# Output growth in a Solowian economy

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?

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.