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 θ
0 (γ, 1). The proof follows from the fact that the specified parameter restriction
allows the intercept (the value of the polynomial at θ
0 = 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

+ Gω


θo γ
bσ(ρ + n∕μ n)


= x (ω) ,


∂x (ω)    / λ (ω) 1 θo γ

∂Gω    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) ∕∂Gω > ∂x (ω) ∕∂Gω 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 [o ω) (γ 1 1) + (G ω)] (θ0 + 1 2Y)(1 θθ) φ^. (A.1.2)

μσ(ρ + n∕μ n) l

Using the Implicit Function Theorem we get:

o ∂F∕∂Ω

'dΩ = ∂F∕∂θ0 =

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

=>> 0

, .n.---τ [(θo Ω) (Γ-1 1) + (G Ω)] + n(θ0-γ) ʌ-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 θ
o(t):

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

22



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