Each unit allows the firm holding the unit to produce

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market. Each unit allows the firm holding the unit to produce one unit of pollution. Assume that the market for pollution rights is perfectly competitive. To derive the demand for pollution rights of firm A solve the following cost minimization problem: min a A (100 - a A ) 3 + p a A (6) Solve the following problem for firm B to derive the demand for permits of firm B: min a B (60 - a B ) 2 + p a B (7) A market equilibrium is achieved when a A = - a B . Compute the market equilibrium. e) Is the market equilibrium socially optimal? Explain.

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Problem 5: Black-White Achievement Gap a) Suppose you are measuring the black-white achievement gap by regressing a stan- dardized test score T i on student characteristics X i and a race dummy variable, D i . T i = βX i + αD i + u i (8) where u i is the error term in the regression. a) Explain how the coefficient α is often used as a measure of the black-white achieve- ment gap. b) What variables are likely to influence test scores and should be included in the vector X i ? c) Provide an example of a variable that is likely to have an impact on test scores, but is very difficult to measure? d) How does the existence of variables discussed in (c) affect the interpretation of the least squares estimate of α ? Problem 6: Police Effectiveness Suppose you analyze the impact of policing on crime. You want to test the hypoth- esis that hiring additional police officers reduces the crime rate. You have collected data on criminal arrests in a city, the number of policy officers as well as a number of socio-economic city characteristics that are related to crime such as poverty and unemployment for a large, random sample of cities. You estimate a linear regression model using least squares. You find that the coefficient that measures the impact of policing on crime is positive and statistically significant from zero. Provide an expla- nation for this empirical result that is not based on the notion that police officers are criminals.
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