172
11 Optimal Control Theory
In the next section we will discuss some of these models, and how they are cast
into a mathematical form [2] that is amenable to the optimization methods we will
discuss further down.
11.1 Macroeconomic Models
First, we consider a simplified version of the Solow model, [1] which considers the
output y of a company with capital base k, and the question what fraction σ of the
output to re-invest in the capital. A larger capital base will enable the company to
produce a higher output in the future, but will leave little profit to distribute to share
holders today. We analyze this model with suitably chosen discrete time steps, labeled
by t. For a company, this could be a three-month period that coincides with common
fiscal reporting practice. We therefore label all quantities with a subscript t to denote
the time step t. The output y t = λ f (k t ) is then some function f (k t ) of the available
capital k t , multiplied by a parameter λ, which describes the technological level of
the company. It is large for a company producing high-tech products and small for a
sweat shop in the third world. A commonly used model for the function f is the CobbDouglas model f (k t ) = k
t with the output elasticity in the range 0 < < < 1.
Since is always smaller than unity, it describes the effect of diminishing returns.
A 100 000 Euro increase in capital will make a huge difference for a small computer
shop but is hardly noticeable in a large company, such as Apple. The output y t is
thus given in terms of the capital base k t at time step t by
y t = λ f (k t ) = λk
t .
(11.1)
But how does the capital k t change from period to period? This is governed by two
effects. First, the capital k t deprecates at a rate δ, for example, due to equipment
that wears down and eventually breaks. Machines used to produce goods lose their
value when used and need to be replaced after some time. This is commonly referred
to as writing off part of the purchasing price for the machines. We describe it by
the deprecation parameter δ. The second effect that changes the capital base are
investments i t during period t, such that the capital base k t+1 for the subsequent
period is given by
k t+1 = (1 − δ)k t + i t .
(11.2)
And here the rate of reinvestment σ comes into play. In this model the investments
i t are given by a fraction σ of the output y t
i t = σ y t = σ λ f (k t ) .
(11.3)
After inserting (11.3) into (11.2), we obtain
k t+1 = (1 − δ)k t + σ λ f (k t ) ,
(11.4)
11 Optimal Control Theory
In the next section we will discuss some of these models, and how they are cast
into a mathematical form [2] that is amenable to the optimization methods we will
discuss further down.
11.1 Macroeconomic Models
First, we consider a simplified version of the Solow model, [1] which considers the
output y of a company with capital base k, and the question what fraction σ of the
output to re-invest in the capital. A larger capital base will enable the company to
produce a higher output in the future, but will leave little profit to distribute to share
holders today. We analyze this model with suitably chosen discrete time steps, labeled
by t. For a company, this could be a three-month period that coincides with common
fiscal reporting practice. We therefore label all quantities with a subscript t to denote
the time step t. The output y t = λ f (k t ) is then some function f (k t ) of the available
capital k t , multiplied by a parameter λ, which describes the technological level of
the company. It is large for a company producing high-tech products and small for a
sweat shop in the third world. A commonly used model for the function f is the CobbDouglas model f (k t ) = k
t with the output elasticity in the range 0 < < < 1.
Since is always smaller than unity, it describes the effect of diminishing returns.
A 100 000 Euro increase in capital will make a huge difference for a small computer
shop but is hardly noticeable in a large company, such as Apple. The output y t is
thus given in terms of the capital base k t at time step t by
y t = λ f (k t ) = λk
t .
(11.1)
But how does the capital k t change from period to period? This is governed by two
effects. First, the capital k t deprecates at a rate δ, for example, due to equipment
that wears down and eventually breaks. Machines used to produce goods lose their
value when used and need to be replaced after some time. This is commonly referred
to as writing off part of the purchasing price for the machines. We describe it by
the deprecation parameter δ. The second effect that changes the capital base are
investments i t during period t, such that the capital base k t+1 for the subsequent
period is given by
k t+1 = (1 − δ)k t + i t .
(11.2)
And here the rate of reinvestment σ comes into play. In this model the investments
i t are given by a fraction σ of the output y t
i t = σ y t = σ λ f (k t ) .
(11.3)
After inserting (11.3) into (11.2), we obtain
k t+1 = (1 − δ)k t + σ λ f (k t ) ,
(11.4)