Labour Productivity and the Chaotic Economic Growth Model: G7

Chaos theory is used to prove that erratic and chaotic fluctuations can indeed arise in completely deterministic models. Chaos theory reveals structure in aperiodic, dynamic systems. The number of nonlinear business cycle models use chaos theory to explain complex motion of the economy. Almost three years after the crisis, the G7 countries continue to be challenged with economic volatility. The global economy has slowed. Growth in the United States has weakened. In Europe, economic instability is generated by the financial and economic imbalances. Europe is gripped with financial strains from the sovereign debt crisis in the euro area periphery. How these G7 economies confront their fiscal challenges will profoundly affect their economic stability. The basic aim of this paper is to provide a relatively simple chaotic economic growth model that is capable of generating stable equilibria, cycles, or chaos. This paper looks in more detail at the GDP growth stability issues in each of the G7 countries in the period 1990-2012 (Retrieved from http://www. imf.org). A key hypothesis of this work is based on the idea that the coefficient           m m p b p b p n i s p  plays a crucial role in explaining local stability of the gross domestic product growth, where, p—the coefficient of labour productivity; pm—the coefficient of the marginal labour productivity, sp—private saving rate; i—investment rate; b—percent of the gross domestic product which belongs to budget deficit; bm—marginal budget deficit coefficient; n—net capital outflow rate.


Introduction 
Chaos theory attempts to reveal structure in aperiodic, unpredictable dynamic systems. Deterministic chaos refers to irregular or chaotic motion that is generated by nonlinear systems evolving according to dynamical laws that uniquely determine the state of the system at all times from a knowledge of the system's previous history. Chaos embodies three important principles: (1) extreme sensitivity to initial conditions; (2) cause and effect are not proportional; and (3) nonlinearity.
The basic aim of this paper is to provide a relatively simple chaotic economic growth model that is capable of generating stable equilibria, cycles, or chaos. This paper looks in more detail at the GDP growth stability issues in each of the G7 countries in the period 1990-2012 (Retrieved from http://www. imf.org).
National saving is the source of the supply of loanable funds. Domestic investment and net capital outflow are the sources of the demand for loanable funds. At the equilibrium interest rate, the amount that people want to save exactly balances the amount that people want to borrow for the purpose of buying domestic capital and foreign capital. Figure 1. The effects of a government budget deficit. When the government runs a budget deficit, it reduces the national saving. The interest rate rises. The higher interest rate reduces investment and labour productivity growth. Further, the higher interest rate reduces net foreign investment (net capital outflow). The real gross domestic product falls as a consequence of the negative open economy multiplier effects and the negative investment multiplier effects.
In an open economy, government budget deficit, as a negative national saving, raises real interest rates, crowds out domestic investment, decreases net capital outflow, causes the domestic currency to appreciate. However this appreciation makes domestic goods and services more expensive compared to foreign goods and services. In this case, net exports fall. Further, the real gross domestic product falls as a consequence of the negative open economy multiplier effects.
Almost three years after the crisis, the G7 countries continue to be challenged with economic volatility. The global economy has slowed. Growth in the United States has weakened. In Europe, economic instability is generated by the financial and economic imbalances. Europe is gripped with financial strains from the sovereign debt crisis in the euro area periphery. How these G7 economies confront their fiscal challenges will profoundly affect their economic movements.
The G7 countries have experienced economic and financial turbulence. High budget deficits and debt, lower national saving, an decrease in labour productivity growth rate between 2000 and 2010, and tensions between saving and investment are weighing on growth in much of the G7 economies (see Figure 2). Also, the G7 countries experienced an decrease in labour productivity growth rate between 2000 and 2010 (see Figure 3 and Figure 4).  Labour productivity growth can be attributed to two causes: (1) technological progress; and (2) capital deepening. Capital deepening is the rate of growth of capital per unit of labor input. When the capital-labor ratio is increasing, each worker can increase production because the amount of capital available is growing.
The tendency in labor productivity growth in the G7 countries since the mid-1990s has attracted significant attentions because the labor productivity growth represents an important factor of economic growth stability.
Productivity growth in Europe exceeded that in the United States up to the mid-1990s, but since then European performance has slackened while the United States has picked up and taken a lead. High-tech sectors have played an important role in the acceleration of productivity growth in the United States. Europe did not benefit to the same degree from the productivity boost that came from the information technology revolution. The downturn in labor productivity growth in Europe may also reflect slower growth in capital-labor ratios. Total factor productivity growth in Europe may also have declined, while in the United States it may have increased. It is important to compare (see Figures 5-11) total investment (percent of GDP), gross national saving (percent of GDP), general government revenue (percent of GDP), general government total expenditure (percent of GDP), and labour productivity growth rate in the G7 countries in the period .

The Model
Irregular movement of the gross domestic product can be analyzed in the formal framework of the chaotic economic growth model: 10) where, S: national saving; Y: the real gross domestic product; NCO: net capital outflow (net foreign investment); Bd: budget deficit; S p : private saving; L: labour ; p: the coefficient of labour productivity; p m : the coefficient of the marginal labour productivity; s p : private saving rate; i: investment rate; b: percent of the gross domestic product which belongs to budget deficit; b m : marginal budget deficit coefficient; n: net capital outflow rate.
By substitution one derives: Further, it is assumed that the current value of the gross domestic product is restricted by its maximal value in its time series. This premise requires a modification of the growth law. Now, the gross domestic product growth rate depends on the current size of the gross domestic product, Y, relative to its maximal size in its time series Y m . We introduce y as y = Y/Y m . Thus y ranges between 0 and 1. Again we index y by t, i.e., write y t to refer to the size at time steps t = 0, 1, 2, 3, ... Now growth rate of the gross domestic product is measured as: This model given by equation (12) is called the logistic model. For most choices of p, b, n, i, s p , b m , and p m there is no explicit solution for equation (12). This is at the heart of the presence of chaos in deterministic feedback processes. Lorenz (1963) discovered this effect-the lack of predictability in deterministic systems. Sensitive dependence on initial conditions is one of the central ingredients of what is called deterministic chaos.
It is possible to show that iteration process for the logistic equation: (13) is equivalent to the iteration of growth model (12) when we use the identification: . Firstly, we transform data on the gross domestic product (Retrieved from http://www.imf.org) from 0 to 1, according to our supposition that actual value of the gross domestic product, Y, is restricted by its highest value in the time-series, Y m . Further, we obtain time-series of y =Y/Y m . Now, we estimate the model (15). The results are obtained in the next section.

Conclusion
This paper suggests conclusion for the use of the chaotic economic growth model in predicting the movement of the gross domestic product in the G7 countries. The model (12) has to rely on specified parameters p, b, n, i, s p , b m , p m , and initial value of the gross domest product, y 0 .
A key hypothesis of this work is based on the idea that the coefficient plays a crucial role in explaining local stability of the gross domestic product growth, where, p-the coefficient of labour productivity, p m -the coefficient of the marginal labour productivity, s p -private saving rate, i-investment rate, b-percent of the gross domestic product which belongs to budget deficit, b m -marginal budget deficit coefficient, n-net capital outflow rate.
An estimated value of the coefficient  was around 1 in the G7 countries in the period 1990-2012. This result confirms stable economic growth in the G7 countries in the observed period. However, if  < 1, then the gross domestic product will be decreased. However, national saving in the G7 countries has been relatively decreased (the decreasing slope of the saving function). The tendency of decreasing saving rates and relatively decreased savings in the G7 countries represent challenges for economic policies. Also, the G7 countries experienced an decrease in labour productivity growth rate between 2000 and 2010. The budget deficits have risen to high levels. In this sense, it is important to increase labour productivity, national saving, private saving, net outflow capital; and to decrease budget deficit through fiscal and monetary reforms.