4. Consider a simple optimal tax model with only two goods: a consumer good x and leisure h. The price of x is p, wage rate is w and time endowment is H. (a) What is the definition of lump-sum tax in public finance? (b) In an optimal income tax model, we assume that equity and efficiency are the policy goals and no lump-sum tax can be levied. However, if lump-sum tax can be used, how should the tax be set and collected in such a model? (c) The government intends to implement progressive income tax. However, suppose that income data is not easily collected and so the government turns to consider taxing good x. Based on the idea of equivalent tax, how can the government design the tax schedule for x so that the effect of progressive. income tax can be equivalently implemented?. (d) Now, suppose that income data is easily collected and so income will be. taxed.. (i) Draw a diagram to illustrate the post-tax budget constraint under ! progressive income tax. The vertical axis is x and the horizontal axis is y. (ii) If an ad valorem tax 10% is also levied on x at the same time, how will this affect your diagram drawn in (i)? [Notice that this ad valorem tax is not the same as the tax in (c).] (iii) Draw a case where originally self-selection constraint is violated under progressive income tax but after a change in the schedule the constraint is satisfied.