2. Consider an economy made up of /V individuals. All individuals have the same preferences over consumption, e, and labour supply, L, represented by the utility function U(c, L) = c- .513 An individual's income is given by y = w/ where w is the individual's wage rate. There are No individuals who are unable to work, N individuals who earn w, and N, individuals who earn w > w, and where No + N + N, = N. The government imposes a linear progressive income tax such that T(y) = ty - b where b > 0 is an income transfer received by all individuals regardless of income. (a) Solve for the optimal labour supply of each type of wage earner. (b) Assume the government wants to maximize the transfer b received by those unable to work, that is, the government wants to maximize total tax revenue raised from income taxes. Write out the government's problem and solve for the optimal tax rate. (c) Suppose instead the government can observe the wage rate earned by each type of individuals and imposes a proportional income tax , on the low-wage indi- viduals and to on the high-wage individuals so the tax liability of an individual with wage w, is given by T(y) = try-b for i = 1, 2. Continue to assume the gov- ernment maximizes tax revenue. Solve for the optimal wage-specific tax rates and interpret what you find