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A dynamical model for real economy and finance


We have studied a discrete time dynamical model with four variables and delays, describing the interaction between a three-sector real economy and a financial market with four assets. Investors and financial intermediaries have heterogeneous beliefs. We show that complexity related to the evolution of state variables emerges and we investigate interdependence among economic fluctuations and assets volatility. By means of stability analysis we have found that real economy influences the existence of equilibrium prices in financial markets and that risky asset prices as well as capital per capita reach zero only when the elasticity of substitution between capital and labour is low enough. Bifurcation analysis shows that an increase of bond return would decrease the price of all the assets, conversely when the bond return decreases fluctuations and complex dynamics may arise. Due to the complexity of the model, computational tools are used to investigate long run dynamics, thus showing that for sufficiently high values of the interest rate bifurcations with repetitive structure emerge. In addition, we show how the total number of shares in each sector influences its price volatility. Finally, when fluctuations appear, economic policy intended to increase employment could stabilise the model only in sufficiently developed economies.

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  1. Parametrisation discussed in appendix.


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Proofs of Propositions

Proof of Proposition 1

The fourth equation in system U, given by (18), is verified for \(k=0\). When capital is equal to zero it has \(\pi ^{(n)}=0\), \(\forall n\in \{1,2,3\}\) in the equations of prices (first equation of system T given by (17)). Therefore \(E_0=\left( 0,0,0,0\right) \) is a solution for system (18) for all parameter values and hence it is a fixed point for map T.

Proof of Proposition 2

The first three equations in system (18), i.e. those related to prices, are linear w.r.t. \(p^{(n)}\), therefore, when a positive fixed point \(k^*\) for capital exists, each price in equilibrium is given by

$$\begin{aligned} {p^{(n)}}^*=\frac{\left( u-x^{(n)}_mr^{(1)}r^{(2)}\right) L{\pi ^{(n)}}^*}{2\left[ (R-1)u+x^{(n)}_mr^{(1)}r^{(2)}\right] x^{(n)}_m}\qquad n=\{1,2,3\} \end{aligned}$$

where \({\pi ^{(n)}}^*\) is the profit of sector n computed in \(k^*\). In order to analyse the existence of a positive fixed point for capital consider that fixed points are solutions of

$$\begin{aligned} k=\sum _{n=1}^{3}\left[ \left( 1-\delta ^{(n)}\right) b^{(n)}k+\left( 1-\nu ^{(n)}_t\right) \left( b^{(n)}\right) ^2k^{s^{(n)}}\left( l^{(n)}+b^{(n)}k^{s^{(n)}}\right) ^{\frac{1}{s^{(n)}}-1}\right] \end{aligned}$$

where previous equation is obtained substituting (4) in (5) and eliminating the subscript t from k. Notice that when price are in equilibrium it has \(\nu ^{(n)}_t=.5\), \(\forall n \in \{1,2,3\}\), hence previous inequality may be written as

$$\begin{aligned} k=\sum _{n=1}^{3}b^{(n)}k -\sum _{n=1}^{3}b^{(n)}\delta ^{(n)}k + 0.5\sum _{n=1}^{3} \left( b^{(n)}\right) ^2k^{s^{(n)}}\left( l^{(n)}+b^{(n)}k^{s^{(n)}}\right) ^{\frac{1}{s^{(n)}}-1}\,. \end{aligned}$$

Dividing both the side of the equation for k and recalling that \(\sum _{n=1}^{3}b^{(n)}=1\) the condition reads

$$\begin{aligned} 2\sum _{n=1}^{3}b^{(n)}\delta ^{(n)}=\sum _{n=1}^{3}\left( b^{(n)}\right) ^2k^{s^{(n)}-1}\left( l^{(n)}+b^{(n)}k^{s^{(n)}}\right) ^{\frac{1}{s^{(n)}}-1} \end{aligned}$$

We define \(db=2\sum _{n=1}^{3}b^{(n)}\delta ^{(n)}\) and

$$\begin{aligned} C(k)=\sum _{n=1}^{3}\left( b^{(n)}\right) ^2k^{s^{(n)}-1}\left( l^{(n)}+b^{(n)}k^{s^{(n)}}\right) ^{\frac{1}{s^{(n)}}-1} \end{aligned}$$

so that fixed points for capital are solution of \(db=C(k)\).

Function (30) first derivative is

$$\begin{aligned} C^\prime (k)=\sum _{n=1}^{3}-\frac{l^{(n)}\left( b^{(n)}\right) ^2\left( 1-s^{(n)}\right) }{k^{2-s^{(n)}}\left( l^{(n)}+b^{(n)}k^{s^{(n)}}\right) ^{2-\frac{1}{^{(n)}}}} \end{aligned}$$

and it is negative for all parameters combinations proving that C(k) is strictly decreasing. Being \(db>0\) and \(C^\prime (k)<0\) no more than one fixed point may exist. Two cases may occur:


if all sectors satisfy \(s^{(n)}<0\) it has \(\lim _{k\rightarrow 0^+}C(k)=\sum _{n=1}^{3}\left( b^{(n)}\right) ^{\frac{1}{s^{(n)}+1}}\) and \(\lim _{k\rightarrow +\infty }C(k)=0\) therefore a fixed point exists iff \(db<\sum _{n=1}^{3}\left( b^{(n)}\right) ^{\frac{1}{s^{(n)}+1}}\);


if \(j\in \{1,2,3\}\) sectors satisfy \(s^{(n)}>0\) it has \(\lim _{k\rightarrow 0^+}C(k)=+\infty \) and \(\lim _{k\rightarrow +\infty }C(k)=\sum _{n=1}^{j}\left( b^{(n)}\right) ^{\frac{1}{s^{(n)}+1}}\) therefore a fixed point exists iff \(db>\sum _{n=1}^{j}\left( b^{(n)}\right) ^{\frac{1}{s^{(n)}+1}}\).

When a fixed point \(k^*\) exists, it can be substituted in the equations of prices to find the equilibrium prices as defined by (29).

Proof of Proposition 3

We consider prices in equilibria therefore \(\left( 1-\nu ^{(n)}_t\right) =0.5\). Then the fixed point \(E_0\) is stable if \(\left| {\mathcal {K}}^\prime \left( 0,p\right) \right| <1\). When \(\exists n\in \{1,2,3\}\) s.t. \(s^{(n)}>0\), it has \(\lim _{k_t\rightarrow 0^+}{\mathcal {K}}^\prime =+\infty \). When \(s^{(n)}<0 \forall n\in \{1,2,3\}\), it has \(\lim _{k_t\rightarrow 0^+}{\mathcal {K}}^\prime =\sum _{n=1}^{3}b^{(n)}\left[ 1-\delta ^{(n)}+0.5\left( b^{(n)}\right) ^\frac{1}{s^{(n)}}\right] \) and \(\lim _{k_t\rightarrow 0^+}{\mathcal {K}}^\prime >1\) as long as condition (i) of Proposition 2 holds. In both cases \(k=0\) is an unstable fixed point for the evolution of capital, therefore \(E_0\) is unstable.


The range of values used for simulations are summarized in the following table.

Variable Range Source
\(l^{(n)}\) (0.01, 0.79) OECD
\(b^{(n)}\) (0.01, 0.83) OECD
\(\delta ^{(n)}\) (0.18, 0.61) OECD
L (289, 42.400) OECD
R (.8, 1.3) OECD
\(\sum _{i=1}^3x^{(i)}_m\) (.5, 6) ECB

For the portion of implied labour force in each sector we considered the percentage of employees in the three sectors Agriculture, hunting and foresty/ Industry/ Services (ISIC rev. 4, groups A/B-F/G-U) for European Countries in 2019, the dataset is provided by OECD (Dataset: Employment by activities and status ALFS).

For the portion of implied capital in each sector we considered the percentage of gross capital formation by activity in the three sectors Agriculture, hunting and foresty/ Industry/ Services (ISIC rev. 4, groups A/B-F/G-U) for European Countries in 2019, the dataset is provided by OECD (Dataset: Capital formation by activity ISIC rev 4).

For the depreciation rate of capital, we considered the depreciation rate that express the age-price profile of fixed capital, considering assets for assets from new to 10 years old. Esteems are given in [24].

For the total number of workers we considered the employed population (aged by 15) in 2019 for European Countries, expressed in thousands. The dataset is provided by OECD (Dataset: Short-term labour market statistics).

For total shares \(x_m\), we based on the Statistical Data Warehouse from European Central Bank and selected the listed shares issued by non-financial corporations (ESA 2010 classification) in EU countries, for the period 2015-2020. The total number of issued shares per country is expressed in million.

For the return R we considered the long-term interest rates for Italy, Germany, France and UK in the period 2015-2020. The dataset is provided by OECD (Dataset: Long-term interest rates).

Free parameters: Parameter \(s^{(n)}\) determines the elasticity of substitution between capital and labour: \(\sigma ^{(n)}=\left( 1-s^{(n)}\right) ^{-1}\). As discussed in Knoblach and Stöckl [18], empirical estimates about elasticity of substitution are heterogeneous due to the specification of the estimation equation. Despite the heterogeneity, the range of esteemed values is dense in the set (-2,2). Due to the differences on the esteems, we let \(s^{(n)}\) as free parameter but, throughout simulations, we focuses on cases in which it results \(\sigma \in (-2,2)\).

Parameters \(\alpha ^{(i)}\), \(r^{(i)}\) and \(z^{(i)}\) refer to risk attitude, beliefs over short-term continuation for cum-dividend-prices and overconfidence. They may not be directly esteemed, despite so, higher values refer to higher risk-aversion and stronger beliefs.

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Grassetti, F., Mammana, C. & Michetti, E. A dynamical model for real economy and finance. Math Finan Econ 16, 345–366 (2022).

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  • Economic growth
  • Asset prices
  • Economic dynamics
  • Heterogeneous beliefs
  • General equilibrium model

JEL Classification

  • C61
  • O41
  • G12