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\begin{appendix}

\chapter{The Dynamic Optimization}   %appear Appendix A and Appendix Title

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The static neoclassical theory of a firm and its dynamization by
dynamic optimization assume profit functions inconsistent with each
other. As a solution to this, we present a dynamic theory of a firm
which is consistent with the static neoclassical theory. We define the
`economic forces' which act upon the production of the firm and show
that the adjustment of production in a profit-seeking way may be
stable\index{stable} or unstable. Explosive unstable production
dynamics may occur due to `economies of scale' or due to the
development of wealth or technology; in stable cases the adjustment
leads to the profit maximizing situation.\index{situation} Our model
provides a micro basis for the modeling of economic growth at
macro-level.

In the theory only equilibrium states are studied, and the adjustment
process from one equilibrium to another is not modeled. In
physics,\index{physics} too, the equilibrium states of various dynamic
systems were known much before Newton defined his dynamic laws of
mechanics. Static analysis, however, is not in accordance with the
observed evolutionary behavior of economies, which requires a dynamic
framework for modeling.\index{modeling}

Most theories\index{theories} of economic growth concentrate on 
steady-states. The equilibrium (steady) state of a dynamic system may
be a fixed point or a fixed time path.\index{path} The first law of
Newton --- {\it A body remains at rest or in a uniform rectilinear
motion unless acted upon by a force} --- covers these both. This
implies that in the modeling of economic dynamics, the equilibrium
(zero-force) assumption has had a dominating role. Our aim is to
extend this tradition by defining the forces which act upon the
production of a firm in a non-equilibrium situation, and to show that
production dynamics can be modeled analogously with Newtonian 
mechanics.\index{mechanics}

The neoclassical theory of a firm assumes that firms' productions in a
given time unit are the profit maximizing ones. In real life, however,
firms' demand change continuously and firms adjust their productions
on this basis.\index{basis} If we now replace the assumption that {\it
firms produce in a profit maximizing way} by assuming that {\it firms
like to better their situation if possible}, with this principle we
can model the adjustment of production so that the modeling covers
also possible steady-states as well as changes in demand and costs. We
believe that the willingness of economic agents (entrepreneurs,
consumers,\index{consumers} workers etc.) to better their situation in
a competitive environment is the fundamental cause of economic
dynamics. The assumption that firms behave in an optimal way prohibits
the understanding of production dynamics because no firm\index{firm}
likes to change an optimal behavior.

Let us now determine a value for torsion in the era based on
formula. Thus by making use of the value for matter density
at the era as $\rho_P = 10^{93}~{\rm g}\cdot{\rm cm}^{-3}$ and
assuming that the angular velocity at the era is the same as
the present value of $\omega_{\rm P} = 10^{-17}$ s$^{-1}$,
substitution of this result into expression 
\begin{equation}  \label{delt}
\frac{\delta\rho}{\rho} = \frac{\omega^3Q}{\rho}\, .
\end{equation} 

Since there is a symmetric part of the torsion tensor $Q$ in
Eq.~(\ref{delt}) torsion definitely contributes to the deviation of
geodesics, which is the main motivation of using torsion in the
paper.\index{paper} This equation reduces to the space when torsion
$Q^\mu_{ij}$ vanishes. Here, we also used the definition of the
rotation tensor as $\omega_{ij} = \partial_{[i} v_{j]}$.

The theory of economic growth studies production dynamics at 
macro-level. This, however, is based on production decisions made at 
micro-level. To study this connection we assume a hypothetical economy
with fixed population where every firm is producing of one type of a
product with a constant velocity of production (a fixed amount in a
given time unit), and every consumer is consuming the goods they
consume with a constant velocity of consumption.

The optimal $S^\ast$ satisfies a marginal rule, Eq.~(\ref{afrg}) 
as in the formula
\begin{eqnarray}    \label{afrg}	
\pi (S^\ast)=\gamma F'(S^\ast)\cdot\pi[F(S^\ast)]\, ,
\end{eqnarray}
which equates the marginal value of the final unit harvested in a
given season to the foregone value of retaining it in the stock, to
contribute to the following year's recruitment.

Technological sources for economic growth are innovations which allow 
firms to decrease their product prices. Possible reasons are research
and development of domestic or foreign firms and private or public
research institutes, improved skills of workers and better education
in schools, colleges and universities. A surplus in balance of trade
demonstrates export led growth and deficit growth of domestic origin.

Every growing economy contains industries with varying growth 
rates. Growing industries benefit others producing input goods or
services to them. In a growing economy, the structure of consumption
changes with consumers' wealth according to their preferences. Demand
may thus be a source of economic growth, and it explains the
structural change of growing economies. These remarks question the
tradition to model growth in an aggregated framework, the neglecting
of demand and the modeling of balanced growth.

Because economic growth has a micro foundation at a firm level, and
with increasing wealth consumers consume different goods according to
their needs, at various stages of development of an economy different
industries have a potential to grow fast (we have no evidence that the
needs of human beings were limited).

In an agricultural economy, technological development in cultivation
may be the most important source of growth, and in a
`post-manufactured' economy, development in information technology may
be most important. Relatively poor economies producing export goods to
wealthier ones may also grow fast. Industries, which may create export
led growth, vary with the relative wealth and technology \hbox{levels}
of economies. The process of economic development (or growth) is the
same in all cases, although the industries and firms which create the
growth may vary. These remarks imply that our approach here has a
potential to unify the theories of economic development and growth.

The accumulated production of the firm till time moment $t$ (the
accumulated kilometers a car has driven) denoted by $Q(t)$ and
measured in units $(unit)$ (a marginal change in time $ds$ is measured
in units $Q(t)$, Eq.~(\ref{tnew}) is
\begin{equation}   \label{tnew}
Q(t)=Q(t_1)+\int_{t_1}^t q(s)ds\,, \quad
Q'(t)=q(t)\,,\quad Q''(t)=q'(t)\, ,
\end{equation}
where $Q(t_1)$ is the accumulated production of the firm from its
foundation till \hbox{moment} $t_1$, $Q'(t)$ with unit (unit/$y$) the
momentous velocity of production and $Q''(t)$ with unit (unit/$y^2$)
the momentous acceleration of production. This kinematics of
production is a necessary prelude for production dynamics analogous to
Newtonian mechanics.

Firms\index{firms!unit!price} like to sell their whole production at a
maximum possible unit price, and they know that the unit price is
either determined according to demand and supply at the market in the
competitive market case, or is set by the firm within the limits that
other firms' pricing policies set in the case of monopolistic
competition\index{competition} or a monopoly firm. Firms know that
they can sell their whole production at a unit price low enough, but
unit price under unit costs creates losses. The ambiguousness of the
planners of a single firm is thus focused on uncertainty about the
maximum unit price the products of the firm can be sold and on unit
production costs.\index{costs}

\end{appendix}


