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1. Facts: A leisure term with an unbounded value function, when added to utility in the Lucas (1988) 'mechanics of economic development', expands enormously the range of data covered by the theory. To explain this we have to ask two questions. First: why leisure would be so much desired? Perhaps because leisure is one's own time and such a leisure term means an unbounded value of individual freedom. But why leisure is economically productive, as implied by the results obtained in this study? Perhaps because cognitive innovations often occur during the time which in economics is registered as leisure? Then an unbounded leisure term would also make room for an unbounded creation of knowledge, as distinguished from the mere transmission of knowledge in education and training. In any case the leisure term seems to act as if it where the'hole' through which strong nonmaterial values affect economics. The ensuing 'extended mechanics' is derived in Chapters 4-6 and proves to involve an extension of growth theory as well as a theory of the causal part of business cycles. Their empirical verification is given by showing (i) that the existence of the two Basic Growth Paths derived from this theory, defining its Growth Type 1 and Growth Type 2, respectively, is verified already by the statistics collected by Solow (1957) but ignored so far (see Chapter 5 of the present study); one of them, viz.



I. The Mathematical Tools

In the present work the classical theory of mathematical dynamics underlying dynamic economics will be applied paying special attention to the parameter conditions of existence of solutions in the applications of dynamics to economic theory. This has not been always done in economic applications, which suggests that a detailled introduction to the mathematical tools as applied here may be useful. It will be given in the present Part I.
Arvid Aulin

II. The Lucas Growth Theory and its Generalization to Business Cycles

The Lucas growth theory (Lucas (1988) has a particular structure that reflects the idea of rational expectations. First there is a situation, in which the households and firms react to what is generally expected to be common knowledge, i.e. a sort of average level of human capital in society. In the theory this level is exogeneously given, just as the exogeneous factor of tech-nological progress in the Solow model. The fundamental equations of the theory are constructed in this situation, which will be here called the ”reaction of the market to common knowledge”. Then market clearing creates a second situation, in which the exogeneously given and the endogeneously produced average levels of human capital coincide. The solution of the fun-damental equations has to take place in the second phase, to be called here the ”market clearing”.
Arvid Aulin

III. The General Theory of Economic Growth and Business Cycles

1. The balanced-growth path: Growth Type 1. Following the algorithm it is easily verified that a special solution of the equations of motion (4.3), (4.5)-(4.12), and of the relations (4.14)-(4.15) and (4.23)-(4.25) associated with them,is given by
$$1 > s = s* = Cons\tan t > 0,\quad thus\quad w = b,$$
$$a > 0\quad i.e\quad \sigma > 1,\quad b = b* = Cons\tan t > 0,\quad \psi = Cons\tan t,$$
$$\alpha = \alpha * = \left( {\beta - s*} \right)b* = Cons\tan t > 0\quad i.e.\quad s* < \beta ,$$
$$\psi = a*,\quad \xi = {\xi _{{0^e}}}^{\left( {\rho + m - \alpha *} \right)t},\quad {\xi _0} > 0,$$
$$Y = Y* = Y_0^*{e^{\lambda t}},\quad K = K* = K_0^*{e^{\lambda t}},\quad h = h* = h_0^*{e^{\nu t}},$$
$$\lambda = s*b*,\quad \nu = \left( {\frac{{1 - \beta }}{{1 - \beta + \kappa }}} \right)\left( {\lambda + m} \right),$$
$$v = v* = v_0^*{e^{ - \left( {m + n} \right)t}},\quad v_0^* = \left( {\alpha * + v} \right)/{k_0}{N_0} < 1,$$
$$1 > u = u* = \alpha */\left( {\alpha * + v} \right) = Cons\tan t > 0,$$
$$p = p* = p_0^*{e^{\left( {p\, - \,\beta b*} \right)t}},$$
$$q = q* = q_0^*{e^{\left( {\rho - \alpha * - v} \right)t}}$$
Arvid Aulin

IV. The Basic Business Cycles as the Causal Part of Business Cycles

The linear approximation (6.4)-(6.5) of the Basic Business Cycles, valid in a neighbourhood of the balanced-growth point \(P\, = \,(s*,b*),\), can be written in the form
$$s - s*\quad \mathop {{\text{ }} = }\limits^{Lin} \quad C{e^{ut}} + \bar C{e^{\bar ut}},C = \left( {\frac{1}{2} + \frac{{i\alpha }}{{4\omega }}} \right)\quad \left[ {s\left( 0 \right) - s*} \right],$$
$$w - b*\quad \mathop = \limits^{Lin} \quad D{e^{ut}} + \bar D{e^{\bar ut}},D = \left[ {\frac{{ia\left( {{\alpha ^2} + 4{\omega ^2}} \right)}}{{8\left( {1 - s*} \right)\omega }}} \right]\left[ {s\left( 0 \right) - s*} \right]$$
Arvid Aulin

V. The Effects of Nonmaterial Values and Other Ignored Factors Upon Economic Growth

The output (Y) P of an economy on a basic growth path P, whether in Growth Type 1 or 2, can be expressed in terms of the average human capital (h) P of population on that growth path:
$${\left( Y \right)_P} = {A^{1/{{\left( {1 - \beta } \right)}_b} - \beta /(1 - \beta )}}\left( {\psi /k} \right)\left( h \right)_P^{\left( {1 - \beta + \kappa } \right)/\left( {1 - \beta } \right)}$$
Arvid Aulin

VI. An Alternative Vision of the Stochastic Element in Business Cycles

Stochastic shocks do not essentially influence the long-term economic development. This is an accomplished fact in current growth theories (of Solow and Lucas, for instance). But ever since an important paper of the Russian mathematician Eugen Slutsky from the year 1927 was translated and published in English in a completed form (Slutsky, 1937), the economists have been fascinated by the idea that the business cycles may be purely stochastic processes. This would surely account for the ragged outlook of most economic time series. What Slutsky showed, however, was something more, viz. that random series are capable of forming cyclic phenomena. In fact this follows already from the symmetry of the Gauss curve around the mean of the series, and the effect can be made more visible by summations of certain sequences in a random series.
Arvid Aulin


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