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Aggregate implications of lumpy investment under heterogeneity and uncertainty: a model of collective behavior

Abstract

Recent studies highlight the effect of uncertainty on investment decisions, i.e., the “wait and see” effect. In the existing literature, however, uncertainty is viewed as just an exogenous variable. In this paper, uncertainty is not exogenously given but endogenously determined, generating a feedback-loop between micro-behaviors and the macroeconomic environment. In particular, we assume that when uncertainty is high, expectations and behaviors of micro-agents are more heterogeneous. Under these settings, we show that on the one hand heterogeneity contributes to the stability at the macroscopic level, but on the other hand this macro-stability is suddenly lost when investors are sensitive to uncertainty. Namely, investors suddenly and simultaneously “wait and see”.

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Notes

  1. Caballero (1999, p.815) argues that “[i]t turns out the changes in the degree of coordination of lumpy actions play an important role in shaping the dynamic behavior of aggregate investment.” On the other hand, Thomas (2002) presents the result that plant-level lumpy investment is irrelevant for aggregate activities in the general equilibrium model: “In contrast to previous partial equilibrium analyses, model results reveal that the aggregate effects of lumpy investment are negligible” (Thomas 2002, p.508). However, Gourio and Kashyap (2007) recalibrate the Thomas’s model and obtain different properties from the standard RBC model.

  2. The relationship between uncertainty and disagreement and related literature are discussed in Sect. 4.

  3. The importance of a feedback mechanism is recognized in the existing literature. For example, Caplin and Leahy (1997, pp.601–602) emphasize that “[o]ne of the most limiting aspects of these models is that they focus exclusively on the impact that microeconomic inertia has on aggregate dynamics. They ignore the feedback from aggregates onto individual behavior”. Thomas (2002) constructs a dynamic stochastic general equilibrium model in order to include a feedback mechanism. We also follow the statement of Caplin and Leahy (1997) by introducing the feedback effect between micro units and macroscopic distributional properties.

  4. In the existing literature, the mean-field effect has been intensively investigated (see, e.g., Opper and Saad 2001), but to the best of our knowledge, the effect of the variance has been ignored. We present an alternative method to analyze collective behavior of this type in this paper.

  5. Examples include Caballero and Engel (1991), Cooper et al. (1999), Caballero and Engel (1999) and Thomas (2002).

  6. It should be noted that while X(t) is treated as a stochastic process in our model, it does not imply that firms’ managers randomly choose their investment. Our assumption of X(t) is not in conflict with the optimal behavior. Their decisions are optimally determined based on X(t) and other variables such as market conditions, the interest rate, the technology of firms, and the personality of entrepreneurs, and if we should obtain these information in detail, investment decision making would be modeled without any randomness. However, the detailed information is not available and, therefore, X(t) is viewed as a stochastic process from a third party’s point of view.

    In this regard, cancer and life insurance give us a good example of how our assumption works. Cancer is a metabolic disease and can be in principle explained by chemical reaction without any random component. However, since its mechanism is too complicated, occurrence of cancer can be dealt with as a random variable, even though it is caused by some deterministic process. It cannot be (and need not be) distinguished with the one generated by the action of chance. Indeed, whether it is generated by a deterministic process or by the action of chance does not yield any difference in the calculation of the premium of life insurance.

  7. See Courrège (1965–1966) and Böttcher et al. (2013).

  8. Here, \(C_c^{\infty }(\mathbb {R})\) denote the set of smooth functions on \(\mathbb {R}\) with compact support. We simply assume \(C_c^{\infty }(\mathbb {R}) \subset \mathcal{D}(A)\), that is, A is defined on the set of test functions that is large enough.

  9. In general, a conservative Markov semigroup (i.e., firms do not disappear) does not imply that the killing rate e is equal to 0. However, if jump size is bounded as discussed in Appendix 1, conservativeness implies \(e=0\) by Lemmas 2.28 and 2.32 in Böttcher et al. (2013). Thus, we assume \(e=0\) in the following.

  10. Recall that the intensity rate \(\lambda (x)\) determines the number of jumps, and \(\nu (x,{\text {d}}y)\) determines the size of independent jumps.

  11. For the functional forms of \(\lambda (x)\) and \(\nu (x,{\text {d}}y)\), and parameters, see Sects. 5 and 6.

  12. For the Krylov–Bogolyubov theorem, see, e.g., Corollary 3.1.2 in Da Prato and Zabczyk (1996).

  13. See, e.g., Section 4.2 in Da Prato and Zabczyk (1996). More precisely, in order to apply the Krylov–Bogolyubov theorem and Doob’s theorem, we need to modify the stochastic behavior at large \(|X_t|\). For example, we have to assume that the depreciation rate \(\delta (x)\), which was simply assumed to be constant, becomes 0 at \(x \ll 0\) and the measure \(\nu (x, \cdot )\) has a compact support, in order that \(\int _{[-E, E]} p_t(x, y){\text {d}}y = 1, \ \forall t \ge 0, \ \forall x \in [-E, E]\) for some \(E >0\). Then, we redefine our stochastic process on \([-E, E]\) and apply the two theorems. As we will see later, the well-posedness and convergence are confirmed by the following stability analysis and numerical simulations.

  14. This stability is related to the concept of “stochastic macro-equilibrium”, which is originally advanced by Tobin (1972) in his attempt to explain the observed Phillips curve. He argues that “a theory of stochastic macro-equilibrium: stochastic, because random inter-sectional shocks keep individual labor markets in diverse states of disequilibrium; macro-equilibrium, perpetual flux of particular markets produces fairly definite aggregate outcomes...” (Tobin 1972, p.9).

  15. Bloom (2014) surveys both empirical and theoretical literatures that study the relationship between uncertainty and economic activity.

  16. Papers in the literature that use a measure of uncertainty close to the one employed here include Zarnowitz and Lambros (1987), Bomberger (1996), Giordani and Söderlind (2003), Fuss and Vermeulen (2008), and Clements (2008), to name a few.

  17. This equation is a natural generalization of linear Markov processes. In fact, if the generator is given by \(B f(x) = b(x) \frac{{\text {d}} f(x)}{{\text {d}}x}+ \frac{1}{2} \sigma ^2(x) \frac{{\text {d}}^2 f(x)}{{\text {d}}x^2}\), which corresponds to an Ito diffusion satisfying a stochastic differential equation of \({\text {d}}X_t = b(X_t) {\text {d}}t + \sigma (X_t) {\text {d}}W_t\) and the adjoint operator \(B^*\) is defined by \((B f, \mu _t) = (f, B^* \mu _t)\), Eq. (6) is reduced to the well-known Fokker–Planck equation.

  18. As will be shown later, we do not need to specify \(\eta (x, {\text {d}}y)\) other than \(m_j(x)\) and \(\sigma _j^2\) under Gaussian approximation.

  19. Of course, since we use the approximation method, \(m_{GA}\) and \(\sigma ^2_{GA}\) deviate from their counterparts \(m^*\) and \({\sigma ^*}^2\). However, as we will see in Sect. 6, Gaussian approximation well describes the evolution of \(m^*\) and \({\sigma ^*}^2\), i.e. that of the probability distribution qualitatively and quantitatively.

  20. Note that the eigenvectors depend on the value of c.

  21. Furthermore, because the number of firms \(N < \infty\) in a real economy, the collective behavior can be induced by the finite size effect, that is, the fluctuation due to \(N < \infty\) without any aggregate shocks.

  22. To simplify the notation, the subscript t representing time is omitted below.

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Correspondence to Yoshiyuki Arata.

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Additional information

This research was conducted as a part of the Project “Sustainable Growth and Macroeconomic Policy” undertaken at Research Institute of Economy, Trade and Industry (RIETI). This paper has greatly benefited from comments by Hiroshi Yoshikawa. We would like to thank Toichiro Asada, Jun-Hyung Ko, Akio Matsumoto as well as many colleagues, especially Masayuki Morikawa and Willem Thorbecke, for very useful comments. We are grateful to participants at the 2015 International Conference on Nonlinear Economic Dynamics (NED) at Chuo University as well as seminars at Aoyama Gakuin University, Chuo University, Kyoto University, Sophia University, and RIETI for helpful and constructive comments. This paper was financially supported by JSPS KAKENHI Grant Numbers 13J10353 (Kimura) and 14J03350 (Murakami).

Appendices

Appendix 1: Feller processes

Here, we discuss Assumption 1 from an economic standpoint. First, we consider the strong continuity. It can be shown that the strong continuity is equivalent to pointwise convergence (e.g., Lemma 1.4 in Böttcher et al. 2013):

$$\begin{aligned} \lim _{t \rightarrow 0} T_t u(x) = u(x) \quad \forall u \in C_{\infty }(\mathbb {R}), \ x \in \mathbb {R}. \end{aligned}$$
(16)

Because \(T_t u(x)\) is the expectation value after time t with an initial point x, (16) simply means that the process \(X_t\) stay at its initial value if t is small. Note that it does not exclude the possibility that the process \(X_t\) jumps at around 0. The property (16) means that this probability converges to 0 as \(t \rightarrow 0\). In this sense, this property is closely related to the stochastic continuity. Since there is no reason to assume that \(X_t\) jumps exactly at 0 with positive probability, the strong continuity is a plausible assumption in an economic sense.

Second, to discuss the Feller property, we introduce an additional assumption:

Assumption 4

For all \(t > 0\) and any increasing sequence of bounded sets \(B_n \in \mathcal{B}(\mathbb {R})\) with \(\cup _{n \ge 1} B_n = \mathbb {R}\), the transition function \(p_t(x, B_n)\) satisfies the following property;

$$\begin{aligned} \lim _{|x| \rightarrow \infty } p_t(x, B_n) = 0, \quad \forall n \ge 1. \end{aligned}$$

Here, \(\mathcal {B}(\mathbb {R})\) is Borel sets. For example, suppose that there exist increasing adjustment costs \(\psi (\Delta X_t)\) such that \(\psi '(\Delta X_t)>0, \psi ''(\Delta X_t)>0\), or financial or physical constraints, and, therefore, capital adjustment to some target level cannot be done at one time when \(X_t\) is extremely small. In such a situation, Assumption 4 can be justified. Then, the following lemma implies the Feller property.

Lemma 5

Assumption 4 implies the Feller property.

Proof

Let \(u \in C_{\infty }(\mathbb {R})\). For an arbitrary \(\epsilon\), there exists N such that \(|u| < \epsilon\) on \(\mathbb {R}\setminus B_n\) for all \(n \ge N\). Therefore,

$$\begin{aligned} |T_t u(x)|\le & {} \int _{B_n}|u(y)|p_t(x, {\text {d}}y) + \int _{\mathbb {R} \setminus B_n} |u(y)|p_t(x, {\text {d}}y) \\\le & {} ||u||_{\infty } p_t(x, B_n) + \epsilon \end{aligned}$$

where \(||\cdot ||_{\infty }\) denotes the sup norm on \(C_{\infty }(\mathbb {R})\).

Since \(\epsilon\) is arbitrary, Assumption 4 implies that \(T_t u \in C_{\infty }(\mathbb {R})\). \(\square\)

Appendix 2: Stability analysis

In this section, we investigate the properties of the following system of differential equations of the first moment (mean) m and the second central moment \(\sigma ^2\ge 0\) obtained in Sect. 5:Footnote 22

$$\begin{aligned} \frac{{\text {d}}m}{{\text {d}}t}= & {} (a-m+\sigma ^2)e^{-m-\sigma ^2/2-c(\sigma ^2-{\sigma ^2}^*)},\end{aligned}$$
(17)
$$\begin{aligned} \frac{{\text {d}}\sigma ^2}{{\text {d}}t}= \; & {} [(a-m)^2-\sigma ^2(\sigma ^2+1)+\sigma _j^2]e^{-m-\sigma ^2/2-c(\sigma ^2-{\sigma ^2}^*)}, \end{aligned}$$
(18)

An equilibrium point of the system of (17) and (18), \((m^*,{\sigma ^2}^*) \in \mathbb {R}\times \mathbb {R}_+,\) is defined as a solution of the following system of simultaneous equations:

$$\begin{aligned}&e^{- m^*+ {\sigma ^2}^*/2} ( a - m^* + {\sigma ^2}^* )=\delta , \end{aligned}$$
(19)
$$\begin{aligned}&(a-m^*)^2 +\sigma _j^2={\sigma ^2}^*({\sigma ^2}^*+1). \end{aligned}$$
(20)

One can easily find that \({\text {d}}m/{\text {d}}t={\text {d}}\sigma ^2/{\text {d}}t=0\) if and only if \((m,\sigma ^2)=(m^*,{\sigma ^2}^*).\) Concerning the existence and uniqueness of an equilibrium \((m^*,{\sigma ^2}^*),\) the following proposition holds.

Proposition 2

There uniquely exists an equilibrium point \((m^*,{\sigma ^2}^*) \in \mathbb {R}\times \mathbb {R}_+\).

Proof

Equation (20) can be solved for \(\sigma ^2\ge 0\) as

$$\begin{aligned} \sigma ^2=\frac{\sqrt{1+4[(a-\sigma ^2)^2+\sigma _j^2]}-1}{2}>0. \end{aligned}$$
(21)

Let \(z=a-m.\) Then, substituting (21) in (19), we have

$$\begin{aligned} g(z)h(z)=\delta e^a, \end{aligned}$$
(22)

where g(z) and h(z) are defined as follows:

$$\begin{aligned} g(z)=\; & {}\; z+\frac{\sqrt{1+4(z^2+\sigma _j^2)}-1}{2},\\ h(z)= & {} \exp \Bigl (z+\frac{\sqrt{1+4(z^2+\sigma _j^2)}-1}{4}\Bigl ) > 0. \end{aligned}$$

Since the right hand side of (22) is positive, for some z to satisfy (22), we must have

$$\begin{aligned} g(z)>0, \end{aligned}$$

or

$$\begin{aligned} z>\frac{3}{8}-\frac{\sigma ^2_j}{2}\equiv \underline{z}. \end{aligned}$$
(23)

Both g and h are positive and strictly increasing in z within the range of (23) and that we have \(g(\underline{z})h(\underline{z})=0\) and \(g(\infty )h(\infty )=\infty .\) Hence, there uniquely exists a \(z^*\) that meets (22). Letting \(m^*=a-z^*\) and \({\sigma ^2}^*=[-1+\sqrt{1+4({z^*}^2+\sigma _j^2)}]/2,\) we can find that \((m^*,{\sigma ^2}^*) \in \mathbb {R}\times \mathbb {R}_+\) is a unique equilibrium point. \(\square\)

Corollary 6

Assume that the following condition is satisfied:

$$\begin{aligned} \sigma _j^2\le \frac{3}{4}, \end{aligned}$$
(24)

Then, we have \(m^*<a.\)

Proof

It is obvious from (23) in Proof of Theorem 2. \(\square\)

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Arata, Y., Kimura, Y. & Murakami, H. Aggregate implications of lumpy investment under heterogeneity and uncertainty: a model of collective behavior. Evolut Inst Econ Rev 14, 311–333 (2017). https://doi.org/10.1007/s40844-017-0074-5

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Keywords

  • Lumpy investment
  • Pseudo-compound Poisson process
  • Uncertainty
  • Collective behavior

JEL Classification

  • E22
  • E32
  • D21