Introduction to Modern Economic Growth the notation k for capital per worker in the next section.. An important implication of this equation is that the human capital investment of each
Trang 1Introduction to Modern Economic Growth the notation k for capital per worker in the next section From the definition of κ, the law of motion of effective capital-labor ratios can be written as
H (t) =
R1
0 bi(t− 1) di
R1
0 hi(t) di . Factor prices are then given by the usual competitive pricing formulae:
(10.31) R (t) = f0(κ (t)) and w (t) = f (κ (t))− κ (t) f0(κ (t)) ,
with the only noteworthy feature that w (t) is now wage per unit of human capital,
in a way consistent with (10.28)
An equilibrium in this overlapping generations economy is a sequence of bequest and consumption levels for each individual,n
[hi(t)]i∈[0,1], [ci(t)]i∈[0,1], [bi(t)]i∈[0,1]o∞
t=0, that solve (10.27) subject to (10.28) a sequence of effective capital-labor ratios, {κ (t)}∞t=0, given by (10.30) with some initial distribution of bequests [bi(0)]i∈[0,1], and sequences of factor prices,{w (t) , R (t)}∞t=0, that satisfy (10.31)
The characterization of an equilibrium is simplified by the fact that the solution
to the maximization problem of (10.27) subject to (10.28) involves
(10.32) ci(t) = ηmi(t) and bi(t) = (1− η) mi(t) ,
and substituting these into (10.27), we obtain the indirect utility function (see Ex-ercise 10.16):
µ
hi(t) a
¶ , which the individual maximizes by choosing hi(t) and recognizing that mi(t) =
w (t) hi(t) + R (t) bi(t− 1) The first-order condition of this maximization gives the human capital investment of individual i at time t as:
µ
hi(t) a
¶ ,
or inverting this relationship, defining γ0−1(·) as the inverse function of γ0(·) (which
is strictly increasing) and using (10.31), we obtain
(10.35) hi(t) = h (t)≡ aγ0−1[a (f (κ (t))− κ (t) f0(κ (t)))]
An important implication of this equation is that the human capital investment of each individual is identical, and only depends on the effective of capital-labor ratio
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