Cobb Douglas production function problem

Question

My question is related to the Cobb-Douglas production function: $Y= A \cdot L^\alpha \cdot K^\beta$

Assumptions:

1. 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 $\alpha+\beta =1$ )
2. $\beta =0.4$
3. labour productivity ($Y/L$) grows by $3\%$
4. 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?

Answer 1

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.

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*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.

Answer 2

Here's a slightly different take using the fact that logarithmic derivatives are percentage growth rates instead of approximation. Put $y = Y/L$ and $k = K/L$. Then after dividing through by $L$ as in Dimitriy Masterov's answer, differentiate:

\begin{align*} \frac{d}{dt}\log y &= \frac{d}{dt}\log A + \beta \frac{d}{dt}\log k\\ \frac{y'}{y} &= \frac{A'}{A} + \beta\frac{k'}{k} \end{align*} Now solve for $A'/A$, the growth rate of TFP.