# Calculating elasticity between terms in a regression equation

Given the following regression: $$ln(w_i)=\beta_1+\beta_2age+\beta_3age_i^2+\beta_4Y_i+\beta_5T_i+\beta_6Mar_i+\epsilon_i$$ I am asked to calculate the elasticity of wages with respect to age. Is the following attempt valid?

$$\frac{\partial ln(w)}{{\frac{\partial age}{age}}}=\frac{\partial ln(w)}{\partial age}age=(\beta_2+2\beta_3age)(age)$$

Is this consistent with the definition of an elasticity as percent change over percentage change? Thank you.

• People usually use a semi-elasticity: dln(w)/dage, which is b2+2*b3*age in your case. It’s perfectly fine to use it. If you must calculate the elasticity, I think your formula works. But it is uncommon to report it. Jan 30 '20 at 4:53

No that would not be correct definition of elasticity. First, mathematically in multivariate function elasticity is defined as follows:

$$EL_x =\frac{ f_x '(x,y)}{f(x,y)}x$$ or in your case it would be: $$\frac{ \partial \ln [w(age,Y,T,Mar)]}{\partial age} \frac{age}{\\ln [w(age,Y,T,Mar)]}$$

However, even if you would plug in the expressions in this formula you would get an elasticity of a log of wage with respect to age - I am not sure if thats what you actually want. You could exponentiate the original function but that would be messy as hell.

One way how to estimate the elasticity directly from the regression would be using log-log form for variables of interest. In your case (assuming there are no people with 0 age in your sample):

$$ln(w_i)=\beta_1+\beta_2 \ln (age) +\beta_4Y_i+\beta_5T_i+\beta_6Mar_i+\epsilon_i$$

In this specification $$\beta_2$$ will already give you an elasticity. This specification also already controls for possible non-linear effect of an age so no quadratic term is necessary. This being said logs are only rarely used for age, but you could estimate this specification as an additional robustness check and still keep the main specification with quadratic term.

$$ln(w_i)=\beta_1+\beta_2age+\beta_3age_i^2+\beta_4Y_i+\beta_5T_i+\beta_6Mar_i+\epsilon_i$$

all the beta coefficients are already semi-elasticities (as also pointed out by @chan1142). So if you would only care about semi-elasticity you will already get your result from your original model as in your case semi elasticity of wages with respect to age is $$\beta_2+2\beta_3age_i$$.

• But what if I want full elasticity of ln(wage) with respect to age, not just semi-elasticity? Jan 30 '20 at 11:15
• @BobCharles I address that in 3rd paragraph (or 4th depending on whether you count that one sentence after 1st paragraph as separate paragraph)
– 1muflon1
Jan 30 '20 at 11:16
• So my expression (𝛽2+2𝛽3𝑎𝑔𝑒)(𝑎𝑔𝑒) is under no circumstance correct? Just wondering because it was listed in the solutions to the worksheet. Jan 30 '20 at 11:23
• @BobCharles I don’t know how exactly was the question formatted and other details but the answer that I gave you was taken from the Essential Mathematics for economic analysis textbook by Sydeater et al which is quite popular textbook. Also the answers that log-log form gives you elasticity and log-level form semi elasticity is based also on the latest edition of Verbeeks guide to modern econometrics and I recall seeing it in Wooldridge textbook as also in Stock and Watson textbook. This being said there might be special cases when that’s the solution, also it might be the solution of just...
– 1muflon1
Jan 30 '20 at 11:31
• @BobCharles Seems like this answer has helped you. It would be great if you could accept it as an answer, or suggest ways that @ 1muflon1 could edit it so that you would accept it.
– Art
Mar 3 '20 at 16:58