The government has to raise a fixed sum K through income tax. It is known that there are two types of worker a and b in the economy and that the output (gross income) produced by each is given by

qj Tj zj , j a, b

where zj is the amount of effort supplied by a person of type j and Ta > Tb. it is also known that workers have preferences given by the function

vj qj - T j + zj

where T j is the tax on a worker of type j and is a decreasing concave function. The government has no interest in the inequality of utility outcomes and so just seeks to maximise

1va + [1 - 1] vb

where 1 is the proportion of a-type workers.

1. What is the government's budget constraint?

2. Write the government's objective function in terms of qa, qb, 1 and K.

3. If the value of Tj for each worker is private and unknown to the gov- ernment, write down the Lagrangean for the government's optimisation problem.

4. Show that the second-best solution in this case is identical to the full- information solution.

5. Set up the problem in an alternative, equivalent way where the govern- ment's budget constraint is modelled as a separate constraint in the La- grangean. What must be the value of the Lagrange multiplier on this con- straint at the optimum?