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Examples: Optimise investment and tax rates in a macro economic growth model

The macro economic growth model was developed by the former Socio-Economic
Research Group; of the Max-Planck-Institute
in Hamburg. It simulates the interaction between two actors: entrepreneurs,
seeking to maximise their income, and the government, seeking to reduce
both unemployment costs and climate change damages, or in other words, maximise
global welfare.

To achieve their respective goals, both actors have to make decisions. While
the government sets a sequence of
tax rates for a fixed period of time, the entrepreneurs choose a sequence
of investment rates, for the same fixed period, that quantify how they can most effectively distribute their investments
into capital stock, labour efficiency, and reduction of greenhouse gas emissions.
In an optimisation framework these rates are termed control variables.
In this example, the optimal values of the control variables are those
for which global welfare is maximised, taking into account that the entrepreneurs
have maximum income.

Technically, these optimal values are found iteratively by varying the
control variables. Extremely powerful gradient algorithms exist for this kind of optimisation problem. In order to apply such an algorithm the user must provide the numerical model that evaluates the function to be
minimised and also a numerical model that evaluates the first derivative,
i.e. the gradient of this function. Fortunately, this task can be handled
automatically by the Tangent linear and Adjoint Model Compiler (TAMC).
Ralf applied TAMC
to generate the first predecessor of the macro economic growth model (see
Hasselmann
et al. 1996a) and was consequently in constant demand to assist the members of
the Socio-Economic Research Group in generating the adjoints of their latest models (see
Hasselmann
et al. 1996b, 1997).