A young intern at Silverman-Flute is given instructions by her manager. Please help her impress the manager.

1. (a) The manager would like to calculate the total tax () paid by a certain individual with income given that the marginal tax rate between and + is (1 ) for [0, (.

   ()= (1) 0

   Calculate the total tax () paid by a certain individual with income . What is the average tax rate faced by this individual, and how does it behave for very high revenues ( going to +)?

2. (b) We assume that the number of individuals with income between and + is for [0, +) in the population. Calculate the total tax revenues within this population.

3. (c) The return R to a certain asset is distributed along the probability density function () = 2/3 for [1, +). The manager suggests calculating the mean of the return. Calculate it!