Two-Dimensional Nonlinear Cobweb Model

Abstract

This chapter extends the one-dimensional model studied in Chap. 2 to include two different types of producers in order to investigate whether slight behavioral heterogeneity could be a factor that drastically changes the dynamical properties of a market. The two types of producers are naive optimizers and cautious adapters. A naive optimizer produces the profit-maximizing quantity instantaneously, while a cautious adapter adjusts his/her output toward the profit-maximizing quantity as a target. We obtain a two-dimensional model, which is more difficult to analyze because mathematical theories of higher-dimensional nonlinear dynamical systems are underdeveloped compared with those of one-dimensional systems. With the aid of mathematical analysis as well as numerical calculations, we show that a single agent may change the complexity of market behavior. In a market of naive optimizers, a single cautious adapter stabilizes the otherwise exploding market. In a market of cautious adapters, a single naive optimizer may destabilize the market. Without him/her, there exists at most one periodic attractor in the market. However, with him/her, many (and even infinitely many) coexisting periodic attractors may appear.

What science cannot discover, mankind cannot know.

Bertrand Russell Religion and Science (1935)

The most incomprehensible thing about the universe is that it is comprehensible.

Albert Einstein Ideas and Opinions (1954)

Appendix

A.1 Proof of Lemma 3.2

Before proving Lemma 3.2, we must prove two lemmas. We first consider the homogeneous case where only adapters exist in the market, namely $m = 1$. In this case, the map $F_{\eta , m}$ given by (3.13) reduces to the singular (thus noninvertible) map $F_{\eta , 1}:\mathbb {R}_{++}^{2} \rightarrow \mathbb {R}_{++}^{2}$, given by

$$\begin{aligned} F_{\eta , 1}(x,y)=(y, f_{\eta }(y)). \end{aligned}$$

The map $F_{\eta , 1}$ is clearly equivalent to the one-dimensional map $f_{\eta }$ in the sense that $f_{\eta }$ on $\mathbb {R}_{++}$ is topologically conjugate to the map $F_{\eta , 1}$ restricted onto its image $\text {Im}(F_{\eta , 1})$ through the conjugacy $\varphi (x)=(x, f_{\eta }(x))$. The dynamics of $f_{\eta }$ are studied in Onozaki et al. (2000) (Chap. 2 in this book), where $f_{\eta }$ is shown to be strictly convex and unimodal with its global minimum at

$$\begin{aligned} \theta = \theta (\eta ) = \left( \frac{\phi \>\!\eta }{1-\phi } \right) ^ {\frac{1}{1+\eta }}. \end{aligned}$$

In other words, $f_{\eta }^{\prime }(\theta )=0$ and $f_{\eta }^{\prime \prime }(x)>0$ for every $x>0$. Furthermore, the Schwarzian derivative of $f_{\eta }$ is given by

$$\begin{aligned} Sf_{\eta }(x)= & {} \frac{f_{\eta }^{\prime \prime \prime }(x)}{f_{\eta }^{\prime }(x)}-\frac{3}{2}\left( \frac{f_{\eta }^{\prime \prime }(x)}{f_{\eta }^{\prime }(x)}\right) ^{2} \nonumber \\= & {} - \frac{\ \phi \>\! \eta \>\!(1 + \eta )\left( \phi \>\! \eta \>\! (\eta -1) + 2(1 - \phi )(2 + \eta )x^{1 + \eta }\right) \ }{2\left( (\phi - 1)x^{2 + \eta } + \phi \>\! \eta x \right) ^{2}}. \end{aligned}$$

(3.16)

We see that

$$\begin{aligned} Sf_{\eta }(x) < 0 \qquad \text{ for } \quad \eta \ge 1 \quad \text{ and } \quad x > 0. \end{aligned}$$

For a large $\eta$, the unstable manifold $W^{u}_{\smash [t] {\eta , 1}}(p)$ on the (x, y) –plane is simply an arc consisting of a compact part of $\text {Im}(F_{\eta , 1})$, which is a part of the graph of $f_{\eta }$.

Lemma 3.3

Let $\eta ^{*}$ be an $\eta$ value such that $f^2_{\smash [t]{\eta ^{*}}}(\theta (\eta ^{*}))=\theta (\eta ^{*})$ and $\theta (\eta )<f^2_{\eta }\left( \theta \left( \eta \right) \right)$ for $\eta >\eta ^{*}$ (such $\eta ^{*}>(2-\phi )/\phi$ exists). Then for $\eta >\eta ^{*}$ and $m=1$, the unstable manifold of the fixed point p is given by $W^{u}_{\smash [t] {\eta , 1}}(p) = \{ (x,y):\,x\in [f_{\eta }(\theta ), f_{\eta }^{2}(\theta )],\, y=f_{\eta }(x) \}$.

Proof

An exercise shows that if $(1-\phi )/\phi <\eta \le (2-\phi )/\phi$, then $1<f^{2}(\theta )<\theta$ (see Appendix A.1 in Chap. 2). In addition, since $f^{2}(\theta )\rightarrow \infty$ and $\theta \rightarrow 1$ as $\eta \rightarrow \infty$ (see Lemma 2.4), there exists an $\eta$ value $\eta ^{*}>(2-\phi )/\phi \,(>1)$ as in the statement of the lemma. Let $\eta >\eta ^{*}$ be given. Then, we have $f(\theta )<1<\theta <f^{2}(\theta )$. Since $f([1,\theta ])=[f(\theta ), 1]$ and $f([f(\theta ), 1])=[1,f^{2}(\theta )]$, the set $W_{\eta , 1}^{u}(p)$ is proved to be the unstable manifold of p if it is shown that for any $z\in (1,\theta )$, there exists $n=n(z)\ge 1$ such that $f^{2n}(z)>\theta$. Clearly, it suffices to show that $f^{2}(x)>x$ for any $x\in (1,\theta ]$. If this does not hold, then there exists $q\in (1,\theta ]$ such that $f^{2}(q)\le q$. Noting that $df^{2}(1)/dx=(f^{\prime }(1))^{2}>1$ and $f^{2}(1)=1$, we can assume $f^{2}(q)=q$ without loss of generality. For notational simplicity, let $g(x)=f^{2}(x)$. Note that g is strictly increasing on $[1,\theta ]$ as $dg(x)/dx=f^{\prime }(f(x))f^{\prime }(x)>0$ for $x\in [1,\theta )$. Again since $dg(1)/dx>1$ and $g(1)=1$, there exists $r\in (1,q)$ such that $r<g(r)<g(q)=q$. Thus,

$$\begin{aligned} 0<\frac{g(q)-g(r)}{q-r}<1. \end{aligned}$$

Since $q<\theta <g(\theta )$, we also have

$$\begin{aligned} 1<\frac{g(\theta )-g(q)}{\theta -q}. \end{aligned}$$

Therefore, by the mean value theorem, $g^{\prime }$ attains a (local) minimum at some $c\in (1,\theta )$, namely $g^{\prime \prime }(c)=0$ and $g^{\prime \prime \prime }(c)\ge 0$. Remember that if the Schwarzian derivative of f, Sf, is negative, then $S(f\circ f)=Sg$ is also negative [cf. Singer (1978)]. Since $Sf(x)<0$ for $x>0$, we then must have

$$\begin{aligned} Sf^{2}(c)=Sg(c)=\frac{g^{\prime \prime \prime }(c)}{g^{\prime }(c)}<0, \end{aligned}$$

which contradicts $g^{\prime }(c)>0$ and $g^{\prime \prime \prime }(c)\ge 0$. $\square$

When $\eta$ is large, the stable manifold $W^{s}_{\smash [t] {\eta , 1}}(p)$ on the (x, y)–plane consists of infinitely many horizontal lines containing $y=1$ and $y=\hat{y}_i(\eta )$ (depending on $\eta$) such that $f^n_{\eta }(\hat{y}_i)=1$ for some integer $n=n(i)$ indexed by i. That is, $\hat{y}_i$ is a preimage of 1 by the one-dimensional map f. We can see that $W^{u}_{\smash [t] {\eta , 1}}(p)$ attains a tangential intersection with a horizontal line $y=\hat{y}_{i}(\eta )$ belonging to $W^{s}_{\smash [t] {\eta , 1}}(p)$, which then unfolds into two transverse intersections as $\eta$ increases. Note here that the map f has two inverses, say $f^{-1}_{L}$ and $f^{-1}_{R}$, giving the preimages of a point y: one on the left of $\theta$ and one on the right of $\theta$. That is, $f^{-1}_{L}(y)=f^{-1}(y)\cap (0,\theta ]$ and $f^{-1}_{R}(y)=f^{-1}(y)\cap (\theta ,\infty )$. Thus, $f^{-1}(1)=1\cup f^{-1}_{R}(1)$ with $f^{-1}_{L}(1)=1$, and so on.

Lemma 3.4

For $m=1$, there exist horizontal lines $\{ y=\hat{y}_{i}(\eta ) \}_{i} \subset W^{s}_{\smash [t] {\eta , 1}}(p)$ (depending smoothly on $\eta$) such that for some $\eta _{1} = \eta _{1}(i)$ and $\eta _{2} = \eta _{2}(i)$ with $\eta _{1} < \eta _{2}$:

• (P1) $W^{u}_{\smash [t] {\eta _{1}, 1}}(p)$ and $y = \hat{y}_{i}(\eta _{1})$ have no intersection;

• (P2) $W^{u}_{\smash [t] {\eta _{2}, 1}}(p)$ and $y = \hat{y}_{i}(\eta _{2})$ have two transverse homoclinic intersections, and

• (P3) $W^{u}_{\smash [t] {\eta _{H}, 1}}(p)$ and $y = \hat{y}_{i}(\eta _{H})$ have a quadratic homoclinic tangency for some $\eta _{H} = \eta _{H(i)} \in (\eta _{1},\eta _{2})$.

Proof

Let $\eta _{1}>\eta ^{*}$ be given, where $\eta ^{*}$ is as in Lemma 3.3. We can see that there is an integer $\bar{k}\ge 1$ such that for $k\ge \bar{k}$, $\theta (\eta _{1})<f_{\eta _{1}}^{2}(\theta (\eta _{1}))<f_{\smash [t]{\eta _{1}, R}}^{-k}(1)\equiv \tilde{y}_{k}(\eta _{1})$, where $f_{\eta , s}^{-k}$ means the kth composite of $f_{\eta , s}^{-1}$($s=R,\, L$) and $\tilde{y}_{k}(\eta )$ is, by the implicit function theorem, a smooth function of $\eta >(1-\phi )/\phi$ with $d\hat{y}_{k}(\eta )/d\eta >0$. Since $f_{\eta }^{2}(\theta (\eta ))\rightarrow \infty$ and $\hat{y}_{k}(\eta )\rightarrow 1/(1-\phi )^{k}$ ($\eta \rightarrow \infty$), there is $\eta _{2}(>\eta _{1})$ such that $f_{\eta _{2}}^{2}(\theta (\eta _{2}))>\tilde{y}_{k}(\eta _{2})>\theta (\eta _{2})$. This implies that $f_{\eta _{1}}(\theta (\eta _{1}))>f_{\smash [t]{\eta _{1}, L}}^{-1}(\tilde{y}_{k}(\eta _{1}))\equiv \hat{y}_{k}(\eta _{1})$ (where $\hat{y}_{k}(\eta )$ is a smooth function of $\eta$) and $f_{\eta _{2}}(\theta (\eta _{2}))<f_{\smash [t]{\eta _{2}, L}}^{-1}(\tilde{y}_{k}(\eta _{2}))=\hat{y}_{k}(\eta _{2})$. Since the horizontal line $y=\hat{y}_{k}(\eta _{j})$ obtained above belongs to $W_{\smash [t]{\eta _{j}, 1}}^{s}(p)$ because $f_{\eta _{j}}^{k+1}(\hat{y}_{k})=1$ and since the point $(\theta (\eta _{j}), f_{\eta _{j}}(\theta (\eta _{j})))$ belongs to $W_{\smash [t]{\eta _{j}, 1}}^{u}(p)$$(j=1,\, 2)$ by Lemma 3.3, the assertions (P1) and (P2) immediately follow. (P3) follows from (P1) and (P2) by continuity. $\square$

We now perturb the singular map $F_{\eta , 1}$ into nonsingular maps by making m slightly smaller.

Proof of Lemma 3.2. (i) By Lemma 3.1, if

$$\begin{aligned} \frac{2-\phi }{\phi }<\eta <\frac{1}{(1-m)(1-\phi )}, \end{aligned}$$

then the fixed point p for $F_{\eta , m}$ is a dissipative hyperbolic saddle. Thus if $\eta >(2-\phi )/\phi$, then for $m\in (\varepsilon _{1}, 1)$ where $\varepsilon _{1}=\varepsilon _{1}(\eta )=1-\frac{1}{\eta (1-\phi )}$, p is a dissipative hyperbolic saddle.

(ii) This part is proved if the following is verified:

Claim

Let $\eta _{1}$ and $\eta _{2}$ be as in Lemma 3.4. Then, there exists $\varepsilon \in (0 , 1)$ such that for every $m\in (\varepsilon , 1)$, the map $F_{\eta , m}$ has arcs $\gamma ^{s}_{\eta ,m}\subset W_{\eta , m}^{s}(p)$ and $\gamma ^{u}_{\eta ,m} \subset W_{\eta , m}^{u}(p)$ satisfying the following:

• (a) $\gamma _{\eta _{1}, m}^{s}\cap \gamma _{\eta _{1}, m}^{u}=\phi$,

• (b) $\gamma _{\eta _{2}, m}^{s}$ and $\gamma _{\eta _{2}, m}^{u}$ have two transverse intersections, and

• (c) For some $\eta _{H}\in (\eta _{1},\eta _{2})$, $\gamma _{\eta _{H}, m}^{s}$ and $\gamma _{\eta _{H}, m}^{u}$ have a quadratic homoclinic tangency that unfolds generically with respect to $\eta$.

Proof

Since $\eta _{2}>\eta _{1}>(2-\phi )/\phi$, the fixed point p for the nonsingular map $F_{\eta , m}$ for every $\eta \in [\eta _{1},\eta _{2}]$ and $m\in (\varepsilon _{1}(\eta _{2}), 1)$ is a dissipative hyperbolic saddle from the result of part (i) above. Thus, by the continuous dependence of the unstable manifold of a hyperbolic fixed point on $F_{\eta , m}$ in the $C^{2}$ topology, the map $F_{\eta , m}$ has an arc $\gamma _{\eta ,m}^{u}\subset W_{\eta , m}^{u}(p)$ which is $C^{2}$-close to $W_{\smash [t]{\eta , 1}}^{u}(p)$ (obtained in Lemma 3.3) for each $\eta \in [\eta _{1},\eta _{2}]$ and for m close to 1. Furthermore, note that each horizontal line $\{y=\hat{y}_{i}\}\subset W_{\smash [t]{\eta , 1}}^{s}(p)$ in the proof of Lemma 3.4 consists of regular points: for every $x\in \{y=\hat{y}_{i}\}\subset \mathbb {R}^{2}$ and for n such that $F_{\smash [t]{\eta , 1}}^{n}(x)=p$, it holds that $\text {Im}(DF_{\smash [t]{\eta , 1}}^{n}(x))+T_{p}(\{y=1\})=\mathbb {R}^{2}$, that is, the vectors $(\varPi _{\smash [t]{j=1}}^{n-1}f^{\prime }(f^{j-1}(\hat{y}_{i})),\varPi _{\smash [t]{j=1}}^{n}f^{\prime }(f^{j-1}(\hat{y}_{i})))$ and (1, 0) span $\mathbb {R}^{2}$ because $f^{j-1}(\hat{y}_{i})\ne \theta$ for $j=1,\dots , n$. Thus, by Proposition 1 in Appendix 4 in Palis and Takens (1993, p. 182), the nonsingular map $F_{\eta , m}$ for m sufficiently close to 1 has an arc $\gamma _{\eta ,m}^{s}\subset W_{\eta , m}^{s}(p)$ which is $C^{2}$-close to a suitable compact line segment of the horizontal line $y=\hat{y}_{i}$. By the stability of transversality and Lemma 3.4, (a) and (b) follow, which are the situation of “inevitable tangency”, a part of Takens’ weakened generic conditions for real-analytic families of diffeomorphisms [see Takens (1992)]. Evidently, the existence of $\eta _{\hat{H}}$ for which $\gamma _{\smash [t]{\eta _{\hat{H}}, m}}^{s}$ and $\gamma _{\smash [t]{\eta _{\hat{H}}, m}}^{u}$ have a quadratic homoclinic tangency follows from (a) and (b). By Takens’ weakened generic conditions, we immediately have the generic unfolding of homoclinic tangency: in fact, the ratio $-\log (|\lambda _{2}(\eta )|)/\log (|\lambda _{1}(\eta )|)$ of eigenvalues $\lambda _{1}$ and $\lambda _{2}$ of $DF_{\eta , m}(p)$ is clearly nonconstant with respect to $\eta$. This proves (c) and thus (ii) of Lemma 3.2. $\square$

A.2 Proof of Proposition 3.2

(i) Since $v_{t}=v_{0}^{\smash [t]{(-\eta )^{t}}}$, we have $\limsup _{t \rightarrow \infty }v_{t}=\infty$ for $v_{0}>0$ and $v_{0}\ne 1$. If $v_{0}=1$, then $v_{t}=1$ for $t\ge 0$ and $u_{t+1}=(1-\phi )u_{t}+\phi$, implying $u_{t}\rightarrow 1$ as $t\rightarrow +\infty$ for any $u_{0}$.

(ii) Suppose first that the (positive) sequence $\{v_{t}\}$ generated by (3.4) and (3.5) is eventually uniformly bounded from above in the sense that $\limsup _{t\rightarrow \infty }v_{t}\le \bar{v}$ for some $0<\bar{v}<\infty$, independent of $(u_{0}, v_{0})\in \mathbb {R}_{++}^{2}$. In other words, there exist an integer K (depending on $v_{0}$ and $u_{0}$) and a uniform $\bar{v}<+\infty$ such that $0<v_{t}\le \bar{v}$ for any $t\ge K$. If so, then $u_{t+1}=(1-\phi )u_{t}+\phi v_{t+1}\le (1-\phi )u_{t}+\phi \bar{v}$ for $t\ge K$. Thus, for $t>K$, $u_{t} \le (1-\phi )^{t-K}u_{K}+\phi \bar{v}\sum _{i=0}^{\smash [t]{t-K-1}}(1-\phi )^{i} \le (1-\phi )^{t-K}u_{K}+\bar{v}$, which implies $\limsup _{t\rightarrow \infty }u_{t} \le \bar{v}$. Evidently, if $v_{t}$ (and thus also $u_{t}$) is eventually uniformly bounded from above, then $\liminf _{t\rightarrow \infty }u_{t}\ge \underline{u}$ and $\liminf _{t\rightarrow \infty }v_{t}\ge \underline{v}$ for some uniform $\underline{u}>0$ and $\underline{v}>0$. The rectangle defined by $R=[\bar{v},\underline{u}]\times [\bar{v},\underline{v}] \subset \mathbb {R}_{++}^{2}$ is then a required compact region.

Therefore, it suffices to show that $\{v_{t}\}$ is eventually uniformly bounded from above. If it does not hold, then, for any $\varepsilon >0$, there exists $L=L(\varepsilon , u_{0}, v_{0})$ such that $v_{L+3}>\varepsilon ^{\smash [t]{-1/\eta }}$ or, equivalently, $0<mu_{L+2}+(1-m)v_{L+2}<\varepsilon$, which implies that $u_{L+2}<\varepsilon /m$ and $v_{L+2}<\varepsilon /(1-m)$. Thus we get

$$\begin{aligned} (1-\phi )u_{L+1}+\frac{\phi }{\left( mu_{L+1}+(1-m)v_{L+1}\right) ^{\eta }}< & {} \frac{\varepsilon }{m}, \end{aligned}$$

(3.17)

$$\begin{aligned} \frac{1}{\left( mu_{L+1}+(1-m)v_{L+1}\right) ^{\eta }}< & {} \frac{\varepsilon }{1-m}. \end{aligned}$$

(3.18)

From (3.17) we obtain

$$\begin{aligned} u_{L+1}<\frac{\varepsilon }{m(1-\phi )}. \end{aligned}$$

(3.19)

Rearranging (3.18) gives

$$\begin{aligned} \left( \frac{1-m}{\varepsilon }\right) ^{\frac{1}{\eta }}<mu_{L+1}+(1-m)v_{L+1}. \end{aligned}$$

(3.20)

By combining (3.19) and (3.20), we obtain

$$\begin{aligned} v_{L+1}>\frac{1}{1-m}\left( \frac{1-m}{\varepsilon }\right) ^{\frac{1}{\eta }}-\frac{\varepsilon }{(1-\phi )(1-m)}\equiv \varDelta (\varepsilon ). \end{aligned}$$

(3.21)

From (3.19) and (3.21), it follows that

$$\begin{aligned} \frac{\varepsilon }{m(1-\phi )}> & {} u_{L+1}=(1-\phi )u_{L}+\frac{\phi }{\left( mu_{L}+(1-m)v_{L}\right) ^{\eta }} \\= & {} (1-\phi )u_{L}+\phi v_{L+1} \\> & {} (1-\phi )u_{L}+\phi \varDelta (\varepsilon ) \\\ge & {} \phi \varDelta (\varepsilon ). \end{aligned}$$

Hence, we obtain $\varepsilon >m\phi (1-\phi )\varDelta (\varepsilon )$. Since $\varDelta (\varepsilon )\rightarrow +\infty$ as $\varepsilon \rightarrow 0$, this is a contradiction. This completes the proof. $\square$

A.3 Proof of Proposition 3.3

(i) Since $f_{\eta }$ is unimodal and the Schwarzian derivative of $f_{\eta }$ given by (3.16) is negative (i.e., $Sf_{\eta }(x)<0)$ for $\eta \ge 1$ and $x>0$, $f_{\eta }$ has, by Singer’s theorem (Singer 1978), at most one periodic attractor for $\eta \ge 1$, and so does $F_{\eta , 1}$. For $\eta \in (0,1)$, the unique fixed point $x=1$ for $f_{\eta }$ has been shown to be globally attracting [see Proposition 2.1], meaning that the fixed point $p=(1,1)$ for $F_{\eta , 1}$ is also globally attracting.

(ii) See (iv) in Proposition 3.1. $\square$