Economic growth and economic development 694
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (158.41 KB, 1 trang )
Introduction to Modern Economic Growth
contrast equilibrium allocation to the Pareto optimal allocation. We will start with
this latter comparison in the next subsection.
15.3.4. Pareto Optimal Allocations. The analysis of Pareto optimal allocation is very similar to the analysis of optimal growth in Chapter 13. For this reason,
we will present only a sketch of the argument. As in that analysis, it is straightforward to see that the social planner would not charge a markup on machines, thus
we have
xSL (ν, t) =
pL (t)1/β L
and xSH (ν, t) =
pH (t)1/β H
.
(1 − β)1/β
(1 − β)1/β
Combining these with the production function and some algebra establish that net
output, which can be used for consumption or research, is equal to (see Exercise
15.6):
S
−1/β
Y (t) = (1 − β)
(15.31)
i
h ¡
¢ σ−1
¢ σ−1
ε¡ S
ε
S
σ
σ
.
β γ NL (t) L
+ (1 − γ) NH (t) H
In view of this, the current-value Hamiltonian for the social planner can be written
as
¢ C S (t)1−θ − 1
¡ S S S S S
S
+µL (t) η L ZLS (t)+µH (t) η H ZH
(t) ,
H NL , NH , ZL , ZH , C , µL , µH =
1−θ
subject to
h ¡
i
¢ σ−1
¢ σ−1
−1/β
ε¡ S
S
ε
S
S
σ
σ
+ (1 − γ) NH (t) H
(t) .
γ NL (t) L
− ZLS (t) − ZH
C (t) = (1 − β)
The necessary conditions for this problem give the following characterization of
the Pareto optimal allocation in this economy.
Proposition 15.5. The stationary solution of the Pareto optimal allocation
involves relative technologies given by (15.27) as in the decentralized equilibrium.
The stationary growth rate is higher than the equilibrium growth rate and is given
by
1
Ê
1
1/
1
1 Ô 1
(1 )
(1 ) (η H H)
+ γ (η L L)
− ρ > g∗,
g =
θ
∗
where g the BGP a clue brim growth rate given in (15.29).
S
Ô
Proof. See Exercise 15.7.
680
