Chapter 11
Complexity of Economic Growth

Economic growth is a complicated topic. It can be examined from many perspectives. The economic growth theory developed in this book addresses some important aspects of economic growth, but neglects many others. For instance, population, resources and environment are essential determinants of global economic growth. We almost omit them on purpose because a proper treatment of these factors would greatly lengthen the book. This book constructed (the key part of) a comprehensive growth theory. From the economic structural point of view, our theoretical framework includes some well established economic systems such as the Arrow-Debreu equilibrium model, the Solow-Swan model, the Uzawa two-sector model, the Kaldor-Pasinetti two-group model, the Arrow learning-by-doing model, the Ricardian model by Pasinetti and Samuelson, as special cases. We can now treat different ideas in growth theory within a single theoretical framework. Indeed, the usefulness of our framework lies not only in that it unifies some theories, but mainly in that it can explain, in a compact way, many economic phenomena which cannot be explained by the traditional theories.

This book covers a large area of economics with a few assumptions. Our main purpose is to construct a theoretical framework which would permit valid generalizations from one special modeling structure to another, and would deepen our understanding of economic evolution as an organic whole. It is easy to extend and generalize most of the models in this book. This book actualizes a general vision about economic evolution. Our investigation is quite limited in scope if one considers obviously possible extensions of the framework proposed in this book and available modern analytical methods and modern computer. We now conclude this book, commenting some aspects of complexity of economic evolution.

11.1 Preference Change with the OSG Model

By preference change, we mean change in attitudes towards savings in different forms, different types of work, food consumption structure, entertainment, different forms and levels of education in different fields, and different kinds of friendships. Preferences change as time passes and socioeconomic conditions vary. Preference and its dynamics play an important role in determination of economic structure and its dynamics. But preference change, growth, and economic structure have been rarely examined within a compact framework. There are a few studies in optimal growth theory, which examine dynamic preference and economic growth.

The traditional growth theory has two main modeling frameworks - the Solow model and the Ramsey model. The Solow model is evidently not proper to deal with preference changes since it does not have a rational mechanism to determine consumer choice. For instance, the Solow model cannot deal with choice between consumption of goods and leisure time, based on rational choice theory. This book introduces various extensions of the Ramsey model. These examples demonstrate that the Ramsey model may not be a proper modeling framework for analyzing various economic issues associated with economic growth not only because of the conceptual issues raised in Chapter 3 but also because of analytical difficulties involved in the modeling framework. The OSG model proposed in this model overcomes the lacking of rational choice theory in the Solow model and simplifies or avoids possible analytical complexity of the Ramsey model.

It is easy to see why the Solow model is not proper for analyzing preference change - human behavior is predetermined. The Ramsey model is not proper for taking account of change in preference structure (for instance, with regards to housing, leisure time and food). The Ramsey model maximizes ${\int }_0^{\infty }$ U(C)exp(-ρt)dt, subject to the dynamic budget constraint of capital accumulation. In the traditional studies of growth and preference change, by preference change it means changes of ρ.¹ Since ρ is defined with respect to utility, it is conceptually not easy to introduce reasonable dynamics of ρ. It is difficult to discuss issues related to preference change and economic structure within the Ramsey framework. For instance, when C(t) is a vector consisting leisure time and consumption levels of agricultural and industrial goods, it is difficult to see how changes in ρ can take account of possible changes in the preference structure for C(t).

This section is concerned with relationships between savings, income, and wealth. In fact, Chapter 3 has already introduced two types of preference changes that make the OSG model to generate the same pattern of behavior as the Solow model and the Ramsey model. The economic system is the same as the OSG model proposed in Chapter 2, except that preference is changeable. The neoclassical production function is F(t) = F(K(t), N^*(j)), where K(t) and F(t) are capital stocks and product, N is fixed labor force, z is a fixed level of human capital, and N^* = zN is the effective labor input. The marginal conditions are r(t) = F_K(t) and w(t) = zF_N. (t).

The current income Y is given by:

$Y\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathit{r}\mathrm{(}\mathit{t}\mathrm{)}\mathit{K}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{+}\mathit{w}\mathrm{(}\mathit{t}\mathrm{)}\mathit{N}\mathrm{=}\mathit{F}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{.}$

Consumers' budget constraints are:

$\mathit{C}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{+}\mathit{S}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\stackrel{\mathrm{ˆ}}{\mathit{Y}}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{\equiv }\mathit{Y}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{+}\mathit{\delta }\mathit{K}\mathrm{(}\mathit{t}\mathrm{)}\text{,}$

where C(t) and S(t) are defined as in Chapter 2. We specify utility functions as follows:

$\mathit{U}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{=}\mathit{C}\mathrm{(}\mathit{t}{\mathrm{)}}^{\mathit{\xi }\mathrm{(}1\mathrm{)}}\mathit{S}\mathrm{(}\mathit{t}{\mathrm{)}}^{\mathit{\lambda }\mathrm{(}t\mathrm{)}},\mathit{\xi }\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{+}\mathit{\lambda }\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathrm{1}\mathrm{,}\mathit{\xi }\mathrm{(}\mathit{t}\mathrm{)},\mathit{\lambda }\mathrm{(}\mathit{t}\mathrm{)}\mathrm{>}\mathrm{0},$

where the propensity to consume goods and to own wealth are changeable in time.

The optimal solution is C = ξŶ and S = λŶ. Capital accumulates by:

$\dot{\mathit{K}}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{=}\mathit{\lambda }\mathrm{(}\mathit{t}\mathrm{)}\stackrel{\mathrm{ˆ}}{\mathit{Y}}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{-}\mathit{K}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathit{\lambda }\mathrm{(}\mathit{t}\mathrm{)}Y\mathrm{(}\mathit{t}\mathrm{)}\mathrm{-}\mathrm{(}\mathrm{1}\mathrm{-}\mathit{\delta }\mathit{\lambda }\mathrm{(}\mathit{t}\mathrm{)}\mathrm{)}K\mathrm{(}\mathit{t}\mathrm{)}\mathrm{.}$(11.1.1)

If λ is constant (which also implies that ξ, is constant because of ξ + λ = 1), the dynamic system is formally identical to the dynamics of the OSG model proposed in Chapter 2. We repeat the following three theorems in Chapter 3 to show how the OSG model can explain the same economic phenomena by the Solow model, the Ramsey model, and the general Solow model with poverty traps.

Theorem 3.2.1

Let the production sectors be identical in the OSG model and the Solow model. If the savings rate ŝ in the Solow model and the propensity to save λ(t) in the OSG model satisfy:

$\mathit{\lambda }\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{=}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{)}\frac{\mathit{\delta }}{\mathit{f}\mathrm{(}\mathit{k}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{)}\mathrm{/}\mathit{k}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{+}\mathit{\delta }},$

then the OSG model is identical to the Solow model in terms of the saving rate (out of current income), the consumption rate, the interest rate, the wage rate, output, income, consumption, and saving.

Theorem 3.6.1

Let the production sectors be identical in the OSG model and the Ramsey model with the temporary utility function:

$\mathit{u}\mathrm{(}\mathit{c}\mathrm{)}\mathrm{=}\frac{\mathit{c}\mathrm{(}\mathit{t}{\mathrm{)}}^{\mathrm{1}-\mathit{\theta }}\mathrm{-}\mathrm{1}}{\mathrm{1}\mathrm{-}\mathit{\theta }}\mathrm{.}$

If the propensity to save λ(t) evolves according to:

$\dot{\mathit{\lambda }}\mathrm{(}t\mathrm{)}\mathrm{=}\frac{{\mathit{f}\prime }^{}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{+}\mathit{\delta }}{\mathit{f}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{+}\delta k\mathrm{(}\mathit{t}\mathrm{)}}\mathit{\xi }\mathrm{(}\mathrm{t}\mathrm{)}\dot{\mathit{k}}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{-}\frac{\mathit{f}\mathrm{\text{'}}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{-}\mathit{\rho }}{\mathit{\theta }}\mathit{\xi }\mathrm{(}\mathrm{t}\mathrm{)}\text{,}$

then the OSG model generates the same dynamics of capital-labor ratio k(t) and per-capita consumption c(t) as the Ramsey model does.

Theorem 3.7.1

Let the production sectors be identical in the OSG model and the generalized Solow model with poverty traps with the following consumption function:

$\mathit{c}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathrm{(}{\mathit{y}}_{\mathrm{0}}\mathrm{-}{\mathit{\delta }}_{\mathit{k}}{\mathit{k}}_{\mathrm{0}}\mathrm{)}\mathrm{+}\overline{\mathit{\xi }}\mathrm{(}\mathit{\gamma }\mathrm{-}{\mathit{c}}_{\mathrm{0}}\mathrm{)}\mathrm{.}$

If the savings rate:

$\stackrel{\mathrm{ˆ}}{\mathit{s}}\mathrm{(}\mathrm{t}\mathrm{)}\mathrm{=}\frac{\mathit{y}\mathrm{(}t\mathrm{)}\mathrm{-}\mathit{c}\mathrm{(}\mathrm{t}\mathrm{)}}{\mathit{y}\mathrm{(}\mathrm{t}\mathrm{)}}\mathrm{=}{\stackrel{\mathrm{ˆ}}{\mathit{s}}}_{\mathrm{0}}\mathrm{-}\frac{{\mathit{\delta }}^{*}}{\mathit{y}\mathrm{(}\mathit{t}\mathrm{)}}\mathrm{,}{\stackrel{\mathrm{ˆ}}{\mathit{s}}}_{\mathrm{0}}\mathrm{\equiv }\mathrm{1}\mathrm{-}\overline{\mathit{\xi }}\mathrm{>}\mathrm{0}\mathrm{,}{\mathit{\delta }}^{*}\mathrm{\equiv }{\stackrel{\mathrm{ˆ}}{\mathit{s}}}_{\mathrm{0}}{\mathit{y}}_{\mathrm{0}}\mathrm{-}{{\mathit{\delta }}_k}_{}{\mathit{k}}_{\mathrm{0}}\mathrm{,}$

in the general Solow model and the propensity to save λ(t) in the OSG model satisfy:

$\mathit{\lambda }\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathrm{1}\mathrm{-}\frac{\overline{\mathit{\xi }}\mathit{y}\mathrm{+}{\mathit{\delta }}^{*}}{\mathit{y}\mathrm{+}\delta k},$

then the OSG model is identical to the generalized Solow model with poverty traps.

This section proposes another type of preference change. As ξ + λ = 1 holds at any point of time, it is sufficient to be concerned with λ(t). In this section, we assume that the propensity λ(t) to own wealth is affected by K(t) and Y(t) as:

$\dot{\mathit{\lambda }}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{=}\mathit{\theta }\mathrm{\{}G\mathrm{(}\mathit{K}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{,}\mathit{Y}\mathrm{(}\mathit{t}\mathrm{)}\mathrm{)}\mathrm{-}\mathit{\lambda }\mathrm{\}}\mathrm{,}1\mathrm{>}\mathit{\lambda }\mathrm{\ge }\mathrm{0}\mathrm{,}\mathrm{\infty }\mathrm{>}\mathit{\theta }\mathrm{\ge }\mathrm{0}\mathrm{,}$(11.1.2)

where θ is a positive adjustment parameter and G is a continuous function of K and Y. If θ = 0, λ is constant. If θ → ∞, G(K, Y) = λ is held at any point of time. The preference changes specified in Theorems 3.2.1 and 3.7.1 belong to this case. This section specifies G as follows:

$\mathit{G}\mathrm{=}\frac{{\mathit{\theta }}_{\mathrm{1}}}{\mathrm{1}\mathrm{+}{\mathit{\theta }}_{\mathrm{1}}{\mathit{K}}^0\mathrm{(}\mathit{Y}\mathrm{-}{\mathit{Y}}_{\mathrm{0}}{{\mathrm{)}}^{}}^b}\mathrm{,}\mathrm{1}\mathrm{>}{\mathit{\theta }}_1\mathrm{>}\mathrm{0}\mathrm{,}{\mathit{\theta }}_{\mathrm{2}}\mathrm{>}\mathrm{0}\mathrm{,}\mathit{Y}\mathrm{\ge }{\mathit{Y}}_{\mathrm{0}}\mathrm{\ge }\mathrm{0..}$(11.1.3)

Here, Y₀ is called the basic (or survival) level. If Y(t) < Y₀, then G = 0. If the level of output is lower than the survival level, the propensity to hold wealth tends to become zero. For convenience, let Y₀ = 0. The requirement of θ₂ > 0 guarantees G < θ₁. If a > (<) 0, the propensity to own wealth tends to be reduced (increased) as capital stock K is increased. If a = 0, the term G is not affected by K. If b > (<) 0, the propensity to own wealth tends to be reduced (increased) as the income Y is increased. If b = 0, the term G is not affected by Y. In this section, we will not specify whether a and b are positive or negative as it is difficult to judge whether an increase in wealth and income will certainly increase or decrease the propensity to own wealth. It should be noted that the function G may be affected by other factors. For instance, preferences for goods may be affected by prices as people may judge quality by price or because of 'snob appeal' effects. We will examine behavior of the system when a and b are taken on different values.

By Y = F and equations (11.1.1)-(11.1.3), the dynamics of K(t) and λ(t) are given by:

$\dot{\mathit{K}}\mathrm{=}\mathit{\lambda }\mathit{F}(\mathit{K}\mathrm{,}{\mathit{N}}^{*})\mathrm{-}\mathrm{(}\mathrm{1}\mathrm{-}\mathit{\delta }\mathit{\lambda }\mathrm{)}\mathit{K}\mathrm{,}\dot{\mathit{\lambda }}\mathrm{=}\mathit{\theta }\mathrm{\{}\frac{{\mathit{\theta }}_{\mathrm{1}}}{\mathrm{1}\mathrm{+}{\mathit{\theta }}_{\mathrm{2}}{{\mathit{K}}^a}^{}\mathit{F}\mathrm{(}\mathit{K}\mathrm{,}\mathit{N}*{\mathrm{)}}^{\mathit{b}}}-\mathit{\lambda }\mathrm{\}}\mathrm{.}$(11.1.4)

For any given K(t) and λ(t) we determine all the other variables in the system at any point of time by the following procedure: F = F(K, N^*) → r and w by the marginal conditions → Y = F → C and S by C = ξŶ and S = λŶ. If we know the motion of dynamic system (11.1.4), the other variables are uniquely determined at any point of time.

We have thus built the model. It should be noted that behavior of the model with the Cobb-Douglas production function is examined by Zhang.²

Equilibrium of the dynamic system is determined by:

$\mathit{\lambda }\mathrm{=}\frac{\mathrm{1}}{\mathit{F}\mathrm{/}\mathit{K}\mathrm{+}\mathit{\delta }}\mathrm{,}\frac{{\theta }_1}{\mathrm{1}\mathrm{+}{\mathit{\theta }}_2{\mathit{K}}^a{{\mathit{F}}^{}}^b}\mathrm{=}\mathit{\lambda }\mathrm{.}$(11.1.5)

For λ < 1 to be satisfied, it is sufficient to require F/K + δ > 1, i.e., F/K > δ_k. Let K₁ stand for the level of capital stock that satisfies F/K = δ_k. Because:

$\frac{\mathit{d}\mathrm{(}\mathit{F}\mathrm{/}\mathit{K}\mathrm{)}}{\mathit{d}\mathit{K}}\mathrm{=}\frac{\mathit{K}{\mathit{F}}_K\mathrm{-}\mathit{F}}{{\mathit{K}}^2}\mathrm{=}\mathrm{-}\frac{\mathit{w}}{{\mathit{K}}^2}\mathrm{<}\mathrm{0}\mathrm{,}\underset{\mathit{K}\mathrm{\to }0}{\lim }\frac{\mathit{F}}{\mathit{K}}\mathrm{\to }\mathrm{+}\mathrm{\infty }\mathrm{,}\underset{\mathrm{A}\mathrm{\to }\infty }{\lim }\frac{\mathit{F}}{\mathit{K}}\mathrm{\to }\mathrm{0}\mathrm{,}$

we identify the existence of a unique positive K₁ at which F(K₁, N^*)/K₁, = δ_k holds. Since F/K decreases in K, F/K < δ_k holds for 0 < K < 0. From conditions (11.1.5), we see that the level of capital stock in equilibrium is determined by:

$\mathrm{\Phi }\mathrm{(}\mathit{K}\mathrm{)}\mathrm{=}\mathit{U}\mathrm{(}\mathit{K}\mathrm{)}\mathrm{-}\mathit{V}\mathrm{(}\mathit{K}\mathrm{)}\mathrm{=}\mathrm{0},\mathrm{0}\mathrm{<}\mathit{K}\mathrm{<}\mathrm{+}{\mathit{K}}_1,$(11.1.6)

where:

$\mathit{U}\mathrm{(}\mathit{K}\mathrm{)}\mathrm{\equiv }\frac{\mathrm{1}}{\mathit{F}\mathrm{/}\mathit{K}\mathrm{+}\mathit{\delta }}\mathrm{,}\mathit{V}\mathrm{(}\mathit{K}\mathrm{)}\mathrm{\equiv }\frac{{\mathit{\theta }}_{\mathrm{1}}}{\mathrm{1}\mathrm{+}{\mathit{\theta }}_{\mathrm{2}}{\mathit{K}}^{\mathit{a}}{\mathit{F}}^{\mathit{b}}}\mathrm{.}$

We always have Φ(K₁) > 0. In the case of K^aF^b → finite as K → 0 (for instance, when a > 0 and b ≥ 0), Φ(0) < 0. We have at least one solution. Since:

${\mathrm{\Phi }\prime }^{}\mathrm{=}\mathit{w}{\mathit{U}}^2\mathrm{+}\mathrm{(}\mathit{a}\mathrm{+}\mathit{b}\mathrm{-}\mathit{b}\mathit{w}\mathrm{/}\mathit{F}\mathrm{)}\frac{{\mathit{\theta }}_{\mathrm{2}}{\mathit{K}}^{}{\mathit{F}}^{\mathit{b}}{V}^{\mathrm{2}}}{{\mathit{\theta }}_{\mathrm{1}}\mathit{K}},$

we see that Φ' > 0 (which guarantees the uniqueness of solution), for instance, in the case of a > 0 and b > 0 or a + b > 0 and b < 0. In the case of K^aF^b → + ∞ as K → 0, the equation may have no solution. We illustrate a few cases when:

$\mathit{F}\mathrm{=}{\mathit{K}}^{\alpha }{{\mathit{N}}^{*}}^{\mathit{\beta }}\mathrm{,}\mathit{a}\mathrm{=}\mathrm{0}\mathrm{.}\mathrm{3}\mathrm{,}{\mathit{N}}^{*}\mathrm{=}\mathrm{1}\mathrm{,}{\theta }_{\mathrm{1}}\mathrm{=}\mathrm{0}\mathrm{.}\mathrm{6}\mathrm{5}\mathrm{,}{\theta }_{\mathrm{2}}\mathrm{=}\mathrm{0}\mathrm{.}\mathrm{2}\mathrm{,}{\mathit{\delta }}_{\mathit{k}}\mathrm{=}\mathrm{0}\mathrm{.}\mathrm{05.}$

We calculate K₁ = 72.213. Figure 11.1 shows two cases with these specified values of the parameter. Figure 11.1a depicts the case of a = 0.4 and b = 0.2. The dashing line denotes the U-curve and the solid line the V-curve. The equilibrium point is determined at the intersection of the two curves within the interval [0, 72.213]. We see that the U-curve rises in K and the V-curve falls in K. There is a unique equilibrium point. The equilibrium (K,λ) (with λ = U = V) is (1.136, 0.5363) as denoted in the figure. Figure 11.1b portrays the case of a = -0.4 and b = 0.2. The equilibrium point (K,λ) is (1.206, 0.5473) as denoted in the figure.

We now show a case that the system has no equilibrium beyond a survival level, for instance, K̄ = 0.004. Figure 11.2 shows the case of:

${\theta }_{\mathrm{1}}\mathrm{=}\mathrm{0}\mathrm{.}\mathrm{6}\mathrm{5}\mathrm{,}{\theta }_{\mathrm{2}}\mathrm{=}\mathrm{0}\mathrm{.}\mathrm{8}\mathrm{,}\mathit{a}\mathrm{=}\mathrm{-}0\mathrm{.}\mathrm{7}\mathrm{,}\mathit{b}\mathrm{=}\mathrm{0}\mathrm{.}\mathrm{01}\mathrm{,}$

when the rest parameters are specified as before. When the propensity to own wealth is strongly negatively affected by wealth, the system cannot sustain any equilibrium as the incentive to save is not sustainable.

It is straightforward to check stability conditions. It can be shown that the system may be either stable or unstable.

We now simulate the dynamics of the model under the following specified values:

$\mathit{F}\mathrm{=}{\mathit{K}}^{\mathit{\alpha }}{\mathit{N}}^{*\mathit{\beta }}\mathrm{,}\mathit{\alpha }\mathrm{=}\mathrm{0}\mathrm{.}\mathrm{3}\mathrm{,}{\mathit{N}}^{*}\mathrm{=}\mathrm{1}\mathrm{,}\mathit{a}\mathrm{=}\mathrm{-}0\mathrm{.}\mathrm{4}\mathrm{,}{\theta }_{\mathrm{1}}\mathrm{=}\mathrm{0}\mathrm{.}\mathrm{6}\mathrm{5}\mathrm{,}{\theta }_{\mathrm{2}}\mathrm{=}\mathrm{0.2},{\mathit{\delta }}_{\mathit{k}}\mathrm{=}\mathrm{0}\mathrm{.}\mathrm{0}\mathrm{5}\mathrm{,}\mathit{b}\mathrm{=}\mathrm{0}\mathrm{.}\mathrm{2}\mathrm{,}\theta \mathrm{=}\mathrm{0}\mathrm{.}\mathrm{2}\mathrm{.}$

Figure 11.3 illustrates the dynamics with the initial values (0.8, 0.4). In initial stage, the propensity to own wealth is relatively low so that the capital stock level declines. The left-hand plot in Figure 11.3 shows changes of capital stocks over time. In the first two years, per-capita wealth declines because of low propensity to own wealth. As the propensity to own wealth rises over 44 percent as shown in the right-hand plot, per capita wealth begins to rise. The economy grows until it reaches the equilibrium. Figure 11.4 shows another case that the system starts with a rich state with high propensity to own wealth state. The two cases may, in a very rough sense (as we omit technological change and many other important dimensions), reflect a possible short-run economic evolution of Japan and China. It can be seen that our model may exhibit different patterns of economic growth for varied possible patterns of technological change and varied possible preference dynamics.

11.2 Economic Chaos as Conclusion

To conclude the book, we show complexity of economic evolution from a new analytical perspective. Over the last few decades, economists - influenced by mathematicians and natural scientists - have applied nonlinear theory to understand complexity of economic evolution. Complex theory has found wide applications in different fields of economics.³ The range of its applications includes many topics, such as catastrophes, bifurcations, trade cycles, economic chaos, urban pattern formation, sexual division of labor and economic development, economic growth, values and family structure, the role of stochastic noise upon socioeconomic structures, fast and slow socioeconomic processes, and relationship between microscopic and macroscopic structures. All these topics cannot be effectively examined by traditional analytical methods, which are concerned with linearity, stability, and static equilibria. Nonlinear economics attempts to provide a new vision of economic dynamics: a vision toward the multiple, the temporal, the unpredictable, and the complexity.

Linearity means that the rule that determines what a piece of a system is going to do next is not influenced by what it is doing now. More precisely, this is referred to in a differential or incremental sense: for a linear growth economy, the increase of GNP is proportional to the value which the economy is producing, with the growth rate exactly independent of how much the economy has already produced. Such an economy can grow/decline arbitrarily rich/poor, and in particular will never oscillate. Linear models enjoy an identical, simple geometry. The simplicity of this geometry allows a relatively easy mental image to apprehend the essence of a problem. For nonlinear problems there is usually no simple and universal geometry. Investigation was case by case. We now use a model to demonstrate that a simple rule can generate chaotic behavior.

The model in this section is based on Section 6.2 in Zhang.⁴ The model is constructed by Haavelmo in continuous form. Its discrete form was examined by Stutzer,⁵ by applying modern mathematics for one-dimensional mappings. First, consider a macroeconomic growth model proposed by Haavelmo:⁶

$\dot{\mathit{N}}\mathrm{=}\mathit{N}(\mathit{a}\mathrm{-}\frac{\mathit{\beta }\mathit{N}}{\mathit{Y}})\mathrm{,}a,\mathit{\beta }\mathrm{>}\mathrm{0}\mathrm{,}\mathit{Y}\mathrm{=}\mathit{A}{{\mathit{N}}^{}}^{\alpha },A\mathrm{>}\mathrm{0}\mathrm{,}\mathrm{0}\mathrm{<}\mathit{\alpha }\mathrm{<}\mathrm{1},$(11.2.1)

where N is the population, Y is real output, and a, β, α and A are constant parameters. Substituting Y = AN^α into the differential equation yields:

$\dot{\mathit{N}}\mathrm{=}\mathit{N}(\mathit{a}\mathrm{-}\frac{\mathit{\beta }{\mathit{N}}^{1-\alpha }}{\mathrm{A}})\mathrm{.}$(11.2.2)

We see that the growth law is a generalization of the familiar logistic form widely used in biological population and economic analysis. It is not difficult to see that the dynamics of this system are very simple. If the initial condition satisfies N(0) > (<) (aA/β)^1/(2-α), then both N and Y will decrease (increase) monotonically until approaching their unique equilibria, respectively. If we replace time derivatives by first differences and accept discrete time, then equation (11.2.2) becomes

${\mathit{N}}_{t+1}\mathrm{=}{\mathit{N}}_t\mathrm{[}\mathrm{(}\mathrm{1}\mathrm{+}\mathit{a}\mathrm{)}\mathrm{-}\frac{\mathit{\beta }{N}_t^{1-\alpha }}{\mathrm{A}}\mathrm{]},$

which can be further simplified as:

${\mathit{x}}_{t\mathrm{+}\mathrm{1}}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{+}\mathit{a}\mathrm{)}{{\mathit{x}}_t}_{}\mathrm{(}1\mathrm{-}{\mathit{x}}_t^{\mathrm{1}/\mathrm{2}}\mathrm{)}\mathrm{=}\mathit{F}\mathrm{(}{\mathit{x}}_t;\mathit{a}\mathrm{,}\mathrm{0.}\mathrm{5}\mathrm{)},$

in which the new variable x_t is defined by the transformation:

${{\mathit{N}}_t}_{}\mathrm{=}\mathrm{[}\mathrm{(}\mathrm{1}\mathrm{+}\mathit{0}\mathrm{)}\frac{\mathrm{A}}{\mathit{\beta }}{\mathrm{]}}^{\mathrm{1}/\mathrm{(}\mathrm{1}-\mathit{\alpha }\mathrm{)}}{{\mathit{x}}_t,}_{}$

with α = 1/2. It can be shown that none of the qualitative properties of the system are affected by the particular choice of 0 < α < 1. The term chaotic dynamics refers to the dynamic behavior of certain equations F which possess: (a) a non-degenerate n-period point for each n ≥ 1, and (b) an uncountable set S ∈ J (= [0,1]) containing no periodic points and no asymptotically periodic points. The trajectories of such points wander around in J randomly.

For each value of a, equilibrium points are given by the intersection of the graph of F(x_t;a) with the 45-degree line. For each value of a, there are two equilibria x₀ = 0 and x₀ = {a/(1 + a)}². The point x₀ = 0 is unstable and repels nearby points. The local stability of the other can be determined by linearization at the equilibrium. We have: F'(x₀;a) = 1 - a/2 - θ(a).

The eigenvalue θ(a) determines the local stability of x₀. When 0 < θ < 1, x₀ attracts nearby points in an exponential, monotonic fashion. When 0 > θ > - 1, x₀ attracts nearby points in a damped oscillatory manner. When θ = 1, x₀ is neither stable nor unstable. Finally, if |θ| > 1, x₀ is unstable. These behaviors occur when 0 < a < 2, 2 < a < 4, a = 4, and 4 < a < 5.57, respectively. When the equilibrium is stable, i.e., a < 4, the trajectory starting at any point always approaches it. In this region a traditional comparative statics analysis shows that an increase in the parameter a will increase x_t for sufficiently large t. If 4 < a < 5.57, trajectories don't approach the equilibrium, but bounded by 0 and 1. In fact, as the parameter a exceeds 4, the unstable equilibrium point bifurcates into two stable points of period two, i.e., into a stable periodic orbit of length 2. The 2 -period cycle becomes unstable for values of a in excess of about 4.8, and each 2 -period point bifurcates into two 4 -period points, producing an stable cycle of length four denoted by:

$\mathrm{\{}{\mathit{x}}_{\mathrm{01}}^4,{\mathit{x}}_{02}^4\mathrm{,}{\mathit{x}}_{\mathrm{03}}^4,{\mathit{x}}_{\mathrm{04}}^4\}\mathrm{.}$

Figure 11.5 illustrates the phenomenon.

This pitchfork bifurcation process continues as the parameter a increases, producing non-degenerate orbits of length 2k(k = 2,...). These orbits are called harmonics of the 2 -period orbit. It can be shown that all the harmonics occur prior to the parameter a reaching 5.54, although how much prior to this value is not known. Thus, the range of a, within which a stable orbit of length k first appears and later becomes unstable and bifurcates to a 2k -period orbit, decreases in length as the parameter a increases to a limiting value a_c < 5.54. The range of a_c < a < 5.75 is termed the chaotic region. As the parameter a enters this region, even stranger behavior can occur. For example, a 3-period orbit exists at values of a near 5.540. This, then gives rise to orbits of periods 3k (k = 2,...) via the pitchfork process just described. In fact, if we can locate the 3 -period orbit, a remarkable theorem of Li and Yorke demonstrates that for any F(x_t;a) in which a non-degenerate 3 -period orbit arises, there must also exist non-degenerate points of all periods, as well as an uncountable set of periodic (not asymptotically periodic) points whose trajectories wander randomly throughout the domain of F.⁷ Our dynamic economic system satisfies the requirements in the LiYorke theorem for some values of a. This guarantees the existence of chaotic behavior as illustrated in Figure 11.6.

The existence of chaos implies that no one can precisely know what will happen in society in the future, except that it will be changing. To illustrate why no one precisely foresee the consequences of the intervention policy, let us try to find out what happen to the chaotic system when it starts from two different but very near states. In Figure 11.7, we consider the case of a = 5.75. Let us consider two cases of x₀ = 0.400 and x₀ = 0.405 over 100 years. It can be seen that the two behaviors are varied over time.

We calculate the difference x_t[0.400] - x_t[0.405] between the path started at x₀ = 0.400 and the one at x₀ = 0.405 over 100 years.

In summary, as the autonomous growth rate a exceeds a certain value, the steady state ceases being approached monotonically, and an oscillatory approach occurs. If a is increased further, the steady state becomes unstable and repels nearby points. As a increases, one can find a value of a where the system possesses a cycle of period k for arbitrary k (see Figure 11.8). Also, there exists an uncountable number of initial conditions from which emanate trajectories that fluctuate in a bounded and aperiodic fashion and are indistinguishable from a realization of some stochastic (chaotic) process.

¹ There are numerous models with endogenous change of time preference, for example, Uzawa (1968), Wan (1970), Boyer (1978), Shi and Epstein (1993), and Das (2003).

² Chapter 9 in Zhang (1999a). It should be noted that there are a large number of the literature on economic growth with bifurcations and chaos (for instance, Day, 1984, Hommes, 1991, 1998, Zhang, 1991, Azaridis, 1993, Boldrin, et al. 2001, Matsuyama, 2001, Shone, 2002). Zhang (2005, 2006) provides an updated treatment of the subject.

³ See Arthur (1989), Puu (1989), Rosser (1991), Zhang (1991), and Lorenz (1993), for earlier development of nonlinear economics. Recent developments are referred to Zhang (2005).

⁴ Section 6.2 in Zhang (1991).

⁵ Stutzer (1980).

⁶ Haavelmo (1954).

⁷ Li and Yorke (1975).