Economic growth and economic development 98
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Introduction to Modern Economic Growth
discussion above, are the constancy of factor shares and the constancy of the capitaloutput ratio, K (t) /Y (t). Since there is only labor and capital in this model, by
factor shares, we mean
αL (t) ≡
w (t) L (t)
R (t) K (t)
and αK (t) ≡
.
Y (t)
Y (t)
By Assumption 1 and Theorem 2.1, we have that αL (t) + αK (t) = 1.
The following proposition is a stronger version of a result first stated and proved
by Uzawa. Here we will present a proof along the lines of the more recent paper
by Schlicht (2006). For this result, let us define an asymptotic path as a path of
output, capital, consumption and labor as t → ∞.
Proposition 2.11. (Uzawa) Consider a growth model with a constant returns
to scale aggregate production function
h
i
˜
Y (t) = F K (t) , L (t) , A (t) ,
with A˜ (t) representing technology at time t and aggregate resource constraint
K˙ (t) = Y (t) − C (t) − δK (t) .
Suppose that there is a constant growth rate of population, i.e., L (t) = exp (nt) L (0)
and that there exists an asymptotic path where output, capital and consumption grow
at constant rates, i.e., Y˙ (t) /Y (t) = gY , K˙ (t) /K (t) = gK and C˙ (t) /C (t) = gC .
Suppose finally that gK + δ > 0. Then,
(1) gY = gK = gC ; and
(2) asymptotically, the aggregate production function can be represented as:
Y (t) = F˜ [K (t) , A (t) L (t)] ,
where
A˙ (t)
= g = gY − n.
A (t)
Proof. By hypothesis, as t → ∞, we have Y (t) = exp (gY (t − τ )) Y (τ ),
K (t) = exp (gK (t − τ )) K (τ ) and L (t) = exp (n (t − τ )) L (τ ) for some τ < ∞.
The aggregate resource constraint at time t implies
(gK + δ) K (t) = Y (t) − C (t) .
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