When estimating wage equations, we expect that young, inexperienced workers will have relatively low wages; with additional experience their wages will rise, but then begin to decline after middle age, as the worker nears retirement. This life$-$cycle pattern of wages can be captured by introducing experience and experience squared to explain the level of wages. If we also include years of education, we have the equation

WAGE

$=$

$1$

$+$

$2$

EDUC

$+$

$3$

EXPER

$+$

$4$

EXPE

$$

$2$

$+$

$$

What is the marginal effect of experience on the mean wage?

What signs do you expect for each of the coefficients $2,$ $3,$ and $4?$ Why?

After how many years of experience does the mean wage start to decline? $($Express your answer in terms of $$s$.)$

Estimating this equation using $600$ observations yields

WAGE

$^$

$=$

$$

$16$

$\mathrm{.}308$

$+$

$2$

$\mathrm{.}329$

EDUC

$+$

$0$

$\mathrm{.}5240$

EXPER

$$

$0$

$\mathrm{.}007582$

EXPE

$$

$2$

se

$2$

$\mathrm{.}745$

$0$

$\mathrm{.}163$

$0$

$\mathrm{.}1263$

$0$

$\mathrm{.}002532$

The estimated covariance between b$3$ and b$4$ is

cov

$$

$$

$3$

$$

$4$

$=$

$$

$0\mathrm{.}00030526$

$.$ Find $95\%$ interval estimates for the following:

The marginal effect of education on mean wage

The marginal effect of experience on mean wage when EXPER $=4$

The marginal effect of experience on mean wage when EXPER $=25$

The number of years of experience after which the mean wage declines