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 and Theorem 2.1, we have that αL (t) + αK (t) = 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 > 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) 84