Why are elasticities defined as logarithmic derivatives?

Question

In my economics class, we often compute the elasticity of $Y$ with respect to $X$, $$\eta = \frac{\partial \log Y}{\partial \log X}.$$ You can compute this from the slope of a line fit to a log-log plot.

Why is it more natural to consider this quantity than the much simpler quantity $$\eta' = \frac{\partial Y}{\partial X}$$ which is just as easy to measure? There seems to be some implicit assumption that $\eta$ is perhaps 'more applicable', maybe that it's less sensitive to the value of $X$ than $\eta'$ is, so it's more useful for extrapolation. For example, I've seen a demand curve extrapolated by assuming that $\eta$ is constant, yielding a power law. But why not assume $\eta'$ is constant, yielding a line? Neither seems particularly more natural to me.

Why is $\eta$ a more useful quantity than $\eta'$ in economics?

Answer 1

If I understand your question, first the elasticity haven't units. The problem with $\partial Y/\partial X$ is that if you change the measure units the result is different. Is less problematic to express it in percentages: $$ El_X(Y)=\frac{\Delta Y/Y}{\Delta X/X}$$ if $\Delta Y \to 0$ when $\Delta X \to 0$ then you obtain $$El_X(Y)=\frac{dY}{dX}\frac{X}{Y}$$ if $Y=Y(X,W,...,Z)$ you change $d$ for $\partial$.

Second, like you know $d \log(x)=\frac{dx}{x}$ thus $$ El_X(Y)=\frac{d Y}{d X}\frac{X}{Y}=\frac{d\log(Y)}{d\log(X)}.$$

Answer 2

First, elasticity measures the responsiveness of quantity demanded or quantity supplied when a change in price occurs.

These measurements are made in percentage change form.

From my perspective, the main reason you are computing elasticity using $log$ is because doing this puts your data in percentage terms. Given that elasticity is a ratio of percentage change in quantity demanded/supplied to the percentage change in price, this would be the most plausible explanation.

You will see using $log$ to transform data into percentage terms in Econometrics courses.

Update: I don't think it is more "natural" to phrase things as percentages. But percentages can give more information. They give relative information. For example, Suppose you have \$100 and I have \$10. A third person gives each of us \$1, now you have \$101 and I have \$11. In absolute terms, we both received a dollar, but in percentage terms, your cash grew by 1% and my cash grew by 10%.

I hope this helps!

Answer 3

One extremely function throughout economics is the Cobb-Douglas functional form (eg. $y = bx_1^a x_2^b$). This won't just show up in utility functions, but also production functions or growth functions (like in the Solow Swan model).

Taking the log-log form here has a few implications. First, it drops $a$ and $b$ from exponents to linear coefficients, which gives you a form $lny = b+alnx_1+blnx_2$. This is important, because you can run a linear regression on this functional form, but not the original one. You can see an example of that in this paper.

It also has applications in empirical microeconomics; you can test to see if a firm has increasing, decreasing or constant returns to scale if you find $a+b > or < 1$. This can have important implications; it can answer some questions like "should we break up these big companies?"

Second it makes the math nicer in some cases. This can actually be useful in some cases; in Maximum likelihood estimation having a more mathematically friendly functional form can save you hours of computation time finding maxima.

Answer 4

First, the interpretation of $\eta=\frac{\partial\log y}{\partial\log x}$ and $\eta'=\frac{\partial y}{\partial x}$ are different. $\eta$ is the ratio of percent change and $\eta'$ is the ratio of absolute change. But you already know that. The real question is not why we define “elasticity” as a ratio of percent changes rather than absolute changes in economics, because that's how we use the word “elasticity” in everyday life: suppose rubber band A is 10 inches long and can be stretched by 1 inch when force F is applied, and rubber band B is 1 inch long and can be stretched by 0.5 inch when the same force F is applied. We would say rubber band B is more “elastic” than A, because we don't care about absolute change, but relative change when defining “elasticity”.

So I think your question is “why is elasticity $\eta$ more applicable/useful than $\eta'$?” My answer is that $\eta$ is not more useful/applicable/natural. Both $\eta$ and $\eta'$ are useful in different applications.

Take the price elasticity example. At unit price \$1, consumer A would buy 10 apples, and consumer B would buy 5 apples. At unit price \$2, consumer A would buy 6 apples, and consumer B would buy 1 apple. That is, when the price of apples increases by \$1, both consumers will reduce their purchase by 4 apples. For both consumers, $\eta'=\frac{\Delta\text{apples}}{\Delta\text{price}}=4$. This is a useful quantity if what we wanted to know is what happens to the sales of apples if price increased by \$1. But if we wanted to know which consumer would respond more dramatically to the price change, $\eta'$ is not a good measure. Because it seems consumer B is more “sensitive” to the price changes: she would cut her apple consumption by 80%, compared with only 40% reduction of consumer A. So $\eta$ is a better measure of this “sensitivity” or “elasticity” with respect to price changes.

In your demand curve extrapolation example, assuming constant elasticity $\eta$ is probably closer to truth than assuming constant $\eta'$. If you assume constant $\eta'$, the demand curve is a straight line. This effectively means price change from \$1 to \$2 will induce same change in quantity demanded as the price change from \$100 to \$101. But this is not supported by either evidence or common sense. Human brain does not seem to work this way. In this sense, relative changes do seem to be relevant in more economic applications than the absolute changes.