Final answer:
The worker's wage in natural logs after 6 years is 7.2. The wage increase percentage from the 6th to the 7th year is approximately 1.72%. The quadratic model predicts the wage is maximized at 25 years of experience.
Step-by-step explanation:
The wage equation given uses experience as the variable to determine the natural logarithm of a worker's wage. To find this worker's wage after working for 6 years in natural logs, we substitute 6 into the equation:
ln(Wage) = 6.9 + 0.05∙6 = 7.2.
In levels, the worker's wage, denoted as W6, is:
W6 = e^7.2 = 1341.64 dollars per month (approximately).
To calculate the wage increase percentage from the 6th to the 7th year, we evaluate the natural logarithm at both points and find their exponential:
ln(Wage) at 6 years: 7.2
ln(Wage) at 7 years: 6.9 + 0.05∙7 = 7.25
Wage at 6 years: e^7.2
Wage at 7 years: e^7.25
The percentage change is then (e^7.25/e^7.2 - 1) × 100% ≈ 1.72%.
When considering a quadratic equation for wage, the derivative of ln(Wage) with respect to Exp shows the marginal effect of experience on wages. For the quadratic equation ln(Wage) = 6.9 + 0.05Exp - 0.001Exp^2, the derivative is:
d(ln(Wage))/dExp = 0.05 - 0.002∙Exp.
The impact of one additional year of experience from 5 to 6 is given by:
d(ln(Wage))/dExp|_{Exp=5} = 0.05 - 0.002∙5 = 0.04.
To find the level of experience at which this worker's wage is maximized, we set the derivative to zero and solve:
0 = 0.05 - 0.002∙Exp,
Exp = 0.05 / 0.002 = 25 years.