Government spending composition, technical change and wage inequality

the lowest price per unit of quality, that is the product of quality j = j^max (ω, t). Hence, the household’s
demand of each product is:

qθ(j, ω, t) =   θ^      for j = j^max(ω, t) and is zero otherwise

p(j, ω, t)

The presence of a lump sum tax does not change the standard Euler equation:

cθ<t) _ -(_rt .
c>(t)" ^r(t) - ^ρ

Individuals are finitely-lived members of infinitely-lived households, being continuously born at
rate β and dying at rate δ, with β - δ = n > 0; D > 0 denotes the exogenous duration of their life¹².
People are altruistic in that they care about their household’s total discounted utility according to
the intertemporally additive functional shown in (1). They choose to acquire education and become
skilled, if at all, at the beginning of their lives, and the (positive) duration of their schooling period,
during which the individual cannot work, is set at T_r < D.

Hence an individual with ability θ decides to acquire education if and only if:

with 0 < γ < 1/2. The ability parameter is defined so that a person with ability θ > γ is able to
accumulate skills (human capital) θ - γ after schooling, while a person with ability below this cut-off
gains no human capital from schooling.

We will focus on the steady-state analysis, in which all variables grow at constant rates and where
wL , wH , and cθ are all constant. It follows that r(t) = ρ at all dates, and that the individual will train
if and only if her ability is higher than

θo = [(1 - e^-ρD) / (e^-ρT^r - e^-ρD)] WL + γ ≡ σWL + γ.

The supply of unskilled labor at time t is:

L(t) ≡ θoN(t) = fσwL + γ^) N(t).
wH

We set wL = 1, so that the unskilled wage becomes our numeraire. A fraction (1 - θo) of the
population decides to receive education. The skilled workforce is represented by those people that
have completed their schooling period, that is individuals born between t - D and t - Tr:

β(1 - θo) N(s)ds = (1 -θo)φN(t).

with 0 < φ = (eⁿ(D^-Tr ) — 1) / (eⁿD — 1) < 1. The uniform distribution of workers abilities implies
that the average skills of workers that have acquired education is [(θo - γ) + (1 - γ)] /2. Hence, the
supply of skilled labor in efficiency units at time s is

H(t) = (θo + 1 — 2γ) (1 — θo) ΦN(t)/2,

¹² As in Dinopoulos and Segerstrom (1999), it is easy to show that the above parameters cannot be chosen independently,
but that they must satisfy δ = _enDn_-1 and β = en⅛D₁ in order for the number of births at time t to match the number
of deaths at t + D.

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