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Introduction to Modern Economic Growth discussion above, are the constancy of factor shares and the constancy of the capital-output ratio, K t /Y t.. The following proposition is a stron

Introduction to Modern Economic Growth discussion above, are the constancy of factor shares and the constancy of the capital-output 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)Y (t) and αK(t)≡ R (t) K (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

Y (t) = Fh

K (t) , L (t) , ˜A (t)i

, 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)

A (t) = g = gY − n

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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