Original Articles
Credit market imperfection, financial market globalization, and catastrophic transition☆,☆☆

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Abstract

We analyze a two-country overlapping generations model with integrated financial markets. We assume heterogeneous countries with respect to the population size, to the technology and to the level of credit market imperfection. We show that a subcritical Neimark–Sacker bifurcation may occur and that, before its destabilization, a stable steady state coexists with two invariant closed curves – one attracting and one repelling. In this way we reinforce existing results on the implications of the credit market imperfection that not only causes amplification and persistence of macroeconomic shocks, but also leads to significant changes in the long run behavior of the economy (i.e., catastrophic transition).

Introduction

Over the past two decades a substantial body of research has been developed to investigate macroeconomic implications of credit market imperfection. These studies have demonstrated that credit market imperfection can be responsible for (a) propagation, amplification, and persistence of macroeconomic shocks to fundamentals (as in [7], [9], [12]); (b) the indeterminacy of equilibria, which can lead to endogenous, self-fulfilling, expectations-driven business cycle fluctuations (as in [21], [5], [6], [4]); (c) the persistence of volatility (as in [10], [17], [18]); and (d) the magnification of between and within country income inequality (as in [15], [16]). The main goal of the present paper is to contribute to our understanding by offering new insights about the macroeconomic implications of credit market imperfection.

Kikuchi and Stachurski in [10] have demonstrated that in the presence of credit market imperfection, financial market globalization not only can induce endogenous inequality via symmetry breaking (as shown in [16]), but can also imply persistent endogenous fluctuations. Periodic trajectories arise due to the occurrence of a Neimark–Sacker (NS) bifurcation. If such a bifurcation is of supercritical type it causes the appearance of an attracting closed curve around an unstable steady state. At the beginning the curve is very small, and gradually grows in size when the parameter moves away from the bifurcation value. Such a result implies that the economy loses stability “softly” and a small change in credit market conditions would not drastically affect the macroeconomic dynamics in both countries. Additionally, the situation is “controllable” because the world economy will return to the original path if the level of credit market imperfection is restored in order to re-stabilize the equilibrium.

Our aim is to show that (a) this is not the case in the present model since an abrupt change in the macroeconomic dynamics may occur when the economy loses its stability and (b) the heterogeneity is not a necessary condition to have endogenous fluctuations. We also extend the model proposed and analyzed in [10], [16] by allowing countries to be heterogeneous not only with respect to relative population size but also technology and the level of credit market imperfection. When the countries are homogeneous, the model reduces to the one proposed and analyzed by Matsuyama in [16], while the model reduces to the one studied by Kikuchi and Stachurski in [10] when countries are heterogeneous only with respect to the population size.

The countries are represented by overlapping generations of identical agents living for two consecutive periods. Agents can become either depositors or entrepreneurs. Entrepreneurs can run an indivisible investment project which is characterized by a minimum investment requirement. The credit markets are characterized by the presence of limited pledgability. As a result, depending on the level of entrepreneurial wealth, the equilibrium in each country may become either credit constrained (so that not all agents are able to borrow and run a project even when it is strictly preferable to become an entrepreneur) or profitability constrained (so that agents are indifferent between becoming an entrepreneur or a depositor). These changes of regime, from credit constrained to profitability constrained and vice-versa, may generate aggregate fluctuations – even in the absence of exogenous shocks to fundamentals.

In this paper we strengthen the results obtained in [10], deepening the analysis of the occurrence of the NS bifurcation. In particular, we demonstrate that two closed invariant curves, one attracting and one repelling, may coexist with the stable steady state, and that the repelling one separates the basins of attraction of the two attractors. By varying the values of the parameters, the attracting closed curve moves away from the fixed point while the repelling shrinks and merges with the fixed point at the subcritical NS bifurcation. After such a bifurcation, the trajectories previously converging to the stable fixed point will converge to the attracting closed curve. This closed curve is quite far from the fixed point and a hysteresis phenomenon can be observed (see [1]). Indeed, once the NS bifurcation has occurred, the re-stabilization of the fixed point may not be sufficient to allow the system to come back to the original equilibrium since the trajectory may be too far from it, belonging to the basin of attraction of the stable closed curve. As a result, a microscopic variation of the credit market conditions may trigger an asymptotic regime macroscopically different. The coexistence of a stable cycle and a stable fixed point also implies that a small shock to the system has no effect on its long run behavior. However, a large shock may lead to another attractor and cause a “sharp” change in long run behavior.1 This result reinforces the existing results by demonstrating that credit market imperfection not only causes amplification and persistence of macroeconomic shocks, but also leads to a significant change in the long run behavior of the economy (i.e., catastrophic transition).2

The rest of the paper is organized as follows. Section 2 introduces the model. Section 3 discusses the concept of equilibrium and identifies relationship between state variables under closed and open economy assumptions. Then, in Section 4, we consider the case of two structurally identical countries with possible differences in population size (quasi-symmetric case). We show that the autarkic steady state can become a Milnor attractor (Section 4.1) and perform the local stability analysis of all the fixed points (Section 4.2), providing also a numerical example (Section 4.3). In Sections 5, we study the appearance of endogenous fluctuations, coming back to heterogeneous countries. We show the occurrence of a subcritical NS bifurcation in 5.1 Subcritical Neimark–Sacker bifurcation, 5.2 Homoclinic bifurcations, we describe the mechanism through which an attractive closed invariant curve emerges around the stable steady state. Section 6 summarizes results and concludes. The proofs of all the propositions are in Appendix A.

Section snippets

Model setup

We consider a world economy consisting of two countries, country 1 and country 2. A single consumption commodity is produced by two factors of production: labor, supplied by young agents, and capital, supplied by old agents. The technology of the consumption commodity producing firm is described by a constant return to scale production function. The output per worker in country-j, j = 1, 2, is yjt=Ajkjtα, where Aj > 0 is the multi factor productivity level, α ∈ (0, 1) is the capital share in

Dynamical system

Let ijt denotes the aggregate investment made by old agents per capita at the beginning of period t, so that kjt = Rjijt. The investment in each country will adjust until either borrowing or profitability constraint binds. Since wjt=Aj(1−α)kjtα and ρjt+1=Ajαkjt+1α−1, it follows from (1) and (2) that for a given pair (ijt, rjt+1), investment in country j is

ijt+1=1rjt+11/(1−α)ϕj(ijt)

where

ϕj(ijt)=λj1−(1−α)TjijtααTj1/(1−α)ifijt<I(λj,Tj)(αTj)1/(1−α)ifijt≥I(λj,Tj).

The parameter Tj=AjRjα summarizes the

The quasi-symmetric case

In this section we assume that the two countries are structurally identical, except the population size. Therefore in (9) we set T1 = T2 = T, λ1 = λ2 = λ while the parameter L ranges in (0, 1) and the map we consider in this section is

Ms:x′=(1−α)TLxα+(1−L)yαLϕ(x)+(1−L)ϕ(y)ϕ(x)y′=(1−α)TLxα+(1−L)yαLϕ(x)+(1−L)ϕ(y)ϕ(y)

where

ϕ(i)=λ1−(1−α)TiααT1/(1−α)ifi<I(λ,T)(αT)1/(1−α)ifi≥I(λ,T)

and I(λ, T) : = (1 − λ/(1 − α)T)1/α.

It is straightforward to verify that Ms(x, y ; L) = Ms(y, x ; 1 − L) and that when x = y the map Ms does not

Endogenous fluctuations

As we have seen in Section 4.1, the world economy with two identical countries can fluctuate when the credit market imperfection are sufficiently strong (λ sufficiently low). Nevertheless, we prefer to analyze the model in a more general setting in order to show that even heterogeneity can be associated with complex dynamics. Then in the following we still assume α = 1/3 in the map M given in (9), but we consider different population size, setting L = 0.35. Moreover we introduce different levels of

Concluding remarks

In the paper we analyzed the model proposed in [16] in case when world economy consists of 2 countries. The same attempt has been made in [10], where countries can differ only in their relative population size. The main result obtained in [10] is that endogenous fluctuation of income is possible when countries are heterogeneous with respect to relative population size. In this paper we strengthen this result and obtain that financial integration can cause endogenous fluctuation even when the

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The authors would like to thank two anonymous Referees for their valuable comments and helpful suggestions. This work was supported by a grant from The City University of New York PSC-CUNY Research Award Program, TRADA-45-120 65033-00 43.

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This work has been performed within the activity of the COST Action IS1104 “The EU in the new complex geography of economic systems: models, tools and policy evaluation”.

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