Choice Continued Interpretation At Z x2 Z p1 MRS x1 x 2 p2 X At Y Y x1 p1 MRS x1 x 2 p2 Consumer Demand Consumer s demanded bundle optimal choice of goods 1 and 2 for given prices and income Consumer s demand functions x1 x1 p1 p 2 m x 2 x 2 p1 p 2 m Cobb Douglas Demand function for good 1 x2 m x1 c p1 Demand function for good 2 x2 x1 x1 m x 2 1 c p2 Perfect Substitutes x2 Demand function for good 1 1 p1 p2 x1 m p1 x1 0 x1 x1 if p1 p 2 if p1 p 2 x1 0 m p1 if p1 p 2 Perfect Complements Optimal choice x2 x 2 x1 x 2 x1 Budget line p1 x1 p 2 x 2 m x2 Demand function for goods 1 and 2 x1 x1 m x1 x 2 p1 p 2 Example Choosing a Tax Q Suppose that the government wants to raise a certain amount of revenue Is it better to raise it via a quantity tax or an income tax Before the Tax x2 Budget line p1 x1 p 2 x 2 m p1 p2 x2 Well behaved preferences o Consumer chooses x1 o x1 x1 x 2 o o Quantity Tax x2 p1 t x1 p 2 x 2 m p1 t p2 x 2o x2 R tx1 x1 x1 o x1 Income Tax New budget line p1 x1 p 2 x 2 m R where R tx1 Questions Q1 What is the slope of the budget line with the income tax Q2 Which kind of tax is this income tax X Q3 Can the consumer afford when he is paying the income tax Answers A1 Slope of the budget line with income tax is p1 A2 Lump sum p2 A3 Yes X Is still Affordable Budget line with income tax Substitute p1 x1 p 2 x 2 m tx Rearrange X 1 p1 x1 p 2 x 2 m tx1 p1 t x 1 Evaluate at Rearrange p 2 x 2 m On the Graph x2 X X o p1 p2 x1 Q Is X Optimal with the Income Tax x2 X X o x1 A No it is not x2 X X o X X x1 X Q Which Tax Does the Consumer Prefer x2 X X o X x1 A Income Tax Gives Higher Utility to the Consumer Caution A uniform income tax for all consumers is not necessarily better than a uniform quantity tax Assumption that income tax is lump sum is key Another Question Suppose that the government can use a uniform value tax on goods 1 and 2 to raise revenue e g a sales tax Does the consumer still prefer an income tax Answer With a uniform value tax the budget consumer line reads 1 t p1 x1 1 t p 2 x 2 m Rearranging m p1 x1 p 2 x 2 1 t Answer Continued Rearranging m p1 x1 p 2 x 2 1 t Budget line with income tax p1 x1 p 2 x 2 m R t R tp1 x1 tp 2 x 2 m 1 t Answer Continued Budget line with income tax m p1 x 1 p 2 x 2 1 t Same as budget line under value tax m p1 x 1 p 2 x 2 1 t