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I use this economics stack exchange for the first time. I have a question regarding the Cobb-Douglas production function. The question is this: GDP in an economy is growing at 3% a year in real terms. Population is constant. The government decides to allow a significant increase in immigration so that the population (and the workforce) starts to grow by 1% a year. Output is produced in the economy according to a Cobb-Douglas production function. The share of labour income in GDP is 70%. How much higher will GDP be as a result of the new immigration policy after 20 years? How much higher will GDP per capita be?

I am not be able to understand how to use the function in this problem. Sorry if I put a question in a wrong way. Thank you.

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I think this approach might be useful.The Cobb douglas production function with constant returns to scale is \begin{equation} Q=K^{\alpha}L^{1-\alpha} \end{equation} For simplicity, we use logarithms for the rates of change of its determinants \begin{equation} ln(Q)=\alpha ln(K)+(1-\alpha)ln(L) \end{equation}

Differentiating with respect to time on both sides of the equation

\begin{equation} \frac{1}{Q}\frac{dQ}{dt}=\alpha\frac{1}{K}\frac{dK}{dt}+(1-\alpha)\frac{1}{L}\frac{dL}{dt} \end{equation}

\begin{equation} \begin{array}{l} \text{Growth rate of GDP:}\frac{1}{Q}\frac{dQ}{dt}=G_{Q} \\ \text{Growth rate of Capital:}\frac{1}{K}\frac{dK}{dt}=G_{K} \\ \text{Growth rate of Labor:}\frac{1}{L}\frac{dL}{dt}=G_{L} \end{array} \end{equation} \begin{equation} G_{Q}=\alpha G_{K}+(1-\alpha)G_{L} \end{equation} Where $\alpha$ is the capital share in the gross domestic product. In your case $(1-\alpha)=70\%$, hence $\alpha=30\%$

Thus, $G_{k}=0\%$ (The capital in this case does not change.), $G_{L}=1\%$ Substituting in the original equation \begin{equation} G_{Q}=0.7\cdot 1\%=0.7\% \end{equation} for further growth rate.

Adding to the previous growth rate ($3\%$), we obtain $3\%+0.7\%=3.7\%$ To find the new GDP after 20 years. \begin{equation} (1+0.037)^{20}=2.068117 \end{equation}

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