My question is related to the Cobb-Douglas production function: $Y= A \cdot L^\alpha \cdot K^\beta$
Assumptions:
- constant-returns to scale, meaning that when $L$ and $ K $ increase
with a factor $\ell$ then $Y$ also increases with a factor $l$: $Y'= A (\ell L)^\alpha (\ell K)^\beta = \ell^{\alpha+\beta} Y$, with $\alpha+\beta =1$ ) - $\beta =0.4$
- labour productivity ($Y/L$) grows by $3\%$
- capital intensity ($K/L$) increases by $4\%$
Question: Calculate the growth rate of total factor productivity ($A$).
I couldn't figure out how to use assumptions 3 and 4 and I think I need that for the rest of the question. Can someone help me please?
Best Answer
Dividing through by $L$ gives you output per worker: $$\frac{Y}{L}= \frac{A L^{\alpha}K^{\beta }}{L}= \frac{A L^{\alpha}K^{\beta }}{L^\alpha L^\beta}=A \left( \frac{K}{L}\right)^\beta.$$
We used the first assumption. Now take logs, which gets you
$$\ln \frac{Y}{L}= \ln A + \beta \cdot \ln \frac{K}{L}.$$
Calculate the difference from adjusting $K$ and $L$: $$\ln \frac{Y}{L}-\ln \frac{Y'}{L'}= \ln A-\ln A' + \beta \left( \ln \frac{K}{L}-\ln \frac{K'}{L'} \right).$$
For small changes, log differences are approximately equal to growth rates*, which leaves you with one equation in one unknown: $$ 3\%= \% \Delta A + 0.4 \cdot 4\%.$$
This uses assumptions 2-4.
*Take a Taylor series expansion of $\ln(1+x)$ to convince yourself that this is true if this is not something that you're intimately familiar with by now.