Government spending composition, technical change and wage inequality

that, if the stated parameter restrictions are satisfied, there exists always one and only one real and
positive solution θ0 ∈ (γ, 1). The proof follows from the fact that the specified parameter restriction
allows the intercept (the value of the polynomial at θ0 = 0) of the LHS polynomial to be bigger than
in intercept of the RHS polynomial. Specifically LHS (0) > RHS(0) implies:

n n n          nγ      Ω - ΓG₄

(1 - 2γ) φ∕2 >   ,    (----_v(-—),

μσ(ρ + n∕μ — n) Γ

which rearranged leads to the parameter restriction. It is easy to see that this condition allows for a
unique solution³⁶. Moreover for Minkowski’s inequality Ω — ΓG < 0, therefore when 1 — 2γ > 0 no
restriction on parameters is needed for a unique solution.

Proof of Proposition 1.a. Solving (18) for x (ω) we get:
and deriving w.r.t. G_ωwe obtain

∂x (ω)    / λ (ω) — 1 ∖ θo — γ

∂G_ω    y λ (ω) j bσ(ρ + n∕μ — n)^,

which is always positive since λ (ω) > 1, θ0 > γ and ρ > n. From this derivative we can also see

that ∂x (ω) ∕∂G_ω > ∂x (ω) ∕∂G_ω∣ when (λ (ω) — 1) ∕λ (ω) > (λ (ω) — 1) ∕λ (ω) which is always true if

λ (ω) > λ (ω).

Proof of Proposition 1.b Rearranging (A11) we get a single polynomial in θo and Ω:

F (θo; ^Ω) = ---j-^θ----г [(θo ^{— ω}) (^γ 1 ^{— 1}) + ^{(G — ω})] ^— (^θ0 + ^{1 — 2}Y)^{(1 —} θθ) ^φ^. (A.1.2)

μσ(ρ + n∕μ — n) ^l

Using the Implicit Function Theorem we get:

dθo —∂F∕∂Ω

'dΩ = ∂F∕∂θ₀ =

n(θθ-γ)
μσ⅛l<√μ^-nl∣'                                                ∩

=>> 0

, .ⁿ.---_τ [(θo — Ω) (Γ^-1 — 1) + (G — Ω)] + n(θ^0-γ) ʌ(Γ^-1 — 1) + φ(θo — γ)

μσ(ρ+n∕μ-n) ^o                                   μσ(ρ+n∕μ-n)          ^{0 o 0 o}

This results follows from the fact that θ0 > γ, ρ > n, Γ^-1 > 1 and finally, from (A1) we know
that the expression inside the square brackets is greater than zero.

10 Appendix II: transitional dynamics

The schooling choice leading to eq. (4) in the paper implies, off steady states, the following ability
threshold θo(t):

³⁶It is easy to check that all parameters restriction are satisfied by the number we use in the calibration excercise.

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