1. Consider a common Cobb-Douglas utility function u(x; y) = x0:5y0:5; suppose that prices are px = 1 and py = 4, while consumer's income is I = 8. Firstly, derive the demand functions and the indirect utility function as functions of px, py and I; secondly, substitute the numbers given above and compute the demand and utility levels for these prices and income.
2. Now assume that a tax of one dollar is imposed on good x; this raises px by one dollar. Compute how much the optimal demand for x and the indirect utility change; compute the total tax revenue as the product of the tax by the new demand x.
3. Now consider an equivalent tax on income that reduces consumer's income by the amount of the tax revenue computed in part 2. Compute the new levels of demand for x and y, as well as the new level of indirect utility. Which tax policy (taxing x or taxing income I) yields a higher utility level for the consumer (given that both yield the same tax revenue)?
4. Draw a graph showing the lump sum principle for the Cobb-Douglas example.