business debt dynamics in a micro-founded stock-flow consistent model

8/22/2019 Business debt dynamics in a micro-founded stock-flow consistent model

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Business debt dynamics in a

micro-founded stock-flow consistent

model

L. de Carvalho, C. Di Guilmi

Sao Paulo School of Economics (EESP), Sao Paulo, Brazil.Business School - University of Technology, Sydney. Australia.

Draft version

May 20, 2013

Abstract

The 2008 financial turmoil has drawn the attention to the crucialrole of credit as a factor leading both to the instability of the sys-tem and to a strengthening of real-financial linkages in the economy.This view was central to the work of Hyman Minsky, who stressed thesystemic effects of financial fragility at the micro-level. This paper in-troduces heterogeneous microeconomic behavior into a demand-drivenstock-flow consistent model, in order to study the joint dynamics ofleverage, income distribution and aggregate demand. The distinctive

feature is in that the aggregation of heterogeneous agents is not per-formed numerically as in traditional agent-based models but by meansof an innovative analytical methodology, originally developed in statis-tical mechanics and recently imported into macroeconomics. The newmethodology opens an original perspective for economic policy model-ing, in particular for addressing situations of coordination failure andexcessive leverage.

Corresponding author: Corrado Di Guilmi. University of Technology, Sydney - POBox 123, Broadway, NSW 2007, Australia. Ph.: +61295147743, Fax: +61295147711.E-mail: [email protected].
mailto:[email protected]:[email protected]


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1 Introduction

The 2008 financial turmoil itself, and the process of de-leveraging by theprivate sector observed in the following years, have drawn the attention tothe crucial role of credit as a factor leading both to the instability of thesystem and to a strengthening of real-financial linkages in the economy. Thisview, which was central to the work of Hyman Minsky, is also supported bythe vast historical evidence presented in Schularick and Taylor (2012), which

highlights that credit booms tend to be followed by deeper recessions whencompared to other financial crises episodes. The macroeconomic frameworkproposed by this paper focuses on the relationships between private sectorleverage, income distribution and aggregate demand as a source of instabilityof the economic system. The theoretical model involves heterogeneous firmsaggregated by means of an innovative analytical methodology.

As it is well known, economic models rooted in general equilibrium andrational expectations, which have dominated theoretical developments inmacroeconomics in the past three decades, have supported the view thatmarkets - albeit being subject to random shocks - are inherently stable, with

all uncertainty as exogenous. The financial crisis has led a larger group ofmacroeconomists to the understanding of the need to add an (inherently)unstable financial sector and its relationship with the real side of the econ-omy to their theories. This challenge has boosted the interest toward threemain alternative approaches to economic modeling.

The first relates to the incorporation of information asymmetries, stickyprices and bounded rationality into dynamic stochastic general equilibrium(DSGE) models (Smets and Wouters, 2007; De Grauwe, 2010). For repre-senting the economy as a self-regulating system in which market forces leadto a stable equilibrium, the new (DS)GE approaches still do not allow for theincorporation of independent financial dynamics and the study of systemicrisk. Moreover, by still treating macroeconomic relationships as reflecting thebehavior of its constitutive parts, these models do not allow for the incorpo-ration of fallacies of composition and other emerging properties of complexsystems.

The stock-flow consistent (SFC) models, first developed by Tobin (1969)and Godley and Lavoie (2007), have also received renewed attention (Khalil and Kinsella,2011; Bezemer, 2010, among others). By taking into account all flows of in-come between different sectors in the economy as well as their accumulationinto financial and tangible assets, this type of models were able to trace the

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flows of credit and stock of debt that could help predict the crisis (Godley,1999). Besides allowing for formal Minskyan analyses of corporate debt andfinancial fragility (Dos Santos, 2005), SFC models have recently been usedto study the macroeconomic effects of shareholder value orientation and fi-nancialization (Treeck, 2009), as well as that of household debt accumulation(Kim and Isaac, 2010).

The limitation of the SFC approach is that of modeling economic be-havior in aggregative terms, thus excluding the heterogeneity of agents as a

source of financial instability. The relevance of a microeconomic analysis inmodeling financial fragility is stressed by Minsky: an ultimate reality in acapitalist economy is the set of interrelated balance sheets among the various

units (Minsky, 2008, 116). Indeed one of the distinctive features of his Fi-nancial Fragility Hypothesis is the heterogeneity of the financial conditionsof economic units. Taylor and OConnell (1985) remark that shifts of firmsamong classes as the economy evolves in historical time underlie much of its

cyclical behavior. This detail is rich and il luminating but beyond the reach

of mere algebra. This lack of proper modeling tools is one of the reasonswhy Taylor and OConnells model and the vast majority of the literature

of Minskyan inspiration, including SFC models, is formulated in aggregativeterms. The recent development of new computational and analytical toolshas now made feasible the solution of macroeconomic models with heteroge-neous agents.

Based on the idea that any aggregate economic system is more than thesum of the microeconomic decisions of (rational) agents, the third strat-egy which gained force after the crisis uses agent-based models to studythe emerging properties of decentralized microeconomic interactions takingplace in complex and adaptive economic and financial systems. As arguedby Delli Gatti et al. (2010), agent-based models can outperform traditional

ones in explaining a wide variety of aggregate phenomena such as fluctuatinggrowth, bankruptcy chains and firms sizes and growth rates distributions.In particular, such models can be found to be very useful for the analysis offinancial instability, which clearly requires an understanding of the economyas an out-of-equilibrium system incorporating heterogeneous economic be-havior (Delli Gatti et al., 2005, 2010; Ussher, 2008).

The agent-based and the SFC approaches to economic modeling can bethought of as complementary in their understanding of the crucial role ofreal-financial linkages for the instability of the economic system, as well asits macroeconomic dynamics. Nevertheless, the fact that agent-based models

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can only be solved numerically has two main drawbacks. The first one isthat, while the micro-behavioral rules are defined and modeled, there is noanalytical definition for the relationships between macro and micro-variables.Second, as a consequence the causality links within the system cannot beclearly identified.

In this context, this paper introduces heterogeneous microeconomic be-havior into a demand-driven SFC model. The model developed here is com-posed of firms, households and a financial sector. Firms have heterogeneous

degrees of financial soundness.The distinctive feature is in that the aggregation of heterogeneous agents

is performed by means of an innovative analytical methodology originally de-veloped in statistical mechanics and recently imported into macroeconomics(see Aoki and Yoshikawa, 2006; Di Guilmi, 2008; Foley, 1994; Weidlich, 2000,among others). This modeling approach builds from the idea that, as theeconomy is populated by a very large number of dissimilar agents, an an-alytical model cannot keep track of the conditions of every single agent ateach point in time. As Aoki and Yoshikawa (2006) remark: the point isthat precise behavior of each agent is irrelevant. Rather we need to recog-

nize that microeconomic behavior is fundamentally stochastic. Therefore,a microfounded analytical model should look at how many agents are in acertain condition, rather than at which agents, and represent their evolutionin probabilistic terms. This approach is particularly suitable to microfoundSFC models, since it is able to endogenously derive the macro-equationsand the dynamics of flows from the microeconomic behavioral rules, withoutimposing ad-hoc constraints.

In order to implement this method in the stock-flow consistent approach,we first simulate the model as agent based with full heterogeneity of firms.These simulations serve the only purpose of generating the values of the

variables that will be subsequently used for the simulation of the system offlows, as done in standard stock-flow consistent models.The proposed framework has two main objectives. First, the closed-

form solution will present the analytical links between the financial micro-variables and the macroeconomy, in order to study the joint dynamics ofleverage, market capitalization, income distribution and aggregate demandin the cyclical pattern of booms and busts which is typical in financial fragilityliterature.

Second, the theoretical structure can assess the effects of the interactionbetween leverage dynamics and income inequality, by studying the shift of

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households between classes of income along the business and leverage cycle.Whereas in standard macro-models the evolution of the economy is de-

termined by aggregate common shocks, in our framework the idiosyncraticmicro-shocks drive the phase transitions of the system. The analysis thus canset the stage for micro and macro policy experiments that will be developedin a further stage.

The paper will be structured as follows. Section 2 presents the behavioralequations and the short-run equilibrium of the model. Section 3 specifies

the firms transition dynamics and mean-field approximation procedure forthe analytical solution of the model. In particular in this section we derivethe system of equations for the aggregate variables, expressed as functionsof firms micro-variables. The dynamics of the micro-, meso- and macro-variables are studied in section 4. Section 5 concludes.

2 The model

The economy described in this paper is composed by firms, households anda financial sector. As in the conventional neo-Kaleckian literature, pricesare set as a mark-up over labor costs, investment behavior is determinedindependently and the degree of capacity utilization of firms adjusts to thequantity they sell. The mark-up and the functional distribution of income areassumed to (exogenously) depend on the degree of industrial concentrationand the relative bargaining power of workers and capitalists.

Firms are divided into two classes and switch between them. While hedgefirms finance all their investment with internal resources, borrowing firmsfinance part or all their investment with stocks and/or bonds. The methodproposed in Section 3 will allow the share of firms in each class to affectmacroeconomic dynamics. The household sector is also divided into two

classes, namely that of wage earners and managers, with the latter receivinga share of firms profits1.

Households allocate their wealth between money and firms shares. Inter-est rates on bonds are assumed to be set exogenously by the Central Bank,with the price of firms shares being determined by supply and demand inthe market for stocks, rather than flows, of these shares. Capital gains (orlosses) then affect consumption levels via wealth effects, thus allowing for

1There is no microfoundation for the household sector, which is treated as an aggregatewith two types of income.

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the study of the role of asset price booms and bursts in aggregate demand.Finally, the financial sector is considered as an aggregate: its basic role isto provide loans, hence holding debt (or bonds) as an asset, and to createmoney deposits endogenously as liabilities.

2.1 The Firms

A single firm is identified by the superscript j, while its state or group by

the subscript z = 1, 2. Thus when a variable is written as xj1, it refers tothe firm j belonging to state 1; a variable with only the subscript indicatesthe mean-field value (the average value for the units in the group). Firmsvariables are represented by small letters, while capital letters indicate macro-variables. The numbers of firms in each group are indicated by N1 and N2with N1(t) + N2(t) = N(t).

Firms prefer to finance their investment with internal resources they havepreviously accumulated in the form of money mj and the flow of retainedprofits aj. If these are not sufficient they issue stocks and bonds. Accordingly,we can define two classes of firms:

z = 1: borrowing firms: that finance part or all their investment withstocks and/or bonds:

mj(t) + aj(t) < ij(t). (1)

z = 2: hedge firms: that finance all their investments with internalresources:

mj(t) + aj(t) ij(t). (2)

As for the investment function we use the valuation ratio (Taylor, 2012),which is the ratio between the values of equity and the value of capital assetsin the economy. In the present treatment we set it equal to

h(t) =P e(t)E(t)

K(t)(3)

where P e is the stock price and K the aggregate stock of capital assets. Theprice of new capital is assumed constant and is therefore normalized to onefor simplicity.

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The investment function for the firm j is given by2

ijz(t) = h(t) + z aj(t) + uj(t) (4)

where i is investment, u is the capacity utilization ratio and , z, > 0.This formulation recalls the investment functions in Delli Gatti et al. (1993)where the sensitivity to internal finance analytically devices the Minskyanborrowers and lenders risk. In their work the price of equity works as the

Tobin q. Following Fazzari et al. (1988) and Delli Gatti et al. (1993), weassume that 1 > 2, that is borrowing firms are more sensitive to internalfinance as they face the risk of bankruptcy and change their behavior inorder to minimize it. Equation (4) involves four factors: a macro-effect (h),a meso-effect (z), a micro-effect (a

j) and a variable combining micro andmacro effects (uj).

All firms adopt the same Leontief-type technology with constant coef-ficients. As a consequence, the demand for labor at full capacity can beresidually quantified once the stock of capital is determined by investmentdecisions in the previous periods. The supply of labor is infinitely elastic.

Accordingly the production function F gives the potential output q

j

for firmjqj(t) = F(kj(t), lj(t)) (5)

with k and l representing, respectively, physical capital and labor.Even if excess capital may exist, the output-labor ratio, which is indicated

by , is constant, so that firms are assumed not to hire excess labor3. Sincetechnology exhibits fixed coefficients, it is then possible to define the potentialoutput only as a function of capital so that

qj(t) = 1/ kj(t) (6)

where the inverse of the capital productivity is a constant parameter.

2For computational needs, in the multi-agent simulations we consider a sequential econ-omy that evolves in discrete time. For this equation investment depends on the previousperiod quantities so that ijz,t = ht1 + z a

jt1 + u

jt1.

3As described in Dutt (1984), the asymmetry which allows excess capital to exist, butthe labor-output ratio to be fixed technologically can be justified based on the simplifyingassumption that labor will not be hired if it does not contribute to production, whereasthe stock of capital is determined by previous investment decisions and may be held inexcess.

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The degree of capacity utilization uj of each firm is defined as the ratioof actual output qj sold by the firm to potential output qj, being equal toone at full capacity and smaller than one with excess capacity, so that

uj(t) =qj(t)

qj(t)=

qj(t)

kj(t) 1 (7)

Hence, with as a positive parameter, fluctuations in the degree of capacity

utilization of the firms will track changes in the actual output-to-capitalratio.4

The quantity actually sold by a firm is subject to a preferential attachmentshock. It comes from the assumption that the total demand pQ(t) is known,as it amounts to the sum of consumption of capitalists and workers, butits distribution among firms needs to be identified. We assume that thisdistribution is partially stochastic. In particular, the demand is allocated onthe base of the relative of size of firms (proxied by their capital) to whicha stochastic idiosyncratic shock s is added. In particular, defining s as auniformly distributed stochastic variable with E[s] = 0, we have

sj(t) = sj(t)

1 kj(t)K(t)

(8)

in order for

qj to be equal to Q. Accordingly the quantity actually soldby the single firm is

qj(t) = Q(t)

1 + sj(t)

kj(t)

K(t)

(9)

where Q(t) is the total demand, given by the consumption of managers andwage earners5. Following Kalecki (1971), firms are assumed to set the priceas a mark-up on the cost of labor, while holding excess capacity6

p = (1 + )w

. (10)

4The capital-output ratio is therefore greater than when there is excess capacity.5Formally Q(t) = (C(t) + C(t))/p, where C, C are quantified below.6See Rowthorn (1982), Dutt (1984) and Taylor (1985) for early models of growth and

distribution in this tradition.

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With taken as a parameter, the mark-up rate is constant and the laborshare of output is given exogenously 7

=w

p

1

=

1

1 + (11)

The gross profit share of aggregate output will then be given by

= 1

=

1 +

(12)

Each firms retained profits are computed as the difference between itsgross profits as given by (12), the net interest it pays8and the portion ofnet profits (defined as gross profits minus interest payments) which is sharedwith firms managers. Since the flow of gross profits is given by a constantshare of each firms output by the mark-up rule, retained profits of eachfirm are given by

aj(t) = (1 ) pqj(t)[1 + sj(t)] r[bj(t) mj(t)] (13)A firm fails if aj

c, where c is a constant. The case c = 0 corresponds to

bankruptcy if the firm is unable to pay interest on bonds without issuing newdebt (no Ponzi scheme assumption). The probability for a bankrupted firmof being replaced is directly proportional to the performance of the economyin the previous period.

Any excess of retained profits over investment will be held by the firmin the form of money m. The law of motion for the stock of money held byfirms is thus given by

mj(t) = aj(t) ij(t) (14)Whenever the flow of investment desired by the firm is higher than the

stock of money it holds plus its retained profits in the period, the firm will

7Again as in Kalecki (1971), the mark-up and the functional distribution of incomeare assumed to depend on structural characteristics of goods markets such as the de-gree of industrial concentration and the relative bargaining power of workers and capi-talists. Distribution can also be endogenized in the short-run if the power of bargain-ing of workers is assumed to depend on the unemployment rate or the rate of capacityutilization, possibly giving rise to Goodwin (1967) type of cycles. Such predator-preydynamics between demand and distribution are studied for instance in Skott (1989) andBarbosa-Filho and Taylor (2006), but will not be allowed for in this version of the model.

8 Firms are assumed to pay interest on bonds b and receive interest on money depositsm at the same interest rate r.

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seek external finance to cover the difference. In particular the firms willfinance its investment for a share with debt and the rest by issuing newequities. The law of motion for the accumulated stock of firms debt b is then

bj1(t) =

ij1(t) aj(t) mj1(t)

(15)

The amount of equities for borrowing firms evolves according to

e

j

1(t) = (1 ) ij

1(t) aj

(t) mj

1(t)

/P e(t). (16)

Borrowing firms that become hedge carry on the quantity of shares previouslyissued.

2.2 The household sector

The household sector is also divided into two sub-categories, namely that ofwage earners (subscript ) and profit earners (subscript ). As described inthe previous subsection, workers earn wages w which add up to a constantshare of total output Q. Managers (profit earners) receive a share out

of firms net profits9. All households allocate their total wealth W betweenmoney M and shares E.

Households disposable income Y is composed by wage or profit earningsplus the interest received on the money deposits they hold M, so that

Y(t) = pQ(t) + rM(t) (17)

Y(t) = [pQ(t) rB(t)] + rM(t) (18)

The wealth of both classes is accumulated as an effect of savings S and

capital gain G W(t) = S(t) + G(t) (19)

where savings S are defined as the difference between households disposableincome and consumption levels S(t) = Y(t) C(t), and G(t) = [P e(t) P e(t)]E(t). Finally, consumption spending by each class will be assumed

9This distinction is needed to allow for some consumption out of profits. There are twooptions here. The first is to have the two types of households with no transition possiblebetween the two. The second is to do the same as for firms, and allow for social mobility.Only the first option will be developed hereafter.

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to be a fixed proportion of both disposable income and the capital gainsobtained on equity:

C(t) = (1 s)Y(t) (20)C(t) = (1 s)Y(t) + (1 )G(t) (21)

where s and s are the propensities to save out of workers and managersdisposable income, and is the propensity to save out of managers capitalgains.

The demand of firms shares money from household is assumed to havethe following functional form

P e(t)E(t)

W(t)=

1

1 exp[rr GG(t dt)] (22)

The demand for money is residually determined as

Mh(t) = W(t) P e(t)E(t) (23)

Given that only profit earners demand for share, we have that

M(t) = Y(t) C(t) (24)

and, accordinglyM(t) = Mh(t) M(t) (25)

2.3 The financial sector

The financial sector is considered as an aggregate. It gives loans to firms,hence holding bonds as an asset, and creates money deposits endogenouslyas liabilities. The interest rate paid on deposits and on loans is considered

to be the same for simplicity, so that the financial sector does not make anyprofits (net worth is zero). Thus, the total stock of money M held by bothhouseholds and firms needs to be equal to the stock of bonds B:

B(t) =j

Mj(t) + M(t) + M(t)

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2.4 Goods market equilibrium

Total output Q is divided between aggregate consumption C and investmentI:

pQ(t) = (1 s)Y(t) + (1 s)Y(t) + (1 )G(t) + I(t)After substituting the labor share and the profit share from expressions(11) and (12) in (17) and (18) and solving for Q we obtain

pQ(t) =1 +

s [1 (1 s)] [I(t) + A(t)] (26)

where

A(t) = r[(1 s)M(t) + (1 s)(M(t) B(t))] + (1 )G(t)

Table 1 provides a visualization of the balance sheets of the productive,household and financial sectors while table 2 illustrates the social accountingmatrix for our economy10.

3 Firms dynamics

As illustrated in sections 1 and 2, the analytical solution method adopted bythis project operates through a reduction in the heterogeneity by grouping theagents in clusters. For each cluster a representative agent can be identified.To this aim, in the present treatment we take the average of the relevantmicro-variables within the group. This procedure is defined as the mean-fieldapproximation, which essentially involves reducing the vector of observationsof a variable over a population to a single value (Brock and Durlauf, 2001).

The following step is presented in subsection 3.1 and concerns the defini-tion of the probabilistic rules for the switching of firms among the differentgroups. The dynamics of the number of agents in each cluster can be repre-sented by a stochastic process of the Markovian type. This class of processescan be analytically described by a master equation which is a stochastic dif-ferential equation. The main steps and the outcome of the solution method

10Physical capital in firms balance sheets is evaluated at its market price and not atthe evaluation price h. We do not therefore consider for the moment the effect of capitalgains on firms balance sheets.

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for the master equation are presented is subsection 3.2. In particular, themaster equation solution can be expressed in compact form by an ordinarydifferential equation plus a stochastic component, given by a Wiener process.Both the ordinary differential equation and the noise term are formulated asfunctions of the micro-variables that determine the transition of agents be-tween the different groups. This result is then used in subsection 3.3 to derivethe laws of motion of the aggregate variables.

3.1 Transition probabilities

The micro-probability for a firm of transitioning from one state to another isdetermined by its capacity of satisfying conditions (1) or(2). These conditionscan be quantified by expressing them as functions of the idiosyncratic shocks, whose distribution is known by assumption. Let us preliminarily introducethe variable z, defining it as

z =iz(t) mz(t) [pqz(t) r(bz(t) mz(t)]

(1 )pqz(t)K(t)

K(t) Kz(t) . (27)

Using equations (1), (2), (9) and (13), it can be demonstrated that a bor-rowing firms becomes hedge if

s(t) 1, (28)

and a hedge firm becomes borrowing if

s(t) < 2. (29)

The first probability is indicated by while the second by . Accordingly,we can write

j(t) = P r[s(t) 1], (30)j(t) = P r[s(t) < 2]. (31)

Assuming s to be uniformly distributed in the interval [0.5, 0.5], we quantifythe two probabilities using the cumulative distribution function of s as

(t) = 1 + 0.5 (32)

(t) = 0.5 2 (33)

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These probabilities concern the transition of an individual firm from onestate to another, thus they can be defined as micro-probabilities. In orderto estimate the number of transitions at aggregate level, we need to referthese quantities to the number of firms in each state. The configurationaltransition rates (Weidlich, 2000) read as

+(t) = N1(t)(t) (34)

and(t) = N2(t)(t) (35)

The master equation quantifies the dynamics of the the probability ofhaving N1 firms in state 1 in a given instant, assuming that the numbers offirms in the two states evolve according to a jump Markov process. It can beformulated as the balance equation between the aggregate transition to andfrom state 1 and expressed as

dP(N1, t)

dt= +(t)P(N11)(t)+(t)P(N1+1)(t)+[+(t) + (t)] P(N1)(t)

(36)

3.2 Master equations solution: stochastic dynamics of

trend and fluctuations

The method for the master equation introduced by Di Guilmi (2008), devel-oping Landini and Uberti (2008), yields a system of two equations. The firstone is an ordinary differential equation which describes the time evolution ofthe trend of the stochastic process. The second one is a partial differentialequation, known as Fokker-Planck equation, whose general solution identifiesthe probability distribution of the fluctuations around the drift component11.

This solution techniques split the state variable in two components (asproposed by Aoki, 2002) according to

N1 = Nm +

N s. (37)

The factor m is the trend and represents the deterministic component; thevariable s is the spread and quantifies the stochastic noise around the trend.

11For a full detail of the solution method we refer the reader to Di Guilmi (2008),Chiarella and Di Guilmi (2011) and Landini and Uberti (2008).

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The solution derives an equation for each of the components. In particular,the trend evolves according to the following ODE

dm

d= m ( + )m2, (38)

where = t/N. The solution for the spread yields the Fokker-Planck equa-tion for the noise, whose stationary distribution is given by the followingGaussian distribution

(s) = Cexp

s

2

22

: 2 =

( + )2. (39)

The dynamics of n1 can be therefore described by

n1(t)

dt= m ( + )m2 + dV(t) (40)

where dV is a stationary Wiener increment and dW is the stochastic fluc-tuation component in the proportion of speculative firms, coming from the

distribution (39).

3.3 Mean-field equations

Having identified the dynamics of the numbers of the two types of firms,we can now use the mean-field values of each micro variable within the sub-population of speculative or hedge firms in order to study the dynamics ofthe aggregate variables. As for the notation, since the population of firms isreduced to just two, the superscript j is no longer needed for the firm-levelvariables.

The total amount of investment is given by

I(t) = K(t) = N h(t) + N1[1a1(t) + u1(t)] + N2[2a2(t) + u2(t)] (41)

Accordingly, the evolution of debt and the amount of equities of the averagespeculative firm can be quantified by re-defining equations (15) and (16),respectively, in the following way

b1(t) = [i1(t) a(t) m1(t)] (42)

e1(t) = (1 ) [i1(t) a(t) m1(t)] /P e(t). (43)

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Consequently, the dynamics of the aggregate debt in the economy is givenby

B(t) = N1 { [i1(t) a(t) m1(t)]} (44)The total number of shares evolves according to

E(t) = N1 {(1 ) [i1(t) a(t) m1(t)] /P e(t)} (45)

4 Analysis

As specified in section 1, the model is initially simulated as agent based withfull heterogeneity of agents. In other words, the behavioral rules introducedin subsection 2.1 are applied on N = 1, 000 potentially heterogeneous firms.At each time step the agent based model provides the mean-field values ofthe micro-variables. In this stage heterogeneity is reduced to two types offirms (one borrowing and one hedge) by taking the average of the relevantvariables within each group of firms. Subsequently, the system composed bythe equations introduced in section 3.3 is numerically analyzed. In particular

we identify a benchmark scenario and study the different dynamics generatedby shocking the parameters.In the benchmark scenario the values of the parameters are = 10; =

0.1; 1 = 0.01; 2 = 0.05; = 0.5; s = 0.05; s = 0.7; = 0.75; r =0.0015; = 0.8; = 1; r = 1; G = 0.000001; N = 1, 000.

Figure 1 shows the evolution of aggregate demand, debt and investmentfor a single simulation. The three variables display the same trend of growth,with short and very regular cycles for demand and investment. The growthof debt is relatively smooth compared to the other series.

4.1 Effects of the heterogeneity of the investment rulesFigure 2 illustrates the evolution of the number of borrowing firms, on a totalof 1,000 firms. It adjusts relatively soon to its steady state level of about 400and then fluctuates quite widely.

Figure 3 provides some insights about the degree of heterogeneity be-tween the two groups of firms. The upper panel reveals that investment isremarkably larger for hedge firms. In particular, it grows exponentially forhedge firms whereas it reaches a steady state level for borrowing ones. This ismainly due to the higher level of accumulated profit for hedge firms (central

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panel) that more than compensate for the lower elastiticity of investment tointernal finance, since 1 > 2 in equation (4) by assumption. The higherrate of growth for firms in this group generates a larger capacity utilization,as shown by the bottom panel of figure 3. This in turn further increases theinvestment of hedge firms compared to the borrowing, in a self-reinforcingmechanism.

In order to further investigate the effects of heterogeneity we shock theparameters 1 and 2 and look at the effect on the aggregate debt to cap-

ital ratio. As shown by figure 4, variations in the coefficient for borrowingfirms does not substantially alter the evolution and the steady state of theratio. However, higher values of 1 cause higher levels of the ratio duringthe adjustment, as one could expect. Figure 5 displays a more interestingpicture for the coefficient of hedge firms. A larger value of this parametercompared to the benchmark case (2 = 0.01) pushes rapidly the steady stateof the debt to capital ratio close to 0. It is worth noting that in the case1 = 2 = 0.05 the heterogeneity of behavioral rules is eliminated, beingthe investment equation identical for both type of firms. The impact onthe system of 2 is confirmed by figure 6, which reveals considerable larger

rates of growth for aggregate demand for a higher sensitivity of hedge firmsinvestment to internal finance. Changes in the parameter have no relevanteffects on the share of borrowing firms.

It is particularly interesting to assess the effect of heterogeneity on thedegree of financialization of the system, defined here as the ratio betweenmarket capitalization and aggregate demand. Figure 7 plots the dynamicsof the ratio between the total market capitalization, measured by P e(t) E(t), and the aggregate demand for different values of2. In the benchmarkscenario (with 2 = 0.01), this ratio displays an increasing trend, since theasset price inflation in the long run is larger than the rate of growth of the

economy. A larger 2 reduces the degree of financialization of the economy,bringing the ratio to a constant level. This steady state level is lower forlarger 2. For 2 = 2 = 0.05 the rate of growth of the economy convergesto the rate of growth of equity price, stabilizing the ratio, despite the largerasset price inflation compared to the benchmark scenario. Heterogeneity, andin particular the financial constraint faced by the borrowing firms, proves tobe a factor increasing the dependence of the real sector to the financial sector.

The other two parameters of the investment function have a lesser impacton the dynamics of the debt to capital ratio as illustrated by figures 8 and 9for, respectively, the elasticity to the evaluation ratio and to the capacity

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utilization . The plots reveal a slightly higher ratio for higher values of bothelasticities.

4.2 Dynamics of leverage

In the benchmark scenario, the ratio between aggregate capital and aggregatedebt tends asymptotically to 0 since the accumulation of capital is fasterthan the accumulation of debt. This outcome can be modified by verying

two parameters.The first one is the propensity to save out of capital gains . Figure

10 shows that the ratio more than doubles for a value of 0.25 compared to0.5 and 0.75 (benchmark case). A higher propensity to save out of capitalgains raises the equity price and, consequently, boosts the level of investmentfor both hedge and borrowing firms in the same amount. This leads to anincrease in the level of debt that is larger than the increase in the stockof capital. Another effect of a higher propensity to save is the rise in thelevel and in the volatility of aggregate demand (figure 11). This is due tothe increase to investment caused by the larger share of income invested in

stocks. This also magnfies the effects of asset price fluctuations on aggregatedemand. High level of savings out of capital gains are realistic in a contextof growing weight of capital gains on total income. This effect is furtherdiscussed below.

The accumulation of debt is also impacted by the bankruptcy thresholdc. Figure 12 shows that lowering the threshold modifies the steady state ofthe aggregate leverage. The figure does not report the simulation with c = 0where the ratio spikes to 300. The higher level of debt has no relevant effecton the size of aggregate demand, as shown by figure 13, since hedge firms,which have no debt, account a progressively larger share of aggregate demandas time passes. Hence the levels of investment and demand are comparableto benchmark case whereas aggregate debt is noticeably larger.

4.3 Changes in the propensities to save, distribution

of income and preference for liquidity

In general, the effect of an increase in the propensity to save for the differentcategories of income is destabilizing for the system. For a propensity tosave of wage earners 0.1, all the firms default at the beginning of

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the simulation because, in the early stages, the biggest fraction of aggregatedemand is provided by salaries. As we show below this is reversed as thesimulation runs.

A larger propensity to save of profit earners has the effect of lowering theaggregate debt to capital ratio, as shown by figure 14, and increasing leveland volatility of aggregate demand, as demonstrated by figure 15. This isdue to the fact that an increase in savings raises the demand for equities andtheir price and, through this channel, the level of investment. The bigger

weight of internal finance on the investment decision causes the the volatilityof profits to affect more significantly the volatility of demand. The study ofthe correlations between the profits and salary bill with aggregate demandhighlights the fact that this economy is profit-led.

The reliance of the system on financial income makes growth possibleeven with a bigger size of the mark-up, and thus a larger share of profits onoverall income. Simulations show that the leverage ratio during the transi-tion toward the steady state is lower for larger . This result is due not onlyto the fact that in our system the only debt is business debt, but also to theincreased relevance of capital gains in sustaining growth. In fact, figure 16

reveals that a larger share of income for profit drives up the ratio betweensize of the equity market and aggregate demand. Also the size of the fluc-tuations of the ratio increases for the higher variance of both equity priceand aggregate demand, proving that the system becomes remarkably morevolatile. The explanation involves the fact that the propensity to save outof profit is larger than the one for salaries and this increases the demand forequities and, as a consequence, the size of the financial sector with respectto the real sector.

As remarked above in commenting figure 10, in our stylized economy theparadox of thrift is avoided by the increasing financialization: the increase

in investment due to higher evaluation ratio and the fact that consumptionis financed for a progressively increasing weight of financial income and thisavoids the shortfall of a lower propensity to consume out of profits and capitalgains. This is consistent with the outcome of the analysis of the sensitivityto the parameter . Our future research will investigate in more detailthe conditions under which this process can sustain itself and over whichthreshold it can lead to a financial collapse and a depression.

Finally, in order to evaluate the role of the preference for liquidity weshock the parameter G, which quantifies the sensitivity of investors to capitalgains. Results are shown in figure 17. The model reproduces the classical

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Keynesian result of higher demand and higher accumulation for lower levelof preference for liquidity. Also this result is evidently impacted by the fastergrowth of the financial sector with respect to the real sector. The three panelsshow that, at the beginning of the simulation, the patterns generated in thebenchmark scenario (G = 10

6) and in the alternative scenario (G = 0.01)are similar. As time passes the two dynamics diverge, with a significantlyhigher growth in the alternative scenario.

5 Concluding remarks

This paper aims to introduce microfoundations in stock flow consistent mod-eling to study how the interaction between financial and real sector affectsgrowth and business cycle. To this aim the model is formulated in a bottom-up fashion identifying heterogeneous micro-behavioral rules for firms andthen deriving the macro-equations for the productive sector using the masterequation.

The research presented here is at a preliminary stage but nevertheless itprovides some insights on the transmission of shocks between the financialand the real side of the economy. The numerical simulations highlight therelevance of the heterogeneity in the investment function. In particular, iffinancially sounder firms are supposed to be less affected by internal financein their investment decision, the aggregate leverage in the economy is higherin the adjustment phase and can stabilize at a higher steady state. A highersensitivity to internal finance, even for sounder firms, increases the level ofgrowth of aggregate demand, reducing the dependence of the real sector tothe financial sector.

In the benchmark setting, firms are not allowed to roll-over interest ondebt. Allowing them to do so makes the aggregate leverage ratio to spike.

A larger propensity to save in each of the three income categories (salaries,profits and capital gains) has the effect of noticeably increasing the instabilityof the system and the size of fluctuations. This effect is particularly relevantfor the propensity to save out of capital gains, which also appears to bepositively related to the aggregate leverage ratio. This result is strictly linkedto the growing weight of the financial sector which affects growth, amplitudeof fluctuations, distribution of income and role of the preference for liquidity.

The next development of this work concerns a more refined study ofthe conditions under which bubbles and busts are generated in the present

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setting. Also, the model will be extended by introducing a variable mark-up, to study the evolution of the shares of income, and the possibility forhouseholds to shift between the two categories of profit-earners and income-earners.

From a techinical point of view, future research will involve a deeper anal-ysis of the analytical solution of the model, to be achieved by the study ofthe dynamical system composed by the dynamic equations for the aggregatevariables. The stability properties of this dynamical system will be investi-

gated in order to provide peculiar insights about the role of heterogeneity inthe dynamics of flows. The steady state analysis will identify a map of thesustainable and unsustainable long run paths of evolution of the economyas a result of the distribution of leverage and income. It will be possibleto appreciate the micro-determinants of the convergent, oscillatory or outof equilibrium dynamics of the economic system. The bifurcation analysiswill illustrate the sensitivity of the dynamics to the behavioral parametersof agents, as the decision of firms for investment and of household for con-sumption.

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Firm Household sector Financial sectork b M + M B Mm P e e P e E

j W

Table 1: Units and sectoral balance sheets

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Households Firms Current Capital

Consumption C +CInvestment +I [N h + N1(1a1 + u1) + N2(2a2 + u2)] Wages +pQ pQProfits +(pQ rB) (N1a1 + N2a2) (1 )(pQ rB) + r(N1m1 + N2m2) Loan interests rB Deposit interests r(M + M) +r(N1m1 + N2m2) Change in loans +N1 [(i1 a m1)]Change in deposits (M + M) (N1m1 + N2m2Change in equities P eE +N1 [(1 )(i1 a m1)] Total 0 0 0

Capital Gains P eE +P eE

Table 2: Matrix of flows.

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Figure 1: Dynamics of aggregate demand, investment and debt.

Figure 2: Dynamics of the number of borrowing firms (total number of firms:1,000).

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Figure 3: Investment (upper panel), accumulated profit (central panel) andcapacity utilization (bottom panel) for the average borrowing and hedgefirms.

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Figure 4: Dynamics of the debt/capital ratio for different values of 1.

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Figure 5: Dynamics of the debt/capital ratio for different values of 2.

Figure 6: Dynamics of aggregate demand for different values of 2.

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Figure 7: Dynamics of equity value to aggregate demand ratio for differentvalues of 2.

Figure 8: Dynamics of the debt/capital ratio for different values of .

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Figure 11: Dynamics of aggregate demand for different values of .

Figure 12: Dynamics of the debt/capital ratio for different values of c.

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Figure 13: Dynamics of debt, investment and aggregate demand for c = 0.01.

Figure 14: Dynamics of the debt/capital ratio for different values of s.

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Figure 15: Dynamics of aggregate demand for different values of s.

Figure 16: Dynamics of equity value to aggregate demand ratio for differentvalues of .

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Figure 17: Dynamics of debt to capital ratio (upper panel), capital accu-mulation (central panel) and aggregate demand (bottom panel)for differentvalues of G.

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business debt dynamics in a micro-founded stock-flow consistent model

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