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Decisions in Economics and Finance

Erschienen in:

28.04.2020

A general equilibrium evolutionary model with two groups of agents, generating fashion cycle dynamics

verfasst von: Ahmad Naimzada, Marina Pireddu

Erschienen in: Decisions in Economics and Finance | Ausgabe 1/2020

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Abstract

We propose a discrete-time exchange economy evolutionary model with two groups of agents. In our setting the definition of equilibrium depends also on agents’ population shares, which affect the market clearing conditions. We prove that, despite such difference with the classical Walrasian framework, for all economies and population shares there exists at least one equilibrium, and we show that for all population shares, generically in the set of the economies, equilibria are finite and regular. We then introduce the dynamic law governing the evolution of the population shares, and we investigate the existence and the stability of the resulting stationary equilibria. We assume that the reproduction level of a group is related to its attractiveness degree, which depends on the social visibility level, determined by the consumption choices of the agents in that group. The attractiveness of a group is described via a generic bell-shaped map, increasing for low visibility levels, but decreasing when the visibility of the group exceeds a given threshold value, due to a congestion effect. The model is able to reproduce the recurrent dynamic behavior typical of the fashion cycle, presenting booms and busts in the agents’ consumption choices, and in the groups’ attractiveness and population shares.

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Oscillatory dynamics in the consumption activities are found, in different contexts, also in Antoci et al. (2004), Bischi and Radi (2012), Matsumoto (2003) and Naimzada et al. (2013).

Cf. also Gori et al. (2014), where an homotopy method is employed to prove existence of equilibria in a model with real assets and restricted participation.

We refer the reader to Naimzada and Pireddu (2019b) for a more detailed bifurcation analysis, conducted for several parameter configurations and for the formulations of the attractiveness maps introduced in Naimzada and Pireddu (2019a), too.

We keep the notation used in Chang and Stauber (2009) and in Naimzada and Pireddu (2016, 2018, 2019a), where \(i\in \{\alpha ,\beta \}\) was the weight assigned to good x in the Cobb–Douglas utility functions by agents in group i, with \(0<\alpha<\beta <1\).

We notice that, if \(u_{\alpha }=u_{\beta }\), then agents are homogeneous since endowments of both goods coincide between groups. In order not to overburden notation and not to excessively complicate the analysis, we will here focus on the case with \(w_{x,\alpha }=w_{x,\beta }=w_x\) and \(w_{y,\alpha }=w_{y,\beta }=w_y\), implicitly assuming that \(u_{\alpha }\ne u_{\beta }\), like it happens in the example considered in Sect. 3.

We remark that the argument above suggests that market equilibria exist for any economy even when \(a=0\) and \(a=1\), although such extreme cases are not encompassed in Proposition 1 due to the need to deal with open sets because of the differential topology kind of proof. We also notice that considering an open interval of the form \((-\varepsilon , 1+\varepsilon )\), with \(\varepsilon >0\) arbitrarily small, would not solve the issue, as some steps in the proof of Proposition 1 would not work anymore.

Actually, along the proof we show the validity of a stronger result, i.e., that in any time period and for every population share, for each choice of the utility functions in \({\mathcal {U}}\), there exists an open and full measure subset of the space of endowments where the generic regularity property holds. In particular, the fact that the smooth dependence of equilibria holds for all share values—for all utility functions in \({\mathcal {U}}\) and for almost all endowment combinations—is crucial, as we will characterize market stationary equilibria in terms of population shares. Indeed, for our purposes a result more in line with the approach in Chapter 8 in Villanacci et al. (2002), where economies are identified with endowments and the utility function vector is taken as given (see in particular Paragraph 8.7 therein), would suffice. Nonetheless, the smooth dependence of equilibria on utility functions comes as a byproduct of the proof of Proposition 2, which is in line with some recent generic regularity results in the general equilibrium literature (see, e.g., Carosi et al. 2009 and Hoelle et al. 2016). We chose to present such version of the proposition, since we believe that a more general result about generic regularity can be of independent interest.

A more detailed bifurcation analysis, performed for several parameter configurations and for the formulations of the attractiveness maps introduced in Naimzada and Pireddu (2019a), too, can be found in the Appendix of Naimzada and Pireddu (2019b).

We remark that \(V_{i}\), and consequently \({\mathscr {A}}_{i}\), also depend on the economy \(\widetilde{{\mathcal {E}}}\), which together with a determines the market equilibrium. However, in order not to overburden notation, we will not make such dependence explicit.

This happens, for instance, with the parameter configuration considered in Sect. 3, when fixing \(c_i=d_i=0\), for \(i\in \{\alpha ,\beta \}\), \(\mu =2\) and letting \(\sigma >0\) free to vary. See Scenario A in Naimzada and Pireddu (2019b) for the corresponding details.

The Stone–Geary utility functions were derived by Geary (1950) in a comment on an earlier work, while Stone (1954) estimated the Linear Expenditure System, arising from the utility functions in (13).

Although such feature is mentioned in Fisher (1972) for Stone–Geary utility functions when the coefficients \(c_i,\,d_i\) are non-negative, for \(i\in \{\alpha ,\beta \}\), a direct proof using the expression for the individual demand functions in (14) shows that the gross substitute property holds also in the case of negative coefficients, as long as their value is not excessively large in absolute value. See the discussion after (14) for more details.

We here consider polarized values for \(v_x\) and \(v_y\), as both of them are positive but one is much larger than the other. As observed in Sect. 2, such case approximates those frameworks in which visibility and attractiveness are produced by the consumption of a single good. However, our results hold true also for more balanced values of \(v_x\) and \(v_y\).

We remark that the symmetry between the bounds of \(c_i\) and \(d_i\) is caused by the fact that, for the parameter configuration we deal with, it holds that \(w_x=(1-\alpha )w_y\). Of course, such peculiarity does not affect the outcomes. Indeed, in the scenario that we will consider below the parameters \(c_i\) and \(d_i\) will not bear any symmetry.

In regard to the second scenario, in Naimzada and Pireddu (2019b) we deal with \(c_{\alpha }=0.3,\,c_{\beta }=0.12,\,d_{\alpha }=0.1,\,d_{\beta }=0.15\).

See Naimzada and Pireddu (2019c) for the corresponding definition and for the proof, in our framework, of the first fundamental theorem of welfare economics, according to which every (stationary) equilibrium allocation is Pareto optimal. Since the employed concept of Pareto optimality concerns one single period at a time, we will use the expression “instantaneous Pareto optimal allocation” in the present work.

We stress that differently from what done in Villanacci et al. (2002), we normalize the price of the first commodity, rather than of the last one. Of course, this change does not affect the validity of the results.

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Titel: A general equilibrium evolutionary model with two groups of agents, generating fashion cycle dynamics
verfasst von: Ahmad Naimzada
Marina Pireddu
Publikationsdatum: 28.04.2020
Verlag: Springer International Publishing
Erschienen in: Decisions in Economics and Finance / Ausgabe 1/2020
Print ISSN: 1593-8883
Elektronische ISSN: 1129-6569
DOI: https://doi.org/10.1007/s10203-020-00280-0

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A general equilibrium evolutionary model with two groups of agents, generating fashion cycle dynamics

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