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Journal of Evolutionary Economics

, Volume 29, Issue 1, pp 265–297 | Cite as

The financial transmission of shocks in a simple hybrid macroeconomic agent based model

  • Tiziana AssenzaEmail author
  • Domenico Delli Gatti
Regular Article

Abstract

Tracking the chain of events generated by an aggregate shock in an Agent Based Model (ABM) is apparently an impossible mission. Employing the methodology described in Assenza and Delli Gatti (J Econ Dyn Control 37(8):1659–1682 2013) (AD2013 hereafter), in the present paper we show that such a task can be carried out in a straightforward way by using a hybrid macro ABM consisting of a IS curve, an Aggregate Supply (AS) curve and a Taylor Rule (TR) in that aggregate investment is a function of the moments of the distribution of firms’ net worth. For each shock (fiscal expansion, monetary tightening, financial shock) we can decompose the change of the aggregate scale of activity (measured by the employment rate) in a first round effect – i.e., the change generated by the shock keeping the moments of the distribution of net worth at the pre-shock level – and a second round effect, i.e., the change brought about by the variation in the moments induced by the aggregate shock. In turn, the second round effect can be decomposed in a term that would show up also in a pure Representative Agent setting (RA component) and a term that is specific to the model with Heterogeneous Agents (HA component). In all the cases considered, the first round effect explains most of the actual change of the output gap. The second round effect is unambiguously negative. The HA component has the same sign of the RA component and explains a sizable fraction of the second round effect.

Keywords

Heterogeneity Financial fragility Aggregation Business cycles 

JEL Classification

C63 E12 E03 E32 E44 E52 

1 Introduction

In a macroeconomic Agent-Based Model (ABM), an aggregate variable such as GDP is determined “from the bottom up” i.e., summing the output of a large number of heterogeneous firms. In other words, GDP is a function of the entire distribution of agents’ characteristics. The dynamic pattern of a macroeconomic variable such as GDP is an emergent property of the model that is determined by the complex microeconomic interactions of myriads of heterogeneous agents. Over the last decade, a fairly large literature has explored the emergent properties of aggregate variables in macro ABMs; see, for instance, Cincotti et al. (2010, 2012); Dawid et al, (2011, 2012); Delli Gatti et al. (2008, 2011); Dosi et al. (2006, 2010, 2013); and Gaffeo et al. (2008). For example, in Delli Gatti et al. (2005) and Assenza et al. (2015), in the presence of a financial friction, investment and output at the firm level are affected by individual net worth. Hence, at the aggregate level, GDP is a function of the distribution of the firms’ net worth.

In an agent-based setting, thinking in macroeconomic terms – i.e., in terms of interrelated changes of aggregate variables – is prima facie impossible. When an aggregate shock occurs, it is extremely difficult to trace the transmission mechanism in a clear and uncontroversial way. In order to understand how a shock trickles down through the web of micro interactions and affects macro variables, it is necessary to rely on “narratives” that may or may not be convincing.

In a previous paper, we have proposed a methodology to deal with this issue (Assenza and Delli Gatti 2013; AD2013 hereafter) that consists in building what we call a Hybrid Macroeconomic Agent Based Model. In our 2013 paper, the Hybrid Macroeconomic ABM is embedded in an optimizing IS-LM framework. In the present paper we follow the same methodology to build a model of an economy populated by households, firms and banks closer in spirit to the contemporary New Keynesian literature. The model consists of a IS curve, an Aggregate Supply (AS) curve (i.e., a Phillips curve) and a Taylor Rule (TR).

In principle, different types and degrees of heterogeneity could be taken into account. For the sake of simplicity, in the following we assume that there is a representative household and introduce heterogeneity only at the firm level. In particular, we assume that the corporate sector consists of a myriad of firms characterized by heterogeneous financial conditions (captured by net worth). In the spirit of Greenwald and Stiglitz (1993), each firm faces an idiosyncratic shock to revenues and decides investment in order to maximize expected profits. We assume that the cost of credit for the borrowing firm decreases with financial robustness. Hence the firm’s optimal expenditure on capital goods is affected by its net worth.

Adopting an appropriate aggregation procedure – that we labeled the Modified-Representative Agent in AD2013 – we approximate the distribution of agents’ net worth by means of the first and second moments of the distribution.1 Therefore, aggregate investment turns out to be a function of an average External Finance Premium (EFP), that in turn is affected by the moments of the distribution of firms’ net worth.2 The moments of the distribution play the role of macroeconomic variables and therefore can be incorporated in a macroeconomic model.3 We thus use this investment equation in the macroeconomic framework described by the IS-AS-TR model.4

In each period, say t, given the moments of the distribution in period t −  1, we determine the macroeconomic equilibrium, i.e., the triple consisting of the equilibrium levels of the employment rate, the inflation rate and the interest rate. Since the moments of the distribution change over time, the average EFP and therefore the macroeconomic equilibrium change as well. The role of heterogeneity (in the corporate sector) in influencing macroeconomic outcomes is captured by the fraction of change in the macroeconomic equilibrium, that can traced back to the change in the cross-sectional variance of the distribution.

In order to assess the quantitative impact of changes in the moments of the distribution on macroeconomic outcomes we develop a simple ABM of the corporate sector. For each firm, we define the law of motion of net worth, that is affected – among other variables – by the interest rate: the higher the interest rate, the lower realized profits and the lower individual net worth. The ABM boils down to a system of non-linear difference equations (one for each firm). From the artificial data obtained through simulations we trace the evolution over time of each and every element of the distribution of net worth. Hence we can retrieve the evolution over time of the cross-sectional mean and variance, that will impact future endogenous macro-variables.

In a nutshell, there is a two-way feed back between the macroeconomic and the agent based submodels: the equilibrium interest rate in t, that is affected by the moments of the distribution in t −  1, will impact on the moments of the distribution in t, that will reverberate on the equilibrium interest rate in t +  1, and so on. Changes over time of the moments drive the evolution of the equilibrium interest rate, the employment rate and inflation.

We use the Hybrid Macroeconomic Agent Based Model to provide an answer to the following question: that is the role of heterogeneity in the transmission mechanism of (fiscal, monetary and financial) shocks to the macroeconomy?

For each shock, we provide a breakdown of the associated change of the employment rate (denoted by x), that can be represented as follows:
$$\begin{array}{@{}rcl@{}} {\Delta} x&=& {\Delta} x_{1st}+{\Delta} x_{2nd} \\ {\Delta} x_{2nd}&=& {\Delta} x_{2nd,RA}+{\Delta} x_{2nd,HA} \end{array} $$

The expression Δx1st is the direct or first round effect i.e., the change in x generated by the shock assuming that the distribution of net worth does not change.

In our setting, however, the distribution does change. There is also an indirect or second round effect represented by Δx2nd that captures the change of the employment rate due to the change in the distribution of net worth, that in turn is generated by the shock.5 The indirect effect captures a financial transmission mechanism, because it is entirely due to the change of net worth, our measure of financial robustness. It can be broken down, in turn, into two components: a Representative Agent (RA) component Δx2nd, RA and a Heterogeneous Agents (HA) component Δx2nd, HA. The former is the indirect change in the employment rate that would occur if the individual EFP coincided with the average EFP (focusing therefore only on the first moment of the distribution) while the latter incorporates also the effect of changes in the variance of the distribution.

Given the chosen parameterization, we are able to quantify these effects. We consider three (permanent) shocks: (i) an expansionary fiscal shock (increase of Government expenditure); (ii) a monetary shock (increase of the exogenous component of the interest rate); (iii) a financial shock (increase of the exogenous component of the individual EFP).

The main results can be summarized as follows:
  • In all the cases considered, the first round effect explains most of the actual change of the output gap.

  • The second round effect is unambiguously negative both in the case of an expansionary fiscal policy and in the case of a contractionary monetary policy. In both cases, in fact, the average EFP goes up.

  • Both in the case of an expansionary fiscal policy and in the case of a contractionary monetary policy, after the shock the cross-sectional mean and variance of the distribution of net worth go down: the first and second moments of the distribution are positively correlated. This is due to the consequences of the increase of the interest rate on the distribution. The reduction of the cross-sectional mean pushes the average EFP up while the reduction of the variance pushes the average EFP down. The first effect prevails so that average EFP goes unambiguously up.

  • The second round effect is negative also in the case of a contractionary financial shock. In this case, the EFP goes up on impact because of the shock itself, and goes further up because of the second round effect. Also in this case, the cross-sectional mean and variance of the distribution of net worth go down, and the first effect prevails so that average EFP goes unambiguously up.

  • The second round effect amplifies the effect of the monetary shock and the financial shock and mitigates the effect of the fiscal shock. In the latter case, in fact, the financial transmission mechanism contributes to crowding out.

  • In the case of the fiscal and monetary shock, the HA component has the same sign of the RA component and explains a sizable part of the second round effect.

Of course, the size of these effects is due to the particular configuration of parameters and to the modelling choices we adopted. Let’s remember that there is only one source of heterogeneity in this model, i.e., the heterogeneity of firms’ financial conditions.

The paper is organized as follows. Section 2 is devoted to the microfoundations of behavior of households, firms and banks. Section 3 presents the IS-AS-TR model. In Section 4 we derive the macroeconomic equilibrium. Section 5 is devoted to the description of the ABM and the discussion of the output of simulations in the baseline scenario. Sections 67 and 8 are devoted to the analysis of the effects of the fiscal shock, the monetary shock and the financial shock, respectively. Section 9 concludes.

2 Microfoundations

In the present section we will describe the main building blocks of the model. In particular, we will describe the microfoudations for the main actors of our simplified economy, i.e., households, firms, and the banking system.

2.1 Households

We examine an economy populated by nH homogeneous households. Each household has m members, of that ma are active (i.e. participating in the labor market). Therefore, total population is H := m × nH, while total labor force is L := manH (both exogenous). Hence we can define the exogenous participation rate as the ratio between total labor force and total population, i.e. L/H = ma/m.

Households’ income consists of the sum of individual income across active members. In particular, each households’ active member earns the wage w if employed and the unemployment subsidy σ if unemployed, with w > σ. We assume that active members are simple Keynesian consumers (i.e. agents who save a fraction of their income) and their propensity to consume out of income is c. By contrast, inactive members are pure “hand to mouth” consumers: they do not save. For instance, a retired member of the household spends her pension entirely in consumption goods. Each inactive member of the family consumes \(\bar c\). All these parameters are exogenous.

As we will show in the following, in our economy total employment (Nt) is endogenous and time varying6 and depends on firms’ production decisions. Hence, the employment rate\(x_{t}=\frac {N_{t}}{L}\) is endogenous and time varying as well. Notice that, since we will assume that technology is linear (see Section 2.2), xt is also the ratio of current output Yt = λNt to full-employment output \(\bar Y=\lambda L\). Of course, unt =  1 − xt is the unemployment rate.

Given the above assumptions, the representative household’s consumption expenditure is:7
$$ C_{ht}=c[wx_{t}+\sigma(1-x_{t})]m_{a}+\bar c (m-m_{a}). $$
(2.1)
Aggregating across the nH households aggregate consumption expenditure, therefore, will be \(C_{t}=n_{H} C_{ht}=c[ w x_{t} + \sigma (1-x_{t})]L +\bar c (H-L)\). Rearranging, we can express aggregate consumption as:
$$ C_{t}=c (w-\sigma) x_{t} L+\bar C $$
(2.2)
where \(\bar C=c\sigma L+\bar c (H-L)\) is exogenous aggregate consumption expenditure. Notice that aggregate consumption expenditure is increasing in the employment rate, moreover being xt time varying it follows that aggregate consumption is time varying as well.

2.2 Firms

The supply side of the economy is populated by nF firms. Each firm, indexed by i =  1, 2, .., nF, produces a homogeneous final good by means of capital and labor in a competitive setting. Firms are heterogeneous with respect to their financial robustness captured by the time varying individual net worthAit. We will denote the mean and variance of the distribution of firms’ net worth in each period t with At := < Ait > and Vt := < (AitAt)2 >, respectively. We will refer to these variables as the cross-sectional mean and variance.8 Since the net worth of each firm is endogenously determined (see Section 5 for details), the cross-sectional mean and variance are also time-varying. We will show that both At and Vt fluctuate around their respective “long run” values, i.e., the average over a long time span of the cross-sectional mean and variance.

The cross-sectional mean in t is obtained by averaging individual net worth across firms in the same period. The long run mean is obtained by averaging the cross-sectional mean across T time periods, with T “large” enough to characterize the long run. We will denote with AL and VL the long run mean and variance of the distribution of net worth.

In symbols, the cross-sectional mean in period t is \(A_{t}={\sum }^{n_{F}}_{i = 1} A_{it}/n_{F}\), while the long run mean (over a time span of T periods) is \(A^{L}={\sum }^{T}_{t = 0} A_{t}/T\). Analogously, the cross-sectional variance in t is \(V_{t}={\sum }^{n_{F}}_{i = 1} (A_{it}-A_{t})^{2}/{n_{F}}\) while the long run variance is \(V^{L}={\sum }^{T}_{t = 0} V_{t}/T\).

We assume that in the case in that firms’ net worth is not sufficient to finance investment then they will rely on bank loans. Therefore, they run the risk of bankruptcy. Banks extend credit to firms at an individual interest rate (rit) that takes this risk into account. They charge to each firm a specific interest rate that includes an external finance premium (Bernanke and Gertler 1989, 1990) decreasing with individual net worth.

Each firm carries on production by means of a Leontief technology that uses labor and capital. The production function of the i-th firm is Yit = min(λNit, νKit) where Yit, Nit and Kit represent output, employment and capital, respectively; ν and λ are parameters that measure the productivity of capital and labor, respectively. Hence the ratio ν/λ is employment per unit of capital.9

The firm faces an idiosyncratic shock uit to revenue due, for instance, to a sudden change in preferences. We assume that uit is a random variable distributed as a uniform over the interval (0, 2) with expected value E(uit) =  1.

We define the profit of the i-th firm in real terms πit as the difference between total revenues uitYit and total costs, that consist of production costs \(C^{p}_{it}=wN_{it}+r_{it}K_{it}\) and adjustment costs \(C^{a}_{it}=\frac {\omega }{2}K^{2}_{it}\).
$$\begin{array}{@{}rcl@{}} \pi_{it}&=& u_{it}Y_{it}-wN_{it}-r_{it}K_{it}-\frac{\omega}{2}K_{it}^{2} \end{array} $$
(2.3)
For simplicity, we assume that the real wage w is given and constant. rit is the real interest rate charged by the bank to firm i. In the following, we will refer to this variable as the individual cost of credit. Assuming 100% depreciation, investment coincides with capital Kit. Adjustment costs are quadratic in investment (as usual in investment theory), ω is a parameter capturing the relative importance of these costs.
Assuming that labor is always abundant, we can write technology as a function of capital only, i.e. Yit = νKit and \(N_{it}=\frac {\nu }{\lambda }K_{it}\). Substituting these expressions into (2.3) and re-arranging we get:
$$\begin{array}{@{}rcl@{}} \pi_{it}&=& \left( u_{it}\nu-w\frac{\nu}{\lambda}-r_{it} \right)K_{it}-\frac{\omega}{2}K_{it}^{2} \end{array} $$
(2.4)
We assume that the risk neutral firm chooses the stock of capital (and therefore employment and output, that are proportional to capital) in order to maximize expected profits E(πit). Recalling that E(uit) =  1, the problem of the firm can be written as
$$\begin{array}{@{}rcl@{}} \max_{K_{it}} E(\pi_{it})&=&\left( \gamma-r_{it} \right)K_{it}-\frac{\omega}{2}K_{it}^{2} \end{array} $$
where \(\gamma =\nu -\nu \frac {w}{\lambda }\) is the difference between expected revenue per unit of capital (ν) and labor cost per unit of capital (\(w\frac {\nu }{\lambda }\)), i.e. earnings before interest payments (and before incurring adjustment costs) per unit of capital. For lack of a better term let’s label this polynomial the “gross rate of profit”.10
The solution to the problem above determines optimal individual investment, i.e.:
$$ K_{it}=\frac{\gamma-r_{it}}{\omega}. $$
(2.5)
Equation 2.5 is an extremely simple individual investment rule, according to that firm’s optimal investment is proportional (being \(\frac {1}{\omega }\) the factor of proportionality) to earnings after interest payments per unit of capital (γrit) that we will refer to as the “net profit rate”. For obvious reasons, we assume:11
$$\begin{array}{@{}rcl@{}} \gamma >r_{it}. \end{array} $$
(2.6)

Notice that, according to Eq. 2.5 optimal individual investment differs across firms due to the different cost of credit (rit). We will explore the determinants of the individual cost of credit in the next section.

2.3 Banks

For the sake of simplicity, the market for credit is described in a very parsimonious way. We assume that the cost of credit, i.e., the interest rate charged to the i-th borrowing firm, is the sum of an aggregate component represented by the interest rate rt – that is set by the central bank according to the Taylor rule and is uniform across firms – and a borrower specific component fit, that we label external finance premium (EFP), set by the banking system.
$$\begin{array}{@{}rcl@{}} r_{it}&=& r_{t}+f_{it}, \end{array} $$
(2.7)
In our model, the banking system plays the crucial role of setting the external finance premium for each borrowing firm. We assume that the EFP is decreasing in a non-linear way with the firm’s internal financial resources (or net worth of the previous period):
$$\begin{array}{@{}rcl@{}} f_{it}=\frac{\alpha}{A_{it-1}} \end{array} $$
(2.8)
where α is the exogenous component of the individual EFP, that is uniform across firms.12

Whatever bank the firm utilises, the EFP will be determined as in Eq. 2.8. Therefore it is not necessary to specify the behavior of each and every bank. The banking system contributes to determining the cost of credit for each borrowing firm along with the central bank.

As far as the quantity of credit is concerned, we assume that the demand of credit is fully satisfied at the given cost of credit. The quantity of credit supplied therefore is demand driven. It is not necessary to be specific on the demand for credit. However, to fix ideas one can think of the demand for credit (for each and every firm) as equal to the financing gap, i.e., the difference between investment and internal finance (e.g., net worth).

The definition of the individual cost of credit in Eq. 2.7 sets the stage for the macro-to-micro feedback: aggregate shocks (e.g. a change in monetary policy) reverberate on the individual cost of credit (and therefore on individual investment) through the risk free interest rate rt.

Assumption (2.8) introduces a non-linearity in the relationship between EFP and net worth at the individual level. This is key for the following analysis, as we will show momentarily.

Substituting (2.7) and (2.8) into (2.5) we obtain:
$$\begin{array}{@{}rcl@{}} K_{it}&=& \frac{1}{\omega}\left[\gamma-\left( r_{t}+\frac{\alpha}{A_{it-1}}\right)\right] \end{array} $$
(2.9)
Optimal individual investment differ across firms due to different interest rates, that in turn are linked (linearly) to different external finance premia, that in turn are linked (in a non-linear way) to different levels of “financial robustness” captured by net worth.13 The heterogeneity of individual investment, in the end, is due exclusively to the heterogeneity of net worth.
Assumption (2.6) implies a lower bound on Ait− 1:
$$ A_{it-1}>A^{min}_{t}=\frac{\alpha}{\gamma-r_{t}} $$
(2.10)
where we γ > rt. This condition provides a straightforward way of introducing exit in the model. It is easy to see that exit is due, in the end, to default on interest payments: when the net worth of the firm reaches the exit threshold\(A^{min}_{t}\) , the individual cost of credit becomes so high as to offset the gross profit rate and the firm defaults on its payments. In order to keep the number of firms constant, in the simulations we will assume that each exiting firm will be replaced by an entrant firm endowed with an initial net worth chosen at random. The fraction of exiting firms is, of course, increasing with the exit threshold.

Notice that the exit threshold is time varying since it is increasing with the risk free interest rate. Any aggregate shock that leads to an increase of the interest rate, therefore, will increase the exit threshold and lead to a higher fraction of exiting firms.

Since the individual EFP is a convex function of net worth, the individual investment is a concave function of net worth defined on \(A_{it-1}>A^{min}_{t}\) . It is easy to see that there is an upper bound on individual investment, that is \(K^{max}_{t}=\frac {\gamma -r_{t}}{\omega }\).

Plugging (2.7) into (2.5) one gets: \(K_{it}=\frac {\gamma -(r_{t}+f_{it})}{\omega }\). Given the linear structure of this equation, the cross-sectional mean of capital Kt := < Kit > turns out to be:
$$\begin{array}{@{}rcl@{}} K_{t}&=& \frac{\gamma-(r_{t}+f_{t})}{\omega} \end{array} $$
(2.11)
where ft = < fit > is the average EFP.
From a second order approximation of Eq. 2.8 around the cross-sectional mean of net worth At− 1, we get the following equation for the average EFP:14
$$ f_{t}\approx \frac{\alpha}{A_{t-1}}+\frac{\alpha V_{t-1}}{A_{t-1}^{3}} $$
(2.12)

The average EFP is the sum of two terms. The first one would be the EFP in the Representative Agent case and depends only on the cross-sectional mean (therefore we will refer to this component as the “RA term”) while the second one captures the role of heterogeneity in the average EFP. The second term (that we will refer to as the“HA term”) is affected both by the cross-sectional mean and the cross-sectional variance of the distribution of firm’s net worth.

Substituting (2.12) in Eq. 2.11, we can specify average investment as:
$$\begin{array}{@{}rcl@{}} K_{t}&=&\frac{1}{\omega}\left[\gamma- \left( r_{t}+\frac{\alpha}{A_{t-1}}+\frac{\alpha V_{t-1}}{A_{t-1}^{3}}\right)\right] \end{array} $$
(2.13)
Therefore, aggregate investment (given the population of firms in our economy equal to nF) will be:
$$\begin{array}{@{}rcl@{}} I_{t}&=&n_{F} \times K_{t}=\frac{n_{F}}{\omega}\left[\gamma- \left( r_{t}+\frac{\alpha}{A_{t-1}}+\frac{\alpha V_{t-1}}{A_{t-1}^{3}}\right)\right] \end{array} $$
(2.14)

3 The macroeconomic model

Given the micro-foundations described in the previous section, in this section we will turn to the macroeconomic set-up representing our economy.

3.1 The IS curve

Given the Leontief production function discussed in Section 2.2 the individual firm’s production is equal to Yit = λNit. Hence, being nF the number of firms in our economy, it follows that aggregate production is Yt = λNt, where \(N_{t}=\sum \limits _{i = 1}^{n_{F}}N_{it}\). Recalling that \(x_{t}=\frac {N_{t}}{L}\) is the employment rate, substituting we get that aggregate production isYt = λxtL. For the sake of simplicity, we assume that public expenditure is exogenous and equal to a certain level G. Hence equilibrium on the market for final goods occurs when Yt = Ct + It + G Plugging (2.2) and (2.14) in the equilibrium condition and rearranging we get:
$$\begin{array}{@{}rcl@{}} x_{t}=x_{0}-x_{1}(r_{t}+f_{t}) \end{array} $$
(3.1)
with \(x_{0}:=\frac {1}{[\lambda -c(w-\sigma )] L}\left (\bar C+G+\frac {n_{F}}{\omega }\gamma \right )\) and \(x_{1}:=\frac {1}{[\lambda -c(w-\sigma )] L}\times \frac {n_{F}}{\omega }\).

Since λ > w by assumption (see Section 2.2 for details), then x0 and x1 are positive parameters. The relation in Eq. 3.1 represents the equation of the IS curve on the (xt, rt) plane. Notice that the (time varying) average EFP ft is a shift parameter of the curve. In particular, given x1 >  0, as the average EFP increases the IS shifts downward on the plane (xt, rt). In words: as conditions to access credit for firms become more stringent on average, then, for each level of the risk free interest rate, they will be able to finance a smaller amount of investment and in turn they will cut production. This effect is captured by the reduction of the employment rate xt.

3.2 The AS curve

We assume inflation is determined by a standard Phillips curve: πt = θyt, where θ is a positive parameter and \(y_{t}:=\frac {Y_{t}-Y_{n}}{Y_{n}}\) is the output gap. Yn is the “natural” level of output, i.e., the level of output associated to the natural rate of unemployment un =  1 − xn, where xn is the natural employment rate, that we assume exogenous. Since technology is linear, Yn = λxnL with \(Y_{n}<\bar Y\). Substituting for Yt and Yn, the output gap turns out to be: yt := ηxt − 1 where η =  1/xn. Hence the Aggregate Supply (AS) curve can be re-written as follows:
$$\begin{array}{@{}rcl@{}} \pi_{t}&=&\theta (\eta x_{t}-1) \end{array} $$
(3.2)
Notice that expected inflation is left out of the picture; it does not show up in the Phillips curve. The reason for this drastic simplification is simplicity. We want to explore the properties of this hybrid model before introducing the complications due to the formation of expectations in a setting characterized by heterogeneous agents. These issues will be dealt with in future developments of our research. Due to this modelling strategy, inflation is an increasing linear function of the output gap. It is positive (negative) when the unemployment rate is higher (smaller) than the natural rate.

3.3 The Taylor Rule (TR)

In order to close the model we need to define a monetary policy rule. We assume that the central bank adopts a standard Taylor rule (with flexible inflation targeting) to set the nominal interest rate:
$$\begin{array}{@{}rcl@{}} i_{t}&=&r_{n}+(1+\alpha_{\pi})\pi_{t}+\alpha_{x}(\eta x_{t}-1) \end{array} $$
(3.3)
with απ, αx >  0. rn is the natural rate of interest. The real interest rate rt is the difference between the nominal interest rate it and inflation πt: rt = itπt.
Hence the real interest rate will be:
$$\begin{array}{@{}rcl@{}} r_{t}&=&r_{n}+\alpha_{\pi}\pi_{t}+\alpha_{x}(\eta x_{t}-1) \end{array} $$
(3.4)
Equation 3.4 represents the Taylor Rule (TR) in our setting.

4 The macroeconomic equilibrium

Given the model described in the previous sections, our economic system is represented by a system of three Eqs. 3.13.2 and 3.4 in three unknowns, i.e., the employment rate (xt), inflation (πt) and the (real) interest rate (rt):15
$$\left\{ \begin{array}{c} x_{t}=x_{0}-x_{1}(r_{t}+f_{t}) \\ \pi_{t}=\theta (\eta x_{t}-1) \\ r_{t}=r_{n}+\alpha_{\pi}\pi_{t}+\alpha_{x}(\eta x_{t}-1) \end{array} \right. $$
This system can be solved for xt, rt and πt given ft.16 In order to solve the model we propose the following strategy. Plugging AS (3.2) into TR (3.4) one gets:
$$ r_{t}=r_{n}+\beta_{x}(\eta x_{t}-1) $$
(4.1)
where βx := αx + απθ. The augmented TR boils down to an upward sloping line on the (x, r) plane. Notice that, when the employment rate is xn both inflation and the output gap are on target, so that the real interest rate targeted by the central bank is the natural rate rn. Solving the system (4.1) (3.1) we get the equilibrium employment rate and interest rate as a function of the time varying average EFP:
$$\begin{array}{@{}rcl@{}} x_{t}^{*}&=&x_{2}- x_{3} f_{t} \end{array} $$
(4.2)
$$\begin{array}{@{}rcl@{}} r_{t}^{*}&=&r_{n}+\beta_{x}[\eta(x_{2}- x_{3} f_{t})-1] \end{array} $$
(4.3)
where \(x_{2}:=\frac {x_{0}-x_{1} (r_{n}-\beta _{x})}{1+x_{1} \beta _{x} \eta }\) and \(x_{3}:=\frac {x_{1}}{1+x_{1} \beta _{x} \eta }\).
Equilibrium inflation can be retrieved by plugging (4.2) into (3.2):
$$ \pi_{t}^{*}=\theta[\eta(x_{2}- x_{3} f_{t})-1] $$
(4.4)
The nominal interest rate can be easily obtained by adding the equilibrium inflation rate to the equilibrium real interest rate.

Equations 4.24.34.4 – i.e., the reduced form of the model – determine the temporary equilibrium of the macroeconomy, i.e., the triple (\(x_{t}^{*}, r_{t}^{*}, \pi _{t}^{*}\)). Equilibrium is temporary becasue the equilibrium values of the endogenous variables are parameterized to the time varying average EFP.

In order to obtain a positive equilibrium employment rate, we impose the following restrictions on parameter values:17
$$\begin{array}{@{}rcl@{}} \beta_{x}>r_{n} \end{array} $$
(4.5)
$$\begin{array}{@{}rcl@{}} \frac{x_{0}}{x_{1}}+\beta_{x}-r_{n}>f_{t} \end{array} $$
(4.6)

While equilibrium output cannot be negative, both the real interest rate and inflation can be negative in equilibrium. Notice that the employment rate plays a leading role in determining the macroeconomic performance: the real interest rate and inflation in fact are increasing linear functions of the employment rate.

The model is characterized by a “natural equilibrium” benchmark (point N in Fig. 1). When the unemployment rate is at the natural level, \(x_{t}^{*}=x_{n}\), \(r_{t}^{*}=r_{n}\), \(\pi _{t}^{*}= 0\) and \(y_{t}^{*}= 0\). The natural benchmark, however, can be reached only by a fluke, i.e., when the average EFP reaches a threshold \(\hat f=(x_{2}-x_{n})/x_{3}\).
Fig. 1

Macroeconomic equilibrium: N: natural equilibrium benchmark, the EFP equals the threshold \(\hat f\) and both the employment rate and the interest rate are on the natural level (and both inflation and output gap are on target). A: temporary equilibrium given \(f_{1}<\hat f\) both the employment rate and the interest rate are above the natural levels (and both inflation and output gap are above the target). L: long run equilibrium, the figure shows a situation in that f1>fL, hence both the employment rate and the interest rate in the long run equilibrium are higher than in the temporary equilibrium in A (and both inflation and the output gap in the long run are above the temporary equilibrium levels)

If average EFP is greater than the threshold (\(f_{t}>\hat f\)), then the economy experiences an episode of recession, with \(x_{t}^{*}<x_{n}\), \(r_{t}^{*}<r_{n}\), \(\pi _{t}^{*}<0\) and \(y_{t}^{*}<0\). If the average EFP is smaller than the threshold, an expansionary episode occurs characterized by \(x_{t}^{*}>x_{n}\), \(r_{t}^{*}>r_{n}\), \(\pi _{t}^{*}>0\) and \(y_{t}^{*}>0\). In Fig. 1, we represent the IS curve in t =  1 given the average external finance premium f1, which in turn depends on the cross-sectional mean A0 and variance V0 (not shown). By construction, the EFP in t =  1 is smaller than the threshold EFP (\(f_{1}<\hat f\)). Hence, the employment rate and the interest rate are greater than their natural counterparts, inflation and output gap are positive (point A in Fig. 1).18

As mentioned above, equilibrium is temporary because the equilibrium values of the endogenous variables are pinned down to the average EFP, that is time varying. In fact, the average EFP is a non linear function of the moments of the distribution of net worth in t −  1, as shown by Eq. 2.12. In a sense, the triple (\(x_{t}^{*}, r_{t}^{*}, \pi _{t}^{*}\)) can be thought as a “frame” that allows one to visualize the macro-economy in each period t. As time goes by, a frame in t +  1 follows the frame in t, and so on. The entire “movie” is shown. The distribution of the firms’ net worth changes and so does the average EFP and x and r and π (and y), which depend on the EFP.

As shown by the simulations described in Section 5.2, the process is ergodic.19 In this case, at the end of the adjustment process (at the end of the movie) there will be a long run distribution of the firms’ net worth. Let’s denote with AL, VL the moments of the long run distribution. AL (respectively VL) is obtained by averaging (over a sufficiently long time interval) the cross-sectional means (variances) of the distribution recorded in the time units belonging to that interval. The long run macroeconomic equilibrium (the final frame of the movie , represented by point L in Fig. 1) therefore will be:
$$\begin{array}{@{}rcl@{}} x^{L}&=&x_{2}- x_{3} f^{L} \end{array} $$
(4.7)
$$\begin{array}{@{}rcl@{}} r^{L}&=&r_{n}+\beta_{x}[\eta(x_{2}- x_{3} f^{L})-1] \end{array} $$
(4.8)
$$\begin{array}{@{}rcl@{}} \pi^{L}&=&\theta[\eta(x_{2}- x_{3} f^{L})-1] \end{array} $$
(4.9)
where
$$ f^{L}=\frac{\alpha}{A^{L}}+\frac{\alpha V^{L}}{({A^{L}})^{3}} $$
(4.10)

Notice that long run equilibrium values are undated. This is due to the fact that, in the long run, the economy has settled in a sort of “statistical steady state”. In Fig. 1, the long run position is denoted by L. The graph is built on the assumption that, after period 1, the cross-sectional mean and variance evolve in such a way as to determine a long run EFP, which is smaller than f1.

5 The agent based model

In the previous sections, we have presented the main building blocks of our model and computed the macroeconomic equilibrium. We have also qualitatively analyzed the impact of changes of the average EFP on the macroeconomy. In particular, we know that, as the net worth distribution evolves over time, this in turn affects the average EFP and therefore the endogenous macro variables, i.e., the output gap, the interest rate and inflation. We have, therefore micro to macro externalities. At the same time as the risk free interest rate changes over time, it in turn affects the individual interest rate on loans charged by the banks to firms. Thus changing firms’ ability to access the credit market. We have indeed macro to micro externalities. In the following section, we go back to the microfoundations of the model and focus on firms in order to develop an agent based model that describes the evolution over time of the key variable in our model, namely, firms’ net worth, and to study the quantitative impact of the externalities described above.

5.1 The law of motion of individual net worth

At the end of period t, once goods have been sold and profits realized, the i-th firm decides on dividends and net worth accumulation. For simplicity, we assume that a fraction δ of the net worth of the previous period Ait− 1 will be paid out as dividends to shareholders, while the firm devotes retained profits to the accumulation of net worth. By definition, therefore, net worth in period t is Ait = (1 − δ)Ait− 1 + πit.20 Plugging (2.3) in the expression above, we get:
$$\begin{array}{@{}rcl@{}} A_{it}&=(1-\delta)A_{it-1}+\left( u_{it}\nu-w\frac{\nu}{\lambda}-r_{it} \right)K_{it}-\frac{\omega}{2}K_{it}^{2} \end{array} $$
(5.1)
Substituting (2.5) for the optimal level of investment, rearranging and simplifying, we get:
$$\begin{array}{@{}rcl@{}} A_{it}&=&(1-\delta)A_{it-1}+\nu \left( u_{it}-\frac{w}{\lambda}\right)\frac{\gamma -r_{it}}{\omega}+\frac{r^{2}_{it}-\gamma^{2}}{2 \omega} \end{array} $$
(5.2)
There is only one endogenous variable in Eq. 5.2, namely, the individual cost of credit rit. It is easy to see that an increase of the individual interest rate has a negative impact on net worth if \(\nu \left (u_{it}-\frac {w}{\lambda }\right )>0\), i.e. if realized revenue per unit of capital (νuit) is higher than labor cost per unit of capital (\(w\frac {\nu }{\lambda }\)).
The individual cost of credit is the sum of the risk free interest rate (controlled by the central bank) and the individual EFP. Taking into account the solution for the equilibrium real interest rate (see Eq. 4.3), we get
$$ r_{it}=r^{*}_{t}+f_{it}=r_{n}+\beta_{x}[\eta(x_{2}- x_{3} f_{t})-1]+f_{it} $$
(5.3)

In the end, therefore, the cost of credit for the firm depends both on the individual and on the average EFP. While an increase of the individual EFP pushes up the individual cost of credit, an increase of the average EFP affects negatively the individual cost of credit by depressing the equilibrium interest rate. This sounds strange but is perfectly understandable, given the context. Other things being equal, in fact, an increase of the average EFP pushes down both the output gap and inflation. Hence the equilibrium interest rate (governed by the Taylor rule) goes down and brings down the individual cost of credit.

Notice now that ft is a function of the cross-sectional mean and variance of the distribution of net worth (see Eq. 2.12) and fit is a function of the individual net worth (see Eq. 2.8). Plugging these equations into (5.3) and rearranging, we get:
$$ r_{it}=r_{0}-r_{1} \left( \frac{\alpha}{A_{t-1}}+\frac{\alpha V_{t-1}}{A_{t-1}^{3}}\right)+\frac{\alpha}{A_{it-1}} $$
(5.4)
where r0 := rn + βx(ηx2 − 1) and r1 := βxηx3. Substituting (5.4) for rit in Eq. 5.2, we conclude that the law of motion of individual net worth is a first-order non-linear difference equation and is affected (in a non linear way) by the mean and the variance of the distribution of net worth. Moreover, it is subject to an idiosyncratic shock. We can succinctly represent this law of motion as follows:
$$ A_{it}=f(A_{it-1},A_{t-1},V_{t-1},u_{it})\\ i = 1,2,...n_{F} $$
(5.5)
Since there is one law for each firm, in the end we have a system of nF non-linear difference equations subject to idiosyncratic shocks.

Given the complexity of the system, it is not possible to compute a closed form solution, and so we need to build a simple Agent Based Model (ABM) and make use of computer simulations to assess the dynamic properties of the economy we are investigating. Notice that the dynamics of net worth is constrained by a lower bound: when Ait reaches \(A^{min}_{t}\) , in fact, the firm exits. The ABM will incorporate this condition. The bankrupt firm will be replaced by a new (entrant) firm with a pre-specified endowment (initial net worth). In the following subsection, we will describe the baseline scenario of the ABM.

5.2 The baseline scenario

We consider an economy populated by H people (household members), of which L are active on the labor market, and nF firms over a time span of T =  1500 periods (the time scale can be thought of as a quarter).

In Table 1 we report the configuration of the 18 parameters in the baseline scenario.
Table 1

Parameters’ value in numerical simulations

Variable

Symbol

Numerical Value

Population

H

2000

Labor force

L

1000

Number of firms

n F

200

Technology: productivity of capital

ν

0.3

Technology: productivity of labor

λ

2

Capital adjustment cost coefficient

ω

0.05

Wage rate

w

1

Unemployment subsidy

σ

0.7

Propensity to consume

c

0.8

Per-capita consumption inactive population

\(\bar c\)

0.3

Natural interest rate

r n

0.0025

Taylor rule: output gap coefficient

α x

0.5

Taylor rule: inflation coefficient

π x

0.5

Phillips Curve: slope

θ

0.04

Natural employment rate

x n

0.96

External finance premium coefficient

α

0.04

Dividend yeld

δ

0.1

Government expenditure

G

400

The first three exogenous variables are calibrated in order to obtain a participation rate (L/H) of 50% and a ratio of firms to active population of 1/5. These figures are roughly in line with the Italian case.

Since technology is linear, by definition the productivity of capital is the reciprocal of the capital/output ratio, that we calibrate – following a well known Kaldorian stylized fact – approximately equal to 3. Hence ν =  0.3.

We normalize the real wage to unity w =  1. The calibration of the productivity of labor (λ =  2) coupled with the normalized real wage, implies a unit labor cost in real terms equal to 1/2. In our setting this is also the fraction of GDP that goes to wages. Therefore gross profits represents 50% of GDP and cover the aggregate cost of credit and adjustment costs. As to capital adjustment costs, we don’t have priors on the parameter ω, that we therefore calibrate in order to obtain a “plausible” level of aggregate investment, more on this later.

We assume a relatively generous unemployment subsidy (70% of the real wage). The value of the propensity to consume out of income for the active population is standard (c =  0.8). Exogenous aggregate consumption is the sum of the consumption of active but unemployed people and of consumption of the inactive population. Consumption of the unemployed is cσL =  560. From the definition and our parametrization, the consumption of the inactive population is \(\bar c (H-L)= 300\). Hence total exogenous consumption is \(\bar C=c\sigma L+\bar c (H-L)= 860\). Government expenditure is also exogenous. In the baseline calibration, it is set at G =  400.

The natural interest rate rn, that is usually associated with the rate of time preference, is set at 0.25% per quarter, i.e., slightly more than 1% per year. The coefficients of the output gap and of the inflation gap in the Taylor rule are the same as in Taylor’s path-breaking paper.21

In our calibration, the Phillips curve is relatively “flat” (θ =  0.04) to replicate recent estimates. The natural unemployment rate is implicitly set at 4% so that xn =  0.96.

The coefficient α, which shows up in the definition of the external finance premium, is relatively low (α =  0.04), in order to keep the latter within acceptable bounds.

We run simulations of the ABM in order to generate, in each time period, a distribution of firms’ net worth. Therefore, in period t we can compute the cross-sectional mean At and variance Vt. This determines also the average EFP in t + 1: ft+ 1. Plugging these numbers into the reduced form of the model (Eqs. 4.24.34.4), we get the employment rate, inflation and the interest rate in t + 1. Using \(r^{*}_{t + 1}\) and Ait, from Eq. 5.2 we get Ait+ 1 for each firm. We are now able to compute the cross-sectional mean At+ 1 and variance Vt+ 1. The iterative procedure is replicated in each period of the horizon of the simulation. We, therefore, can retrieve the time series of the cross-sectional mean and variance of the net worth over 1500 periods, and the associated values for the employment rate, inflation and the interest rate.

Notice that a certain fraction of the population of firms will go out of business every period and will be replaced. Exits and replacements, of course, will impact on the shape of the distribution of net worth, and thus on the moments and the average EFP.

In Fig. 2 we show the results of Montecarlo simulations for the benchmark scenario. We run 1000 independent simulations. In the panels, we show the HP filtered time series of (a): the employment rate, (b): mean and variance of the distribution of net worth, (c): interest rate, (d): average EFP, (e): inflation rate, (f): number of bankruptcies.
Fig. 2

The benchmark scenario: Monte Carlo simulations. Trend component of HP-filtered time series (solid lines) and long run averages (dotted horizontal lines) of the following variables: a: employment rate; b1: cross-sectional mean of net worth; b2: cross-sectional variance of net worth; c: interest rate; d: average EFP; e: inflation rate; f: number of exits. Light gray area: 95% confidence Interval, Dark gray area: mean ± one standard deviation

Each time series fluctuates irregularly around a long run mean or statistical equilibrium (represented in Fig. 2 by a horizontal dashed line).

Comparing panels (a)(c)(e), notice that inflation and the interest rate are perfectly correlated with the employment rate. This is not suprising because, by construction, inflation and the interest rate are increasing linear functions of the employment rate, (see Eqs. 3.2 and 3.4, respectively). Moreover, comparing panels (a) and (f) we see that the employment rate is negatively correlated whit the average EFP. This is also not surprising because, in equilibrium, the employment rate is a decreasing linear function of the average EFP (see Eq. 4.2).

From panel (b), we can infer that the cross-sectional mean and variance of the distribution of net worth are positively correlated. This conclusion can be reached only by means of the ABM, that, in our setting, is in charge of tracking the evolution of the distribution over time. Therefore when, on average, firms become more financially fragile, the dispersion of firms increases (and vice versa). The correlated dynamics of the cross-sectional mean and variance have opposite effects on the average EFP: an increase of the cross-sectional mean pushes the EFP down, while the associated increase in the variance pushes the EFP up. From panel (d), however, we conclude that the effect of the mean prevails: when the cross-sectional mean increases, the average EFP generally decreases. The increase in variance has a positive effect on average EFP but this is offset by the negative effect of the increase of the cross-sectional mean.

Finally, from panel (f) we infer that the number of bankruptcies is positively correlated with the average EFP (and therefore negatively correlated with the cross-sectional mean).

Notice that the cross-sectional mean and variance of the distribution also fluctuate around a long run average. Therefore there is a long run distribution of net worth. We can compute the long run averages of the cross-sectional mean and variance as well as those of the finance premiun, employment rate, interest rate and inflation and consider them as a good approximation of the long run equilibrium around that the economy fluctuates. The long run averages of the variables of interest (denoted by the apex L) are shown in Table 2.22 The symbols have the usual meaning. We also report the coefficient of variation, i.e., the ratio of the long run standard deviation to the long run mean of net worth (over the chosen time span), denoted by cvL, and the long run mean of the fraction of firms that go bankrupt, denoted by Av.bank.
Table 2

Long run averages

A L

V L

f L

x L

r L

π L

cv L

Av. bank.

1.6545

0.7694

0.031

0.9701

0.0071

0.00036

0.5302

3.2%

In the long run, the employment rate is xL =  0.97, higher than the natural employment rate. Therefore, the current unemployment rate is lower than the natural unemployment rate and inflation is positive (albeit very close to zero). The real interest rate is greater than the natural interest rate, being set at 0.7% per quarter. The coefficient of variation is slightly higher than 50%. The number of bankruptcies is low (3.2%).

Given our parameterization and the long run employment rate, in the long run GDP is YL = λxLL =  1940. Total consumption is \(C^{L}=c(w-\sigma )x^{L} L+\bar C = 1093\). Hence consumption represents approximately 57% of GDP. Government expenditure in the baseline scenario is 21% of GDP. Hence investment is 19% of GDP. These features are not too far from reality (taking also into account that there is no net foreign demand).

In the next three sections, we will show the effects of a fiscal shock (namely, a permanent increase of Government expenditure), a monetary shock (a permanent increase of the natural interest rate) and a financial shock (a permanent increase of the α parameter in the definition of the EFP). We will focus on the effects on the employment rate (and, therefore, on the output gap). As we said, in fact, inflation and the interest rate are increasing linear transformations of the employment rate.

6 A fiscal shock

In this section, we explore the effects of an expansionary permanent fiscal policy shock captured by an increase of Government expenditure from G0 =  400 (as in the baseline scenario) in t0 =  599 to G1 =  450 from t0 + 1 = 600 on. The other parameters remain unchanged.

The results are shown in Fig. 3. Long run averages of the variables of interest after the shock are reported in Table 3 and compared with the values obtained in the baseline scenario. Arrows denote the direction of change of the variables generated by the shock.
Fig. 3

The effects of a fiscal shock: Monte Carlo simulations. Trend component of HP-filtered time series (solid lines) and long run averages (dotted horizontal lines) of the following variables: a: employment rate; b1: cross-sectional mean of net worth; b2: cross-sectional variance of net worth; c: interest rate; d: average EFP; e: inflation rate; f: number of exits. Light gray area: 95% confidence Interval, Dark gray area: mean ± one standard deviation

Table 3

Long run averages

 

A L

V L

f L

x L

r L

π L

cv L

Av. bank.

G0 = 400

1.6545

0.7694

0.031

0.9701

0.0071

0.00036

0.5302

3.2%

G1 = 450

1.5372 ↓

0.6605 ↓

0.033 ↑

0.9804 ↑

0.0127 ↑

0.00079 ↑

0.5287 ↓

3.6% ↑

A change in regime occurs in t0 +  1 = 600. The employment rate, that fluctuated around 97% before the shock, jumps up and starts fluctuating around 98%. Therefore GDP increases by 1% approximately as a consequence of an increase of Government expenditure of 12.5%. The inflation rate and the interest rate move in the same direction. The increase of the interest rate is sizable: from 0.7% to 1.3%. It affects negatively the accumulation of individual net worth, and so the distribution changes also. Both the long run cross-sectional mean and variance of net worth go down. The net effect of this change in the moments of the distribution is an increase of the EFP, from 3.1% to 3.3%. The tendency of the EFP to increase due to the reduction of the average net worth is mitigated by the reduction of the cross-sectional variance. Finally, as the EFP increases, bankruptcies increase as well.

In the following, we describe the transmission mechanism of the shock. In order to understand the rationale behind it, let’s start from the definition of the fiscal multiplier, i.e., the derivative of the long run GDP with respect to G: \(\frac {dY^{L}}{dG}=\lambda L \frac {dx^{L}}{dG}\). The fiscal multiplier is proportional to the change in the long run employment rate due to a change in G. From the reduced form Eq. 4.2, we infer:
$$ dx^{L}=\frac{dx_{2}}{dG}dG-x_{3}\frac{df^{L}}{dG}dG $$
(6.1)
The first term in Eq. 6.1 is the first round or direct effect of a fiscal shock, that can be computed analytically. In fact
$$ \frac{dx_{2}}{dG}=\frac{1}{1+x_{1} \beta_{x} \eta} \frac{dx_{0}}{dG} $$
(6.2)
where
$$ \frac{dx_{0}}{dG} =\frac{1}{[\lambda-c(w-\sigma)]L} $$
(6.3)
The first round effect is therefore unambiguously positive. In the following, we will write:
$$ {\Delta} x_{1st} =m_{G} {\Delta} G $$
(6.4)
where
$$ m_{G}:=\frac{dx_{2}}{dG}=\frac{1}{(1+x_{1} \beta_{x} \eta)[\lambda-c(w-\sigma)]L} $$
(6.5)

With our parameterization, it is easy to see that mG =  2.549 × 10− 4 so that Δx1st =  2.549 × 10− 4 × 50 = 0, 0127. Following the shock, if the distribution of net worth remained unchanged, the employment rate would increase approximately by 1.3 percentage points and the increase of GDP would be ΔY1st =  25.

The second term in Eq. 6.1 is the second round or indirect effect, that depends on the response of the EFP to changes in the cross-sectional mean and variance. This response will be provided by the emergent properties of the ABM. In fact we can infer from the simulations that the cross-sectional mean and variance go down. The first effect prevails so that the average EFP goes up: the indirect effect is negative.

We can quantify these effects using the long run averages of simulated data before and after the fiscal policy shock shown in Table 3.

As a consequence of the fiscal policy shock (ΔG =  50), the increase of the long run employment rate is Δx ≈  0.01. Therefore, overall GDP increases by ΔY =  0.01 × 2000 = 20.

Starting from the reduced form of the model we can decompose the effect of an increase of Government expenditure on the employment rate as follows:
$$ {\Delta} x=\underset{\Delta x_{1st}}{\underbrace{m_{G}{\Delta} G}}\underset{\Delta x_{2nd}}{\underbrace{-x_{3} {\Delta} f}} $$
(6.6)
The second round effect depends on the change in the average EFP. It is easy to decompose the change in average EFP as follows:
$$ {\Delta} f=\underset{\Delta f_{RA}}{\underbrace{\frac{\alpha }{A_{1}}-\frac{ \alpha }{A_{0}}}}+\underset{\Delta f_{HA}}{\underbrace{\frac{\alpha V_{1}}{\left( A_{1}\right)^{3}}-\frac{\alpha V_{0}}{\left( A_{0}\right)^{3}}}} $$
(6.7)
The first component of Eq. 6.7 is the change in EFP, which would occur in a Representative Agent world where higher moments of the distribution are irrelevant; hence the definition ΔfRA. The second component incorporates also the change in the second moment of the distribution, that is playing a role when heterogeneity is taken duly into consideration; hence the label ΔfHA where HA stands for Heterogeneous Agents. Substituting (6.7) into (6.6) we get:
$$ {\Delta} x=\underset{\Delta x_{1st}}{\underbrace{m_{G}{\Delta} G}}\underset{\Delta x_{2nd,RA}}{\underbrace{-x_{3}\left( \frac{\alpha }{A_{1}}-\frac{ \alpha }{A_{0}}\right) }}\underset{\Delta x_{2nd,HA}}{\underbrace{ -x_{3}\left[ \frac{\alpha V_{1}}{\left( A_{1}\right)^{3}}-\frac{\alpha V_{0}}{\left( A_{0}\right)^{3}}\right] }} $$
(6.8)
where A0 =  1.6545 is the long run average of the cross-sectional mean before the shock (i.e. up to period t0 =  599) and A1 =  1.5372 is the long run average of the cross-sectional mean after the shock, i.e., between t0 + 1 = 600 and t1 =  1500. A similar notation applies to the variance.
Given the numerical values of the parameters and of the long run averages moments of the distribution before and after the shock, we can assess quantitatively the size of each effect:
  • The first round effect is positive and sizable: Δx1st =  0.0127;

  • The RA component of the second round effect is negative, since the steady state cross-sectional mean decreases: Δx2nd, RA = − 0.0019;

  • The HA component of the second round effect is also negative: Δx2nd, HA = − 4.8748 × 10− 4.

The last result may sound strange. After all, the variance has gone down, contributing to depress the EFP. Hence one would expect the HA component of the change in EFP to be negative and the HA component of the change in x to be positive. Notice, however, that a reduction of the long term variance (i.e. V1 < V0) that parallels a reduction of the long term cross-sectional mean (A1 < A0), makes the HA component of the change in EFP negative only if \(\frac {V_{1}}{V_{0}}<\frac {{A^{3}_{1}}}{{A^{3}_{0}}}\). This condition is not satisfied in our case.

The first round effect explains most of the change in the employment rate, being one order of magnitude bigger than the indirect effect. The direct effect is driven by the size of the shock: a 20% increase of government expenditure. The HA component of the second round effect represents about 20% of the entire second round effect.

The transmission mechanism of a fiscal shock can be graphically represented as in Fig. 4. Suppose initially the system is in long run equilibrium L0. An increase of Government expenditure (G1 > G0) makes the IS curve shift up along the upward sloping TR curve so that a new equilibrium M will be reached characterized by a higher x and r. Since x > xn inflation has also increased. Comparing L0 to M, one gets an idea of the first round effect of the fiscal shock. The fiscal shock generates a crowding out effect due to the monetary tightening engineered by the central bank: when x and π increase, the central bank reacts by increasing r, as shown by the Taylor rule. As a consequence, investment shrinks and output goes up less than in the absence of monetary tightening.23 As in all Keynesian models, the expansionary effect of an increase of Government expenditure is bigger in size than the standard crowding out effect.
Fig. 4

Temporary and long run equilibria: the transmission mechanism of a fiscal policy shock

The novelty of the present model is the indirect effect of the fiscal shock that is due to the impact of the increase of the interest rate on the distribution of net worth. The increase of the interest rate, in fact, hits the accumulation of net worth for each and every surviving firm. Moreover, the number of bankruptcies increases (as shown in panel (f) of Fig. 3) because some firms that were already on the verge of bankruptcy end up with a net worth below the exit threshold. These effects make the cross sectional mean of net worth A to decrease, pushing up the average EFP f. At the same time, however, the variance decreases because the probability mass of the distribution moves towards the exit threshold.24 The reduction of the long run variance tends to reduce f. All in all, however, the external finance premium goes up (from \({f_{0}^{L}}\) to \({f_{1}^{L}}\)) because the impact of the reduction of A on f is greater than the effect of the reduction of V. The increase of the average EFP makes the IS curve shift down. We can conclude that the positive impact on the employment rate of an increase in Government expenditure is mitigated by the “second round crowding out effect” following the increase of the EFP.25 In the graphical representation of the model, the second round effect is captured by comparing L1 to M.

7 A monetary shock

In this section, we explore the effects of a contractionary permanent monetary policy shock captured by an increase of the exogenous component of the Taylor rule, i.e., the natural rate, from \({r^{0}_{n}}= 0.0025\) (as in the baseline scenario) before t0 =  599 to \({r^{1}_{n}}= 0.005\) from t0 + 1 = 600 on. The results are shown in Fig. 5. Long run averages of the variables of interest after the shock are reported in Table 4 and compared with the values obtained in the baseline scenario.
Table 4

Long run averages

 

A L

V L

f L

x L

r L

π L

cv L

Av. ban.

\({r^{0}_{n}}= 0.0025\)

1.6545

0.7694

0.031

0.9701

0.0071

0.00036

0.5302

3.2%

\({r^{1}_{n}}= 0.0050\)

1.6353 ↓

0.7537 ↓

0.0314 ↑

0.9672 ↓

0.008 ↑

0.00023 ↓

0.5309 ↑

3.3% ↑

As a consequence of the shock, in t0 + 1 = 600, the interest rate goes up while the employment rate goes down and starts fluctuating around a lower long run mean. The inflation rate moves in the same direction. The distribution changes also. The long run cross-sectional mean and variance of net worth both go down. The EFP increases, meaning that the impact of the decrease of the cross-sectional mean on the EFP prevails on the impact of the decrease of the cross-sectional variance.
Fig. 5

The effects of a monetary shock: Monte Carlo simulations. Trend component of HP-filtered time series (solid lines) and long run averages (dotted horizontal lines) of the following variables: a: employment rate; b1: cross-sectional mean of net worth; b2: cross-sectional variance of net worth; c: interest rate; d: average EFP; e: inflation rate; f: number of exits. Light gray area: 95% confidence Interval, Dark gray area: mean ± one standard deviation

In order to understand the rationale behind the monetary transmission mechanism, let’s start from the derivative of the long run GDP with respect to rn: \(\frac {dY^{L}}{dr_{n}}=\lambda L \frac {dx^{L}}{dr_{n}}\). From Eq. 4.2, we infer that the change in the long run employment rate due to a change in rn is:
$$ dx^{L}=\frac{dx_{2}}{dr_{n}}dr_{n}-x_{3}\frac{df^{L}}{dr_{n}}dr_{n} $$
(7.1)
The first term in Eq. 7.1 is the first round or direct effect of a monetary shock, that can be computed analytically. In fact,
$$ \frac{dx_{2}}{dr_{n}}=-\frac{x_{1}}{1+x_{1} \beta_{x} \eta}=-x_{3}. $$
(7.2)
The first round effect is therefore unambiguously negative. In the following, we will write
$$ {\Delta} x_{1st} =m_{r} {\Delta} r_{n} $$
(7.3)
where
$$ m_{r}:=-x_{3} $$
(7.4)
With our parameterization, it is easy to see that mr = − 1.0196 so that Δx1st = − 1.0196 × 0.0025 = − 0.0026. Following the shock, if the distribution of net worth remained unchanged, the employment rate would decrease by 0.26 percentage points and the decrease of GDP would be ΔY1st = − 5.1.

The second term in Eq. 7.1 is the second round or indirect effect, that depends on the response of the EFP to changes in the cross-sectional mean and variance. We can infer from the simulations that the cross-sectional mean and variance go down and the average EFP goes up.

In order to quantify these effects, we proceed as in the previous section using the long run averages from simulated data before and after the monetary policy shock shown in Table 4.

From the long run values we notice that the employment rate decreases slightly (Δx = − 0.0029) as a consequence of the shock (\({\Delta } r_{n}={r^{1}_{n}}-{r^{0}_{n}}= 0.0025\)). We can break down this change as follows:
$$ {\Delta} x=\underset{\Delta x_{1st}}{\underbrace{m_{r}{\Delta} r_{n}}}\underset{\Delta x_{2nd,RA}}{\underbrace{-x_{3}\left( \frac{\alpha }{A_{1}}-\frac{ \alpha }{A_{0}}\right) }}\underset{\Delta x_{2nd,HA}}{\underbrace{ -x_{3}\left[ \frac{\alpha V_{1}}{\left( A_{1}\right)^{3}}-\frac{\alpha V_{0}}{\left( A_{0}\right)^{3}}\right] }} $$
(7.5)
Given the numerical values of the parameters, we can conclude that an increase of the natural interest rate Δrn =  0.0025 leads to the following effects:
  • A negative direct effect (Δx1st = − 0.0026);

  • A negative indirect RA effect (Δx2nd, RA = − 2.8941 × 10− 4) due to the decrease of the cross-sectional mean;

  • A negative indirect HA effect (Δx2nd, HA = − 1.0050 × 10− 4).

The first round effect explains most of the change in the employment rate. The second round effect is one order of magnitude smaller than the first round effect. The HA component of the second round effect is of the same order of magnitude of the RA effect and represents about 26% of the entire second round effect.

The transmission mechanism of a monetary shock can be represented as in Fig. 6. An increase of the natural rate of interest (\({r^{1}_{n}}>{r^{0}_{n}}\)) makes the TR curve shift up along the downward sloping IS curve so that a new equilibrium M will be reached, characterized by a lower x and higher r. Inflation goes down as well, but it is still positive (since x > xn even after the shock). Comparing L0 to M, one gets an idea of the first round effect of the monetary shock. The increase of the interest rate has a depressing effect on investment.
Fig. 6

Temporary and long run average equilibria: the transmission mechanism of a monetary policy shock

The increase of the interest rate, however, has a second round effect. The long run cross sectional mean A decreases inducing an increase of the average EFP f. This effect is mitigated (but not offset) by the reduction of V. The increase of average EFP has an additional adverse effect on investment and the employment rate. Graphically, the IS curve shifts down along the new TR curve. The final equilibrium position is L1. All in all, we can conclude that the negative impact on the employment rate (and output gap) of a contractionary monetary policy is exacerbated by the increase of the EFP f. In the graphical representation of the model, the second round effect is captured comparing M to L1.

8 A financial shock

In this section we explore the effects of a contractionary permanent financial shock captured by an increase of the exogenous component of the EFP from α0 =  0.04 until t0 =  599 to α1 =  0.05 from t0 +  1 = 600 on.

The results are shown in Fig. 7 and Table 5. As a consequence of the shock, in t0 + 1 = 600 the employment rate goes down and starts fluctuating around a lower long run mean. The inflation rate and the interest rate go in the same direction. The distribution changes as well. The long run level of both the cross-sectional mean and the variance of net worth go down. The EFP goes up and so do bankruptcies. This indirect effect exacerbates the negative repercussions of the financial shock on the employment rate and inflation.
Fig. 7

The effects of a financial shock: Monte Carlo simulations. Trend component of HP-filtered time series (solid lines) and long run averages (dotted horizontal lines) of the following variables: a: employment rate; b1: cross-sectional mean of net worth; b2: cross-sectional variance of net worth; c: interest rate; d: average EFP; e: inflation rate; f: number of exits. Light gray area: 95% confidence Interval, Dark gray area: mean ± one standard deviation

Table 5

Steady state values

 

A L

V L

f L

x L

r L

π L

cv L

Av. bank.

α = 0.04

1.6545

0.7694

0.031

0.9701

0.0071

0.00036

0.5302

3.2%

α = 0.05

1.6411 ↓

0.7626 ↓

0.0391 ↑

0.9618 ↓

0.0027 ↓

0.000001 ↓

0.5321 ↑

3.7% ↑

The long run employment rate goes down from approximately 97% (with α0 =  0.04) to 96% (with α1 =  0.05): Δx = − 0.008.

From the reduced form Eq. 4.2, noticing that the parameter alpha shows up only in the EFP, we obtain:
$$\begin{array}{@{}rcl@{}} {\Delta} x=-x_{3}{\Delta} f \end{array} $$
In the case of a financial shock, differently from the case of changes in fiscal and monetary policy, the shock propagates through changes in the EFP only.26 The EFP, in turn, changes not only because of the change in α, but also because the distribution of firms’ net worth changes. Therefore, we can break down the change in f into two components:
$${\Delta} f=\underset{\Delta f_{1st}}{\underbrace{\left( \alpha_{1}-\alpha_{0} \right) \frac{f_{0}}{\alpha_{0}}}}+\underset{\Delta f_{2nd}}{\underbrace{ \alpha_{1}\left( \frac{f_{1}}{\alpha_{1}}-\frac{f_{0}}{\alpha_{0}}\right) }} $$
The first round effect of the shock on the average EFP Δf1st is computed keeping the mean and the variance unchanged (f0 is a function of A0 and V0). Recalling the definition of EFP we obtain:
$${\Delta} f_{1st}=\left( \alpha_{1}-\alpha_{0} \right) \frac{1}{A_{0}} \left( 1+\frac{V_{0}}{{A_{0}^{3}}}\right) $$
The second round effectΔf2nd takes changes in the distribution into account. It can be decomposed as follows:
$${\Delta} f_{2nd}=\underset{\Delta f_{2nd,RA}}{\underbrace{\alpha_{1}\left( \frac{1}{ A_{1}}-\frac{1}{A_{0}}\right) }}+\underset{\Delta f_{2nd,HA}}{\underbrace{\alpha_{1}\left( \frac{V_{1}}{{A_{1}^{3}}}-\frac{V_{0}}{{A_{0}^{3}}} \right) }} $$
The second round effect can be decomposed in a RA component and a HA component. The interpretation is straightforward.
The effects of the shock on the employment rate are proportional to the effects on the EFP. It is easy to see that:
$$ {\Delta} x=\underset{\Delta x_{1st}}{-\underbrace{x_{3}{\Delta} f_{1st}}}\underset{\Delta x_{2nd,RA}}{-\underbrace{x_{3} {\Delta} f_{2nd,RA} }}-\underset{\Delta x_{2nd,HA}}{\underbrace{x_{3}{\Delta} f_{2nd,HA} }} $$
(8.1)
Given the numerical values of the parameters, we can conclude that an increase of the parameter αα =  0.01), i.e., a contractionary financial shock, leads to the following change of the employment rate:
  • a negative direct effect (Δx1st = − 0.0072);

  • a negative indirect RA effect (Δx2nd, RA = − 2.5159 × 10− 4)

  • a negative indirect HA effect (Δx2nd, HA = − 1.3545 × 10− 4)

As in the case of previous shocks, the first round effect has a predominant role in determining the overall change in the employment rate. The second round effect is one order of magnitude smaller than the first round effect. Both the RA and the HA effect are negative, but the latter is smaller than the former. The HA effect accounts for approximately 35% of the second round effect.

The transmission mechanism of a financial shock can be characterized as follows. As α increases all the firms experience an increase of the cost of credit due to the increase of the individual EFP (fi); thus investment goes down and so does aggregate demand, the employment rate and inflation. Graphically, as shown in Fig. 8, the shock leads to a downward shift of the IS curve due to an exogenous increase of the EFP for each and every firm (in other words, the change in α is an aggregate shock). The new short run equilibrium will be point M. Hence the first round effect will be captured by comparing L0 to M.
Fig. 8

Temporary and long run average equilibria: the transmission mechanism of a financial shock

In M, it is clear that the interest rate has been steered down by the central bank. There are two opposite effects on the individual cost of credit: fi goes up on impact and the interest rate (r) goes down as a consequence of the reaction of the central bank. Notice that we are keeping the distribution unchanged. However, the former effect offsets the latter. In fact we observe that the cross-sectional mean of the cost of credit (represented by the sum r + f) goes up. Moreover, in the period in that α rises, the number of bankruptcies increases (as shown in panel (f) of Fig. 7).

The increase of α leads to a reduction of A, which pushes up f. This tendency is mitigated (but not offset) by the slight reduction in V. The increase in f leads to a further decrease in the employment rate. The second round effect is captured graphically by a further downward shift of the IS curve. The final equilibrium will be point L1. We can conclude that the negative impact on the output gap of an increase in the individual EFP is amplified by the further increase in the average EFP.

9 Conclusion

In this paper, we have pursued further a line of research on Hybrid Macroeconomic ABMs that allows us to resume macroeconomic thinking in a multi-agent context. We consider a population of firm characterized by heterogeneous financial conditions. Each firms chooses the optimal level of investment in the presence of a financial friction. Hence individual investment depends on individual financial robustness captured by net worth. We aggregate individual investment by means of a stochastic procedure that resorts to the first and second moments of the distribution of net worth. Aggregate investment, therefore, will be affected by the interest rate and by the first and second moments of the distribution. We use this behavioral aggregate equation in the context of an IS-AS-TR framework, where the IS curve is augmented by the moments of the distribution. Therefore, in equilibrium, the interest rate, inflation and the employment rate (and output gap) will be functions of the moments mentioned above. The evolution over time of individual net worth turns out to be a function of the cross-sectional mean and variance (through the equilibrium interest rate). We simulate the model to understand the statistical properties of the results. We explore the consequences of three types of shocks. Thanks to our modelling strategy, we are able to disentangle the first round effect of a shock (keeping the distribution unchanged) and the second round effect and to distinguish the specific role played by heterogeneity in the latter. Our results can be summarized as it follows:
  • In all the scenarios considered (fiscal shock, monetary shock, financial shock), the first round effect explains most of the actual change of the output gap.

  • The second round effect amplifies the effect of a contractionary monetary shock and of the financial shock and mitigates the effect of the expansionary fiscal shock. In the latter case, in fact, the financial transmission mechanism contributes to crowding out.

  • In the case of the fiscal and monetary shock, heterogeneity explains about 30% of the second round effect.

  • In the case of the financial shock, the entire second round effect is due to heterogeneity.

The benchmark model lends itself to a wide range of possible extensions, such as the explicit consideration of income and wealth inequality among households.

Footnotes

  1. 1.

    The procedure has already been used, e.g. in Agliari et al. (2000). It is thoroughly discussed and compared with other aggregation procedures in Gallegati et al. (2006) where it is labelled the Variant-Representative-Agent methodology.

  2. 2.

    In the following we show, in fact, that the aggregate EFP is decreasing with the cross-sectional mean of the net worth distribution and increasing with the cross-sectional variance.

  3. 3.

    A similar procedure, within a different setup, is adopted by Krusell and Smith (1998).

  4. 4.

    The moments of the distribution of firms’ net worth, therefore, enter as (predetermined) state variables in the reduced form of the IS-AS-TR model.

  5. 5.

    More precisely, the second round effect is determined by the repercussion of the shock on the average EFP, that in turn is determined by the change of the moments.

  6. 6.

    The time indicator is t =  0, ... T (with T =  1500 in our simulations).

  7. 7.

    Where c[wxt + σ(1 − xt)]ma represents consumption of active members of the household, respectively employed and unemployed. While \(\bar c (m-m_{a})\) represents consumption of unemployed members of the household.

  8. 8.

    Notice that the net worth of the representative firm coincides with the cross-sectional mean when the variance is zero. In other words, the representative firm is the zero-variance average firm.

  9. 9.

    All the parameters must be intended as positive unless otherwise specified.

  10. 10.

    We assume that λ > w so that γ >  0.

  11. 11.

    Equation 2.6 guarantees that individual expected profits are positive; realized profits maybe positive or negative depending on the realization of the idiosyncratic shock uit.

  12. 12.

    This assumption can be thought of as a reduced form for the EFP generated by an optimal debt contract in the presence of a financial friction, as in Bernanke and Gertler (1989, 1990).

  13. 13.

    The specification (2.9) for the individual investment function can be thought of as a simplified version of the specification in AD2013, where the optimization procedure at the firm level involved the probability of bankruptcy in a setting reminiscent of Greenwald and Stiglitz (1993).

  14. 14.
    First, we approximate (2.8) around the cross-sectional mean of net worth At− 1 by means of a second order Taylor expansion:
    $$\begin{array}{@{}rcl@{}} f_{it}&\approx& \frac{\alpha}{A_{t-1}}+\left.\frac{\partial f_{it}}{\partial A_{it-1}}\right\vert_{A_{t-1}}(A_{it-1}-A_{t-1})+\left.\frac{1}{2}\frac{\partial^{2} f_{it}}{\partial A_{it-1}^{2}}\right\vert_{A_{t-1}}(A_{it-1}-A_{t-1})^{2} \end{array} $$
    where \(\left .\frac {\partial f_{it}}{\partial A_{it-1}}\right \vert _{A_{t-1}}=-\frac {\alpha }{A_{t-1}^{2}}\) and \(\left .\frac {\partial ^{2} f_{it}}{\partial A_{it-1}^{2}}\right \vert _{A_{t-1}}=\frac {2\alpha }{A_{t-1}^{3}}\). Substituting these derivatives in the expression above, taking the expected value of the RHS and recalling that < Ait− 1At− 1 >= 0 and < (Ait− 1At− 1)2 >= Vt− 1, we obtain (2.12).
  15. 15.

    Notice that the output gap, as shown in subsection 3.2, is yt = ηxt − 1.

  16. 16.

    The nominal interest rate will be determined residually from Eq. 3.3.

  17. 17.

    These restrictions imply an upper bound on the average EFP, that must be enforced across simulations.

  18. 18.

    From Eq. 4.3 we infer that the interest rate goes down when the EFP goes up. Hence an increase of the EFP has opposite effects on the cross-sectional mean of the individual interest rates: < rit >= rt + ft, which is the sum of the risk free interest rate and of the average EFP. On impact, by construction, the cross-sectional mean of the individual interest rates increases one to one with an increase of the average EFP but the equilibrium risk free rate decreases. that effect prevails? Using the definition of the equilibrium interest rate, we get: < rit >= rnβx(ηx2 − 1) + (1 − βxηx3)ft. Notice that, from the definition of x3 it follows that 1 − βxηx3 >  0. Hence we can conclude that the direct effect prevails: an increase of the EFP unambiguously increases the average cost of credit. Since investment in the aggregate is \(K_{t}=\frac {\gamma - (r_{t}+f_{t})}{\omega }\) (see Eq. 2.11), investment unambiguously declines when the EFP goes up.

  19. 19.

    In the simulations, the cross-sectional mean and variance of the distribution of net worth tend to fluctuate around a lsong run value. Therefore, there is a long run mean and a long run variance of the distribution of net worth. We infer from this that the process is ergodic, meaning that there is a tendency for the distribution to achieve a long run shape captured, in our simple framework, by the long run cross-sectional mean and variance.

  20. 20.

    The assumption on dividend distribution is technical. By assuming that a fraction δ of net worth goes wasted in dividend distribution, we slow down the process of net worth accumulation.

  21. 21.

    In fact we set αx = απ =  0.5. Moreover, the inflation target is set, for simplicity, at πT =  0.

  22. 22.

    Notice that the long run averages are determined as the average of the time series shown in Fig. 2, that in turn are the time series derived from the Monte Carlo simulations.

  23. 23.

    The fiscal multiplier in the absence of monetary tightening would be \(\frac {dx_{0}}{dG}\).

  24. 24.

    The empirical distribution of net worth is endogenously determined (by means of the artificial data generated by the ABM) and evolving over time. It has support (\(A_{t}^{min}\), \(A_{t}^{max}\)) where \(A_{t}^{min}\) is the minimum level of net worth and coincides with the exit threshold (that is the same for all the firms) and \(A_{t}^{max}\) is the maximum level of net worth, i.e., the net worth of the wealthiest firm. The distribution is affected by the fiscal expansion through the effect of the latter on the interest rate rt. Since the interest rate goes up, \(A_{t}^{min}\) will increase. Some firms that would survive if the shock did not occur will go bankrupt. Moreover, profit and net worth of the surviving firms will go down. Hence the first moment of the distribution (the cross-sectional mean) goes down. The probability mass shifts towards the exit threshold. In this case it is perfectly possible that \(A_{t}^{max}\) will go down, i.e., even the wealthiest firm will be poorer. The variance therefore will go down.

  25. 25.

    The first round crowding out effect is due to the increase of the interest rate when we keep the distribution unchanged, as shown by comparing M to L0.

  26. 26.

    Fiscal and monetary parameters are incorporated in x3.

Notes

Acknowledgments

Earlier versions of this paper have been presented at conferences and seminars in Ancona, Roma, Guildford, New York, Castellon, Nice and Berlin. We would like to thank participants for useful comments and discussions. We are also grateful to two anonymous referees and the Editor for their detailed comments on an earlier draft, that have led to significant improvements. None of the above are responsible for errors in this paper.

Funding

The authors declare they have received no funding.

Compliance with Ethical Standards

Conflict of interests

The authors declare that they have no conflict of interest.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Complexity Lab in Economics (CLE), Department of Economics and FinanceUniversità Cattolica del Sacro CuoreMilanoItaly
  2. 2.Amsterdam School of EconomicsUniversity of Amsterdam, CeNDEFAmsterdamNetherlands
  3. 3.CESifo Group MunichMunichGermany

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