Minimum investment requirement, financial market imperfection and self-fulfilling belief

Abstract

We develop a model in which a strategic complementarity in saving decisions arises due to a minimum investment requirement and financial market imperfection. We explore the role of self-fulling beliefs in determining the long run dynamics. The model exhibits a wide range of dynamic phenomena such as a poverty trap, a big push and a sunspot equilibrium, depending on the level of financial market imperfection. They account for excessive volatility and a sudden change in the saving rate and its macroeconomic consequences without any shocks to fundamentals.

Appendices

Appendix A: Alternative specifications

We have assumed a fixed investment size and a linear utility function in our model above. These assumptions reduced the notational burden, made the model analytically tractable and helped to represent ideas by simple graphs and formulas. This section shows that our key results remain robust under a log utility function and as long as there is a minimum investment requirement.

A.1 Log utility function

One might think that the existence of multiple equilibria depends on the assumption of a linear utility function, which implies an infinitely elastic saving function. This section demonstrates that our results on multiple equilibria and thus the strategic complementarity are robust even when $u(c)=\log c$.

If s _t ≥ 1 − λ, then, from Eqs. 2 and 5, the young agent’s optimal saving is

$$ {s_{t}^{i}}=\frac{\beta}{1+\beta}w_{t}. $$

(21)

If s _t < 1 − λ, then, from Eqs. 3 and 5, the young agent’s optimal saving must satisfy

$$ \frac{1}{w_{t}-{s_{t}^{i}}}=\displaystyle\frac{\beta s_{t} r_{t+1}}{Rf^{\prime}(Rs_{t})-(1-{s_{t}^{i}})r_{t+1}}+\frac{\beta(1-s_{t})}{{s_{t}^{i}}}. $$

(22)

Equations 21 and 22 with

$$ \begin{array}{ccc} {s_{t}^{i}}=s_{t} & \text{and} & r_{t+1}(1-{s_{t}^{i}})=\lambda Rf^{\prime}(Rs_{t}) \end{array} $$

(23)

imply that the equilibrium pair (w _t , s _t ) satisfies

$$ w_{t}={\Gamma}(s_{t},\lambda) $$

(24)

where

$$ {\Gamma}(s,\lambda)=\left\{ \begin{array}{ccc} \frac{1}{\beta}\frac{(1-\lambda)s(1-s)}{\lambda s^{2}+(1-\lambda)(1-s)^{2}}+s & \text{if} & s<1-\lambda \\ [2.00ex] \frac{1+\beta}{\beta}s & \text{if} & s \geq 1-\lambda. \end{array} \right. $$

(25)

One can easily verify that s ↦ Γ(s, λ) is a continuous function satisfying boundary conditions Γ(0, λ) = 0, ${\Gamma }(1-\lambda ,\lambda )=\frac {(1+\beta )(1-\lambda )}{\beta }$ and ${\Gamma }(1,\lambda )=\frac {1+\beta }{\beta }$.

Lemma A.1

If $\lambda <\frac {1}{2+\beta }$, then there exists a unique $\hat {s} \in (0, 1-\lambda )$ which solves Γ₁(s, λ) = 0.

Proof

It follows from Eq. 25that

$${\Gamma}\left( \frac{x}{1+x},\lambda\right)= \left\{\begin{array}{ll} \frac{1}{\beta}\frac{(1-\lambda)x}{1-\lambda+\lambda x^{2}}+\frac{x}{1+x} & \text{if} \ x<\frac{1-\lambda}{\lambda} \\ \frac{1+\beta}{\beta}\frac{x}{1+x} & \text{if} \ x \geq \frac{1-\lambda}{\lambda}. \end{array}\right. $$

Onecan easily verify that

$$\begin{array}{ccc} {\Gamma}_{1}\left( \frac{x}{1+x},\lambda\right)=0 & \Leftrightarrow & \frac{\beta}{1-\lambda}=\frac{(1+x)^{2}}{\lambda x^{2}+1-\lambda}\frac{\lambda x^{2}-1+\lambda}{\lambda x^{2}+1-\lambda}. \end{array} $$

Since$x \mapsto \frac {(1+x)^{2}}{\lambda x^{2}+1-\lambda }$and$x\mapsto \frac {\lambda x^{2}-1+\lambda }{\lambda x^{2}+1-\lambda }$are both increasingfunctions for $x \in (0,\frac {1-\lambda }{\lambda })$,${\Gamma }_{1}\left (\frac {x}{1+x},\lambda \right )=0$admits a unique solution.Since $x \mapsto \frac {x}{1+x}$is a monotonictransformation, ${\Gamma }_{1}\left (s,\lambda \right )=0$also admits at most a unique solution. Since

$$\begin{array}{ccc} \lim_{x \downarrow 0}\frac{(1+x)^{2}}{\lambda x^{2}+1-\lambda}\frac{\lambda x^{2}-1+\lambda}{\lambda x^{2}+1-\lambda}<0 & \text{and} & \lim_{x \uparrow \frac{1-\lambda}{\lambda}}\frac{(1+x)^{2}}{\lambda x^{2}+1-\lambda}\frac{\lambda x^{2}-1+\lambda}{\lambda x^{2}+1-\lambda}=\frac{1-2\lambda}{\lambda(1-\lambda)} \end{array} $$

aunique solution exists if and only if

$$ \begin{array}{ccc} \frac{\beta}{1-\lambda}<\frac{1-2\lambda}{\lambda(1-\lambda)} & \Leftrightarrow & \lambda<\frac{1}{2+\beta}. \end{array} $$

(26)

□

Figure 12a shows the equilibrium pair (w _t , s _t ) when the utility function is logarithmic and $\lambda <\frac {1}{2+\beta }$. The figure indicates that there exist multiple equilibria s _t for $w_{t} \in [{\Gamma }(1-\lambda ,\lambda ),{\Gamma }(\hat {s},\lambda )]$.

Fig. 12

Equilibrium pair (w _t , s _t )

Proposition A.1

If $\lambda <\frac {1}{2+\beta }$, then there exists multiple equilibria for $w(Rs_{t-1}) \in ({\Gamma }(1-\lambda ,\lambda ),{\Gamma }(\hat {s},\lambda ))$ and a unique equilibrium for $w(Rs_{t-1}) \in (0,{\Gamma }(1-\lambda ,\lambda )) \cup ({\Gamma }(\hat {s},\lambda ),\infty )$.

Proof

The proof of the proposition is a direct consequence of Lemma A.1. If λ (2 + β) < 1, then s ↦ Γ(s, λ) is a non-monotonic function achieving its local maximum at $s=\hat {s}$ and a local minimum at s = 1 − λ. If λ (2 + β) ≥ 1 then s ↦ Γ(s, λ) is a monotonically increasing function. This impliesthat there exist multiple equilibria if and only if λ (2 + β) < 1 and $w(Rs_{t-1}) \in ({\Gamma }(1-\lambda ,\lambda ),{\Gamma }(\hat {s},\lambda ))$. Otherwise there always exists a unique equilibrium. □

Let us now compare the linear utility case and the logarithmic utility case. When the utility is logarithmic, then the pair (w _t , s _t ) solves (24) and (25). Figure 12(b) shows the equilibrium pair (w _t , s _t ) when the utility function is linear and λ ∈ (ψ ₁ (R), ψ ₂ (R)).

A.2 CES production function

This section demonstrates that our results on multiple equilibria and thus the strategic complementarity are robust even when the production function is the following CES function

$$ f(k)=\left\{ \begin{array}{ccc} \left( 1-\alpha+\alpha k^{\frac{\sigma-1}{\sigma}}\right)^{\frac{\sigma}{\sigma-1}} & \text{if} & \sigma \neq 1 \\ k^{\alpha} & \text{if} & \sigma=1, \end{array} \right. $$

(27)

where α ∈ (0, 1) and σ ∈ (0, ∞) denotes the elasticity of substitution between capital and labor. When σ = 1, we obtain the Cobb-Douglas case.

When the utility function is logarithmic and the production function is CES, then the equilibrium pair (s _t−1, s _t ) satisfies w(R s _t−1) = Γ(s _t , λ) where

$$ w(k)=(1-\alpha)\left( 1-\alpha+\alpha k^{\frac{\sigma-1}{\sigma}}\right)^{\frac{1}{\sigma-1}} $$

(28)

is the wage function.

Figure 13 plots the the equilibrium pair (s _t−1, s _t ) satisfying w(R s _t−1) = Γ(s _t , λ) for different values of σ. The figure shows that multiple equilibria remain a robust phenomenon.

Fig. 13

Equilibrium pair (w _t , s _t ) with logarithmic utility and CES production: α = 0.4, β = 0.7, λ = 0.17 and R = 29

A.3 The role of minimum investment requirement

We have assumed that the young agents can run at most one project, which required one unit of consumption goods. This section demonstrates that the model remains essentially the same as long as there is a minimum investment requirement. Suppose investment technology is described by

$$F(i_{t})= \left\{\begin{array}{ll} 0 & \text{if}\ i_{t}<I \\ Ri_{t} & \text{if}\ i_{t} \geq I \end{array}\right. $$

where i _t is the investment of the consumption goods, F(i _t ) is the produced amount of capital, and I is the minimum investment requirement.

It follows from Eq. 5 that the equilibrium interest rate is

$$r_{t+1}= \left\{\begin{array}{ll} \frac{\lambda I}{I-s_{t}}Rf^{\prime}(Rs_{t}) & \text{if} \ s_{t}<I(1-\lambda) \\ Rf^{\prime}(Rs_{t}) & \text{if} \ s_{t} \geq I(1-\lambda). \end{array}\right. $$

This with Eq. 6 implies that

$$ r_{t+1}= \left\{\begin{array}{ll} \frac{1}{I}H\left( \frac{s_{t}}{I},\lambda, R I\right) & \text{if} \ s_{t}<I(1-\lambda) \\ Rf^{\prime}(Rs_{t}) & \text{if} \ s_{t} \geq I(1-\lambda). \end{array}\right. $$

(29)

If I = 0, then $r_{t+1}=Rf^{\prime }(Rs_{t})$ and the model reduces to a standard OLG model. If I > 0, then, from Eq. 29, we can see that the property of the investment curve H(., λ, R) remains essentially unchanged with respect to s _t .

Appendix B: Remaining proofs

Proof of Lemma 3.1

Differentiating both sides of $H(s,\lambda ,R)=\frac {\lambda \alpha R^{\alpha }s^{\alpha -1}}{1-s}$with respect to s we obtain that s ^csolves H _s (s, λ, R) = 0 ⇔ $s^{c}=\frac {1-\alpha }{2-\alpha }$. Since $\lim _{s \downarrow 0}H(s,\lambda ,R)=\infty$ and $\lim _{s \uparrow 1}H(s,\lambda ,R)=\infty$, it follows that H (., λ, R)obtains its minimum at s = s ^c. □

Lemma B.1

The two functions ψ ₁ : (0, R ⁺] → [0, 1) and $\psi _{2}:(0,R^{+}] \to \mathbb R_{++}$ defined in Eq. 10 satisfy

• (a) ψ ₁ is strictly decreasing with $\lim _{R \downarrow 0}\psi _{1}(R)=1$ and ψ ₁ (R ⁺) = 0.

• (b) ψ ₂ is strictly decreasing.

• (c) ψ ₁ and ψ ₂ are tangent to each other at R ^c ∈ (0, R ⁺) and ψ ₁ (R) < ψ ₂ (R) for any R ≠ R ^c.

Proof of Lemma B.1

(a) The monotonicity and boundary behavior follow directly from the definition of $\psi _{1}(R):=1-(\alpha \beta R^{\alpha })^{\frac {1}{1-\alpha }}$ and $R^{+}:=(\frac {1}{\alpha \beta })^{\frac {1}{\alpha }}$.

(b) The monotonicity and boundary behavior follow directly from the definition of $\psi _{2}(R):=\frac {1}{\alpha \beta R^{\alpha }}\frac {(1-\alpha )^{1-\alpha }}{(2-\alpha )^{2-\alpha }}$ and $R^{+}:=(\frac {1}{\alpha \beta })^{\frac {1}{\alpha }}$.

(c) We observe that

$$H(1-\psi_{1}(R),\psi_{1}(R),R) \equiv H(s^{c},\psi_{2}(R),R)<H(1-\psi_{1}(R),\psi_{2}(R),R), $$

if and onlyif 1 − ψ ₁ (R)≠s ^c. Because H(s,⋅, R)is increasing forany given (s, R), thisimplies that ψ ₁ (R) < ψ ₂(R)if 1 − ψ ₁ (R)≠s ^c, and ψ ₁ (R) = ψ ₂ (R)if 1 − ψ ₁ (R) = s ^c. Thus solving ψ ₁ (R) = ψ ₂ (R)is equivalentto solving 1 − ψ ₁ (R) = s ^cor

$$ \begin{array}{ccc} R=R^{c}:=\left( \frac{1}{\alpha\beta}\right)^{\frac{1}{\alpha}}\left( \frac{1-\alpha}{2-\alpha}\right)^{\frac{1-\alpha}{\alpha}} & \text{and} & \psi_{1}(R^{c})=1-s^{c}=\frac{1}{2-\alpha}. \end{array} $$

(30)

Since

$$ \psi_{1}^{\prime}(R^{c})=\psi_{2}^{\prime}(R^{c})=-\frac{\alpha}{1-\alpha}(\alpha\beta)^{\frac{1}{\alpha}}\left( \frac{1-\alpha}{2-\alpha}\right)^{\frac{2\alpha-1}{\alpha}} $$

(31)

it follows that ψ ₁ and ψ ₂ tangenteach other at R = R ^c. □

Proof of Proposition 3.1

The proposition follows directly from Lemma B.1. □

Proof of Proposition 3.2

The proposition follows directly from Proposition 3.1. □

Lemma B.2

$\psi _{3}:(0,R^{+}]\to \mathbb R_{++}$ defined in Eq. 14 satisfy

• (a) ψ ₃ is strictly decreasing.

• (b) ψ ₂ and ψ ₃ are tangent to each other at R = R ^cc and ψ ₃ (R) < ψ ₂ (R) for any R≠R ^cc.

Proof of Lemma B.2

(a) It directly follows from $\psi _{3}(R):=\frac {1-\alpha }{\alpha \beta }(1-[(1-\alpha )R^{\alpha }]^{\frac {1}{1-\alpha }})$.

(b) We observe that

$$H(s^{*},\psi_{3}(R),R) \equiv H(s^{c},\psi_{2}(R),R)<H(s^{*},\psi_{2}(R),R), $$

if and only if s ^∗ ≠ s ^c. Since H(s,⋅, R) is increasing for any given (s, R), ψ ₃ (R) < ψ ₂ (R) if s ^∗ ≠ s ^c, and ψ ₃ (R) = ψ ₂ (R) if s ^∗ = s ^c. Thus, solving ψ ₃ (R) = ψ ₂ (R)is equivalentto solving s ^∗ = s ^c or

$$ \begin{array}{ccc} R=R^{cc}:=\frac{1}{(1-\alpha)(2-\alpha)^{\frac{1-\alpha}{\alpha}}} & \text{and} & \psi_{2}(R^{cc})=\psi_{3}(R^{cc})=\frac{1}{\alpha\beta}\frac{1-\alpha}{2-\alpha}. \end{array} $$

(32)

Since $\psi _{2}^{\prime }(R^{c})=\psi _{3}^{\prime }(R^{c})$, ψ ₂ and ψ ₃ are tangent toeach other at R = R ^cc. □

Proof of Proposition 4.1

It can be easily verified that, if $(\beta ,\lambda , R)\in {\mathcal E}$and either R < R ^ccor λ < ψ ₃ (R),then there exists a unique steady state $\min \{s_{L},s^{*}\}$. The necessary and sufficient condition for existence of multiple steady states is s _M < s ^∗.Note that $\beta <\underline {\beta }(\alpha )\Leftrightarrow s_{H}<s^{*}$. Lemma B.2 establishes that $(\beta ,\lambda , R)\in {\mathcal E}_{1}$ and $(\beta ,\lambda , R)\in {\mathcal E}_{2}$ are necessary and sufficient conditions for s _M < s ^∗. □

Proof of Lemma 4.1

If $(\beta ,\lambda , R)\in {\mathcal E}_{1}\cup {\mathcal E}_{2}$,then s _L < s ^c < s _M < s ^∗.If λ is sufficiently close to ψ ₂ (R),then s _L is sufficiently close to s _M , s _L ≈ s _M .This implies that s _L ≈ s ^c ≈ s _M < Ψ(s _L ) since s _L < s ^∗.Hence, s _M < Ψ(s _L ). Continuity of both s _L and s _M in parameters ensures the claim. □

Proof

of Proposition 4.3 First, we consider the parameter set${\mathcal E}_{1}$ corresponding to Proposition 4.1 (b), where $s_{H}<s^{*}\Leftrightarrow \beta <\underline {\beta }(\alpha )$.As all possible equilibria have positive probability, for any initial condition s ₀ ∈ (0, 1),the trajectory will end up in a few iterations in one of the two constants s _L or s _M and, from that point on, the possible states are only those of the finite set given by Eq.17. The assumption of i.i.d for the stochastic process easily guarantees existence anduniqueness of the stationary distribution. Since the probability of returning to states s _L , s _M and s _H is p _L , p _M and p _H ,respectively, it follows that the stationary probability distribution overall possible states is given by Eq. 18. In this case, the interval [s _L , s _H ] is a forward invariant set.

Second, we consider the parameter set ${\mathcal E}_{2}$ corresponding to Proposition 4.1 (c), where $s_{H} \geq s^{*}\Leftrightarrow \beta \geq \underline {\beta }(\alpha )$. For any initial condition s ₀ ∈ (0, 1),the trajectory will end up in a few iterations in one of two constants s _L or s _M and, from that point on, the possible states are only those ofthe countable set given by Eq. 19. It is worthwhile to note that s ^∗ does not belong to the above set but belongs toits closure. Since the probability of returning to state s _L and s _M is p _L and p _M ,respectively, and both p _L and p _M are independent of the current state, it follows that the stationary probabilitydistribution over all possible states is given by Eq. 20. In this case, the interval [s _L , s ^∗)is a forward invariant set. □

Appendix C: Data

Let λ _{i t} = DRate _{i t} /LRate _{i t} where DRate _{i t} and LRate _{i t} are the deposit rate and the lending rate in country i at time t and λ _i be the sample average of λ _{i t} . We consider two alternative measures of the saving rate volatility σ _i , namely, $\sqrt {\frac {1}{T}{\sum }_{t=0}^{T}(s_{it}-s_{it-1})^{2}}$ and $\frac {1}{T}{\sum }_{t=0}^{T}\left | s_{it}-s_{it-1}\right |$ where s _{i t} is the gross saving rate as a percentage of GDP in country i at time t. The data are obtained from the World Bank Development Indicators for the period 1960-2015. First, we calculate the sample average standard deviation of the time series of the saving rate for all 34 OECD countries (Table 1). Second, we calculate the sample average of the deposit to lending rate ratio for 32 countries (Table 2). The deposit rate data are missing for the United States and the lending rate data are missing for Turkey. The deposit rate is measured as the rate at which commercial banks pay on demand, time or savings deposits, while the lending rate is measured as the rate commercial banks charge on loans to prime customers.

Table 1 Annual data for the saving rate

Table 2 The lending rate to deposit rate ratio

The two time series data might not perfectly match for some countries. For example, France has the saving rate from 1975 to 2014 (40 observations) and the lending rate to deposit rate ratio from 1966 to 2004 (39 observations). In this case, 1975-2004 is the overlapping time period (30 observations) and we define the overlap coefficient as 30/40=75%. Among 32 countries, which are included in the sample, the overlap coefficient is 100% for Japan. Table 3 shows that the overlap coefficient is at least 50% for 29 countries, which is 91% of all counties included in the sample.

Table 3 Overlap coefficient