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”.

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. 5:²²

$$\begin{aligned} \frac{{\text {d}}m}{{\text {d}}t}= & {} (a-m+\sigma ^2)e^{-m-\sigma ^2/2-c(\sigma ^2-{\sigma ^2}^*)},\end{aligned}$$

(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. $\square$