Economic growth and economic development 99
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Introduction to Modern Economic Growth
Since the left-hand side is positive by hypothesis, we can divide both sides by
exp (gK (t − τ )) and write date t quantities in terms of date τ quantities to obtain
(gK + δ) K (τ ) = exp ((gY − gK ) (t − τ )) Y (τ ) − exp ((gC − gK ) (t − τ )) C (τ )
for all t. Differentiating with respect to time implies that
(gY − gK ) exp ((gY − gK ) (t − τ )) Y (τ )−(gC − gK ) exp ((gC − gK ) (t − τ )) C (τ ) = 0
for all t. This equation can hold for all t either if gY = gK = gC or if gY = gC
and Y (τ ) = C (τ ). However the latter condition is inconsistent with gK + δ > 0.
Therefore, gY = gK = gC as claimed in the first part of the proposition.
Next, the aggregate production function for time τ can be written as
h
i
˜
exp (−gY (t − τ )) Y (t) = F exp (−gK (t − τ )) K (t) , exp (−n (t − τ )) L (t) , A (τ ) .
Multiplying both sides by exp (gY (t − τ )) and using the constant returns to scale
property of F , we obtain
i
h
Y (t) = F exp ((t − τ ) (gY − gK )) K (t) , exp ((t − τ ) (gY − n)) L (t) , A˜ (τ ) .
From part 1, gY = gK , therefore
h
i
˜
Y (t) = F K (t) , exp ((t − τ ) (gY − n)) L (t) , A (τ ) .
Moreover, this equation is true for t irrespective of the initial τ , thus
Y (t) = F˜ [K (t) , exp ((t − τ ) (gY − n)) L (t)] ,
= F˜ [K (t) , A (t) L (t)] ,
with
A˙ (t)
= gY − n
A (t)
establishing the second part of the proposition.
Ô
A remarkable feature of this proposition is that it was stated and proved without
any reference to equilibrium behavior or market clearing. Also, contrary to Uzawa’s
original theorem, it is not stated for a balanced growth path (meaning an equilibrium
path with constant factor shares), but only for an asymptotic path with constant
rates of output, capital and consumption growth. The proposition only exploits the
definition of asymptotic paths, the constant returns to scale nature of the aggregate
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