in the following that workers have a positive savings rate and that they hold theirsavings in the form of short-term bonds solely which is mirrored here in the accumulationand finance account in a straightforward way.
I
Production Account of Hor¡seholds (\Morkers) :
Uses Resources
Income Account of Households (IVorkers):
Accumulation Account of Households (\Morkers):
Uses Resources
Financial Surplus FS lp,K Savings Si,lp,X
Financial Account of Households (Workers):
Uses Resources
Short-te¡rn bond accumulation .Ê,,å,, Financial Surplus F S I p"K
Tøble 5
2,3.4 The public sector: Government and Monetary Authority
There are finally the accounts of the fiscal and monetary authority (see table 6) which dueto the many taxation schemes and transfer pa)¡ments that are assumed are more volumi_nous than the preceding accounts - at least with respect to the income account. There isfirst however a fictitious production account where the supply of public goods is valued atproduction costs which consist of government expenditures for goods and labor.
Uses Resources
Taxes r, [ur" ld. + u"" (1. - 1..) I w," a¿I$] * r"rb_
Consumption c- ¡ p¡clf pu
Savings Si,lp,K
Wages w"Id"
Unemployment benefits w""(1. - l-.)
Pensions w"a¿\fi
rå- Interest payments
10
The resources of government income consist of taxes on the various forms of workers, income(taxed at a uniform rate), of taxes on the various forms of profit, interest and rent income(again taxed at a uniform rate), payroll taxes, value added taxes and import taxes. Usesof this tax income of the government are interest pa¡rments, transfers to the unemployedand the retired persons and the costs of the mention"á gorrurrr-ent ,production,. In genðralall these uses of the tax income of the government will exceed its income so that there willresult a negative amount of nominal savings Si lnrK which balances the income account ofthe government.
There is no accumulation of real assets in the government sector which means that we onlyhave to look into the financial account of the gorr"rrr*"rt to see how the excess of golrer¡¡munioutlays over government revenue is financed through short- or long-term debt. Note thatthere is some type of accounting money in the
".ono*y that however orrly fuels the economy
druing the transactions period, but does not appear as flow in the financial accounts of assetowners' and workers, and the government, but instead returns to the banking sector at theend of each transaction period ú by the settlement of atì budget restrictions in the economy.In striking contrast to a cash in advance constraint we thus a^ssume in this paper, followingChiarella and Flaschel (1999c), that agents can obtain all money they need for transactionpurposes during the transaction period ú (as intra-day credit in one form or another), but thatthey have to satisfy their budget constraint at the end of each such period ú whåre moneyholdings are not needed and thus not present in them. Instead af üquid asset holdingsconcern the short term bonds of the government as some form of interest bearing savirìgdeposit.
Production Account of Fiscal and Monetary Authorities:
Uses
Government expenditure for goods gd : (l - ùU"
Government expenditure for services ,r.l!.
Resources
Costless Provision
of public goods
11
Income Account of Fiscal and Monetary Authorities:
Accumulation Account of Fiscal and Monetary Authorities:
Uses Resources
Savings Si ln,K
Financial Deficit FD lp,K
Financial Account of Fiscal and Monetary Authorities:
Uses Resources
Financial deficit F D lp"K Short-term debt Bô
Long-term debt pbBtbl
Table 6
2.3.5 The balance of payments
We finally describe in table 7 the balance of payments of the economy under consideration.This will be done from the viewpoint of the foreign sector which is thus a fifth agent ofthe economic structure considered in this paper. The description of the behavior ãf tru,agent will however be confined to steady state behavior in the present investigation of thestructural equation-s of the model.
Uses Resources
Interest payment rå
Interest payment ó¿
Pensions wr"a¡\ft
Unemployment benefits .""(1" - l-.)
Government consumption gd
Salaries .be¡de
Savings Si ln,K
Wage income taxation r-fw.ld" * w".(1. - P") I w,"a¿Ifil
Profit and interest talcation r.l,p"pa lp, + rb + btl
Rent income taxation r"(p¡cftlprK - 6nknpalp,)
Payroll taxes row.ld"
Value added tax r,(c. * c" + gd)pa lp,
Import taxes r*epijd f pu
L2
The balance of payments:lO
Uses Resources
eP* r/pu
e(t - ri)bl,
e1.at. ¡rt
eP* jd lpu
(7 - r")b\.
"a[UIr¡rt.
Table 7
This balance of pa¡rments shows the trade account (exports ø and imports jo p", unit ofcapital), the international component of interest payàerrts (to foreigners and from abroad)that are all assumed to cross borders and the rutflãw and the inflow oirr"* capital (long-teribonds)' Note that this account does not show any reserve changes ofthe central bank due toforeign exchange market operations. This is possible in the approach chosen in this paper,despite a temporaliy given exchange rate e, since the supply of bonds of the go.rernmentdomestically and abroad of the equities of firms are channeled here into the savings decisionsof households without readjustments, while the excess of domestic private savings is goinginto foreign bonds, which in turn implies that the balance of payments must be balancedwithout any intervention from the central bank, see Chiarella and Flaschel (1gg9b,c) formore details.
This conciudes our intensive form description of the four accounts of the three typical sectorsof a small open economy (here with heterogeneous agents in the household sector) plus aforeign sector that is here represented via the balance of payments solely.
3 The implied B4D Dynamics
In order to study the dynamics of our stylized disequilibrium growth model analyticallyand numerically it is necessary to reduce the equations of the model presented in Chiarellaand Flaschel (1999c) to intensive or per (value) unit of capital form. thir t* already beenindicated and discussed in the preceding section from the viewpoint of the system of nationalaccounts by dividing all (nominal) level magnitudes through (the value of) the capital stocklf of firms (measured at consumer prices pr) and by taking note of the fact that the modelexhibits Harrod neutral technological change which means that all variables involving labormust be meastued in efficiency units, i.e., they are to be multiplied with the term exp[n¿t) inaddition to remove the trend in labor producti 'ity from them. Note that this procedure mustalso be applied to the nominal wage u/ which is to be replaced by the term uã : r/exp(n¿ú)since nominal wages ø will rise with labor productivity in the steady state and must theieforåbe detrended and replaced by the wage rate per efficiency unit nf lubor in order to get avariable that in principle allows for stationarity (tî," : û - nù.
from the foreign economy and in terms of the domestic currency,loviewed
13
Note here again that the model has been formulated in a way that implies zero price in-flationin the steady state if it is a^ssumed that the target inflation rate of the central bank: ø., equalszero. This assumption will be made for the remainder of the paper.
The variable-s w',pr,p¡, etc. therefore need not be detrendud *y fiuther, but represent statevariables of the dynamical system to be formulated below, The first two laws of motion ofthese state variabl module 5b. of Chiarella and Flaschet (1g99c)by inserting there IIî",U : alap in particular and by replacin!level variables by _ ,es. In the same \Ã/ây we obtain the dynamiãlaws for long-run in-flationary expectations zrl by making use of Yi: K, and also the law ofmotion for the price of dwelling services pä, see module 2b of Chiarella and Flaschel (1g9gc).
Note finally that the magnitude c" : C"lK, the consumption of asset holders of the domesticgood per unit of capital, is a given magnitude in the steady state of the model (but not offthe steady state), due to the assumption C": l made in Chiarella and Flaschel (1gg9c).
3.1 The laws of motion
Let us start oru presentation of the model in intensive form by considering the quantitydynamics it implies first:11
1. The Dynamics (7 laws of motion)
ù' : 00"(ao - a) + 0 - Øo - 6))a"u : y-ad-bo-6),
: aiþ'-/e." - 1) + aiV - V) + "y - bk - 6),: a!(cf; lcf- 1) + "r(v -V) + t - kt, - 6),: ^t-(gk-6),: þ,(vi-VT¡tt-Øo-o¡: þr(V - V)
cu;dçh
ô",7Derl:V
(1)
(2)
(3)
(4)
(5)
(6)
(7)
These formulae are obtained from the extensive form presented in Chiarella and Flaschel
!1999")- by. ther.rsual growth rate formula for intensive expressions, for example Vlt< :
Y" - K with -Il : g* - ó by reformulating such expression in terms of time derivativeswhenever necessary. The dynamical laws for quantities describe sales expectations dynam-ics, actual inventory dynamics, three types of consumption demand dynamics for workers(domestic goods and housing services) and asset holders (domestic gooãr including houses),the dynamic employment policy of firms and finally the dynamics of the NAIRU rate ofemplol'rnent.
llThese dynaurics, as well a.s the growth dynamics, will be considerably more complicated if substitutionis allowed for in the production possibilities of firms, see Chiarella, Flaschel, Groh, Köper and Semmler(1999a,b) on this matter,
t4
Next we describe the price dynamics as far as real markets are concerned.:
2. Wage / Price Dynamics (4 laws of motion):u":
+Po:frr:Pn:
þ.,(v -v) + p_,(vi -vî¡rc-(ps + nù + (L - rc,)(trt + ñ,) - n,
þo(alao - U) + noû" i (L - nr)rt
Bnr(an,(p, - nt) + (1 - a,,r)(0 - "t))nd
0^(å- uh) + nnþo + (1 - *n)n'
(8)
(e)
(10)
(11)
These equations for wage and price dynamics (including medium mn inflationary expecta-tions) and rental price d¡mamics are straightforward consequences of the laws of motion asthey were formulated in Chiarella and Flaschel (lgggc).
We have next the dynamics of asset prices, expectations about this dynamics and the dy-namics of certain long-run concepts of interest and profits. Note that we here make use ofTobin's Q : pux@ as an aggregate expression for theloint dynamics of equity prices p" and.the number of equities ,8. Due to the formulation of the model in Chiarella and Flaschel(1999c) it suffices to describe the dynamics of q in the intensive form of the model whichmoreover, due to the lack of wealth effects and the like, does not feed back into the rest ofthe dynamics. Note however that the expression for þ" from Chiarella and Flaschel (1gg9c):
Pe: 1-0n.(1 -o,) l0-r.)p"f q*a¡r.,- ((1 - r")rl +16)],
with aggregate expectations z16 being determined by a,r6r+ (1 - ar)þa, has to be insertedinto the law of motion for Tobin's q in order to get a description of these dynamics that iscomplete. Due to the isolated nature of these dynamics it is however not necessary here togo into these dynamics in more detail.
3. Asset Prices and Expectations (8 laws of motion ):
Po L-þeo(l -o,) [(L - r")rl * d"qtb, - ((1 - r")r * €)], pu: r/rtiro, : þno"(pu-¡rar)
â : þ.-þot S* - 6 * y - ad - (þ"(þ"dy" - u ) + l1"oa")q
fr., : þn."(þ.-¡r"r)^þ'þ - [(t-rj)rita,ê,-((1 -r")rt+16)]
è, : þr"(ê-er)þ' : þ0,(p'-p')þL : Þo¡Øn-pL)
-@o-0¡
(r2)
( 13)
(14)
( 15)
(16)
(17)
(18)
(1e)
15
We have first the law of motion for the price of long-term bonds pu : I/rt which is hereexpressed in terms of the interest rate these bond prìces (consols) ,åpres"nt. This interestrate adjusts in the direction of the risk free interest rate on short- term bonds after taxes(I - r")r, augmented by a risk and liquidity premium { for long-term bonds. Note that wehave removed the perfect foresight expectations from the left side of this adjustment equation
Flaschel (lgggc) - gives rise to the fraction in front of thethat only the law of motion of.'razy' expectations is thentructure determinate, but that aggregate expectations on
these bond price changes 16 Ð'rê needed in the subsequent laws of motion of asset prices.
Making use of the formula for the rate of change of equity prices zr" expected on average wecan transform the lau¡ of motion for equity prices just as the law of motion for long--termbond prices and remove the explicit representation of 'ambitious' agents from it too - asshown in the representation of the p"-dynamics. This law is again tã ¡e supplemented bythe law of motion for the expectations of ,lazy, agents.
The nex Tobin's q which, as already shown, is measured by ratiobetween and the producer price of ihe existing capital stoci: q :ffi'W" .Ê where the first two inflation rates have already beendete¡mined in the formulae preceding block 3. For the remaining expression fiK : E - kwe get
t-K: E-k:4 4-oPsK P"E: lgr- 0*y -ad - (p*(p^dy" -u) +yB,oy")]lq- Øo- 6¡,
which implies the law of motion for Tobin's g shown in module B. Note again that theexpression f.ot þ can be inserted into the q dynamics and thus gives rise to one law ofmotion in Tobin's q solely.
The method used for the describing the dynamics of p" also applies to the law of motionfor the exchange tate e by removing again the correct expectations of the ,ambitious, agentsfrom its right hand side after having inserted the expression for average expectations € :o,€, * (1 - a')ê into this formula. Again this is to be supplemented by the law that describesthe evolution of the expectations of.'lazy' agents in the postulated adaptive way.
There follow the two laws of motion for expected long-run profitabilit¡: pt, which is usedin the investment equation for the capital stock, and, pt¡, which is used in the investmentequation for the capital stock in housing. Both of these measures follow their short-runequivalents with some time delay.
Next we consider the laws of growth that apply to the economy under consideration:
4. Growth Dynamics (6 laws of motion
: þn-(ñ - n), n: n(V,V): n*n¿-(so-0¡
):
iL
î"(20)
(2r)
16
gx: þnrþl-gn)îtn: 9n-6n-(rlo-O¡gn: þn^GÍ,-gn)n¿ : 0",çñt - nù, ñt : ítt(gn)
5. Monetarv and Fiscal Rules (3 laws of motion):
(22)
(23)
(24)
(25)
The growth equations represent the time rate of change of the so-called natural rate ofgrowth, of law of motion for the full employment laboiintensity (in efficiency units), thetime rate of change of gross investment per unit of capital (also ftr the housin! sector), thegrowth rate of the relative magnitude of the stock of hãuses to the capital stocklmptoyãa Uyfirms and finally the time rate of change of the rate of Harrod neutrál technological change.There is no further comment needed with respect to the above presentations of the growthlaws of the economy which again use the forrn-ula k : g*- ó in the formulation of intensiveexpressions.
Next the dynamic feedback rules for government behavior are collected, concerning thesteering of the short-term nominal rate of interest by the central bank, the rate wage taxationrule based on the evolution of government debt d and the motion of the tax rate on importswhich is here used to establish a balanced trade account in the steady state.
t - -0,,(r+€- ri) + þ,,(po-î) + þ,,@lao _U), î :0î.: a,-r(dfd-1) +dr-râ, d-
y'A Prr - P^jdt nL (lr^-
sl
(26)
(27)
(28)
Note with respect to module b that the a of government bonds cannotbe divorced from the real sector (even are not yet included in themodel) due to the assumed wage income the governm""t
"it"*pìri"steer a certain ratio for government debt to a
There remains the dynamics of aggregate and individual asset holdings which represents themost involved block in our d¡mamical system, We have already stressed that the individual
workers and asset holders and also throughout thening dynamics, since only total government bondsthe absence of wealth effects in consumption and
due to the independence of consumption of workers and asset holders from their interestincome. In the current version of the dynamics only the laws of motion for å, ö, feed baciinto the real pa.rt of the dynamics via the wage tax collection rule of the government.
6. Assets 6 laws of motion)
b : af[ka"trb+bltn + (w"" f p")(1. - 1.") + (r," lpu)o,t"z + @b" lpòtl.l(pr+g*-6)b
L7
(2e)
i,\
ui:
r¿ alor(t - "!)lga. + rb + bt
t" + (w"" lpò(f - p") + (r," lpu)o,tfi+ (wb" lp)t!.)(pr+9*-6)b\
"'(t - "frXt - on)[ga" + rb + b¿
t" + (w"" /p,)(1. - 1,.) + (*," lpo)o,t; + çwb" lp)t!.1@o+s*-6)b!iB!rl@"K) - (po + gk - Ðbt2, where we have
(30)
(31)
(32)
br, :(33)
(34)
according to the flow budget constraint of asset holders.
This part of the d¡mamics is to some extent missing in the Murphy model, see powell andflurphy (1997), with which we compared otu moáel in detail in Chiarella and Flaschel(1999c), due to the lack of a complete treatment of the budget equations of the three sectorsthat form the basis of this model. As indicated above the laws of motion of the individualassets that are held by the household sector in our model do however not feed back into therest of the dynamics, since they do not show up in the real part of the economy. Therefore,onJy the two laws of motion for short-term and long-term government debt are really neededat present in the discussion of the growth pattern and the fluctuations that may occr'," aroundthem which are implied by the disequilibrium growth model under consideration.
Note with respect to the right hand sides of these stock accumulation equations that theyare based on a fixed ratio of describing the allocation between the shoit- and long-termfinancing of government debt done by the government (as in the Mruphy model) *ã tfrutthe term in square brackets, the government budget equation:
ga" + rb + bt - t" + (*"" lpò(t" - 1.") l- (w," lpòaúí t (rr" lpòttin both cases represents the sum of government expenditure for goods, labor, interest andtransfer payments to the ruremployed and the retired persons minus ún, the sum of all taxesthat are raised by the government (per value unit of capital).12
l2Note that all wage concepts in the above intensive form of the GBR are in efficiency units (to allow forstationarity) and are deflated by consumer prices pr.
18
The total amount of debt financing is thus releading to å + b,¡rt in sum, and it is dividedthe GBR in the above intensive form, intoö¿/r¿. Note thar (p, -t gn - ó)ö and (ps -t gnexpression in addition, due to the fact thatare thus divided by p,K.
The dynamics of real savings of workers per unit of capital, b_ : B.l@rK), follow fromthe definition of this expression and from the definition of the ãisporuú1" income of workerhouseholds and their saving plans, while the law of motion for short-term debt of asset holdersper capital, b": B"l(prK), is a simple consequence of the two laws of motion assumed forthe expressions ö, är.
The next two dynamic laws for the distribution of new domestic long-term debt throughoutthe world basically flow again from the above shown budget restriction of the goverriment(expressed in intensive form), since there is also a fixed proplortion of, assumed to apply withrespect to the distribution of long-term debt between domestic anà foreign asset holders.
financing that go to domestic and foreign(p,K) are thus obtained by applying the
ote of the fact that now (po + On _ 6)bt, andsulting expressions (due to the intensive form
formulation) in the place of the former expression (p, +-go_ ó)6r. '
There remains the law of motion of foreign assets held by ue tothe definition of the intensive variable bt, : B!rl@,,K) basi ationofthe variable BLI@,K) in.terms of inténsive-ffioriorrr. Iy byreferring to the fact that Bl is given by the dispàsable income of asset holders minus theirconsumption minus all other asset accumulation that these households undertake. Thisprovides the last law of motion shown in the above block of asset accumulation equations,which is thus purely residual in nature,
3.2 Static relationships
There are a variety of definitions and static relationships used in the above collection of thelaws of motion of our disequilibrium model of monetary growth. These abbreviations arecollected in the next six blocks of the intensive presentation of the model and are gener-ally immediate consequences of the corresponding equations in extensive form presented inChiarelia and Flaschel (199gc).
7. Output and Demand on the market for Services)
ad : c-* cc* gn+ gnlenl ga"
y : A" * þ"(þn"A" - u) + yBnoy"
fr:raujo : js'a
(35)
(36)
(37)
(38)
19
goods (including Housing
c"u)
^doLh
: cl(1 - r-)[w'ld" * w""(1" - l-") * w,"a¿tfi]f p,:
"r(l - r.)lw"ld. * w""(1" - 1.") -f u,.a¿lfilf p¡
: ol((t - r.)p:n- r,) + rrrîI -(" +€)) * "!(#^- uh) i t t 6n
(3e)
Note with respect to block 7 of the static equations that all variables are obtained here bydividing the extensive expressions by K, giving rise for example to le¡ : xnlK. The samäprocedure applies to block 8 of the algebraic equations underþing our d¡mamic model:
8. Emplovment , Labor Su and Retired Persons:
t!' : lva see block
Fs': lï":asga"1. for the law of motion for lf"
P' : tdr" + t!" , vi : t|" ltl-' : li"+li": V:l-./1.t; : (Lr(0)IL1(0))t la¿
Note here again that all magnitudes concerning labor inputs and supply are expressed inefficiency nnits - due to the technological condition Lj : to exp(_n¿tif tnut representsHarrod neutral technical change. When multiplied with wages measured in efficiency
'nitsthe term exp(-n¿t) just cancels from the resulting expression and gives the corresponding\¡/age payments in both efficiency units and original levels. Note finally, that we need initialconditions in order to relate the sizes of the intensive expression for fotential labor supplyand retired people.
L Desired Growth Rates of the Capital Stocks:al ((r-r")p'-r,))+a!(r¿ -("+€)) + "g@lf -û)+j+6 (46)
(47)
(40)
(41)
(42)
(43)
(44)
(45)
sÍ
st
The growth rates of the capital stocks for fixed investment of firms and housing investmentof asset holders (here shown as gross rates) are an immediate consequence of their originalformulation in Chiarella and Flaschel (1999c), and simply state that these ratios are assumedto be influenced by long-mn profitability mea^sures, by the interest rate spread and by rateof capacity utilization. The same immediate correspondence to what has been introducedin Chiarella and Flaschel (1999c) holds true for the following definitions of rates of returnneeded for the dynamical laws or just for the discussion of the steady state of the model,
20
10
a' - 6 + (p,/pr), - (ro" lpùto¡" - @*lpùjoUuP
ffi, an:UaP
adp - 6 + (p,lpo)rra" - ru"loa" - (p^lpòjna"(L-r")rt -rt(pnlpò"llten - 6n
p":
ad' :p":p':Pn:
(48)
(4e)
(50)
(51)
The various prices we use inthat n the dynamics are alsodefin of the model in Chiarellais again taken of the fact that \¡/ages have to be expressed in efficiency units in order to gettheir stationarity in the steady state: w" : w/exp(n¿t).
11. Consumer Prices , Gross and TlansfersPs
Prn
1nb'
'r-Du'
'u"Pu
: ep_,"
: (I+r^)epi: (L + rr)w": auu)"
: e,''u': (L + r")po
(52)
(53)
(54)
(55)
(56)
(57)
Finally the short-cut expressions for the intensive forms of disposable incomes and taxes areeasily obtained from their extensive form analogs and are of the form shown b.l;. i;;;that the expression for disposable income of woikers and asset holders are only needed intheir bond accumulation equations and are not yet involved in the consumption decisions ofthe two types of households considered as the model is currently formulatéd.
L2. Disposable Incomes and Government ta:ces per value unit of capital:(l - r.)[w'lù + u""(1" - l*') -f w,"a¡tfi]f pa + (r - r")rb*(l - r-)lwe¿d'e ¡ w""(1" - y') * w,.a¿Ii)f p,(L - ,òlp"(pu/p") * rb.+ öl + (pnlpòcÊ - (pulpò\nknle(l - r")bl2
r-[w'ld" * w""(1" - l-') * w,"a¿}fi]f p" * ro(w" f p,)ld"ro(ad - gt - gnkn)(po/p,)
r.lp"(palp,) + rU + ó¿ + (pnlpòpnknl + r^e(pi,lpò jd
YD, :yil:a!:
+lfLL-
++
(58)
(5e)
(60)
(61)
21
In closing the discussion of the intensive form of our disequilibrium model of monetarygrowth we stress again that the law of motion for equity prices p" is not needed, since it canbe substituted into the law of motion for Tobin'r q. li can thus be removed from explicitrepresentation, since there is here no feedback from Tobin's q on the real part of the economy,due to the lack of wealth effects in the present version of the model.
4 Steady State Analysis
In this section of the paper we show that there is an, up to the level of nominal variables,uniquely determined economically meaningful balanced growth path or steady state solutionof our model which provides us with a useful reference path fãr the dynamical evolutionsimplied by the model, which may or may not converge to this steady staie solution, not evenif long-run moving averages are used in the place of th. temporary positions the economywill nrn through.
The calculation of this interior, economically meaningful, steady statels of the full model(which up to the level of nominal magnitudes for wagJs and goods prices is uniquely deter-mined) is in many respects simple due to the given growth rate of the world economy and thegiven interest rate (on consols) abroad. Note that we only consider expressions for the totalsupply of domestic bonds in the following, and not their distribution at home and abroadwhich can be easily obtained from the savings decisions of workers and pure asset owners.
To simplify subsequent presentations of the dynamics of the model and also its steady statesolution somewhat we assume in the remainder of this paper for the consum of assetowners and for the liquidity premium applied to long-term debt
not restrict the dynamical behavior of the system in anThese two
portant way
The first set of steady state conditions presented below concerns the growth rates of orusmall open economy:
.t: gn-6:gÍ-6 -gk:gl:1i6 $2).y : gn-6n:gl-6n -gh:gÍ:l*6n (63)'t : n$) + nt(gn) : ñ,(V) + ñ¿úl + 6) -, V, n: ñ,nt : ñ,t (64)
These equations state that capital (and thus also output) will grow with the external rate7, to which also the natural rate_of growth adjusts. This means that the steady state valueof NAIRU rate of employment 7þ v) ha^s to adjust such rhat "'r : ñ(v) + ñ,(go) holds,This determines a unique NAIRE V : V for the long-run of the model under suitãbie - butsimple - assumptions on the function ñ.
The next set of steady state conditions concerns inflation and expected inflation (for allprices that exist in ou model, except wage rates) and also the vari<¡us rates of interest and
lswhere we thus neglect as alternative steady state positions all zeros which can be obtained mathematicallyfrom the grow our model employs, Note also the steady state depends parametricallyon the initial c"(o) ¡5"¡ characterize the initial composition of the labor force, thlinitial output Íåtlào"o tæces) and a relative expression for the consumption of assetownels,
C.:0 €:0
22
profit of our model
Pu : þg :þn: T l :4fbs :7tb: 7(es:.¡re:€s : 6- 0ri: rI:p':r:L/puri : pt:pl:pn:ph:pL
a : UPt , c!n: lentln
tj': lî":loa4': lï":asga"l'" : F":lT.+lï"l" : Io" lV IV : V), ll: l.lat, tt: (L2(0), ¿1(0))¿ít : fraU, jd : jsyd : (b +b¿lri)ly. : ã
- p:,r - ph,idtn'L: T
(65)
(66)
(67)
These conditions state that there is no steady state inflation and thus no nonzero expectationsof it, which is due to the interest rate policy rule of the central bank. F\rrthermore, all(expected) rates of return are equalized in the steady state and determined by the worldrate of interest on long-term bonds rf.
The next block concerns the steady state determination of various quantities, of the steadystate ratio of government debt to aggregate demand and of the tax rate on imports:
(68)
(6e)
(70)
(71)
(72)
(73)
(74)
(75)
These equations still depend on the steady state value of A" - which will be given below -and they reveal certain supply side influences on the steady behavior of our economy.
F\rrther steady state relationships on the side of quantities are:
vI l1B.aUt, u: þnoA": cßP,t, c* : czprU?t/pn
(L - r.)fwe¡de ¡ u""(l' - y') * w'"a¿|fi)f po
(76)
(77)
(78)
(7e)
a':ad:^ou7D
aìt
By using the distribution laws for government bonds we furthermore get:
þ - olãA"b¿ : ril-on)ãa"
(80)
(81)
from which the individual distribution of bonds can be derived if desired.
Next, one can determine the following nominal steady state expressions on the basis of an(arbitrarily) given price level p,. This indeterminacy of the general price level of domestically
23
produced goods is due to the fact that the central banl< has adopted an interest rate policyrule and allows for intraday deviations from the budget constraints of the two types ofhouseholds, the firms and the government and that therefore, in our present model, moneyis not held as cash balance by the agents of ou¡ economv.
(82)
(83)
(84)
(85)
The equation for the determination of gross \Mages in efficiency units holds since we havebalanced trade in the steady state (see the above), and thus no influence of the internationaltrade in goods on the rate of profit in the steaáy state. Note here that net real wages ø"(excluding payroll taxes, but before wage taxation), measured by in terms of consumer-pricespu and efficiency units, are given by
')":u" y"-6-ri
pu (L + r")(L + r) lsaand thus do y determined nominal price level p,. This observationalso applies implying that there are no real efiects of shocks whichlead to a di p, in the long_run.
The equations cf :[Jnkn and ri :pncf,,l(pskn) - ó¿, on the one ha.rìd, and
cza7t: phcdhlp,
U" : ctyDu * I + 6+ (f +6r,)È¡ * gA",
on the other hand,la allow us next to determine the price ratio pnlps and k¿, ult, r. asfollows:
'trb' u'ia rl : a"- ó - tÊ. :,,.Db" - a" - 6 -'î ^ps r -- :6oo'tÙt : wbtl(I * ro), y¿ue : eu1u", ,tDr. : e,rlJ).
Pu:Py:
indetermined and thus a parameterp"l(L + r")
(ri + 6n)/tn
a'G-g)-t-6"r(
ri+6¡)lþr0+r"))+1*6n: p¡U¡k¡f (c2p")
1a3'
lwelde ¡ *ue(le - l-") + uredtlglf pu
Pnf ps
k¡
a3t
T1J)
(86)
(87)
(88)
(8e)
Note a.lso that this determination of yfl, and, k¿ must be used above for the determinationof consumption plans per unit of capital.
There remains the determination of the steady state value of the exchange rate e,15 sincethe price of long term bondspb: Llri has already been determined and since the price ofequities does not matter in the core 34D dynamics of the model.l6 The calculation of the
lawhere the last one stands for goods-market equilibrium in the steady statelswhich in turn determines the prices patp¡n,16The same holds true for the disposable income of asset holders y"D.
24
steady state exchange rate is economically seen complex, but mathematically still simple.Mathematically it is provided by the following impliãit equations for the va¡iables t, , e onthe basis of the expression for ú" shown below:
0 - ga" +rb+bt -t- -t" +[w""(1" -1.")lw,"a¿If+wb.t!"]lp,_úlasb0 - r-vl¡lQ - r.) I ro(w" fp,)to' + r"(pylpò@o - gÍ - slkh) -t r*epi¡d lpu - t'
where
t" : r.[p"(ps/p,) + riu + ä¿+ (pn/pò"í- @rlpò6nkn]: r.[(p' + rik¡) lQ * ,,) + rib + b¿)
which give rise to:
t' : ga. +rb+bt -t" +1r".(1" _p") iw,"a¿l$+.b.ton1/n, _ úlagot' - î.y]rl(r - r.) * rr(w"fp,)lo"+ro (polp" )(vo - gÍ- gdhkn))
(e0)
(e1)r^Ph,id /P"
We see that the long-mn rate of exchange is neither determined in the market for goods, northrough the trade balance nor by international capital flows, but is a complicatðd expres-sion of many parameters and steady state values of the model and is in particular heàvilydependent on the form of the GBR and its components.
Note finally that the dynamics of equity prices implies that g : 1 must hold true in thesteady state, i.e', p"E : PyK and that the steady state distribution of long-term bonds canbe derived from block 6. of Chiarella and Flaschel (1gg9c), while the steidy state value ofô., (and ó") follows by setting eq. (81) equal to zero. This gives rise to:12
b*: (L-"r-@n /p,)"2)y3,7-(1-rt)ri
due to the difference that exists between our definitions of yD*, and, y!.Summarizing, we see that the steady state of the considered economy heavily depends onthe data assumed to apply to the rest of the world (generally in a iairty straightforwardand simple way) and that the steady values of the wage taxation rate and in particular theexchange rate are complicated functions of the parameters and various other steady statesolutions of the d¡mamics. F\rrthermore, since the steady rate of profit p. and, thus also thereal wage rate u' : u" /pr, see the above expressions, â,re determined through the foreignrate of interest, we get for the real wage u"" (after taxes, at consumer prices *ã itr efficienãyunits) the expression:
u)n" : (L - r*)u. : L-r* a"-6-ri(1 + r,)(t + ro) lyU
17This solution implies by economic reasoning that 7 > (1 - rj)ri should hold true in the world economy(because of the particular interest income policy of workers).
25
which implies that all increa'ses in the three ta>< rates shown (value added tax, payroll taxesand (endogenously- determined) wage income tæ<es) fall on real wages of workers imeasuredin this paper by the consumer price of domestic goods solely). õh"." is however also anin-fluence of the value added tax on overall interest rate income per value unit of capital,besides of course a dependence of this income on the capital taxation rate r".
Output levels (per unit of capital) and also steady employment (per unit of capital andmeasured in efficiency units) are basically determined through supply side considerations(technology and desired steady rates of capacity utilization), tut tnäfårmer also depend ongovernment expenditures g per unit of capital and thus will be changing with this ratio.'We
stress again that the external growth rate 7 determines the NAIRU rate of employmentvia
t:n(v)+ñ,t(t+6)which implies that an increase in 7 will increase the NAIRU rate of employment V if. ñt >o,ñt € (0, 1) holds.
The above has also shown that there is no inflation occu¡ring in the steady state of 34Ddynamics as far as goods and asset prices are concerned. It is of course important to alsoconsider economies which allow for (moderate) price inflation in the steady state, in theplace of stationary price levels that here prevail in the steady state. Just as rates of growth,the steady levels of interest and profit rates are here all fixed and given through the externairate of interest rf so that there is no possibility that the considerãd country may exhibit anextraordinary level of profitability in the steady state.
5 The 18D Core Dynamics of the Model
In order to get (as a starting point for our dynamical investigations of the model in subsequentpapers) a dynamical system that is, on the one hand, as close as possible in spirit to theone of the general model of this paper and, on the other hand, also as low dimensional aspossible, we consider in this section a simplified structure for our Keynesian disequitibriumdynamics of monetary growth which reduces its 34D representation to the 18D coredynamics. This reformulation makes use of the a^ssumptions already used inthe preceding section (there used to simplify slightly the the interior steadystate of the model), *d it removes furthermore certain delayed adjustment processes fromthe considered dynamics, which gives the dynamic model an outlook that is not too far arü/ayfrom the theoretical models introduced and analyzed in Chiarella and Flaschel (1999a, Ch.s6,7, 1999b, part III). F\rrthermore, the natural rate of growth, of Harrod neutral technologicalchange, and of employment are assumed as constant in the following. Finally, the parameter0"-, is set equal to zero, implying that there is only a proportional influence of governmentdebt on the wage taxation rate (and no longer an additional derivative one),
C":0,€ : 0
26
5.1 The laws of motion
1. The Quanti tv Dynamics
ù" : þ,"(ao _ y.) + 0 _ Øl _ O))y.ù : a-ad-Gi-6),
iy" : þ,(tl' - vyn+ [r - Øi - 6)]rî"
ú)" : 0,,(l'"f l" -V) + p.r(Idr.ltî" -UJ,) I n.þy+ (1 _ ^_)n,
Po : þo@ lao - Ú) + Kpû' + (I - no)rlfrt : Bnt(ant(po - nI) + (I - c'nt)(0 - "¿))ndopn : Pr(t - uÌ,) +
^nþo+ (1 - *o)n'
(e2)
In the quantity dynamics we have removed the adjustment equations for the actual consump-tion of workers (for goods and housing services) and have thus replaced these consumptionplans by the desired consumption targets which are now immediately realized (withoui anylag). F\üthermore, the NAIRU rate of employment, V, is no longei considered as endoge-nously given, but now a parameter of the model. There remain the equations for the adju-st-ment of sales expectations, for inventories and for the workforce employed by firms.
2. The Price Dynamics:
(e3)
(e4)
(e5)
(e6)
(e7)
(e8)
There is no direct change of the dynamics with respect to the various types of price ad-justment rules of the model, on the market for labor, goods a¡rd housing services (includingthe expectations formation mechanism for medium nrn inflation rate of tn" doàesticail|produced good), but an indirect on due to the following assumptions on natural growth anäemplo¡rment (which are norÃ/ considered as given: 1 : n * n¿ : ñ, * ñ,¿,V all constant).
3. The Growth Dynamics
î" : -t-(gÍ-6)îro: gÍ-6n-(gÍ-o)
In block 3 we have assumed that the natural rates of growth and of technical change, n, n¿,are given exogenously and equal in sum to 7. F\rrthermore actual accumulation rates areassumed to be adjusted with infinitc speed to their desired targets and thus are no longerrepresented as lagged adjustment rules. We have furthermore removed from this btock 3of the dynamics the adjustment equations for the rates of rettun pt, pL by assuming that
(ee)
(1oo)
27
these rates are adjusted with in-finite speed to their short-run equivalents, nolrr' used in thecorresponding behavioral equations in their place.
Next, we present the set of equations that represent the dynarnics of asset accumulation andasset prices that are needed for the analysis of the real part of the dynamics of the model:
4. Asset Market
b - o!u[kA'+ rb + bI
t- - t' * (w"'lp,)(t" - t-") * (r," lpòa¡Ii -l @b" lp)t!")(pr+sÍ-6)b
bt : ,'(L-oÐ[ga"+rb+btt' - t" * (w".lp,)(t - t-") * (w," lpòa¿\fi i_ (wb" lp)I!.)(pr+sÍ-qbt
ô^ : þoo [(1 - n\r, J- ^ -. - r/1 - \-\r 1ru r- þoo(r-aJt(l -r")rt*d¡r6' - ((1 - r")r)l' '': àiu, : þno"(þu - Tos), lrb: e.sltbs + (1 - ar)pø
ê- 0"L - p"0-
""li(t - r")ri * a,e" - ((1 - r")rt + nu)l
¿" : 0r"(ê-e,¡
(101)
(102)
(103)
(104)
(105)
The price adjustment rules in block 4 concern the nominal value of long-term bonds p6 andthe nominal exchange rate e and they are based (as explained in Chiarella and Flaschel(1999c) and in this paper) on heterogeneous expectationsof the pure wealth owners of thismodel. Note that - as in the larger model - the dynamics of equity prices and of Tobin,s qare not needed in the investigation of the core dynamics of the model.
Next, the dynamic policy rules are presented which are basicalty the same aß in the largerdynamics' Note however that we have removed the derivative term from the right hand sideof the wage tax rate dynamics.
5. The Feedback Policv Rules:
(106)
(107)
(108)
Summing up we thus have arrived at a 18D dynamical system by setting certain adjustmentspeeds equal to infinity or equal to zero, by cutting certain feedback effects of the individualdistribution of bonds on aggregate demand (based on the facts that C.:0 holds and thatworkers save all their interest income), and by assuming constant 'natural' rates of growth
i - -Þ,,(r-ri) +þ,"(pn-0) +þ,,@lao-ú)
î. : a,-r(d,fd, - L), -b+bt/rIa"
î*: or^2"-P^idPst
28
as 'well as of emplo)rynent. Note that the housing services sector feeds back into this coredynamics of the model through the investment áemand for dwellings (on the ma¡ket fordomestic goods) and through its rate of return which is in particular determined by the lawof motion for the rent price of housing services.
5.2 Static Relationships
As abbreviations and static relationships we now have the following reduced and modifiedlist of equations underlying this lgD dynamical system.
1, Output and Demand:
ad : c"-+gÍ+glko*ga",ci : cg7tcfl' : p,czUDølpn
a : A'* p"(þ.dA' - u) + 1Bnoy"r:rgu
:d.J -JaUand Labor
lj" : lau
tt" : Iï":asga.ld' : tj"+I!'I'" : li'+li"t3 : @2(0) I L1 (0))¿í : (L2(0)/ L1(0))t" la¡
3. Growth Rates of the Capital Stocks of Firms and Asset Owners:gÍ: al (Q - r")p" - p')) + af(rt - r) + a!(alap - u) + "y + 6
2
(10e)
(110)
(111)
(L12)
(1 13)
(r14)
(115)
(116)
(117)
(118)
(11e)
(120)
(121)
(122)
(123)
(124)
(125 )
(126)
(r27)(128)
(12e)
(130)
sÍ, :4.
p':p':Pn:
ð.
P,:Pm:,nb" :?Du" :
u)" :Pu:
"f((1 - r.)pn - p,) + or?t - Ò + rg14 - uù *.,r * 6n
Rates of Return:
a'- 6 + (p,lpùr - (ro"/pùt|" - @^lpòjo(L-r")rt-nI@nlpò"d,f lkn - 6n
Prices, Wages and Tbansfers:
eP,
(r + r^)epi(L + rr)w"e.u'tr"
d,''rr'
(L + r")po
29
a3,y3
t-
6. Disposable Income of workers and taxes per value unit of capital: (l - ,.)l*"lo' * w""(1" - l-") * w,"a¿tfilf p,: ah + (L - r")rb*: r-lw'ld' I w"'(1" - p.) -f w'.a¡tfilf pu I ro(w.lp,)to.+ r"(palpò(ao - goo - gÍkh) * r^epijd f pu
t" : r"lp"(polp") +rø+bI+ (pnlpòcdn- @olpò6nkn)l
(131)
(132)
(133 )
(134)
The steady state of the model is the same as before, see the preceding section. The dimensionof this dynamical system can be fruther reduced, to dimension 16, if the housing sector isremoved from the model.
6 Outlook: Feedback structures and stability issues
We have presented a structural mode of disequilibrium growth which is fairly complete withrespect to markets, sectors and agents. We believe it is sufficiently detailed to cÀpture theessential dynamic features of modern macroeconomies, whilst at the same time abstractingfrom the welter of detail that inevitably must cha¡acterize large scale macroeconometricmodels such as the Fair model of the US economy of the Mruphy model of the Australianeconomy.
We considered the model from the point of view of national accounts, and then discussedthe development of its extensive form and finally expressed its dynamic structure in termsof intensive form state variables. 'We
saw in particular that in intensive form we are deal-ing with a 34-D dynamical system. We further found that with a small number of furtherassumptions (concerning consumption of asset holders and certain secondary delayed adjust-ment processes) the dynamics reduce to an 18-D dynamical system which we call the coremodel.
These turn out to be eight main partial feedback mechanisms contained in the 18D coremodel' First, the labor and goods market interaction whose tendency to become destabiliz-ing is determined by an interplay between wage and price flexibility. Second, the expectedsales and inventory accumulation interaction. The tendency to instability of this mechanismis determined by the relative values of speeds of adjustment of expected sales and inventorychanges' Third, the dynamics of the housing sector is determined largely by the strength ofinvestment into this sector and the speed of adjustment of the prices for housing ..Ãi....This partial feedback mechanism is always stabilizing. Fourth, the dynamic interaction be-tween the level of economic activity and the nominal interest rate (the so--called Keyneseffect). The stabilizing/destabilizing tendency of this mechanism is very dependent on th"sensitivity of the nominal interest rate to the price level. We note that when considered inconjunction with the Taylor-style interest rate rule used in ou model this mechanism is byand large stabilizing. Fifth, the inflation / expected inflation mechanism. This is essentiallya destabilizing effect (associated with the narnes of Mundell and Cagan) determined by thlinterplay of speeds of adjustment of prices and inflationaxy expectations. Sixth, the bondand stock market dynamics (originally considered by Blanchard (1981)) first in isolation and
30
then their interaction, are driven by rates of retrun and expectations feedback. The mainlydestabilizing tendency of this mechanism is driven by the speeds of a-djustment of bond pricesand expectations of bond price inflation. Seventh,lhe dynamics of itre government budgetrestraint. This mechanism, when considered in isolation, is stabilizing provided the rate ofgrowth of government debt is restricted in certain \'/ays. Eight, the excúange rate/expectedexchange rate (Dornbusch) mechanism where stabilizing/aestaUitizing tendencies essentiallydepend upon the relationship between the speed of aJjustment of exchange rates and ex-change rate expectations.
The 18D core model of this paper will be studied in a series of further papers, see Chiarellaand Flaschel (1999d), Chiarella, Flaschel and Zhu (19gga,b), Chiarella, Fiaschel, Groh, Köperand Semmler (1999a,b). These papers discuss the feedback structure of the 18D core ày-namics (discussed above) in great detail with respect to partial stabilizing or destabilizingfeedback chains there present. They also present further numerical investigations of thémodel both from the local and the global point of view (adding also extrinsic nonlinearitiesto achieve global boundedness) and extending the theoretical basis of the disequilibriumgrowth model employed so far (by allowing for smooth input and output substitution andother flexibilities).
Prototype subdynamics of this 18D system which are often discussed in the literature inisolation, and which we have briefly discussed above, will be derived and analyzed in Chiarellaand Flaschel (1999d). This paper will provide more insights into the stability properties ofthe 18D dynamics by investigating the partial feedback chains it contains (often well-knownfrom comparative static analysis) with respect to the partial dynamics to which they giverise. The interaction of these partial dynamics will then be studied in Chia¡ella, Flasãheland Zhu (1999a), and further papers, there basically from the numerical point of view asin Barnett and He (1998) who - as we have seen in section 7 - use a L4D second orde¡dynamical representation of the UK economy for studying the bifurcation loci to which sucha model type can give rise.
7 References
B.tRruprr, W. and Y. Hp (1998): Bifurcations in continuous-time macroeconomic systems.Washington University in St. Louis: Mimeo.
BpRcsrRoM, 4.R., K.B. Nowtr,tRlv and S. WRxoesrpwtcz (lgga): Monetary and fiscalpolicy in a second-order continuous time macroeconometric model of the Uniteá Kingdom.Joumal of Economic Dynamics and, Control,18, Zgl-261.
BsncsrRoM' A.R. (1996): Survey of continuous time econometrics. In: W. Ba¡nett, W.,G. Gandolfo and C. Hillinger: Dynamic Disequilibrium Mod,eling: Theory and, Applications,Cambridge, UK: Cambridge University Press.
BlRNcu¡,nD, O.J. (1981): Output, the stock market, and interest rates. American Eco-nom'ic Reuiew 7L, 132-L43.
CHI¡,Rpr,l,A, C. and P. FlRscHpl, (1999a): The Dgnamics of Keynesian Monetary Growth:Macrofoundations. Cambridge University Press, forthcoming.
31
cHlRRpl'l'A, c. and P. Flescupl, (1999b): open Economy Di,sequi,librium Growth Theorg.To appear: Springer Verlag.
cuteRpu'A, c. and P. FlAsc¡roi, (1999c): Towards Applied Disequilibrium Growth The-ory: L The starting model. UTS Sydn.y, iir.rrrsion paper.
cHleRpu,A, c. and P' FlascHpl, (1999d): Towards Applied Disequilibrium Growth The-ory: III' Basic partial feedback structures and stability issues. UiS Sydney, DiscussionPaper.
cHreRpu,A, c,, P. FlescHEL, G. Gnou, c. Köppn and w. spuvu,pR (199ga):: Tewards Applied Disequilibrium Growth Theory: VL Substitution, money holdings, wealtheffects and further extensions. urs sydney: óiscussion paper,
GnoH, C. Köppn and W. Spurr¿l,pR (1g9gb):: To-Theory: VII. Intensive form and steady state analysisey; Dis< ussion Paper.
culeRell.a,, c., P. FlnscHpl, and P. Zuu (1999a): Towards Applied DisequilibriumGrowth Theory: IV. Numerical investigations or ine coíe 18D model. úrS syarrey: Discus-sion Paper.
cHtARell.t, c., P. FuscHpl, and P. zuv (1999b): Towards Applied DisequilibriumGrowth rheory: V. Global considerations. urs bydr.f, Discussion eup"r.A' Pow¡ll' and Mqn¡_Hv, C. (1997): Insid,e a Mod,em Mauoeconometric Mod,el. A Gu¿d,eto the Murphy Model Heidelberg: Springer.
Appendix: Notation
The following listmagnitudes arescriptions we shall(A$) and the 'US
A. Statically or dynamically endogenous variables:
uudgp
adP
uny"yI,a?,el,l,eL2
t6
ld"td.e,lIt":Ltueo!
I-.
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 K) 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(n¿ú), per .I{)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 force
deI
32
via,¿
v:ld.lr"-("'.)c"(c'.)c:c-*c.cicfl
sÍsî,I" lK(1"" lK)TIKNIKvd
Employment rate of those employed in the private sectorParticipation rate of the potential work forceRate of employment (7 ttre employment-rcomplement of the NAIRU)Real (equilibrium) goods consumption of workers (per /l)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 K)Gross fixed housing investment (per K)Gross (net) actual total investment (per K)Planned inventory investment (per K)Actual inventories (per K)Desired inventories (per K)Nominal short-term rate of interest (price of bonds p6 : 1)Nominal long-term rate of interest (price of bonds p6 : tlrt)expected appreciation in the price of long-term domestic bondsRequired rate of interestPrice of equitiesexpected appreciation in the price of equitiesS$ ln"K + Si ln"K + Si ln"KTotal nominal savings (per p,K)Si,lp,K + S! lp"KNominal savings of households (per p,.I{)Nominal savings of firms (: prY¡ lp"K, the income of firms) per p,KGovernment nominal savings (per puK)Nominal (real) taxes prK,KReal government expenditure (per K)Expected short-run rate of profit 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 servicesExpected long-run rate of return for housing servicesCapital stockCapital stock in the housing sector (per K)Nominal wages including payroll ta>< (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 cuuencyPrice level of import goods in domestic currency including ta><ationRent per unit of dwellingExpected rate of inflation (over the long run)Exchange rate (units of domestic currency per unit offoreign currency: A$/$)Expected ¡ate 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"lp,K)Stock of domestic long-term bonds, of which b! are held (- Bllp"K)by domestic a.sset-holders (index d: demand)and ö¿r. by foreigners (index d: demand)
r-t1tb : þ3p'Pe
71. : p".
s" fpuK :Si ln"K :Si ln,Ksi ln,KT" lp,K(TlK)
þi
e: êe
Ip"po
pn
PI
Pn
PLKk¡,ub.
w"rou"'tD"
Pu
Pv
P"Pm
Pn*I_
e
Pb
b-b"
6t
33
bL Foreign bonds held by domestic a.sset-holders (index d: demand) (: BLlp,K)€ Equities (index d: dernand) (: Elp,K)n Natural growth rate of the labor force (adjustment towards ñTr¿ Rate of Harrod neutral technical change (a-djustment towards ñ,¿Tn Tax rates on imported commoditiesr Exports (per K)id Imports (i,* ¡rjnr-: Qt,r - epk jd)lp$et exports in terms of the domestic currenc (per puK)nlr Net factor export payments (per prK)ncr Net capital exports (per puK)Tu tax rate on wages, pensions and unemployment benefitsd Actual public debt / output ratio
B. Parameters of the model
Kp
6
6¡
4þ,.Y
UUnKuK,AP
talv
isdÈ
cSe
lC
TU
TP
Cl
tz
Depreciation rate of the capital stock of firmsDepreciation 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_rc, I I): (1 - n.Kp)-lOutput-capital ratioExport-output ratioLabor-output ratio (labor in efficiency units)Import-output ratioDesired public debt I output ratioRisk and liquidity premium of long-term over short-te¡m debtRisk premium of long-term foreign debt over long-term domestic debtTa¡< rates on profit, ¡ent and interestValue added tarc ratePayroll taxPropensity to consume goods (out of wages)Propensity to consume housing services (out of wages)
34
