Government spending composition, technical change and wage inequality

t              Γ^t+D e-ft^s r(^τ)d^τds

^θ0(^t) = ^{γ +} t+D - rs _r(_τ)d_τ          ■

_t+Tr e- _t r(τ)dτ w_{H (}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(_s^pt)—FTT
_t+Tr e-ρ(s-t)w_H(s)_ds

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 Tr^P_p 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 e_e F.w ■

Let us remind that population growth rate n, birth rate β are linked by: β = enenD₁ ■ Unskilled
labor supply is:

where β is the birth rate, N_t 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) ≡

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

β₂ (t) = (1 — ψ)(G2 — β₂(t)),

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