Janusz Kudła, Katarzyna Kopczewska, Agata Kocia, Robert Kruszewski and Konrad Walczyk
variables are given by equations (3) and (4) . ρ is the rate of time preference and can be considered as a positive number, p > 0. Then, we proceed according to the well-established methodology of dynamic optimisation ( Pontryagin et al., 1962 ; Seierstad, Sydsæter, 1987 ; Léonard, Long, 1992 ; Schättler, Ledzewicz, 2012 ).
The necessary condition for the problem (5) under the constraints (3) and (4) can be expressed using the current-value Hamiltonian function H . Let Ф 1 and Ф 2 denote the right-hand sides of equations (3) and (4) , respectively