ideas needed to model national GDP given different sector growth rates subject to some contraints
Given:
- GDP equations for $n$ industries depend on growth rates and time i.e. $g(r_1,t), g(r_2, t), \ldots, g(r_n,t)$
- initial conditions - GDP growth rates for $n$ industries i.e. $r_1, r_2, \ldots, r_n$
- initial conditions - GDP (in the base year) for $n$ industries i.e. $g(r_1,0), g(r_2,0),\ldots g(r_n,t)$
- such that, the sum of all $n$ industries is total national GDP i.e. $G(r_1, r_2,\ldots,t)$
I don't know what form this relationship takes - happy to hear ideas/leads of how this is typically modelled - but I assume $g(r,t) = g(r,t-1)\cdot[r^t + 1]$ and sum of $g_n(r_n, t)$ over all industries gives $G(r_1,r_2,\ldots,r_n,t)$.
But I know this isn't right because I'd like to have $r$ vary with time to run some scenarios i.e $r(t)$
In time period, $t$, is there a way to add the following contraints:
- in time period, $t$, total GDP $G(r_1, r_2,\ldots, r_n,t)$ is a given constant $G_0$
- $dG(r_1,r_2,\ldots,t)/dt$ is a constant
- what method do you use to solve it?
Any thoughts, references or links to other resolved questions I may have missed will be great