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Steady State and Stability
Steady State Solution
The search for the steady state solution is conducted by means of the undetermined coefficients method through the definition of the expressions for the exogenous variables and the solution for the endogenous ones. The functional form we try is exponential. ρ and µ indicate, respectively, the steady state rates of growth for exogenous and endogenous variables. Starred variables identify the initial conditions.
Clearly the steady state solution depends on the constraints we impose on coefficients. As there are several possibilities regarding constraints, also dependent on economic-policy experiments we will carry out later, it seems reasonable at this stage to solve the steady state solution for the more general unconstrained case.
The steady state solution will be characterised by the equality of rate of growth of flows and of the stock of technology in order to ensure the constancy of all rates of growth in the steady state.
Another possibility is -other than consider solely the technology stock--the possibility to reach the steady state only in the, let’s say, “very long term”--i.e. in the limit as time tends to infinity.
Alternatively different r.o.g.’s for variables for patents are admitted and the stock of steady state of technology will grow at a pace given by the highest rate of growth of the flows involved. In the limit this will be the relevant one. As an example, consider the following decomposition of t h e s t o c k o f k n o w l e d g e w i t h T 0 b e i n g t h e i n i t i a l l e v e l :