What would be the effective MTR for individuals entitled to SSI?

Problem 2: Optimal Income Taxation
Consider an economy with two agents who differ in their innate earning
ability, denoted by w and w for the high-ability individual and the lowability individual, respectively, where w > w ³ 0. Both individuals share
the same preferences given by ( , ) / 2 2 U c l = c -l , where c denotes
consumption and l denotes labor (the price of c is normalized to 1).
Income is subject to a proportional income tax denoted by 0 < t <1.
Moreover, the government is enacting a supplementary-security-income
policy (SSI) according to which, any individual whose (net) income is
lower than a specified level of income, I, is entitled to supplementary
income support up to this level of income (I is a minimal level of
income guaranteed by the government).
1) Draw the budget constraint faced by both individuals.
2) What would be the effective MTR for individuals entitled to SSI?
Demonstrate how the SSI program may adversely affect
individuals’ incentives to participate in the labor market.
3) Suppose the government is seeking to maximize the well-being of
the least well-off individual in the economy. Formulate the
constrained optimization problem of the government. Assume that
a fiscal deficit is not allowed. (Hint: There are two incentive
constraints for the two individuals. Each individual can choose
either a level of income that is entitled to SSI or a higher level of
income that is not entitled to SSI. The choice is optimal!)
4) Show that the incentive constraint of the low-ability individual is
satisfied as a strict inequality.
5) Show that the revenue constraint of the government is satisfied as
equality.
6) Show that the incentive constraint of the high-ability individual is
satisfied as equality; hence, the optimal tax rate is given by:
t =1/3. Calculate social welfare.
7) Show that when w = 0, an optimal linear income tax system (as
discussed in class) is dominated by the SSI program. Explain!