help with question 3

3. [30 points total] In an economy, all people have the same utility function u (c - Itb 1 1 1+6 ) , b > 0, (2) with u' > 0 > u" . However, people differ in labor productivity. An individual with type j has labor productivity w; and can earn income y; using labor L; by the production function y; = wjLj. Individuals with income y should pay income tax T (y) = Ty - R with a tax rate T E (0, 1) and the basic income R. Given the income tax, individuals' budget constraint can be formulated as c = (1 - T)y + R. (3) In setting r and R, the government should satisfy the government budget constraint R= T Tjyj = TY, (4) where 7; denotes the population share of type j, and n is the number of types. (a) [10 points] Taking (7, R) as given, each individual chooses (c, L) to maximize util- ity (2) subject to the budget constraint (3). Solve the individual's problem and express L; and y; in terms of 1 - 7 and wj. (b) [5 points] Define s; as the elasticity of y, with respect to 1 - 7. Show that &, = for all j. (Based on this result, define = = ;.) (c) [15 points] The government wants to maximize R = TY, taking the individuals' behaviors into consideration. Draw the Laffer curve, which displays the relation- ship between 7 and R. Prove that 7 = 14 should hold to maximize R