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Quasi-periodic motions in a two-class economy with technology choice: an extreme case

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This paper constructs a simple overlapping generations (OLG) model with the working and capitalist classes and two types of production technologies. The behavior of agents belonging to the working class is basically the same as that in the standard Diamond (Am Econ Rev 55:1126–1150, 1965) type OLG model, whereas agents belonging to the capitalist class face two available technologies, select the one with a higher return on capital, and bequeath their assets to the next generation without supplying labor. Using techniques concerning the circle map in dynamical systems theory, we show that in an extreme case in which one technology is linear and the other is of the Leontief type, the economy exhibits bounded, non-periodic but non-chaotic motions for a large set of parameter values. We provide explicit formulas for the rotation number and the absolutely continuous invariant probability measure of the model.

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  1. Although Pintus et al. [29] analyze the circle dynamics using a standard infinite horizon agent model, their analyses depend on numerical simulations.

  2. Woodford [37] and Pintus et al. [29] also assume that the economy consists of two classes: workers and capitalists.

  3. See also Aghion et al. [1], Iwaisako [18], and Matsuyama [25] for graphical analyses of endogenous business cycle models due to endogenous technology choice, and Kunieda and Shibata [20], Asano et al. [2], Matsuyama et al. [24], Asano and Yokoo [4], and Umezuki and Yokoo [35] for more rigorous mathematical analyses.

  4. Here we simply assume this technology choice criterion. However, this kind of behavior can be derived from agents’ explicit optimization under some circumstances. See Appendix A.

  5. In this model, the capitalists are assumed to consume only when they are young. It is easy to extend this setting to the case in which the capitalists consume both when young and old. Under the Cobb–Douglas utility specification, the introduction of utility from old age consumption causes only a reduction in the rate of wealth accumulation at a fixed rate, and the basic structure of dynamics derived below remains unchanged.

  6. For this argument, see, for example, Turer [34].

  7. For a given map \(g:X \rightarrow X\), \(x\in X\) is said to be periodic if \(g^q(x)=x\) for some integer \(q\ge 1\), where \(g^q\) denotes the composition of g with itself q times. Furthermore, if \(g^q(x)=x\) and \(g^n(x)\ne x\) for \(1\le n< q\), then the point x is called a periodic point of period q.

  8. In the literature, it is also called almost periodic.

  9. Several economic models in which the dynamics can be characterized by a piecewise continuous map with a discontinuity of the interval can be identified with some circle maps, but they are not necessarily continuous as circle maps. For instance, every economic model developed in Ishida and Yokoo [17], Asano et al. [2], and Umezuki and Yokoo [35] could be treated as a circle map. However, in contrast to ours, all these models have a discontinuity as a circle map. For its dynamic consequences, see Keener [19].

  10. Note that \(\text {gcd}(p,q)\) denotes the greatest common divisor of p and q.

  11. This also says that the Lyapunov exponent \(\lim _{n \rightarrow \infty }(1/n) \sum _{t=0}^{n-1} \log |T'_{a,b}(x_t)|=\log a^{1-\alpha }b^{\alpha }\) is zero, which indicates that there is neither expansion nor contraction on average for the trajectories generated by the map. This applies to the case where the rotation number of \(T_{a,b}\) is rational as well. In fact, each periodic point of \(T_{a,b}\) is neither repelling nor attracting.


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The authors would like to thank two anonymous referees for their useful comments and suggestions.


Takao Asano received research grants from the Grants-in-Aid for Scientific Research, JSPS (20K01745, 20H01507, 17K03806). Akihisa Shibata received research grants from the Grants-in-Aid for Scientific Research, JSPS (20H05631). Masanori Yokoo received research grants from the Grants-in-Aid for Scientific Research, JSPS (21K01388). All of the three authors received research Grants from International Joint Research Center of Advanced Economic Research of KIER (Kyoto University).

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This paper was previously circulated under the title “Polarization and Permanent Fluctuations: Quasi-Periodic Motions in a Two-Class OLG Model”.


Appendix A: A microfoundation of the technology choice behavior

In the main text, we assume that agents select the technology giving the higher return from capital. In Appendix, we show that such technology choice behavior can be derived under some circumstances.

Our basic setup follows Matsuyama [25]. There are N types of production technologies in this economy. A type i technology converts \(e_{i}\) units of the final goods into \(e_{i}V_{i}\) units of capital and the final good is produced by \(F_{i}(K_{t},L_{t})\), where \(K_{t}\) and \(L_{t}\) are capital and labor at time t, respectively. The final good production functions in per agent terms are

$$\begin{aligned} \frac{F_{i}(K_{t},L_{t})}{L_{t}}=F_{i}(k_{t},1)=f_{i}(k_{t}),\,\,\,i=1, \ldots ,N \end{aligned}$$

where \(k_{t}=K_{t}/L_{t}\).

Agents have log-linear utility; their saving rate is constant and independent of the return from saving. Any saver has two options in managing their saving, namely becoming either a lender or an entrepreneur. An agent selecting to be a lender lends his/her saving \(\eta _{t}\) and obtains \( r_{t+1}\eta _{t}\) when old, where \(r_{t+1}\) denotes the real interest rate. An agent becoming an entrepreneur selects one technology from the N types of technologies. Because an entrepreneur’s wealth is equal to his/her saving, if \(e_{i}>\eta _{t}\), the entrepreneur has to borrow \(e_{i}-\eta _{t}\) . However, due to the presence of capital market frictions, each entrepreneur can pledge only up to a constant fraction of the project revenue for the repayment, \(\lambda _{i}e_{i}V_{i}f_{i}^{\prime }(k_{t+1}),\) where \(0\le \lambda _{i}\le 1.\) The fraction, \(\lambda _{i},\) differs between the N types of projects. The entrepreneur’s borrowing constraint is represented by

$$\begin{aligned} \lambda _{i}e_{i}V_{i}f_{i}^{\prime }(k_{t+1})\ge r_{t+1}(e_{i}-\eta _{t}) \text { for }i=1,\ldots ,N. \end{aligned}$$

As \(\lambda _{i}\) becomes smaller, the credit constraint becomes stronger.

Because an entrepreneur is always able to choose to become a lender, earnings from investment will not be smaller than those from lending:

$$\begin{aligned} f_{i}^{\prime }(k_{t+1})e_{i}V_{i}-r_{t+1}(e_{i}-\eta _{t})\ge r_{t+1}\eta _{t}, \end{aligned}$$

that is:

$$\begin{aligned} r_{t+1}\le f_{i}^{\prime }(k_{t+1})V_{i}\text { for }i=1,\ldots ,N. \end{aligned}$$

(18) can be rewritten as:

$$\begin{aligned} r_{t+1}\le \frac{V_{i}f_{i}^{\prime }(k_{t+1})}{\left( 1-\frac{\eta _{t}}{ e_{i}}\right) /\lambda _{i}}\text { for }i=1,\ldots ,N. \end{aligned}$$

By defining:

$$\begin{aligned} \Phi _{i}\equiv \frac{V_{i}f_{i}^{\prime }(k_{t+1})}{\max \left\{ 1,\left( 1- \frac{\eta _{t}}{e_{i}}\right) /\lambda _{i}\right\} }, \end{aligned}$$

we can summarize (18) and (19) as:

$$\begin{aligned} r_{t+1}\le \Phi _{i}\text { for }i=1,\ldots ,N. \end{aligned}$$

Let us assume here that \(r_{t+1}<\Phi _{i}\). Then, all agents become entrepreneurs and adopt type i technology and there is no lender in this economy. Obviously, this cannot be an equilibrium as we have \(r_{t+1}\ge \Phi _{i}.\) Next, let us suppose that \(r_{t+1}>\Phi _{i}\). Then, at least one of (18) and (19) for i is not satisfied, and thus, type i is not adopted. In equilibrium, because we must have positive investment, it follows that:

$$\begin{aligned} r_{t+1}=\underset{}{\max }\left\{ \Phi _{1},\Phi _{2},\ldots ,\Phi _{N}\right\} , \end{aligned}$$

showing that the technology yielding the highest value on the right-hand side of (20) is adopted.

Let us consider a special case of (20), that is:

$$\begin{aligned} N=2,{ }V_{1}=V_{2}=V,{ }\lambda _{1}=\lambda _{2}=\lambda \text { and }e_{1}=e_{2}=e. \end{aligned}$$

In this case, (20) reduces to

$$\begin{aligned} r_{t+1}= & {} \underset{}{\max }\left\{ \frac{Vf_{1}^{\prime }(k_{t+1})}{\max \left\{ 1,\left( 1-\frac{\eta _{t}}{e}\right) /\lambda _{{}}\right\} },\right. \\&\quad \left. \frac{ Vf_{2}^{\prime }(k_{t+1})}{\max \left\{ 1,\left( 1-\frac{\eta _{t}}{e}\right) /\lambda _{{}}\right\} }\right\} \\= & {} \frac{V}{\max \left\{ 1,\left( 1-\frac{\eta _{t}}{e}\right) /\lambda \right\} }\underset{}{\max }\left\{ f_{1}^{\prime }(k_{t+1}),f_{2}^{\prime }(k_{t+1})\right\} . \end{aligned}$$

Thus, we can confirm that agents select the technology with a higher marginal productivity of capital.

Appendix B: Proofs

Proof of Proposition 1

For case (i), we first notice that \(k_{t_0}=x_{t_0} \le 1\) for some \(t_0\). As \(a>1\) in \(F_L\), which makes \(x_t\) increase; it follows that \(k_{t_1}=x_{t_1}\in (1,a]\) for some \(t_1>t_0\). As \(b\in (0,1)\) in \(F_R\), which makes \(x_t\) decrease, we have that \(x_t\le a\) for \(t \ge t_1\). Moreover, there is the smallest integer \(n\ge 1\) such that \(k_{t_1+n}=c+x_{t_1+n}\le 1\) or \(x_{t_1+n}\le 1-c\) and that \(k_{t_1+n-1}=c+x_{t_1+n-1}> 1\) or \(x_{t_1+n-1}>1-c\). Thus, it follows that \(x_t\ge b(1-c)\) for \(t\ge t_1+n\). For case (ii), the statement is evident from (9). \(\square \)

Proof of Proposition 3

See de Faria and Tresser [9] for the proof for an equivalent mathematical model. However, we provide a proof for self-containedness.

Since \(T_{a,b}\) is a circle homeomorphism by Proposition 2, the rotation number exists, which we denote by \(\alpha \). We claim that it must satisfy:

$$\begin{aligned} a^{1-\alpha }b^{\alpha }=1. \end{aligned}$$

There are two cases to consider. First, when \(\alpha \) is irrational, then we know from Theorem 1 in Coelho et al. [7] that \(T_{a,b}\) is uniquely ergodic and we let its unique invariant measure be \(\mu \). By definition, \(\alpha =\mu (b,T_{a,b}(b))=\mu (b,ab)\). Since \(\mu \) is invariant under \(T_{a,b}\), it follows that \(\alpha =\mu (b,ab)=\mu (T^{-1}_{a,b}(b,ab))=\mu (1,a)\). By the Ergodic Theorem and the fact that \(|DT^n_{a,b}|\), where D denotes the derivative, is bounded away from 0 and infinity (see for this point, Coelho et al. [7], Proposition 2]), we have

$$\begin{aligned} 0=\lim _{n \rightarrow \infty } \frac{1}{n}\log |DT^n_{a,b}(x)|=\int \log |DT_{a,b}|\text {d}\mu (x), \end{aligned}$$

which implies

$$\begin{aligned} 0= & {} \mu (b,1) \log a + \mu (1,a) \log b\\= & {} (1-\alpha ) \log a + \alpha \log b. \end{aligned}$$

ThisFootnote 11 gives (21).

Next, when \(\alpha \) is rational or \(\alpha =p/q\) where p and q are prime integers, there is some \(x_0\in [b, a]\) such that \(T^q_{a,b}(x_0)=x_0\), implying \(a^{(1-\alpha ) q} b^{\alpha q} x_0=x_0\), and thus, we obtain (21). This means that \(DT^q_{a,b}\) is identically unity and therefore, every point in [ba] is a periodic point of period q.

Taking logarithm of (21) and solving for \(\alpha \), we obtain:

$$\begin{aligned} \alpha =\log a/\log (a/b). \end{aligned}$$

For the topological conjugacy (see, e.g., the proof of Theorem 1 in de Faria and Tresser (2014) ), let \(h:I=[0,1] \rightarrow I_{a,b}=[b,a]\) be given by:

$$\begin{aligned} h(t)=b(a/b)^t, \end{aligned}$$

which is clearly a homeomorphism. To prove conjugacy, it suffices to check that \(h\circ R_{\alpha }=T_{a,b}\circ h\). There are two cases to consider: (i) \(0 \le t <1-\alpha \), and (ii) \(1-\alpha <t \le 1\).

For case (i), as \(R_{\alpha }(t)=t+\alpha \), it follows that

$$\begin{aligned} h\circ R_{\alpha }(t)= & {} h(t+\alpha ) \\= & {} b\left( \frac{a}{b}\right) ^{t+\alpha }\\= & {} \left( \frac{a}{b}\right) ^{\alpha }h(t)\\= & {} ah(t) =T_{a,b}\circ h(t). \end{aligned}$$

Similarly, for case (ii), as \(R_{\alpha }(t)=t+\alpha -1\), it follows that

$$\begin{aligned} h\circ R_{\alpha }(t)= & {} h(t+\alpha -1) \\= & {} b\left( \frac{a}{b}\right) ^{t+\alpha -1}\\= & {} \left( \frac{a}{b}\right) ^{\alpha }\frac{b}{a}h(t)\\= & {} bh(t) =T_{a,b}\circ h(t). \end{aligned}$$

Thus, the topological conjugacy between \(R_{\alpha }\) and \(T_{a,b}\) is proven.

Finally, the absolutely continuous invariant measure \(\mu \) for \(T_{a,b}\) in \(I_{a,b}\), when \(\alpha \) is irrational, can be expressed as the push-forward of the Lebesgue measure \(\lambda \) in [0, 1] via the homeomorphism h. See again the proof of Theorem 1 in de Faria and Tresser (2014) . That is, for a Borel measurable set \(E\subset I_{a,b}\):

$$\begin{aligned} \mu (E)=\lambda (h^{-1}(E))=\int _{h^{-1}(E)}dt=\int _{E}(h^{-1}(x))'\text {d}x. \end{aligned}$$

From (22), we have:

$$\begin{aligned} (h^{-1})'(x)=\left( \frac{\log (x/b)}{\log (a/b)}\right) '=\frac{1}{x\log (a/b)}. \end{aligned}$$

Thus, we obtain:

$$\begin{aligned} d\mu (x) =\frac{dx}{x \log (a/b)}, \end{aligned}$$

as desired. \(\square \)

Proof of Proposition 4

We first show that for any initial condition \((k_0,x_0)\), the trajectory of \(x_t\) (\(t=0,1,2, \dots \)) generated by the iteration of the map (7) is eventually trapped in the interval \(I_{a,b,c}=[(1-c)b,(1-c)a]\) and that its dynamics is governed by (15). That is, there is some \(t_0\ge 0\) (depending on the initial condition) such that for \(t\ge t_0\), \(x_t\in [(1-c)b,1-c)\) if and only if \(k_t<1\) and \(x_t\in I_{a,b,c}\).

From Proposition 1 and the inevitable occurrence of technology change for \(c\in [0,1)\), it suffices to assume \(x_0=k_0\in L=[(1-c)b,1)\subset [(1-c)b,a]\). By partitioning \(L=L_1\cup L_2\), where \(L_1=[(1-c)b,(1-c))\) and \(L_2=[1-c,1)\), we have two cases to examine.

Case (i): Let \(x_0=k_0\in L_1\). Then, \(F(k_0,x_0)=F_L(x_0,x_0)\in [ab(1-c), a(1-c)]^2\subset [1,a(1-c)]^2\). The last inclusion is followed by the assumption that \(c\le 1-1/ab\) in (16). Thus, \(x_1\ge 1\) and \(k_1\ge 1\). Then there is the smallest integer \(n\ge 1\) such that \(F^{n+1}(k_0,x_0)=F^{n}_R\circ F_L(k_0,x_0)=F^{n}_R\circ F_L(x_0,x_0)=(b^{n}x_1+c, b^{n}x_1)=(k_{n+1}, x_{n+1})\) with \(x_{n+1}< 1\). There are two subcases to follow. That is, subcase (i-1): \(k_{n+1}\ge 1\) and subcase (i-2): \(k_{n+1}< 1\).

For subcase (i-1), from \(k_{n+1}-x_{n+1}=c\), we have \(x_{n+1}\in L_2\). As \(k_{n+1}\ge 1\), we have \(F(k_{n+1},x_{n+1})=F_R(k_{n+1},x_{n+1})=(bx_{n+1}+c, bx_{n+1})\!=\!(k_{n+2},x_{n+2})\). As \(x_{n+1}\in [(1-c),1)=L_2\), we have \(x_{n+2}=bx_{n+1}\in L_1\). We also obtain \(k_{n+2}\in L\), that is, \(k_{n+2}<1\), because \(k_{n+2}=x_{n+2}+c\) and \(|L_2|=c\). Thus, \(F(k_{n+2},x_{n+2})=F_L(x_{n+2},x_{n+2})\), which brings us to the beginning of case (i) and we are done for \(k_0=x_0\) and \(t\ge 0 \). For subcase (i-2), \(k_{n+1}< 1\) implies \(x_{n+1}< 1-c\) and hence \(x_{n+1}\in L_1\). Thus, again, this subcase goes back to the beginning of case (i). Therefore, the argument above shows that the dynamics of \(x_t\) is governed by (15) for case (i) for any \(t\ge 0\).

Case (ii): Let \(x_0=k_0\in L_2\). As \(k_0< 1\), we have \(F(k_0,x_0)=F_L(x_0,x_0)=(ax_0, ax_0)\). Thus, \(k_1=x_1\in (a(1-c), a]\), which implies \(x_1\notin I_{a,b,c}\). However, as \(c\le 1-1/ab\) by (16), we have \(a(1-c)>ab(1-c)\ge 1\) and hence \(x_1=k_1>1\). Therefore, there is the smallest integer \(m\ge 1\) such that \(F^{m+1}(k_0,x_0)=F^{m}_{R}\circ F_L(k_0,x_0)=F^{m}_R(x_1,x_1)=(b^mx_1+c,b^mx_1)=(k_{m+1},x_{m+1})\) with \(x_{m+1}< 1\), which implies that the argument reduces to that of case (i). Thus, we are also done for \(k_0=x_0\in L_2\) and for \(t\ge m+1\).

Thus, the above argument shows that for any initial conditions \((k_0, x_0)\), \(x_t\in I_{a,b,c}\) for \(t\ge t_0\) for some \(t_0\) and that the sequence of \(x_t\) can be described by (15) for \(t\ge t_0\).

For conjugacy, let \(h_c:I_{a,b} \rightarrow I_{a,b,c}\) with \(h_c(x)=(1-c)x\). Then, we see that for \(x\in I_{a,b}\), \(h_c\circ T_{a,b}(x)=\tau _{a,b,c}\circ h_c(x) \in I_{a,b,c}\). Because \(\tau _{a,b}\) is topologically conjugate to the rigid rotation by Proposition 3, so is \(\tau _{a,b,c}\) by the chain of topological conjugacy.

For the invariant measure, we can take a homeomorphism \(\varphi : [0,1] \rightarrow I_{a,b,c}=[(1-c)b, (1-c)a]\) such that \(\varphi (t)=h_c\circ h(t)=(1-c)b(a/b)^t\) and use the same argument as in Proposition 3. This completes the proof. \(\square \)

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Asano, T., Shibata, A. & Yokoo, M. Quasi-periodic motions in a two-class economy with technology choice: an extreme case. Nonlinear Dyn 110, 945–961 (2022).

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