elasticity calculation

[Liu Qiang]

dear all, I now face an easy question that's confusing me. There are three variables as follows:
Age: the age of mother in a household
Child: the number of children in a household
ln_ninc: ln(husband income)
I use this code to estimate the income impact of fathers on the fertility(the number of children)

areg child ln_ninc, robust a(age)

And then report the elasticity: the percentage change effect of husband income on the percentage change of fertility.
There are two ways to get it:
1.

margins,eydx(ln_ninc)

2. since the estimated coefficient equals dy/dlnx=dy/dx*x, e=dy/dx*x/y, I can also use this code to calculate the elasticity

local b=_b[ln_ninc]
sum child
local y=r(mean)
dis "e=" `b'/`y'

The two results are quite similar. But what confused me is that the paper reported this number as _b[ln_ninc]*r(mean) [my guess as the results show]
Can anyone help explain why I am wrong.
for your convenience, you can get the paper here: https://core.ac.uk/download/pdf/6543120.pdf
the data here: https://dataverse.harvard.edu/datase...l:1902.1/21475

* Example generated by -dataex-. To install: ssc install dataex
clear
input int age float child double ln_ninc
42 2 11.290756438652245
49 4  11.27935271816548
43 1 10.610660904829166
43 1  9.751559401061966
42 2 10.308952660644293
43 0 11.589432699447986
42 2  11.32506459446695
46 3 10.400406931127549
45 1 11.404226048046887
45 2 11.040999990161794
42 2 11.245725527459099
41 2 10.761004448831892
42 2 10.408526465540385
40 2 10.916015307290108
40 3 10.696480068065789
42 3 10.460643173884725
49 1 11.010844825456925
44 3 11.079184331510326
43 1 10.620009404412524
41 3  10.50701085558772
45 0 11.372191049896736
40 2 11.195471234184778
41 2 10.487740202378962
40 2 10.314371286671152
43 1 10.126631103850338
47 1  9.187583384853566
50 0  10.72326738402944
46 2 11.396739902673568
40 0 11.082142548877775
46 3 11.726527641797215
43 2  10.82774645405946
44 3 10.968198289528557
42 2 10.911445472936107
45 2 10.894384703747138
45 2 11.049301442688533
45 3 10.705489138008156
46 2  10.72326738402944
40 2 11.088155205804059
41 2 10.858998997563564
43 1   9.94515718770752
44 1  9.135616825780247
41 1 11.324618028491475
47 2 10.463131911491967
49 1   11.2708539037705
44 4 10.858998997563564
48 0  11.01613423016804
49 4 11.242257822677576
47 3 10.959540226785442
42 1 10.121015365064702
40 2 11.745440962083432
42 3 10.491274217438248
42 2  9.848978567035735
40 2  10.82377030567982
43 8 11.486930504693143
48 4 10.961277846683982
44 0  7.824046010856292
44 3  11.46831505345168
49 2 11.300746581380709
40 3  8.699514748210191
40 4 11.187818399427712
42 3  11.02679245379461
40 2 10.856110213657674
40 5 10.222631954177766
49 1 11.344506813345266
43 2 10.336892029333534
43 3   10.9244805834911
50 2  10.68311055003235
48 1  9.128370809108931
48 2  9.852194258148577
40 1 10.927771331913476
50 3  10.46310334047155
42 0 10.522880544507412
42 2 11.330611908144274
42 3  11.09093455490473
47 2 11.273690640756074
50 3  9.913437883389296
40 2 10.308952660644293
43 5 10.842517771379773
43 0  9.845593574117226
40 1 11.048888002702997
47 4 11.122191321462058
47 3 10.783114300038692
41 0 10.776870783399007
44 0 11.677804108152463
43 1 10.132056360495403
40 1 10.915088464214607
41 1 11.222452835867818
41 0 11.156250521031495
49 4 10.491274217438248
48 2  9.392661928770137
49 1 11.401993904262946
40 1 10.596634733096073
41 2 11.961315953929825
46 4 11.762601885782415
47 3 10.641034320392757
44 3  11.27236867948514
45 2 11.016545005635734
40 2 10.961555585888657
40 1 11.107510355612106
50 4 10.941995917134532
end
label values age agelbl
label def agelbl 40 "40", modify
label def agelbl 41 "41", modify
label def agelbl 42 "42", modify
label def agelbl 43 "43", modify
label def agelbl 44 "44", modify
label def agelbl 45 "45", modify
label def agelbl 46 "46", modify
label def agelbl 47 "47", modify
label def agelbl 48 "48", modify
label def agelbl 49 "49", modify
label def agelbl 50 "50", modify

------------------ copy up to and including the previous line ------------------

Listed 100 out of 422427 observations

[Rebecca Water]

I can't give any advice regarding your code but what comes to my mind is that if you are interested in the elasticity you can e.g. use a usual regression with the OLS method and have both your variables of interest as logarithms while one of them is the regressand. The interpretation of the OLS estimates of something like

reg lnchild ln_ninc othervars, Options

would be that a 1% increase in ninc in-/decreases child by xx%.

[Andrew Musau]

I have not looked at the paper, and if no one has the time to look at it and respond, it may be better to directly contact the authors. Your definition of the elasticity is correct, as easily derived below:

$$Y= \beta_{0} + \beta_{1} \;\text{ln(X)}$$

The marginal effect is the first-order derivative:

$$\frac{\partial Y}{\partial X} =\frac{\partial \left(\beta_{0} + \beta_{1} \;\text{ln(X)}\right)}{\partial X} = \frac{\beta_{1}}{X}$$

and the elasticity is

$$\frac{X}{Y}\cdot \frac{\beta_{1}}{X} = \frac{\beta_{1}}{Y}$$

local b=_b[ln_ninc]
sum child
local y=r(mean)
dis "e=" `b'/`y'

However, your code using margins is equivalent to

margins,eydx(ln_ninc) atmeans

[Liu Qiang]

I have not looked at the paper, and if no one has the time to look at it and respond, it may be better to directly contact the authors. Your definition of the elasticity is correct, as easily derived below:

$$Y= \beta_{0} + \beta_{1} \;\text{ln(X)}$$

The marginal effect is the first-order derivative:

$$\frac{\partial Y}{\partial X} =\frac{\partial \left(\beta_{0} + \beta_{1} \;\text{ln(X)}\right)}{\partial X} = \frac{\beta_{1}}{X}$$

and the elasticity is

$$\frac{X}{Y}\cdot \frac{\beta_{1}}{X} = \frac{\beta_{1}}{Y}$$





However, your code using margins is equivalent to

margins,eydx(ln_ninc) atmeans

Thank you for your reply and nice advice. I have got the answer from the author who's admitted he may have made a mistake and is checking it now. As this paper has been published on a top journal, I have no reason to convince myself with the basic knowledge. But this is quite a small problem that will not influence the main results. I am very honored to get my questions solved with the author's quick reply.