Deriving Cobb–Douglas Growth Rates with Logarithms

Summary

The note explains how to turn a Cobb–Douglas production function into a relation between proportional changes in output, productivity, capital, and labor. Taking logarithms converts the product of powered terms into a sum; differentiating, or using small-change approximations, then gives each input’s contribution weighted by its exponent. The result is commonly used to decompose output growth and interpret the exponents as elasticities.

The explanation sketches the derivation rather than presenting empirical evidence or a trading application. Its key caveat is that the proportional-change formula is exact for infinitesimal changes; for finite changes, differences in logarithms are not generally identical to simple percentage changes. The note also assumes the stated Cobb–Douglas form and fixed exponents, so it does not address model estimation, changing returns to scale, or whether the production function fits a particular dataset.

Key ideas

• Taking logarithms turns the Cobb–Douglas product into a sum.
• The exponents weight the contributions of capital and labor to output growth.
• Log changes correspond approximately to percentage changes when changes are small.
• The decomposition depends on the assumed production function and its parameters.

Tags

# What is the gross accounting relation of Cobb-Douglas function?


# What is the gross accounting relation of Cobb-Douglas function?












We have Cobb-Douglas function like this $Y=AK^\alpha L^{1-\alpha}$, in one of the book, it deduce like this:

How can we get this formula? $$\frac{\Delta Y}Y = \frac{\Delta A}A+\alpha\frac{\Delta K}K+(1-\alpha)\frac{\Delta L}L$$

## Answer by Kiwiakos (score 2)

https://quant.stackexchange.com/a/17146

Take logs of both sides, i.e. $$\log Y=\log A+ a \log K +(1-a)\log L$$ This gives: $$\Delta\log Y = \Delta\log A + a \Delta\log K +(1-a) \Delta\log L$$

Then use that $\frac{d}{dx}\log x= 1/x$, which yields $\Delta\log x=\Delta x/x$. Apply that to each log-diff above.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.