Question:

Using data from the Global Environmental Monitoring System, Grossman and Krueger (QJE, 1995) estimate a cubic relationship between a country's income per capita in thousands of dollars (Y) and the concentration level (in μg per cubic meter) of a variety of pollutants in its cities. They find that the relationship between the concentration of smoke (S), income per capita, and other factors not related to income (X) is 

S(Y) = 0.13Y3 - 4.2Y2 + 37.1Y + X.
To simplify, suppose X is a parametric constant in the above equation.  

1. Find the levels of income at which there is (local) minimum and maximum concentrations of smoke. (Be sure to check your second-order condition.) 
2. Find the (local) minimum and maximum values of smoke. 
3. Based on your answers to parts i) and ii) above, draw a rough diagram and mark the values you have found in this diagram. 
4. Explain how these results make sense. [Hint: What does your diagram tell you? Can you tell a story based on what you see?]