Question

lm(formula = lnwage ~ female + exper + exper.sq + (female * exper) + 
##     (female * exper.sq) + education + married, data = df)
## Coefficients:
##                   Estimate Std. Error t value        Pr(>|t|)    
## (Intercept)      0.8032332  0.0200735   40.01         < 2e-16 ***
## female          -0.0542592  0.0210303   -2.58          0.0099 **       dummy => female=1    male=0 
## exper            0.0456059  0.0014216   32.08         < 2e-16 ***
## exper.sq        -0.0007692  0.0000301  -25.53         < 2e-16 ***
## education        0.1156786  0.0010345  111.82         < 2e-16 ***
## married          0.1342694  0.0074061   18.13         < 2e-16 ***     dummy => married=1    otherwise=0
## female:exper    -0.0178926  0.0021015   -8.51         < 2e-16 ***
## female:exper.sq  0.0003147  0.0000461    6.83 0.0000000000086 ***

(a) At what year of experience does men's return to experience start to become negative?

(b) At what year of experience does women's return to experience start to become negative?

(c) What is the earnings gap (difference in log wages between men and women) at 5 years of experience?

Answer 1

The regression equation is:

log(wage) = 0.803 - 0.054 * Female + 0.046 * exper - 0.0008 * exper² + 0.116 * education + 0.134 * married - 0.0179 * female * exper + 0.0003 * female * exper²

a) For men, Female = 0

log(wage) = 0.803 - 0.054 * Female + 0.046 * exper - 0.0008 * exper² + 0.116 * education + 0.134 * married

The men's return to experience will be positive at first and then grab a maximum value and go negative.
Return = 0.046 * exper - 0.0008 * exper²

dR/de = 0.046 - 2*0.0008*exper = 0

exper = 0.046/0.0016 = 28.75

Hence, At 29 years of experience the men's return to experience starts to become negative

b) For a women, female = 1

log(wage) = 0.803 - 0.054 * Female + 0.046 * exper - 0.0008 * exper² + 0.116 * education + 0.134 * married - 0.0179 * exper + 0.0003 * exper²

log(wage) = 0.803 - 0.054 * Female + 0.0281 * exper - 0.0011 * exper² + 0.116 * education + 0.134 * married

Return = 0.0281 * exper - 0.0011 * exper²

dR/dE = 0.0281 - 0.0011*2*exper = 0

exper = 0.0281/0.0022 = 12.78

Hence, at 13 years of experience the women's return to experience starts to become negative.

c) At exper = 5

log(wage) = 0.803 - 0.054 * Female + 0.046 * 5 - 0.0008 * 25 + 0.116 * education + 0.134 * married - 0.0179 * female * 5 + 0.0003 * female * 25

log(wage) = 0.803 + 0.046*5 - 0.0008*25 + Female*(-0.054 - 0.0179*5 + 0.0003*25) + 0.116*education + 0.132*married

log(wage) = 1.013 - Female*0.136 + 0.016*education + 0.132*married

Hence, earnings gap (difference in log wages between men and women) at 5 years of experience is the coefficient of Female which is 0.136