t Γt+D e-fts r(τ)dτds
θ0(t) = γ + t+D - rs r(τ)dτ ■
t+Tr e- t r(τ)dτ wH (s)ds
In our numerical simulations we have chosen to set r(t) = ρ, which facilitates the convergence
of the numerical algorithm. In fact, though the analysis could be generalized, since in our relatively
long time period the exogenous policy variables G modify substantially from initial to the final steady
state, convergence is much for difficult to achieve than in a more standard local analysis. Hence the
previous equation becomes:
1-e-Dρ
θo(t) = y + ft+D p(spt)—FTT
t+Tr e-ρ(s-t)wH(s)ds
(25)
Defining: Ws(t) = Jtt+D. e p(s t)u,H(s)ds and differentiating with respect to t:
• ____ _____
WS(t) = e-pDwH(t + D) - e-pTrwH(t + Tr) +ρWS(t), (A 2.4)
which in the steady state implies: Ws = e TrPp e D w∏■ In light of the previous definitions, we can
rewrite eq. (25) as:
1 - e-Dp
θ0(t) = Y + ρWs(t) ■ (A 2.5)
In the steady state: θo = γ + (e Λ ee F.w ■
Let us remind that population growth rate n, birth rate β are linked by: β = enenD1 ■ Unskilled
labor supply is:
M(t) = βN (t)
t-D
en(s-t)θ0(s)ds,
where β is the birth rate, Nt is the population at time t, and θ0 (s) is the education ability threshold
at time s_ We stationarize unskilled labour supply by dividing it by the population level, m(t) ≡
M (t)
N(t) ■
Differentiating with respect to time:
m(t) = βθo(t) — βe nDθo(t — D~) — nm(t) (A 2∙6)
The unskilled labour market equilibrium (where 1 stands for low tech 2 for high tech - using equal
weight for each) is:
m(t) = 1((c(t) + βι(t)c(t))∕λι) + ((c(t) + β2(t)c(t))∕λ2), (A 2,7)
where β1(t) is the government expenditure in low tech products as a fraction of private consumption
and β2(t) is the government expenditure in high tech products as a fraction of private consumption
These shares change according to differential equations:
β1(t) = (1 — ψ)(G1 — β1(t)), and
(A 2∙8a)
(A 2∙8b)
β2 (t) = (1 — ψ)(G2 — β2(t)),
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