I solved number 1 that lⁱ = [(1-tau)(gamma)ⁱ]¹ but can't figure out number 2

3 Macro, Inequality and Taxation [21 Points] Consider a country with two groups of people i = 1, 2.. They all live only for one period and have a utility function of the form Til") uw', r) = log (a (1) where 7] 2 0 is a parameter. The budget constraint of each household reads as cl = (1 Thrill + T (2) where T is a proportional labor income tax and T is a lump-sum transfer. 50% of the population is poor and denoted by i = 1, and 50% is rich (i = 2). Poor households have labor productivity 0 S '71 1. Average labor productivity in the economy is 1, that is 0.571 + 0.572 : 1 (3) The government budget constraint reads as 0.5\"."71l1 + 0.5'r'y1l2 = T (4) 1. Determine optimal labor supply of both types of households, as a function of taxes, by maximizing (1) subject to (2). 2. Use your result in the previous question to determine the tax revenue- maximizing tax rate, i.e. the tax rate '7' that maximizes T in equation (4). How does the revenue-maximizing rate depend on 7]? Interpret your result. 3. Now assume 1} > 0. If the government can choose the degree of inequality by chosing ('71, 72) subject to average productivity remaining at 1 (i.e. subject to equation (3) being satised) what is the tax-revenue maximizing degree of inequality (71, 72)? Interpret your result