Output growth in a Solowian economy

Question

Consider a Solowian economy governed by the Cobb-Douglas production function. Further assume the owners of capital receive two-thirds of the national income and the labourers receive the remaining one-third. If the labour force increases by $5\%$, what is the growth in output?

I get different answers when I attempt it in different ways.

1. $Y = K^{\alpha} L^{1-\alpha} = K^{2/3}L^{1/3}$ is the initial set of (output, capital stock, labour force). The new labour strength is $L' = (1.05)L$ such that the new output is $Y' = K^{2/3}(L')^{1/3} = (1.05)^{1/3}Y$.

   The output growth will be $\frac{\Delta Y}{Y} = (1.05)^{1/3}-1$

2. $\frac{\dot{Y}}{Y} = (2/3) \frac{\dot{K}}{K} + (1/3) \frac{\dot{L}}{L} = (1/3) \frac{\dot{L}}{L} = 5/3 = 1 \frac{2}{3}$ as capital stock remains constant and labour growth is $5\%$.

Why is there a difference in the two methods? I think it's because of assuming continuous time and discrete time models, but I could be wrong.

Which one is correct?

Answer 1

1. Seems to be a discrete time model, as you assume an instantaneous jump of $5\%$ in the size of the labor force.

2. Seems to be a formula for the continuous time model. Here you assume a constant $5\%$ growth rate for the labor force. As a side note, you are thinking in percentages here, but make no explicit mention of this.

The two growth values will be relatively close as long as the growth percentage is small.

Neither one is correct; the family of Solow models are models; any of them applied to any real situation will be wrong in some way. (Though, according to George Box, the wrong model may still be useful.)