Can someone walk me through this problem? I questions 1 & 2 very easy... However question 3 & 4's math doesn't make sense to me... also where did the 40k come from in question 4? Thanks :)

Consider a society of 100 inhabitants composed by two types of individuals. Half of them are poor and have a yearly income of 10,000 dollars, and half of them are rich, with an annual income of 90,000 dollars. The Government plans to tax annual income in order to redistribute it with a lump sum transfer ?' per person that is equal for all members of the society. However, for a positive tax rate of r over annual income, there is a ~ leak in the bucket. Thus, for one dollar of annual income, the total tax revenue collected is equal to 7 - 72. Assume 7 2 0. 1. Write down in terms of r and ?', the total income that a poor individual has after taxes are collected and transfers are made? [3 points] The total income of a poor individual in this case is given by: 10, 000 * (1 - 7) + T 2. Write down in terms of ~ and T, the total income that a rich individual has after taxes are collected and transfers are made? [3 points] The total income of a poor individual in this case is given by: 90, 000 * (1 - 7) + T 3. Assuming that the Government needs to balance the budget, write down its budget constraint of the govern- ment, and solve for the lump sum transfer ?' in terms of r. [6 points] We know that the budget constraint of the Government must satisfy that tax revenues equal transfers. Thus, we should have that: 100 * T = 50 * 10, 000 * T * (1 - T) + 50 = 90, 000 * T * (1 -T) T = 0.5 * 10, 000 * T * (1 - T) + 0.5 * 90, 000 * T * (1 - T) T = T * (1 - 7) (50, 000) 4. Using what you found in 3, what is the optimal tax rate 7poor that maximizes the income of a poor individual. (6 points] We have that the total income of a poor individual is given by: 10, 000 * (1 - 7) + 50, 000 # T * (1 - T) = 10, 000 + 40, 000 # T - 50, 00072 If we marimize this with respect to + we have that the first order condition of this problem is given by: 40, 000 = 100, 000T