Dynamic Macroeconomic Modeling with Matlab

4 Numerical Simulation of Macroeconomic Models
competition in the research sector. Different to Romer (1990), we assume ηA < 1, which reflects
the fact that spill-overs, known as the ‘standing on shoulders’ effect, do not fulfill the knife-edge
condition to be linear. Workers are allowed to enter the R&D sector freely, wherefore they are
paid according to their marginal product
w = VaαJA ηA [(1 − φ)L] ηL−1
(106)
with the value of one blueprint Va. A no arbitrage condition forces the discounted stream of
profits to equal the patent’s value
Va(t) =
∞
Differentiating with respect to time yields
Finally, households maximize intertemporal utility according to
subject to
t
∞
max
c
0
π(τ)e − R τ
t r(s)ds dτ (107)
˙Va = rVa − π (108)
(C/L) 1−θ
e
1 − θ
−ρt dt (109)
˙K = rK + wL − Va ˙ A + Aπ − C (110)
with consumption C and the relative risk aversion equal to 1.
First order condition of the
θ
consumer’s maximization problem is
˙C = C
(r − ρ − n) + nC (111)
θ
We transform the variables into stationary ones by expressing the system in scale adjusted
variables, which are defined by y := Y/L βK , k := K/L βK , c := C/L βK , a := A/L βA , j := J/L βA
and va := v/L with βK = 1−ηA+ηL
1−ηA , βA = ηL . The dynamic system which governs the
1−ηA
evolution of the economy under study, can be summarized as follows:
˙k = y − c − δk − βKnk (112)
˙a = j − βAna (113)